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International crises are modeled as a political “war of attrition” in which state leaders choose at each moment whether to attack, back down, or escalate. A leader who backs down suffers audience costs that increase as the public confrontation proceeds. Equilibrium analysis shows how audience costs enable leaders to learn an adversary’s true preferences concerning settlement versus war and thus whether and when attack is rational. The model also generates strong comparative statics results, mainly on the question of which side is most likely to back down. Publicly observable measures of relative military capabilities and relative interests prove to have no direct effect once a crisis begins. Instead, relative audience costs matter: the side with a stronger domestic audience (e.g., a democracy) is always less likely to back down than the side less able to generate audience costs ( a nondemocracy). More broadly, the analysis suggests that democracies should be able to signal their intentions to other states more credibly and clearly than authoritarian states can, perhaps ameliorating the security dilemma between democratic states.
国际危机被模拟为一场政治 "消耗战"，在这场战争中，国家领导人每时每刻都在选择是进攻、退让还是升级。退缩的领导人会付出观众成本，这种成本会随着公开对抗的进行而增加。均衡分析表明了观众成本如何使领导者了解对手在和解与战争方面的真实偏好，从而知道进攻是否合理以及何时进攻合理。该模型还得出了强有力的比较静态结果，主要是在哪一方最有可能退让的问题上。事实证明，一旦危机开始，公众可观察到的相对军事能力和相对利益的衡量标准并没有直接影响。相反，相对受众成本很重要：国内受众较强的一方（如民主国家）总是比受众成本较低的一方（非民主国家）更不可能退让。更广泛地说，分析表明，民主国家应该比专制国家更可信、更明确地向其他国家表明自己的意图，这或许能改善民主国家之间的安全困境。

An international crisis occurs when one state resists a threat or demand made by another, with both taking actions that suggest that the dispute might be decided militarily. Crises are frequently characterized as “wars of nerves.” Measures such as troop deployments and public threats make crises public events in which domestic audiences observe and assess the performance of the leadership. For reasons linked to this public aspect of crises, state leaders often worry about the danger that they or their adversary might become locked into their position and so be unable to back down, make concessions, or otherwise avoid armed conflict.
当一个国家抵制另一个国家的威胁或要求时，双方都会采取行动，暗示可能会通过军事手段解决争端，这就是国际危机。危机经常被描述为 "神经战"。部署部队和公开威胁等措施使危机成为公共事件，国内受众可在其中观察和评估领导层的表现。出于与危机的公开性相关的原因，国家领导人经常担心他们或他们的对手可能会被锁定在自己的立场上，从而无法退让、做出让步或以其他方式避免武装冲突。

In this article I model an international crisis as a political “war of attrition.” The formalization is motivated by an empirical claim, namely that crises are public events carried out in front of domestic political audiences and that this fact is crucial to understanding why they occur and how they unfold. I characterize crises as political attrition contests with two defining features. First, at each moment a state can choose to attack, back down, or escalate the crisis further. Second, if a state backs down, its leaders suffer audience costs that increase as the crisis escalates. These costs arise from the action of domestic audiences concerned with whether the leadership is successful or unsuccessful at foreign policy (Fearon 1990, 1992; Martin 1993).
在本文中，我将国际危机模拟为一场政治 "消耗战"。这种形式化是出于一种经验性主张，即危机是在国内政治受众面前发生的公共事件，而这一事实对于理解危机为何发生以及如何发展至关重要。我将危机定性为政治消耗竞赛，它有两个显著特点。首先，在每一时刻，国家都可以选择进攻、退让或进一步升级危机。其次，如果一个国家退缩，其领导人就会付出观众成本，这种成本会随着危机的升级而增加。这些成本来自于国内受众对领导层外交政策成败的关注（Fearon 1990, 1992; Martin 1993）。
The formalization has three major benefits. First, it helps answer an important question about the origins of war that is missed in the informal literature and begged by existing formal models of crisis bargaining. Briefly, if fighting entails any cost or risk, then rational leaders would not choose war until they had concluded that attack was justified by a sufficiently low chance of an acceptable diplomatic settlement. Thus another way to ask the question “Why do wars occur?” is to ask what leads states to abandon the hope of a cheaper, nonmilitary resolution. A theoret-
正式化有三大好处。首先，它有助于回答一个关于战争起源的重要问题，这个问题在非正式文献中被忽略，而现有的危机讨价还价的正式模型也没有给出答案。简而言之，如果战争需要付出任何代价或承担任何风险，那么理性的领导人就不会选择战争，除非他们认为攻击是合理的，因为达成可接受的外交解决方案的几率足够低。因此，"为什么会发生战争？"这个问题的另一种问法是，是什么导致国家放弃以更低成本、非军事方式解决问题的希望。一种理论
ical answer requires us to explain how a state with rational leaders would learn. During a crisis, how do leaders come to revise their beliefs about an opponent so that attack is preferred to holding out for concessions? I shall argue that neither the informal nor the formal literature on international conflict supplies satisfactory answers.
要回答这个问题，我们需要解释一个拥有理性领导人的国家是如何学习的。在危机期间，领导人如何修正他们对对手的看法，从而选择进攻而不是坚持让步？我将论证，关于国际冲突的非正式或正式文献都没有提供令人满意的答案。

The answer suggested here is that audience costs are an important factor enabling states to learn about an opponent’s willingness to use force in a dispute. At a price, audience costs make escalation in a crisis an informative although noisy signal of a state’s true intentions. They do so in part by creating the possibility that leaders on one or both sides will become locked into their position and so will be unable to back down due to unfavorable domestic political consequences. I find that in the model, a crisis always has a unique horizon-a level of escalation after which neither side will back down because both are certainly locked in, making war inevitable. Before the horizon is reached, the fear of facing an opponent who may become committed to war puts pressure on states to settle. The model thus captures a common informal story about international crises-that their danger and tension arise from the risk of positions hardening to the point that both sides prefer a fight to any negotiated settlement.
这里提出的答案是，受众成本是一个重要因素，使国家能够了解对手在争端中使用武力的意愿。作为代价，受众成本使危机升级成为一个国家真实意图的信息信号，尽管这种信号很嘈杂。受众成本的部分作用是创造了一种可能性，即一方或双方的领导人会锁定自己的立场，从而因不利的国内政治后果而无法退让。我发现，在这个模型中，危机总是有一个独特的地平线--在这个地平线之后，双方都不会退让，因为双方肯定都被锁定了，战争不可避免。在达到这个水平线之前，由于担心面对的对手可能会投入战争，各国都会面临和解的压力。因此，该模型捕捉到了关于国际危机的一个常见的非正式故事--危机的危险性和紧张局势源于立场趋于强硬的风险，以至于双方都倾向于战斗而非任何谈判解决。

The second major benefit of the formal analysis is a set of comparative statics results that provide insights into the dynamics of international disputes. The strongest and most striking of these bear on the question of which state is more likely to concede in a confrontation. I find that regardless of the initial conditions, the state more sensitive to audience costs is always less likely to back down in disputes that become public contests. The intuition is that the greater the domestic cost for escalating and then backing down, the more informative is the signal of escalation and the less escalation is required to con-
正式分析的第二大益处是一系列比较统计结果，这些结果提供了对国际争端动态的洞察力。其中最有力、最引人注目的是关于哪个国家更有可能在对抗中让步的问题。我发现，无论初始条件如何，对观众成本更敏感的国家总是不太可能在成为公开竞争的争端中让步。其直觉是，升级然后退让的国内成本越高，升级信号的信息量就越大，就越不需要升级来让步。
vey intentions. A stronger domestic audience thus allows a state to signal its true preferences concerning negotiated versus military settlements more credibly and more clearly.
窥探意图。因此，更强大的国内受众可以让一个国家更可信、更明确地表明其在谈判解决与军事解决方面的真实倾向。

This result and the audience cost mechanism underlying it suggest hypotheses about how state structure might influence crisis bargaining. For example, if actions such as mobilizing troops create larger audience costs for democratic than for authoritarian leaders, then democratic states should be less inclined to bluff or to try “limited probes” in foreign policy-to make military threats and then back off if resistance is met. More broadly, stronger domestic audiences may make democracies better able to signal intentions and credibly to commit to courses of action in foreign policy than nondemocracies, features than might help ameliorate “the security dilemma” (Herz 1950; Jervis 1978) between democratic states.
这一结果及其背后的受众成本机制提出了关于国家结构如何影响危机谈判的假设。例如，如果动员军队等行动给民主国家领导人带来的受众成本大于专制国家领导人，那么民主国家在外交政策中就不会那么倾向于虚张声势或尝试 "有限试探"--发出军事威胁，然后在遇到抵抗时退缩。更广泛地说，与非民主国家相比，更强大的国内受众可能使民主国家更有能力发出信号表明意图，并在外交政策中可信地承诺行动方针，这可能有助于改善民主国家之间的 "安全困境"（赫茨，1950 年；杰维斯，1978 年）。

The comparative statics results also speak to the question of how relative military capabilities and relative interests influence the outcomes of international disputes. Conventional wisdom suggests that the state with inferior military capabilities, or with fewer “intrinsic interests” at stake, is more likely to back down (e.g., George and Smoke 1974, 556-61; Jervis 1971; Snyder and Diesing 1977, 189-95). Surprisingly, in the model, neither the balance of forces nor the balance of interests has any direct effect on the probability that one side rather than the other will back down once both sides have escalated. The reason is that in choosing initially whether to threaten or to resist a threat, rational leaders will take into account observable indices of relative power and interest in a way that tends to neutralize their impact if a crisis ensues. For example, a militarily weak state will choose to resist the demands of a stronger one only if it happens to be quite resolved on the issues in dispute and so is relatively willing to escalate despite its military inferiority. The argument implies that observable aspects of capabilities and interests should strongly influence who gets what in international politics but that their impact should be seen more in uncontested positions and faits accomplis than in crises. Which side backs down in a crisis should be determined by relative audience costs and by unobservable, privately known elements of states’ capabilities and resolve.
比较统计结果还涉及相对军事能力和相对利益如何影响国际争端结果的问题。传统观点认为，军事能力较弱或 "内在利益 "较少的国家更有可能退让（如 George and Smoke 1974, 556-61; Jervis 1971; Snyder and Diesing 1977, 189-95）。令人惊讶的是，在该模型中，一旦双方的冲突升级，力量平衡和利益平衡都不会直接影响一方而非另一方退让的概率。原因在于，在最初选择威胁还是抵制威胁时，理性的领导者会考虑可观察到的相对实力和利益指数，以便在危机发生时抵消它们的影响。例如，一个军事上的弱国只有在其对争议问题已经有了相当大的决心，因此尽管在军事上处于劣势，但仍相对愿意采取升级行动的情况下，才会选择抵制强国的要求。这一论点意味着，在国际政治中，能力和利益的可观察方面会对谁得到什么产生重大影响，但其影响更多体现在无争议的立场和既成事实上，而非危机中。在危机中，哪一方退让应由相对的受众成本以及国家能力和决心中无法观察到的、私下已知的因素决定。

