Games with Simultaneous Moves I :Discrete Strategies

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## Games with Simultaneous Moves I :Discrete Strategies

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**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**• Simultaneous moves? Strategy vs. action • Discrete/Continuous strategy**Games in normal (strategic) form.**• Games Table/ Game Matrix/ Payoff Table**Nash Equilibrium**• What is Equilibrium? • Cell-by-cell-inspection or enumeration • Best Response Analysis**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**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**• 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**MiniMax Method for Zero-Sum Games**• MiniMax/Maximin**Three Players**TALIA chooses C TALIA chooses NC**Three Players**TALIA chooses D TALIA chooses ND**Multiple Equilibria in Pure Strategies**• Coordination Games • Battle of the Sexes**Focal Point**• Convergence of expectation • Assurance Games**No Equilibrium in Pure Strategies**• Rock-Paper-Scissors • 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, 2) R 2 1 D L (1, 3) R (4, 1) 2