Game Theory

Sinem BulkanPhD in Organisational Behaviour

OutlineO Introduction: Founders of the Game TheoryO Introduction: Application Areas in Branches of ScienceO Glossary of Terms in TheoryO Game Theory DefinedO Assumptions of the TheoryO Types of GamesO Nash EquilibriumO Zero-Sum and Non-Zero Sum GamesO Finite and Infinite GamesO Prisoner’s Dilemma and Tit-For-Tat StrategyO Social DilemmasO CooperationO Co-Opetition O References for Further Reading

John von Neumann (1903-1957)

O Hungarian-born American (Jewish)

O Received his PhD in Mathematics at the age of 22.O Gottinghen University – Rockfeller scholarship

O First mastery paper ‘Mathematical Formulation of Quantum Mechanics’ published at the age of 23.

O Invited to Princeton University in 1930 he remained there as a mathematics professor until his death.

O During the war worked for Military, (MANIAC, NORC)O He was included in the target selection committee responsible for

choosing the Japanese cities of Hiroshima and Nagasaki as the first targets of the atomic bomb.

O Assigned to United Atomic Atom Energy Program in 1954.O 150 published papers in his life; 60 in pure mathematics, 20 in

physics, and 60 in applied mathematics.

O Speculated that the reason for his death is a cancer due to the nuclear tests.

John von Neumann (1903-1957)

O The inspiration for game theory was poker.

O Poker was not only guided by Probability Theory but also Bluffing.

O Wanted to formalize the idea of ‘bluffing’ , a strategy that is meant to deceive the other players and hide information from them.

O John von Neumann & Oskar Morgenstern (1944) Theory of Games and Economic Behaviour

John Nash (1928)• American Mathematician

• His works of game theory, differential geometry, and partial differential equations have provided insight into the forces that govern chance and events inside complex systems in daily life.

• In 1978, Nash was awarded the John von Neumann Theory Prize for his discovery of non-cooperative equilibria, now called Nash equilibria.

• The PhD Thesis ‘Game Theory’ he published at the age of 21 enabled him to be awarded with Nobel Memorial Prize in Economic Sciences in 1994. (27 pages!!!)

• Paranoid schizophrenia• Creator of the games of “Hex” and “So Long Sucker”

• Now lives in New Jersey.

What is Game Theory? (The Math of Competition)

O The study of mathematical models of conflict and cooperation between intelligent rational decision-makers – (Myerson,2007)

O Game theoretic concepts apply when the actions of players are interdependent. Concepts of the theory provide a language to to formulate, structure, understand and analyze strategic scenarios (by Turocy and Stengel, 2001)

O It arises whenever 2 or more individuals, with different values or goals, compete to try to control the course of events.

O Uses mathematical tools, called games, to study situations that involve both conflict and cooperation.

Application Areas

O In the 1950s and 1960s, game theory was broadened theoretically and applied to problems of war and politics.

O Since the 1970s, it has driven a revolution in economic theory (competitive markets, economic negotiations).

O Additionally, it has found applications in sociology and psychology, and established links with evolution and biology.

O Game theory received special attention in 1994 with the awarding of the Nobel prize in economics to Nash, John Harsanyi, and Reinhard Selten.

O At the end of the 1990s, a high-profile application of game theory has been the design of auctions.

4 Elements of Game Theory

O The players• How many players are there?• People, organisations, countries

O They choose from a list of options available to them, called strategies.

O A complete description of the information available to players at each decision made.

O A description of the consequences (payoffs) for each player for every possible profile of strategy choices of all players.

Assumptions of Game Theory

O Payoffs are known and fixed.

O All players behave rationally. They understand and seek to maximize gains and

minimize losses. They are flawless in calculating which actions will

maximize their payoffs.

O The rules of the game are common knowledge: Each player knows the set of players, strategies and

payoffs from all possible combinations of strategies: call this information ‘X’.

Common Knowledge means that each player knows that all players know X, that all players know that all players know X, that all players know that all players know that all players know X and so on,....., ad infinitium.

Outcome/PayoffO The outcome depends on the choices of all the players.

O Many strategy choices are non-cooperative, such as those between combatants in warfare or in sports.

O Some strategy choices are co-operative, aiming joint gains. (economics and politics)

O Some applications are in bargaining tactics in labour-management disputes, resource-allocation decisions in political campaigns, military choices in international crises, and the use of threats by animals in habitat acquisition and protection.

Strategic Interaction – Game Example

O Players: x and yO Strategies: Advertise or Not AdvertiseO Payoffs: Companies’ Profits

O Strategic Landscape:O Each firm earns $50 million from its customersO Advertising costs a firm $20 millionO Advertising captures $30 million from competitor

O How to represent this game?

