Military Power, the Nuclear Revolution, and Brinkmanship

The balance of military power, much less the trade-off between power and risk, plays virtually no role in nuclear brinkmanship. Indeed, the essence of the nuclear revolution is often thought to be that it transformed contests of relative military strength into contests of resolve.Footnote ¹² Crises become a competition in risk taking in which resolve and a willingness to run risks are more important in determining the outcome than the balance of military power is.

During the late 1950s and early 1960s, strategists and policy-makers anticipated the arrival of a technological condition of mutual assured destruction (MAD) in which both the United States and the Soviet Union could launch a devastating nuclear second strike even after absorbing a massive nuclear first strike.Footnote ¹³ Secure, second-strike forces would render defense impossible because neither state could physically protect itself from an attack. It was the advent of MAD and not simply the development of nuclear weapons that marked the nuclear revolution.Footnote ¹⁴

A state's assured-destruction capability gives it the ability to impose costs on an adversary that outweigh any possible gains the adversary might hope to achieve. If, therefore, a state could make its threat to impose these costs sufficiently credible, an adversary would prefer backing down to continuing the conflict. In particular, a state that was losing a contest of military strength on the battlefield could compel an adversary to back down if the losing state could make its threat to impose these costs sufficiently credible. Thus the ability to exert coercive pressure would seem to turn on the credibility of the threat, not the balance of military power. In this way, crises become contests of resolve rather than contests of relative military strength.

But how can a state make the threat to impose a sanction credible when carrying it out would subsequently result in its own destruction? Given that both states have second-strike capabilities and can therefore make the costs outweigh any gains, why would either state be able to exert more coercive pressure on its adversary than its adversary could exert on it? Why do these capabilities not simply cancel each other out?

Schelling's notion of brinkmanship provided an explanation.Footnote ¹⁵ Even if a state cannot credibly threaten to deliberately launch an all-out nuclear attack, it can credibly make “threats that leave something to chance.” That is, a state may be able to credibly threaten and actually engage in a process—a crisis or a limited war—that raises the risk that events will go out of control and end in a catastrophic nuclear exchange. How much risk a state could credibly threaten to run would depend on what was at stake in the political conflict. The higher the stakes, the more risk a state would be willing to run.

In a brinkmanship crisis, states exert coercive pressure on each other by taking steps that raise the risk that events will go out of control. This is a real and shared risk that the confrontation will end in a catastrophic nuclear exchange. States do not bid up the risk eagerly or enthusiastically. Rather, a state faces a series of terrible choices throughout the conflict. A state can back down or it can decide to hang on a little longer and accept a somewhat greater risk in the hope that its adversary will find the situation too dangerous and back down. If neither state backs down, the crisis continues with each state bidding up the risk until one state eventually finds the risk too high and backs down or until events actually do spiral out of control.

It is important to be clear about what precisely is at risk in “threats that leave something to chance.” Most narrowly, one can think of there being what Snyder and Diesing call an “autonomous” risk that the crisis will end in a large, counter-value, nuclear exchange without any national authority ordering such an attack.Footnote ¹⁶ This is the most literal reading of the nature of the risk in Schelling's description of brinkmanship in which two actors are tied together by a rope and standing near a brink. While neither can credibly threaten to push the other over the brink deliberately, “loose gravel, gusty winds, and a propensity toward dizziness” make it possible for “one to credibility threaten to fall off accidentally.”Footnote ¹⁷ It is also the nature of the risk in Schelling's use of a roll of a die to impose the disastrous outcome in other illustrations of brinkmanship.Footnote ¹⁸

Another somewhat broader interpretation of what is at risk is also possible. Suppose there are situations that are so dire that the states are sure to escalate and thereby incur very high costs.Footnote ¹⁹ These costs could come through a process of deliberate decisions, inadvertence, or accident.Footnote ²⁰ Because the expected costs of getting into these situations are so high for both states, neither state can credibly threaten to deliberately put itself or its adversary in them. For example, no state can credibly threaten to deliberately cross another state's nuclear-use threshold if the states believe the crisis is sure to spiral out of control once this threshold has been crossed. However, the states still can make threats that leave something to chance. What is left to chance in this broader interpretation is the prospect of getting into one of these dire situations. For example, a state might inadvertently cross another's nuclear threshold because of uncertainty about that threshold.Footnote ²¹

Military forces do play a role in brinkmanship, but the balance of military power does not. Military actions provide a way of generating the risk that events will go out of control. As Schelling framed the issue,

According to Jervis,

In other words, military forces are to be judged primarily in terms of the risks these forces generate when used and the coercive pressure inherent in that risk. A weak force is effectively as good as a strong force as long as both are equally good at generating the risk that the crisis will go out of control.

In brief, there is no trade-off between power and risk. Deciding to take some steps and not others because of the risk of escalation is an essential element of brinkmanship. But the central focus is on the level of risk each step generates, not on any possible effect that these steps may also have on the probability of prevailing if events do not go out of control.

The risk of nuclear escalation is also at the heart of the stability-instability paradox that has framed much of the discussion of nuclear proliferation's effects on the likelihood of war in South Asia. Kapur summarizes the basic logic:

That is, the less likely a conventional war is to escalate to a nuclear war, the lower the expected cost of launching a conventional war and the more likely states are to start them.

It will be useful in what follows to restate the paradox somewhat more precisely. Let ε be the probability that a conventional war escalates to the nuclear level, and take κ to be the probability of a conventional war.Footnote ²⁵ Then the stability-instability paradox asserts that a decrease in ε leads to an increase in κ.Footnote ²⁶

Note that the risk of nuclear escalation is completely endogenous in brinkmanship. The states choose how much risk to run and, as Ganguly and Wagner point out, brinkmanship models typically assume that the states are physically able to generate as high a risk as they want.Footnote ²⁷ Because the more resolute state is always able to outbid the other state, the balance of military power does not matter in the preceding discussion of brinkmanship. A weak but resolute state that was losing a contest of strength could always outbid its adversary in a contest of resolve.

