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IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Games Theory Appliedto Economics and Political Sciences
Ahmed Doghmi
National Institute of Statistics and Applied Economics,Madinat Al Irfane, Rabat Institutes, 10100 Rabat, Morocco
,Max Planck Institute of Economics, Strategic Interaction Group,
Kahlaische Straße 10, D-07745, Jena, Germanyand
HEC - Rabat, Centre de Rabat 67, rue Jaafar EssadikAgdal 10080 Rabat, Morocco
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Introduction and Definitions
Game theory is a mathematical approach that is applied toseveral areas:
1 in economics and business;2 in biology (particularly evolutionary biology and ecology);3 engineering;4 political science;5 international relations;6 computer science;7 philosophy.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Introduction and Definitions
Game theory is a mathematical approach that is applied toseveral areas:
1 in economics and business;2 in biology (particularly evolutionary biology and ecology);3 engineering;4 political science;5 international relations;6 computer science;7 philosophy.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Introduction and Definitions
It is considered as new discipline since the publication of thebook “Theory of Games and Economic Behavior” by J. V.Neumann and Morgenstern (1944);
The goal of games theory is to model the behavior of agents(or players) by studying their strategic interactions;
A player can be a individual, a firm, a political party...
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Introduction and Definitions
It is considered as new discipline since the publication of thebook “Theory of Games and Economic Behavior” by J. V.Neumann and Morgenstern (1944);
The goal of games theory is to model the behavior of agents(or players) by studying their strategic interactions;
A player can be a individual, a firm, a political party...
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Introduction and Definitions
It is considered as new discipline since the publication of thebook “Theory of Games and Economic Behavior” by J. V.Neumann and Morgenstern (1944);
The goal of games theory is to model the behavior of agents(or players) by studying their strategic interactions;
A player can be a individual, a firm, a political party...
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Introduction and Definitions
Branches of games theory: We distinguish two branches ofgames:
1 Cooperative games: cooperative games theory describes onlythe outcomes (without detail) that result when the playerscome together in different combinations;
2 Non-cooperative games: a game is a detailed model of all themoves available to the players.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Introduction and Definitions
Branches of games theory: We distinguish two branches ofgames:
1 Cooperative games: cooperative games theory describes onlythe outcomes (without detail) that result when the playerscome together in different combinations;
2 Non-cooperative games: a game is a detailed model of all themoves available to the players.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Introduction and Definitions
Branches of games theory: We distinguish two branches ofgames:
1 Cooperative games: cooperative games theory describes onlythe outcomes (without detail) that result when the playerscome together in different combinations;
2 Non-cooperative games: a game is a detailed model of all themoves available to the players.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Introduction and Definitions
Now, we are going to look at the non-cooperative branch.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Introduction and Definitions
A non-cooperative game is a situation where the consequencesof actions (or strategies) of each depend on those other;
To predict the outcome of a game, several concept solutionmay be considered depending on the type of studied games;
For example, for simultaneous (static) games:1 Where the information is complete, the concept of solution is
represented by the notion of Nash equilibrium: it is to find alist (or profile) of individual strategies such that no agent hasincentive to unilaterally deviate from its strategy, if all othersdo not change theirs;
2 In the case of incomplete information, games are generallyresolved by the notion of Bayesian equilibrium.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Introduction and Definitions
A non-cooperative game is a situation where the consequencesof actions (or strategies) of each depend on those other;
To predict the outcome of a game, several concept solutionmay be considered depending on the type of studied games;
For example, for simultaneous (static) games:1 Where the information is complete, the concept of solution is
represented by the notion of Nash equilibrium: it is to find alist (or profile) of individual strategies such that no agent hasincentive to unilaterally deviate from its strategy, if all othersdo not change theirs;
2 In the case of incomplete information, games are generallyresolved by the notion of Bayesian equilibrium.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Introduction and Definitions
A non-cooperative game is a situation where the consequencesof actions (or strategies) of each depend on those other;
To predict the outcome of a game, several concept solutionmay be considered depending on the type of studied games;
For example, for simultaneous (static) games:1 Where the information is complete, the concept of solution is
represented by the notion of Nash equilibrium: it is to find alist (or profile) of individual strategies such that no agent hasincentive to unilaterally deviate from its strategy, if all othersdo not change theirs;
2 In the case of incomplete information, games are generallyresolved by the notion of Bayesian equilibrium.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Introduction and Definitions
A non-cooperative game is a situation where the consequencesof actions (or strategies) of each depend on those other;
To predict the outcome of a game, several concept solutionmay be considered depending on the type of studied games;
For example, for simultaneous (static) games:1 Where the information is complete, the concept of solution is
represented by the notion of Nash equilibrium: it is to find alist (or profile) of individual strategies such that no agent hasincentive to unilaterally deviate from its strategy, if all othersdo not change theirs;
2 In the case of incomplete information, games are generallyresolved by the notion of Bayesian equilibrium.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Introduction and Definitions
A non-cooperative game is a situation where the consequencesof actions (or strategies) of each depend on those other;
To predict the outcome of a game, several concept solutionmay be considered depending on the type of studied games;
For example, for simultaneous (static) games:1 Where the information is complete, the concept of solution is
represented by the notion of Nash equilibrium: it is to find alist (or profile) of individual strategies such that no agent hasincentive to unilaterally deviate from its strategy, if all othersdo not change theirs;
2 In the case of incomplete information, games are generallyresolved by the notion of Bayesian equilibrium.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Introduction and Definitions
Types of games: We distinguish two types of games:
1 Simultaneous games: the players take their strategiessimultaneously, ie, a player chooses his strategy withoutknowing the other and vice versa.
2 Dynamic games: the players involved one after the other.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Introduction and Definitions
Types of games: We distinguish two types of games:
1 Simultaneous games: the players take their strategiessimultaneously, ie, a player chooses his strategy withoutknowing the other and vice versa.
2 Dynamic games: the players involved one after the other.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Introduction and Definitions
Types of games: We distinguish two types of games:
1 Simultaneous games: the players take their strategiessimultaneously, ie, a player chooses his strategy withoutknowing the other and vice versa.
2 Dynamic games: the players involved one after the other.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Introduction and Definitions
Types of games: We distinguish two types of games:
1 Simultaneous games: the players take their strategiessimultaneously, ie, a player chooses his strategy withoutknowing the other and vice versa.
2 Dynamic games: the players involved one after the other.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Introduction and Definitions
Information:
1 Complete but imperfect information: the players know allrelevant parameters of games as strategies profile Si and payofffunctions, but the players do not know what others will choose;
2 Perfect information: For dynamic games where players haveseen what others do;
3 Incomplete information: there is private information.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Introduction and Definitions
Information:
1 Complete but imperfect information: the players know allrelevant parameters of games as strategies profile Si and payofffunctions, but the players do not know what others will choose;
2 Perfect information: For dynamic games where players haveseen what others do;
3 Incomplete information: there is private information.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Introduction and Definitions
Information:
1 Complete but imperfect information: the players know allrelevant parameters of games as strategies profile Si and payofffunctions, but the players do not know what others will choose;
2 Perfect information: For dynamic games where players haveseen what others do;
3 Incomplete information: there is private information.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Plan1 Introduction2 Simultaneous Games with complete - imperfect information
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium3 Dynamic (Extensive form) games with perfect information
The concept of subgamesEconomic application: Stackelberg Model (1934)
4 Dynamic (Sequential) games with complete and imperfectinformation
5 Nash ImplementationIntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rules
Borda rulePlurality ruleAnti-plurality rule
Danilov’s - Yamato’s theorems6 Nash Implementation under Domain Restrictions with
IndifferencesNash implementation in exchange economies withsingle-plateaued preferences
New sufficient Conditions
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Example: Prisoner Dilemma
Assume that two accomplices i and j of a crime areinterviewed separately;
They have the choice between the strategy of confession andthat the accusation;
If i admits accusing j and j does not confess, i is released andj is punishable by 10 years in prison;
If both confess, they each have a term of 6 years prison;
If neither confesses, they will each have one year in prison.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Example: Prisoner Dilemma
Assume that two accomplices i and j of a crime areinterviewed separately;
They have the choice between the strategy of confession andthat the accusation;
If i admits accusing j and j does not confess, i is released andj is punishable by 10 years in prison;
If both confess, they each have a term of 6 years prison;
If neither confesses, they will each have one year in prison.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Example: Prisoner Dilemma
Assume that two accomplices i and j of a crime areinterviewed separately;
They have the choice between the strategy of confession andthat the accusation;
If i admits accusing j and j does not confess, i is released andj is punishable by 10 years in prison;
If both confess, they each have a term of 6 years prison;
If neither confesses, they will each have one year in prison.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Example: Prisoner Dilemma
Assume that two accomplices i and j of a crime areinterviewed separately;
They have the choice between the strategy of confession andthat the accusation;
If i admits accusing j and j does not confess, i is released andj is punishable by 10 years in prison;
If both confess, they each have a term of 6 years prison;
If neither confesses, they will each have one year in prison.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Example: Prisoner Dilemma
Assume that two accomplices i and j of a crime areinterviewed separately;
They have the choice between the strategy of confession andthat the accusation;
If i admits accusing j and j does not confess, i is released andj is punishable by 10 years in prison;
If both confess, they each have a term of 6 years prison;
If neither confesses, they will each have one year in prison.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Example: Prisoner Dilemma
Prisoner j
Prisoner iConfess (C ) Not Confess (NC )
Confess (C ) (-6,-6) (0,-10)Not Confess (NC ) (-10,0) (-1,-1)
1 Players: prisoner i and prisoner j ;
2 Strategies: {C ,NC};3 Payoffs: let ui (., .) denote the utility function of a player i .
- For prisoner i : ui (C ,C ) = −6; ui (C ,NC ) = 0;ui (NC ,C ) = −10; ui (NC ,NC ) = −1;- For prisoner j : uj (C ,C ) = −6; ui (C ,NC ) = 0;ui (NC ,C ) = −10; ui (NC ,NC ) = −1
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Example: Prisoner Dilemma
Prisoner j
Prisoner iConfess (C ) Not Confess (NC )
Confess (C ) (-6,-6) (0,-10)Not Confess (NC ) (-10,0) (-1,-1)
1 Players: prisoner i and prisoner j ;
2 Strategies: {C ,NC};3 Payoffs: let ui (., .) denote the utility function of a player i .
- For prisoner i : ui (C ,C ) = −6; ui (C ,NC ) = 0;ui (NC ,C ) = −10; ui (NC ,NC ) = −1;- For prisoner j : uj (C ,C ) = −6; ui (C ,NC ) = 0;ui (NC ,C ) = −10; ui (NC ,NC ) = −1
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Example: Prisoner Dilemma
Prisoner j
Prisoner iConfess (C ) Not Confess (NC )
Confess (C ) (-6,-6) (0,-10)Not Confess (NC ) (-10,0) (-1,-1)
