Building Reputation in a War of Attrition Game: Hawkish or Dovish Stance?

2.1 Timing

Two players, 1 and 2, are in dispute over a prize worth vi>0 for each player i∈{1,2}. The dispute (war of attrition game) is a three-stage infinite-horizon, continuous-time game. All stages start and the first two stages end at time 0. In stage 1, players simultaneously choose and announce their messages, regimes, or tones mi∈{h,d}:=M for i=1,2. Each player can choose either the hawkish(h) or the dovish(d) regime.

A player knows that he will never be forced to commit to a hardline policy, but is uncertain about his opponent. Therefore, each player believes that nature sends one of two messages {r,f} to his opponent in stage 2. A player who receives the message r, resolved, is constrained to follow a hardline policy; he never quits and never attacks. If a player receives the message f, flexible, then he will continue to play the game with no commitment. The players share the same belief that player i receives the message r with probability zi where zi∈(0,1). Therefore, the probability of being resolved is independent of the chosen messages.

Upon the beginning of the third stage (still at time 0), the players begin to play a war of attrition game, where at all times t≥0 before the dispute ends, each player can choose to escalate (wait), quit (back down), or attack. The dispute ends when one or both players attack or quit.

2.2 Payoffs

If either player attacks before the other quits, then the dispute ends with “war” and each player i receives the (net expected) payoff −wi<0. It indicates all the risks and gains resulting from war. The case −wi<0 indicates that staying away from the dispute is more desirable for a player than attacking once the player is involved in it. If player i picks message mi∈M and quits at time t≥0 before the other quits or attacks, then its opponent j receives the prize while player i suffers escalation costs equal to cimi(t), a continuous and strictly increasing function of the amount of escalation. I assume that 0≤cimi(t), cimi(0)<vi, and cid(t)≤cih(t) for all i and t. The last condition implies that the hawkish stance leads to a (weakly) higher escalation costs.

If one player chooses to attack at time t and the other chooses to quit or attack at the same time, then both flexible players receive their war payoffs −wi. However, if both quit at time t, then flexible player i receives vi2−cimi(t) given that i’s regime choice was mi in the first stage. These particular assumptions are not crucial because in equilibrium, simultaneous concessions or attacks occur with probability 0. Finally, if players escalate the conflict indefinitely, each flexible player gets a payoff that is strictly less than his war payoff. ^[4] Call this war of attrition game where all the parameters are common knowledge G.

2.3 Strategies

The only source of uncertainty is the players’ actual types, which matters only in the third stage of the game. In the first stage, all players are flexible (rational) in the sense that they choose their strategies, given their beliefs, to maximize their expected payoffs. At the beginning of the game each player knows that his future self will be flexible. However, each player is uncertain about his opponent’s future self: following the second stage, the opponent is either flexible or resolved.

The strategy of the resolved type is simple: never quit and never attack during the war of attrition game. Resolved types’ payoffs are ignored because they are redundant for the analysis. What motivates and interests the subsequent analyses are the optimal (i. e., equilibrium) strategies of the flexible types. Equilibrium strategies of the third stage of the game are in mixed strategies, and so they are intricate objects. For pedagogical reasons, I first present what pure strategies may look like and the payoffs that they correspond to. Then I provide the general description for the strategies and payoffs.

A pure strategy for player i in the third stage of the game G is a tuple (Qi(mˉ),Ai(mˉ))∈R+2 for each (m1,m2):=mˉ∈M, where Qi(mˉ) and Ai(mˉ) denote the times at which player i quits and attacks, respectively. Therefore, the strategies of the resolved type of player i are Qi(mˉ)=Ai(mˉ)=∞. Suppose that players choose (m1,m2)=mˉ∈M in stage 1 and players i and j, for i,j=1,2 and j≠i, are flexible.

