Lecture 11

Lecture 11 - Static Games with Incomplete Information An...

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Static Games with Incomplete Information An incomplete information game is a game where a party knows something that some other party does not know. For instance, a player may not know another player’s utility function, while the player himself knows his utility function. Such situations are modeled through games where Nature moves and some players can distinguish certain moves of nature while some others cannot. Example: Firm hiring a worker, the worker can be high or low ability and the firm does not know which. These notes are heavily based on the notes by Professor Daron Acemoglu, who taught this course in previous years. 1 Nature High p Low 1-p 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) In this game W knows whether he is of high (who will work) or low (who will shirk) ability worker, while the Firm does not know this; the Firm believes that the worker is of high ability with probability p. And all these are common knowledge. We call a player’s private information as his “type”. For instance, here W has two types — high and low — while Firm has only one type (as everything firm knows is common knowledge). Note that we represent the incomplete information game in an extensive form very similar to imperfect information games. Formally, a static game with incomplete information is as follows. First, Nature chooses some t = (t 1 , t 2 , . . . , t n ) T, where each t T is selected with probability p (t). Here, t i T i is the type of player i N = {1, 2, . . . , n}. Then, each player observes his own type, but not the others’. finally, players simultaneously choose their actions, each player knowing his own type. We write a = (a 1 , a 2 , . . . , a 2 ) A for any list of actions taken by all the players, where a i A i is the action taken by player i. Such a static gamewith incomplete information is denoted by (N, T, A, p).
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Notice that a strategy of that player determines which action he will take at each information set of his, represented by some t i T i . That is, a strategy of a player i is a function
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Lecture 11 - Static Games with Incomplete Information An...

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