1 Players: prisoner i and prisoner j ;
2 Strategies: {C ,NC};3 Payoffs: let ui (., .) denote the utility function of a player i .
- For prisoner i : ui (C ,C ) = −6; ui (C ,NC ) = 0;ui (NC ,C ) = −10; ui (NC ,NC ) = −1;- For prisoner j : uj (C ,C ) = −6; ui (C ,NC ) = 0;ui (NC ,C ) = −10; ui (NC ,NC ) = −1
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Example: Prisoner Dilemma
Prisoner j
Prisoner iConfess (C ) Not Confess (NC )
Confess (C ) (-6,-6) (0,-10)Not Confess (NC ) (-10,0) (-1,-1)
1 Players: prisoner i and prisoner j ;
2 Strategies: {C ,NC};3 Payoffs: let ui (., .) denote the utility function of a player i .
- For prisoner i : ui (C ,C ) = −6; ui (C ,NC ) = 0;ui (NC ,C ) = −10; ui (NC ,NC ) = −1;- For prisoner j : uj (C ,C ) = −6; ui (C ,NC ) = 0;ui (NC ,C ) = −10; ui (NC ,NC ) = −1
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Example: Prisoner Dilemma
Prisoner j
Prisoner iConfess (C ) Not Confess (NC )
Confess (C ) (-6,-6) (0,-10)Not Confess (NC ) (-10,0) (-1,-1)
1 Players: prisoner i and prisoner j ;
2 Strategies: {C ,NC};3 Payoffs: let ui (., .) denote the utility function of a player i .
- For prisoner i : ui (C ,C ) = −6; ui (C ,NC ) = 0;ui (NC ,C ) = −10; ui (NC ,NC ) = −1;- For prisoner j : uj (C ,C ) = −6; ui (C ,NC ) = 0;ui (NC ,C ) = −10; ui (NC ,NC ) = −1
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Plan1 Introduction2 Simultaneous Games with complete - imperfect information
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium3 Dynamic (Extensive form) games with perfect information
The concept of subgamesEconomic application: Stackelberg Model (1934)
4 Dynamic (Sequential) games with complete and imperfectinformation
5 Nash ImplementationIntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rules
Borda rulePlurality ruleAnti-plurality rule
Danilov’s - Yamato’s theorems6 Nash Implementation under Domain Restrictions with
IndifferencesNash implementation in exchange economies withsingle-plateaued preferences
New sufficient Conditions
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominated strategies:Example 1
Prisoner j
Prisoner iConfess (C ) Not Confess (NC )
Confess (C ) (-6,-6) (0,-10)Not Confess (NC ) (-10,0) (-1,-1)
Prisoner i C -6 0NC -10 -1
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominated strategies:Example 1
Prisoner j
Prisoner iConfess (C ) Not Confess (NC )
Confess (C ) (-6,-6) (0,-10)Not Confess (NC ) (-10,0) (-1,-1)
Prisoner i C -6 0NC -10 -1
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominatedstrategies:Example 1
Prisoner i C -6 0NC -10 -1
I −6 > −10 and 0 > −1 ⇒ strategy NC is strictly dominated bystrategy C .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominatedstrategies:Example 1
Prisoner i C -6 0NC -10 -1
I −6 > −10 and 0 > −1 ⇒ strategy NC is strictly dominated bystrategy C .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominatedstrategies:Example 1
Prisoner j
Prisoner iConfess (C ) Not Confess (NC )
Confess (C ) (-6,-6) (0,-10)Not Confess (NC ) (-10,0) (-1,-1)
Prisoner j
C NC-6 -100 -1
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominatedstrategies:Example 1
Prisoner j
Prisoner iConfess (C ) Not Confess (NC )
Confess (C ) (-6,-6) (0,-10)Not Confess (NC ) (-10,0) (-1,-1)
Prisoner j
C NC-6 -100 -1
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominatedstrategies:Example 1
Prisoner j
C NC-6 -100 -1
I −6 > −10 and 0 > −1 ⇒ strategy NC is strictly dominated bystrategy C ;For two players, strategy NC is strictly dominated by strategy C .Thus, the combination (C ,C ) is a solution of game.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominatedstrategies:Example 1
Prisoner j
C NC-6 -100 -1
I −6 > −10 and 0 > −1 ⇒ strategy NC is strictly dominated bystrategy C ;For two players, strategy NC is strictly dominated by strategy C .Thus, the combination (C ,C ) is a solution of game.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominatedstrategies:Notations and generalization
Let N = {1, 2, ..., n} be a set of players;
Each player has a set of strategies;
Let Si be strategies set of player i , the strategies profile set isdenoted by S = S1 × S2 × ...× Sn.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominatedstrategies:Notations and generalization
Let N = {1, 2, ..., n} be a set of players;
Each player has a set of strategies;
Let Si be strategies set of player i , the strategies profile set isdenoted by S = S1 × S2 × ...× Sn.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominatedstrategies:Notations and generalization
Let N = {1, 2, ..., n} be a set of players;
Each player has a set of strategies;
Let Si be strategies set of player i , the strategies profile set isdenoted by S = S1 × S2 × ...× Sn.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominatedstrategies:Notations and generalization
The elements of S are denoted bys = (s1, s2, ..., sn) = (si , s−i ), wheres−i = (s1, ..., si−1, si+1, ..., sn);
When s ∈ S and bi ∈ Si ,(bi , s−i ) = (s1, ..., si−i , bi , si+1, ..., sn) is obtained afterreplacing si by bi , and g(Si , s−i ) is the set of results whichagent i can obtain when the other agents choose s−i fromS−i = Πj∈N,j 6=i Sj ;
Payoff function: Each player has a payoff functionUi : S =
∏n1 Si → R.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominatedstrategies:Notations and generalization
The elements of S are denoted bys = (s1, s2, ..., sn) = (si , s−i ), wheres−i = (s1, ..., si−1, si+1, ..., sn);
When s ∈ S and bi ∈ Si ,(bi , s−i ) = (s1, ..., si−i , bi , si+1, ..., sn) is obtained afterreplacing si by bi , and g(Si , s−i ) is the set of results whichagent i can obtain when the other agents choose s−i fromS−i = Πj∈N,j 6=i Sj ;
Payoff function: Each player has a payoff functionUi : S =
∏n1 Si → R.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominatedstrategies:Notations and generalization
The elements of S are denoted bys = (s1, s2, ..., sn) = (si , s−i ), wheres−i = (s1, ..., si−1, si+1, ..., sn);
When s ∈ S and bi ∈ Si ,(bi , s−i ) = (s1, ..., si−i , bi , si+1, ..., sn) is obtained afterreplacing si by bi , and g(Si , s−i ) is the set of results whichagent i can obtain when the other agents choose s−i fromS−i = Πj∈N,j 6=i Sj ;
Payoff function: Each player has a payoff functionUi : S =
∏n1 Si → R.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominatedstrategies:Generalization
Definition
Let Γ = {N, (Si )i∈N , (ui )i∈N} be a normal form game. Ifs ′i , s
′′i ∈ Si , then s ′i is strictly dominated by s ′′i if for all strategies
combination, the payoff of player i is smaller with s ′i than that withs ′′i . Formally,ui (s1, s2, ..., si−1, s
′i , si+1, ..., sn) < ui (s1, s2, ..., si−1, s
′′i , si+1, ..., sn).
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominatedstrategies:Example 2
Player 2
Player 1L M R
U (1,0) (1,2) (0,1)D (0,3) (0,1) (2,0)
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominatedstrategies:Example 2
Player 1L M R
U 1 1 0D 0 0 2
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominatedstrategies:Example 2
Player 1L M R
U 1 1 0D 0 0 2
I For player 1, 1 > 0 and 1 > 0 but 0 < 2 ⇒ player 1 does notadmit strictly dominated strategy.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominatedstrategies:Example 2
Player 2
Player 1L M R
U (1,0) (1,2) (0,1)D (0,3) (0,1) (2,0)
Player 2
L M R0 2 13 1 0
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominatedstrategies:Example 2
Player 2
L M R0 2 13 1 0
I 2 > 1 and 1 > 0 ⇒ strategy R is strictly dominated by strategyM.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominatedstrategies:Example 2
⇒ The game become:
Player 2
Player 1L M
U (1,0) (1,2)D (0,3) (0,1)
For player 1, we have:
Player 1L M
U 1 1D 0 0
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominatedstrategies:Example 2
Player 1L M
U 1 1D 0 0
I strategy D is strictly dominated by strategy U;I Player 2 doses not admet strictly dominated strategy.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominatedstrategies:Example 2
Player 1L M
U 1 1D 0 0
I strategy D is strictly dominated by strategy U;I Player 2 doses not admet strictly dominated strategy.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominatedstrategies:Example 2
⇒ The game will be reduced and we obtain:
Player 2
Player 1 L MU (1,0) (1,2)
1 Player 1 is indifferent between L and M;
2 For player 2, L is strictly dominated by M;
The combination (U,M) of payoff (1, 2) is a solution of game.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominatedstrategies:Example 2
⇒ The game will be reduced and we obtain:
Player 2
Player 1 L MU (1,0) (1,2)
1 Player 1 is indifferent between L and M;
2 For player 2, L is strictly dominated by M;
The combination (U,M) of payoff (1, 2) is a solution of game.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominatedstrategies:Example 2
⇒ The game will be reduced and we obtain:
Player 2
Player 1 L MU (1,0) (1,2)
1 Player 1 is indifferent between L and M;
2 For player 2, L is strictly dominated by M;
The combination (U,M) of payoff (1, 2) is a solution of game.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominatedstrategies:Example 2
⇒ The game will be reduced and we obtain:
Player 2
Player 1 L MU (1,0) (1,2)
1 Player 1 is indifferent between L and M;
2 For player 2, L is strictly dominated by M;
The combination (U,M) of payoff (1, 2) is a solution of game.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominatedstrategies:Example 3
Player 2
Player 1L C R
T (0,4) (4,0) (5,3)M (4,0) (0,4) (5,3)B (3,5) (3,5) (6,6)
In this game, The method of iterative elimination of strictlydominated strategies can not give us a solution;
Thus, we proceed to another concept of solution in purestrategy: it is Nash Equilibrium.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominatedstrategies:Example 3
Player 2
Player 1L C R
T (0,4) (4,0) (5,3)M (4,0) (0,4) (5,3)B (3,5) (3,5) (6,6)
In this game, The method of iterative elimination of strictlydominated strategies can not give us a solution;
Thus, we proceed to another concept of solution in purestrategy: it is Nash Equilibrium.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Iterative elimination of strictly dominatedstrategies:Example 3
Player 2
Player 1L C R
T (0,4) (4,0) (5,3)M (4,0) (0,4) (5,3)B (3,5) (3,5) (6,6)
In this game, The method of iterative elimination of strictlydominated strategies can not give us a solution;
Thus, we proceed to another concept of solution in purestrategy: it is Nash Equilibrium.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Plan1 Introduction2 Simultaneous Games with complete - imperfect information
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium3 Dynamic (Extensive form) games with perfect information
The concept of subgamesEconomic application: Stackelberg Model (1934)
4 Dynamic (Sequential) games with complete and imperfectinformation
5 Nash ImplementationIntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rules
Borda rulePlurality ruleAnti-plurality rule
Danilov’s - Yamato’s theorems6 Nash Implementation under Domain Restrictions with
IndifferencesNash implementation in exchange economies withsingle-plateaued preferences
New sufficient Conditions
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Nash Equilibrium: Example
Player 2
Player 1L C R
T (0,4) (4,0) (5,3)M (4,0) (0,4) (5,3)B (3,5) (3,5) (6,6)
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Nash Equilibrium: Example
Player 2
Player 1L C R
T (0,4) (4,0) (5,3)M (4,0) (0,4) (5,3)B (3,5) (3,5) (6,6)
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Nash Equilibrium: Example
Player 2
Player 1
L C RT (0,4) (4,0) (5,3)M (4,0) (0,4) (5,3)B (3,5) (3,5) (6,6)
We obtain a Nash Equilibrium (B,R) of payoffs (6, 6).
