# lecture12 - Lecture XII Analysis of Infinite Games Markus M...

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Unformatted text preview: Lecture XII: Analysis of Infinite Games Markus M. M¨obius April 7, 2004 • Gibbons, chapter 2.1.A,2.1.B,2.2.A • Osborne, sections 14.1-14.4, 16 • Oxborne and Rubinstein, sections 6.5, 8.1 and 8.2 1 Introduction - Critique of SPE The SPE concept eliminates non-credible threats but it’s worth to step back for am minute and ask whether we think SPE is reasonable or in throwing out threats we have been overzealous. Practically, for this course the answer will be that SPE restrictions are OK and we’ll always use them in extensive form games. However, it’s worth looking at situations where it has been criticized. Some of the worst anoma- lies disappear in infinite horizon games which we study next. 1.1 Rationality off the Equilibrium Path Is it reasonable to play NE off the equilibrium path? After all, if a player does not follow the equilibrium he is probably as stupid as a broomstick. Why should we trust him to play NE in the subgame? Let’s look at the following game to illustrate that concern: 1 1 L R 1 A B 2 x y 2 x y 10 1 1 5-100 4 Here ( L,A,x ) is the unique SPE. However, player 2 has to put a lot of trust into player 1’s rationality in order to play x. He must believe that player 1 is smart enough to figure out that A is a dominant strategy in the subgame following R. However, player 2 might have serious doubts about player 1’s marbles after the guy has just foregone 5 utils by not playing L. 1 1.2 Multi-Stage Games Lemma 1 The unique SPE of the finitely repeated Prisoner’s Dilemma game in which players get the sum of their payoffs from each stage game has every player defecting at each information set. The proof proceeds by analyzing the last stage game where we would see defection for sure. But then we would see defection in the second to last stage game etc. In the same way we can show that the finitely repeated Bertrand game results in pricing at marginal cost all the time. Remark 1 The main reason for the breakdown of cooperation in the finitely repeated Prisoner’s Dilemma is not so much SPE by itself by the fact that there is a final period in which agents would certainly defect. This raises the question whether an infinitely repeated PD game would allow us to cooperate. Essentially, we could cooperate as long as the other player does, and if there 1 After all any strategy in which L is played strictly dominates any strategy in which R is played in the normal form. 2 is defection, we defect from then on. This still looks like a SPE - in any subgame in which I have defected before, I might as well defect forever. If I haven’t defected yet, I can jeopardize cooperation by defection, and therefore should not do it as long as I care about the future sufficiently....
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lecture12 - Lecture XII Analysis of Infinite Games Markus M...

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