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Assessing competing defense postures: the strategic implications of "flexible response."

The concept of flexible response, originally formulated by NATO, can be traced to the dissatisfaction of some strategic thinkers with the Eisenhower administration's New Look defense policy and its implementation in the doctrine of massive retaliation. Critics charged that New Look, stressing as it did "more bang for the buck," placed undue reliance on strategic weapons to deter Soviet aggression in Europe and elsewhere, leaving little room for maneuver during periods of acute crisis.(1) To avoid the stark choice of all-out nuclear war or capitulation, they proposed that United States conventional forces be strengthened and augmented with an arsenal of tactical nuclear weapons.

When the Democrats came to power in 1961, these changes were pursued under the policy labeled flexible response. After extensive debate and compromise, NATO formally adopted flexible response in 1967.(2) Today, even as it moves toward a new deployment policy reflecting recent dramatic changes in Eastern Europe and the former Soviet Union, NATO continues to maintain both a tactical and a strategic capability.

This is not to suggest that flexible response is a well-articulated policy. In fact, Daalder argues that NATO deployments have been "deliberately ambiguous" in order to mask "differences among the allies concerning the role and relative weight to be accorded to theater nuclear forces in support of [its extended deterrence] strategy."(3) Consequently, there are a number of competing defense postures, all of which claim consistency with NATO'S loosely articulated declaratory policy.(4)

In this paper we model the strategic relationship implied when one state adopts a deployment policy--such as flexible response--that permits a range of credible responses to a probe or challenge. We then contrast this relationship with one that relies instead on strategic weapons and a more restricted set of response options associated with them. With this model we offer a new and explicit evaluation of rival flexible response policies, identifying when and how substrategic deployments make limited war possible and total war less--or more--likely. The goal ultimately is to gauge the policy implications of various mixes of tactical and strategic weapons.

One might object that the demise of the Warsaw Pact, the disintegration of the Soviet Union, and the consequent reorientation of NATO itself render the present model a historical curiosity. We think not. Our model is not limited to nuclear situations, nor are any restrictions placed on player preferences that confine its empirical domain to the superpower relationship. In fact, as we argue below, the model is applicable to any situation, nuclear or otherwise, in which the players believe that certain response options are qualitatively different from others and that the choice of these options involves a serious escalation of the conflict. For instance, the model could apply to a relationship, such as Iran and Iraq's, in which chemical weapons play a role, or to a relationship like Greece and Turkey's, in which invasion is an enormous concern but only conventional weapons are realistically involved. Other examples abound.

I. ASSUMPTIONS AND PREVIOUS RESEARCH

To explore the strategic relationships implied by a flexible response deployment policy, we begin,with the Asymmetric Escalation Game, a generic asymmetric two-stage escalation model, shown in Figure 1. In so doing, we assume that a status quo outcome (cc) exists, and that one player, Challenger (Ch), must decide (at Node 1) whether to cooperate (C) and accept it, taking no aggressive action, or to defect (D) and attempt to overturn it.(5) In this model Challenger could defect by precipitating a crisis, by launching a conventional military attack, or by taking some aggressive action other than a direct strategic nuclear assault.(6) Note that it is precisely this broad range of substrategic challenges that flexible response options are designed to prevent.

[Figure 1 ILLUSTRATION OMITTED]

If Challenger supports the status quo, the game ends at cc and the payoffs to Challenger and the other player, Defender (Def), are ([c.sub.CC],[d.sub.CC]), respectively. (The notation for the players' payoffs at the other outcomes is similar.)