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Nash Equilibrium: Example
Player 2
Player 1
L C RT (0,4) (4,0) (5,3)M (4,0) (0,4) (5,3)B (3,5) (3,5) (6,6)
We obtain a Nash Equilibrium (B,R) of payoffs (6, 6).
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Nash Equilibrium: Example
In this example, we have:
For player 1:U1(B,R) ≥ U1(T ,R) for T ∈ {T ,M,B};U1(B,R) ≥ U1(M,R) for M ∈ {T ,M,B};For player 2:U2(B,R) ≥ U2(B, L) for L ∈ {L,C ,R};U2(B,R) ≥ U2(B,C ) for C ∈ {L,C ,R}.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Nash Equilibrium for two-player: Generalization
Concept of stability: situation where no player has interest todeviate unilaterally of his strategy.
Definition
A Nash equilibrium for two-player in pure strategies of a normalform game Γ = {N = {1, 2}, (Si )i=1,2, (ui )i=1,2} is a profile ofstrategies (s∗1 , s
∗2 ) such that the strategy of each player s∗i is a
better response to the strategies chosen by the other player (s∗i−1),i.e., u1(s∗1 , s
∗2 ) ≥ u1(s1, s
∗2 ) for all strategy s1 ∈ S1;
u2(s∗1 , s∗2 ) ≥ u2(s∗1 , s2) for all strategy s2 ∈ S2;
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Nash Equilibrium n Players
Definition (Nash 1951)
A Nash equilibrium for n players in pure strategies of a normalform game Γ = {N, (Si )i∈N , (ui )i∈N} is a profile of strategies(s∗1 , s
∗2 , ..., s
∗n ) such that the strategy of each player s∗i is a better
response to the strategies chosen by the other players(s∗1 , s
∗2 , ..., s
∗i−1, ..., s
∗n ), i.e.,
ui (s∗1 , s∗2 , ..., s
∗i−1, s
∗i , s∗i+1, ..., s
∗n ) ≥ ui (s∗1 , s
∗2 , ..., s
∗i−1, si , s
∗i+1, ..., s
∗n )
for all strategy si ∈ Si .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Nash equilibrium and iterative elimination (IE) method ofstrictly dominated strategies
Proposition
If IE method eliminates all strategies except one then this strategyis a Nash equilibrium;If a strategy is a Nash equilibrium, then this strategy survives withIE method.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Multiplicity of Nash Equilibrium
Player 2
Player 1O F
O (2,1) (0,0)F (0,0) (1,2)
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Multiplicity of Nash Equilibrium
Player 2
Player 1O F
O (2,1) (0,0)F (0,0) (1,2)
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Multiplicity of Nash Equilibrium
Player 2
Player 1O F
O (2,1) (0,0)F (0,0) (1,2)
In this game, we have two Nash equilibrium! they are thestrategies profile (O,O) of payoffs (2,1) and the strategies profile(F ,F ) of payoffs (1,2).
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Multiplicity of Nash Equilibrium
Player 2
Player 1O F
O (2,1) (0,0)F (0,0) (1,2)
In this game, we have two Nash equilibrium! they are thestrategies profile (O,O) of payoffs (2,1) and the strategies profile(F ,F ) of payoffs (1,2).
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
A Game without a Nash Equilibrium
Player 2
Player 1H T
H (-1,1) (1,-1)T (1,-1) (-1,1)
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
A Game without a Nash Equilibrium
Player 2
Player 1H T
H (-1,1) (1,-1)T (1,-1) (-1,1)
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
A Game without a Nash Equilibrium
Player 2
Player 1H T
H (-1,1) (1,-1)T (1,-1) (-1,1)
This game does not admit a Nash equilibrium.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
A Game without a Nash Equilibrium
Player 2
Player 1H T
H (-1,1) (1,-1)T (1,-1) (-1,1)
This game does not admit a Nash equilibrium.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Existence of Nash Equilibrium: Some definitions andtheorems
Definition (Upper hemicontinuous correspondence)
Let X be a compact set. A correspondence F : X ⇒ X is said tobe upper hemicontinuous if its graphg(F ) ≡ {(x , y) : x ∈ X , y ∈ F (x)} is a closed set (in the usualtopology). An equivalent requirement is that given any x∗ ∈ Xand some sequence {xq}∞q=1 convergent to x∗, every sequence{y q}∞q=1 with y q ∈ F (xq) has a limit point y∗ ∈ F (x∗).
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Existence of Nash Equilibrium: Some definitions andtheorems
Theorem (Kakutani’s Fixed Point Theorem)
Let X ⊂ Rm(m ∈ N) be a compact, convex, and nonempty set,and F : X ⇒ X an upper hemicontinuous correspondence withconvex and noempty images (i.e., ∀x ∈ X ,F (x) is a nonempty andconvex subset of X ). Then, the correspondence F has a fixedpoint, i.e., there exists some x∗ ∈ X such that x∗ ∈ F (x∗).
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Existence of Nash Equilibrium: Some definitions andtheorems
Definition (Quasi-concavity)
The function ui : S1 × S2 × ...× Sn → R is quasi-concave en si iffor all strategies profile s−i , the set {si : ui (si , s−i ) ≥ a} is convex.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Existence of Nash Equilibrium: Debreu, Fan, Glicksberg’sTheorem, 1952
Theorem (Debreu, 1952; Fan, 1952; Glicksberg, 1952)
Let Γ be a game in strategic form such that, for eachi = 1, 2, ..., n, Si ⊂ Rm is compact and convex, and the functionui : S1 × S2 × ...× Sn → R is continuous in s = (s1, s2, ..., sn) andquasi-concave in si . Then, the game Γ has a Nash equilibrium inpure strategies.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Economic application: Cournot’s duopoly model
Let two firms 1 and 2;
Let q1 = firm 1 quantity and q2 = firm 2 quantity;
Let P(Q) = a− Q be inverse demand function of price Pwhere Q is the total quantity produced (Q = q1 + q2) and ais the absorption capacity limit of market;
P(Q) =
{a− Q if Q ≤ a;0 otherwise.
Firm i have the cost structure Ci (qi ) = cqi , the marginal costsof production are equal and fixed, they are represented by c .’
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Economic application: Cournot’s duopoly model
Following Cournot, firms strategically play the quantityvariable;
Players ⇒ two firms;
Payoffs ⇒ profit of each firm, i.e., ui (qi , qj ⇒ Πi (qi , qj
1 Πi (qi , qj = qi [a− (qi + qj )− c];2 The vector (q∗i , q
∗j ) is a Nash equilibrium if q∗i resolves
Maxqi∈[0,a] Πi (qi , q∗j ) = Maxqi∈[0,a] qi [a− (qi + q∗j )− c].
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Economic application: Cournot’s duopoly model
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Plan1 Introduction2 Simultaneous Games with complete - imperfect information
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium3 Dynamic (Extensive form) games with perfect information
The concept of subgamesEconomic application: Stackelberg Model (1934)
4 Dynamic (Sequential) games with complete and imperfectinformation
5 Nash ImplementationIntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rules
Borda rulePlurality ruleAnti-plurality rule
Danilov’s - Yamato’s theorems6 Nash Implementation under Domain Restrictions with
IndifferencesNash implementation in exchange economies withsingle-plateaued preferences
New sufficient Conditions
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Pareto Optimality: Example
Prisoner j
Prisoner iC NC
C (-6,-6) (0,-10)NC (-10,0) (-1,-1)
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Pareto Optimality: Example
Prisoner j
Prisoner iC NC
C (-6,-6) (0,-10)NC (-10,0) (-1,-1)
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Pareto Optimality: Example
Prisoner j
Prisoner iC NC
C (-6,-6) (0,-10)NC (-10,0) (-1,-1)
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Pareto Optimality: Example
Prisoner j
Prisoner iC NC
C (-6,-6) (0,-10)NC (-10,0) (−1,−1︸ ︷︷ ︸)
1 The combinaison (C ,C ) is a Nash Equilibrium;
2 The combinaison (NC ,NC ) is a Pareto Otimality Equilibrium.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Pareto Optimality: Example
Prisoner j
Prisoner iC NC
C (-6,-6) (0,-10)NC (-10,0) (−1,−1︸ ︷︷ ︸)
1 The combinaison (C ,C ) is a Nash Equilibrium;
2 The combinaison (NC ,NC ) is a Pareto Otimality Equilibrium.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Pareto Optimality: Example
Prisoner j
Prisoner iC NC
C (-6,-6) (0,-10)NC (-10,0) (−1,−1︸ ︷︷ ︸)
1 The combinaison (C ,C ) is a Nash Equilibrium;
2 The combinaison (NC ,NC ) is a Pareto Otimality Equilibrium.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Pareto Optimality: Definition
Definition
A set of strategies is Pareto-optimal iff it is impossible to make atleast one person better off without making anyone else worse off.Pareto solution selects the feasible strategies which are not weaklydominated by an other strategy for all agents and not strictlydominated for at least one player. Formally, it is defined as follows.Let u ∈ U, P(u) = {s ∈ S : @s ∈ S such that for all i ∈ N,ui (si , s−i ) ≥ ui (si , s−i ), and for some i ∈ N,ui (si , s−i ) > ui (si , s−i )}.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Pareto Optimal Nash Equilibrium: Definition
Definition
A Pareto optimal Nash equilibrium in pure strategies of a normalform game Γ = {N, (Si )i∈N , (ui )i∈N} is any Nash equilibrium(s∗1 , s
∗2 , ..., s
∗n ) such that there does not exist another Nash
equilibrium (s1, s2, ..., sn) with ui (s∗1 , s∗2 , ..., s
∗n ) < ui (s1, s2, ..., sn)
for all i ∈ N.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Pareto Optimal Nash Equilibrium: Example
Firm j
Firm iWait Invest
Wait (400,400) (0,300)Invest (300,0) (200,200)
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Pareto Optimal Nash Equilibrium: Example
Firm j
Firm iWait Invest
Wait (400,400) (0,300)Invest (300,0) (200,200)
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Pareto Optimal Nash Equilibrium: Example
Firm j
Firm iWait Invest
Wait (400,400) (0,300)Invest (300,0) (200,200)
The combinations (Wait,Wait) and (Invest,Invest) are twoNash equilibria;
The combination (Wait,Wait) is Pareto Optimal NashEquilibrium.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Pareto Optimal Nash Equilibrium: Example
Firm j
Firm iWait Invest
Wait (400,400) (0,300)Invest (300,0) (200,200)
The combinations (Wait,Wait) and (Invest,Invest) are twoNash equilibria;
