Auction Theory: Formats and Strategies

Auctions as Allocation Mechanisms

Auctions have existed for thousands of years: slaves in Babylon, fish in Japanese harbors, Treasury bonds, spectrum frequencies, internet advertising. Modern auction theory is precise mathematics about how participants place bids and how much revenue the seller earns.

The key question: with the same preferences among participants, do different auction formats yield the same or different revenue for the seller? The intuition “higher bids = more revenue” is not always correct — the auction format deeply affects strategies.

Main Auction Formats

First-Price Sealed-Bid Auction (FPSB): Sealed bids; the highest bid wins, the winner pays their own bid. Strategy: no dominant strategy — one should "shade" the bid below their own valuation. The optimal shading depends on the number of participants and the distribution of valuations.

Second-Price Auction (Vickrey, 1961): Sealed bids; the highest bid wins, the winner pays the second-highest bid. Dominant strategy: bid your true value θᵢ! This is a DSIC mechanism. Why: when bidding above θᵢ — there’s a risk of winning and overpaying. When bidding below — there’s a risk of missing a good deal. Bidding θᵢ is always optimal regardless of the opponents.

English Auction: Open ascending bidding. Equivalent to the second-price auction in the case of independent private values (the winner pays slightly more than the second-highest). In practice, it allows participants to update beliefs as the auction progresses.

Dutch Auction: The price falls from high until the first bid is made. Strategically equivalent to FPSB for independent private values.

Revenue Equivalence Theorem

Revenue Equivalence Theorem (RET, Myerson 1979; Mirrlees–Vicklund 1981): Under the following conditions — (i) symmetric risk-neutral participants; (ii) independent private valuations θᵢ ~ F on [0,θ̄]; (iii) the auction is "standard" (the item is allocated to the participant with the highest valuation); (iv) a participant with zero valuation gets zero expected surplus — the seller’s expected revenue is identical for all standard auctions.

Numerical confirmation: Two participants, θ ~ U[0,1].

FPSB: equilibrium strategy b*(θ) = θ/2. Expected revenue: E[max(b₁, b₂)] = E[max(θ₁/2, θ₂/2)] = (1/2)E[max(θ₁,θ₂)] = (1/2)·(2/3) = 1/3.

Vickrey: the maximum wins, pays the second-highest. E[revenue] = E[min(θ₁,θ₂)] = 1/3.

Coincides! RET is satisfied ✓.

Myerson's Optimal Auction

What if the seller wants to maximize revenue (not necessarily sell to the highest valuer)?

Virtual value: ψ(θ) = θ − (1−F(θ))/f(θ). With monotonic virtual value: the optimal auction sells to the participant with the highest positive ψ(θ); if ψ(θ) < 0 for all — doesn’t sell.

Reserve price: In the symmetric case, it is optimal to set a reserve price r* where ψ(r*) = 0: r* − (1−F(r*))/f(r*) = 0.

For θ ~ U[0,1]: F = θ, f = 1, ψ(θ) = 2θ − 1 = 0 → r* = 1/2.

One buyer, θ ~ U[0,1]: without a reserve price E[revenue] = 0 (no competitor to "push up" the price). With r* = 0.5: if θ > 0.5, sell at r*; if θ < 0.5 — don’t sell. E[revenue] = ∫₀.₅^1 0.5 dθ = 0.25. The optimal reserve price significantly increases revenue!

Winner’s Curse

With interdependent valuations (oil field, artwork): valuation θᵢ = V + εᵢ, where V is the true value, εᵢ is a private signal. The winner had the highest signal, thus on average overestimated V.

Rational correction: bid below θᵢ by the "curse" amount. Winning with bid b means: "my signal was the highest" — the conditional distribution of V is lower than the unconditional mean of θᵢ.

Practice: companies in oil tenders systematically overpaid during the 1960s–70s. After realizing the winner’s curse — they started adjusting for the selection effect.

GSP Auction (Google, 2002): Search ads are traded via the generalized second-price auction: position k goes to the participant with the k-th bid, who pays the (k+1)-th bid times the click-through rate for their position. GSP is not a true VCG and not DSIC, but in symmetric equilibrium yields the same expected revenue as VCG (revenue equivalence theorem). Billions of such auctions are held daily — the largest market designed by game theory.

Auction Theory in Business Valuation and Procurement

Auction formats and strategies have direct implications for corporate finance and public procurement. In mergers and acquisitions (M&A), the sale of a company is often conducted as a first-price sealed bid auction: the advisor bank collects “indicative” offers from potential buyers and then accepts the best one. Auction theory predicts bid aggressiveness, the “winner’s curse”, and optimal reserve prices. In public procurement (defense, infrastructure) the reverse auction (winner pays the lowest) is the WTO and OECD standard: the government chooses the lowest bid, conditional on equal quality. First-price auction theory predicts that suppliers underbid costs proportionally to the level of competition. Central bank credit auctions (REPO, QE operations) allocate liquidity through multiple formats: the Federal Reserve uses second-price auctions, while the ECB uses both first- and second-price auctions depending on the instrument. Empirical data confirms the revenue equivalence theorem (Myerson, 1981): with symmetric agents and independent private values, the seller's expected revenue is the same for all standard auction formats.

In investment banking practice, the “bid shading” strategy in FPSB is described exactly by the equilibrium formula b*(θ) = θ(n−1)/n: as the number of participants n → ∞, bid shading disappears and the market approaches competition. This explains why auctions with a limited number of participants yield lower revenue for the seller, and why expanding the pool of auction participants is a standard recommendation when selling assets.

Assignment: (a) Three participants in FPSB, θ ~ U[0,1]. Find b*(θ). Calculate expected revenue. Does it coincide with Vickrey? (b) One buyer θ ~ U[0,2], optimal reserve price r*. (c) Explain in your own words why in a second-price auction truthful bidding is a dominant strategy.