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Unformatted text preview: 14.126 GAME THEORY MIHAI MANEA Department of Economics, MIT, 1. Sequential Equilibrium In multi-stage games with incomplete information, say where payoffs depend on initial moves by nature, the only proper subgame is the original game, even if players observe one anothers actions at the end of each period. Thus the refinement of Nash equilibrium to subgame perfect equilibrium has no bite. Since players do not know the others types, the start of a period can only be analyzed as a separate subgame when the players posterior beliefs are specified. The concept of sequential equilibrium proposes a way to derive plausible beliefs at every information set. Based on the beliefs, one can test whether the continuation strategies form a Nash equilibrium. The complications that incomplete information causes are easiest to see in signaling gamesleader-follower games in which only the leader has private information. The leader moves first; the follower observes the leaders action, but not the leaders type, before choos- ing his own action. One example is Spences (1974) model of the job market. In that model, the leader is a worker who knows her productivity and must choose a level of education; the follower, a firm (or number of firms), observes the workers education level, but not her productivity, and then decides what wage to offer her. In the spirit of subgame perfection, the optimal wage should depend on the firms beliefs about the workers productivity given the observed education. An equilibrium needs to specify not only contingent actions, but also beliefs. At information sets that are reached with positive probability in equilibrium, beliefs should be derived using Bayes rule. However, there are some theoretical issues about belief update following zero-probability events. Date : March 2, 2010. 2 MIHAI MANEA Refer for more motivation to the example in FT, figure 8.1 (p. 322). The strategy profile ( L, A ) is a Nash equilibrium, which is subgame perfect as player 2s information set does not initiate a proper subgame. However, it is not a very plausible equilibrium, since player 2 prefers playing B rather than A at his information set, regardless of whether player 1 has chosen M or R . So, a good equilibrium concept should rule out the solution ( L, A ) in this example and ensure that 2 always plays B . The problem with the considered equilibrium is that player 2 does not play a best response to any possible belief at his information set. For most definitions, we focus on extensive form games of prefect recall with finite sets of decision nodes. We use some of the notation introduced earlier. A sequential equilibrium (Kreps and Wilson 1982) is an assessment ( , ), where is a (behavior) strategy profile and is a system of beliefs . The latter component consists of a belief specification ( h ) over the nodes at each information set h . The definition of sequential equilibrium is based on the concepts of...
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- Spring '05
- Game Theory