The combination (Wait,Wait) is Pareto Optimal NashEquilibrium.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Pareto Optimal Nash Equilibrium: Example
Firm j
Firm iWait Invest
Wait (400,400) (0,300)Invest (300,0) (200,200)
The combinations (Wait,Wait) and (Invest,Invest) are twoNash equilibria;
The combination (Wait,Wait) is Pareto Optimal NashEquilibrium.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Plan1 Introduction2 Simultaneous Games with complete - imperfect information
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium3 Dynamic (Extensive form) games with perfect information
The concept of subgamesEconomic application: Stackelberg Model (1934)
4 Dynamic (Sequential) games with complete and imperfectinformation
5 Nash ImplementationIntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rules
Borda rulePlurality ruleAnti-plurality rule
Danilov’s - Yamato’s theorems6 Nash Implementation under Domain Restrictions with
IndifferencesNash implementation in exchange economies withsingle-plateaued preferences
New sufficient Conditions
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Example
Player 2
Player 1H T
H (-1,1) (1,-1)T (1,-1) (-1,1)
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Example
Player 2
Player 1H T
H (-1,1) (1,-1)T (1,-1) (-1,1)
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Example
Player 2
Player 1H T
H (-1,1) (1,-1)T (1,-1) (-1,1)
In this game, there is not a Nash equilibrium in pure strategies.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Example
Player 2
Player 1H T
H (-1,1) (1,-1)T (1,-1) (-1,1)
In this game, there is not a Nash equilibrium in pure strategies.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Mixed strategy
Mixed strategy is the uncertainty which a player has on whatthe other can make (Harsanyi 1973);
A mixed strategy is a probability distribution (q, 1− q) whereq is probability to play H and 1− q is probability to play T ;
The mixed strategy (0,1) is simply the pure strategy T .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Mixed strategy: example
Player 2
Player 1L M R
U (1,0) (1,2) (0,1)D (0,3) (0,1) (2,0)
A mixed strategy for player 2 is a probability distribution (q,r,1-q-r);
For example, the probability distribution ( 13 ,
13 ,
13 ) is equal for
L,M and R strategies.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Mixed strategy: example
Player 2
Player 1L M R
U (1,0) (1,2) (0,1)D (0,3) (0,1) (2,0)
A mixed strategy for player 2 is a probability distribution (q,r,1-q-r);
For example, the probability distribution ( 13 ,
13 ,
13 ) is equal for
L,M and R strategies.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Mixed strategy: example
Player 2
Player 1L M R
U (1,0) (1,2) (0,1)D (0,3) (0,1) (2,0)
A mixed strategy for player 2 is a probability distribution (q,r,1-q-r);
For example, the probability distribution ( 13 ,
13 ,
13 ) is equal for
L,M and R strategies.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Mixed strategy: Definition
Definition (Mixed strategy)
Let si = (si1, si1, ..., sij , ..., sik) be the strategies set of a player i . Amixed strategy for this player is a distribution probabilityPi = (Pi1,Pi1, ...,Pij , ...,Pik ) with Pij is the realization probability
to choose sij and∑k
j=1 Pij = 1.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Expected utility function: Example
Player 2
Player 1H T
H (-1,1) (1,-1)T (1,-1) (-1,1)
If player 1 thinks that player 2 play strategy H withprobability q and play strategy q with probability 1− q, thenthe expected utility of player 1 is:
1 q(−1) + (1− q)1 = 1− 2q if player 1 play H;2 q(1) + (1− q)(−1) = 2q − 1 if player 1 play T .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Expected utility function: Example
Player 2
Player 1H T
H (-1,1) (1,-1)T (1,-1) (-1,1)
If player 1 thinks that player 2 play strategy H withprobability q and play strategy q with probability 1− q, thenthe expected utility of player 1 is:
1 q(−1) + (1− q)1 = 1− 2q if player 1 play H;2 q(1) + (1− q)(−1) = 2q − 1 if player 1 play T .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Expected utility function: Example
Player 2
Player 1H T
H (-1,1) (1,-1)T (1,-1) (-1,1)
If player 1 thinks that player 2 play strategy H withprobability q and play strategy q with probability 1− q, thenthe expected utility of player 1 is:
1 q(−1) + (1− q)1 = 1− 2q if player 1 play H;2 q(1) + (1− q)(−1) = 2q − 1 if player 1 play T .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Expected utility function: Example
Player 2
Player 1H T
H (-1,1) (1,-1)T (1,-1) (-1,1)
If player 1 thinks that player 2 play strategy H withprobability q and play strategy q with probability 1− q, thenthe expected utility of player 1 is:
1 q(−1) + (1− q)1 = 1− 2q if player 1 play H;2 q(1) + (1− q)(−1) = 2q − 1 if player 1 play T .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Expected utility function: Notation
Let ∆(Si ) denote the set of probability distributions over Si ;
We identify this set with the simplex: Pi ∈ ∆(Si ) implies∑si∈Si
Pi (si ) = 1 and Pi (si ) ≥ 0;
We refer to Pi (si ) as a mixed strategy, while the members ofSi are pure strategies;
The support of a probability measure µ is defined as x :µ(x) > 0. For P ∈ ∆(Si ) the support of Pi is all si such thatPi (si ) ≥ 0.
The probability of obtaining a specific (pure) strategies profiles = (sj )j∈N is
∏j∈N Pj (sj ).
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Expected utility function: Notation
Let ∆(Si ) denote the set of probability distributions over Si ;
We identify this set with the simplex: Pi ∈ ∆(Si ) implies∑si∈Si
Pi (si ) = 1 and Pi (si ) ≥ 0;
We refer to Pi (si ) as a mixed strategy, while the members ofSi are pure strategies;
The support of a probability measure µ is defined as x :µ(x) > 0. For P ∈ ∆(Si ) the support of Pi is all si such thatPi (si ) ≥ 0.
The probability of obtaining a specific (pure) strategies profiles = (sj )j∈N is
∏j∈N Pj (sj ).
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Expected utility function: Notation
Let ∆(Si ) denote the set of probability distributions over Si ;
We identify this set with the simplex: Pi ∈ ∆(Si ) implies∑si∈Si
Pi (si ) = 1 and Pi (si ) ≥ 0;
We refer to Pi (si ) as a mixed strategy, while the members ofSi are pure strategies;
The support of a probability measure µ is defined as x :µ(x) > 0. For P ∈ ∆(Si ) the support of Pi is all si such thatPi (si ) ≥ 0.
The probability of obtaining a specific (pure) strategies profiles = (sj )j∈N is
∏j∈N Pj (sj ).
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Expected utility function: Notation
Let ∆(Si ) denote the set of probability distributions over Si ;
We identify this set with the simplex: Pi ∈ ∆(Si ) implies∑si∈Si
Pi (si ) = 1 and Pi (si ) ≥ 0;
We refer to Pi (si ) as a mixed strategy, while the members ofSi are pure strategies;
The support of a probability measure µ is defined as x :µ(x) > 0. For P ∈ ∆(Si ) the support of Pi is all si such thatPi (si ) ≥ 0.
The probability of obtaining a specific (pure) strategies profiles = (sj )j∈N is
∏j∈N Pj (sj ).
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Expected utility function: Notation
Let ∆(Si ) denote the set of probability distributions over Si ;
We identify this set with the simplex: Pi ∈ ∆(Si ) implies∑si∈Si
Pi (si ) = 1 and Pi (si ) ≥ 0;
We refer to Pi (si ) as a mixed strategy, while the members ofSi are pure strategies;
The support of a probability measure µ is defined as x :µ(x) > 0. For P ∈ ∆(Si ) the support of Pi is all si such thatPi (si ) ≥ 0.
The probability of obtaining a specific (pure) strategies profiles = (sj )j∈N is
∏j∈N Pj (sj ).
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Expected utility function: Definition
Ui (P) =∑
s∈S (∏
j∈N Pj (sj ))ui (s).
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Two player mixed Nash equilibrium: Definition
Definition
A mixed strategy profile P∗ is a mixed strategy Nash Equilibriumfor two player ifU1(P∗1 ,P
∗2 ) ≥ U1(P1,P
∗2 ) for all distribution P1 on S1;
U2(P∗1 ,P∗2 ) ≥ U2(P∗1 ,P2) for all distribution P2 on S2.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Two player mixed Nash equilibrium: Example
Player 2
Player 1H T
H (-1,1) (1,-1)T (1,-1) (-1,1)
If player 1 thinks that player 2 play strategy H withprobability q and play strategy T with probability 1− q, thenthe expected utility of player 1 is:
1 q(−1) + (1− q)1 = 1− 2q if player 1 play H;2 q(1) + (1− q)(−1) = 2q − 1 if player 1 play T .
The best response of player 1 in pure strategy is H if q < 12 , is
T if q > 12 , and player is indifferent between H and T if
q = 12 .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Two player mixed Nash equilibrium: Example
Player 2
Player 1H T
H (-1,1) (1,-1)T (1,-1) (-1,1)
If player 1 thinks that player 2 play strategy H withprobability q and play strategy T with probability 1− q, thenthe expected utility of player 1 is:
1 q(−1) + (1− q)1 = 1− 2q if player 1 play H;2 q(1) + (1− q)(−1) = 2q − 1 if player 1 play T .
The best response of player 1 in pure strategy is H if q < 12 , is
T if q > 12 , and player is indifferent between H and T if
q = 12 .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Two player mixed Nash equilibrium: Example
Player 2
Player 1H T
H (-1,1) (1,-1)T (1,-1) (-1,1)
If player 1 thinks that player 2 play strategy H withprobability q and play strategy T with probability 1− q, thenthe expected utility of player 1 is:
1 q(−1) + (1− q)1 = 1− 2q if player 1 play H;2 q(1) + (1− q)(−1) = 2q − 1 if player 1 play T .
The best response of player 1 in pure strategy is H if q < 12 , is
T if q > 12 , and player is indifferent between H and T if
q = 12 .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Two player mixed Nash equilibrium: Example
Player 2
Player 1H T
H (-1,1) (1,-1)T (1,-1) (-1,1)
If player 1 thinks that player 2 play strategy H withprobability q and play strategy T with probability 1− q, thenthe expected utility of player 1 is:
1 q(−1) + (1− q)1 = 1− 2q if player 1 play H;2 q(1) + (1− q)(−1) = 2q − 1 if player 1 play T .