By contrast, the simplest formulation of the stability-instability paradox takes the risk of nuclear escalation ε to be a completely exogenous constant. The risk of nuclear escalation is not something the states can manipulate to exert coercive pressure on each other. It is simply a risk that has to be accepted if a state starts a conventional war.

Taking the risk of nuclear escalation to be completely exogenous is a very strong assumption. More generally, it reflects the basic idea that the risk of nuclear escalation depends at least in part on something exogenous—on something beyond the control of the states, at least in the short run of a crisis. More concretely, the risk of escalation is determined at least partially by the characteristics of the states' nuclear forces, postures, and doctrines, or by the actions taken during a conflict, for example, a policy of launching on warning. Indeed, an exogenous influence on the risk of escalation is the basis of arms control that seeks to reduce these sources of risk.

The model developed here draws from both of these approaches to risk. The trade-off between power and risk is exogenous (at least during the time frame of the model) and affects the maximum amount of risk the states can generate. The actual amount of risk the states run is determined endogenously and depends on the stakes.

A Model

The model formalizes the trade-off between military power and the risk of escalation in a very simple, reduced-form game. A challenger, C, is considering the use of force to achieve an objective. To fix ideas, suppose the challenger is deciding whether to try to take some territory from a defender, D. The more power the challenger brings to bear, the higher the probability of taking the territory if the conflict does not escalate to an unlimited nuclear war. But bringing more power to bear also increases the potential risk of escalation and widens the scope for transforming the contest of strength into one of resolve. After the challenger decides how much power to bring to bear, the defender decides whether to exploit this potential and, if so, how much risk to generate.Footnote ²⁸ This discussion assumes there is complete information.

The game is illustrated in Figure 1. The challenger begins by accepting or challenging the status quo. The game ends if there is no challenge, and the defender retains control of the territory worth v _D > 0. The challenger's payoff is normalized to zero.

Figure 1.

A model of nuclear brinkmanship and military power.

The challenger challenges the status quo by bringing military power to bear and starting to fight. Formally, the challenger chooses $p \in \lpar 0\comma \; \bar p\rsqb $ where p is the probability that the challenger prevails if the crisis remains under control and does not escalate to an unlimited nuclear exchange.Footnote ²⁹ The defender prevails with probability 1–p. The higher p, the more military might the challenger is bringing to bear and the better off it will be as long as the crisis remains a contest of military strength. The upper bound $\bar p$ is the challenger's probability of prevailing if it brings all of its military power to bear. Fighting (that is, choosing any p > 0) imposes costs k _C and k _D on the challenger and defender.

Bringing more power to bear (selecting a larger p) may include the “tactical” use of nuclear weapons, that is, the use of nuclear weapons against an adversary's military forces to defeat an attack. Examples include NATO's possible use of tactical nuclear strikes to disrupt a Soviet invasion or Pakistan's use of nuclear weapons on its own territory intended to blunt an Indian attack.Footnote ³⁰ Of course bringing more power to bear in this way may also substantially increase the potential risk of escalation to all-out war with nuclear strikes against cities.

Once the challenger has set p, the defender decides how much risk and coercive pressure to generate. Formally, the defender chooses the risk r that the crisis will go out of control if neither state subsequently backs down. This is a simple, reduced-form way of modelling the idea that the defender can respond to the challenger's actions in many different (unmodelled) ways that generate varying degrees of risk. For example, whether or not the Kremlin authorized the use of tactical nuclear weapons in Cuba had a large effect on the probability that an American invasion would ultimately escalate to an all-out war. Indeed, it was this effect that dissuaded the Kremlin from authorizing the use of tactical nuclear weapons.

The model's defining feature is that the amount of power the challenger brings to bear affects the stability of the conflict. More specifically, how much power the challenger brings to bear limits how much risk the defender can generate. If the challenger brings p to bear, the defender must choose a risk r between $\underline r \lpar p\rpar $ and $\bar r\lpar p\rpar $ with $\underline r \lpar p\rpar \lt \bar r\lpar p\rpar $ .

The upper bound $\bar r\lpar p\rpar $ is the potential risk inherent in bringing p to bear. If the defender chooses, it can bid the actual risk up to this level in an effort to coerce the challenger into backing down. The higher $\bar r\lpar p\rpar $ , the larger the potential risk and the less stable the conflict.

We assume that the more power the challenger brings to bear, the less stable the conflict and the larger the potential risk. This formalizes the trade-off I highlighted in the discussion of the Kargil War and the Cuban missile crisis. Actions that seemed to offer a higher probability of military success, like crossing the Line of Control, using air power in Kargil, or attacking the missiles in Cuba, also seemed to create a larger potential risk. More precisely, assume the potential risk $\bar r\lpar p\rpar $ is zero if the challenger exerts no power and increases at an increasing rate as the challenger brings more power to bear. In symbols, $\bar r\lpar 0\rpar = 0$ with $\bar r^{\prime}\lpar p\rpar \gt 0$ and $\bar r^{{\prime}{\prime}}\lpar p\rpar \gt 0$ (as long as $\bar r\lpar p\rpar \lt 1$ ).

The lower bound $\underline r \lpar p\rpar $ is the “existential” risk generated by bringing p to bear. That is, $\underline r \lpar p\rpar $ is the minimal or inescapable risk that fighting at p generates. Once the fighting starts, there is no avoiding this risk.Footnote ³¹ We assume this risk is zero at p = 0 and weakly increases at a weakly increasing rate: $\underline r \lpar 0\rpar = 0$ with $\underline r ^{\prime}\lpar p\rpar \ge 0$ and $\underline r ^{{\prime}{\prime}}\lpar p\rpar \ge 0$ as long as $\underline r \lpar p\rpar \lt 1$ . Taking $\underline r $ to be weakly increasing allows for the possibility that there is no existential risk, that is, $\underline r \lpar p\rpar = 0$ for all p. Finally, the scope for escalation, $\bar r\lpar p\rpar - \underline r \lpar p\rpar $ , is also assumed to increase at an increasing rate. A simple example satisfying these conditions is $\underline r \lpar p\rpar = \underline h {p^2}$ and $\bar r\lpar p\rpar = \bar h{p^2}$ with $\underline h \lt \bar h$ . Figure 2 illustrates the choice of p and r at the start of the game.Footnote ³²

Figure 2.

