Lecture 1:Introduction

Introduction to Game Theory

Robert Menkyna

SyllabusContact: Robert Menkyna

rmenkyna@cerge-ei.czhome.cerge-ei.cz/menkyna/teaching (will be

updated)

Office hours: Friday 16:30 – 17:30 in NB 339

Literature: Osborne, M.J. – An Introduction to Game TheoryGibbons, R. – A Primer in Game Theory

Room: 228

SyllabusGrading:

Final – 50%Midterm – 30%Homework – 20%

Important dates:Hmw1: Oct., 26th - Nov., 2nd (lecture on Tue.

or office NB349)Midterm: Nov., 11th – in classHmw2: Dec., 3rd – Dec., 10th (lecture on Fri.

or office NB349)Final: to be announced

IntroductionEconomic Model

Abstraction made in order to understand our observations and experience

Simple but useful - Keep purpose in mindAssumptions should capture the essence of

the problem and drop irrelevant details – Good enough approximation

Model is not wrong neither correctTrade off between generality and simplicity

(solvability)

Game Theoretic ModelsModels of situations where decision makers

interact – strategic situationsPayoff to decision makers depends NOT only on

their decisions but on decisions done by opponents.

Classic modelsPerfect competition – firms are price takers, decision of

one firm does not affect payoff to the other firm.Monopoly – Monopolist considers demand as given,

independent of his decision.What if the market environment is somewhere

between perfect competition and monopolist?Strategic decisions

Strategic Situations – ExamplesEconomy:

Imperfect competition – Windows cares what Apple does.

Sociology:Bargaining within family – Do we go to opera

or watch on TV Politics:

Political candidates competing for votesSport:

Penalty shot…

Example: Prisoners’ Dilemma

Assumptions of the Model of PDSet of players (decision makers) – {Prisoner A, Prisoner

B}Set of actions – {Confess, Silent}

Simultaneous decisionsAction profiles – combinations of all players’ actions

{CC,CS,SC,SS}Preferences over set of action profiles – less is better

Completeness – For every pair of actions player knows which action is better for her or if she is indifferent between them (tie).

Transitivity – If action A is preferred to B and B is preferred to C then A is preferred to C.

Rational choice – Action chosen by player is at least as good based on her preferences as any other action available.

Consistency – If A is preferred to B when {A,B} is available, then B cannot be preferred to A when{A,B,C} is available.

Payoff functionsPayoff functions represent preferences.Just ORDINAL significance , no cardinal

u1(C,S) > u1(S,S) > u1(C,C) > u1(S,C)u1(C,S) –> 3; u1(S,S) –> 2; u1(C,C) –> 1;

u1(S,C) –> 0Values of payoff functions carry information

only on order, not on distances nor on proportions.

Any other assignment of numbers is possible as long as order is unchanged.

Representation of Game by Table

Confess

Silent

Confess

1 3

Silent 0 2

Confess

Silent

Confess

1 0

Silent 3 2

1 2

Confess Silent

Confess 1,1 3,0

Silent 0,3 2,2

Payoff Table Prisoner 1 Payoff Table Prisoner 2

Game in Normal Form

ClassificationPrisoners’ dilemma is Static game of

complete information.Static – Players simultaneously chose

actions.Complete information –Payoff functions are

known to all playersOther games:

Games of perfect information – All players know all moves that have taken place.

Dynamic games – Players moves sequentially

Another Game of Prisoners’ Dilemma TypeWorking on the joint project – hard work vs.

goof offDuopoly – low production vs. high

productionCommon property – low consumption vs.

high consumption1 2 Confess Silent

Confess 1,1 3,0

Silent 0,3 2,2

1 2 Little Hard

Little 1,1 3,0

Hard 0,3 2,2

1 2 Low P High P

Low P 1,1 3,0

High P 0,3 2,2

1 2 Bomb No Bomb

Bomb 1,1 3,0

No Bomb 0,3 2,2

1 2

A Lot Little

A Lot 1,1 3,0

Little 0,3 2,2

Battle of Sexes Set of players – {Woman, Man}Set of Actions – {Football, Opera}Set of action profiles – {FF, FO, OF, OO}Preferences