The best response of player 1 in pure strategy is H if q < 12 , is
T if q > 12 , and player is indifferent between H and T if
q = 12 .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Two player mixed Nash equilibrium: Example
Player 2
Player 1H T
H (-1,1) (1,-1)T (1,-1) (-1,1)
If player 1 thinks that player 2 play strategy H withprobability q and play strategy T with probability 1− q, thenthe expected utility of player 1 is:
1 q(−1) + (1− q)1 = 1− 2q if player 1 play H;2 q(1) + (1− q)(−1) = 2q − 1 if player 1 play T .
The best response of player 1 in pure strategy is H if q < 12 , is
T if q > 12 , and player is indifferent between H and T if
q = 12 .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Two player mixed Nash equilibrium: Example
The best responses in mixed strategies:1 For player 1: If player 1 play H with probability r and T with
probability 1− r , then for each value of q, we will seek thevalue of r noted r∗(q): (r∗, 1− r∗) is better response to(q, 1− q);X U1(P1,P2) = r(1− 2q) + (1− r)(2q − 1)= (2q − 1) + r(2− 4q), this function is increasing in r if2− 4q > 0 and it is decreasing if 2− 4q < 0;X The best response r = 1 (i.e. H) if q < 1
2 , and the bestresponse r = 0 (i.e. T ) if q > 1
2 ;
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Two player mixed Nash equilibrium: Example
The best responses in mixed strategies:1 For player 1: If player 1 play H with probability r and T with
probability 1− r , then for each value of q, we will seek thevalue of r noted r∗(q): (r∗, 1− r∗) is better response to(q, 1− q);X U1(P1,P2) = r(1− 2q) + (1− r)(2q − 1)= (2q − 1) + r(2− 4q), this function is increasing in r if2− 4q > 0 and it is decreasing if 2− 4q < 0;X The best response r = 1 (i.e. H) if q < 1
2 , and the bestresponse r = 0 (i.e. T ) if q > 1
2 ;
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Two player mixed Nash equilibrium: Example
The best responses in mixed strategies:1 For player 1: If player 1 play H with probability r and T with
probability 1− r , then for each value of q, we will seek thevalue of r noted r∗(q): (r∗, 1− r∗) is better response to(q, 1− q);X U1(P1,P2) = r(1− 2q) + (1− r)(2q − 1)= (2q − 1) + r(2− 4q), this function is increasing in r if2− 4q > 0 and it is decreasing if 2− 4q < 0;X The best response r = 1 (i.e. H) if q < 1
2 , and the bestresponse r = 0 (i.e. T ) if q > 1
2 ;
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Two player mixed Nash equilibrium: Example
The best responses in mixed strategies:1 For player 1: If player 1 play H with probability r and T with
probability 1− r , then for each value of q, we will seek thevalue of r noted r∗(q): (r∗, 1− r∗) is better response to(q, 1− q);X U1(P1,P2) = r(1− 2q) + (1− r)(2q − 1)= (2q − 1) + r(2− 4q), this function is increasing in r if2− 4q > 0 and it is decreasing if 2− 4q < 0;X The best response r = 1 (i.e. H) if q < 1
2 , and the bestresponse r = 0 (i.e. T ) if q > 1
2 ;
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Two player mixed Nash equilibrium: Example
The best responses in mixed strategies:1 We make the same reasoning for player 2: for each value of r ,
we will seek the value of q noted q∗(r): (q∗, 1− q∗) is betterresponse to (r , 1− r);X If r < 1
2 , the best response is T (q∗(r) = 0;X If r > 1
2 , the best response is H (q∗(r) = 1;X If r = 1
2 , indifference.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Two player mixed Nash equilibrium: Example
The best responses in mixed strategies:1 We make the same reasoning for player 2: for each value of r ,
we will seek the value of q noted q∗(r): (q∗, 1− q∗) is betterresponse to (r , 1− r);X If r < 1
2 , the best response is T (q∗(r) = 0;X If r > 1
2 , the best response is H (q∗(r) = 1;X If r = 1
2 , indifference.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Two player mixed Nash equilibrium: Example
The best responses in mixed strategies:1 We make the same reasoning for player 2: for each value of r ,
we will seek the value of q noted q∗(r): (q∗, 1− q∗) is betterresponse to (r , 1− r);X If r < 1
2 , the best response is T (q∗(r) = 0;X If r > 1
2 , the best response is H (q∗(r) = 1;X If r = 1
2 , indifference.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Two player mixed Nash equilibrium: Example
The best responses in mixed strategies:1 We make the same reasoning for player 2: for each value of r ,
we will seek the value of q noted q∗(r): (q∗, 1− q∗) is betterresponse to (r , 1− r);X If r < 1
2 , the best response is T (q∗(r) = 0;X If r > 1
2 , the best response is H (q∗(r) = 1;X If r = 1
2 , indifference.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Two player mixed Nash equilibrium: Example
The best responses in mixed strategies:1 We make the same reasoning for player 2: for each value of r ,
we will seek the value of q noted q∗(r): (q∗, 1− q∗) is betterresponse to (r , 1− r);X If r < 1
2 , the best response is T (q∗(r) = 0;X If r > 1
2 , the best response is H (q∗(r) = 1;X If r = 1
2 , indifference.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Two player mixed Nash equilibrium: Example
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium
Many players mixed Nash equilibrium
Theorem (Nash, 1951)
Let Γ = {N, (Si )i∈N , (ui )i∈N} be strategic form game. If N is finiteand Si is finite for all i , then, there is at least a Nash equilibrium inmixed strategies.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
Plan1 Introduction2 Simultaneous Games with complete - imperfect information
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium3 Dynamic (Extensive form) games with perfect information
The concept of subgamesEconomic application: Stackelberg Model (1934)
4 Dynamic (Sequential) games with complete and imperfectinformation
5 Nash ImplementationIntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rules
Borda rulePlurality ruleAnti-plurality rule
Danilov’s - Yamato’s theorems6 Nash Implementation under Domain Restrictions with
IndifferencesNash implementation in exchange economies withsingle-plateaued preferences
New sufficient Conditions
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
The concept of subgames
The players intervene the ones after the others in a preciseorder;
We suppose the finite number of strategies;
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
The concept of subgames
The players intervene the ones after the others in a preciseorder;
We suppose the finite number of strategies;
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
The concept of subgames
The presentation of such game is to trace a tree (tree ofKuhn):
If the number of blows is finite, the tree ends with sequencesof numbers that give the payoff of each player.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
The concept of subgames
The presentation of such game is to trace a tree (tree ofKuhn):
If the number of blows is finite, the tree ends with sequencesof numbers that give the payoff of each player.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
The concept of subgames: Example
Suppose that we have a firm in monopoly situation (M);
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
The concept of subgames: Example
Suppose that we have a new firm (NF ) who want to make adecision to enter or not to the market;
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
The concept of subgames: Example
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
The concept of subgames: Backward induction method
Monopoly firm M decision:1 if NF does not enter, it suffices for M to continue to produce
the same quantity;2 if NF enters, M may find it beneficial to yield since it ensures a
positive profit equalizes to 4.
New firm NF decision: the firm NF which anticipates thechoice of M by putting itself at the place of M:
1 NF decides to enter;
Solution: NF enters and M yields ⇒ payoff (4,4).
This method is called: Backward induction method: itconsists in reasoning starting from the end.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
The concept of subgames: Backward induction method
Monopoly firm M decision:1 if NF does not enter, it suffices for M to continue to produce
the same quantity;2 if NF enters, M may find it beneficial to yield since it ensures a
positive profit equalizes to 4.
New firm NF decision: the firm NF which anticipates thechoice of M by putting itself at the place of M:
1 NF decides to enter;
Solution: NF enters and M yields ⇒ payoff (4,4).
This method is called: Backward induction method: itconsists in reasoning starting from the end.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
The concept of subgames: Backward induction method
Monopoly firm M decision:1 if NF does not enter, it suffices for M to continue to produce
the same quantity;2 if NF enters, M may find it beneficial to yield since it ensures a
positive profit equalizes to 4.
New firm NF decision: the firm NF which anticipates thechoice of M by putting itself at the place of M:
1 NF decides to enter;
Solution: NF enters and M yields ⇒ payoff (4,4).
This method is called: Backward induction method: itconsists in reasoning starting from the end.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
The concept of subgames: Backward induction method
Monopoly firm M decision:1 if NF does not enter, it suffices for M to continue to produce
the same quantity;2 if NF enters, M may find it beneficial to yield since it ensures a
positive profit equalizes to 4.
New firm NF decision: the firm NF which anticipates thechoice of M by putting itself at the place of M:
1 NF decides to enter;
Solution: NF enters and M yields ⇒ payoff (4,4).
This method is called: Backward induction method: itconsists in reasoning starting from the end.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
The concept of subgames: Backward induction method
Monopoly firm M decision:1 if NF does not enter, it suffices for M to continue to produce
the same quantity;2 if NF enters, M may find it beneficial to yield since it ensures a
positive profit equalizes to 4.
New firm NF decision: the firm NF which anticipates thechoice of M by putting itself at the place of M:
1 NF decides to enter;
Solution: NF enters and M yields ⇒ payoff (4,4).
This method is called: Backward induction method: itconsists in reasoning starting from the end.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
The concept of subgames: Backward induction method
Monopoly firm M decision:1 if NF does not enter, it suffices for M to continue to produce
the same quantity;2 if NF enters, M may find it beneficial to yield since it ensures a
positive profit equalizes to 4.
New firm NF decision: the firm NF which anticipates thechoice of M by putting itself at the place of M:
1 NF decides to enter;
Solution: NF enters and M yields ⇒ payoff (4,4).
This method is called: Backward induction method: itconsists in reasoning starting from the end.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
The concept of subgames: Backward induction method
Monopoly firm M decision:1 if NF does not enter, it suffices for M to continue to produce
the same quantity;2 if NF enters, M may find it beneficial to yield since it ensures a
positive profit equalizes to 4.
New firm NF decision: the firm NF which anticipates thechoice of M by putting itself at the place of M:
1 NF decides to enter;
Solution: NF enters and M yields ⇒ payoff (4,4).
This method is called: Backward induction method: itconsists in reasoning starting from the end.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
Backward induction method: Example
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
Backward induction method: Example
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
Backward induction method: Example
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
Backward induction method: Example
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
Backward induction method: Example
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
Backward induction method: Example
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
Backward induction method: Example
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
Backward induction method: Example
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
Backward induction method: Example
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
Backward induction method: Example
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
Backward induction method: Example
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
Backward induction method: Example
Solution strategies:
Payoff:
Remark: This solution is not Pareto optimal < (5,5,5)
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
Backward induction method: Example
Solution strategies:
Payoff:
Remark: This solution is not Pareto optimal < (5,5,5)
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
Kuhn’s Theorem, 1953
Theorem (Kuhn, 1953)
Every dynamic game with perfect information has a Nashequilibrium.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
Plan1 Introduction2 Simultaneous Games with complete - imperfect information
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium3 Dynamic (Extensive form) games with perfect information
The concept of subgamesEconomic application: Stackelberg Model (1934)
4 Dynamic (Sequential) games with complete and imperfectinformation
5 Nash ImplementationIntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rules
Borda rulePlurality ruleAnti-plurality rule
Danilov’s - Yamato’s theorems6 Nash Implementation under Domain Restrictions with
IndifferencesNash implementation in exchange economies withsingle-plateaued preferences
New sufficient Conditions
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
The concept of subgamesEconomic application: Stackelberg Model (1934)
Economic application: Stackelberg Model (1934)
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Definitions
4 Players
Players 1 and 2 choose simultaneously two strategies s1 ∈ S1
and s2 ∈ S2;
Players 3 and 4 observe s1 and s2 and choose simultaneouslys3 ∈ S3 and s4 ∈ S4;
ui (s1, s2, s3, s4), i = 1, ..., 4.
