Auction Theory: The Ultimate Guide

Auction Theory, a vital branch of Game Theory, analyzes strategic bidding and auction design to optimize resource allocation. Pioneered by economists like William Vickrey (1996 Nobel Prize winner), it explores how bidders’ private valuations and incomplete information shape outcomes. Auctions drive markets, from ancient trade to modern platforms like eBay, with global auction revenue exceeding $400 billion in 2023.

This MathMultiverse guide covers auction types, optimal strategies, mathematical models, and applications in economics and technology. Whether designing spectrum auctions or bidding on Google Ads, Auction Theory provides tools to maximize efficiency and revenue.

Auction Types and Mechanics

Different auction formats influence bidder behavior and outcomes.

English Auction

Open ascending bids, winner pays their bid:

\[ p = b_{\text{max}} \]

Dutch Auction

Price descends until accepted:

\[ p = b_{\text{first}} \]

First-Price Sealed-Bid

Highest secret bid wins, pays their bid:

\[ p = \max \{ b_1, b_2, \ldots, b_n \} \]

Vickrey Auction

Highest bid wins, pays second-highest:

\[ p = \max \{ b_{-i} \} \]

All-Pay Auction

All pay bids, highest wins:

\[ u_i = v_i - b_i \text{ if win, } -b_i \text{ if lose} \]

Bidding Strategies

Strategies vary by auction type and bidder valuation.

First-Price Auction

Bid shading for \( v_i \sim U[0,1] \), \( n \) bidders:

\[ b_i(v_i) = \frac{n-1}{n} v_i \]

Vickrey Auction

Truthful bidding is optimal:

\[ b_i = v_i \]

English Auction

Bid up to valuation:

\[ b_i(t) = \min \{ v_i, p(t) + \epsilon \} \]

Revenue Equivalence

Expected revenue for risk-neutral bidders:

\[ E[R] = E[\text{second-highest } v_i] \]

Applications

Auction Theory drives real-world markets.

Spectrum Auctions

FCC’s multi-round auctions maximize allocation:

\[ \max \sum b_{i,k} x_{i,k}, \quad \sum_i x_{i,k} \leq 1 \]

Online Advertising

Google Ads’ second-price model:

\[ p = \max \{ b_{-i} \} + \delta \]

Art Auctions

English auctions at Sotheby’s:

\[ R = v_{(2)} \]

Charity Auctions

All-pay auctions boost donations:

\[ R = \sum b_i \]