The risk functions

The risk functions can be seen as a generalization of the stability-instability paradox. If $\bar r\lpar p\rpar = \underline r \lpar p\rpar = \varepsilon $ for all $p \le \bar p$ , then fighting at any p generates a constant risk ε of nuclear escalation. Letting the potential and existential risks vary with p generalizes the stability-instability paradox in two ways. First, if we think of different values of p as representing different types or levels of conflict, then how states fight, or the level of violence at which they fight, affects the risk of all-out war. Second, letting the potential risk differ from the existential risk means that the amount of power brought to bear limits and influences the risk of escalation to general nuclear war, but it does not determine it. The actual level of risk the states run is endogenously determined as it is in brinkmanship.

Once the challenger has chosen p and the defender has chosen r, the challenger decides whether to quit or stand firm. The advantage of quitting is that the state only has to hazard the existential risk $\underline r \lpar p\rpar $ rather than the possibly higher risk r. The disadvantage is that the challenger gives up any chance of acquiring the territory whereas it would prevail with probability p if the states fight and the conflict does not escalate out of control. It follows that the challenger's expected payoff to quitting is $[1 - \underline r \lpar p\rpar ]\lpar 0 - {k_C}\rpar- \underline r \lpar p\rpar \lpar {k_C} + {n_C}\rpar $ where n _C is the additional cost of an all-out nuclear war if there is one.Footnote ³³ The defender keeps the territory if the challenger quits and events remain under control, and this leaves the defender with $[ 1 - \underline r \lpar p\rpar ]$ $\times\, \lpar {v_D} - {k_D}\rpar - \underline r \lpar p\rpar [{k_D} + {n_D}]$ .

If the challenger stands firm, the defender has to decide whether to quit. Quitting again trades off a lower risk of disaster for a higher probability of prevailing if the conflict remains under control. The defender's and challenger's respective payoffs are $[1 - \underline r \lpar p\rpar ]\lpar 0 - {k_D}\rpar - \underline r \lpar p\rpar \lpar {k_D} + {n_D}\rpar $ and $[ 1 - \underline r \lpar p\rpar ]\lpar {v_C} - {k_C}\rpar - \underline r \lpar p\rpar \lpar {n_C} + {k_C}\rpar $ .

If both states stand firm, the challenger prevails with probability p in the contest for the territory. That is, the challenger obtains $p\lpar {v_C} - {k_C}\rpar + \lpar 1 - p\rpar \lpar 0 - {k_C}\rpar = p{v_C} - {k_C}$ as long as the contest remains under control which it does with probability 1−r. As a result, the challenger's payoff if both states stand firm is $\lpar 1 - r\rpar \lpar p{v_C} - {k_C}\rpar- r\lpar {n_C} + {k_C}\rpar $ . The defender obtains $\lpar 1 - r\rpar [\lpar 1 - p\rpar \lpar {v_D} - {k_D}\rpar + p\lpar 0 - {k_D}\rpar ]- r\lpar {n_D} + {k_D}\rpar $ .Footnote ³⁴

The fundamental assumption and key point of departure for analysis is that the amount of power brought to bear affects the conflict's stability. This assumption is based most directly on the apparent trade-off decision makers have actually faced during nuclear crises. We can, however, also frame the trade-off in terms of Schelling's notions of “rocking the boat” and “fighting in a canoe.” Schelling sometimes used boat rocking as an example of a threat that leaves something to chance.

Now suppose the boat is a canoe and one of the two parties in the boat is physically stronger than the other. The trade-off between power and risk is inherent in Schelling's image of fighting in the canoe. “Limited war ... is like fighting in a canoe. A blow hard enough to hurt is in some danger of overturning the canoe. One may stand up to strike a better blow, but if the other yields it may not have been the harder blow that worried him.”Footnote ³⁶ That is, taking steps like standing up, which bring more power to bear and increase the probability that the stronger will prevail in the contest of physical strength, also raises the risk of capsizing. The risk functions formalize this trade-off.

To push the analogy further and foreshadow some of the results, consider three cases: calm, moderate, and heavy seas. If the seas are calm, neither party is able to rock the boat hard enough to generate much risk of tipping over even if the parties stand up. There is some risk, but it is not high enough to compel either to back down nor even to deter them from bringing their full power to bear by standing up. In these circumstances, the physically stronger party seems more likely to prevail. In moderate seas, the weaker party cannot rock the boat hard enough to generate much risk of capsizing as long as the stronger does not stand up. The stronger may still choose to fight, but may also decide not to stand up and thus not bring its full power to bear. In heavy seas, by contrast, both parties can rock the boat hard enough to make capsizing likely. The balance of physical strength is now largely irrelevant, and the side willing to run the greatest risk will prevail—unless, of course, the boat flips over and both drown.

However rough the seas, bringing more power to bear by standing up raises the risk of tipping over. But the intensity of the trade-off between power and risk varies with the seas. If the seas are calm, standing up does not raise the risk much. If $\bar r\lpar p\rpar = \bar h{p^2}$ , calm seas correspond to a low $\bar h$ . If the seas are moderate, $\bar h$ is larger, the trade-off is sharper, and standing up is riskier. When the seas are heavy, $\bar h$ is still larger and standing up generates even more risk.Footnote ³⁷

The Brinkmanship Subgame

The first step in characterizing the subgame perfect equilibria of the game is to analyze play starting with the defender's choice of r given p. The analysis shows that the defender has no incentive to engage in a contest of resolve when the balance of resolve favors the challenger. Rather, the defender minimizes the risk of escalation by setting r to the lowest level, $r = \underline r \lpar p\rpar $ . By contrast, the defender transforms the contest of strength into one of resolve in which it is sure to prevail when the balance of resolve favors the defender and the potential for raising the risk $\bar r\lpar p\rpar $ is high enough. That is, the defender generates a level of risk r it is willing to run but the challenger is not. This risk compels the challenger to back down.