Definition
Suppose that (s∗1 , s∗2 ) is the unique Nash equilibrium in the first
stage and (s∗3 ((s∗1 , s∗2 )), s∗4 ((s∗1 , s
∗2 ))) a Nash equilibrium in the
second stage. The solution (s∗1 , s∗2 , s∗3 ((s∗1 , s
∗2 )), s∗4 ((s∗1 , s
∗2 ))) is
subgame perfect Nash equilibrium.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Definitions
4 Players
Players 1 and 2 choose simultaneously two strategies s1 ∈ S1
and s2 ∈ S2;
Players 3 and 4 observe s1 and s2 and choose simultaneouslys3 ∈ S3 and s4 ∈ S4;
ui (s1, s2, s3, s4), i = 1, ..., 4.
Definition
Suppose that (s∗1 , s∗2 ) is the unique Nash equilibrium in the first
stage and (s∗3 ((s∗1 , s∗2 )), s∗4 ((s∗1 , s
∗2 ))) a Nash equilibrium in the
second stage. The solution (s∗1 , s∗2 , s∗3 ((s∗1 , s
∗2 )), s∗4 ((s∗1 , s
∗2 ))) is
subgame perfect Nash equilibrium.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Definitions
4 Players
Players 1 and 2 choose simultaneously two strategies s1 ∈ S1
and s2 ∈ S2;
Players 3 and 4 observe s1 and s2 and choose simultaneouslys3 ∈ S3 and s4 ∈ S4;
ui (s1, s2, s3, s4), i = 1, ..., 4.
Definition
Suppose that (s∗1 , s∗2 ) is the unique Nash equilibrium in the first
stage and (s∗3 ((s∗1 , s∗2 )), s∗4 ((s∗1 , s
∗2 ))) a Nash equilibrium in the
second stage. The solution (s∗1 , s∗2 , s∗3 ((s∗1 , s
∗2 )), s∗4 ((s∗1 , s
∗2 ))) is
subgame perfect Nash equilibrium.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Definitions
4 Players
Players 1 and 2 choose simultaneously two strategies s1 ∈ S1
and s2 ∈ S2;
Players 3 and 4 observe s1 and s2 and choose simultaneouslys3 ∈ S3 and s4 ∈ S4;
ui (s1, s2, s3, s4), i = 1, ..., 4.
Definition
Suppose that (s∗1 , s∗2 ) is the unique Nash equilibrium in the first
stage and (s∗3 ((s∗1 , s∗2 )), s∗4 ((s∗1 , s
∗2 ))) a Nash equilibrium in the
second stage. The solution (s∗1 , s∗2 , s∗3 ((s∗1 , s
∗2 )), s∗4 ((s∗1 , s
∗2 ))) is
subgame perfect Nash equilibrium.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Definitions
4 Players
Players 1 and 2 choose simultaneously two strategies s1 ∈ S1
and s2 ∈ S2;
Players 3 and 4 observe s1 and s2 and choose simultaneouslys3 ∈ S3 and s4 ∈ S4;
ui (s1, s2, s3, s4), i = 1, ..., 4.
Definition
Suppose that (s∗1 , s∗2 ) is the unique Nash equilibrium in the first
stage and (s∗3 ((s∗1 , s∗2 )), s∗4 ((s∗1 , s
∗2 ))) a Nash equilibrium in the
second stage. The solution (s∗1 , s∗2 , s∗3 ((s∗1 , s
∗2 )), s∗4 ((s∗1 , s
∗2 ))) is
subgame perfect Nash equilibrium.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: international economy
Two countries 1 and 2
In each country we find:1 A government which selected a rate;2 A firm which produces a good for internal consumption and for
export;3 Consumers who buy from the inside market compounding
resident firm and nonresident firm.
Let Pi (Qi ) = a− Qi be inverse demand function of price Pi
where Qi is the total quantity in market i and a is theabsorption capacity limit of market.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: international economy
Two countries 1 and 2
In each country we find:1 A government which selected a rate;2 A firm which produces a good for internal consumption and for
export;3 Consumers who buy from the inside market compounding
resident firm and nonresident firm.
Let Pi (Qi ) = a− Qi be inverse demand function of price Pi
where Qi is the total quantity in market i and a is theabsorption capacity limit of market.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: international economy
Two countries 1 and 2
In each country we find:1 A government which selected a rate;2 A firm which produces a good for internal consumption and for
export;3 Consumers who buy from the inside market compounding
resident firm and nonresident firm.
Let Pi (Qi ) = a− Qi be inverse demand function of price Pi
where Qi is the total quantity in market i and a is theabsorption capacity limit of market.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: international economy
Two countries 1 and 2
In each country we find:1 A government which selected a rate;2 A firm which produces a good for internal consumption and for
export;3 Consumers who buy from the inside market compounding
resident firm and nonresident firm.
Let Pi (Qi ) = a− Qi be inverse demand function of price Pi
where Qi is the total quantity in market i and a is theabsorption capacity limit of market.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: international economy
Two countries 1 and 2
In each country we find:1 A government which selected a rate;2 A firm which produces a good for internal consumption and for
export;3 Consumers who buy from the inside market compounding
resident firm and nonresident firm.
Let Pi (Qi ) = a− Qi be inverse demand function of price Pi
where Qi is the total quantity in market i and a is theabsorption capacity limit of market.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: international economy
Two countries 1 and 2
In each country we find:1 A government which selected a rate;2 A firm which produces a good for internal consumption and for
export;3 Consumers who buy from the inside market compounding
resident firm and nonresident firm.
Let Pi (Qi ) = a− Qi be inverse demand function of price Pi
where Qi is the total quantity in market i and a is theabsorption capacity limit of market.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: international economy
Two countries 1 and 2
In each country we find:1 A government which selected a rate;2 A firm which produces a good for internal consumption and for
export;3 Consumers who buy from the inside market compounding
resident firm and nonresident firm.
Let Pi (Qi ) = a− Qi be inverse demand function of price Pi
where Qi is the total quantity in market i and a is theabsorption capacity limit of market.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: international economy
A firm in country i produces a quantity hi for internalconsumption and a quantity ei for export;
Qi = hi + ej , where ej is the exported quantity by the foreignfirm j to the market of country i ;
Firms have constants marginal costs c and no fixed cost,ci (hi , ei ) = c(hi + ei );
Each firm i pays a tax tj if it exports ei .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: international economy
A firm in country i produces a quantity hi for internalconsumption and a quantity ei for export;
Qi = hi + ej , where ej is the exported quantity by the foreignfirm j to the market of country i ;
Firms have constants marginal costs c and no fixed cost,ci (hi , ei ) = c(hi + ei );
Each firm i pays a tax tj if it exports ei .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: international economy
A firm in country i produces a quantity hi for internalconsumption and a quantity ei for export;
Qi = hi + ej , where ej is the exported quantity by the foreignfirm j to the market of country i ;
Firms have constants marginal costs c and no fixed cost,ci (hi , ei ) = c(hi + ei );
Each firm i pays a tax tj if it exports ei .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: international economy
A firm in country i produces a quantity hi for internalconsumption and a quantity ei for export;
Qi = hi + ej , where ej is the exported quantity by the foreignfirm j to the market of country i ;
Firms have constants marginal costs c and no fixed cost,ci (hi , ei ) = c(hi + ei );
Each firm i pays a tax tj if it exports ei .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: international economy
The game:
1 Governments play in non-cooperative way in variable tax;
2 Firms observe tax rates and produce for internal consumptionand for export ;
3 The payoff functions for firms: Πi (ti , tj , hi , ei , hj , ej ) =[a− (hi + ej )]hi + [a− (ei + hj )]ei − c(hi + ei )− tj ei
For governments;payoffs= the total welfare;
= consumer surplus + profit of the resident firm + tax;= 1
2 Q2i + Πi + ti ej .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: international economy
The game:
1 Governments play in non-cooperative way in variable tax;
2 Firms observe tax rates and produce for internal consumptionand for export ;
3 The payoff functions for firms: Πi (ti , tj , hi , ei , hj , ej ) =[a− (hi + ej )]hi + [a− (ei + hj )]ei − c(hi + ei )− tj ei
For governments;payoffs= the total welfare;
= consumer surplus + profit of the resident firm + tax;= 1
2 Q2i + Πi + ti ej .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: international economy
The game:
1 Governments play in non-cooperative way in variable tax;
2 Firms observe tax rates and produce for internal consumptionand for export ;
3 The payoff functions for firms: Πi (ti , tj , hi , ei , hj , ej ) =[a− (hi + ej )]hi + [a− (ei + hj )]ei − c(hi + ei )− tj ei
For governments;payoffs= the total welfare;
= consumer surplus + profit of the resident firm + tax;= 1
2 Q2i + Πi + ti ej .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: international economy
The game:
1 Governments play in non-cooperative way in variable tax;
2 Firms observe tax rates and produce for internal consumptionand for export ;
3 The payoff functions for firms: Πi (ti , tj , hi , ei , hj , ej ) =[a− (hi + ej )]hi + [a− (ei + hj )]ei − c(hi + ei )− tj ei
For governments;payoffs= the total welfare;
= consumer surplus + profit of the resident firm + tax;= 1
2 Q2i + Πi + ti ej .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: international economy
The game:
1 Governments play in non-cooperative way in variable tax;
2 Firms observe tax rates and produce for internal consumptionand for export ;
3 The payoff functions for firms: Πi (ti , tj , hi , ei , hj , ej ) =[a− (hi + ej )]hi + [a− (ei + hj )]ei − c(hi + ei )− tj ei
For governments;payoffs= the total welfare;
= consumer surplus + profit of the resident firm + tax;= 1
2 Q2i + Πi + ti ej .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: international economy
The game:
1 Governments play in non-cooperative way in variable tax;
2 Firms observe tax rates and produce for internal consumptionand for export ;
3 The payoff functions for firms: Πi (ti , tj , hi , ei , hj , ej ) =[a− (hi + ej )]hi + [a− (ei + hj )]ei − c(hi + ei )− tj ei
For governments;payoffs= the total welfare;
= consumer surplus + profit of the resident firm + tax;= 1
2 Q2i + Πi + ti ej .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: international economy
The game:
1 Governments play in non-cooperative way in variable tax;
2 Firms observe tax rates and produce for internal consumptionand for export ;
3 The payoff functions for firms: Πi (ti , tj , hi , ei , hj , ej ) =[a− (hi + ej )]hi + [a− (ei + hj )]ei − c(hi + ei )− tj ei
For governments;payoffs= the total welfare;
= consumer surplus + profit of the resident firm + tax;= 1
