Games with Simultaneous Moves I :Discrete Strategies

Outline Games with simultaneous moves Nash Equilibrium Dominance Minimax in Zero-sum Game Three Players Multiple/Zero Equilibria in pure strategies Tree Forms into Strategic Forms

Games with Simultaneous Moves Strategy vs. action Discrete/Continuous strategy

Games in normal (strategic) form. Games Table/ Game Matrix/ Payoff Table Column Left Middle Right Row Top 3, 1 2, 3 10, 2 High 4, 5 3, 0 6, 4 Low 2, 2 5, 4 12, 3 Bottom 5, 6 9, 7

Zero-sum Game Column Run Pass Blitz OFFENSE 2 5 13 Short Pass 6 5.6 10.5 Medium Pass 4.5 1 Long Pass 10 3 -2

Nash Equilibrium What is Equilibrium? Cell-by-cell-inspection or enumeration Best Response Analysis Column Left Middle Right Row Top 3, 1 2, 3 10, 2 High 4, 5 3, 0 6, 4 Low 2, 2 5, 4 12, 3 Bottom 5, 6 9, 7

Column Left Middle Right Row Top 3, 1 2, 3 10, 2 High 4, 5 3, 0 6, 4 Low 2, 2 5, 4 12, 3 Bottom 5, 6 9, 7

The N.E is (Low, Middle) →(5, 4) A Nash Equilibrium in a game is a list of strategies, one for each player, such that no player can get a better payoff by switching to some other strategy that is available to her while all the other players adhere to the strategies specified for them in the list.

A N.E. does not have to be jointly best for the players. EX: Prisoner’s dilemma WIFE Confess (Defect) Deny (Cooperate) HUSBAND 10yr, 10yr 1yr, 25yr 25yr, 1yr 3yr, 3yr

Nash Equilibrium as a system of beliefs Nash Equilibrium is a set of strategies, such that (1)each player has correct beliefs about the strategies of the others (2)the strategy of each is the best for herself, given her beliefs about the strategies of the others

Dominance WIFE Confess (Defect) Deny (Cooperate) HUSBAND 10yr, 10yr Dominant strategy is an action clearly best for a player, no matter what the others might be doing. “Confess” is a dominant strategy for the husband, while “Confess” is also a dominant strategy for the wife. NE: (Confess, Confess) →(10yr, 10yr)

(A,B,…..) are strategies for Player 1. A is a dominant strategy for Player 1. A dominates B/C/D…. B/C/D… is a dominated strategy for Player 1.

Both (all) Players Have Dominant Strategies One Player has a Dominant Strategy FED Low interest rate High interest rate CONGRESS Budget balance 3, 4 1, 3 Budget deficit 4, 1 2, 2

Successive (Iterated) Elimination of Dominated Strategies Left Middle Right Up 5, 2 2, 3 3, 4 Medium 4, 1 3, 2 4, 0 Down 3, 3 1, 2 2, 2

MiniMax Method for Zero-Sum Games MiniMax/Maximin Column Run Pass Blitz OFFENSE 2 5 13 Short Pass 6 5.6 10.5 Medium Pass 4.5 1 Long Pass 10 3 -2

Three Players TALIA chooses C TALIA chooses NC NINA C NC EMILY 5, 5, 5 3, 6, 3 6, 3, 3 4, 4, 1 NINA C NC EMILY 3, 3, 6 1, 4, 4 4, 1, 4 2, 2, 2

Three Players TALIA chooses D TALIA chooses ND NINA D ND EMILY 2, 7, 2 3, 2, 2 5, 1, 2 2, 5, 2 NINA D ND EMILY 3, 3, 3 1, 2, 3 2, 1, 1 2, 2, 1

Multiple Equilibria in Pure Strategies Coordination Games Battle of the Sexes SALLY Starbucks Banyan HARRY 2, 1 0, 0 1, 2

Convergence of expectation Assurance Games Focal Point Convergence of expectation Assurance Games DEAN Swerve (Chicken) Straight (Tough) JAMES 0, 0 -1, 1 1, -1 -2, -2

No Equilibrium in Pure Strategies Rock-Paper-Scissors Rock Paper Scissors 0, 0 -1, 1 1, -1 N.E exists in mixed Strategies

Homework, Ch4 question 3, 6, 11, and 12(a & b)

Tree form into a strategic form L (3, 1) U 2 R (2, 2) 1 D L (1, 3) R (4, 1) 2 LL LR RL RR U 3, 1 2, 2 D 1, 3 4, 1 LL LR RL RR U D

Minimax/Maximin vs. Nash Equilibrium Revisited