The states' levels of resolve are critical to solving the game. Define a state's resolve to be the maximum risk it would be willing to run in order to win the “prize” of fighting for the territory given that the challenger prevails in this fight with probability p. Formally, consider the last node in the tree where the defender chooses between quitting or standing firm and continuing to fight. It stands firm when the payoff to doing so is at least as large as the payoff to quitting: $\lpar 1 - r\rpar [\lpar 1 - p\rpar {v_D} - {k_D}]$ $- r[{k_D} + {n_D}] \ge [1 - \underline r \lpar p\rpar ]\lpar - {k_D}\rpar - \underline r \lpar p\rpar \lpar {k_D} + {n_D}\rpar $ or

$$r \le \displaystyle{{\lpar 1 - p\rpar {v_D} + \underline r \lpar p\rpar {n_D}} \over {\lpar 1 - p\rpar {v_D} + {n_D}}} \equiv {R_D}\lpar p\rpar$$

where R _D (p) is defined to be the defender's resolve at p.

As for the challenger's resolve, suppose the challenger is choosing between standing firm and quitting, given that the defender will subsequently stand firm. Standing firm in these circumstances yields $\lpar 1 - r\rpar \lpar p{v_C} - {k_C}\rpar - r\lpar {k_C} + {n_C}\rpar $ while quitting brings $ - {k_C} - \underline r \lpar p\rpar {n_C}$ . Hence, the challenger prefers standing firm whenever $\lpar 1 - r\rpar \lpar p{v_C} - {k_C}\rpar - r\lpar {k_C} + {n_C}\rpar \ge - {k_C} - \underline r \lpar p\rpar {n_C}$ or

$$r \le \displaystyle{{\,p{v_C} + \underline r \lpar p\rpar {n_C}} \over {\,p{v_C} + {n_C}}} \equiv {R_C}\lpar p\rpar$$

where R _C (p) is the challenger's resolve at p.

A crucial consequence of the trade-off between power and risk is that the states' levels of resolve vary with p. As illustrated in Figure 3, the challenger's resolve is increasing in p and the defender's resolve, subject to some qualifications, is decreasing. The intuition here is that the value of the prize of fighting at p increases for the challenger and decreases for the defender as p increases. As a result, the challenger is willing to run a higher risk in order to win this prize whereas the defender is willing to run a lower risk only.Footnote ³⁸

Figure 3.

Resolve and risk

Turning to the subgame perfect equilibrium of the brinkmanship subgame, observe first that the defender never bluffs by bidding more than its resolve, that is, the defender never chooses $r \gt {R_D}\lpar p\rpar $ , when there is complete information. To see why, suppose that it did. As long as there is complete information, the challenger knows that this is a bluff. Reasoning backwards from the end of the tree, the challenger knows that the defender is sure to quit when $r \gt {R_D}\lpar p\rpar $ . As a result, the challenger stands firm.

Given that the challenger stands firm and the defender quits when $r \gt {R_D}\lpar p\rpar $ , the defender's payoff to bidding any $r \gt {R_D}\lpar p\rpar $ is $ - {k_D} - \underline r \lpar p\rpar {n_D}$ . It follows that bidding $r \gt {R_D}\lpar p\rpar $ cannot be part of equilibrium play, since the defender is sure to do better by bidding $r = \underline r \lpar p\rpar $ and then standing firm. This brings $[1 - \underline r \lpar p\rpar ]{v_D} - {k_D} - \underline r \lpar p\rpar {n_D}$ if the challenger quits and $[1 - \underline r \lpar p\rpar ]\lpar 1 - p\rpar {v_D} - {k_D} - \underline r \lpar p\rpar {n_D}$ if the challenger stands firm. Both are strictly larger than the defender's payoff to bidding r. Hence, the defender never bids more than its resolve, and consequently $r \le {R_D}\lpar p\rpar $ in any equilibrium.

In light of this, there are three cases to consider: the defender is unwilling to outbid the challenger in a contest of resolve; the defender is both willing and able to outbid the challenger; and, finally, the defender is willing but unable to outbid the challenger. Suppose the defender is unwilling to outbid the challenger in a contest of resolve. That is, the balance of resolve favors the challenger: ${R_C}\lpar p\rpar \gt {R_D}\lpar p\rpar $ as at ${p_1}$ in Figure 3. This unfavorable balance implies that the challenger is willing to run any risk the defender is willing to run. In symbols, $r \le {R_D}\lpar p\rpar \lt {R_C}\lpar p\rpar $ . Because the defender is unwilling to bid the risk up high enough to compel the challenger to back down, generating any lesser risk only makes things more dangerous and raises the expected costs. As a result, the defender runs the minimal risk possible by setting $r = \underline r \lpar p\rpar $ .Footnote ³⁹

To specify the payoffs associated with this case, observe that both the challenger and defender stand firm after a bid of $r = \underline r \lpar p\rpar $ . Doing so entails no additional risk and offers at least some chance of prevailing if events remain under control. Thus, the states' payoffs at p when the balance of resolve favors the challenger are $[1 - \underline r \lpar p\rpar ]\lpar p{v_C} - {k_C}\rpar - \underline r \lpar p\rpar [{k_C} + {n_C}]$ and $[1 - \underline r \lpar p\rpar ][\lpar 1 - p\rpar {v_D} - {k_D}]- \underline r \lpar p\rpar [{k_D} + {n_D}] $ .

In the second case, the defender is both willing and able to outbid the challenger. That is, the balance of resolve favors the defender and the potential risk is high enough that the defender can generate more risk than the challenger is willing to run. In symbols, ${R_D}\lpar p\rpar \gt {R_C}\lpar p\rpar $ and $\bar r\lpar p\rpar \gt {R_C}\lpar p\rpar $ as, for example, at ${p_2}$ . In these circumstances, the defender maximizes its payoff by transforming the contest of strength into a contest of resolve which it is sure to win. It does this by bidding the risk up to a level that generates enough coercive pressure to compel the challenger to back down. Formally, the defender bids an r between ${R_C}\lpar p\rpar $ and $\min \lcub {R_D}\lpar p\rpar \comma \; \bar r\lpar p\rpar \rcub $ .