2 Q2i + Πi + ti ej .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: international economy
We solve this game:
If (t∗1 , t∗2 ) is a Nash equilibrium in stage 1 and if
(h∗1, e∗1 , h∗2, e∗2 ) is a Nash equilibrium in stage 1, then (h∗i , e
∗i )
must solve Maxhi ,ei≥0 Πi (ti , tj , hi , ei , h∗j , e∗j ) (*);
Πi (ti , tj , hi , ei , h∗j , e∗j ) =
hi [a− (hi + e∗j )− c] + ei [a− (ei + h∗j )− c]− tj ei
= profit in market i+ profit in market j .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: international economy
We solve this game:
If (t∗1 , t∗2 ) is a Nash equilibrium in stage 1 and if
(h∗1, e∗1 , h∗2, e∗2 ) is a Nash equilibrium in stage 1, then (h∗i , e
∗i )
must solve Maxhi ,ei≥0 Πi (ti , tj , hi , ei , h∗j , e∗j ) (*);
Πi (ti , tj , hi , ei , h∗j , e∗j ) =
hi [a− (hi + e∗j )− c] + ei [a− (ei + h∗j )− c]− tj ei
= profit in market i+ profit in market j .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: international economy
We solve this game:
If (t∗1 , t∗2 ) is a Nash equilibrium in stage 1 and if
(h∗1, e∗1 , h∗2, e∗2 ) is a Nash equilibrium in stage 1, then (h∗i , e
∗i )
must solve Maxhi ,ei≥0 Πi (ti , tj , hi , ei , h∗j , e∗j ) (*);
Πi (ti , tj , hi , ei , h∗j , e∗j ) =
hi [a− (hi + e∗j )− c] + ei [a− (ei + h∗j )− c]− tj ei
= profit in market i+ profit in market j .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: international economy
(*) becomes:
Maxhi≥0 hi [a− (hi + e∗j )− c] ⇒ h∗i = a−c+ti3 ;
Maxei≥0 ei [a− (ei + h∗j )− c]− tj ei ⇒ e∗i =a−c−2tj
3 .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: international economy
(*) becomes:
Maxhi≥0 hi [a− (hi + e∗j )− c] ⇒ h∗i = a−c+ti3 ;
Maxei≥0 ei [a− (ei + h∗j )− c]− tj ei ⇒ e∗i =a−c−2tj
3 .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: international economy
(*) becomes:
Maxhi≥0 hi [a− (hi + e∗j )− c] ⇒ h∗i = a−c+ti3 ;
Maxei≥0 ei [a− (ei + h∗j )− c]− tj ei ⇒ e∗i =a−c−2tj
3 .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: international economy
After solving the second stage, we will solve the first stage:
Maxti≥0 Πi (ti , tj , h∗i , e∗i , h∗j , e∗j );
After calculates: Totalwelfare= (2(a−c)−ti )
2
18 + (a−c+ti )2
9 +(a−c−2tj )
2
9 + ti(a−c−2ti )
3 ;
⇒ t∗i = a−c3 , i = 1, 2;
⇒ h∗i = 4(a−c)9 , e∗i = a−c
9 .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: international economy
After solving the second stage, we will solve the first stage:
Maxti≥0 Πi (ti , tj , h∗i , e∗i , h∗j , e∗j );
After calculates: Totalwelfare= (2(a−c)−ti )
2
18 + (a−c+ti )2
9 +(a−c−2tj )
2
9 + ti(a−c−2ti )
3 ;
⇒ t∗i = a−c3 , i = 1, 2;
⇒ h∗i = 4(a−c)9 , e∗i = a−c
9 .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: international economy
After solving the second stage, we will solve the first stage:
Maxti≥0 Πi (ti , tj , h∗i , e∗i , h∗j , e∗j );
After calculates: Totalwelfare= (2(a−c)−ti )
2
18 + (a−c+ti )2
9 +(a−c−2tj )
2
9 + ti(a−c−2ti )
3 ;
⇒ t∗i = a−c3 , i = 1, 2;
⇒ h∗i = 4(a−c)9 , e∗i = a−c
9 .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: international economy
After solving the second stage, we will solve the first stage:
Maxti≥0 Πi (ti , tj , h∗i , e∗i , h∗j , e∗j );
After calculates: Totalwelfare= (2(a−c)−ti )
2
18 + (a−c+ti )2
9 +(a−c−2tj )
2
9 + ti(a−c−2ti )
3 ;
⇒ t∗i = a−c3 , i = 1, 2;
⇒ h∗i = 4(a−c)9 , e∗i = a−c
9 .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Example: international economy
After solving the second stage, we will solve the first stage:
Maxti≥0 Πi (ti , tj , h∗i , e∗i , h∗j , e∗j );
After calculates: Totalwelfare= (2(a−c)−ti )
2
18 + (a−c+ti )2
9 +(a−c−2tj )
2
9 + ti(a−c−2ti )
3 ;
⇒ t∗i = a−c3 , i = 1, 2;
⇒ h∗i = 4(a−c)9 , e∗i = a−c
9 .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
IntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rulesDanilov’s - Yamato’s theorems
Plan1 Introduction2 Simultaneous Games with complete - imperfect information
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium3 Dynamic (Extensive form) games with perfect information
The concept of subgamesEconomic application: Stackelberg Model (1934)
4 Dynamic (Sequential) games with complete and imperfectinformation
5 Nash ImplementationIntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rules
Borda rulePlurality ruleAnti-plurality rule
Danilov’s - Yamato’s theorems6 Nash Implementation under Domain Restrictions with
IndifferencesNash implementation in exchange economies withsingle-plateaued preferences
New sufficient Conditions
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
IntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rulesDanilov’s - Yamato’s theorems
1 Introduction2 Simultaneous Games with complete - imperfect information
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium3 Dynamic (Extensive form) games with perfect information
The concept of subgamesEconomic application: Stackelberg Model (1934)
4 Dynamic (Sequential) games with complete and imperfectinformation
5 Nash ImplementationIntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rules
Borda rulePlurality ruleAnti-plurality rule
Danilov’s - Yamato’s theorems6 Nash Implementation under Domain Restrictions with
IndifferencesNash implementation in exchange economies withsingle-plateaued preferences
New sufficient Conditions
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
IntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rulesDanilov’s - Yamato’s theorems
Definition of implementation
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
IntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rulesDanilov’s - Yamato’s theorems
Definition of the implementation
A social choice correspondence (SCC): is a correspondence Ffrom a set of class of admissible preference < into the set ofalternatives (or options) A, that associates with everypreference profile R a nonempty subset of A;
A solution concept: is represented by the set of Nashequilibria N(g ,R,S) of the game (Γ,R).
A mechanism (or form game): is given by Γ = (S , g) whereS = Πi∈NSi ; Si denotes the strategy set of the agent i and gis a function from S to A;
Implementability:F (R) = g(N(g ,R,S)).
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
IntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rulesDanilov’s - Yamato’s theorems
Maskin’s Mechanism (1977,1999)
- Ri : a preference relation of the individual i (complete, transitive,and reflexive);- L(a,Ri ) = {b ∈ A | aRi b} is the lower contour set for agent i atalternative a;Rule 1: If for each i ∈ N, si = (R, a,m) and a ∈ F (R), theng(s) = a.Rule 2: If for some i , sj = (R, a,m) for all j 6= i , a ∈ F (R) andsi = (R i , ai ,mi ) 6= (R, a,m), then:
g(s) =
{ai if ai ∈ L(a,Ri ),a otherwise.
Rule 3: In any other situation, g(s) = ai∗ , where i∗ is the index ofthe player of which the number mi∗ is largest. If there are severalindividuals who check this condition, the smallest index i will bechosen. Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
IntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rulesDanilov’s - Yamato’s theorems
Plan1 Introduction2 Simultaneous Games with complete - imperfect information
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium3 Dynamic (Extensive form) games with perfect information
The concept of subgamesEconomic application: Stackelberg Model (1934)
4 Dynamic (Sequential) games with complete and imperfectinformation
5 Nash ImplementationIntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rules
Borda rulePlurality ruleAnti-plurality rule
Danilov’s - Yamato’s theorems6 Nash Implementation under Domain Restrictions with
IndifferencesNash implementation in exchange economies withsingle-plateaued preferences
New sufficient Conditions
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
IntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rulesDanilov’s - Yamato’s theorems
Maskin’s theorems (1977,1999)
Definition (Maskin monotonicity)
A SCC F satisfies monotonicity if for all R,R ′ ∈ <, for anya ∈ F (R), if for any i ∈ N, L(a,Ri ) ⊆ L(a,R ′i ), then a ∈ F (R ′).
Example 1: F satisfies Maskin monotonicity.
R: R1 R2 R3
b b aa c cc a b
,R ′: R ′1 R ′2 R ′3
a b ab a cc c b
F (R) = {a} F (R ′) = {a}
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
IntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rulesDanilov’s - Yamato’s theorems
Maskin’s theorems (1977,1999)
Example 2: F does not satisfy Maskin monotonicity.
R: R1 R2 R3
b b aa c cc a b
,R ′: R ′1 R ′2 R ′3
b a aa b cc c b
F (R) = {a, b} F (R ′) = {b}
Theorem (Maskin 1977,1999)
If a SCC F is Nash implementable, then F satisfies monotonicity.
Proof.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
IntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rulesDanilov’s - Yamato’s theorems
Maskin’s theorems (1977,1999)
Definition (No veto power)
A SCC F satisfies no veto power if for i , R ∈ <, and a ∈ A, ifL(a,Rj ) = A for all j ∈ N\{i}, then a ∈ F (R).
Example 3:
R: R1 R2 R3
a a bb c cc b a
F (R) = {a}
Theorem (Maskin 1977,1999)
If n ≥ 3, and if a SCC satisfies monotonicity and no veto powerconditions, then F is Nash implementable.Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
IntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rulesDanilov’s - Yamato’s theorems
Plan1 Introduction2 Simultaneous Games with complete - imperfect information
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium3 Dynamic (Extensive form) games with perfect information
The concept of subgamesEconomic application: Stackelberg Model (1934)
4 Dynamic (Sequential) games with complete and imperfectinformation
5 Nash ImplementationIntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rules
Borda rulePlurality ruleAnti-plurality rule
Danilov’s - Yamato’s theorems6 Nash Implementation under Domain Restrictions with
IndifferencesNash implementation in exchange economies withsingle-plateaued preferences
New sufficient Conditions
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
IntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rulesDanilov’s - Yamato’s theorems
Borda rule
For this rule, each individual assigns points to alternativesaccording to its ranking, for example, for the m alternativesavailable, the favorite alternative obtains m points, the next m − 1and so on. The Borda rule selects alternatives which have thehighest score.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
IntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rulesDanilov’s - Yamato’s theorems
Borda rule
Example. Let N = {1, 2, 3} and X = {x , y , z}. Let R,R ′ ∈ < bedefined by:
R: R1 R2 R3
x y zy x xz z y
R ′: R ′1 R ′2 R ′3x,y y zz x x,y
zB(R) = {x} B(R ′) = {y}
We have x ∈ B(R), it is very easy to see that the inclusion ofMaskin monotonicity is checked, but x /∈ B(R ′).
