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Unformatted text preview: NON-COOPERATIVE GAMES MIHAI MANEA Department of Economics, MIT, 1. Normal-Form Games A normal (or strategic ) form game is a triplet ( N, S, U ) with the following properties • N = { 1 , 2 , . . . , n } is a finite set of players • S i is the set of pure strategies of player i ; S = S 1 × . . . × S n u i : S R is the payoff function of player i ; u = ( u 1 , . . . , u n ). • → Player i ∈ N receives payoff u i ( s ) when s ∈ S is played. The game is finite if S is finite. The structure of the game is common knoweldge : all players know ( N, S, U ), and know that their opponents know it, and know that their opponents know that they know, and so on. 1.1. Detour on common knowledge. Common knowledge looks like an innocuous as- sumption, but may have strong consequences in some situations. Consider the following story. Once upon a time, there was a village with 100 married couples. The women had to pass a logic exam before being allowed to marry. The high priestess was not required to take that exam, but it was common knowledge that she was truthful. The village was small, so everyone would be able to hear any shot fired in the village. The women would gossip about adulterous relationships and each knows which of the other husbands are unfaithful. However, no one would ever inform a wife about her cheating husband. The high priestess knows that some husbands are unfaithful and decides that such im- morality should not be tolerated any further. This is a successful religion and all women agree with the views of the priestess. The priestess convenes all the women at the temple and publicly announces that the well- being of the village has been compromised—there is at least one cheating husband. She Date : May 13, 2010. 2 MIHAI MANEA also points out that even though none of them knows whether her husband is faithful, each woman knows about the other unfaithful husbands. She orders that each woman shoot her husband on the midnight of the day she finds out. 39 silent nights went by and on the 40 th shots were heard. How many husbands were shot? Were all the unfaithful husbands caught? How did some wives learn of their husband’s infidelity after 39 nights in which nothing happened? Since the priestess was truthful, there must have been at least one unfaithful husband in the village. How would events have evolved if there was exactly one unfaithful husband? His wife, upon hearing the priestess’ statement and realizing that she does not know of any unfaithful husband, would have concluded that her own marriage must be the only adulterous one and would have shot her husband on the midnight of the first day. Clearly, there must have been more than one unfaithful husband. If there had been exactly two unfaithful husbands, then each of the two cheated wives would have initially known of exactly one unfaithful husband, and after the first silent night would infer that there were exactly two cheaters and her husband is one of them. (Recall that the wives are all perfect logicians.) The unfaithful her husband is one of them....
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This note was uploaded on 11/28/2011 for the course ECONOMICS - taught by Professor Muhammadyildiz during the Spring '05 term at University of Massachusetts Boston.

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