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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 3/23. 3/9. Review. Review Terms Counting Strategies Prudent v. Best-Response Strategies Graduate Assignment. Review.

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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 3/23 3/9

  2. Review • Review Terms • Counting Strategies • Prudent v. Best-Response Strategies • Graduate Assignment

  3. 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.

  4. 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.

  5. Counting Strategies Player 1 Player 2 has 4 strategies: Left Right L R L R -2 4 2 -1 Player 2 LL RR LR RL -2 4 -2 4 2 -1 -1 2 L R GAME 2: Button-Button

  6. Counting Strategies Player 1 Player 2 has 4 strategies: Left Right L R L R -2 4 2 -1 Player 2 LL RR LR RL -2 4 -2 4 2 -1 -1 2 L R GAME 2: Button-Button

  7. Counting Strategies Player 1 Player 2 has 4 strategies: Left Right L R L R -2 4 2 -1 Player 2 LL RR LR RL -2 4 -2 4 2 -1 -1 2 L R GAME 2: Button-Button

  8. Counting Strategies Player 1 Player 2 has 4 strategies: Left Right L R L R -242 -1 Player 2 LL RR LR RL -2 4 -2 4 2 -1 -1 2 L R GAME 2: Button-Button

  9. Counting Strategies Player 1 If Player 2 cannot observe Player 1’s choice … Player 2 will have fewer strategies. Left Right L R L R -2 4 2 -1 Player 2 GAME 2: Button-Button

  10. Counting Strategies L R L R Player 1 -2 4 2 -1 Left Right L R L R -2 4 2 -1 Player 2 GAME 2: Button-Button

  11. Prudence v. Best Response 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 Find all the NE of the game.

  12. Prudence v. Best Response 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); }

  13. Prudence v. Best Response 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 = {(O,O); (F,F); }

  14. Battle of the Sexes Prudence v. Best Response 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

  15. Battle of the Sexes Prudence v. Best Response OF Let (p,1-p) = prob1(O, F) (q,1-q) = prob2(O, F) 2, 1 0, 0 0, 0 1, 2 O F NE = {(1, 1); (0, 0); (MNE)}

  16. Battle of the Sexes Prudence v. Best Response O F Let (p,1-p) = prob1(O, F) (q,1-q) = prob2(O, F) Then EP1(Olq) = 2q EP1(Flq) = 1-1q EP2(Olp) = 1p EP2(Flp)= 2-2p 2, 1 0, 0 0, 0 1, 2 O F NE = {(1, 1); (0, 0); (MNE)}

  17. Battle of the Sexes Prudence v. Best Response 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)} }

  18. Battle of the Sexes Prudence v. Best Response q=1 q=0 Player 1’s Expected Payoff EP1 2/3 0 2 p=1 2,1 0,0 0, 0 1, 2 p=1 p=0 q NE = {(1, 1); (0, 0); (2/3,1/3)}

  19. Battle of the Sexes Prudence v. Best Response 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)}

  20. Battle of the Sexes Prudence v. Best Response 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)}

  21. Battle of the Sexes Prudence v. Best Response 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)}

  22. Battle of the Sexes Prudence v. Best Response 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)}

  23. Battle of the Sexes Prudence v. Best Response 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 = 1/3 q NE = {(1, 1); (0, 0); (2/3,1/3)}

  24. Battle of the Sexes Prudence v. Best Response 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 = 1/3 q NE = {(1, 1); (0, 0); (2/3,1/3)}

  25. Battle of the Sexes Prudence v. Best Response 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)}

  26. Battle of the Sexes Prudence v. Best Response O F Find the prudent strategy for each player. 2, 1 0, 0 0, 0 1, 2 O F

  27. Battle of the Sexes Prudence v. Best Response 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

  28. Battle of the Sexes Prudence v. Best Response 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)}

  29. Battle of the Sexes Prudence v. Best Response 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 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)}

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

  31. Battle of the Sexes Prudence v. Best Response 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)}

  32. Battle of the Sexes Prudence v. Best Response 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)}

  33. Battle of the Sexes Prudence v. Best Response BATNA: Best Alternative to a Negotiated Agreement O F P2 2 1 2/3 2, 1 0, 0 0, 0 1, 2 NE (0,0) O F BATNA NE (1,1) 2/31 2 P1 NE = {(1, 1); (0, 0); (2/3,1/3)} Prudent: {(1/3, 2/3)}

  34. Battle of the Sexes Prudence v. Best Response 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?

  35. Battle of the Sexes Prudence v. Best Response NO: 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)}

  36. 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).

  37. 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

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

  39. 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?

  40. 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?

  41. Bargaining Games Bargaining games are fundamental to understanding the price determination mechanism in “small” markets. The central issue in all bargaining games is credibility: the strategic use of threats, bluffs, and promises. When information is asymmetric, profitable exchanges may be “left on the table.” In such cases, there is an incentive to make oneself credible (e.g., appraisals; audits; “reputable” agents; brand names; lemons laws; “corporate governance”).

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