The third major benefit of the analysis is slightly more technical. The model clarifies how international crises differ structurally from the classical war of attrition studied by economists and theoretical biologists (Maynard Smith 1982, chap. 3; Tirole 1989, chap. 8). In the classical case, two firms (or animals) compete for control of a market (or territory) that is not large enough to support both at a profit. The competition lasts until one or both players “quit.” International crises are analyzed here as a war of attrition that differs from the classical model in two important respects. First, in crises state leaders possess an additional option beyond continuing the contest or quitting-they can always choose to attack. Second, whereas in the classical war of attrition both sides pay
分析的第三个主要益处在技术上略有不同。该模型阐明了国际危机在结构上与经济学家和理论生物学家所研究的经典消耗战有何不同（梅纳德-史密斯，1982 年，第 3 章；蒂罗尔，1989 年，第 8 章）。在传统的情况下，两家公司（或动物）争夺一个市场（或领地）的控制权，而这个市场（或领地）的规模不足以支撑两家公司都获利。竞争一直持续到一方或双方 "退出 "为止。这里将国际危机分析为一场消耗战，它在两个重要方面不同于经典模型。首先，在危机中，国家领导人除了继续竞争或退出之外，还有一个选择--他们可以随时选择进攻。其次，在经典的消耗战中，双方都要付出代价。
costs for continuing the contest, in international crises it is empirically more plausible to assume that only the side that backs down suffers audience costs. ${}^1$${}^1$^(1){ }^{1} The existence of a military “outside option” along with audience costs proves to have major consequences for strategic behavior. Together they create the possibility of “lock in” and thus give crises a horizon. More technically, whereas the classical war of attrition has an infinity of (asymmetric) equilibria involving delay, the game studied here has a unique equilibrium distribution on outcomes up to the horizon.
在国际危机中，只有退让的一方才会付出观众成本，这在经验上更为合理。 ${}^1$${}^1$^(1){ }^{1} 事实证明，军事 "外部选项 "和观众成本的存在会对战略行为产生重大影响。它们共同创造了 "锁定 "的可能性，从而使危机有了地平线。更严格地说，经典的消耗战有无穷多个涉及延迟的（非对称）均衡，而本文研究的博弈在地平线之前有唯一的均衡结果分布。

First, I briefly review the relevant formal literature and also elaborate the theoretical puzzle: Given incentives to misrepresent, how can states involved in a dispute rationally reach the conclusion that the opponent would prefer war to backing down? I then informally discuss possible answers, arguing for the centrality of domestic audience costs, model an international crisis as a political attrition contest to examine the logic of equilibrium behavior, and, finally, draw some general conclusions.
首先，我简要回顾了相关的正式文献，并阐述了理论难题：既然存在歪曲事实的动机，卷入争端的国家如何才能理性地得出 "对手宁愿战争也不愿让步 "的结论？然后，我非正式地讨论了可能的答案，论证了国内受众成本的中心地位，将国际危机建模为一场政治消耗竞赛，以考察均衡行为的逻辑，最后得出一些一般性结论。

The costs and risks of war supply states with strong incentives to locate nonmilitary settlements that both sides would prefer to a fight. Most often, it seems, their efforts are successful: very few international disagreements become wars. This may seem unsurprising at first glance. One might expect that given the incentives to avoid war, state leaders who disagree on some issue could simply tell each other what they would be willing to accept rather than fight, and then choose a mutually acceptable bargain. The problem, however, is that states can also have strong incentives to misrepresent their willingness to fight in order to gain a better deal. Given these incentives, quiet diplomatic exchanges may be rendered uninformative about a state’s preferences. For example, in the Cuban Missile Crisis Kennedy did not ask Khrushchev what he would do if the United States were to impose a blockade or to attack the missile sites in Cuba: answers would have been almost worthless as indicators, due to Khrushchev’s incentives to misrepresent (and Kennedy may also have had an incentive not to tip his hand) (cf. Wagner 1989, 197).
战争的代价和风险为各国提供了强烈的动机，促使它们寻找双方都愿意选择的非军事解决办法。看起来，他们的努力往往是成功的：很少有国际分歧演变成战争。乍一看，这似乎不足为奇。人们可能会认为，在避免战争的激励机制下，在某些问题上存在分歧的国家领导人可以简单地告诉对方，与其打仗，他们愿意接受什么，然后选择一个双方都能接受的谈判方案。但问题是，国家也可能有强烈的动机来歪曲自己的战争意愿，以获得更好的交易。鉴于这些动机，平静的外交交流可能会使一国的偏好变得不明朗。例如，在古巴导弹危机中，肯尼迪并没有问赫鲁晓夫如果美国对古巴实施封锁或攻击导弹基地，他将会怎么做：由于赫鲁晓夫有动机歪曲事实，因此回答几乎没有任何参考价值（肯尼迪可能也有动机不给自己通风报信）（参见 Wagner 1989, 197）。

States in a dispute thus face a dilemma. They have strong incentives to learn whether there are agreements both would prefer to the use of force, but their incentives to misrepresent mean that normal forms of diplomatic communication may be worthless. I argue that international crises are a response to this dilemma. States resort to the risky and provocative actions that characterize crises (i.e., mobilization and deployment of troops and public warnings or threats about the use of force) because less-public diplomacy may not allow them credibly to reveal their own preferences concerning international interests or to learn those of other states.
因此，争端中的国家面临两难境地。它们有强烈的动机去了解是否存在双方都倾向于使用武力的协议，但它们又有动机歪曲事实，这意味着正常形式的外交沟通可能毫无价值。我认为，国际危机是对这一困境的回应。国家采取危机所特有的高风险和挑衅性行动（即动员和部署军队，公开警告或威胁使用武力），是因为不那么公开的外交可能无法让它们可信地揭示自己在国际利益方面的偏好，也无法了解其他国家的偏好。
To support this claim it must be shown how such actions can credibly reveal that a state would prefer
为了支持这一主张，必须证明这些行动如何能够令人信服地揭示一个国家更愿意
using force to making concessions. In particular, how is it that actions like mobilization and public warnings allow learning? If states can have incentives to misrepresent their willingness to use force, why should such actions be taken as credible indicators?
从使用武力到做出让步。特别是，像动员和公开警告这样的行动是如何允许学习的？如果国家有动机歪曲其使用武力的意愿，那么为什么要把这些行动作为可信的指标呢？

For the most part, the informal literature on international conflict and the causes of war takes it as unproblematic that actions such as mobilization “demonstrate resolve.” The literature has focused instead on how psychological biases may impair a leader’s ability to interpret crisis signals (e.g., Lebow 1981; Snyder and Diesing 1977, chap. 4; Jervis, Lebow, and Stein 1985). The prior question of how a rationally led state would learn in a crisis, given incentives to misrepresent, has not been answered in a theoretically thorough or satisfactory way.
在大多数情况下，关于国际冲突和战争原因的非正式文献认为，动员等行动 "表明决心 "是没有问题的。相反，这些文献关注的是心理偏差如何损害领导者解读危机信号的能力（例如，Lebow，1981 年；Snyder 和 Diesing，1977 年，第 4 章；Jervis、Lebow 和 Stein，1985 年）。4; Jervis, Lebow, and Stein 1985）。对于一个理性领导的国家在危机中如何学习的问题，由于存在虚假陈述的动机，还没有从理论上给出一个全面或令人满意的答案。

Consider the inference problem faced by a state whose adversary in a dispute has just mobilized troops. If rational, the state’s leaders should increase their belief that the adversary will fight only if a high-resolve adversary is more likely to mobilize than an adversary that in fact prefers backing down to war. Thus, if mobilization is to convey information and allow learning, it must carry with it some cost or disincentive that affects low-resolve more than highresolve states. In Spence’s (1973) terms, mobilization (or any other move in a crisis) must be a costly signal if it is to warrant revising beliefs. Costless signals, which often include private diplomatic communication and sometimes more public measures, will be so much “cheap talk,” since a state with low resolve may have no disincentive to sending them. ${}^2$${}^2$^(2){ }^{2}
考虑一个国家面临的推论问题，其争端中的对手刚刚调动了军队。如果是理性的，那么只有在高决心的对手比事实上更愿意退让的对手更有可能动员军队的情况下，国家的领导人才会增加他们对对手会参战的信念。因此，如果动员是为了传递信息和促进学习，就必须付出一定的代价或抑制因素，而这种代价或抑制因素对低决心国家的影响要大于对高决心国家的影响。用斯彭斯（Spence，1973）的话来说，动员（或危机中的任何其他举动）必须是一种有代价的信号，才能保证人们修正自己的信念。无代价的信号通常包括私下的外交沟通，有时也包括更公开的措施，但这些信号都是 "廉价的空谈"，因为决心低的国家可能没有动力去发出这些信号。 ${}^2$${}^2$^(2){ }^{2}