1. If Ai(mˉ)≤min{Aj(mˉ),Qi(mˉ),Qj(mˉ)}, then the payoffs to players i and j are −wi<0 and −wj<0, respectively.

2. If Qi(mˉ)<min{Ai(mˉ),Aj(mˉ),Qj(mˉ)}, then the payoffs to players i and j are −cimi(Qi(mˉ))<0 and vj>0, respectively.

3. If Qi(mˉ)=Qj(mˉ)<min{Ai(mˉ),Aj(mˉ)}, then the payoffs to players i and j are vi2−cimi(Qi(mˉ)) and vj2−cjmj(Qj(mˉ)), respectively.

4. If Qi(mˉ)=Qj(mˉ)=Ai(mˉ)=Aj(mˉ)=∞, then the payoffs to players i and j are some −M∈R−, where max{wi,wj}<M.

More formally, a mixed strategy of flexible player i in the first stage is a function μi:M→[0,1], where μi(m) denotes the probability that flexible player i chooses the message m∈M. Note that μi(h)+μi(d)=1 for i=1,2.

Player i’s third stage (mixed) strategy, {Qimˉ,Aimˉ}mˉ∈M2, has two parts. For each message profile mˉ=(m1,m2)∈M2, a right-continuous distribution function Qimˉ(t):R+∪{∞}→[0,1] represents the probability that player i quits by time t (inclusive). Similarly, for each message profile mˉ, a continuous distribution function Aimˉ(t):R+∪{∞}→[0,1] represents the probability that player i attacks by time t (inclusive). If player i’s strategy, following the message profile mˉ, is Qimˉ(t), then flexible player i’s strategy is Qimˉ(t)/(1−zi) because the resolved type never quits. The same arguments hold for Aimˉ(t). Note that Qimˉ(t)+Aimˉ(t)≤1 for all 0≤t and limt→∞Aimˉ(t)≤1−zi for all i and mˉ.

Given a strategy profile σ=μi,{Qimˉ,Aimˉ}mˉ∈M2i=12, in the subgame following the message realization mˉ, flexible player i’s expected payoff of quitting at time t is

with Qjmˉ(t−)=limy↑tQjmˉ(y). Similarly, flexible player i’s expected payoff of attacking at time t is

Therefore, flexible player i’s expected payoff in the subgame following the message realization mˉ is

Finally, flexible player i’s expected payoff in the game is

where for all mˉ=(m1,m2)∈M2, μ(mˉ)=μ1(m1)μ2(m2).

2.4 Discussion

Escalation costs can be interpreted as the players’ opportunity costs of quitting. That is, it is the total value of the opportunities that are missed or not used effectively by backing down from the hardline policy. Public commitments that players make during a dispute are expected to serve an agenda the players may form. This agenda may include items that are not directly related to the prize (e. g., increasing audiences’ support and the likelihood of winning upcoming elections, building reputation for future negotiations, discouraging potential rivals, and preventing future disputes etc.). Therefore, a player’s expected value of following a hardline policy may increase with time because as time passes, the player is more likely to convince his audiences about his resolve, and persuaded audiences would increase the likelihood of the successful execution of his agenda. As a result, if the expected benefit of following the hardline policy increases in time, then the opportunity cost of backing down is expected to increase in time as well. Alternatively, escalation costs may represent a player’s psychological disutility – due to social disapproval or embarrassment – of quitting. ^[5]

A player may commit not to back down in a dispute for reasons other than the prospects of the dispute. For example, a leader may be resolved because of his firm belief that backing down, and thus giving up for the prize, is simply a decision that will not be ratified by his supporters. Therefore, the positive priors (i. e., zi’s) can be interpreted as the players’ initial beliefs on the existence of such motives that may force their opponents to be resolved. Resolved types closely resemble the commitment types that are first defined by Myerson (1991) (r-insisting types) and studied further by Abreu and Gul (2000) and Kambe (1999). These strategy types are first used in games by Kreps and Wilson (1982) and Milgrom and Roberts (1982), where commitments are modeled as behavioral types that exist in society so that rational players can mimic them if it is optimal to do so. In a bilateral negotiation, commitment types always demand a particular share and accept an offer if and only if it weakly exceeds that share. ^[6]