Man: u(FF)>u(OO)>u(FO)=u(OF)Woman: u(OO)>u(FF)>u(FO)=u(OF)

1 2

Opera Football

Opera 2,1 0,0

Football 0,0 1,2

Battle of Sexes modifiedDriver vs. pedestrian:Two players – {Driver, Pedestrian}Actions – Driver{Go, Stop}, Pedestrian {Wait,

Walk}Action profiles – {WaitG, WaitS, WalkG, WalkS}

PreferencesD:

{Wait,Go}>{Wait,Stop}={Walk,Stop}>{Walk,Go}

P: {Walk,Stop}>{Wait,Go}={Wait,Stop}>{Go,Walk}

P D

Stop Go

Walk 2,1 0,0

Wait 1,1 1,2

Matching Pennies or Penalty ShotTwo PlayersTwo same actionsPlayer 1 prefers to copy action of Player 2.Player 2 prefers to play different action

than Player 1.Penalty shot, Choice of new appearance

1 2

Left Right

Left 1,-1 -1,1

Right -1,1 1,-1

Solution of the GameGiven the assumptions, what is the

plausible outcome of the game?Prisoners’ dilemma:

Whatever Prisoner 1 plays, Prisoner 2 is better off playing Confess:1=u(CC)>u(SC)=0 and 3=u(CS)>u(SS)

1 2

Confess Silent

Confess 1,1 3,0

Silent 0,3 2,2

Strictly Dominated StrategyPlayer i’s action a strictly dominates her action

b if ui(a,a−i) > ui(b,a−i) for every list a−i of other players’ actions.• ui is a payoff function that represents player i’s

preferences.• a−i ={a1, … , ai-1, ai+1,…, aN} - actions of others

players.If any action strictly dominates the action b, we say

that b is strictly dominated.Assumption of rational players:

Rational player never plays dominated strategy.Stable outcome in Prisoners’ dilemma game:

{Confess,Confess}

Strictly Dominated Strategies

Right is dominated by Center If Player 1 knows that Player 2 is rational, he considers

only Left and RightDown is dominated by UPPlayer 1 knows that 2 knows that 1 is rational and 1

knows that 2 is rationalLeft is dominated by Center{Up, Center}

1 2

Left Center Right

Up 1,0 1,2 0,1

Down 0,3 0,1 2,0

Iterative Elimination of Str. Dom. Str.Additional assumption: Rationality is

common knowledgeAll players know that all players are

rational. All players know that all players know that all players are rational. All players know … etc.

Order of the elimination of strictly dominated strategies does not matter.

Iterative elimination of strictly dominated strategies sometimes give ambiguous results

Students’ GameClass of students is having examination next

week. Lecturer has decided to play following game with students. In the case when nobody participates at exam, everybody is graded by A. However if at least one of the students comes, everybody missing is graded F and those coming are graded based on their performance. Define a game in normal form.

Students’ Game – ModelThree types of students – Good, Average,

BadActions – Skip, ComeGrading:

If non of the students come everybody gets AIf at least one comes, missing students get FThose coming get grades order by students’

skills such that the best attending student gets A, second best attending gets B and the third attending (if any) gets C

Preferences: Better grade is preferred

Students’ Game – Normal FormBest

Come

Skip

Avg. Bad

Come Skip

Come 1,0,2 1,0,2

Skip 0,1,2 0,0,2

Avg. Bad

Come Skip

Come 2,1,0 2,0,0

Skip 0,2,0 2,2,2

Students’ Game – Discussion Is model good enough approximation?What is the solution by iterative elimination

of strictly dominated strategies?What should lecturer change in order to

enforce participation on the exam?

Students’ Game – Revised Non of the strategy is dominatedIterative elimination of dominated

strategies has ambiguous result

Come

Skip

Avg. Bad

Come Skip

Come 1,0,2 1,0,2

Skip 0,1,2 0,0,2

Avg. Bad

Come Skip

Come 2,1,0 2,0,0

Skip 0,2,0 1,1,1

SummaryEconomic model

Good enough approximation of real worldGame Theory models

Economic models where decision makers interact