Course Games Theory INSEA 2012-2013

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

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.

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.

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...

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.

Now, we are going to look at the non-cooperative branch.

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.

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.

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.

Information:

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

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

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.

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

Prisoner j

Iterative elimination of strictly dominated strategies:Example 1

Prisoner j

Prisoner i C -6 0NC -10 -1

Iterative elimination of strictly dominated strategies:Example 1

Prisoner j

Iterative elimination of strictly dominatedstrategies:Example 1

I −6 > −10 and 0 > −1 ⇒ strategy NC is strictly dominated bystrategy C .

Prisoner j

C NC-6 -100 -1

Prisoner j

C NC-6 -100 -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.

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.

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.

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.

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).

Player 2

Player 1L M R

U (1,0) (1,2) (0,1)D (0,3) (0,1) (2,0)

Player 1L M R

U 1 1 0D 0 0 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.

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

Player 2

L M R0 2 13 1 0

I 2 > 1 and 1 > 0 ⇒ strategy R is strictly dominated by strategyM.

⇒ 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

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.

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.

⇒ 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.

Player 2

Player 1 L MU (1,0) (1,2)

Player 2

Player 1 L MU (1,0) (1,2)

Player 2

Player 1 L MU (1,0) (1,2)

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.

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)

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)

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)

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)

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).

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).

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}.

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;

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 .

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.

Multiplicity of Nash Equilibrium

Player 2

Player 1O F

O (2,1) (0,0)F (0,0) (1,2)

Player 2

Player 1O F

O (2,1) (0,0)F (0,0) (1,2)

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).

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).

A Game without a Nash Equilibrium

Player 2

Player 1H T

H (-1,1) (1,-1)T (1,-1) (-1,1)

Player 2

Player 1H T

H (-1,1) (1,-1)T (1,-1) (-1,1)

Player 2

Player 1H T

H (-1,1) (1,-1)T (1,-1) (-1,1)

This game does not admit 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.

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∗).

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∗).

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.

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.

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 .’

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].

Pareto Optimality: Example

Prisoner j

Prisoner iC NC

C (-6,-6) (0,-10)NC (-10,0) (-1,-1)

Prisoner j

Prisoner iC NC

C (-6,-6) (0,-10)NC (-10,0) (-1,-1)

Prisoner j

Prisoner iC NC

C (-6,-6) (0,-10)NC (-10,0) (-1,-1)

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.

Prisoner j

Prisoner iC NC

C (-6,-6) (0,-10)NC (-10,0) (−1,−1︸︷︷︸)

Prisoner j

Prisoner iC NC

C (-6,-6) (0,-10)NC (-10,0) (−1,−1︸︷︷︸)

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 )}.

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.

Pareto Optimal Nash Equilibrium: Example

Firm j

Firm iWait Invest

Wait (400,400) (0,300)Invest (300,0) (200,200)

Firm j

Firm iWait Invest

Wait (400,400) (0,300)Invest (300,0) (200,200)

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.

Firm j

Firm iWait Invest

Wait (400,400) (0,300)Invest (300,0) (200,200)

Firm j

Firm iWait Invest

Wait (400,400) (0,300)Invest (300,0) (200,200)

Example

Player 2

Player 1H T

H (-1,1) (1,-1)T (1,-1) (-1,1)

Example

Player 2

Player 1H T

H (-1,1) (1,-1)T (1,-1) (-1,1)

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.

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.

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 .

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 ) is equal for

L,M and R strategies.

Player 2

Player 1L M R

U (1,0) (1,2) (0,1)D (0,3) (0,1) (2,0)

13 ) is equal for

Player 2

Player 1L M R

U (1,0) (1,2) (0,1)D (0,3) (0,1) (2,0)

13 ) is equal for

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.

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 .

Player 2

Player 1H T

H (-1,1) (1,-1)T (1,-1) (-1,1)

Player 2

Player 1H T

H (-1,1) (1,-1)T (1,-1) (-1,1)

Player 2

Player 1H T

H (-1,1) (1,-1)T (1,-1) (-1,1)

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 ).

Pi (si ) = 1 and Pi (si ) ≥ 0;

∏j∈N Pj (sj ).

Pi (si ) = 1 and Pi (si ) ≥ 0;

∏j∈N Pj (sj ).

Pi (si ) = 1 and Pi (si ) ≥ 0;

∏j∈N Pj (sj ).

Pi (si ) = 1 and Pi (si ) ≥ 0;

∏j∈N Pj (sj ).

Expected utility function: Definition

Ui (P) =∑

s∈S (∏

j∈N Pj (sj ))ui (s).

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.

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:

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 .

Player 2

Player 1H T

H (-1,1) (1,-1)T (1,-1) (-1,1)

q = 12 .

Player 2

Player 1H T

H (-1,1) (1,-1)T (1,-1) (-1,1)

q = 12 .

