One-Sided Games in a War of Attrition

1.1 Related Literature

The war-of-attrition model with complete information was generalized by Bishop and Cannings (1978) and Hendricks et al. (1988). Various versions of the model with incomplete information were developed by Bishop et al. (1978), Riley (1980), Milgrom and Weber (1985), Nalebuff and Riley (1985), Ponsati and Sákovics (1995), Bulow and Klemperer (1999), and Hörner and Sahuguet (2011), just to name a few. However, these studies do not consider the possibility that a player could be defeated, and thus, only deem the wars over when one player concedes.

Some studies suppose that the war has an exogenous (and possibly random) end period (Ordover and Rubinstein, 1986; Kim and Xu Lee, 2014). In this scenario, though, both players may be able to obtain positive benefits at the end of the war, which does not imply that one of the players is defeated. The possibility of defeat is explored by Langlois and Langlois (2009). In their model, players’ resources decrease over time, and if a player’s resources reach zero, they are defeated; consequently, both players can be defeated. Thus, to the best of my knowledge, this study is the first that analyzes one-sided games in a war of attrition where one player can be defeated by the other, but not vice versa.

On the other hand, some studies analyze wars of attrition between asymmetric players who have different benefits, costs, or discount factors (Kambe, 1999; Abreu and Gul, 2000; Myatt, 2005). One can infer that a player who has a higher benefit, lower cost, or higher discount factor is stronger than the other since such a player has a higher incentive to fight. My model provides a different description of asymmetric robustness; that is, only one player can be defeated by the other.

The rest of the paper proceeds as follows: Section 2 develops a model with complete information. Section 3 further extends the model to include player 1’s asymmetric information regarding player 2’s robustness, and Section 5 concludes.

2.1 Settings

The game involves two players, 1 and 2, who are at war. The model assumes that time is continuous, t∈0,∞]. Players 1 and 2 strategically choose times T1 and T2, respectively, to settle the war. Similar to a standard war-of-attrition model, these strategic decisions are made simultaneously at the beginning of the game. Therefore, the equilibrium concepts are a Nash equilibrium with complete information and a Bayesian–Nash equilibrium with incomplete information.

Player 2 faces a risk of defeat, where he/she is not strong enough to overcome player 1. Player 2 is defeated when t=τ, where τ∈[0,∞) is a random variable with the cumulative distribution function Fτ≡1−exp−rτ.^[2] The parameter r∈0,1 denotes player 2’s robustness in fighting. Because the expected timing of player 2’s defeat is 1/r, a larger r implies that player 2 is more likely to be defeated early.

If player 1 concedes before player 2 gives in or is defeated (T1<min{τ,T2}), player 2 wins a one-shot benefit, b2>0, at t=T1, while player 1 gains nothing. By contrast, if player 2 gives in or is defeated before or at the same time as player 1’s concession (T1≥min{τ,T2}), player 1 secures benefit b1 at t=min{τ,T2}, while player 2 gains nothing.^[3] The war inflicts costs c1 and c2 on player 1 and player 2, respectively, per unit of time.

The players’ expected payoffs at the game’s onset can be obtained as follows:

where I⋅ is an indicator that equals 1 if its condition holds and 0 otherwise.

Following the standard war-of-attrition model, we assume that the players are risk-neutral and that there is no time discounting. Even if risk aversion and time discounting were to be introduced, the main implications of our model would not change, though the duration of the war would be shorter.

2.2 Equilibrium

The following proposition summarizes the Nash equilibria of the model with complete information.^[4]

Player 1’s rational choice of T1 depends on the relative sizes of the marginal benefit (b1r) and the marginal cost (c1) of extending the war. The marginal benefit and cost are independent of T1 because the exponential distribution is memoryless. That is, Prτ>t+Δt|τ>t=Prτ>Δt. If b1r is sufficiently low and c1 is sufficiently high (Proposition 1 (ii)), similar to the canonical war-of-attrition model, both players will receive a negative payoff for extending the fight. Thus, the equilibria resemble those of the standard war-of-attrition model. In the pure-strategy equilibria (a,b), the game immediately ends. It is only in the mixed-strategy equilibrium (c) that the war can last for an indeterminate length of time. On the other hand, if b1r is sufficiently high and c1 is sufficiently low (Proposition 1 (i)), there exists a unique type of equilibrium where player 2 concedes immediately. This is because player 1 has a positive payoff for extending the fight (because of high b1r and low c1), so he/she has an incentive to wait until player 2 is defeated. Thus, as player 2 has no hope of obtaining b2, he/she will give in at the beginning of the game.

