All-Pay Auction with Common Known Value

• Content map: SMU H3 Game Theory Map

Setup

Definition:

All-Pay Auction with Common Known Value

• Players: $n$ bidders, where $n\geq 2$.
• Strategies: Each bidder chooses a bid $x\geq 0$; In the symmetric mixed-strategy equilibrium, each bidder draws $x$ from the same distribution $F$.

Rules

• Start with a common-value prize; Players simultaneously choose non-negative bids and pay their bids regardless of outcome.
• The player who submits the highest bid wins the prize.
• The prize value is $1$; Every bid is paid.
• Only the highest bidder receives the prize.
• With a continuous mixed strategy, ties occur with probability $0$.

Derivation (Verification by Expected Payoff FOC)

Candidate distribution

• Claim that the symmetric mixed-strategy equilibrium CDF is

• Let $F(x)$ be the probability that one opponent’s bid is at most $x$.
• If bidder $1$ bids $x$, bidder $1$ wins when all $n-1$ opponents bid at most $x$.
• Independent mixing gives

• Expected payoff is

• Substituting the candidate:

• Thus every bid in $[0,1]$ gives the same expected payoff.

First-order condition

• Since expected payoff is constant on the support,

• Equivalently,

• Hence

• The candidate $F(x)=x^{1/(n-1)}$ satisfies this FOC.

Support

• At the lower endpoint, $F(a)=0$, so

• Since a bidder can always bid $0$ and get payoff at least $0$, the lower endpoint must be

• Therefore $k=0$.
• The indifference condition becomes

• Hence

• At the upper endpoint, $F(b)=1$, so

• Thus

Nash Equilibrium

Result:

The symmetric mixed-strategy equilibrium has support $[0,1]$ and cumulative distribution function

Social Optimum

• Because the prize has common known value, every bidder values winning at $1$.
• Equilibrium expected payoff is $0$ for each bidder.
• Expected payment by each bidder is

• Total expected payments are therefore $1$.

Diagram (Best Response Functions)

Insights

Insight:

• All bidders pay regardless of whether they win.
• Mixing makes every bid in $[0,1]$ give the same expected payoff.
• Competition dissipates the full prize value in expectation.
• Any bid above $1$ wins for sure but gives negative payoff.