Player 2

Player 1H T

H (-1,1) (1,-1)T (1,-1) (-1,1)

q = 12 .

Player 2

Player 1H T

H (-1,1) (1,-1)T (1,-1) (-1,1)

q = 12 .

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

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.

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.

The concept of subgames

The players intervene the ones after the others in a preciseorder;

We suppose the finite number of strategies;

The players intervene the ones after the others in a preciseorder;

We suppose the finite number of strategies;

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.

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.

The concept of subgames: Example

Suppose that we have a firm in monopoly situation (M);

Suppose that we have a new firm (NF ) who want to make adecision to enter or not to the market;

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.

Backward induction method: Example

Solution strategies:

Payoff:

Remark: This solution is not Pareto optimal < (5,5,5)

Solution strategies:

Payoff:

Remark: This solution is not Pareto optimal < (5,5,5)

Kuhn’s Theorem, 1953

Theorem (Kuhn, 1953)

Every dynamic game with perfect information has a Nashequilibrium.

Economic application: Stackelberg Model (1934)

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.

Definitions

4 Players

and s2 ∈ S2;

ui (s1, s2, s3, s4), i = 1, ..., 4.

Definition

stage and (s∗3 ((s∗1 , s∗2 )), s∗4 ((s∗1 , s

∗2 )), s∗4 ((s∗1 , s

∗2 ))) is

Definitions

4 Players

and s2 ∈ S2;

ui (s1, s2, s3, s4), i = 1, ..., 4.

Definition

stage and (s∗3 ((s∗1 , s∗2 )), s∗4 ((s∗1 , s

∗2 )), s∗4 ((s∗1 , s

∗2 ))) is

Definitions

4 Players

and s2 ∈ S2;

ui (s1, s2, s3, s4), i = 1, ..., 4.

Definition

stage and (s∗3 ((s∗1 , s∗2 )), s∗4 ((s∗1 , s

∗2 )), s∗4 ((s∗1 , s

∗2 ))) is

Definitions

4 Players

and s2 ∈ S2;

ui (s1, s2, s3, s4), i = 1, ..., 4.

Definition

stage and (s∗3 ((s∗1 , s∗2 )), s∗4 ((s∗1 , s

∗2 )), s∗4 ((s∗1 , s

∗2 ))) is

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.

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 .

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 .

The game:

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 .

We solve this game:

∗i )

Πi (ti , tj , hi , ei , h∗j , e∗j ) =

We solve this game:

∗i )

Πi (ti , tj , hi , ei , h∗j , e∗j ) =

(*) 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

(*) becomes:

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 )

18 + (a−c+ti )2

9 +(a−c−2tj )

9 + ti(a−c−2ti )

⇒ t∗i = a−c3 , i = 1, 2;

⇒ h∗i = 4(a−c)9 , e∗i = a−c

18 + (a−c+ti )2

9 +(a−c−2tj )

9 + ti(a−c−2ti )

⇒ t∗i = a−c3 , i = 1, 2;

⇒ h∗i = 4(a−c)9 , e∗i = a−c

18 + (a−c+ti )2

9 +(a−c−2tj )

9 + ti(a−c−2ti )

⇒ t∗i = a−c3 , i = 1, 2;

⇒ h∗i = 4(a−c)9 , e∗i = a−c

18 + (a−c+ti )2

9 +(a−c−2tj )

9 + ti(a−c−2ti )

⇒ t∗i = a−c3 , i = 1, 2;

⇒ h∗i = 4(a−c)9 , e∗i = a−c

18 + (a−c+ti )2

9 +(a−c−2tj )

9 + ti(a−c−2ti )

⇒ t∗i = a−c3 , i = 1, 2;

⇒ h∗i = 4(a−c)9 , e∗i = a−c

IntroductionMaskin’s theorems (1977,1999)Applications to political sciences: voting rulesDanilov’s - Yamato’s theorems

1 Introduction2 Simultaneous Games with complete - imperfect information

Definition of implementation

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)).

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

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}

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.

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

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.

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 ′).

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.

Plurality rule

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 ′).

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.

Anti-plurality rule

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 ′) .

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.

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).

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 ′).

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}

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}

Danilov’s - Yamato’s theorems

Theorem (Danilov (1992), Yamato (1992))

If n ≥ 3, and if a SCC satisfies strong monotonicity, then F isNash implementable.

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.

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.

What happens about the implementability of the SCCs inthese restricted domains when there are indifferentpreferences?

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

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 ) = ∅.

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 ) = ∅.

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 ′).

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

R ′: R ′1 R ′2 R ′3a,b,c,d,e b a,b

f ad c,dc e,f

F (R) = {a, f } F (R ′) = {a, b, c}

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.

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.

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).

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.

Course Games Theory INSEA 2012-2013

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