Game Theory(Economics)Nimish

Game TheoryGame Theory

The study of rational behavior among The study of rational behavior among interdependent agentsinterdependent agents

Agents have a common interest to make the pie Agents have a common interest to make the pie as large as possible, butas large as possible, but

Agents have competing interests to maximize Agents have competing interests to maximize their own share of the pie.their own share of the pie.

An agent’s rational decisions require An agent’s rational decisions require anticipating rivals’ responsesanticipating rivals’ responses

These expectations are not perfect, so These expectations are not perfect, so uncertainty is a necessary feature of gamesuncertainty is a necessary feature of games

Prisoner’s DilemmaPrisoner’s Dilemma

During the Stalinist EraDuring the Stalinist EraRussian Conductor studying Tchaikovsky Russian Conductor studying Tchaikovsky

score on the train to Minskscore on the train to Minsk

Arrested by the KGBArrested by the KGB

Thrown into prisonThrown into prison

For three days, he is told nothingFor three days, he is told nothing

…………………….Then…………….Then……………

Prisoner’s DilemmaPrisoner’s Dilemma

““We have your friend Tchaikovsky and We have your friend Tchaikovsky and he is starting to talk”he is starting to talk”

Should the conductor confess?Should the conductor confess?

Prisoner’s DilemmaPrisoner’s Dilemma

Confess Don’t ConfessConductor

Tchaikovsky

Confess

Don’t Confess

( -8, -8)

( -15, 0)

( 0, -15)

( -1, -1)

Prisoner’s DilemmaPrisoner’s Dilemma

Confess Don’t ConfessConductorConductor

Tchaikovsky

ConfessConfess

Don’t Confess

( -8-8, -8)

( -15, 0)

( 0, -15)

( -1, -1)

Prisoner’s DilemmaPrisoner’s Dilemma

Confess Don’t ConfessConductor

Tchaikovsky

Confess

Don’t Confess

( -8, -8)

( -15, 0)

( 00, , -15)

( -1, -1)

Prisoner’s DilemmaPrisoner’s Dilemma

Conclusion:Conclusion:

The Conductor will confessThe Conductor will confess

And Tchaikovsky?And Tchaikovsky?

Prisoner’s DilemmaPrisoner’s Dilemma

ConfessConfess Don’t ConfessConductor

TchaikovskyTchaikovsky

Confess

Don’t Confess

( -8, -8-8))

( -15, 0)

( 0, -15)

( -1, -1)

Prisoner’s DilemmaPrisoner’s Dilemma

ConfessConfess Don’t ConfessConductorConductor

TchaikovskyTchaikovsky

ConfessConfess

Don’t Confess

( -8, -8-8, -8))

( -15, 0)

( 0, -15)-15)

( -1, -1)

Prisoner’s DilemmaPrisoner’s Dilemma

Conclusion:Conclusion:

Tchaikovsky confesses alsoTchaikovsky confesses also

Both get 8 years, even though if they Both get 8 years, even though if they cooperated, they could get off with one cooperated, they could get off with one year eachyear each

For both, confession is a dominant For both, confession is a dominant strategy: a strategy that yields a better strategy: a strategy that yields a better outcome regardless of the opponent’s outcome regardless of the opponent’s choicechoice

Prisoner’s DilemmaPrisoner’s DilemmaWhat would the Conductor and What would the Conductor and Tchaikovsky decide if they could Tchaikovsky decide if they could negotiate?negotiate?

They could both become better off if they They could both become better off if they reached the cooperative solution….reached the cooperative solution….

which is why police interrogate suspects in which is why police interrogate suspects in separate rooms.separate rooms.

Equilibrium need not be efficient. Equilibrium need not be efficient. Noncooperative equilibrium in the Prisoner’s Noncooperative equilibrium in the Prisoner’s dilemma results in a solution that is not the best dilemma results in a solution that is not the best possible outcome for the parties.possible outcome for the parties.

EquilibriumEquilibrium

Nash Equilibrium: Neither player Nash Equilibrium: Neither player has an incentive to change strategy, has an incentive to change strategy, given the other player’s choicegiven the other player’s choice

Both confess is a Nash EquilibriumBoth confess is a Nash Equilibrium

Both don’t confess is not a Nash Both don’t confess is not a Nash Equilibrium, rival will always want to Equilibrium, rival will always want to renegerenege

Dominant Firm GameDominant Firm Game

Two firms, one large and one smallTwo firms, one large and one small

Either firm can announce an output Either firm can announce an output level (lead) or else wait to see what level (lead) or else wait to see what the rival does and then produce an the rival does and then produce an amount that does not saturate the amount that does not saturate the market. market.

