(continued) The equilibrium price in the first-price auction is:

Question

BlackTom AI · Accepted Answer

For n symmetric bidders in a first-price sealed-bid auction the symmetric equilibrium bidding function is b(v) = (n − 1)/n · v, so with three bidders and a value v = 90 the equilibrium bid is (2/3)·90 = 60.

1. Relevant concepts
- First-price sealed-bid auction: each bidder submits one bid, highest bidder wins and pays her bid.
- Private-value, symmetric bidders: each bidder’s value v is her private signal and bidders use the same bidding strategy.
- Equilibrium bid function: a monotone strategy b(v) giving the optimal bid for value v when others use the same function.

2. Step-by-step reasoning and derivation
Assume values are i.i.d. on [0, V] with continuous cdf F and density f. Suppose others use a linear strategy b(v) = αv (monotone). If you with value v bid b, you win iff all other values v_i satisfy α v_i ≤ b, i.e. v_i ≤ b/α. Probability of winning = F(b/α)^{n−1}. Your expected payoff:
U(b) = (v − b) · F(b/α)^{n−1}.
First-order condition for an interior maximum (differentiate w.r.t. b and set to zero):
dU/db = −F(b/α)^{n−1} + (v − b)(n − 1)F(b/α)^{n−2} f(b/α)·(1/α) = 0.
For uniform[0, V], F(x) = x/V and f = 1/V. Substitute and simplify (cancellations occur) to obtain:
−(b/α)^{n−1} + (v − b)(n − 1)(b/α)^{n−2}·(1/α) = 0
→ −b + (v − b)(n − 1) = 0
→ b = (n − 1)/n · v.
Thus α = (n − 1)/n.

3. Calculation for n = 3, v = 90
b = (3 − 1)/3 · 90 = (2/3)·90 = 60.

4. Why 60 is correct
This follows directly from the equilibrium formula b(v) = (n − 1)/n · v with n = 3. It is the bid that makes a bidder indifferent at the margin and is the best response when others follow the same rule.

5. Why the other options are incorrect
- 20 would be (2/9)·90, not the equilibrium multiplier (2/3).
- 30 equals (1/3)·90, which would be wrong α for n = 3.
- 90 is bidding one’s full value (b = v); that is not an equilibrium in first-price auctions because one can profitably shade the bid below v.

6. Conclusion
Using the equilibrium bid b(v) = (n − 1)/n · v with n = 3 and v = 90 gives b = 60, matching the provided correct answer: 60.

(continued) The equilibrium price in the first-price auction is:单项选择题

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