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Unformatted text preview: ECON 212 Game Theory Prof. Andrew Postlewaite Fall 2010 University of Pennsylvania Suggested Solution for Problem Set 6 1. a. Announcing his true valuation is a dominant strategy. Let v be his valuation, and b and s be the buyer's and the seller's announcement, respectively. The buy'er payo is v s if b ≥ s and otherwise. Suppose b < v . If b ≥ s of v ≤ s , then the buyer is indi erent between announcing b and v . Otherwise, the buyer gets nothing by announceing b , while he gets v s by announcing v . Similarly, for b > v , the buyer is indi erent if s > b or v ≥ s , but becomes worse o if b ≥ s > v . Hence, announceing the true valuaion is a (weakly) dominant strategy. b. Suppose the seller's valuaion is v S . Then her optimal announcement s * solves the following problem. s * ∈ arg max s ∈ [0 , 1] (1 s )( s v S ) . The rst component 1 s is the probability for the buyer to announce more than s and the second is the seller's payo in case she sells at s . The foc is s * + v s + 1 s * = 0 s * = 1 + v S 2 . Therefore, the seller's strategy in the Bayes equilibrium in which the buyer plays the strategy in part a is s ( v S ) = 1+ v S 2 . c. The buyer's strategy is independent of the seller's distribution. Hence the seller's problem is the same as before. Nothing changes. 2. This game is a canonical signalling game which is conveniently represented by the following game tree. Player 1's strategy is s i : { L,R } → { U,D } . Player 2's strategy is s 2 : { 1 , 2 } → { L,R } ....
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 Spring '12
 gg
 Equilibrium, Game Theory, player, announceing b

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