All-Pay Auction Symmetric Equilibrium

Theorem (MSZ Thm 12.19; Krishna Prop 3.1). In the symmetric IPV all-pay auction — where every bidder pays their bid, but only the highest bidder receives the object — the unique symmetric strictly-increasing Bayes–Nash equilibrium bid is $$ \beta^{\mathrm{AP}}(t) \;=\; \int_0^t z\, (n-1) F(z)^{n-2} f(z)\, dz \;=\; \mathbb{E}\bigl[\, Y_1 \cdot \mathbf{1}\{Y_1 \le t\} \,\bigr], $$ i.e. the expected value of the highest opponent type conditional on that opponent type being below $t$ — un-divided by the probability $F(t)^{n-1}$.

The expected revenue from this equilibrium equals the expected revenue of any other standard auction format under the same symmetric IPV hypothesis ([[mechanism_design.myerson.revenue_equivalence]]).

Setup

• $n$ symmetric IPV bidders, types $t_i \sim F$ i.i.d. on $[0, \omega]$.
• All-pay auction: each bidder submits a sealed bid; the highest bidder wins the object; every bidder (winner and losers) pays their own bid. Interpret as effort or rent-seeking expenditure that is sunk before the outcome is determined.

Equilibrium derivation

The interim expected payment of a type-$t$ bidder using equilibrium strategy $\beta$ is $m(t) = \beta(t)$ (everyone pays their bid, no conditioning on winning). The interim allocation probability is $q(t) = F(t)^{n-1}$.

By the Myerson envelope identity for the same allocation rule as first-price/second-price (highest-bidder-wins): $$ m(t) - m(0) \;=\; t \cdot q(t) \;-\; \int_0^t q(z)\, dz \;=\; t F(t)^{n-1} - \int_0^t F(z)^{n-1} dz. $$ Integration by parts converts the right-hand side to $\int_0^t z\, (n-1) F(z)^{n-2} f(z)\, dz$, giving the formula above.

Comparison to first-price

• Winners pay less, losers pay more. In first-price, only the winner pays $\beta^{\mathrm{FP}}(t_{\mathrm{winner}})$; in all-pay, every bidder pays $\beta^{\mathrm{AP}}(t_i)$. To equalise expected revenue (per the revenue-equivalence theorem [[mechanism_design.myerson.revenue_equivalence]]), bidders shade more aggressively in all-pay than in first-price for low types, with $\beta^{\mathrm{AP}}(t) < \beta^{\mathrm{FP}}(t)$ pointwise except at the endpoints.
• Total expected payment equals expected second-highest type, just as in first-price and second-price under symmetric IPV.

Why it matters

The all-pay auction is the canonical model of:

• Rent-seeking and lobbying contests, where effort is sunk before the prize is awarded.
• R&D races and patent contests, where participation costs are paid by all contenders.
• Political campaigns, where campaign spending occurs regardless of victory.

The revenue equivalence with first-price and second-price is the most striking consequence of the general revenue-equivalence theorem: three auction formats with very different payment structures generate the same expected revenue under symmetric IPV.

Lean port (deferred)

Planned Lean module: EconCSLib/Auction/AllPay.lean.

Planned declarations:

• AllPayAuction (mechanism with transfers: payment rule paymentRule b i = b i for every $i$, regardless of winner).
• symmetricAllPayBid (the function $\beta^{\mathrm{AP}}$).
• symmetricAllPayBid_bne (the BNE certification).
• allPay_revenue_eq_firstPrice (revenue-equivalence corollary specialised to all-pay).

Dependencies: the symmetric IPV environment from [[mechanism_design.auction.bayesian.single_item_framework]], the symmetric first-price equilibrium ([[mechanism_design.auction.bayesian.symmetric_first_price_equilibrium]]), and the Myerson envelope identity from EconCSLib/MechanismDesign/Auction/Myerson.lean. Tracked in the MSZ Ch.12 auction gap review.

References

• [MSZ Chapter 12, Thm 12.19] Maschler, Solan, and Zamir, Game Theory. All-pay auction equilibrium.
• [Krishna, Chapter 3, Section 3.5] Vijay Krishna, Auction Theory, 2nd ed.. All-pay auction and comparison to first-price.
• [Riley-Samuelson 1981] John Riley and William Samuelson, "Optimal Auctions", AER 71(3):381–392. Revenue equivalence including all-pay.