All-Pay Auction Expected Revenue (Symmetric) Calculator

Overview

An all pay auction is a format where every participant pays their bid whether they win or not, and the highest valuation bidder receives the good or prize. Under the standard independent private value model, risk neutral and symmetric bidders follow the same strategy. Revenue equivalence then implies the seller’s expected revenue matches the expected second highest valuation for the chosen value distribution, provided there is no reserve price and tie breaking is fair.

How expected revenue is defined

Expected revenue is the average payment the seller obtains before the auction is run. Because all players pay their bids, the total payment equals the sum of all bids. In symmetric equilibrium that total equals the benchmark revenue from any standard highest bid auction, so you can compute it from the distribution of values alone. For a uniform distribution on an interval, the exact formula is especially simple and highlights how competition among bidders raises revenue.

Why it matters for design and strategy

Knowing expected revenue helps a seller compare mechanisms, set participation rules, and judge whether costly promotion is worthwhile. It also helps analysts evaluate contests, grant competitions, lobbying races, and innovation tournaments that naturally behave like all pay auctions. For bidders, understanding the equilibrium structure clarifies how aggressive bidding translates into costs and probabilities of winning, especially when the field is large and small differences in value become decisive.

What this calculator does

This calculator implements the symmetric benchmark. Enter the number of bidders and select a value distribution. For a uniform model the tool returns the exact closed form. For exponential values it uses a clean harmonic number formula. You can also run a Monte Carlo simulation that draws synthetic values, computes the second highest order statistic, and averages the results.

Visualization and downloads

To build intuition, the calculator plots expected revenue against the number of bidders so you can see how competition scales the outcome. The CSV export lets you move results into spreadsheets and reporting tools. The PDF snapshot preserves your assumptions, inputs, and the chart for quick sharing with colleagues or students.

Assumptions and scope

The symmetric model assumes independent private values, identical distributions, and risk neutrality. It abstracts from reserve prices, entry costs, and budget constraints. Those features can change outcomes, so treat the numbers as a clean baseline. Use the simulation option to stress test alternative parameter choices and sample sizes when analytical formulas are unavailable.

Practical tips

Start with a uniform model to benchmark results, then vary bidder count to see marginal gains. Switch to exponential when skew matters. Always record assumptions, show the chart, and compare scenarios side by side.