Dominant Firm GameDominant Firm Game

Lead FollowSubordinate

Dominant

Lead

Follow

( 0.5, 4)

( 1, 8)

( 3, 2)

( 0.5, 1)

Dominant Firm GameDominant Firm Game

Lead FollowSubordinate

DominantDominant

Lead

Follow

( 0.5, 44)

( 1, 8)

( 3, 2)

( 0.5, 1)

Dominant Firm GameDominant Firm Game

Lead FollowSubordinate

DominantDominant

Lead

Follow

( 0.5, 4)

( 1, 88)

( 3, 2)

( 0.5, 1)

Dominant Firm GameDominant Firm Game

Conclusion:Conclusion:

Dominant Firm will always lead…..Dominant Firm will always lead…..

But what about the Subordinate But what about the Subordinate firm?firm?

Dominant Firm GameDominant Firm Game

Lead FollowSubordinateSubordinate

Dominant

Lead

Follow

( 0.5, 4)

( 1, 8)

( 33, 2)

( 0.5, 1)

Dominant Firm GameDominant Firm Game

Lead FollowSubordinateSubordinate

Dominant

Lead

Follow

( 0.5, 4)

( 11, 8)

( 3, 2)

( 0.5, 1)

Dominant Firm GameDominant Firm Game

Conclusion:Conclusion:

No dominant strategy for the Subordinate No dominant strategy for the Subordinate firm.firm.

Does this mean we cannot predict Does this mean we cannot predict what they will do?what they will do?

Dominant Firm GameDominant Firm Game

Lead FollowSubordinateSubordinate

DominantDominant

Lead

Follow

( 0.5, 4)

( 11, 88)

( 3, 2)

( 0.5, 1)

Dominant Firm GameDominant Firm Game

Conclusion:Conclusion:

Subordinate firm will always follow, Subordinate firm will always follow, because dominant firm will always lead.because dominant firm will always lead.

EquilibriumEquilibrium

Nash Equilibrium: Neither player Nash Equilibrium: Neither player has an incentive to change strategy, has an incentive to change strategy, given the other player’s choicegiven the other player’s choice

Dominant: Lead & Subordinate Follow Dominant: Lead & Subordinate Follow is a Nash Equilibriumis a Nash Equilibrium

A player’s best option may be A player’s best option may be dictated by anticipating the rival’s dictated by anticipating the rival’s best optionbest option

Timing and EndingTiming and EndingTwo Stage Game between A and B who Two Stage Game between A and B who

are dividing $1.00are dividing $1.00

1)1) Player A moves first and proposes Player A moves first and proposes how to split the dollar. Player B how to split the dollar. Player B either accepts the split in which case either accepts the split in which case the game ends, or we move to round the game ends, or we move to round 2.2.

2)2) The pie drops to $.90. Player B The pie drops to $.90. Player B proposes a split. Player A accepts or proposes a split. Player A accepts or the game ends and both get 0.the game ends and both get 0.

Timing and EndingTiming and Ending

What should A do?What should A do?

Timing and EndingTiming and Ending

The timing of the end of the game can The timing of the end of the game can dictate the strategy employed. If dictate the strategy employed. If the game went past round 2, A’s the game went past round 2, A’s strategy would change.strategy would change.

Go back to Prisoner’s Dilemma:Go back to Prisoner’s Dilemma:Is there a way to generate the cooperative Is there a way to generate the cooperative solution?solution?

Confess Don’t ConfessDon’t ConfessConductorConductor

TchaikovskyTchaikovsky

Confess

Don’t ConfessDon’t Confess

( -8, -8)

( -15, 0)

( 0, -15)-15)

( -1, -1)( -1, -1)

Go back to Prisoner’s Dilemma:Go back to Prisoner’s Dilemma:Is there a way to generate the cooperative Is there a way to generate the cooperative solution?solution?

