Memoryless-Strategy Equilibria of a N-Player War of Attrition Game with Complete Information

Proof of Proposition 1.

Assuming the players adopt mixed strategies with exponential distributions, an equilibrium λ ⃗ must satisfy ∑ n = − i λ n V i = c i , i = 1 , … , N , as shown in text. The candidate equilibrium strategies can be solved from them, which are the λ 1 * , … , λ N * as stated in the proposition. All the λ′s are nonnegative if and only if the parameters satisfy V N c N ≥ 1 − 1 N S ̃ , since V 1 c 1 ≥ ⋯ ≥ V N c N . For the strategies to be equilibrium strategies, we need to check whether player i is indifferent to any pure strategy of exiting at t ∈ [0, +∞). Indeed, player i’s utility from pure strategy t ∈ [0, +∞) is

Hence, λ ⃗ * is a mixed-strategy equilibrium of the game conditional on V N c N ≥ 1 − 1 N S ̃ . And, the equilibrium payoffs of the players are zero. ■

Proof of Proposition 2.

With the equilibrium strategy profile (G ₁, …, G _N ), player i ∈ {1, …, N} faces a suppositional rival, denoted as −i, whose strategy is represented by distribution

Since the support of his strategy is [0, +∞), player i is indifferent to any pure strategy t ∈ [0, +∞), i.e.,

where C stands for a constant. Differentiating both sides of the equation with respect to t, it becomes

Since the distributions are atomless, we have G _−i (0) = 0. The differential equation implies

i.e., the suppositional player −i must adopt an exponential strategy with rate c i V i . Hence,

From the N equations above, we have

Hence, the equilibrium strategy profile (G ₁, …, G _N ) must be the λ 1 * , … , λ N * given in Proposition 1. ■

Proof of Proposition 3.

Let the last N − K + 1 players that stay in the game be players 1, 2, …, and N − K + 1. The other K − 1 players concede immediately. Suppose the game played by the last N − K + 1 players has a nondegenerate equilibrium (λ ₁, λ ₂, …, λ _N−K+1) as characterized in Proposition 1. For the proposed strategy profile to be an equilibrium of the game ( N , N − K , V ⃗ , c ⃗ ) , we need to show no player wishes to deviate. We shall first notice that none of the players in {1, 2, …, N − K + 1} can do strictly better by switching to a pure strategy since all pure strategies result in zero expected payoff for them.

If a player i > N − K + 1 deviates to a pure strategy x > 0, he wins only if two of the players in {1, 2, …, N − K + 1} concede before x. Denote the second concession of the players 1, 2, …, N − K + 1 as a random variable x ₂. Then x ₂’s CDF, written as F ₂(x, N − K + 1), is

In particular, if all the players have equal strengths, the CDF becomes

On the other hand, we already know that it is not desirable for a player in {1, 2, …, N − K + 1}, say player 1, deviates to x > 0. In that case, he wins when one of the players in {2, …, N − K + 1} concedes before x. Denote the first concession of the players {2, …, N − K + 1}, who choose the exponential strategies, as random variable x ₁. Its CDF, written as F ₁(x, N − K), is

In particular, if the players have the same strengths,

Since

when the players have the same strengths, we have

which means F ₂(x, N − K + 1) first-order-stochastically-dominates F ₁(x, N − K). Therefore, if no player in {1, 2, …, N − K + 1} wishes to deviate, a player i > N − K + 1 strictly prefers not to deviate. Since all the functions involved are continuous, the player still does not want to deviate when the players’ strengths are heterogeneous but are close enough to each other.

Hence, the proposed strategy profile is an equilibrium of the game. ■

Proof of Proposition 4.

Since the support of an exponential strategy is [0, +∞), we only need to show that given the other players’ strategies λ ⃗ − i * , player i is indifferent to any pure strategy t ∈ [0, +∞). If player i exits at t, since he faces a suppositional player with strategy ∑ n = − i λ n * , his expected payoff is

Hence, λ * ⃗ = c ⃗ ⋅ U − 1 is an equilibrium mixed-strategy profile of the game, conditional on λ ⃗ * ≥ 0 . And, the equilibrium payoffs to the players are all zero. ■

Proof of Proposition 5.

From the first order conditions we immediately obtain the λ 1 * , … , λ N * described in the proposition. We only need to show that given the other players’ strategies λ − i * , player i is indifferent to any pure strategy t ∈ [0, +∞). Under strategy profile λ 1 * , … , λ N * , player i faces a suppositional player with strategy ∑ n = − i λ n * + η n . To simplify the exposition, write

If player i exits at t ∈ [0, +∞), his expected payoff is

Hence, λ ⃗ * is an equilibrium mixed-strategy profile of the game, conditional on λ ⃗ * ≥ 0 . And the players’ expected payoffs are zero in the equilibrium. ■