In a number of countries, government contracts are awarded by means auctions in which the winner is the bidder whose bid is closest to the average bid. Consider such an auction with three bidders competing to win a single object. Bidder i has a valuation v; for the object, where v₁ > V₂ > V3 > 0. Bidders simultaneously submit bids b; ≥ 0. The bidder whose bid is closest to the average of b₁,b2, and b3 wins the object and pays her own bid (thus if i is the winning bidder, her payoff is v; − b;). In case of a tie, the bidder with the highest valuation wins the object. Losing bidders receive a payoff of 0. (a) Find all (pure strategy) Nash equilibria of this game. Solution: Given by and b3, note that 1 can win the auction by bidding min{b2, b3}. If min{b2, b3} > v₁, then only losing bids are best responses for 1. Hence either 2 or 3 is winning and paying more than her valuation, which cannot be a best response. Hence we must have min{b2, b3} ≤ v₁. Similarly, min{b2, b3} = v₁ can only be a best response for 2 and 3 if 1 is winning, which can only be a best response for 1 if b₁ = v₁. The profiles (v₁,b2, b3) such that min{b2, b3} = v₁ are indeed Nash equilibria. If min{b2, b3} < v₁, then the unique best response for 1 is to choose b₁ = min{b₂, b3}. If b₁ = b₂ = b3 < V₁, then neither 2 nor 3 can win by deviating and if 1 deviates she loses. Therefore, these are Nash equilibria. If b₁ = min{b2, b3} < v₁ and b₂ < b3 then we have a Nash equilibrium if and only if b2 ≥ v2. If b₁ = min{b2, b3} < v₁ and b3 < b2 then we have a Nash equilibrium if and only if b3 ≥ v3. (b) Show that in every Nash equilibrium in which the three players do not submit identical bids, at least one player plays a weakly dominated strategy. Solution: In any such Nash equilibrium, there is at least one player i who bids b¡ ≥ v₂. A bid b; ≥ v; is weakly dominated by any bid b € (0, v₁).
In a number of countries, government contracts are awarded by means auctions in which the winner is the bidder whose bid is closest to the average bid. Consider such an auction with three bidders competing to win a single object. Bidder i has a valuation v; for the object, where v₁ > V₂ > V3 > 0. Bidders simultaneously submit bids b; ≥ 0. The bidder whose bid is closest to the average of b₁,b2, and b3 wins the object and pays her own bid (thus if i is the winning bidder, her payoff is v; − b;). In case of a tie, the bidder with the highest valuation wins the object. Losing bidders receive a payoff of 0. (a) Find all (pure strategy) Nash equilibria of this game. Solution: Given by and b3, note that 1 can win the auction by bidding min{b2, b3}. If min{b2, b3} > v₁, then only losing bids are best responses for 1. Hence either 2 or 3 is winning and paying more than her valuation, which cannot be a best response. Hence we must have min{b2, b3} ≤ v₁. Similarly, min{b2, b3} = v₁ can only be a best response for 2 and 3 if 1 is winning, which can only be a best response for 1 if b₁ = v₁. The profiles (v₁,b2, b3) such that min{b2, b3} = v₁ are indeed Nash equilibria. If min{b2, b3} < v₁, then the unique best response for 1 is to choose b₁ = min{b₂, b3}. If b₁ = b₂ = b3 < V₁, then neither 2 nor 3 can win by deviating and if 1 deviates she loses. Therefore, these are Nash equilibria. If b₁ = min{b2, b3} < v₁ and b₂ < b3 then we have a Nash equilibrium if and only if b2 ≥ v2. If b₁ = min{b2, b3} < v₁ and b3 < b2 then we have a Nash equilibrium if and only if b3 ≥ v3. (b) Show that in every Nash equilibrium in which the three players do not submit identical bids, at least one player plays a weakly dominated strategy. Solution: In any such Nash equilibrium, there is at least one player i who bids b¡ ≥ v₂. A bid b; ≥ v; is weakly dominated by any bid b € (0, v₁).
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:In a number of countries, government contracts are awarded by means auctions in which the
winner is the bidder whose bid is closest to the average bid. Consider such an auction with
three bidders competing to win a single object. Bidder i has a valuation vi for the object,
where v₁ > V₂ > V3 > 0. Bidders simultaneously submit bids b; ≥ 0. The bidder whose bid is
closest to the average of b₁,b2, and b3 wins the object and pays her own bid (thus if i is the
winning bidder, her payoff is vi- bi). In case of a tie, the bidder with the highest valuation
wins the object. Losing bidders receive a payoff of 0.
(a) Find all (pure strategy) Nash equilibria of this game.
Solution: Given by and b3, note that 1 can win the auction by bidding min{b2, b3}. If
min{b2, b3} > v₁, then only losing bids are best responses for 1. Hence either 2 or 3 is
winning and paying more than her valuation, which cannot be a best response. Hence we
must have min{b₂2, b3} ≤ v₁. Similarly, min{b2, b3}: = v₁ can only be a best response for
2 and 3 if 1 is winning, which can only be a best response for 1 if b₁ = v₁. The profiles
(v₁, 62, 63) such that min{b2,b3} = v₁ are indeed Nash equilibria. If min{b2, b3} < v₁,
then the unique best response for 1 is to choose b = min{b2, b3}. If b₁ = b₂ = b3 < 1,
then neither 2 nor 3 can win by deviating and if 1 deviates she loses. Therefore, these are
Nash equilibria. If b₁ = min{b2, b3} <v₁ and b₂ < b3 then we have a Nash equilibrium
if and only if b2 ≥ v₂. If b₁ = min{b2, b3} < v₁ and b3 < b2 then we have a Nash
equilibrium if and only if b3 ≥ v3.
(b) Show that in every Nash equilibrium in which the three players do not submit identical
bids, at least one player plays a weakly dominated strategy.
Solution: In any such Nash equilibrium, there is at least one player i who bids bį ≥ vį.
A bid bi vi is weakly dominated by any bid b € (0, v₂).
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