Subsequent play and payoffs following this bid are easy to determine. Since the risk is less than the defender's resolve, the defender is sure to stand firm if it has to choose between standing firm and quitting at the end of the game. This means that the challenger will have to run risk r if it stands firm. However, this risk exceeds the challenger's resolve, so it prefers to back down. Thus, bidding any r between ${R_C}\lpar p\rpar $ and $\min \lcub {R_D}\lpar p\rpar \comma \; \bar r\lpar p\rpar \rcub $ yields payoffs of $ - {k_C} - \underline r \lpar p\rpar {n_C}$ and ${v_D} - {k_D} - \underline r \lpar p\rpar \lpar {v_D} + {n_D}\rpar $ .

Finally, suppose the defender is willing but unable to outbid the challenger. More precisely, the balance of resolve favors the defender, but the situation is sufficiently stable that the challenger is willing to run the highest risk that can be generated: ${R_D}\lpar p\rpar \gt {R_C}\lpar p\rpar $ but ${R_C}\lpar p\rpar \gt \bar r\lpar p\rpar $ as at ${p_3}$ . In these circumstances, the defender, although more resolute than the challenger, is unable to compel the challenger to back down. Any risk the defender can run (that is, any $r \le \bar r\lpar p\rpar \rpar $ , the challenger is also willing to run. Thus, generating any risk above $\underline r \lpar p\rpar $ only makes things more dangerous and raises the expected costs. Hence, the defender runs the minimal risk possible by setting $r = \underline r \lpar p\rpar $ . Both states are willing to run this risk ( $r \le \bar r\lpar p\rpar \lt {R_C}\lpar p\rpar \lt {R_D}\lpar p\rpar $ ), so each subsequently stands firm. This yields payoffs of $p{v_C} - {k_C} - \underline r \lpar p\rpar [p{v_C} + {n_C}]$ and $\lpar 1 - p\rpar {v_D} - {k_D} - \underline r \lpar p\rpar [\lpar 1 - p\rpar {v_D} + {n_D}]$ .

Figure 4 summarizes these results, showing how the defender's bid varies with the power the challenger brings to bear. Lemma 1 states the results formally.

Proof: The lemma is a direct consequence of backwards induction.

Figure 4.

D's bid as a function of power p

A surprising implication of Lemma 1 is that the challenger may at times bring more rather than less power to bear in order to prevent the defender from transforming the contest of strength into one of resolve. Let $\widehat{p}$ denote the point at which ${R_C}\lpar p\rpar $ and ${R_D}\lpar p\rpar $ cross. If $\widehat{p}$ is less than the maximum amount of power the challenger can bring to bear, that is, if $\widehat{p} \lt \bar p$ as in Figure 4, then the defender is no longer willing to outbid the challenger when the challenger brings more than $\widehat{p}$ to bear and the risk the defender runs drops to the existential risk ${r^{ \ast} } = \underline r \lpar p\rpar $ . This gives the challenger an incentive to bring more power to bear. By contrast, the challenger cannot induce the defender to bid the minimal risk by bringing more power to bear when $\widehat{p} \gt \bar p$ as it is in Figure 5 when the defender's resolve is given by ${R_D}^{\prime}\lpar p\rpar $ .

Figure 5.

More and less resolute defenders

Complete-Information Equilibria

There are many different equilibria depending on the relationship assumed to hold among the parameters. We focus on two cases of substantive interest. Roughly, the balance of resolve favors the challenger in the first and the defender in the second. The key result is that when there is complete information and balance of resolve favors the defender, the defender's latent threat to transform the contest of strength into a test of resolve and then outbid the challenger induces the challenger to bring less power to bear. India, for example, limited its operations in Kargil to its side of the LoC, trading military advantage for greater stability.

Before formalizing the cases, it is useful to define $\widetilde{p}$ to be the level of power at which the maximal risk the challenger is willing to run equals the maximal risk that the defender can generate, that is, $\widetilde{p}$ satisfies ${R_C}\lpar \widetilde{p}\rpar = \bar r\lpar \widetilde{p}\rpar $ .Footnote ⁴¹ If the challenger brings less power to bear than $\widetilde{p}$ , the defender cannot outbid the challenger in a contest of resolve. The situation is sufficiently stable and the potential risk sufficiently low that the challenger is willing to run any risk the defender can generate: $p \lt \widetilde{p}$ implies $\bar r\lpar p\rpar \lt {R_C}\lpar p\rpar $ . By contrast, the defender can outbid the challenger when $p \gt \widetilde{p}$ .

To specify the first case in which the balance of resolve favors the challenger, suppose the stakes for the defender ${v_D}$ are sufficiently small that the defender is never both willing and able to outbid the challenger. This is the situation in Figure 5 when the defender's resolve is ${R_D}\lpar p\rpar $ . The defender is unable to outbid the challenger when $p \lt \widetilde{p}$ . When $p \gt \widetilde{p}$ , the defender can generate enough risk to outbid the challenger but is unwilling to do so (that is, ${R_D}\lpar p\rpar \lt {R_C}\lpar p\rpar $ ).

