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Lecture 12 - Dynamic Games with Incomplete Information In...

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Dynamic Games with Incomplete Information In these lectures, we analyze the issues arise in a dynamics context in the presence of incomplete information, such as how agents should interpret the actions the other parties take. We define perfect Bayesian Nash equilibrium, and apply it in a sequential bargaining model with incomplete information. As in the games with complete information, now we will use a stronger notion of rationality – sequential rationality. 2 Perfect Bayesian Nash Equilibrium Recall that in games with complete information some Nash equilibria may be based on the assumption that some players will act sequentially irrationally at certain information sets off the path of equilibrium. In those games we ignored these equilibria by focusing on subgame perfect equilibria; in the latter equilibria each agent’s action is sequentially rational at each information set. Now, we extend this notion to the games with incomplete information. In these games, once again, some Bayesian Nash equilibria are based on sequentially irrational moves off the path of equilibrium. Consider the game in Figure 1. In this game, a firm is to decide whether to hire a worker, who can be hard-working (High) or lazy (Low). Under the current contract, if the worker is hard-working, then working is better for the worker, and the firm makes profit of 1 if the worker works. If the worker’s lazy, then shirking is better for him, and the firm will lose 1 if the worker shirks. If the worker is sequentially rational, then he will work if he’s hard-working and shirk if he’s lazy. Since the firm finds the worker 1 Nature High .7 Low .3 Firm Hire W Work Shirk Do not hire (1, 2) (0, 1) (0, 0) Do not hire Hire W Work Shirk (1, 1) (-1, 2) (0, 0) Figure 1: more likely to be hard-working, the firm will hire the worker. But there is another Bayesian Nash equilibrium: the worker always shirks (independent of his type), and therefore the firm does not hire the worker. This equilibrium is indicated in the figure by the bold lines. It is based on the assumption that the worker will shirk when he is hard-working, which is sequentially irrational. Since this happens off the path of equilibrium, such irrationality is ignored in the Bayesian Nash equilibrium–as in the ordinary Nash equilibrium. We’ll now require sequential rationality at each information set. Such equilibria will be called perfect Bayesian Nash equilibrium. The official definition requires more
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details. For each information set, we must specify the beliefs of the agent who moves at that information set. Beliefs of an agent at a given information set are represented by a probability distribution on the information set. In the game in figure 1, the players’ beliefs are already specified. Consider the game in figure 2. In this game we need to specify the beliefs of player 2 at the information set that he moves. In the figure, his beliefs are summarized by μ, which is the probability that he assigns to the event that player 1 played T given that 2 is asked to move.
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