To explain how states learn in a crisis, we need to know what makes escalation or delay costly for a low-resolve state that in fact prefers making concessions to military conflict. It is tempting to answer “the risk of war”, but this would beg the question since we are trying to establish how this risk arises in the first place. I shall argue that the role of domestic political audiences has typically been crucial for generating the costs that enable states to learn. First, however, I briefly review how the published formal literature on crisis bargaining has addressed the issue.
要解释国家如何在危机中学习，我们需要知道是什么让低决断力的国家付出升级或拖延的代价，而这些国家事实上更愿意做出让步而不是军事冲突。我们很想回答 "战争风险"，但这是在自寻烦恼，因为我们首先要确定的是这种风险是如何产生的。我将论证，国内政治受众的作用通常对于产生使国家能够学习的成本至关重要。首先，我简要回顾一下已出版的关于危机谈判的正式文献是如何处理这个问题的。

A number of studies have developed models in which states rationally update their beliefs about an adversary’s resolve in the course of a crisis (Bueno de Mesquita and Lalman 1992; Fearon 1990, 1992; Kilgour 1991; Morrow 1989; Nalebuff 1986; Powell 1990; Wagner 1991). Though this is not always apparent, the mechanism that enables learning in each case is costly signaling.
许多研究都建立了这样的模型：在危机过程中，国家会理性地更新其对对手决心的看法（布埃诺-德梅斯基塔和拉尔曼，1992；费伦，1990，1992；基尔古，1991；莫罗，1989；纳勒布夫，1986；鲍威尔，1990；瓦格纳，1991）。尽管这一点并不总是很明显，但在每种情况下促成学习的机制都是代价高昂的信号传递。

While updating of beliefs occurs in these models, they actually do not address the question of how and why states might rationally come to conclude that fighting was preferable to holding out for concessions. The reason is that almost all of the models have finite horizons: the modeler exogenously determines that one of the states in the game will have a final choice between backing down or fighting. In effect, one player will ultimately have no choice but to “take it or leave it,” and this restriction creates a cost for escalation. In actual crises, by contrast, whenever a
虽然在这些模型中会出现信念的更新，但它们实际上并没有解决国家如何以及为什么会理性地得出结论，认为战斗比坚持让步更可取。原因在于，几乎所有的模型都有有限的视野：建模者外生地决定，博弈中的一个国家将在退让或战斗之间做出最终选择。实际上，一方最终将别无选择，只能 "要么接受，要么放弃"，而这一限制造成了升级的成本。相比之下，在实际危机中，只要有一方
state has the option of attacking it also has the option of delaying or doing nothing. If there are horizons in actual crises they arise endogenously, as a consequence of the fact that for some reason waiting ultimately becomes an undesirable choice. Models that exogenously fix a horizon cannot explain why a state would choose to use force (and thus why wars occur) because they cannot explain what makes force preferable to holding out for concessions by the other side.
国家可以选择进攻，也可以选择拖延或什么都不做。如果在实际危机中存在地平线，那么地平线是内生的，是由于某种原因等待最终成为不可取的选择这一事实的结果。外生地固定地平线的模型无法解释为什么一个国家会选择使用武力（因此也无法解释为什么会发生战争），因为它们无法解释是什么使得使用武力比等待对方让步更可取。

There are two partial exceptions to this argument. Nalebuff’s (1986) and Powell’s (1990) models of nuclear brinkmanship have something like an infinite horizon: they allow states to escalate in a crisis indefinitely, until one side backs down or nuclear war occurs. However, in these models states never choose to attack. Instead, war can occur only as a result of an accident beyond either side’s control. Thus these formalizations cannot and were never intended to explain why states would consciously choose to abandon peace for war.
这一论点有两个部分例外。纳勒巴夫（1986）和鲍威尔（1990）的核边缘政策模型具有类似于无限的视野：它们允许国家在危机中无限升级，直到一方退让或核战争发生。然而，在这些模型中，国家从不选择进攻。相反，战争的发生只能是任何一方无法控制的意外事件的结果。因此，这些形式化无法也从未打算解释为什么国家会有意识地选择放弃和平而选择战争。

Historically, war has virtually always followed from the deliberate choices of state leaders, if not always as the result they originally intended (Blainey 1973; Howard 1983). Since this pattern seems likely to continue even in the nuclear age, it makes sense to ask how states could reach the conclusion that attack was worth choosing.
从历史上看，战争几乎总是源于国家领导人的深思熟虑的选择，即使并非总是他们最初想要的结果（布莱尼，1973 年；霍华德，1983 年）。即使在核时代，这种模式似乎也会继续下去，因此，我们有理由问一问，国家是如何得出结论，认为攻击是值得选择的？

I shall consider several types of costs that could serve to make the actions that states take in crises informative about their actual willingness to fight. I argue that while there are several plausible candidates that may play a role in specific cases, audience costs are probably most important and characteristic of crisis bargaining. I shall then discuss variations in audience costs across regime types, suggesting that they may be most significant in states where foreign policy is conducted by an agent on behalf of a principal, as in democracies.
我将考虑几种类型的成本，它们可以使国家在危机中采取的行动让人了解其实际的战斗意愿。我认为，虽然有几种看似合理的候选成本可能会在特定情况下发挥作用，但观众成本可能是最重要的，也是危机讨价还价的特点。然后，我将讨论受众成本在不同制度类型中的差异，指出受众成本在外交政策由代理人代表委托人执行的国家中可能最为重要，如民主国家。

Two sorts of costs that leaders face for backing down in a crisis should be distinguished. First, there is the domestic and international price for conceding the issues at stake, which is the same regardless of when concessions are made or after how much escalation. Second, there are whatever additional costs are generated in the course of the crisis itself. By the costly signaling argument, only such added costs can convey new information about a state’s resolve. To ask what enables learning in a crisis-and thus why some states ultimately choose to attack-is to ask what makes escalating and then backing down worse for a leader than simply conceding at the outset.
应区分领导人在危机中退缩所面临的两种代价。首先，是在事关重大的问题上让步所要付出的国内和国际代价，无论何时让步或让步升级的程度如何，代价都是一样的。其次，危机本身会产生额外的代价。根据代价高昂的信号传递论点，只有这种额外的代价才能传递有关国家决心的新信息。要想知道是什么促成了危机中的学习--也就是为什么有些国家最终会选择进攻--就要问是什么让领导者先升级再退让比一开始就让步更糟糕。

There are a number of possible mechanisms,
有多种可能的机制、
grouped here into three categories: physical costs, costs linked to the risk of accidental or preemptive war, and international and domestic audience costs.
在此，我们将这些成本分为三类：实际成本、与意外战争或先发制人的战争风险相关的成本，以及国际和国内观众成本。

The first class includes the financial and organizational costs of mobilizing and deploying troops and also simple impatience (“time preferences”) on the part of state leaders. The economic burden of mobilization is sometimes significant enough to convey information about resolve. The fact that Israel’s economy cannot bear full mobilization for very long may make Israeli mobilization unusually informative (Shimshoni 1988, 110). One could also argue that the enormous costs of Desert Shield, given well-known U.S. budget constraints, helped make the deployment a (partially) credible indicator of Bush’s preferences (Fearon 1992, 153-54). But since the early nineteenth century, the financial costs of mobilization rarely seem the principal concern of leaders in a crisis, particularly in comparison to how their performance looks to domestic and foreign audiences. In few cases do financial costs seem to be what makes crises into political “wars of nerves.” ${}^3$${}^3$^(3){ }^{3}
第一类包括动员和部署部队的财政和组织成本，以及国家领导人简单的不耐烦（"时间偏好"）。动员的经济负担有时很重，足以传递有关决心的信息。以色列的经济无法长期承受全面动员，这一事实可能使以色列的动员具有不同寻常的信息量（Shimshoni 1988, 110）。我们也可以说，鉴于众所周知的美国预算限制，"沙漠盾牌 "行动的巨大成本有助于使这次部署成为布什偏好的（部分）可信指标（Fearon 1992, 153-54）。但自 19 世纪初以来，动员的财政成本似乎很少成为领导人在危机中的主要考虑因素，尤其是与他们在国内外受众眼中的表现相比。在极少数情况下，财政成本似乎是使危机成为政治 "神经战争 "的原因。 ${}^3$${}^3$^(3){ }^{3}

For similar reasons, pure time preferences appear less significant a signaling mechanism in crises than in buyer-seller bargaining and other economic examples. ${}^4$${}^4$^(4){ }^{4} Under time preferences, delay in a crisis would be a costly signal because a leader with a high value for settlement versus war is relatively more impatient to enjoy whatever benefits a resolution would allow. If state leaders are sometimes impatient for a deal, it seems more often due to domestic political pressures (e.g., American elections or Gorbachev or Yeltsin’s need for cash) than to a pure preference by the leader for “territory today rather than next month.”
出于类似的原因，与买卖双方讨价还价和其他经济实例相比，纯粹的时间偏好在危机中似乎不是那么重要的信号机制。 ${}^4$${}^4$^(4){ }^{4} 在时间偏好下，危机中的延迟将是一种代价高昂的信号，因为相对于战争而言，领导人更看重和解，相对而言，他们更急于享受和解所带来的任何好处。如果说国家领导人有时会对达成协议感到不耐烦，那么这似乎更多是因为国内的政治压力（如美国大选或戈尔巴乔夫或叶利钦对现金的需求），而不是因为领导人纯粹倾向于 "今天就占领领土，而不是下个月"。

The second class of signaling costs concerns risks of war that are generated in some direct way by crisis escalation. Schelling’s famous notion of the “threat that leaves something to chance” falls into this category (1960, chap. 8). Schelling suggested that in nuclear crises, at least, escalation or delay might create a risk of war resulting from something other than the deliberate choices of state leaders (e.g., a mechanical mishap or an unauthorized launch). If such risks exist, then escalation in a crisis will be a costly signal of resolve, since the risks are less worth running for a state with low interest in the issues at stake. ${}^5$${}^5$^(5){ }^{5}
第二类信号成本涉及危机升级以某种直接方式产生的战争风险。谢林著名的 "留有余地的威胁 "概念就属于这一类（1960 年，第 8 章）。谢林认为，至少在核危机中，升级或延迟可能会造成战争风险，而这种风险并非由国家领导人的蓄意选择造成（如机械故障或未经授权的发射）。如果存在这种风险，那么危机升级将是一种代价高昂的决心信号，因为对于一个对所涉问题兴趣不大的国家来说，这种风险更不值得去冒。 ${}^5$${}^5$^(5){ }^{5}