These results suggest that the war can only be maintained in the mixed-strategy equilibrium in which player 1 has a higher probability of conceding earlier when b1=b2 and c1=c2.^[5] This seems unrealistic as it requires an invincible player to concede more quickly than a vulnerable one. A similar problem can be found in the standard war-of-attrition model with asymmetric benefits (b1≠b2) and/or costs (c1≠c2): The player with a higher bi and a lower ci has a higher probability of conceding earlier.^[6]Kornhauser et al. (1989) assert that “the weaker player ... should concede immediately.” Therefore, even though the conflict is maintained in a mixed-strategy equilibrium, it is difficult to justify these strategies.^[7] Proposition 1 (i) shows the unique type of equilibria proposed by Kornhauser et al. (1989) in which the weak player concedes immediately. However, this cannot explain why such wars occur.

3.1 Settings

Next, I introduce incomplete information about player 2’s robustness into the model. Suppose that player 1 is uncertain about player 2’s robustness, r, while player 2 may know it. Although player 1 does not know the true value of r, he/she knows that player 2 is either a strong (S) type with r=rS or a weak (W) type with r=rW>rS, and that this characteristic is distributed according to prior probabilities Pr(rS)∈(0,1) and Pr(rW)=1−Pr(rS). If Player 2 has rW, he/she is more likely to be defeated early.

In order to rule out uninteresting cases that resemble that with complete information (Proposition 1), I impose the following restrictions:

If rS>c1/b1, the equilibria are identical to Proposition 1 (i); furthermore, if Pr(rS)rS+PrrWrW<c1/b1, the equilibria are identical to Proposition 1 (ii).^[8]

3.2 Equilibrium with an Ongoing War

This section shows an equilibrium where player 1 chooses a pure strategy to fight until a certain period (i.e. T1∈(0,∞)), because these novel equilibria do not appear in the standard war-of-attrition model or the model with complete information. The other equilibria will be discussed in Section 3.3.

When player 1 chooses a pure strategy, T1∈(0,∞), player 2 (who is some type-R∈{S,W}) may adopt a mixed strategy between fighting on (T2>T1) and giving in immediately (T2=0); namely, with probability σR∈[0,1], type R intends to fight until player 1 gives in (T2>T1), and with probability 1−σR, type R immediately gives in (T2=0). I focus only on T2=0 and T2>T1, because any T2∈0,T1, which is costly but never wins b2, is strictly dominated by T2=0. I define σR as each type of R’s mixed strategy and Σ as the set of the two types of mixed strategies (Σ≡(σS,σW)).

3.3 Other Equilibria

1. Equilibrium III: Player 2 immediately gives in (T2=0) regardless of his/her type (Σ=0,0), and player 1 fights until t=T1 such that VS(T1)<0.

2. Equilibrium IV: A weak player 2 immediately gives in. Player 1 and a strong player 2 back down probabilistically such that

   PrT1<t=1−exp−c2b2+rSt,

   PrT2<t=1−exprS−c1b1t.

These equilibria are the same as those in the model with complete information (Proposition 1 (ii-b) and (ii-c)) and the standard war-of-attrition model.

One important difference between my model and the standard war-of-attrition model is that there is no equilibrium in which player 1 concedes immediately. Both strong and weak types of player 2 would have an incentive to fight because they could get b2 immediately. However, under Assumption 1, if both types fight, player 1 also has an incentive to fight (T1∗(ΣI)>0).

Note that these equilibria (in Propositions 2 and 4) satisfy the conditions of a perfect Bayesian equilibrium. This is because the model assumes continuous and infinite periods, thus, all periods are identical except in regards to the revised beliefs, which are the same as those in the Bayesian–Nash equilibrium.^[14] I employ Bayesian–Nash equilibria to facilitate comparisons between my model and the standard war-of-attrition model, which uses them as well.

3.4 Differences from the Standard Model

There are two significant differences between my model’s implications and those of the classical war-of-attrition model. First, my model allows for a unique equilibrium in which player 2, who can be defeated by player 1, concedes immediately. This occurs when the probability of defeat is sufficiently high (r>c1/b1 in the complete-information model and rS>c1/b1 in the asymmetric-information model). Moreover, when Assumption 1 holds in the asymmetric-information model, there is no equilibrium in which player 1 (who is invincible) concedes immediately. These equilibria confirm the suggestion of Kornhauser et al. (1989) who argued that a weak player (i.e. a vulnerable player in my model) should concede immediately.

Second, and more importantly, an ongoing war can occur in equilibrium. In Equilibria I and II, player 1 and player 2 (strong types and some weak types) choose to fight for a certain period (T1∗(Σ)). During these periods, the war will end if and only if player 2 is defeated. These equilibria simply show (i) why conflicts between players who have asymmetric power (such as a war against terrorism) occur, (ii) why a vulnerable player (such as a terrorist group) decides to attack even though it faces the risk of defeat, and (iii) why an invincible player (such as the targeted state) decides to compromise in a war. To my knowledge, such an equilibrium has not been found in any past extensions of the classical war-of-attrition model.