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 Subject: game theory in microeconomics Category: Business and Money > Economics Asked by: vitaminc-ga List Price: \$10.00 Posted: 08 Oct 2002 19:42 PDT Expires: 07 Nov 2002 18:42 PST Question ID: 74262
 ```1)Consider the following two-person game: (a1,a2 b1,c2) (c1,b2 d1,d2)......actually all braket where for i=1,2, ai>ci and bi>di. Compute the best response functions for the two players. What are the Nash equilibria of this game? 2)There are 10 students in a game theory class. The teacher proposes the following to the students. Each student has to write simultaneously either "yes" or "no" on a slip of paper. For every student who write "yes", the teacher will pay \$1 to all other students; and, in addition, if there are at least two "yes"-sayers her will pay one additional dollar to all "yes"-sayers. a)What are the pure strategy Nash equilibria of the above game? b)What strategy would you adopt if you were playing this game? Explain your choice.``` Request for Question Clarification by rbnn-ga on 08 Oct 2002 20:26 PDT `what does "actually all braket" mean?` Clarification of Question by vitaminc-ga on 08 Oct 2002 21:40 PDT ```Left Right Up a1,a2 b1,c2 Down c1,b2 d1,d2 i mean there is a big bracket for all these four items. well, you can ignore it. just check the arrangement above(that's what i mean)``` Clarification of Question by vitaminc-ga on 08 Oct 2002 21:42 PDT `it's payoff for the game theory`
 ```1. If player 1 chooses UP, then player 2s best response is LEFT since a2>c2. If player 1 chooses DOWN, then player 2s best response if LEFT since b2>d2 If player 2 chooses LEFT then player 1s best response is UP since a1>c1 If player 2 chooses RIGHT then player 1's best response is UP since b1>d1 . The only Nash equilibrium for the game is therefore (UP,LEFT). 2. a) We examine the number of students who choose yes, from 0 through 10, systematically. First, suppose no students choose YES. Can any student improve his payoff by changing his vote to YES? The answer is no; even though, if the one student does this, all the other students' payoffs increase, the definition of Nash equilibrium is that if a player can improve his OWN value by changing only his OWN strategy then there is no equilibrium. Hence, the state (also called strategy profile) in which all students choose NO is a Nash equilibrium. In general, suppose there are k students choosing YES and 10-k students choosing NO. If k>=2 then each YES student gets k dollars (k-1 from the other YES students and 1 because there are two YES students); the NO students each get k dollars as well. Hence, the higher k is the more money the students get, so that the only Nash equilibrium for k>=2 is when all the students choose YES and they each get \$10.00. The state with one YES, however, is not a NASH equilibrium, because a NO student gets only \$1, but could increase his payoff to \$2 by switching his vote. So, the only Nash equilibria are those states in which all students make the same vote. (Note that the phrase in the problem "for every student who write "yes", the teacher will pay \$1 to all other students" has to be read carefully - I misread it at first as implying only the NO voters get the extra cash. 2b) This is quite complex because my *utility* function might be different from my *game payoff*. I might want my fellow students to get money; or I might not to. I might want the teacher to lose money; or I might not to. I also might want to SIGNAL to the other students, or to the teacher, what my utility function is for future reference. I could also use a mixed strategy (here I am only consider pure strategy equilibria, where there is no randomization). Still, I think the answer sought, and what I would do is: choose YES. In no case do I ever LOSE money by choosing YES rather than NO; no matter what the other students do I either stay the same or gain money by choosing YES. Search Strategy: -------------- For this problem I used a rather enjoyable book I'd read not long ago called Game Theory Evolving, by Herbert Gintis, Princeton, 2000.``` Request for Answer Clarification by vitaminc-ga on 11 Oct 2002 14:32 PDT ```well,i figure it out that i got question wrong.(for Q1) it should be bidi, then (UP,LEFT) and DOWN,RIGHT are both pure-strategy Nash equilibrium.```
 vitaminc-ga rated this answer: ```now i got it thanks a lot ^^```