In the Cuban Missile Crisis, American decision makers did indeed worry about the risk of war stemming from a mechanical or a command-andcontrol accident (e.g., Blight and Welch 1990, 109, 311). But even in this most intense of all nuclear crises (a “best case” for the threat-that-leaves-something-to-chance argument), the key decision makers were much more concerned about risks of war connected to what the other side would choose to do. While the risk of accidental war may contribute to crisis learning, it rarely, if ever, seems to provide the main cost of escalating a dispute. ${}^6$${}^6$^(6){ }^{6}
在古巴导弹危机中，美国决策者确实担心机械或指挥控制事故引发的战争风险（如 Blight 和 Welch 1990, 109, 311）。但即使在这场最激烈的核危机中（"威胁--留有机会 "论点的 "最佳案例"），主要决策者也更担心与对方的选择有关的战争风险。虽然意外战争的风险可能有助于危机学习，但它似乎很少（如果有的话）成为争端升级的主要代价。 ${}^6$${}^6$^(6){ }^{6}

First-strike advantages, or incentives for preemptive war, provide a more appealing explanation in this class. If escalating a crisis entails a risk the other side will conclude that concessions are unlikely
先发制人的优势，或先发制人的战争动机，为这一类问题提供了更有吸引力的解释。如果危机升级会带来风险，另一方会得出让步不太可能的结论
enough to justify seizing a first-strike advantage, then escalation might be a costly signal of resolve. By running a real risk of preemption, delay in a crisis might credibly reveal a high willingness to fight rather than concede.
如果说 "先发制人 "足以证明有理由夺取先发优势，那么 "升级 "则可能是一种代价高昂的决心信号。通过冒着被先发制人的实际风险，在危机中拖延可能会令人信服地显示出战斗而非让步的高度意愿。

This mechanism appears to have figured in some historical cases. For example, part of what made the Russian mobilization in 1914 an informative signal of Russia’s willingness to fight was that it was undertaken in the knowledge that it would increase Germany’s incentive to choose preemptive war (Fearon 1992, chap. 5). ${}^7$${}^7$^(7){ }^{7} In theory, however, first-strike advantages could also have the opposite effect. A state might conclude that since the adversary has not so far availed itself of a first strike, it must be more willing to back down than initially anticipated. Further, major concerns about loss of first-strike advantage do not seem common in case-evidence on international crises, and even when such concerns are present, as in 1914, they often compete with worries about the political disadvantages of going first. ${}^8$${}^8$^(8){ }^{8}
这种机制似乎在某些历史案例中有所体现。例如，1914年的俄国动员之所以成为俄国愿意参战的信息信号，部分原因在于它是在知道这会增加德国选择先发制人的战争的动机的情况下进行的（Fearon 1992, chap. 5）。 ${}^7$${}^7$^(7){ }^{7} 然而，从理论上讲，先发制人的优势也可能产生相反的效果。一国可能会得出这样的结论：既然对手至今还没有利用先发制人的优势，那么它一定比最初预期的更愿意退让。此外，在有关国际危机的案例证据中，对丧失先发优势的重大担忧似乎并不常见，即使存在这种担忧（如在1914年），它们也常常与对先发制人的政治劣势的担忧相竞争。 ${}^8$${}^8$^(8){ }^{8}

While each of the preceding mechanisms may well foster crisis learning in some cases, I would argue that none fits with our intuitive sense of what it is that makes international crises into political wars of nerves. The reason is that none of these mechanisms recognizes the public aspect of crises, the fact that they are carried out in front of political audiences evaluating the skill and performance of the leadership. In prototypical cases (e.g., the standoff leading to the 1991 Gulf War, the Cuban Missile Crisis, and July 1914), a leader who chooses to back down is (or would be) perceived as having suffered a greater “diplomatic humiliation” the more he had escalated the crisis. Conversely, our intuition is that the more a crisis escalates, the greater the perception of diplomatic triumph for a leader who “stands firm” until the other side backs down.
虽然上述每种机制在某些情况下都可能促进危机学习，但我认为，没有一种机制符合我们的直觉，即是什么使国际危机成为政治神经战。原因在于，这些机制都没有认识到危机的公共性，即危机是在政治受众面前发生的，而政治受众则在评估领导层的技能和表现。在典型案例中（如 1991 年导致海湾战争的对峙、古巴导弹危机和 1914 年 7 月），选择退让的领导人被认为（或将被认为）遭受了更大的 "外交羞辱"，因为他将危机升级得越厉害。相反，我们的直觉是，危机越升级，"坚守阵地 "直至对方退让的领导人在外交上取得胜利的可能性就越大。

Political audiences need not and do not always have this pattern of perceptions and reactions: they are social conventions that are at times resolved differently. For example, leaders of small states may be rewarded for escalating crises with big states and then backing down, where they would be castigated for simply backing down. Standing up to a “bully” may be praised even if one ultimately retreats. ${}^9$${}^9$^(9){ }^{9} Nonetheless, at least since the eighteenth century leaders and publics have typically understood threats and troop deployments to “engage the national honor,” thus exposing leaders to risk of criticism or loss of authority if they are judged to have performed poorly by the relevant audiences. Two illustrations follow, both taken from eighteenth-century diplomacy. While a wealth of similar examples are available from the nineteenth and twentieth centuries, these earlier cases suggest that political audiences have mattered in international confrontations for a long time.
政治受众不需要也不一定总是有这种认知和反应模式：它们是社会习俗，有时会以不同的方式加以解决。例如，小国领导人可能会因为与大国的危机升级然后退缩而受到奖励，而如果只是退缩则会受到指责。面对 "恶霸 "挺身而出可能会受到赞扬，即使最终退缩。 ${}^9$${}^9$^(9){ }^{9} 尽管如此，至少自18世纪以来，领导人和公众通常将威胁和出兵理解为 "维护国家荣誉"，因此，如果相关受众认为领导人表现不佳，他们就有可能受到批评或失去权威。下面有两个例子，都来自十八世纪的外交。虽然 19 世纪和 20 世纪有大量类似的例子，但这些早期的案例表明，政治受众在国际对抗中的重要性由来已久。

The Seven Years War (1757-64) between France and Britain was preceded by several years of “crisis” diplomacy-threats, warnings, and troop mobilizations and deployments (Higonnet 1968; Smoke 1977, chap. 8). In response to French demands on the Ohio
在英法七年战争（1757-64 年）之前，双方进行了数年的 "危机 "外交--威胁、警告、部队动员和部署（Higonnet，1968 年；Smoke，1977 年，第 8 章）。为了回应法国对俄亥俄河的要求

River Valley, the Duke of Newcastle chose in late 1755 to send two regiments to America to impress the French with British resolve. The decision distressed several of Newcastle’s colleagues and ambassadors, who seem to have felt that the action engaged the honor of the king and so committed the cabinet to a warlike course, perhaps unnecessarily. One wrote, “It requires great dexterity to conduct [these diplomatic and military moves] in such a manner to maintain the honor of King and Nation” (Higonnet 1968, 79). In a later interview with Rouillé, the French minister of foreign affairs, the British ambassador reported that the minister “complain’d very much of the licentiousness of our Publick papers in exaggerating things beyond measure which only served to irritate and stir up animosity amongst the lower sort of People in both Nations without a just cause” (p. 80). This complaint suggests that even in nondemocratic, eighteenth-century France, a minister could be concerned with what I have called domestic audience costs: it seems that British pamphlets could have the effect of increasing Rouillé’s costs for acceding to British demands (as they increased Newcastle’s domestic costs for ceding French demands).
1755年末，纽卡斯尔公爵决定派遣两个团前往美洲，让法国人对英国的决心刮目相看。纽卡斯尔的几位同僚和大使对这一决定感到不安，他们似乎认为这一行动涉及国王的荣誉，因此内阁可能不必要地走上了战争的道路。其中一位写道："要想以这样的方式进行（这些外交和军事行动），以维护国王和国家的荣誉，需要极大的灵活性"（Higonnet，1968 年，第 79 页）。英国大使后来在采访法国外交部长鲁瓦耶（Rouillé）时报告说，这位部长 "非常抱怨我们的报纸放荡不羁，夸大其词，只会无端刺激和挑起两国下层人民的敌意"（第 80 页）。这一抱怨表明，即使是在非民主的 18 世纪法国，大臣也会关心我所说的国内听众成本：英国的小册子似乎会增加鲁瓦耶满足英国要求的成本（正如它们会增加纽卡斯尔满足法国要求的国内成本一样）。

About 35 years later, Britain and Spain nearly went to war over an obscure incident involving alleged Spanish insults to British seamen who had landed on Vancouver Island, along with competing claims on the territory (Manning 1904). Once again, both states resorted to troop mobilizations, forceful diplomatic notes, and public threats. There is strong evidence that these moves created significant domestic audience costs for Prime Minister William Pitt: “With an election imminent, the Opposition was ready to make the most of any of the Government’s mistakes in negotiating” (Norris 1955, 572). Pitt’s vote for navy credits in Parliament and his government’s publication of an account of the Vancouver incident led opposition politicians almost to clamor for appropriate satisfaction of British honor and right. Pitt would have faced serious domestic political costs for backing down, much larger than if he had chosen initially to pursue a less public and aggressive line in the dispute. ${}^{10}$${}^{10}$^(10){ }^{10}
大约 35 年后，英国和西班牙差点因为一起不起眼的事件而开战，该事件涉及据称西班牙侮辱登陆温哥华岛的英国海员，以及对该领土的竞争性主权要求（Manning，1904 年）。两国再次诉诸军队调动、强硬的外交照会和公开威胁。有确凿证据表明，这些举动给首相威廉-皮特造成了巨大的国内观众成本："由于选举在即，反对党随时准备利用政府在谈判中的任何失误"（Norris 1955, 572）。皮特在议会中投票支持海军信用额度，他的政府发表了关于温哥华事件的报道，这使得反对派政客们几乎鼓噪起来，要求对英国的荣誉和权利进行适当的补偿。如果皮特退缩，他将面临严重的国内政治代价，这比他最初在争端中选择不那么公开和激进的路线要大得多。 ${}^{10}$${}^{10}$^(10){ }^{10}

