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UNIT II: The Basic Theory

UNIT II: The Basic Theory. Zero-sum Games Nonzero-sum Games Nash Equilibrium: Properties and Problems Bargaining Games Review Midterm 3/19. 3/12. Review. Prudent v. Best-Response Strategies Problem Sets 1 & 2 Graduate Assignment. Review. Battle of the Sexes. O F O F.

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UNIT II: The Basic Theory

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  1. UNIT II: The Basic Theory • Zero-sum Games • Nonzero-sum Games • Nash Equilibrium: Properties and Problems • Bargaining Games • Review • Midterm 3/19 3/12

  2. Review • Prudent v. Best-Response Strategies • Problem Sets 1 & 2 • Graduate Assignment

  3. Review Battle of the Sexes O F O F Player 1 2, 1 0, 0 0, 0 1, 2 Opera Fight O F O F (2,1) (0,0) (0,0) (1,2) Player 2 Compare best response and prudent strategies.

  4. Review Battle of the Sexes O F O F Player 1 2, 1 0, 0 0, 0 1, 2 Opera Fight O F O F (2,1) (0,0) (0,0) (1,2) Player 2 NE = {(1, 1); (0, 0); } Both are correct Find all the NE of the game. NE = {(O,O); (F,F); }

  5. Battle of the Sexes Review O F P2 2 1 2, 1 0, 0 0, 0 1, 2 NE (0,0) O F NE (1,1) 1 2 P1 NE = {(1, 1); (0, 0); (MNE)} Mixed Nash Equilibrium

  6. Battle of the Sexes Review O F Let (p,1-p) = prob1(O, F) (q,1-q) = prob2(O, F) Then EP1(Olq) = 2q EP1(Flq) = 1-1q q* = 1/3 EP2(Olp) = 1p EP2(Flp)= 2-2p p* = 2/3 2, 1 0, 0 0, 0 1, 2 O F NE = {(1, 1); (0, 0); (2/3,1/3)} Prudent: {1/3, 2/3)}

  7. Battle of the Sexes Review q=1 q=0 Player 1’s Expected Payoff EP1 2/3 0 2 EP1 = 2q +0(1-q) 2,1 0,0 0, 0 1, 2 p=1 p=0 q NE = {(1, 1); (0, 0); (2/3,1/3)}

  8. Battle of the Sexes Review q=1 q=0 Player 1’s Expected Payoff EP1 1 2/3 0 2 0 p=1 2,1 0,0 0, 0 1, 2 p=1 p=0 p=0 q NE = {(1, 1); (0, 0); (2/3,1/3)}

  9. Battle of the Sexes Review q=1 q=0 Player 1’s Expected Payoff EP1 1 2/3 0 2 0 p=1 2,1 0,0 0, 0 1, 2 p=1 p=0 EP1 = 0q+1(1-q) q NE = {(1, 1); (0, 0); (2/3,1/3)}

  10. Battle of the Sexes Review q=1 q=0 Player 1’s Expected Payoff EP1 1 2/3 0 2 0 Opera 2,1 0,0 0, 0 1, 2 p=1 p=0 Fight q NE = {(1, 1); (0, 0); (2/3,1/3)}

  11. Battle of the Sexes Review q=1 q=0 Player 1’s Expected Payoff EP1 1 2/3 0 2 0 2,1 0,0 0, 0 1, 2 p=1 p=0 p=1 0<p<1 p=0 0<p<1 q NE = {(1, 1); (0, 0); (2/3,1/3)}

  12. Battle of the Sexes Review If Player 1 uses her (mixed) b-r strategy (p=2/3), her expected payoff varies from 1/3 to 4/3. q=1 q=0 EP1 2/3 1/3 2 0 2,1 0,0 0, 0 1, 2 p = 2/3 p=1 p=0 4/3 p=1 p=0 q NE = {(1, 1); (0, 0); (2/3,1/3)}

  13. Battle of the Sexes Review If Player 2 uses his (mixed) b-r strategy (q=1/3), the expected payoff to Player 1 is 2/3, for all p. q=1 q=0 EP1 2/3 1/3 2/3 2 0 2,1 0,0 0, 0 1, 2 p=1 p=0 4/3 p=1 p=0 q = 1/3 q NE = {(1, 1); (0, 0); (2/3,1/3)}

  14. Battle of the Sexes Review O F Find the prudent strategy for each player. q* = 2/3 2, 1 0, 0 0, 0 1, 2 O F Prudent strategies: 1/3; 2/3

  15. Battle of the Sexes Review O F Let (p,1-p) = prob1(O, F) (q,1-q) = prob2(O, F) Then EP1(Olp) = 2p EP1(Flp) = 1-1p p* = 1/3 EP2(Oiq) = 1q EP2(Flq)= 2-2q q* = 2/3 2, 1 0, 0 0, 0 1, 2 O F Prudent strategies: 1/3; 2/3

