What is game theory? Study of interacting decision makers – emphasis on cold-blooded, “rational” decision making. Session 11 - Chapter 15 Game Theory Generalisation of standard, one person decision theory – Matias Järnefelt & Tuukka Salminen how should a rational expected utility maximiser behave in a situation in which his payoff depends on the choices of another expected utility maximiser? S ystems S ystems Analysis Laboratory Analysis Laboratory Session 11 - M. Järnefelt & T. Salminen Session 11 - M. Järnefelt & T. Salminen Helsinki University of Technology Helsinki University of Technology Seminar on Microeconomics - Fall 1998 / 1 Seminar on Microeconomics - Fall 1998 / 2 � � Description of a game Presentation outline � Description of a game, strategic form • Strategic form includes: � Simultaneous-move games – Set of players (= agents) – Zero-sum & variable-sum games – Set of strategies (= choices) – Nash equilibrium – Set of payoffs (= outcomes = utilities) – Repeated games • Can be depicted in a game matrix: � Sequential games Player Column – Game tree ~ extensive form Head Tail – Subgames Head 1,-1 -1,1 – Information set Player � Bayes-Nash equilibrium (incomplete information) Row Tail -1,1 1,-1 S ystems S ystems Analysis Laboratory Analysis Laboratory Session 11 - M. Järnefelt & T. Salminen Session 11 - M. Järnefelt & T. Salminen Helsinki University of Technology Helsinki University of Technology Seminar on Microeconomics - Fall 1998 / 3 Seminar on Microeconomics - Fall 1998 / 4 � � Simultaneous-move games Zero-sum game Assumptions: Decisions under uncertainty • Common knowledge in complete information games: involving two or more – The description of the game intelligent opponents in which – Each player is “fully rational” each opponent aspires to – Each player knows that the other player knows this optimise his own decision at the expense of other opponents. • Not known in advance: – Other player’s actual choice of strategies S ystems S ystems Analysis Laboratory Analysis Laboratory Session 11 - M. Järnefelt & T. Salminen Session 11 - M. Järnefelt & T. Salminen Helsinki University of Technology Helsinki University of Technology Seminar on Microeconomics - Fall 1998 / 5 Seminar on Microeconomics - Fall 1998 / 6
Zero-sum game Variable-sum game • Game of two players (Row & Column), where • Example: The Prisoner’s Dilemma payoff C = - payoff R (two prisoners, duopoly, …) • Example: Matching pennies • Aumann (1987) version: Player Column Player Column Co-op Defect Head Tail Co-op 3,3 0,4 Player Head 1 -1 Row Player Defect 4,0 1,1 Row Tail -1 1 S ystems S ystems Analysis Laboratory Analysis Laboratory Session 11 - M. Järnefelt & T. Salminen Session 11 - M. Järnefelt & T. Salminen Helsinki University of Technology Helsinki University of Technology Seminar on Microeconomics - Fall 1998 / 7 Seminar on Microeconomics - Fall 1998 / 8 Example: Cournot duopoly Example: Bertrand duopoly • Similar to Prisoner’s Dilemma • Players: Two firms i =1, 2 producing a good at • Players: Two firms i=1, 2 facing market zero cost facing inverse demand curve p ( x ) demand curve x ( p ) • Strategy: Production level x i • Strategy: Price p i of the good • Payoff: Profit p i = p ( x 1 + x 2 ) x i • Payoff: π 1 ( p 1 , p 2 ) = p 1 x ( p 1 ) if p 1 < p 2 p 1 x ( p 1 ) /2 if p 1 = p 2 0 if p 1 > p 2 S ystems S ystems Analysis Laboratory Analysis Laboratory Session 11 - M. Järnefelt & T. Salminen Session 11 - M. Järnefelt & T. Salminen Helsinki University of Technology Helsinki University of Technology Seminar on Microeconomics - Fall 1998 / 9 Seminar on Microeconomics - Fall 1998 / 10 Pure & mixed strategies & beliefs Objectives • R = set of pure strategies r available to Row • Mixed strategy: probability of playing strategy r is p r ∑∑ = π max ( , ) • Set of mixed strategies available to Row is the set of p u r c • Row’s objective: r c r ( p ) r c all probability distributions ( p r ) over R r • Payoff to Row: u r ( r , c ) ∑∑ = π max p u ( r , c ) • Column’s objective: c r c • Row’s subjective probability distribution over ( ) p c r c Column’s choices: ( π c ) • Similarly: C , c , ( p c ), u c ( r , c ), ( π r ) S ystems S ystems Analysis Laboratory Analysis Laboratory Session 11 - M. Järnefelt & T. Salminen Session 11 - M. Järnefelt & T. Salminen Helsinki University of Technology Helsinki University of Technology Seminar on Microeconomics - Fall 1998 / 11 Seminar on Microeconomics - Fall 1998 / 12
Nash equilibrium Nash Equilibrium • A Nash equilibrium consists of probability beliefs • Existence: ( π r , π c ) over strategies, and probability of choosing – Yes – with a finite number of agents and a finite number strategies ( p r , p c ) such that: of pure strategies 1 The beliefs are correct: p r = π r and p c = π c ∀ r , c ; and • Uniqueness: 2 Each player is choosing ( p r ) and ( p c ) so as to maximise – Very unlikely in general his expected utility given his beliefs – Further criteria to choose among Nash equilibria • A Nash equilibrium in pure strategies is a pair ( r * , c * ) = refinements such that u r ( r * , c * ) ≥ u r ( r , c * ) ∀ r and u c ( r * , c * ) ≥ u c ( r * , c ) ∀ c S ystems S ystems Analysis Laboratory Analysis Laboratory Session 11 - M. Järnefelt & T. Salminen Session 11 - M. Järnefelt & T. Salminen Helsinki University of Technology Helsinki University of Technology Seminar on Microeconomics - Fall 1998 / 13 Seminar on Microeconomics - Fall 1998 / 14 Nash Equilibrium Nash Equilibrium Refinement: If there is no dominant strategy equilibrium? • Dominant strategy: Refinement: r 1 , r 2 ∈ R r 1 strictly dominates r 2 if u r ( r 1 , c ) > u r ( r 2 , c ) ∀ c . • Elimination of strictly dominated strategies : r 1 weakly dominates r 2 if u r ( r 1 , c ) ≥ u r ( r 2 , c ) ∀ c First eliminate all strategies that are strictly dominated and then calculate the Nash equilibria of the remaining game and ∃ c ’ so that u r ( r 1 , c ’) > u r ( r 2 , c ’) • Example: Player Column • Dominant strategy equilibrium: A choice of strategies by each player such that each Left Right Top 2,2 0,2 strategy (weakly) dominates every other strategy Player available to that player Row Bottom 2,0 1,1 S ystems S ystems Analysis Laboratory Analysis Laboratory Session 11 - M. Järnefelt & T. Salminen Session 11 - M. Järnefelt & T. Salminen Helsinki University of Technology Helsinki University of Technology Seminar on Microeconomics - Fall 1998 / 15 Seminar on Microeconomics - Fall 1998 / 16 About mixed strategies Repeated games • May be an unrealistic or the only sensible choice • For any mixed strategy equilibrium, if one party believes that the other player will play the • A repeated game ≠ a repetition of a one- equilibrium mixed strategy, then he is indifferent as shot game to whether he plays his equilibrium mixed strategy, • Each player can determine his choice at or any pure strategy that is part of the mixed strategy some point as a function of the entire • Reason: The expected utility function is linear in history of the game until that point probabilities • Often mixed strategy probabilities = population frequencies S ystems S ystems Analysis Laboratory Analysis Laboratory Session 11 - M. Järnefelt & T. Salminen Session 11 - M. Järnefelt & T. Salminen Helsinki University of Technology Helsinki University of Technology Seminar on Microeconomics - Fall 1998 / 17 Seminar on Microeconomics - Fall 1998 / 18
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