The notion that troop movements and public demands or threats “engage the national honor”-thus creating audience costs that leaders would pay if they backed down-continues strongly through the nineteenth and twentieth centuries. Such costs can be classified according to whether the audience that imposes them is domestic or international. Relevant domestic audiences have included kings, rival ministers, opposition politicians, Senate committees, politburos, and, since the mid-nineteenth-century, mass publics informed by mass media in many cases. Relevant international audiences include a state’s opponent in the crisis and other states not directly involved, such as allies. Here the costs of escalating and then backing down would be felt indirectly through injury to the state’s reputation for threatening the use of force only when serious.
军队调动和公众要求或威胁 "关系到国家荣誉"--因此会产生受众成本，如果领导人退缩就会付出代价--这一概念在 19 世纪和 20 世纪一直延续至今。这种成本可以根据施加成本的受众是国内还是国际来划分。相关的国内受众包括国王、对立的大臣、反对派政治家、参议院委员会、政治局，以及自 19 世纪中叶以来，在许多情况下由大众媒体提供信息的大众。相关的国际受众包括一国在危机中的对手和其他未直接参与的国家，如盟国。在这种情况下，只有在事态严重时才威胁使用武力的国家声誉会受到损害，从而间接感受到升级后又退缩的代价。

Leaders engaged in disputes appear to worry about
参与争端的领导人似乎担心
both international and domestic audiences. Domestic audience costs may be primary, however. Backing down after making a show of force is often most immediately costly for a leader because it gives domestic political opponents an opportunity to deplore the international loss of credibility, face, or honor. ${}^{11}$${}^{11}$^(11){ }^{11} Because governments are far more likely to be deposed or to lose authority due to internal political developments than due to foreign conquest and because opposition groups frequently condition their activities on the international successes and failures of the leaders in power, domestic audiences may provide the strongest incentives for leaders to guard their states’ “international” reputations. Audience costs thus figure in a domestic system of incentives that encourages leaders to have a realist concern with their state’s “honor” and reputation before international audiences.
国际和国内受众。然而，国内受众的代价可能是最主要的。对领导人来说，炫耀武力后的退缩往往会立即造成巨大损失，因为这会给国内政治对手一个机会，让他们对在国际上丧失公信力、颜面或荣誉表示遗憾。 ${}^{11}$${}^{11}$^(11){ }^{11} 由于政府因国内政治事态发展而被废黜或失去权威的可能性远远高于因外国征服而被废黜或失去权威的可能性，而且由于反对派团体经常以当权领导人在国际上的成败作为其活动的条件，因此国内受众可能为领导人维护其国家的 "国际 "声誉提供了最强有力的激励。因此，受众成本在国内激励体系中占有一席之地，该体系鼓励领导人在国际受众面前以现实主义的方式关注国家的 "荣誉 "和声誉。

As noted, audience costs have a strongly conventional aspect; how they are felt and implemented depends on shared perceptions and expectations in a society. Nevertheless, the historical norm seems to have domestic political audiences punishing a leader who concedes after having deployed troops more than one who concedes outright. Why this norm? My theoretical results will suggest a possible explanation, which I anticipate here in order to develop some broader points about how audience costs vary across types of states.
如前所述，受众成本具有强烈的传统性；受众成本的感受和实施取决于社会的共同看法和期望。然而，根据历史惯例，国内政治受众对出兵后认输的领导人的惩罚似乎要比直接认输的领导人更严厉。为什么会出现这种规范？我的理论结果将提出一个可能的解释，我在此对其进行预测，以便就不同类型国家的受众成本如何不同提出一些更广泛的观点。

Equilibrium analysis of the crisis model reveals that a state’s ex ante expected payoff in a dispute is increasing in the degree to which escalatory moves create audience costs for the state’s leadership. The reason, in brief, is that greater audience costs improve a state’s ability to commit and to signal resolve.
对危机模型的均衡分析表明，国家在争端中的事前预期收益会随着升级行动给国家领导层带来受众成本的程度而增加。简而言之，原因在于更高的受众成本会提高国家的承诺能力和发出决心信号的能力。

Thus both democratic and nondemocratic leaders should have an incentive to represent that they will pay added domestic political costs for “engaging the national honor” and subsequently backing down. The extent to which such representations are believable, however, depends on the nature of the domestic political institutions that the leadership faces. In democracies, foreign policy is made by an agent on behalf of principals (voters) who have the power to sanction the agent electorally or through the workings of public opinion. By contrast, in authoritarian states the principals often conduct foreign policy themselves. The result here suggests that in the former case, if the principal could design a “wage contract” for the foreign policy agent, the principal would want to commit to punishing the agent for escalating a crisis and then backing down. On the other hand, principals who conduct foreign policy themselves may not be able credibly to commit to self-imposed punishment (such as leaving power) for backing down in a crisis. ${}^{12}$${}^{12}$^(12){ }^{12}
因此，无论是民主领导人还是非民主领导人，都应该有动力表明，如果 "为国争光 "并随后退缩，他们将付出额外的国内政治代价。然而，这种表述的可信程度取决于领导人所面对的国内政治体制的性质。在民主国家，外交政策是由代理人代表委托人（选民）制定的，委托人有权通过选举或舆论对代理人进行制裁。与此相反，在专制国家，委托人往往自己制定外交政策。这里的结果表明，在前一种情况下，如果委托人可以为外交政策代理人设计一份 "工资合同"，委托人就会希望对代理人将危机升级然后又退缩的行为进行惩罚。另一方面，自己执行外交政策的委托人可能无法令人信服地承诺对在危机中退缩的代理人进行自我惩罚（如下台）。 ${}^{12}$${}^{12}$^(12){ }^{12}
Examples of the apparent effect and import of forceful public speeches by democratic and nondemocratic leaders suggest that this argument is at least
民主和非民主领导人强有力的公开演讲所产生的明显效果和影响的实例表明，这一论点至少是正确的。

A Schematic Representation of the Crisis Game
危机游戏示意图

plausible. Repeatedly, leaders in democratic countries have been able credibly to jeopardize their electoral future by making strong public statements during international confrontations. A few prominent examples are Lord Salisbury’s speeches during the Fashoda crisis of 1898, Lloyd George’s Mansion House speech during the Agadir crisis of 1911, Kennedy’s televised speech announcing the presence of Soviet missiles in Cuba, and Bush’s many declarations on Kuwait in 1990 (including the “this will not stand” remark). ${}^{13}$${}^{13}$^(13){ }^{13} By contrast, even more forceful public bluster by authoritarian leaders (e.g., Hitler, Khrushchev, Mao, and Saddam Hussein) appears to create fewer credible audience costs, and to have correspondingly lower value as signals of intent. For example, it was quite difficult for Western observers to know what to conclude about Saddam Hussein’s willingness to fight from his many strong public refusals to pull out of Kuwait in the Fall of 1990. One reason, I would argue, is that it was difficult to know what, if any, added domestic political costs such a tyrant would suffer for making concessions at the last minute.
可信。民主国家的领导人曾多次在国际对抗中发表强硬的公开声明，从而危及自己的选举前途。几个突出的例子是索尔兹伯里勋爵在1898年法绍达危机中的演讲、劳合-乔治在1911年阿加迪尔危机中的大厦演讲、肯尼迪宣布苏联在古巴部署导弹的电视讲话，以及布什在1990年就科威特问题发表的多次声明（包括 "这将是站不住脚的 "言论）。 ${}^{13}$${}^{13}$^(13){ }^{13} 相比之下，专制领导人（如希特勒、赫鲁晓夫、毛泽东和萨达姆-侯赛因）在公开场合发出的更有力的虚张声势似乎造成的可信受众成本更低，作为意图信号的价值也相应更低。例如，1990 年秋天，萨达姆-侯赛因多次公开强烈拒绝撤出科威特，西方观察家很难从他的拒绝中得出关于萨达姆-侯赛因作战意愿的结论。我认为，其中一个原因是，很难知道这样一个暴君在最后一刻做出让步会付出什么额外的国内政治代价（如果有的话）。

This is not to suggest that authoritarian states are completely unable to generate audience costs in international confrontations or that democracies can invariably do so. On the one hand, nondemocracies may evolve institutional arrangements in foreign policy that give domestic audiences an ability to sanction decision makers. For example, the politburo after Stalin could sanction the paramount Soviet leader, and eighteenth- and nineteenth-century monarchs could replace unsuccessful ministers. Moreover, since the price of losing power is often greater for a dictator than for an elected leader, a weak or unstable authoritarian regime might be able to create significant expected audience costs in a crisis. On the other hand, in democracies the existence of multiple politically relevant audiences may make it difficult for foreign leaders to gauge the costs created by public statements or actions, particularly before elections. The idea that democratic leaders on average have an easier time generating audience costs is advanced here as a plausible working hypothesis that has interesting theoretical and empirical implications.
这并不是说专制国家完全无法在国际对抗中产生受众成本，也不是说民主国家一定能做到这一点。一方面，非民主国家可能会在外交政策上做出制度安排，让国内受众有能力制裁决策者。例如，斯大林之后的政治局可以制裁苏联最高领导人，十八和十九世纪的君主可以撤换不成功的大臣。此外，由于独裁者失去权力的代价往往比民选领导人更大，因此一个软弱或不稳定的独裁政权可能会在危机中产生巨大的预期受众成本。另一方面，在民主国家，由于存在多个政治相关受众，外国领导人可能很难衡量公开声明或行动所造成的成本，尤其是在选举之前。民主国家的领导人平均来说更容易产生受众成本，这一观点在此作为一个合理的工作假设提出，具有有趣的理论和实证意义。

I shall describe a model of a crisis as a political attrition contest, informally discuss equilibrium behavior when the states know each other’s values for conflict, and then develop the main results on equilibrium with uncertainty about resolve.
我将描述一个作为政治消耗竞赛的危机模型，非正式地讨论当国家知道彼此对冲突的价值观时的均衡行为，然后得出在决心不确定的情况下的均衡的主要结果。

