Economics 202: Intermediate Microeconomic Theory

1. HW #6 on website. Due Tuesday.

2. Second test covers up through today’s material, and will be “pseudo-cumulative” (to be explained).

Game Theory• “The Dating Game”

– Multiple Nash equilibria

– Nash equilibrium concept loses appeal

• “Copycat Game”– No Nash equilibrium

– Players want to outguess the other

– Introduce mixed strategies (in contrast to pure strategies)

Timing

Static Nash Equilibrium

Information

Dynamic

Complete Incomplete

• Mixed Strategy = a probability distribution over some or all of a player’s pure strategies

• Mixed strategies can add Nash equilbria

• Result: Any game with finite # players who have finite # pure strategies has a Nash equilibrium (possibly utilizing mixed strategies)

Backward Induction

Bayesian Nash Equilibrium

Perfect BayesianEquilibrium

2, 1 0, 0

0, 0 1, 2Chicken

Steak

Red WhitePat

Chris

Jack

Dating Game

-1, 1 1, -1

1, -1 -1, 1

Outside

Outside

InsideJill

InsideCopycat Game

Game Theory• Dynamic, complete 2-player sequential move game

• Order of play– Player 1 chooses action a1

– Player 2 observes a1 and then chooses a2

– Players receive their payoffs U1(a1,a2) & U2(a1,a2)

• Examples– Stackelberg-version of Cournot duopoly

– Trust Game -- equilibrium?

Timing

StaticNash Equilibrium

Information

Dynamic

Complete Incomplete

• Dynamic, simultaneous move (or infinite horizon) games requires an extension of backward induction called subgame-perfect Nash equilibrium

Backward Induction

Bayesian Nash Equilibrium

Perfect BayesianEquilibrium

1, 1 -1, 2

0, 0 0, 0Not trust

Trust

Honor BetrayPlayer 2

Player 1

Player 1

Trust Game(normal form)

Honor

Not trust

BetrayPlayer 2

Trust

0,0

1,1 -1, 2

Trust Game (extensive form)

Game Theory• “The Dormitory Game”

– Write extensive form if simultaneous game

– Write extensive & normal forms if A chooses first 6, 3 6, 4

5, 4 7, 5Softly

Loudly

Loudly SoftlyB

A

Game Theory• “Vote by Alternating Veto”

– Player 1 prefers X to Y to X, Player 2 prefers Z to Y to X

YXX YXY YZX YZY ZXX ZXY ZZX ZZY

X 0,2 0,2 0,2 0,2 1,1 1,1 1,1 1,1

Y 0,2 0,2 2,0 2,0 0,2 0,2 2,0 2,0

Z 1,1 2,0 1,1 2,0 1,1 2,0 1,1 2,0

• Find Nash equilibria and subgame perfect Nash equilbria