Not a Nash EquilibriumNot a Nash EquilibriumIf Conductor commits to “Don’t Confess”, Tchaikovsky has If Conductor commits to “Don’t Confess”, Tchaikovsky has

an incentive to confessan incentive to confessIf Tchaikovsky commits to “Don’t Confess”, Conductor has If Tchaikovsky commits to “Don’t Confess”, Conductor has

an incentive to confessan incentive to confess

Role of a contract—to commit parties to actions they would Role of a contract—to commit parties to actions they would not undertake voluntarilynot undertake voluntarily

Alternative: Implied contractAlternative: Implied contractif there were a long relationship between the parties—if there were a long relationship between the parties—

(partners in crime) are more likely to back each other(partners in crime) are more likely to back each other

Application to Collective BargainingApplication to Collective Bargaining

Two agent gameTwo agent gameUncertaintyUncertaintyEach party has to anticipate what the Each party has to anticipate what the

other will doother will doTime limit mattersTime limit mattersAbility to contract affects outcomeAbility to contract affects outcomeA long, continuing relationship can A long, continuing relationship can

enhance the efficiency of the outcomeenhance the efficiency of the outcome

Complicating the gameComplicating the game

Suppose there is a range of outcomesSuppose there is a range of outcomes

Wage

Union

Firm

Contract Zone

Wm

WM

WM : Firm maximum wage

Wm : Union minimum wage

Where we end up in the contract zone Where we end up in the contract zone depends on bargaining powerdepends on bargaining power

Bargaining power depends on Bargaining power depends on Alternative opportunities if no bargain is Alternative opportunities if no bargain is

reached (outside option)reached (outside option) Union: alternative employmentUnion: alternative employment Firm: Substitute for union workersFirm: Substitute for union workers

Relative cost of delayRelative cost of delay Union: Strike fundUnion: Strike fund Firm: Inventory, strength of sales demandFirm: Inventory, strength of sales demand

Commitment strategyCommitment strategy Extent to which you can make the other party believe Extent to which you can make the other party believe

you will not budgeyou will not budge

Example of a more complicated Bargaining Example of a more complicated Bargaining SimulationSimulation

Two teams: Union(U); Management (M)Two teams: Union(U); Management (M)Each team gets a suit of 13 cardsEach team gets a suit of 13 cardsThe cards correspond to wage thresholds:The cards correspond to wage thresholds:

WageWage MM UU 1515 22 1414 3-43-4 1313 5-75-7 1212 8-108-10 1111 J-KJ-K AA 1010 AA J-KJ-K 99 8-108-10 88 5-75-7 77 3-43-4 66 22

Each team draws one card that will set their reservation wage. This Each team draws one card that will set their reservation wage. This must not be shown to the other team except for the full information must not be shown to the other team except for the full information game.game.

Through the remaining rounds, teams will reveal additional cards at Through the remaining rounds, teams will reveal additional cards at randomrandom

Teams can reveal other cards if they wishTeams can reveal other cards if they wish Teams can make statements about their bargaining objectives to the Teams can make statements about their bargaining objectives to the

other teamother team

Bargaining TechniquesBargaining Techniques

Distributive Bargaining: View Distributive Bargaining: View bargaining as a zero sum gamebargaining as a zero sum game ——split of the pie: what one party gains, split of the pie: what one party gains,

the other losesthe other loses Example—our gameExample—our game Often accompanied by pressure tacticsOften accompanied by pressure tactics

ThreatsThreats BluffsBluffs Bullying/Bullying/

Bargaining TechniquesBargaining Techniques

Interest-Based BargainingInterest-Based Bargaining Attempt to arrive at efficient outcomeAttempt to arrive at efficient outcome

No relevant information privilegedNo relevant information privileged Focus on solving mutual problem of making Focus on solving mutual problem of making

the firm as successful as possiblethe firm as successful as possible Requires long-standing bargaining Requires long-standing bargaining

relationship between partiesrelationship between parties

Bargaining TechniquesBargaining Techniques

Principled NegotiationsPrincipled Negotiations Both parties start with perceptions of Both parties start with perceptions of

the economic climate and their goalsthe economic climate and their goals Alternative mechanisms to reach these Alternative mechanisms to reach these

goals are presentedgoals are presented Options are evaluated against some Options are evaluated against some

objective criteria or by a third party objective criteria or by a third party expertexpert

No private informationNo private information

Bargaining TechniquesBargaining Techniques

Collective Bargaining by ObjectivesCollective Bargaining by Objectives Each party lists its objectivesEach party lists its objectives Objectives are prioritizedObjectives are prioritized Areas of agreement are identified, while Areas of agreement are identified, while

the remaining areas of disagreement the remaining areas of disagreement are ranked by their importance to the are ranked by their importance to the partiesparties

Parties can withhold informationParties can withhold information

Bargaining TechniquesBargaining Techniques

Note the similarity between these Note the similarity between these techniques and the gamestechniques and the games Role of informationRole of information Role of historyRole of history Role of the other party’s objectives, Role of the other party’s objectives,

actionsactions Role of cooperative vs competitive Role of cooperative vs competitive

bargaining environmentbargaining environment