As Lemma 1 shows, the defender has no incentive to add to the existential risk and sets $r = \underline r \lpar p\rpar $ in these circumstances. This leaves the challenger with ${W_C}\lpar p\rpar = p{v_C} - {k_C} - \underline r \lpar p\rpar \lpar p{v_C} + {n_C}\rpar $ for $p \in \lpar 0\comma \; \bar p]$ . If the challenger decides not to use force (p = 0), it obtains ${W_C}\lpar 0\rpar \equiv 0$ . It follows that the challenger brings the unique level of power ${p^{ \ast} }$ to bear that maximizes this payoff where ${p^{ \ast} }$ balances the marginal gain of a higher probability of prevailing against the marginal cost of a higher existential risk of disaster. If there is no existential risk ( $\underline r \lpar p\rpar = 0$ for all p) and the balance of resolve favors the challenger, the challenger brings its full power to bear with $p = \bar p$ .Footnote ⁴²

Now consider the case in which the balance of resolve favors the defender. Assume more specifically the defender's stakes are so large that the balance of resolve favors the defender even if the challenger brings its full power to bear. That is, ${R_D}^{\prime}\lpar \bar p\rpar \gt {R_C}\lpar \bar p\rpar $ where ${R_D}^{\prime}$ in Figure 5 denotes the defender's resolve.Footnote ⁴³

To focus on the substantively interesting case, suppose further that ${p^{ \ast} }$ is larger than $\widetilde{p}$ . Were this not so, the challenger could not be outbid at its optimal level of power ${p^{ \ast} }$ and the balance of resolve would not matter. That is, ${p^{ \ast} } \le \widetilde{p}$ means $\overline r \lpar {p^{ \ast} }\rpar \le {R_C}\lpar {p^{ \ast} }\rpar $ and hence that the challenger cannot be outbid at ${p^{ \ast} }$ .

When the balance of resolve favors the defender with ${p^{ \ast} } \gt \widetilde{p}$ , the defender's latent threat to turn a contest of military strength into a contest of resolve induces the challenger to bring less power to bear. In particular, the challenger either foregoes the use of force (p = 0) or brings $\widetilde{p}$ rather than ${p^{ \ast} }$ to bear.

To verify that the challenger's equilibrium choice is either zero or $\widetilde{p}$ , note that if the challenger brings more power than $\widetilde{p}$ to bear, the defender is willing and able to outbid the challenger.Footnote ⁴⁴ According to Lemma 1, the challenger backs down in these circumstances and obtains ${W_C}\lpar p\rpar = - {k_C} - \underline r \lpar p\rpar {n_C}$ for $p \in \lpar \widetilde{p}\comma \; \bar p]$ . If, by contrast, the challenger brings $\widetilde{p}$ or less to bear, it cannot be outbid and the defender sets $r = \underline r \lpar p\rpar $ . This gives the challenger ${W_C}\lpar p\rpar = p{v_C} - {k_C} - \underline r \lpar p\rpar \lpar p{v_C} + {n_C}\rpar $ for $p \in \lpar 0\comma \; \widetilde{p}]$ . The maximum occurs at either 0 or $\widetilde{p}$ .Footnote ⁴⁵ If ${W_C}\lpar \widetilde{p}\rpar \gt 0$ , the challenger brings $\widetilde{p}$ to bear. If, however, the chances of prevailing $\widetilde{p}$ are sufficiently small, then ${W_C}\lpar \widetilde{p}\rpar \lt 0$ and the challenger foregoes the use of power (p = 0).

Proof: The proposition follows directly from Lemma 1 and backwards induction.

The higher the challenger's payoff to prevailing, the more risk it is willing to run and the more power it is willing to bring to bear ( $\partial \widetilde{p}/\partial {v_C} \gt 0$ and $\partial {p^{ \ast} }/\partial {v_C} \ge 0$ ). Conversely, the higher the costs of fighting or of escalation, the less power the challenger is willing to bring to bear ( $\partial {p^{ \ast} }/\partial {k_C} \lt 0$ , $\partial \widetilde{p}/\partial {n_C} \lt 0$ and $\partial {p^{ \ast} }/\partial {n_C} \le 0$ ).Footnote ⁴⁶

The Risk Functions and Nuclear Deterrence Theory

My analysis formalizes the stability of the strategic environment in a very simple and highly stylized way. Even so, the risk functions provide a framework for analyzing and clarifying important issues in nuclear deterrence theory and policy.

The stability-instability paradox revisited

The simplest version of the stability-instability paradox asserts that the more likely conventional conflict is to escalate to a nuclear conflict, the less likely states are to engage in conventional conflict. However, the implications of the paradox and the underlying mechanism are less clear when we allow for multiple levels of conflict—low, medium, and high-intensity conflict. Does greater instability at the strategic-nuclear level make violence at all lower levels less likely? Does it make conflict at some lower levels less likely and more likely at others? Snyder seems to suggest the former, “the greater the instability of the ‘strategic’ balance of terror, the lower the stability of the overall balance at its lower levels.”Footnote ⁴⁷ Jervis too suggests this. “To the extent that the military balance is stable at the level of all-out war, it will become less stable at lower levels of violence.”Footnote ⁴⁸

My analysis leads to a different conclusion. Increasing instability in the form of a sharper trade-off between power and risk induces the challenger to bring less power to bear. This makes conflict at higher levels of violence less likely but conflict at lower levels more likely.

To develop this linkage, note that the size or severity of the trade-off between power and risk can be seen as a measure of stability. That is, the sharper the trade-off between power and risk, the higher the risk associated with bringing a given amount of power to bear and the less stable the conflict. To fix ideas, assume again that $\bar r\lpar p\rpar = \bar h{p^2}$ and $\underline r \lpar p\rpar = \underline h {p^2}$ with $\underline h \lt \bar h \lt 1$ .Footnote ⁴⁹ The parameters $\bar h$ and $\underline h $ formalize the severity of the trade-off between power and risk. The larger $\bar h$ , the greater the potential risk of bringing p to bear ( $\partial \bar r/\partial \bar h \gt 0$ ) and the sharper the trade-off between power and risk ( ${\partial ^2}\bar r/\partial \bar h\partial p \gt 0$ ). Less formally, the larger $\bar h$ , the rougher the seas and the higher the risk of capsizing the weaker party can generate by rocking the boat. The larger $\underline h $ , the greater the existential risk ( $\partial \underline r /\partial \underline h \gt 0$ ) and the sharper the trade-off ( ${\partial ^2}\underline r /\partial \underline h \partial p \gt 0$ ). Figure 6 illustrates the effects of increasing $\overline h $ from ${\overline h _0}$ to a higher ${\overline h _1}$ .

Figure 6.

