Lecture 13

# Lecture 13 - Reputation Consider a game in which a player i...

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Reputation Consider a game in which a player i has two types, say A and B. Imagine that if the other players believe that i is of type A, then i’s equilibrium payoff will be much higher than his equilibrium payoff when the other players believe that he is of type B. If there’s a long future in the game and i is patient, then he will act as if he is of type A even when his type is B, in order to convince the other players that he is of type A. In other words, he will try to form a reputation for being of type A. This will change the equilibrium behavior dramatically when the other players assign positive probability to each type. For example, if a seller thinks that the buyer does not value the good that much, then he will be willing to sell the good at a low price. Then, even if the buyer values the good highly, he will pretend that he does not value the good that much and will not buy the object at higher prices–although he could have bought at those prices if it were common knowledge that he values the object highly. (If the players are sufficiently patient, then in equilibrium the price will be very low.) Likewise, in the entry deterrence game, if it is possible that the incumbent gains from a fight in case of an entry, if this is the incumbent’s private information, and if there is a long future in the game, then he will fight whenever the entrant enters, in order to form a reputation for being a fighter and deter the future entries. In that case, the entrants will avoid 1 entering even if they are confident that the incumbent is not a fighter. We will now illustrate this notion of reputation formation on the centipede game. Consider the centipede game in figure 1. In this game, a player prefers to exit (or 100 100 98 101 99 99 97 100 98 98 11 03 22 1 2 1 1 2 1 2 Figure 1: to go down) if he believes that the other player will exit at the next node. Moreover, player 2 prefers exiting at the last node. Therefore, the unique backward induction outcome in this game is that each player goes down at each node. In particular, player 1 goes down at the first node and the game ends. This outcome is considered to be very counterintuitive, as the players forego very high payoffs. We will see that it is not robust

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to asymmetric information, in the sense that the outcome would change dramatically if there were a slight probability that a player is of a certain “irrational” type. In figure 2, we consider such a case. Here, player 2 assigns probability .999 to the event that player 1 is a regular rational type, but she also assigns probability .001 to the event that player 1 is a “nice” irrational type who would not want to exit the game. We index the nodes by n starting from the end. Sequential rationality requires that player 1 goes
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Lecture 13 - Reputation Consider a game in which a player i...

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