A seller offers to sell an object to a buyer. The buyer and seller's valuations for the object, v and vs, are independent and uniformly distributed on [0, 1]. Each knows her own valuation but not the valuation of the other. Trade proceeds as follows. The buyer and seller simultaneously choose prices på and på, respectively. You may think of på as the amount the buyer is offering to pay for the object, and p, as the minimum amount the seller will accept. If pb ≥ ps, then trade occurs at price på. If på < P、, then no trade occurs. If no trade occurs, the payoff to the buyer is 0 and to the seller is vs. If trade occurs at price p, the payoff to the buyer is v-p and to the seller is p. (a) Find a Nash equilibrium of this game in which both players use strategies that are linear in their valuation and the seller's strategy is not weakly dominated. Solution: By the same argument as in a second-price auction with private values, the strategy ps(vs) = vs is weakly dominant for the seller. If the seller uses that strategy, a buyer of type v, who chooses pt = [0, 1] receives a payoff of Pr(Pb ≥ Vs) (vb – Pb) = Pb(vb — Pb). Since this is maximized when pb = v₁/2, the strategy profile (pb(v₁), Ps(vs)) given by Pb(vb) = vb/2 and ps(vs) = v¸ is a Nash equilibrium.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter18: Asymmetric Information

Section: Chapter Questions

Problem 18.7P

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Transcribed Image Text:A seller offers to sell an object to a buyer. The buyer and seller's valuations for the object, vb and us, are independent and uniformly distributed on [0, 1]. Each knows her own valuation but not the valuation of the other. Trade proceeds as follows. The buyer and seller simultaneously choose prices po and ps, respectively. You may think of p, as the amount the buyer is offering to pay for the object, and ps as the minimum amount the seller will accept. If pb ≥ ps, then trade occurs at price pь. If p < Ps, then no trade occurs. If no trade occurs, the payoff to the buyer is 0 and to the seller is vs. If trade occurs at price p, the payoff to the buyer is up and to the seller is p. (a) Find a Nash equilibrium of this game in which both players use strategies that are linear in their valuation and the seller's strategy is not weakly dominated. Solution: By the same argument as in a second-price auction with private values, the strategy ps(vs) = Us is akly dominant for the seller. If the seller uses that strategy, a buyer of type v, who chooses p = [0, 1] receives a payoff of Pr(pb ≥ Vs) (vb - Pb) = Pb(vb - Pb). Since this is maximized when pb = vb/2, the strategy profile (pi(vb), Ps(vs)) given by Pb(vb) = vb/2 and ps(vs): = vs is a Nash equilibrium.

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