Two states-1 and 2 -are in dispute over a prize worth $v>0$$v>0$v > 0v>0. The crisis occurs in continuous time, starting at $t=0$$t=0$t=0t=0. For every finite $t\ge 0$$t\ge 0$t >= 0t \geq 0 before the crisis ends, each state can choose to attack, quit, or escalate. The crisis ends when one or both states attack or quit. “Escalate” can be thought of either as simply waiting or as taking actions such as mobilizing or preparing troops.
两个国家--1 和 2 --为价值 $v>0$$v>0$v > 0v>0 的奖品发生争执。危机在连续时间内发生，从 $t=0$$t=0$t=0t=0 开始。在危机结束前的每个有限时间 $t\ge 0$$t\ge 0$t >= 0t \geq 0 内，每个国家都可以选择攻击、退出或升级。当一个或两个国家攻击或退出时，危机结束。"升级 "可以理解为简单的等待，也可以理解为采取行动，如动员或准备军队。
Payoffs are given as follows. If either state attacks before the other quits, both receive their expected utilities for military conflict, $w_1$$w_1$w_(1)w_{1} and $w_2$$w_2$w_(2)w_{2}. These are the states’ values for war, incorporating military expectations, values for objects in dispute, and costs for fighting. They can be thought of as levels of “resolve,” since the higher $w_i$$w_i$w_(i)w_{i}, the higher the risk of war state $i$$i$ii is willing to run in hope of attaining the prize. Throughout, I shall suppose that neither $w_1$$w_1$w_(1)w_{1} nor $w_2$$w_2$w_(2)w_{2} is greater than $0.{}^{14}$$0.{}^{14}$0.^(14)0 .{ }^{14}
收益情况如下。如果任何一个国家在另一个国家退出之前发动进攻，那么双方都会得到其对军事冲突的预期效用 $w_1$$w_1$w_(1)w_{1} 和 $w_2$$w_2$w_(2)w_{2} 。这些是国家的战争价值，包括军事预期、争议对象的价值和战斗成本。它们可以被看作是 "决心 "的水平，因为 $w_i$$w_i$w_(i)w_{i} 越高， $i$$i$ii 就愿意冒更大的战争风险，希望获得战利品。在整个过程中，我将假设 $w_1$$w_1$w_(1)w_{1} 和 $w_2$$w_2$w_(2)w_{2} 都不大于 $0.{}^{14}$$0.{}^{14}$0.^(14)0 .{ }^{14} 。
If state $i$$i$ii quits the crisis before the other has quit or attacked, then its opponent $j$$j$jj receives the prize while $i$$i$ii suffers audience costs equal to $a_i(t)$$a_i(t)$a_(i)(t)a_{i}(t), a continuous and strictly increasing function of the amount of escalation with $a_i(0)=0$$a_i(0)=0$a_(i)(0)=0a_{i}(0)=0. I will often consider the linear case $a_i(t)=a_{i}t$$a_i(t)=a_{i}t$a_(i)(t)=a_(i)ta_{i}(t)=a_{i} t, where $a_i>0$$a_i>0$a_(i) > 0a_{i}>0 is a parameter indicating how rapidly escalation creates audience costs for state $i$$i$ii. Figure 1 schematically depicts the structure of the contest, with payoffs indicated in the case that state 1 quits or attacks first at time $t_1.{}^{15}$$t_1.{}^{15}$t_(1).^(15)t_{1} .{ }^{15}
如果国家 $i$$i$ii 在对方退出或进攻之前退出危机，那么其对手 $j$$j$jj 将获得奖金，而 $i$$i$ii 则会遭受相当于 $a_i(t)$$a_i(t)$a_(i)(t)a_{i}(t) 的观众成本，这是升级量与 $a_i(0)=0$$a_i(0)=0$a_(i)(0)=0a_{i}(0)=0 的一个连续且严格递增的函数。我通常会考虑线性情况 $a_i(t)=a_{i}t$$a_i(t)=a_{i}t$a_(i)(t)=a_(i)ta_{i}(t)=a_{i} t ，其中 $a_i>0$$a_i>0$a_(i) > 0a_{i}>0 是一个参数，表示升级给状态 $i$$i$ii 带来观众成本的速度。图 1 以示意图的形式描述了竞赛的结构，并标出了国家 1 在时间 $t_1.{}^{15}$$t_1.{}^{15}$t_(1).^(15)t_{1} .{ }^{15} 时首先退出或攻击的情况下的报酬。
Call this game, with common knowledge of all parameters, $G$$G$GG and the subgame beginning at $t,G(t)$$t,G(t)$t,G(t)t, G(t). A pure strategy in $G$$G$GG is a rule $s_i$$s_i$s_(i)s_{i} specifying a finite time $t\ge t^{\prime }$$t\ge t^{\prime }$t >= t^(')t \geq t^{\prime} to quit or attack in every $G\left(t^{\prime }\right)$$G\left(t^{\prime }\right)$G(t^('))G\left(t^{\prime}\right), for all $t^{\prime }\ge 0$$t^{\prime }\ge 0$t^(') >= 0t^{\prime} \geq 0. I shall write $\{t$$\{t${t\{t, attack $\}$$\}$}\} for the subgame strategy “escalate up to $t$$t$tt, then attack” and $\{t$$\{t${t\{t, quit $\}$$\}$}\} for “escalate up to $t$$t$tt, then quit.”
在所有参数均为常识的情况下，将此博弈称为 $G$$G$GG ，子博弈始于 $t,G(t)$$t,G(t)$t,G(t)t, G(t) 。 $G$$G$GG 中的纯策略是一条规则 $s_i$$s_i$s_(i)s_{i} ，它指定了一个有限的时间 $t\ge t^{\prime }$$t\ge t^{\prime }$t >= t^(')t \geq t^{\prime} ，在每个 $G\left(t^{\prime }\right)$$G\left(t^{\prime }\right)$G(t^('))G\left(t^{\prime}\right) 中退出或攻击，对于所有 $t^{\prime }\ge 0$$t^{\prime }\ge 0$t^(') >= 0t^{\prime} \geq 0 。我将用 $\{t$$\{t${t\{t , attack $\}$$\}$}\} 来表示子博弈策略 "升级到 $t$$t$tt , 然后攻击"，用 $\{t$$\{t${t\{t , quit $\}$$\}$}\} 来表示 "升级到 $t$$t$tt , 然后退出"。

If the states knew each other’s levels of resolve, they could in principle look ahead and anticipate what would happen if they were to escalate and create a crisis. For example, in the linear case where $a_i(t)=a_{i}t$$a_i(t)=a_{i}t$a_(i)(t)=a_(i)ta_{i}(t)=a_{i} t, they would see that ultimately the audience costs for backing down would be so large that neither would quit: doing so would be strictly worse than attacking. In other words, both states would eventually become “locked in.”
如果各国都知道对方的决心，那么原则上它们就可以提前预知如果事态升级并造成危机会发生什么。例如，在 $a_i(t)=a_{i}t$$a_i(t)=a_{i}t$a_(i)(t)=a_(i)ta_{i}(t)=a_{i} t 的线性情况下，它们会发现最终退让的受众成本会非常高，以至于双方都不会放弃：这样做严格来说比进攻更糟糕。换句话说，两种状态最终都会被 "锁定"。

However, they would also notice that one side would become locked in before the other (excepting certain symmetric cases). Suppose, for example, that audience costs increase linearly and at the same rate on both sides, and that state 1’s value for war is higher than state 2’s. Then if both sides escalated the crisis, state 1 would reach the point where its leaders preferred conflict to backing down sooner than state 2 would. At this point, state 1 has in effect committed itself not to back down, while state 2 still prefers making concessions to a fight. Thus a rational state 2 would have to back down at this point. Anticipating this at the outset, state 2 would quit immediately rather than pay the larger audience costs that would go with publicly observable escalation. ${}^{16}$${}^{16}$^(16){ }^{16}
不过，他们也会注意到，一方会先于另一方被锁定（某些对称情况除外）。例如，假设双方的受众成本都以相同的速度线性增长，而国家 1 的战争价值高于国家 2。那么，如果双方都将危机升级，国家 1 的领导人会比国家 2 更早达到宁愿冲突也不愿退让的地步。此时，国家 1 实际上已经承诺不会退让，而国家 2 仍倾向于做出让步而不是开战。因此，理性的国家 2 此时不得不退让。国家 2 一开始就预料到了这一点，因此会立即退出，而不是付出更大的受众成本，因为公开观察到的升级会带来更大的受众成本。 ${}^{16}$${}^{16}$^(16){ }^{16}

Thus with complete information about resolve, no public crisis occurs. Instead, if audience costs increase at the same rate on both sides, the state with the lower value for conflict immediately cedes the prize rather than incur costs above those associated with the loss on the issue. If audience costs increase at different rates, then the side with weaker audience effects may be forced to concede even if it has a higher value for war (since it may require more escalation to commit itself to fight).
因此，在完全了解决心的情况下，不会出现公共危机。相反，如果双方的受众成本以相同的速度增长，冲突价值较低的国家就会立即放弃战利品，而不会承担与该问题上的损失相关的成本。如果受众成本以不同的速度增长，那么受众效应较弱的一方即使战争价值较高（因为它可能需要更多的升级才能投入战斗），也可能被迫让步。

This equilibrium logic mirrors a logic found in many analytic and diplomatic historical discussions of international disputes. It is often argued that in crises that do not become wars, states look ahead and the side expecting to do worse in military conflict then backs down. But the standard argument does not work by itself: audience costs are required. If it were known that the state with the higher value for war nonetheless preferred making some concessions to a fight, why should the state with the lower value for war necessarily back down? If both prefer concessions to a fight, how can either make a credible threat to go to war and why should this be related to their values for military conflict? Increasing audience costs supply an answer. The state with the higher value for war may be able, in a public crisis, to reach more quickly the point where it prefers conflict to paying the audience costs of backing down.
这种平衡逻辑反映了许多关于国际争端的分析和外交历史讨论中的逻辑。人们通常认为，在没有演变成战争的危机中，国家会向前看，而预期在军事冲突中表现更差的一方则会退缩。但这一标准论点本身并不成立：受众成本是必要条件。如果我们知道战争价值较高的国家宁愿做出一些让步也不愿开战，那么战争价值较低的国家为什么一定要退让呢？如果双方都更倾向于让步而不是开战，那么任何一方如何才能发出可信的开战威胁呢？观众成本的增加提供了答案。在公共危机中，战争价值较高的国家可能会更快地达到宁愿冲突也不愿付出退让的受众成本的地步。

