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Lecture 1:Introduction
Introduction to Game Theory
Robert Menkyna
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SyllabusContact: Robert Menkyna
[email protected]/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
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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
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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)
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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
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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…
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Example: Prisoners’ Dilemma
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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.
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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.
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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
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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
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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
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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
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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
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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
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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
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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}
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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
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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
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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.
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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
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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
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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?
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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
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SummaryEconomic model
Good enough approximation of real worldGame Theory models
Economic models where decision makers interact