Auction Theory

Study of resource allocation via bidding mechanisms.

Ask the Game theory assistant 1 min read · Updated September 9, 2026

Definition

Standard auction formats include first-price sealed-bid, second-price (Vickrey), English (ascending), and Dutch (descending).

With private values $v_i$ i.i.d.\ from $F$ , a symmetric equilibrium bid in the first-price auction is $b(v)=\mathbb E[\max_{j\ne i}v_j\mid v_i=v]$ .

Intuition

Different formats trade off revenue, efficiency, and strategic complexity.

Under symmetric risk-neutral private values all four standard auctions yield the same expected revenue — the Revenue Equivalence Theorem.

Worked example

In a Vickrey auction, bidding $b_i=v_i$ is weakly dominant and the highest value wins paying the second-highest bid.

In eBay-style ascending auctions, proxy bidding implements the Vickrey outcome under IPV.

The math

Revenue Equivalence (Myerson-Riley-Samuelson): any symmetric IPV auction allocating to the highest-valuation bidder with the losers paying zero has the same expected revenue.

Myerson's optimal auction uses virtual valuations $\psi(v)=v-\frac{1-F(v)}{f(v)}$ and allocates to the bidder with highest $\psi$ if $\ge 0$ .

Where it is used

Used in spectrum auctions, online ad exchanges (GSP, VCG), treasury bonds, and art sales.

Basis for algorithmic mechanism design.

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