With complete information, no public war of nerves occurs because the ultimate outcome can be seen in advance. In a rationalist framework, international crises occur precisely because state leaders cannot
有了完整的信息，就不会发生公开的神经战，因为最终的结果可以事先看到。在理性主义框架下，国际危机的发生正是因为国家领导人不能
anticipate the outcome, owing to the fact that adversaries have private information about their willingness to fight over foreign policy interests and the incentive to misrepresent it. I now consider equilibrium in the model in which the states have private information about their willingness to go to war.
由于对手拥有关于其是否愿意为外交政策利益而开战的私人信息，并且有动机歪曲这些信息，因此我们无法预测结果。现在，我将考虑在国家拥有关于其战争意愿的私人信息的模型中的均衡问题。

Three main theoretical results are developed. Proposition 1 establishes that in any equilibrium in which a crisis may occur, the crisis has a unique horizon-an amount of escalation after which neither state will back down and war is certain. Proposition 2 characterizes the set of equilibria in which a crisis may occur. Proposition 3 asserts that in this set, the probability distribution on outcomes is unique up to the horizon time. Throughout, the equilibrium concept is a modification of perfect Bayesian equilibrium (Fudenberg and Tirole 1991) for an infinite game, with an additional restriction ruling out strongly optimistic off-equilibrium-path inferences. Details on the solution concept, along with all proofs, are given in the Appendix. ${}^{17}$${}^{17}$^(17){ }^{17}
本文提出了三个主要理论结果。命题 1 证明，在任何可能发生危机的均衡中，危机都有一个唯一的范围--在这个范围内，任何国家都不会退让，战争必将爆发。命题 2 描述了可能发生危机的均衡的特征。命题 3 断言，在这个集合中，结果的概率分布在地平线时间之前是唯一的。在整个过程中，均衡概念是对无限博弈的完全贝叶斯均衡（Fudenberg 和 Tirole，1991 年）的修改，并增加了一个限制条件，即排除强烈乐观的非均衡路径推断。关于求解概念的细节以及所有证明都在附录中给出。 ${}^{17}$${}^{17}$^(17){ }^{17}

Preliminaries. Formally, suppose that each state knows its own level of resolve $w_i$$w_i$w_(i)w_{i} but knows only the distribution of its adversary’s resolve $w_j$$w_j$w_(j)w_{j}. For $i=1,2$$i=1,2$i=1,2i=1,2, let $w_i$$w_i$w_(i)w_{i} be distributed on the interval $W_i\equiv [{\underset{―}{w}}_i,0],{\underset{―}{w}}_i<$$W_i\equiv \left[{\underset{\_}{w}}_i,0\right],{\underset{\_}{w}}_i<$W_(i)-=[w__(i),0],w__(i) <W_{i} \equiv\left[\underline{w}_{i}, 0\right], \underline{w}_{i}< 0 , according to a cumulative distribution function $F_i$$F_i$F_(i)F_{i} that has continuous and strictly positive density $f_i{}^{18}{}_I$$f_i{}^{18}{}_I$f_(i)^(18)_(I)f_{i}{ }^{18}{ }_{I} refer to the crisis game with this information structure and all other elements assumed to be common knowledge as $\mathrm{\Gamma }$$\mathrm{\Gamma }$Gamma\Gamma.
前言。形式上，假设每个国家都知道自己的决心水平 $w_i$$w_i$w_(i)w_{i} ，但只知道对手的决心分布 $w_j$$w_j$w_(j)w_{j} 。对于 $i=1,2$$i=1,2$i=1,2i=1,2 ，让 $w_i$$w_i$w_(i)w_{i} 分布在区间 $W_i\equiv [{\underset{―}{w}}_i,0],{\underset{―}{w}}_i<$$W_i\equiv \left[{\underset{\_}{w}}_i,0\right],{\underset{\_}{w}}_i<$W_(i)-=[w__(i),0],w__(i) <W_{i} \equiv\left[\underline{w}_{i}, 0\right], \underline{w}_{i}< 0 上，根据累积分布函数 $F_i$$F_i$F_(i)F_{i} ，该函数具有连续且严格为正的密度 $f_i{}^{18}{}_I$$f_i{}^{18}{}_I$f_(i)^(18)_(I)f_{i}{ }^{18}{ }_{I} 将具有此信息结构和所有其他假设为共同知识的元素的危机博弈称为 $\mathrm{\Gamma }$$\mathrm{\Gamma }$Gamma\Gamma 。

In informal terms, $f_1$$f_1$f_(1)f_{1} and $f_2$$f_2$f_(2)f_{2} represent the states’ precrisis beliefs about each other’s value for war on the issue in dispute. For example, the more weight $f_1$$f_1$f_(1)f_{1} puts on values of $w_1$$w_1$w_(1)w_{1} that are close to 0 (as opposed to very negative), the greater is state 2 's initial belief that 1 has a relatively high willingness to fight rather than make concessions.
用非正式术语来说， $f_1$$f_1$f_(1)f_{1} 和 $f_2$$f_2$f_(2)f_{2} 代表国家在危机前对彼此在争议问题上的战争价值的看法。例如， $f_1$$f_1$f_(1)f_{1} 对 $w_1$$w_1$w_(1)w_{1} 中接近于 0（而不是非常负值）的值所赋予的权重越大，国家 2 的初始信念就越强，即国家 1 有相对较高的战争意愿，而不是做出让步。

Crises in the Model have a Unique Horizon. This proposition is developed by way of two lemmas, which also help to make clear the logic of strategic choice in the model. I begin with a definition. A crisis has a horizon if there is a level of escalation such that neither state is expected to quit after this point is reached. Formally, let $Q_i(t)$$Q_i(t)$Q_(i)(t)Q_{i}(t) be the probability that state $i$$i$ii quits on or before $t$$t$tt in some equilibrium. Then $t_h>0$$t_h>0$t_(h) > 0t_{h}>0 is a horizon for $\mathrm{\Gamma }$$\mathrm{\Gamma }$Gamma\Gamma if in this equilibrium $t_h$$t_h$t_(h)t_{h} is the minimum $t$$t$tt such that neither $Q_1(t)$$Q_1(t)$Q_(1)(t)Q_{1}(t) nor $Q_2(t)$$Q_2(t)$Q_(2)(t)Q_{2}(t) increase for $t>t_h$$t>t_h$t > t_(h)t>t_{h}. If a horizon exists, war has become inevitable by the time it is reached.
模型中的危机具有独特的地平线。这一命题是通过两个悖论提出的，这两个悖论也有助于明确模型中战略选择的逻辑。首先是定义。如果存在这样一种升级水平，即在达到该点后，两国都不会退出，那么危机就具有地平线。形式上，假设 $Q_i(t)$$Q_i(t)$Q_(i)(t)Q_{i}(t) 是在某个均衡中，国家 $i$$i$ii 在 $t$$t$tt 当日或之前退出的概率。那么，如果在这个均衡中， $t_h$$t_h$t_(h)t_{h} 是最小的 $t$$t$tt ，使得 $Q_1(t)$$Q_1(t)$Q_(1)(t)Q_{1}(t) 和 $Q_2(t)$$Q_2(t)$Q_(2)(t)Q_{2}(t) 对 $t>t_h$$t>t_h$t > t_(h)t>t_{h} 来说都没有增加，那么 $t_h>0$$t_h>0$t_(h) > 0t_{h}>0 就是 $\mathrm{\Gamma }$$\mathrm{\Gamma }$Gamma\Gamma 的地平线。如果存在地平线，那么在达到地平线时，战争已经不可避免。
Lemma 1 establishes that in any equilibrium in which a crisis may occur, there must exist a horizon. Horizons are thus shown to arise endogenously, as a consequence of the equilibrium choices by the states involved. The intuition for this result is straightforward in the case of linearly increasing audience costs: eventually the price of backing down will be so great that even the least-resolved type of state would prefer to attack rather than quit. But the result holds even if “maximum escalation” (arbitrarily large $t$$t$tt ) will not
定理 1 证明，在任何可能发生危机的均衡中，都必然存在地平线。由此可见，地平线是内生的，是相关国家均衡选择的结果。在受众成本线性增长的情况下，这一结果的直观性是显而易见的：最终，退缩的代价会如此之大，以至于即使是最没有决心的国家也会选择进攻而不是退出。但是，即使 "最大升级"（任意大的 $t$$t$tt ）也不会使结果成立。
create large-enough audience costs to commit every type of state to war.
创造足够大的受众成本，让每一类国家都投入战争。

Lemma 1. In any equilibrium of $\mathrm{\Gamma }$$\mathrm{\Gamma }$Gamma\Gamma in which both states choose to escalate with positive probability, there must exist a finite horizon ${\mathrm{t}}_{\mathrm{h}}<\mathrm{\infty }$${\mathrm{t}}_{\mathrm{h}}<\mathrm{\infty }$t_(h) < oo\mathrm{t}_{\mathrm{h}}<\infty.
定理 1.在 $\mathrm{\Gamma }$$\mathrm{\Gamma }$Gamma\Gamma 的任何均衡中，如果两个状态都以正概率选择升级，那么一定存在一个有限的视界 ${\mathrm{t}}_{\mathrm{h}}<\mathrm{\infty }$${\mathrm{t}}_{\mathrm{h}}<\mathrm{\infty }$t_(h) < oo\mathrm{t}_{\mathrm{h}}<\infty 。
Lemma 2 characterizes the behavior of states that choose a strategy that could lead to war. The first part establishes that in equilibrium no state will choose to attack before a horizon time. When there is no advantage to striking first, a state unwilling to make concessions will want to delay attack as long as there is any chance that the other side will back down, thus avoiding the risk of an unnecessary military conflict while maximizing the chance of a “foreign policy triumph.”
定理 2 描述了选择可能导致战争的策略的国家的行为特征。第一部分证明，在均衡状态下，没有国家会选择在地平线时间之前发动进攻。当先发制人没有任何好处时，只要对方有可能退让，不愿让步的国家就会希望推迟进攻，从而避免不必要的军事冲突风险，同时最大限度地增加 "外交政策胜利 "的机会。

