# PS135: Game Theory in the Social Sciences Quiz 5 UC Berkeley · Department of Political Science

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PS135: Game Theory in the Social Sciences Quiz 5 UC Berkeley · Department of Political Science Summer 2016 · Professor Sean Gailmard Scores are out of 100 points possible. Each numbered question has equal weight toward that total. Each lettered subpart has equal weight in the point value of its question. 1. What is a pure strategy in a Bayesian game? What is a pure strategy Bayesian Nash equilibrium in a Bayesian game? 2. Consider the following Bayesian game. Nature decides whether payoﬀs re given by the strategic game on the left or on the right: a L R L R T 1, 1 0, 0 0, 0 0, 0 T B B 0, 0 0, 0 0, 0 2, 2 Nature is equally likely to pick either game, and player 1 knows which game is being played but player 2 does not. They move simultaneously, and payoﬀs are given by the strategic game Nature chose. What are the pure strategy Bayesian Nash equilibria of this Bayesian game? 3. Consider a ﬁrst-price, sealed-bid auction in which a bidder’s valuation can take one of three values: 5, 7, and 10, occurring with probabilities .2, .5, and .3, respectively. There are two bidders, whose valuations are independently drawn by Nature. After each bidder learns her valuation, they simultaneously choose a bid that is required to be a positive integer. A bidder’s payoﬀ is zero if she loses the auction and is her valuation minus her bid if she wins it. (a) Determine whether it is a symmetric Bayes Nash equilibrium for a bidder to bid 4 when her valuation is 5, 5 when her valuation is 7, and 6 when her valuation is 10. (b) Determine whether it is a symmetric Bayes Nash equilibrium for a bidder to bid 4 when her valuation is 5, 6 when her valuation is 7, and 9 when her valuation is 10. 1 4. Two U.S. senators are considering entering the race for the Democratic nomination for U.S. president. Each candidate has a privately known personal cost to entering the race. Assume that the probability of having a low entry cost, fL, is p and the probability of having high entry cost, fH , is 1 − p. Thus, the type space has just two values. A candidate’s payoﬀ depends on whether he enters the race and whether the other senator enters as well. Let v2 be a candidate’s payoﬀ when he enters and the other senator does as well (so that there are two candidates), v1 be a candidate’s payoﬀ when he enters and the other senator does not (so there is one candidate), and 0 be the payoﬀ when he does not enter. Assume that v1 = 25 > v2 = 10 > 0, (1) fH = 15 > fL = 5 > 0, (2) v2 − fL > 0 > v2 − fH , (3) v1 − fH > 0. (4) (a) Is it a symmetric Bayes Nash equilibrium for a candidate to enter if and only if she has a low cost from doing so? (b) Suppose v1 = 18. Now is it a BNE for a candidate to enter if and only if she has a low cost from doing so? 5. Players 1, 2, and 3 are involved in a game requiring some coordination. Each chooses among two options: A and B. Nature determines which of these options is the best one to coordinate on, picking A and B with equal probability. If a majority chooses the option that Nature deems best, then all three receives a payoﬀ of 1. If a majority chooses the option that is not the one that Nature deems best, then each receives a payoﬀ of 0. Player 1 learns which option is best (i.e., she learns Nature’s choice). The three players then simultaneously choose an option. Find a Bayes Nash equilibrium in which a majority is guaranteed to pick the choice Nature deems best. 6. A seller oﬀers a good with value v ∈{0,…, 10}, i.e., v is one of these 11 possible values. The seller knows the exact value of v, but a potential buyer only knows that each value of v is equally likely (it is uniformly 2 distributed). The buyer makes an oﬀer b and the seller accepts the oﬀer if and only if b ≥ v. (a) What is the expected value of v? (b) What is the expected value of v, given that the seller accepts the oﬀer? (c) What is the buyer’s equilibrium value of b? (d) Explain this result.

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