Game Theory Expected Utility-1

8/22/2019 Game Theory Expected Utility-1

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Game Theory and ExpectedUtility Theory

POLS 331: Introduction to Word Politics

(Professor Tarar)


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Game Theory

What is it?

A set of tools used to study interactive

decision-making When there are at least 2 decision-makers


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Game Theory

Why is it useful for studying international

relations? Because a lot of what is important in

international relations involves decisionsby two or more leaders

And hence we want to understand howtheir preferences and interactions lead tooutcomes


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3 , 3 0 , 5

5 , 0 1, 1

C

D

1

C D

2Prisoners Dilemma

2 players each has to choose C or D

C = cooperate, D = defect they choose simultaneously payoffs are given in table Payoffs: DC > CC > DD > CD


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3 , 3 0 , 5

5 , 0 1, 1

C

D

1

C D

2

What should each player do?

Dominant strategy a strategythat is best for you no matter what youexpect the opponent to do

Individual rationality leads to suboptimal outcomehence the dilemma

Prisoners Dilemma


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3 , 3 0 , 5

5 , 0 1 , 1

C

D

C DPrisoners Dilemma

can represent arms races:D = build new arms, C = dont

can represent trade preferences:

D = impose tariffs, C = dont

PD illustrates realist idea thatcooperation is difficult underanarchy

what if binding agreementspossible?

DC > CC > DD > CD


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3 , 3 0 , 5

5 , 0 1 , 1

C

D

C D

Prisoners Dilemma

What if there is an international institution that imposes apenalty of 3 points for D?

3 , 3 0 , 2

2 , 0 -2 , -2

C

D

C D


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5 , 5 0 , 3

3 , 0 1 , 1

C

D

1

C D

2Stag Hunt

same as PD, but now eacheach player prefers mutualcooperation to exploiting

Payoffs: CC > DC > DD > CD

does either player have adominant strategy?


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5 , 5 0 , 3

3 , 0 1 , 1

C

D

1

C D2

Nash equilibrium: a set of strategies, one for each

player, such that no playercan do better by changing itsstrategy, given the strategiesof the others

each player is choosing a best responseto what the others are doing

Pareto-dominant equilibrium

Stag Hunt


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Sequential-Move Games

Extended Deterrence:

ATTACKER

DEFENDER

attacknot

defendnot

3

1

2

3

1

2

Game tree

Backwardsinduction

Subgame-perfect equilibrium


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Security Council Game

BUSH ADM.

resolutionno resolution

SECURITY COUNCIL

approvereject

FRANCE

vetonot

?

??

??

????

???


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Inspectors Game

BUSH ADM.

attack

IRAQ

BUSH ADMBUSH ADM

not

hing

send inspectors

cooperate not

attacknot

attack

not

?? ?

?

??

??

??

??


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Expected Utility Theory

Now let us consider single-playerdecision-

makingThe player has a set of actions

Each action leads to a certain outcome

The player has preferences over the outcomes

These preferences can be represented using

numbers that are called payoffs or utilities A higher number indicates a more preferred outcome


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For example, the actor can choose:

Attack another country Do not attack

Suppose the payoffs are as follows:

Attacking: 5

Not attacking: 3

The actor will choose to attack Principle of rational choice: you choose an

action that leads to a most preferred outcome


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But sometimes the link between actions

and outcomes is uncertainTwo (or more) possible outcomes may

result from a given action, and you are notsure which one will result

Hence you are not sure which action tochoose


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The solution: assign probabilities to the

different outcomes occurring, and thencalculate expected utilities for the differentactions

Suppose that if you attack the othercountry, either you will win or lose the war

Two possible outcomes can result fromattacking


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Suppose utilities are as follows:

Not attack: 3 Attack and win the war: 6

Attack and lose the war: 2 Not clear whether it is rational to attack or

not

Depends on how likely you are to win thewar


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Suppose you win the war with probability 1>p>0,

and lose with probability 1 pTo calculate the expected utility for an action, we

multiply the probability of each outcome (that

can result from that action) by the utility for thatoutcome, and then add them all up

EU(attack) = (p)(6) + (1 p)(2) = 4p + 2

EU(not attack) = 3

Rational to attack if 4p + 2 > 3, or p >


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Principle of rational choice: you choose an

action that gives you the highest expectedutility

Principle is the same if there are 3 or moreoutcomes that can result from a givenaction: multiply probabilities by utilities,

and then add them all up


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Expected utility theory is also used in

game-theoretic models of incompleteinformation

Game Theory Expected Utility-1

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