The second part of the lemma shows that a state will choose the strategy of escalating to the horizon and then attacking if and only if its privately known level of resolve, $w_i$$w_i$w_(i)w_{i}, is sufficiently large. Thus crises in the model separate states according to their unobservable willingness to fight over the issues: a highly resolved state credibly reveals its motivation by choosing an unyielding crisis-bargaining strategy. This gives it a greater chance of prevailing if the crisis ends peacefully but at the (unavoidable) price of a greater risk of war (cf. Banks 1990). It follows that for outside observers, crisis outcomes must be unpredictable to a significant degree. Comparative resolve strongly influences the outcome, but the fact that resolve is privately known (and unobservable) gives rise to public crises in the first place.
该定理的第二部分表明，当且仅当一个国家私下已知的决心水平 $w_i$$w_i$w_(i)w_{i} 足够大时，它才会选择升级到地平线然后发动攻击的策略。因此，模型中的危机会根据国家在问题上的斗争意愿将其区分开来：一个决心很大的国家会选择一种不屈不挠的危机谈判策略，从而可信地揭示其动机。这使它在危机和平结束时有更大的胜算，但代价是（不可避免的）更大的战争风险（参见班克斯，1990 年）。因此，对于外部观察者来说，危机的结果在很大程度上是不可预测的。比较决心在很大程度上影响着结果，但决心是私下可知的（无法观察到）这一事实首先导致了公共危机。
Lemma 2. In any equilibrium of $\mathrm{\Gamma }$$\mathrm{\Gamma }$Gamma\Gamma with ${\mathrm{t}}_{\mathrm{h}}$${\mathrm{t}}_{\mathrm{h}}$t_(h)\mathrm{t}_{\mathrm{h}} as the horizon and in which escalation may occur, (1) if state i chooses $\{\mathrm{t}$$\left\{\mathrm{t}\right.${t:}\left\{\mathrm{t}\right., attack}, it must be the case that $\mathrm{t}\ge {\mathrm{t}}_{\mathrm{h}}$$\mathrm{t}\ge {\mathrm{t}}_{\mathrm{h}}$t >= t_(h)\mathrm{t} \geq \mathrm{t}_{\mathrm{h}}; and (2) state i will choose $\{\mathrm{t}$$\left\{\mathrm{t}\right.${t:}\left\{\mathrm{t}\right., attack} where $\mathrm{t}\ge {\mathrm{t}}_{\mathrm{h}}$$\mathrm{t}\ge {\mathrm{t}}_{\mathrm{h}}$t >= t_(h)\mathrm{t} \geq \mathrm{t}_{\mathrm{h}} if ${\mathrm{w}}_{\mathrm{i}}>-{\mathrm{a}}_{\mathrm{i}}\left({\mathrm{t}}_{\mathrm{h}}\right)$${\mathrm{w}}_{\mathrm{i}}>-{\mathrm{a}}_{\mathrm{i}}\left({\mathrm{t}}_{\mathrm{h}}\right)$w_(i) > -a_(i)(t_(h))\mathrm{w}_{\mathrm{i}}>-\mathrm{a}_{\mathrm{i}}\left(\mathrm{t}_{\mathrm{h}}\right) and only if ${\mathrm{w}}_{\mathrm{i}}\ge -{\mathrm{a}}_{\mathrm{i}}\left({\mathrm{t}}_{\mathrm{h}}\right)($${\mathrm{w}}_{\mathrm{i}}\ge -{\mathrm{a}}_{\mathrm{i}}\left({\mathrm{t}}_{\mathrm{h}}\right)($w_(i) >= -a_(i)(t_(h))(\mathrm{w}_{\mathrm{i}} \geq-\mathrm{a}_{\mathrm{i}}\left(\mathrm{t}_{\mathrm{h}}\right)( for $\mathrm{i}=1,2)$$\mathrm{i}=1,2)$i=1,2)\mathrm{i}=1,2).
定理 2：在以 ${\mathrm{t}}_{\mathrm{h}}$${\mathrm{t}}_{\mathrm{h}}$t_(h)\mathrm{t}_{\mathrm{h}} 为地平线的 $\mathrm{\Gamma }$$\mathrm{\Gamma }$Gamma\Gamma 的任何均衡中，都可能发生升级：（1）如果国家 i 选择 $\{\mathrm{t}$$\left\{\mathrm{t}\right.${t:}\left\{\mathrm{t}\right. ，攻击}，则一定是 $\mathrm{t}\ge {\mathrm{t}}_{\mathrm{h}}$$\mathrm{t}\ge {\mathrm{t}}_{\mathrm{h}}$t >= t_(h)\mathrm{t} \geq \mathrm{t}_{\mathrm{h}} ；（2）国家 i 将选择 $\{\mathrm{t}$$\left\{\mathrm{t}\right.${t:}\left\{\mathrm{t}\right. ，攻击}，其中 $\mathrm{t}\ge {\mathrm{t}}_{\mathrm{h}}$$\mathrm{t}\ge {\mathrm{t}}_{\mathrm{h}}$t >= t_(h)\mathrm{t} \geq \mathrm{t}_{\mathrm{h}} 如果 ${\mathrm{w}}_{\mathrm{i}}>-{\mathrm{a}}_{\mathrm{i}}\left({\mathrm{t}}_{\mathrm{h}}\right)$${\mathrm{w}}_{\mathrm{i}}>-{\mathrm{a}}_{\mathrm{i}}\left({\mathrm{t}}_{\mathrm{h}}\right)$w_(i) > -a_(i)(t_(h))\mathrm{w}_{\mathrm{i}}>-\mathrm{a}_{\mathrm{i}}\left(\mathrm{t}_{\mathrm{h}}\right) ，并且只有 ${\mathrm{w}}_{\mathrm{i}}\ge -{\mathrm{a}}_{\mathrm{i}}\left({\mathrm{t}}_{\mathrm{h}}\right)($${\mathrm{w}}_{\mathrm{i}}\ge -{\mathrm{a}}_{\mathrm{i}}\left({\mathrm{t}}_{\mathrm{h}}\right)($w_(i) >= -a_(i)(t_(h))(\mathrm{w}_{\mathrm{i}} \geq-\mathrm{a}_{\mathrm{i}}\left(\mathrm{t}_{\mathrm{h}}\right)( 对于 $\mathrm{i}=1,2)$$\mathrm{i}=1,2)$i=1,2)\mathrm{i}=1,2) 。
Lemma 2 implies that the ex ante (precrisis) probability that state $i$$i$ii will choose a strategy involving attack is $1-F_i(-a_i\left(t_h\right))$$1-F_i\left(-a_i\left(t_h\right)\right)$1-F_(i)(-a_(i)(t_(h)))1-F_{i}\left(-a_{i}\left(t_{h}\right)\right), the prior probability, that $w_i\ge$$w_i\ge$w_(i) >=w_{i} \geq $-a_i\left(t_h\right)$$-a_i\left(t_h\right)$-a_(i)(t_(h))-a_{i}\left(t_{h}\right). Thus we can write state $i$$i$ii 's ex ante expected utility for escalating up to a horizon time $t_h$$t_h$t_(h)t_{h} and then backing down as $u_i\left(t_h\right)=F_j(-a_j\left(t_h\right))v+(1-$$u_i\left(t_h\right)=F_j\left(-a_j\left(t_h\right)\right)v+(1-$u_(i)(t_(h))=F_(j)(-a_(j)(t_(h)))v+(1-u_{i}\left(t_{h}\right)=F_{j}\left(-a_{j}\left(t_{h}\right)\right) v+(1- $F_j(-a_j\left(t_h\right)))(-a_i\left(t_h\right)){}^{19}$$\left.F_j\left(-a_j\left(t_h\right)\right)\right)\left(-a_i\left(t_h\right)\right){}^{19}${:F_(j)(-a_(j)(t_(h))))(-a_(i)(t_(h)))^(19)\left.F_{j}\left(-a_{j}\left(t_{h}\right)\right)\right)\left(-a_{i}\left(t_{h}\right)\right){ }^{19} The function $u_i(\cdot )$$u_i(\cdot )$u_(i)(*)u_{i}(\cdot) proves to play an important role in defining equilibrium strategies and establishing uniqueness. It is easily shown that $u_i(t)$$u_i(t)$u_(i)(t)u_{i}(t) is continuous and strictly decreasing and that if audience costs increase “enough” with $t$$t$tt there is a unique level of escalation $t_i^{\ast }>0$$t_i^{\ast }>0$t_(i)^(**) > 0t_{i}^{*}>0 such that $u_i\left(t_i^{\ast }\right)=0$$u_i\left(t_i^{\ast }\right)=0$u_(i)(t_(i)^(**))=0u_{i}\left(t_{i}^{*}\right)=0. Loosely, if the opponent $j$$j$jj can generate sufficient audience costs by escalating a crisis, there will be a unique level of escalation such that state $i$$i$ii would be indifferent between backing down at $t=0$$t=0$t=0t=0 and at $t_i^{\ast }$$t_i^{\ast }$t_(i)^(**)t_{i}^{*}, were this the horizon. I assume in what follows that the states are able to use escalation to generate audience costs at least this large. ${}^{20}$${}^{20}$^(20){ }^{20}
谚语 2 意味着国家 $i$$i$ii 选择攻击策略的事前（危机前）概率为 $1-F_i(-a_i\left(t_h\right))$$1-F_i\left(-a_i\left(t_h\right)\right)$1-F_(i)(-a_(i)(t_(h)))1-F_{i}\left(-a_{i}\left(t_{h}\right)\right)