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Unformatted text preview: Economics 101, UCLA Fall 2010 Jernej Copic Lecture 4, 10/5/10. Game Theory: Mixed strategies. Introducing Pareto optimality. When Grunf left, No1 was still scribbling furious notes on random pieces of paper. Among the more notable ones, there were some rather unpronouncible phrases, composed mainly of vulgarities, and there was also a write-up about a game. Ironically, it’s title was “Prisoners’ Delight”. ‘ A prisoners’ delight. This is like a prisoners’ dilemma, but the payoffs are different – when they are silent, they get a large payoff. The reason is that Pai Mei and the monk have bonded so that they care for each other – even if the tea ceremony they would have to endure when they are both silent is very long, dull, and tedious, remarkably enough, each one prefers going through that to seeing his companion rot in jail for a very long time and then going berserk with revenge upon release. The payoffs are all the same as in the prisoners’ dilemma between Pai Mei and the monk (see Lecture 3), except that if they are both silent, then the monk gets a payoff of 6 utils and Pai Mei a payoff of 5. Okay, so now, to figure out the equilibria, let’s ask the question, when would neither of them have an incentive to change their behavior. 1 • If Pai Mei is silent and the monk is silent, then none of them could benefit by changing their behavior - if nothing else, each of them is getting the most he can get. So this is an equilibrium! • If Pai Mei is silent and the monk rats, then Pai Mei would in fact prefer to change to rat; also the monk would then prefer to be silent. 2 • If Pai Mei rats and the monk is silent, then again, either would have wanted to change their behavior, so this doesn’t work. 1 Exercise. Write down this game in a matrix form. 2 Exercise. Confirm this by looking at the matrix. Note, when for instance Pai Mei makes his consideration of whether he could’ve benefited by changing his behavior, he can’t affect the behavior of the monk. So he compares his payoffs for the case when the monk rats. 1 • If they both rat, then... Oh, then again none of them could benefit by a change. For example, Pai Mai, if he instead remains silent, he is getting a payoff of- 1 rather than 1, so he wouldn’t want to do anything differently. ’ No1 was still staring at the Prisoners’ Delight, and he was determined he would get EVERYTHING right this time. Something was bothering him. “What about if Pai Mei decides for some reason to flip a coin between his two possible actions? I remember from a book that this is called a mixed strategy , and they wrote it like 1 2 silent + 1 2 rat . Hmmm... hold on, hold on... Then, lemme compute how much the monk gets out of playing either of his two actions. Okay, so if he decides to be silent, he is now facing a lottery. With probability 1 2 he will find himself in an outcome where Pai Mei was silent (and the monk is silent) in which case he gets a 6, and with probability 1 2 he will find himself in an outcome where Pai...
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- Fall '08
- Game Theory, Boulder, payoﬀs, Pai Mei