  16. Battle of the Sexes Review If Player 1 uses her prudent strategy (p=1/3), expected payoff is 2/3 no matter what player 2 does O F EP1 2/3 1/3 2/3 2 0 2, 1 0, 0 0, 0 1, 2 p = 2/3 O F 4/3 p=1 p=0 p = 1/3 q NE = {(1, 1); (0, 0); (2/3,1/3)} Prudent:{(1/3, 2/3)}

  17. Battle of the Sexes Review If both players use (mixed) b-r strategies, expected payoff is 2/3 for each. O F EP1 2/3 1/3 2/3 2 0 2, 1 0, 0 0, 0 1, 2 p = 2/3 O F 4/3 p=1 p=0 p = 1/3 q = 1/3 2/3 q NE = {(1, 1); (0, 0); (2/3,1/3)} Prudent:{(1/3, 2/3)}

  18. Battle of the Sexes Review If both players use (mixed) b-r strategies, expected payoff is 2/3 for each. O F P2 2 1 2/3 2, 1 0, 0 0, 0 1, 2 NE (0,0) O F NE (1,1) 2/31 2 P1 NE = {(1, 1); (0, 0); (2/3,1/3)} Prudent:{(1/3, 2/3)}

  19. Battle of the Sexes Review If both players use prudent strategies, expected payoff is 2/3 for each. O F P2 2 1 2/3 2, 1 0, 0 0, 0 1, 2 NE (0,0) O F NE (1,1) 2/31 2 P1 NE = {(1, 1); (0, 0); (2/3,1/3)} Prudent: {(1/3, 2/3)}

  20. Battle of the Sexes Review O F P2 2 1 2/3 2, 1 0, 0 0, 0 1, 2 NE (0,0) O F NE (1,1) 2/31 2 P1 NE = {(1, 1); (0, 0); (2/3,1/3)} Prudent:{(1/3, 2/3)} Is the pair of prudent strategies an equilibrium?

  21. Battle of the Sexes Review Player 1’s best response to Player 2’s prudent strategy (q=2/3) is Opera (p=1). O F EP1 2/3 1/3 2/3 2 0 Opera 2, 1 0, 0 0, 0 1, 2 p = 2/3 O F 4/3 p=1 p=0 p = 1/3 q = 1/32/3 q NE = {(1, 1); (0, 0); (2/3,1/3)} Prudent:{(1/3, 2/3)} Therefore not an equilibrium!

  22. Review [I]f game theory is to provide a […] solution to a game-theoretic problem then the solution must be a Nash equilibrium in the following sense. Suppose that game theory makes a unique prediction about the strategy each player will choose. In order for this prediction to be correct, it is necessary that each player be willing to choose the strategy predicted by the theory. Thus each player’s predicted strategy must be that player’s best response to the predicted strategies of the other players. Such a prediction could be called strategically stable or self-enforcing, because no single player wants to deviate from his or her predicted strategy (Gibbons: 8).

  23. Review SADDLEPOINT v. NASH EQUILIBRIUM STABILITY: Is it self-enforcing? YES YES UNIQUENESS: Does it identify an unambiguous course of action? YES NO EFFICIENCY: Is it at least as good as any other outcome for all players? --- (YES) NOT ALWAYS SECURITY: Does it ensure a minimum payoff? YES NO EXISTENCE: Does a solution always exist for the class of games? YES YES

  24. Review Problems of Nash Equilibrium • Indeterminacy: Nash equilibria are not usually unique. 2. Inefficiency: Even when they are unique, NE are not always efficient.

  25. Review Problems of Nash Equilibrium T1 T2 Multiple and Inefficient Nash Equilibria S1 S2 5,5 0,1 1,0 3,3 When is it advisable to play a prudent strategy in a nonzero-sum game? What do we need to know/believe about the other player?

  26. Review Problems of Nash Equilibrium T1 T2 Multiple and Inefficient Nash Equilibria S1 S2 5,5 -99,1 1,-99 3,3 When is it advisable to play a prudent strategy in a nonzero-sum game? What do we need to know/believe about the other player?

  27. Review Dominant Strategy: A strategy that is best no matter what the opponent(s) choose(s). Prudent Strategy:A prudent strategy maximizes the minimum payoff a player can get from playing different strategies. Such a strategy is simply maxsmintP(s,t) for player i. Mixed Strategy:A mixed strategy for player i is a probability distribution over all strategies available to player i. Best Response Str’gy: A strategy, s’, is a best response strategy iff Pi(s’,t) > Pi(s,t) for all s. Dominated Strategy: A strategy is dominated if it is never a best response strategy.

  28. Review Saddlepoint:A set of prudent strategies (one for each player), s. t. (s*, t*) is a saddlepoint, iff maxmin = minmax. Nash Equilibrium: a set of best response strategies (one for each player), (s’,t’) such that s’ is a best response to t’ and t’ is a b.r. to s’. Subgame: a part (or subset) of an extensive game, starting at a singleton node (not the initial node) and continuing to payoffs. Subgame Perfect Nash Equilibrium (SPNE): a NE achieved by strategies that also constitute NE in each subgame. eliminates NE in which the players threats are not credible.

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