War of Attrition / Second-Price All-Pay Auction

• Content map: SMU H3 Game Theory Map

Setup

Definition:

War of Attrition / Second-Price All-Pay Auction

• Players: Two players: player $1$ and player $2$.
• Strategies: Each player chooses a stopping time or bid $x_i\in[0,\infty)$; In equilibrium, each player uses a continuous mixed strategy.

Rules

• Start with two players contesting a prize; Players simultaneously choose stopping times or bids $x_i\in[0,\infty)$.
• The player who waits longer wins the prize.
• Player $i$ values the prize at $v_i>0$; The player with the larger bid wins.
• The winner receives the prize and pays the lower bid.
• The loser pays own bid; Ties occur with probability $0$ under continuous mixing.

Derivation (Verification by Expected Payoff FOC)

• Let $F_1$ and $F_2$ be the mixed-strategy CDFs of players $1$ and $2$.
• If player $1$ bids $x$ and player $2$ bids $y<x$, player $1$ wins and receives payoff $v_1-y$.
• If player $2$ bids $y>x$, player $1$ loses and receives payoff $-x$.
• Player $1$‘s expected payoff from bid $x$ is

• Expanding:

• In a mixed-strategy equilibrium, $E(\pi_1)$ must be constant on the support.
• Differentiate with respect to $x$:

• Simplify:

• Since $f_2(x)=F_2'(x)$,

Derivation (Nash Equilibrium)

• Let

• Using

the differential equation becomes

• Integrate from lower bound $a$ to $x$:

• Since $F_2(a)=0$,

• The lower bound is $a=0$ and the upper bound is $\infty$.
• Therefore

• By the same logic,

Nash Equilibrium

Result:

The mixed-strategy equilibrium is

Social Optimum

• The prize should be allocated without costly delay.
• War of attrition dissipates surplus because players pay waiting costs.

Insights

Insight:

• Each player’s distribution is chosen to make the opponent indifferent.
• If $v_1>v_2$, then $F_2(x)<F_1(x)$ for $x>0$.
• Player $1$ bids less on average:

• Equivalently, expected bids are $v_2$ for player $1$ and $v_1$ for player $2$.
• Player $1$ wins with probability

• If $v_1>v_2$, then $P(\text{P1 wins})<\frac{1}{2}$.