Representing a Game

12

PLAYERS

STRATEGIESPAYOFFS

y

No Ad Ad

x No Ad 50 , 50 20 , 60

Ad 60 , 20 30 , 30

What to Do?

If you are advising x, what strategy do you recommend?

O Best reply for x:O If y advertises: AdO If y does not advertise: Ad

13

y

No Ad Ad

xNo Ad 50 , 50 20 , 60

Ad 60 , 20 30 , 30

Dominance (Baskın Strateji)

O A strategy for one player dominates another strategy if it yields a higher payoff no matter what the other players choose.

O Strategy U dominates strategy D for the player 1 below

O A strategy is dominant if dominates all other strategies.O No rational player would ever play a dominated

strategy, so if both players have dominant strategies, we can make a unique prediction.

You / Opponent

L R

U 3,9 3,8

D 0,0 2,1

Nash Equilibrum (Nash Dengesi)Ignore The Blonde – Go for the Brunettes

If everyone competes for the blond, we block each other and no one gets her. So then we all go for her friends. But they give us the cold shoulder, because no one likes to be second choice. Again, no winner. But what if none of us go for the blond. We don’t get in each other’s way, we don’t insult the other girls. That’s the only way to win. That’s the only way we all get a girl (from A Beautiful Mind)

Adam Smith ‘ the best result comes from everyone in the group doing what's best for himself‘John Nash ‘the best result will come where everyone in the group does what is best for himself ... and the group.' 

Nash Equilibrum (Nash Dengesi)

O Solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally.

O If each player has chosen a strategy and no player can benefit by changing strategies while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium.

O if every player prefers not to switch (or is indifferent between switching and not) then the set of strategies is a Nash equilibrium.

O Example: if Amy is making the best decision she can, taking into account Wili's decision, and Wili is making the best decision he can, taking into account Amy's decision. Likewise, a group of players are in Nash equilibrium if each one is making the best decision that he or she can, taking into account the decisions of the others.

Nash Equilibrum (Nash Dengesi)

Coordination Game

Player 2 adopts strategy A

Player 2 adopts strategy B

Player 1 adopts strategy A

4,4 1,3

Player 1 adopts strategy B

3,1 2,2

Prisoners’ Dilemma Game (Tutsak İkilemi Oyunu)

O Two players, prisoners 1, 2. There is no physical evidence to convict either one, so the prosecuter seeks a confession.

O Each prisoner has two strategies.O Prisoner 1: Don’t Confess, ConfessO Prisoner 2: Don’t Confess, ConfessO Payoff consequences are quantified in prison years.

O More years=worse payoffs.

O Information about strategies and payoffs is complete; both players (prisoners) know the available strategies and the payoffs from the intersection of all strategies.

O Strategies are chosen by the two Prisoners simultaneously and without communication.

Prisoners’ Dilemma Game (Tutsak İkilemi Oyunu)

• Think of the payoffs as prison terms/years lost.• Blue section shows the best strategy for both

prisoners.

Prisoner 1

Prisoner 2

Don’t Confess

Confess

Don’tConfess

-1,-1 -15,0

Confess 0,-15 -5,-5

Can Communication Help?

O Suppose we recognize the Prisoner’s Dilemma and we can talk to one another in advance, for instance, make promises to not confess.

O If these promises are non-binding and / or there are little consequences from breaking these promises (They are ‘cheap talk’) the the ability of the prisoners to communicate prior to choosing their strategies may not matter.

TIT-FOR-TAT Strategy for Prisoners’ Dilemma

O Consists of just 2 simple maneuvers:

1. Cooperate on the first move. 2. On following moves, do whatever the other player did.

O Example: Defect, defect, cooperate, defect, cooperate

O Tit-for-that: Cooperate, defect, defect, cooperate, defect, cooperate

TIT-FOR-TAT Strategy for Prisoners’ Dilemma

The strategy is distinguished by the following characteristics:

O It is nice. Those who employ the tit-for-tat strategy are never the first to defect. This is critical to building a trustful environment, and is the biggest predictor of success.

O It is provocable. When the other party defects, the first party responds immediately and clearly.

O It is forgiving. If the other party backs off, then the first party responds in kind. "Let bygones be bygones."

O It is simple. This is not computer chess. The tit-for-tat strategy requires little strategic planning.

Prisoners’ Dilemma is an example of a Non-Zero Sum Games

O A zero-sum game is one in which the players’ interests are in direct conflict, e.g. in football, one team wins and the other loses.