4.1 One-Sided Communication Under the Absence of War

Let the game G1 be the same as the game G with only two differences: (1) in the first stage, only player 1 can choose his stance and (2) in the third stage, players cannot choose to attack. Furthermore, I suppose that c1d<c2<c1h. Therefore, the game G1 will end at time tm∗=min{t1m,t2m}, where t1m=v2(1−z1)z1c2 and t2m=v1(1−z2)z2c1m.

Moreover, player 1’s expected payoff is

if player 1 is advantaged in the subgame following m (i. e., t1m<t2m). Note that U1m is a decreasing function of c1m, and so player 1 can increase his game payoff by choosing a message that generates smaller escalation costs. Clearly, if player 1 can never be advantaged (i. e., t2m≤t1m for all m∈M), then player 1’s equilibrium payoff will be 0 regardless of his message.

If two players are similar in the sense that vi=v and zi=z for i=1,2, then the above inequality would become c1d<c2, which is automatically satisfied by our assumption. Therefore, we could conclude that if two players are playing the war of attrition game under the absence of war and if only player 1 can choose his regime, then player 1 prefers to choose the dovish regime such that he keeps his escalation costs minimal.

4.2 One-Sided Communication Under the Shadow of War

Let G2 be the same as the game G with only one difference: in the first stage, only player 1 can choose his regime. Similar to the previous section, I will continue to assume that c1d<c2<c1h. Furthermore, let vi=v, wi=w, and zi=z hold for each i. According to Lemma 2, the equilibrium horizon of the game is a function of the players’ war cost w. In particular, for each m∈M, the game G2 will end at time tm∗=min{t1m,t2m}, where t1m=minwc1m,v(1−z)zc2 and t2m=minwc2,v(1−z)zc1m. As a result, in equilibrium, player 1 should choose the hawkish regime when the benefit-cost ratio of the game is higher than the relative likelihood of a player being the resolved type. The next result formally proves this point.

If only player 1 has had the option of attacking, and so player 2 can never attack, then there would exist a unique equilibrium where player 1 was advantaged and choosing the dovish regime when vw<z1−z. However, there would exist a continuum of equilibrium – in all of which player 1 is advantaged – when z1−z<vw because player 1 would be indifferent between the dovish and the hawkish regimes.

These two subsections show that player 1, who is the only player that can choose his escalation costs in the game, is the advantaged player in equilibrium, and this conclusion is correct whether or not players have the option to attack. However, the advantaged player’s regime varies depending on the benefit-cost ratio of the dispute and the relative likelihood of a player being the resolved type.

4.3 Two-Sided Communication

In this section, I consider the war of attrition game G, where both players can choose a regime. For sharper results, I will suppose that c2d<c1d<c2h<c1h, vi=v, wi=w, and zi=z for each i.

However, ifvw<z1−z, then in any sequential equilibrium of the game G, player 2 is advantaged and choosing the dovish regime in the first stage.

4.4 Continuous Message Space and Intensity

Similar to the previous section, I suppose that both players can choose escalation costs coefficients (ci), and vi=v, wi=w, and zi=z hold for each i. Furthermore, I suppose that message space is continuous, and so for i=1,2, ci∈[cmin,cmax] with 0<cmin<cmax<∞. The lower (or the upper) bound represents the most dovish (hawkish) tone a player can pick.

Therefore, more dovish tone increases a player’s payoff as long as the dovish regime is optimal and the opponent has not chosen the most dovish regime cmin. On the other hand, a more hawkish tone never benefits a player if it is the optimal regime for the player. Nevertheless, in all sequential equilibrium of the war of attrition game G, at least one of the players will certainly choose the most hawkish regime cmax whenever z1+z<vw.