1. 1

2. problem set 6 from Osborne’s Introd. To G.T. p.210 Ex. 210.1 p.234 Ex. 234.1 p.337 Ex. 26,27 from Binmore’s Fun and Games

3. 3 Minimax & Maximin Strategies Minimax & Maximin Strategies Given a game G(, ) and a strategy s of player 1: is the worst that can happen to player 1 when he plays strategy s. He can now choose a strategy s for which this ‘worst scenario’ is the best

4. 4 A strategy s is called a maximin (security) strategy if { {

5. 5 A strategy s is called a maximin (security) strategy if These can be defined for mixed strategies as well. Similarly, one may define If the game is strictly competitive then this is the best of the ‘worst case scenarios’ of player 2.

6. 6 where s,t are mixed strategies Lemma: Take the matrix to be the matrix of player 1’s payoffs of a game G, i.e. G 1 For any matrix G:

7. 7 For any matrix G: Proof: For any two strategies s,t : ?? where s,t are mixed strategies hence:

8. 8 Theorem: (von Neumann) For any matrix G: Lemma: If s is a maximin strategy and t is a minimax strategy of a strictly competitive game, then (s,t) is a Nash equilibrium. Proof: The max & min is taken over mixed strategies No proof is provided in the lecture

9. 9 = but hence maxmin = minmax

10. 10 t is a best response against s s is a best response against t ( s, t ) is a Nash Equilibrium.

11. 11 Mixed Strategies Equilibria in Infinite Games The ‘All Pay’ Auction TTwo players bid simultaneously for a good of value K the bids are in [0,K]. EEach pays his bid. TThe player with the higher bid gets the object. IIf the bids are equal, they share the object. There are no equilibria in pure strategies

12. 12 There are no equilibria in pure strategies

13. 13 Equilibrium in mixed strategies a b0 K F 1 a b0 K f x F(x)

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17. 17 Rosenthal’s Centipede Game 1 2 0, 101, 0 1 2 0, 10 3 10 2, 0 1 2 0, 10 5 10 4, 0 0, 0 D A ‘Exploding’ payoffs due to P. Reny ‘Centipede’ due to K.G.Binmore

18. 18 Rosenthal’s Centipede Game 1 2 0, 101, 0 1 2 0, 10 3 10 2, 0 1 2 0, 10 5 10 4, 0 0, 0 D A Sub-game perfect equilibrium

19. 19 Rosenthal’s Centipede Game 1 2 1, 32, 0 1 2 3, 54, 2 1 2 5, 76, 4 8, 6 D A Sub-game perfect equilibrium different payoffs 1 2 0, 101, 0 1 2 0, 10 3 10 2, 0 1 2 0, 10 5 10 4, 0 0, 0 D A

20. 20 1 2, 2 Quiet evening A Variation of the Battle of the Sexes Noisy evening BX B3, 10, 0 X 1, 3 Player 1 has 4 strategies Player 2 has 2 strategies

21. 21 1 2, 2 Quiet evening A Variation of the Battle of the Sexes Noisy evening BX B3, 10, 0 X 1, 3 Nash Equilibria BX B3, 10, 0 X 1, 3 [ (N,B), B ] BX B3, 10, 0 X 1, 3 [ (Q,X), X ] BX B3, 10, 0 X 1, 3 [ (Q,B), X ]

22. 22 1 2, 2 Quiet evening A Variation of the Battle of the Sexes Noisy evening BX B3, 10, 0 X 1, 3 Nash Equilibria [ (N,B), B ] [ (Q,X), X ] [ (Q,B), X ] not a sub-game perfect equilibrium !!! These S.P.E. guarantee player 1 a payoff of at least 2 7