A seller offers to sell an object to a buyer. The buyer and seller's valuations for the object, t and u,, 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 ps and ps, respectively. You may think of ps as the amount the buyer is offering to pay for the object, and p, as the minimum amount the seller will accept. If ps2 ps, then trade occurs at price ps. If ps < Ps, then no trade occurs. If no trade occurs, the payoff to the buyer is 0 and to the seller is v.. If trade occurs at price p, the payoff to the buyer is t-p and to the seller is p.
A seller offers to sell an object to a buyer. The buyer and seller's valuations for the object, t and u,, 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 ps and ps, respectively. You may think of ps as the amount the buyer is offering to pay for the object, and p, as the minimum amount the seller will accept. If ps2 ps, then trade occurs at price ps. If ps < Ps, then no trade occurs. If no trade occurs, the payoff to the buyer is 0 and to the seller is v.. If trade occurs at price p, the payoff to the buyer is t-p and to the seller is p.
Chapter18: Asymmetric Information
Section: Chapter Questions
Problem 18.7P
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![A seller offers to sell an object to a buyer. The buyer and seller's valuations for the object,
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 p, and p₁, respectively. You may think of p, as the amount the buyer is offering
to pay for the object, and p, as the minimum amount the seller will accept. If po≥ Ps, then
trade occurs at price ps. If p < Ps, then no trade occurs. If no trade occurs, the payoff to
the buyer is 0 and to the seller is us. If trade occurs at price p, the payoff to the buyer is
U-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) = U, is weakly 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(p ≥ vs) (vb - Pb) = Pb(v₁ - Pb).
Since this is maximized when p = vb/2, the strategy profile (ps(vb), p.(vs)) given by
ps(vb) = vb/2 and p.(vs) = v, is a Nash equilibrium.
(b) Explain whether the equilibrium you found is efficient.
Solution: The efficient outcome is for trade to occur whenever > vs. In the equilib-
rium, trade only occurs if v 2vs. Therefore, the equilibrium involves less trade than
would be efficient.
(c) Find a Nash equilibrium in which each player always chooses one of two prices (note
that the two prices may be different for the two players).
Solution: One such equilibrium is for the seller to choose ps = 1 if vs > 1/2 and
Ps= = 1/2 if v, ≤ 1/2, and the buyer to choose p = 0 if v < 1/2 and pt = 1/2 if
2 ≥ 1/2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0542aa5b-cdbd-4921-99ff-0dadb1c30f45%2Fad060485-7e86-48c2-b7ed-d6f2880eaf1a%2Fadv4sse_processed.png&w=3840&q=75)
Transcribed Image Text:A seller offers to sell an object to a buyer. The buyer and seller's valuations for the object,
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 p, and p₁, respectively. You may think of p, as the amount the buyer is offering
to pay for the object, and p, as the minimum amount the seller will accept. If po≥ Ps, then
trade occurs at price ps. If p < Ps, then no trade occurs. If no trade occurs, the payoff to
the buyer is 0 and to the seller is us. If trade occurs at price p, the payoff to the buyer is
U-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) = U, is weakly 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(p ≥ vs) (vb - Pb) = Pb(v₁ - Pb).
Since this is maximized when p = vb/2, the strategy profile (ps(vb), p.(vs)) given by
ps(vb) = vb/2 and p.(vs) = v, is a Nash equilibrium.
(b) Explain whether the equilibrium you found is efficient.
Solution: The efficient outcome is for trade to occur whenever > vs. In the equilib-
rium, trade only occurs if v 2vs. Therefore, the equilibrium involves less trade than
would be efficient.
(c) Find a Nash equilibrium in which each player always chooses one of two prices (note
that the two prices may be different for the two players).
Solution: One such equilibrium is for the seller to choose ps = 1 if vs > 1/2 and
Ps= = 1/2 if v, ≤ 1/2, and the buyer to choose p = 0 if v < 1/2 and pt = 1/2 if
2 ≥ 1/2.
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