Auction Theory

Hard4–5 hrsAuctionsBiddingWinner's CurseRevenue Equivalence

Overview

Auction theory teaches bidding strategies. The second-price auction's key insight: your bid only affects WHETHER you win, not HOW MUCH you pay (you pay the second-highest). So you might as well bid truthfully. In first-price, you set your own price — shade below your value to create profit margin. The winner's curse applies in common-value settings: if everyone bids their estimate of the true value, the winner systematically overpaid (they were the most optimistic estimator).

How to Recognize

→'How much should you bid in a sealed-bid auction?'
→Multiple buyers, unknown valuations, competitive bidding
→Common in market-making: 'how do you price this lot?'
→Winner's curse questions

Step-by-Step Approach

1.Second-price (Vickrey): dominant strategy = bid your true value v. Always.
2.First-price: bid below true value. With n bidders and Uniform[0,v]: bid b(v) = v·(n−1)/n
3.Revenue equivalence: expected revenue is same across standard auction formats
4.Common value: shade more aggressively — the winner is the most optimistic bidder

Key Formulas

01Second-price: bid your true value

v

(dominant strategy)

02First-price,

n

bidders, Uniform

[0,1]

b(v) = v \cdot \frac{n-1}{n}

E[\text{highest of } n \text{ Uniform}[0,1]] = \frac{n}{n+1}

04Revenue equivalence:

E[\text{revenue}_{\text{1st-price}}] = E[\text{revenue}_{\text{2nd-price}}]

Worked Examples

0Quick Example

10 bidders, first-price, valuations Uniform[0,100]. Optimal bid: b(v) = v·9/10. If v=

70, bid

63.

Problem

In a second-price sealed-bid auction, prove that bidding your true value v is dominant.

Solution

Your value: v. Highest competing bid: h. You pay h if you win.

Case 1: your bid > h → you win, pay h. Profit = v − h.

Case 2: your bid < h → you lose. Profit = 0.

If v > h: you should win. Bidding v → you win (v > h), profit = v − h > 0. Bidding below h → you lose, profit = 0. So bid ≥ h is better, and v ≥ h means bidding v wins at the right times.

If v < h: you should lose. Bidding v → you lose (v < h), profit = 0. Bidding above h → you win but pay h > v, loss! Bidding v correctly avoids winning when you shouldn't.

Conclusion: bidding v is optimal in all cases. This is a DOMINANT strategy — optimal regardless of others' bids.

Answer

Bidding true value $v$ is dominant in second-price: it wins when profitable (v>h), doesn't win when unprofitable (v<h). You cannot improve by overbidding (might win at a loss) or underbidding (might miss a profitable win).

Common Mistakes

!
Applying first-price optimal bidding to second-price auctions (and vice versa).
!
Forgetting the winner's curse in common value settings. In private value auctions, your value is your value — no adjustment needed. In common value (IPO, oil lease), you must correct for selection bias.
!
Thinking more competition always means lower prices. In first-price, more bidders → less shading → higher revenue for the seller.

Practice Problems

Click "Show Answer" to reveal

You're the only bidder in a first-price auction (n=1). Your value is $100. What should you bid?

Hint: n=1: b*(v) = v·(n−1)/n = v·0/1 = 0.

Vickrey auction: you value the item at $75. Others are bidding around$ 50– $80. Your friend says 'bid$ 90 to be safe.' Is this good advice?

Hint: In second-price, overbidding can only hurt, never help.

Back to Study Plan

Nash Equilibrium & Dominant Strategies Bid-Ask Spread & Edge