The effects of changes in the potential-risk function

We can now use Proposition 1 to link changes in stability to the levels of power brought to bear and the likelihood of conflict. According to Proposition 1, the challenger brings ${p^{ \ast} }$ to bear when the balance of resolve favors the challenger and $\widetilde{p}$ to bear when the balance of resolve favors the defender. Moreover, the levels of power $\widetilde{p}$ and ${p^{ \ast} }$ depend on the properties of the risk functions and especially on the trade-off between power and risk. The sharper the trade-off, the lower $\widetilde{p}$ and ${p^{ \ast} }$ and the less power the challenger brings to bear. As Figure 6 indicates, an increase in $\bar h$ from ${\bar h_0}$ to a larger ${\bar h_1}$ lowers $\widetilde{p}$ from ${\widetilde{p}_0}$ to ${\widetilde{p}_1}$ .Footnote ⁵⁰

Thus a strong but less-resolute challenger brings less and less power to bear as the situation becomes less stable. As the trade-off between power and risk becomes sharper, the more resolute defender's latent threat to transform the contest of strength into a test of resolve “binds” at a lower level of violence, that is, ${R_C}\lpar p\rpar = \bar r\lpar p\rpar $ at a lower p. This in turn lowers the challenger's payoff to challenging the status quo ( $\partial {W_C}\lpar \widetilde{p}\rpar /\partial \widetilde{p} \lt 0$ ).Footnote ⁵¹ Indeed, if the level of instability is high enough, the chances of prevailing at $\widetilde{p}$ are too low to make the use of force worthwhile and there is no challenge. That is, ${W_C}\lpar \widetilde{p}\rpar \lt 0$ when $\bar h$ is high enough. In sum, the framework developed here leads to the following stability-instability paradox: greater strategic instability leads to less power being brought to bear and thus less large-scale (high p) conflict, often more lower-level conflict (that is, when ${W_C}\lpar \widetilde{p}\rpar \gt 0$ ), and sometimes no conflict at all (when ${W_C}\lpar \widetilde{p}\rpar \lt 0$ ).

This is the essence of what Kapur argues has happened in South Asia.Footnote ⁵² A high degree of instability deters India from bringing its conventional superiority to bear against Pakistan. This in turn enables Pakistan to pursue low-level conflict which otherwise would be deterred by the threat of a large-scale Indian conventional retaliation. In this way, “the existence of a substantial degree of strategic instability has fueled lower-level violence”Footnote ⁵³

The (ir)relevance of the balance of military power

As observed earlier, a common argument during the Cold War was that the balance of military power between the Soviet Union and United States was relatively unimportant. Virtually any use of force would quickly become a contest of resolve.Footnote ⁵⁴ The model shows that whether this is the case depends on an important implicit assumption about stability and the trade-off between power and risk.

The military balance becomes irrelevant if bringing even a small amount of power to bear raises the potential risk to very high levels. Standing up to fight in these circumstances is too dangerous and the outcome will be determined by which party is willing to run the bigger risk by rocking the boat the hardest. More formally, when $\bar h$ is large, the states can bid the risk up to very high levels if they choose to do so, and the balance of resolve determines the outcome. Figure 7 illustrates this situation. Once $p \gt p^{\prime}$ , neither state is willing to bid the risk up to the highest level possible, that is, to $\bar r\lpar p\rpar $ . The potential risk is no longer a “binding constraint,” and the risks the states run are determined solely by their levels of resolve.Footnote ⁵⁵

Figure 7.

High potential risk and MAD during the Cold War

Put another way, suppose the large nuclear and conventional forces the United States and Soviet Union and their respective NATO and Warsaw Pact allies had during the Cold War, along with their command and control capabilities and the existing technology, combined to create a strategic environment in which the potential risk generated by any significant use of force was high. Then contests of strength would become contests of resolve. But it is important to emphasize that even if this assumption held to a reasonable degree between the superpowers during the Cold War, it may or may not hold in other cases. Those cases, such as India and Pakistan, China and the United States, the United States and a nuclear-armed “rogue,” must be evaluated in their own right.

Uncertainty about the Balance of Resolve

How does uncertainty about the balance of power affect the dynamics of escalation? More specifically, how does uncertainty affect the amount of power the challenger brings to bear? How likely is it that the defender will try to transform a contest of strength into one of resolve by bidding the risk up above the minimal level $\underline r $ and how high will the risk go?

When the balance of resolve is uncertain, the challenger faces a trade-off when bringing more than $\widetilde{p}$ to bear. The more power it brings to bear, the more likely it is to prevail if events remain under control. But bringing more power to bear also increases the challenger's resolve (recall that ${R_C}\lpar p\rpar $ is increasing in p). As a result, the possibly more resolute defender has to bid the risk of escalation up to a higher level in order to compel the challenger to back down. The optimal amount of power the challenger brings to bear in equilibrium balances this trade-off.

We focus on the case in which the challenger is unsure whether the balance of resolve is favorable or unfavorable. More precisely, the challenger believes it is facing one of two possible types of defender, a resolute defender $D^{\prime}$ or an irresolute defender D, whose respective payoffs to prevailing are ${v_D}^{\prime}$ and ${v_D}$ with ${v_D}^{\prime} \gt {v_D}$ . The probability of facing the “tougher” type $D^{\prime}$ is $\tau $ and the chances of facing D are $1 - \tau $ . Let ${R_D}^{\prime}\lpar p\rpar $ and ${R_D}\lpar p\rpar $ denote the resolve of $D^{\prime}$ and D.

