**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 •
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• 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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