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
IntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rulesDanilov’s - Yamato’s theorems
Plurality rule
For this rule, each individual assigns points to top-rankingalternatives. The Plurality rule selects alternatives which have thehighest score. The plurality rule does not satisfy Maskinmonotonicity as shown in the following example.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
IntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rulesDanilov’s - Yamato’s theorems
Plurality rule
Example. Let N = {1, 2, 3} and X = {x , y , z}. Let R,R ′ ∈ < bedefined by:
R: R1 R2 R3
y x xx y zz z y
R ′: R ′1 R ′2 R ′3y x,y x,y
x,z z z
P(R) = {x} P(R ′) = {y}We have x ∈ P(R), it is very easy to see that the inclusion Maskinmonotonicity is checked, but x /∈ P(R ′).
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
IntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rulesDanilov’s - Yamato’s theorems
Anti-plurality rule
For this rule, each individual assigns the equal points to all rankingalternatives, except for bottom-ranked alternative, it assigns zeropoint. The Anti-plurality rule selects alternatives which have thehighest score. This rule does not satisfy Maskin monotonicity asshown in the following example.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
IntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rulesDanilov’s - Yamato’s theorems
Anti-plurality rule
Example. Let N = {1, 2, 3} and X = {x , y , z}. Let R,R ′ ∈ < bedefined by:
R: R1 R2 R3
z y zy x xx z y
R ′: R ′1 R ′2 R ′3z y zy x,z xx y
A(R) = {x , y , z} A(R ′) = {z}We have x , y ∈ A(R), it is very easy to see that the inclusion ofMaskin monotonicity is checked, but x , y /∈ A(R ′) .
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
IntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rulesDanilov’s - Yamato’s theorems
Related literature: other works
Moore and Reppulo (1990): necessary and sufficientconditions;
Sjostrom(1991): algorithm;
Danilov (1992), Yamato (1992): strong monotonicity(sufficient);
Ziad (1997, 1998): version of strong monotonicity (necessary)+ algorithm;
Benoıt and Ok (2006): Maskin monotonicity + limited veto;
Doghmi and Ziad (2008), variants of monotonicity + variantsof no-veto power + unanimity.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
IntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rulesDanilov’s - Yamato’s theorems
Plan1 Introduction2 Simultaneous Games with complete - imperfect information
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium3 Dynamic (Extensive form) games with perfect information
The concept of subgamesEconomic application: Stackelberg Model (1934)
4 Dynamic (Sequential) games with complete and imperfectinformation
5 Nash ImplementationIntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rules
Borda rulePlurality ruleAnti-plurality rule
Danilov’s - Yamato’s theorems6 Nash Implementation under Domain Restrictions with
IndifferencesNash implementation in exchange economies withsingle-plateaued preferences
New sufficient Conditions
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
IntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rulesDanilov’s - Yamato’s theorems
Danilov’s - Yamato’s theorems: Essential options
Definition (Essential options)
Let i be a player and B ⊂ A. An alternative b ∈ B is essential for iin set B if b ∈ F (R) for some preference profile R such thatL(b,Ri ) ⊂ B. The set of all essential elements is denoted asEssi (F ,B).
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
IntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rulesDanilov’s - Yamato’s theorems
Danilov’s - Yamato’s theorems: Strong monotonicity
Definition (Strong monotonicity)
A SCC F satisfies strong monotonicity if for all R,R ′ ∈ < and forall a ∈ F (R), if for all i ∈ N, Essi (F , L(a,Ri )) ⊂ L(a,R ′i ), thena ∈ F (R ′).
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
IntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rulesDanilov’s - Yamato’s theorems
Danilov’s - Yamato’s theorems: Example of strongmonotonicity
Example 1: F satisfies strong monotonicity.
R: R1 R2 R3
a c cb b bc a a
,R ′: R ′1 R ′2 R ′3
a c cc a ab b b
F (R) = {a, b} F (R ′) = {a}
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
IntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rulesDanilov’s - Yamato’s theorems
Danilov’s - Yamato’s theorems: Example of strongmonotonicity.
Example 2: F does not satisfy strong monotonicity
R: R1 R2 R3
a c cb a ac b b
,R ′: R ′1 R ′2 R ′3
b c ca a ac b b
F (R) = {a, c} F (R ′) = {c}
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
IntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rulesDanilov’s - Yamato’s theorems
Danilov’s - Yamato’s theorems
Theorem (Danilov (1992), Yamato (1992))
If n ≥ 3, and if a SCC satisfies strong monotonicity, then F isNash implementable.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
IntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rulesDanilov’s - Yamato’s theorems
Yamato’s Mechanism
Rule 1: If for each i ∈ N, si = (R, a, 0) and a ∈ F (R), theng(s) = a.Rule 2: If for some i , sj = (R, a, 0) for all j 6= i , a ∈ F (R) andsi = (R i , ai ,mi ) 6= (R, a, 0), then:
g(s) =
{ai if ai ∈ Essi (F , L(a,Ri )),a otherwise.
Rule 3: In any other situation, g(s) = ai , wherei = (
∑j∈N nj )(mod n) + 1.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
IntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rulesDanilov’s - Yamato’s theorems
Applications in exchange economies with single-peakedpreferences
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
IntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rulesDanilov’s - Yamato’s theorems
Applications in exchange economies with single-peakedpreferences
Thomson (1990): intersections of the monotonic SCCs satisfyneither no veto power nor strong monotonicity. Heimplemented these correspondences by difficult algorithm;
Doghmi and Ziad (2008) identified new simple sufficientconditions to implement these correspondences.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
IntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rulesDanilov’s - Yamato’s theorems
What happens about the implementability of the SCCs inthese restricted domains when there are indifferentpreferences?
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Nash implementation in exchange economies with single-plateaued preferences
Plan1 Introduction2 Simultaneous Games with complete - imperfect information
Example: prisoner’s dilemma gameIterative elimination of strictly dominated strategiesNash EquilibriumNash Equilibrium and Pareto Optimality
Mixed Nash Equilibrium3 Dynamic (Extensive form) games with perfect information
The concept of subgamesEconomic application: Stackelberg Model (1934)
4 Dynamic (Sequential) games with complete and imperfectinformation
5 Nash ImplementationIntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rules
Borda rulePlurality ruleAnti-plurality rule
Danilov’s - Yamato’s theorems6 Nash Implementation under Domain Restrictions with
IndifferencesNash implementation in exchange economies withsingle-plateaued preferences
New sufficient Conditions
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Nash implementation in exchange economies with single-plateaued preferences
Single-plateaued preferences
Figure: Single-plateaued preferences.Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Nash implementation in exchange economies with single-plateaued preferences
Indifferent options subset
Definition (Indifferent options subset)
For any agent’s i preference Ri , any alternative a ∈ F (R), for somesingleton “operator” {o} ∈ LI (a,Ri ) with o 6= a, the indifferentoptions subset is the subset I (a, o,Ri ) = {b ∈ A \ {a, o} s.t.a ∼i b ∼i o}.
Remark
I (a, o,Ri ) 6= ∅ if |LI (a,Ri )| ≥ 3, otherwise I (a, o,Ri ) = ∅.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Nash implementation in exchange economies with single-plateaued preferences
Indifferent options subset
Definition (Indifferent options subset)
For any agent’s i preference Ri , any alternative a ∈ F (R), for somesingleton “operator” {o} ∈ LI (a,Ri ) with o 6= a, the indifferentoptions subset is the subset I (a, o,Ri ) = {b ∈ A \ {a, o} s.t.a ∼i b ∼i o}.
Remark
I (a, o,Ri ) 6= ∅ if |LI (a,Ri )| ≥ 3, otherwise I (a, o,Ri ) = ∅.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Nash implementation in exchange economies with single-plateaued preferences
I-monotonicity
Definition (I-monotonicity)
A SCC F satisfies I-monotonicity if for all R,R ′ ∈ <, for anya ∈ F (R), if for any i ∈ N, LS(a,Ri ) ∪ I (a, o,Ri ) ⊆ L(a,R ′i ), thena ∈ F (R ′).
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Nash implementation in exchange economies with single-plateaued preferences
Example
Example 1: A = {a, b, c, d , e, f }, N = {1, 2, 3} and < = {R,R ′}are defined by:
R: R1 R2 R3
e b b,da,c,d a,c a,c
f d fb f e
e
R ′: R ′1 R ′2 R ′3a,b,c,d,e b a,b
f ad c,dc e,f
e,f
F (R) = {a, f } F (R ′) = {a, b, c}
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Nash implementation in exchange economies with single-plateaued preferences
I -weak no veto power
Definition (I -weak no veto power)
A SCC F satisfies I -weak no veto power if for i , R ∈ <, anda ∈ F (R), if for R ′ ∈ <, b ∈ LS(a,Ri ) ∪ I (a, o,Ri ) ⊆ L(b,R ′i ) andL(b,R ′j ) = A for all j ∈ N\{i}, then b ∈ F (R ′).
Remark
If I (a, o,Ri ) = ∅, then I -weak no veto power becomes equivalentto strict weak no veto power. Otherwise, there is no-logicalrelationship between the two conditions.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Nash implementation in exchange economies with single-plateaued preferences
I -weak no veto power
Definition (I -weak no veto power)
A SCC F satisfies I -weak no veto power if for i , R ∈ <, anda ∈ F (R), if for R ′ ∈ <, b ∈ LS(a,Ri ) ∪ I (a, o,Ri ) ⊆ L(b,R ′i ) andL(b,R ′j ) = A for all j ∈ N\{i}, then b ∈ F (R ′).
Remark
If I (a, o,Ri ) = ∅, then I -weak no veto power becomes equivalentto strict weak no veto power. Otherwise, there is no-logicalrelationship between the two conditions.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Nash implementation in exchange economies with single-plateaued preferences
Unanimity
Definition (Unanimity)
An SCC F satisfies unanimity if for any a ∈ A, any R ∈ <, and forany i ∈ N, L(a,Ri ) = A implies a ∈ F (R).
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences
IntroductionSimultaneous Games with complete - imperfect information
Dynamic (Extensive form) games with perfect informationDynamic (Sequential) games with complete and imperfect information
Nash ImplementationNash Implementation under Domain Restrictions with Indifferences
Nash implementation in exchange economies with single-plateaued preferences
Doghmi and Ziad’s theorem
Theorem (Doghmi and Ziad (2012))
Let n ≥ 3. If a SCC F satisfies I-monotonicity, I -weak no vetopower and unanimity, then F can be implemented in Nashequilibria.
Ahmed Doghmi Games Theory Applied to Economics and Political Sciences