Two assumptions formalize the idea that the resolute defender $D^{\prime}$ is more resolute than the challenger who in turn is more resolute than the less resolute defender. Assume (1) ${R_D}^{\prime}\lpar \bar p\rpar \gt \bar r\lpar \bar p\rpar \gt {R_C}\lpar \bar p\rpar $ and (2) ${R_C}\lpar \widetilde{p}\rpar \gt {R_D}\lpar \widetilde{p}\rpar $ . Both of these hold in Figure 5. Assumption (1) ensures that the resolute defender $D^{\prime}$ is always willing to outbid the challenger and is actually able to do so for some $p \lt \bar p$ .Footnote ⁸⁰ Assumption (2) implies that the irresolute defender is never both willing and able to outbid the challenger.Footnote ⁸¹

As in the complete-information game, the key to analyzing the game when there is uncertainty about the balance of resolve is characterizing the equilibrium of the brinkmanship game given p. When information is incomplete, this continuation game is effectively a signaling game between the defender who sends signal r and the challenger who decides whether to quit or stand firm after seeing r. This signaling game has an essentially unique equilibrium that satisfies two selection criteria.Footnote ⁸²

To describe this equilibrium, let $\rho \lpar p\rpar $ and $\rho ^{\prime}\lpar p\rpar $ denote the risks D and $D^{\prime}$ bid. Uncertainty about the defender's resolve does not matter if $p \lt \widetilde{p}$ . Regardless of its resolve, the defender cannot outbid the challenger because the potential risk $\bar r\lpar p\rpar $ is too low (that is, ${R_C}\lpar p\rpar \lt \bar r\lpar p\rpar $ ). Running a risk greater than $\underline r \lpar p\rpar $ therefore only adds to the danger and increases the expected costs without bringing enough coercive pressure to bear to compel the challenger to back down. As a result, the defender, regardless of its resolve, prefers to generate no additional risk. Formally, both D and $D^{\prime}$ run the minimal risk: $\rho \lpar p\rpar = \rho ^{\prime}\lpar p\rpar = \underline r \lpar p\rpar $ . This leaves the challenger with a payoff of ${U_C}\lpar p\rpar \equiv \lpar 1 - \underline r \rpar p{v_C} - {k_C} - \underline r {n_C}$ for $p \lt \widetilde{p}$ .

If the challenger brings more than $\widetilde{p}$ to bear, the defender can generate more risk than the challenger is willing to run. This results in a separating equilibrium. The resolute defender $D^{\prime}$ transforms the contest of strength into one of resolve by bidding a risk just high enough to compel the challenger to back down, namely $\rho ^{\prime}\lpar p\rpar = {R_C}\lpar p\rpar $ .Footnote ⁸³ The challenger stands firm with probability p after this bid. By contrast, the less resolute defender D runs the minimal risk by setting $\rho \lpar p\rpar = \underline r \lpar p\rpar $ , and both states subsequently stand firm.Footnote ⁸⁴ (See the appendix for the derivation of the equilibrium.)

To specify the challenger's payoff to $p \gt \widetilde{p}$ , observe that the challenger must be indifferent between quitting and standing firm since it randomizes over these two alternatives after a bid of R _C (p) by standing firm with probability p. If the challenger were not indifferent between standing firm and quitting, mixing could not be equilibrium behavior. Thus the challenger's payoff to standing firm also equals payoff to quitting which is $ - {k_C} - \underline r \lpar p\rpar {n_C}$ . As a result, the challenger is sure to get $ - {k_C} - \underline r \lpar p\rpar {n_C}$ when the defender bids R _C (p) after the challenger brings p to bear. It follows that the challenger's payoff to any $p \gt \widetilde{p}$ is

$$\eqalign{{U_C}{\rm (}p{\rm )} & = (1 - \tau )\left[ {{\rm (}1 - \underline{r} (\,p{\rm ))}[\,p{v_C} - {k_C}{\rm ]} - \underline{r} (\,p{\rm )}({k_C} + {n_C}{\rm )}} \right] \cr & \quad + \tau [ - {k_C} - \underline{r} (\,p{\rm )}{n_C}] \cr & = {\rm (}1 - \tau {\rm )}[1 - \underline{r} {\rm (}p{\rm )]}p{v_C} - {k_C} - \underline{r} {\rm (}p{\rm )}{n_C}.} $$

The first term on the right side of the first equation is the challenger's payoff to fighting at the minimal risk $\underline r \lpar p\rpar $ weighted by the probability of this bid which is the probability of facing the irresolute defender. The second term is the challenger's payoff given a bid of ${R_C}\lpar p\rpar $ weighted by the probability of this bid.

The challenger's equilibrium choice of p follows immediately. It chooses the unique p that maximizes the challenger's payoff U _C (p) over $0 \le p \le \overline p $ . Let ${p^{ \ast \ast }}$ denote this maximizer.Footnote ⁸⁵ If ${p^{ \ast \ast }} = 0$ , the challenger acquiesces to the status quo and foregoes the use of force. If ${p^{ \ast \ast }} \in \lpar 0\comma \; \widetilde{p}]$ , both D and $D^{\prime}$ run as little risk as possible by bidding $\underline r \lpar {p^{ \ast \ast }}\rpar $ . If the challenger brings ${p^{ \ast \ast }} \gt \widetilde{p}$ to bear, D again bids $\underline r \lpar {p^{ \ast \ast }}\rpar $ . By contrast, the resolute defender $D^{\prime}$ transforms the contest of strength into a test of resolve by bidding ${R_C}\lpar {p^{ \ast \ast }}\rpar $ . The challenger stands firm with probability ${p^{ \ast \ast }}$ after a bid of ${R_C}\lpar {p^{ \ast \ast }}\rpar $ , and $D^{\prime}$ stands firm. Hence the probability that a conflict becomes a contest of resolve is $\tau $ , and the probability of a contest of resolve in which both states stand firm and run risk ${R_C}\lpar {p^{ \ast \ast }}\rpar $ is $\tau {p^{ \ast \ast }}$ .

The more likely it is that the balance of resolve favors the defender (the higher $\tau $ ), the less power the challenger brings to bear ( $\partial {p^{ \ast \ast }}/\partial \tau \lt 0$ ). By contrast, the higher the stakes for the challenger, the more power it brings to bear. That is, ${p^{ \ast \ast }}$ is increasing in the challenger's payoff to prevailing, ${v_C}$ , and decreasing in the cost of disaster, ${n_C}$ . Higher stakes also make contests of resolve in which both sides stand firm more likely and more dangerous: $\tau {p^{ \ast \ast }}$ and $\rho ^{\prime}\lpar {p^{ \ast \ast }}\rpar = {R_C}\lpar {p^{ \ast \ast }}\rpar $ , are increasing in ${v_C}$ and decreasing in ${n_C}$ .Footnote ⁸⁶