O A game is non-zero sum, if players’ interests are not always in conflict, so that there are opportunities for both to gain.

O Example: When both players choose don’t confess in Prisoners’ Dilemma, they both gain relative to both choosing Confess.

Zero Sum Games (Sıfır Toplamlı Oyunlar)

O For every combination of strategies, the total benefit to all players in the game always adds to zero

O A player benefits at the expense of others such as poker and chess

Zero Sum Games (Sıfır Toplamlı Oyunlar)

YOU/OPPONENT

A B C

1 30, -30 -10,10 20,-20

2 10,-10 20,-20 -20,20

Non-Zero Sum Games

O Some outcomes have net results greater or less than zero

O A gain by one player does not necessarily correspond with a loss by another

You / Opponent

A B

A 3,3 0,5

B 5,0 1,1

NON-ZERO SUM GAMES: BATTLE OF THE SEXES ( a two player

coordination game)

WIFE

HUSBAND

OPERA FOOTBALL

OPERA 3,2 0,0

FOOTBALL 0,0 2,3

Chicken Dilemma Game

O In the prototypical chicken dilemma two rebellious teens race towards each other in the car

O whoever swerves is considered a ‘chicken’ and loses.

O If neither turns away a bad crash happens and both loose

Chicken Dilemma Game

A payoff matrix of Chicken

Chicken with numerical payoffs

Swerve Straight

Swerve Tie, Tie Lose, win

Straight Win, Lose

Crash, Crash

Swerve Straight

Swerve 0, 0 -1, +1

Straight +1, -1 -10,-10

Simultaneous-Sequential Move Games

O Games where players choose actions simultaneously are simultaneous move games.Examples: Prisoners’ Dilemma, Sealed-Bid Auctions- Must acticipate what your opponent will do right now,

recognising that your oppenent is doing the same.

O Games where players choose actions in particular sequence are sequential move games.- Examples: Chess, Bargaining/Negotiations- Must look ahead in order to know what action to

choose now.

Finite-Infinite Games

Finite games O definite beginning and endingO played with the goal of winningO Rules exist to ensure the game is finiteO Ex. receiving a degree from an educational institution

Infinite gamesO Do not necessarliy have a knowable end point to reason

backwardO Repeated interaction continuse for infinite amount of timeO The rational long term behavior is affected by endless

moves of the playersO Ex. Repeated prisoner’s dilemma

Perfect Information-Imperfect Information Games

O Players have perfect information if they know exactly what has happened every time a decision needs to be made, e.g. in chess.

O Otherwise, the game is one of imperfect information

- Example: In the investment game, the sender and receiver might be differentially informed about the investment outcome. For example, the receiver may know that the amount invested is always tripled, but the sender may not be aware of this fact.

Conflict and Game Theory

O Game theory is a good way of starting to analyse conflicts

O It sets out broad patterns of resource allocation

O Shows how conflict arises from the allocations

O Highlights potential solutions to those conflicts.

Conflict and Game Theory in a Daily Life

Limitations to Game Theory

O Simple games can be modelled, such games are rare in real life. This is the reason why books uses the same examples.

O Most real conflicts involve multiple games and multiple strategies.

‘In-Group Love’ and ‘Out-Group Hate’ as Motives for Individual Participation in

Intergroup ConflictNir Halevy, Gary Bornstein, and Lilach Sagiv

The Hebrew University of JerusalemAssociation for Psychological Science Vol.19

No.4, 405-411

ARTICLE

Introduction

O Groups that fail to mobilize sufficient participation will not survive the aggresion of other groups,

and

O They will have to forgo the benefits of victory or, worse yet, bear the costs of defeat.

Introduction

O Importance of solidarity mechanisms

O Groups with more effective means of instilling self-sacrifice in their members have prevailed over groups with less effective solidarity mechanisms.

Modelling Intergroup Conflict

O Consideration of the internal tension between group welfare and individual welfare.

O The relative success of the two groups in overcoming this intragroup conflict determines the outcome of the intergroup competition.

O A basic model of this two-level structure is the intergroup prisoner’s dilemma game (IPD) game (Bornstein, 1992, 2003; Bornstein & Ben Yossef, 1994)

Intergroup Prisoner’s Dilemma (IPD) Game

O Game is played with 2 groups.O 3 members in each group.O Each player receives and endowment of 10

tokens and can contribute any number of these tokens to the group’s pool.

O For each token contributed by a member of the in-group, each of its members, gaind 1 money unit (MU), and each member of the out-group loses 1MU.

