Review Lecture

Review Lecture - Midterm Material •  Ch. 1 ­11...

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Unformatted text preview: Midterm Material •  Ch. 1 ­11 •  Problem Set 1 ­4 •  “Represen:ng Games” & “Analysis of Sta:c SeCngs” posted notes •  Lectures corresponding to posted notes Midterm Material •  Normal form, dominance, best response, iterated dominance, common knowledge, ra:onalizability, beliefs, expected payoffs, Nash equilibrium, efficiency •  Construct extensive form representa:on from narra:ve •  Convert extensive form to normal form •  Matrix representa:on: finding ra:onalizable strategies, pure and mixed strategy Nash equilibria •  Find ra:onalizable strategies and Nash equilibria in oligopoly, partnership, loca:on 1 What is NOT on the exam •  Congruity, congruous sets (pg. 90 ­91) •  Contracts (Ch. 13) Ch 7: #5 and #6 •  Suppose that in some two ­player game, s1 is a ra:onalizable strategy for player 1. If s2 is a best response to s1, is s2 a ra:onalizable strategy for player 2? •  Suppose that in some two ­player game, s1 is a ra:onalizable strategy for player 1. If s1 is a best response to s2, can you conclude that s2 is a ra:onalizable strategy for player 2? 2 Ch 7: #5 Suppose that in some two ­player game, s1 is a ra:onalizable strategy for player 1. If s2 is a best response to s1, is s2 a ra:onalizable strategy for player 2? R={U, M, D} x {L, R} 2 1 L U 1, 1 M 2, 1 D 1, 2 C 1, 0 2, 0 1, 0 R 2, 2 1, 1 2, 1 L is a best response to D. D may ra:onally be played. Thus L is ra:onalizable. Similarly, R is a best response to U. U may ra:onally be played. Thus R is ra:onalizable. Ch 7: #6 Suppose that in some two ­player game, s1 is a ra:onalizable strategy for player 1. If s1 is a best response to s2, can you conclude that s2 is a ra:onalizable strategy for player 2? R={U, M, D} x {L, R} 2 1 L U 1, 1 M 2, 1 D 1, 2 C 1, 0 2, 0 1, 0 R 2, 2 1, 1 2, 1 M is a best response to C, but we cannot conclude from that that the C is ra:onalizable. M is a best response to L, and just happens to also be best response to C, which will not be ra:onally played. 3 Ra:onalizability and Nash Equilibrium •  Every Nash equilibrium is in the set of ra:onalizable strategies. –  Nash equilibrium strategy profile is always ra:onalizable. •  Every ra:onalizable strategy profile is NOT a Nash equilibrium. •  Nash equilibrium is a refinement of the ra:onalizable strategies. Ra:onalizability and Nash Equilibrium The set of ra:onalizable strategies have mutual consistency in terms of the en:re set of ra:onalizable strategies. But there may not be mutual consistency in the beliefs of a specific strategy profile. A Nash equilibrium will have mutual consistency in the beliefs of that specific strategy profile. 4 Ra:onalizability and Nash Equilibrium •  •  •  •  Communica:on Social ins:tu:ons Historical factors Culture … facilitate coordina:on of beliefs and behavior. •  In a Nash equilibrium, beliefs are coordinated Ra:onalizability •  Each player’s ra:onalizable strategies includes every best response to her opponents’ ra:onalizable strategies. •  Players may have doubts about what others will do. •  Players beliefs may not accurately describe the others actually do. 5 Solu:on Concepts •  Dominance –  Assump:on: Ra:onality –  Never play a strategy that is never a best response to anything. •  Ra:onalizability –  Assump:on: Common Knowledge of Ra:onality –  Never play a strategy that is only a best response to a strategy the other player will not ra:onally select, (with common knowledge of ra:onality). •  Nash Equilibrium –  Assump:on: Common Knowledge of Ra:onality, correct beliefs in equilibrium –  Never play a strategy that is only a best response to a strategy the other player does not actually select. Indifference as Key to MSNE •  In an MSNE, a mixed strategy is a best response for at least one player. •  If a mixed strategy is a best response, it must be that each of the pure strategies involved in the mix must itself be a best response. •  Each of the pure strategies involved in the mix must yield the same expected payoff. 6 Indifference as Key to MSNE 1 2 (q) X (1 ­q) Y (p) A 2,3 6,0 (1 ­p) B 2,1 0,2 In an MSNE of this game, one player plays a pure strategy and the other mixes over his two strategies. Which player can play a pure strategy in an MSNE? Player 2 playing X. If a player is playing a mix, while the other is playing a pure, the player playing a mix must be indifferent between at least two strategies. Strategic Tensions •  First: conflict b/w individual and group incen:ves –  Prisoner’s Dilemma •  Second: strategic uncertainty –  Stag Hunt •  Third: inefficient coordina:on –  Pareto Coordina:on 7 Strategic Tensions •  First: conflict b/w individual and group incen:ves –  Prisoner’s Dilemma •  Second: strategic uncertainty –  Stag Hunt •  Third: inefficient coordina:on –  Pareto Coordina:on Bertrand Model NE when calculus does not apply •  There are a small number of firms in the market. –  All firms have constant marginal costs. •  Firms produce a homogeneous good. –  Consumers choose the cheapest product. •  Firms choose price, –  Quan:ty, Q(p) is determined by the market demand func:on. 8 Bertrand (Price Compe::on) Example Example: QD(p) = 500 – p; c1 = c2 = 100 Firm 1’s profit func:on: Best responses are not well ­defined Example: QD(p) = 500 – p; c1 = c2 = 100 Suppose p2=$200. $199.99 is not the highest possible price lower than 100. $399.99 for 2 ­> p1=$199.995 $599.99 for 3 ­> p1=$199.9967 … Gas sta:ons also use frac:onal cents. There can always be a number smaller than epsilon. f p2>pm where pm , BR is pm 9 Best Responses in Bertrand Bertrand Model Nash equilibrium with 2 firms: •  If both firms have iden:cal marginal costs, c, then the only Nash equilibrium is that both firms set p = c 10 Best Response Func:ons Consider a two ­player game. S1 = [0, 1] and S2 = [0, 1]. The graph illustrates each player’s best response func:on. Which line is player 1’s best response func:on? (the solid or striped)? Which line is player 2’s best response func:on? (the solid or striped)? A best response func:on for Player 2 must specify a strategy for any belief about player 1. It must reach from the lex side to right side. A best response func:on for Player 1 must specify a strategy for any belief about player 2. It must reach from the boyom to the top. The dashed line could be a BR for either player as it reaches from boyom to top and lex to right. However, the solid line can only be a BR for player 2. It cannot be a BR func:on for player 1 because it does not specify a strategy for all beliefs of what player 2 may play. It does not reach from the boyom to the top. 11 ...
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This document was uploaded on 11/30/2011.

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