O For each token kept, the player is paid 2 MU.This means contributing a token yields a return 1MU, but at a cost of 2 MU.

Intergroup Prisoner’s Dilemma (IPD) Game

O Dominant Individual Strategy: The strategy that yields the highest personal payoffs regardless of what all the other in-group and out-group members do – is to contribute nothing.

O Dominant Group Strategy: The strategy that yields the highes payoffs for each group regardless of what the other group does – for all group members to contribute all their tokens.

O Collectively Optimal Strategy: The strategy that maximizes the payoff of all players in both groups – is for all players to withhold contribution.

Intergroup Prisoner’s Dilemma – Maximizing Difference (IPD-MD)

GameO The same game rules with IPD.

O However contributions can be made to two different pools.

O Contributing a token to the within-group pool (Pool W) increases the payoff for each in-group member, by 1 MU, without affecting the out-group. (a Cooperative motivation)

O Contributing a token to the between-group pool (Pool B) increases the payoff for each in-group member, by 1 MU, and at the same time decreases the payoff for each out-group member. (aggresive motivation to hurt the out-group, or a competitive motivation to increase the in-group’s advantage over the out-group.

Intergroup Prisoner’s Dilemma – Maximizing Difference (IPD-MD)

Game

• Corner cells – IPD-MD• Thick outline box – two independent 3-person PD game• Triple outline box – Maximizing difference or ‘spite’ game

The Experiment: Group and Individual Behaviour in the IPD and

IPD-MD GamesO A laboratory experiment that compared behaviour

in the IPD and IPD-MD games.

O Made decisions without communicating with the other in-group members (half the experimental sessions).

O Group member met for a short discussion before making their decisions (other half).

O Communication increases cooperation by enhancing group identity and commitment.

MethodO Participants 240 male students (mean age=24 years) Promised monetary reward for participation.

O Design and Procedure Cohorts of 6. 4 conditions (IPD or IPD MD with or without intragroup communication). Each cohort were divided into two 3-person groups. Informed on rules and payoffs. Pools were labeled ‘A’ and ‘B’. Explained how each player’s own decision and the decision of other players would affect the payoffs.

Method

O Design and Procedure

‘Decision Form’ 10 tokens – Endowment, Pool W, Pool B Pool W – 1 New Israeli Shekel (NIS) to each in-group member Pool B – Subtracted 1 NIS from the payoff to each out-group member For each token a player kept, he was paid 2 NIS Each player was paid a flat bonus of 30 NIS to endure that payoffs would be positive Participants could earn between 10NIS and 70 NIS. Postexperimental questionnaire

ResultsO Across the two communication conditions, the

participants contributed 63% of their endowment in the IPD-MD Game, as compared with 51% in the IPD game.

O Within group communication increased the overall contribution rate from 44% to 70%.

O Game type and communication did not have an interactive effect on the overall contribution rate.

ResultsO In the IPD MD game, contributions made almost

exclusively to the cooperative, within-group pool.O Following communication, contributions to pool W

increased from 47% to 68%.

O In the IPD game, about 7% of the participants in the no-communication condition contributed all 10 tokens to pool B, and 33% contributed nothing.

O Following group discussion, 57% of the individuals contributed everything but 18% contributed nothing.

ResultsO In general, efficiency was higher in the IPD-MD.

O In both games, group communication enhanced individual contributions.

O IPD-MD Communication increased contributions to the cooperative, within ingroup pool, thus enhancing efficiency.

O IPD Communication raised contributions to the competitive, between- group pool, thereby diminishing collective efficiency.

ResultsO IPD-MD 6 players could have earned a total of 180

NIS if they all had contributed their entire endowment to pool W.

O They actually earned about 78% of this amount in the no-communication condition and almost 87% in the communication condition.

O IPD 6 players could have earned a total of 120 NIS if they had all kept their entire endowment.

O The average efficiency rate was 65% in the no-communication, and only 32% in the communication condition.

DiscussionO The peaceful group coexistence observed in the

IPD-MD game was utterly shattered in the IPD game, in which maximizing the in-group’s gain was necessarily at the expense of the out-group.

O Under these circumstances, group members did not hesitate to compete.

O One obvious explanation for this behaviour is that the in-group members placed more weight on the gains their contribution produced for the in-group than on the loses it inflicted on the out-group.

Implications for Conflict Resolution

O IPD-MD game suggest that intergroup conflicts can be resolved by channeling group members’ alturism toward internal group causes.

O Whereas in the IPD game, ‘peace’ is achieved only if all members of both groups defect, in the IPD-MD game, groups, can avoid war while maintaining their ability to mobilize collective action.

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