But if Challenger chooses to defect, Defender must select (at Node 2) one of three alternatives: concede (C) to Challenger's demand by doing nothing; defy (D) Challenger by responding in kind; or escalate (E) the conflict. Defender's choice of c ends the game (at outcome DC), while the choice of D or E leads to a second move for Challenger (Nodes 3a or 3b, respectively): either concede (C) by sticking with its prior action choice, thereby engaging Defender at a restrained level of conflict (outcome DD or DE), or escalate. If at the previous move Defender had selected D, escalation by Challenger provides Defender with an additional opportunity to concede (outcome ED) or (counter-) escalate (Node 4). As Figure 1 shows, the game ends if there is no challenge, or if some player backs down by choosing C, or as soon as both players escalate (outcome EE).(7)

Note that the Asymmetric Escalation Game provides an escalation model that applies to any situation in which the defender may decide to respond by crossing a threshold, thereby inducing a (psychologically) distinct level of conflict.(8) For example, the lower level of conflict could be thought of as conventional, and the higher as nuclear; or the distinction could rest on a perceived difference between tactical (theater) and strategic nuclear weapons; or there could be some other mutually understood boundary. The model applies whenever the two actors concur that there exists a saliency (in the sense of Schelling)(9) and that crossing this barrier--whether real or psychological--represents a serious escalation of the conflict. Like NATO'S description and implementation of its flexible response deployment, then, the model is "deliberately ambiguous" about the nature of Defender's limited-response option.

In this context, it is important to point out that the sequence of choices we postulate can lead to two distinct symmetric conflict outcomes--limited conflict at outcome DD, and all-out conflict at outcome EE. This possibility distinguishes the Asymmetric Escalation Game from other game-theoretic models of interstate escalation, and it enables us to offer a more realistic assessment of the conditions associated with successful extended deterrence.(10) Our objective is to gain insight into the precise set of circumstances in which substrategic deployments make limited war possible and total war either more or less likely. Additional levels of deployment are possible but might reduce tractability without substantially improving verisimilitude, in view of the severely limited vocabulary of credible signals available to states.(11) Note that the model is not meant to apply to deterrence relationships in which high-level initiations are salient.(12)

Like most other recent attempts to model the escalation process, we postulate players with incomplete information. In our model, the principal source of uncertainty is each player's lack of information about the other's relative preference between certain critical outcomes that we identify below. We relate this uncertainty to the credibility of each player's final stage escalation threat. This connection allows us to explore the relationship between threat credibility and the dynamics of escalation; it also distinguishes our model from those models that postulate players who know each other's preferences but are uncertain about the consequences of their actions.(13) This approach, we believe, bypasses the credibility problem associated with thermonuclear war because it assumes that all endgame escalation threats run counter to the interests of the players and are imposed probabilistically by "nature."(14) Rather than prejudging the question, our model permits analysis of the strategic implications of any configuration of credible threats. This is not to say that we hold that nature plays no role in the way conflicts evolve; rather, we assume that the risks associated with war and other conflict outcomes are reflected in the prayers' preferences. In other words, we ask what rational players do in the uncertain and risky environment characteristic of superpower crises.

To explore the strategic implications of a flexible response deployment policy, we make several assumptions about the preferences of the players. First, we assume that Challenger always strictly prefers DC to CC; that is, it prefers to initiate given that Defender does not respond. This assumption provides Challenger with an immediate incentive to upset the status quo.

Next, we assume that once conflict has been initiated, Defender prefers to respond in kind rather than capitulate (that is, prefers DD to DC). This assumption is consistent with the stated rationale of flexible response: to provide a defender with a credible substrategic response to a challenge.(15) Or as Helmut Schmidt put it in his argument for a strong conventional defense capability in Europe: NATO must... have troops and weapons on a scale ample to make non-nuclear aggression appear hopeless, and sufficient in an emergency to force one of two courses on the aggressor--to halt or to extend the conflict."(16) Note that it is precisely this choice that Challenger faces at Node 3a after Defender chooses to respond in kind at Node 2.

Similarly, we assume that Defender prefers to escalate (that is, prefers DE to DD and therefore to DC), provided that Challenger does not respond by also choosing E. To be sure, this is a strong assumption. We make it, however, to explore those situations in which the incentive to escalate is strongest. As well, this premise is implicit in a massive retaliation or a flexible response deployment policy: under massive retaliation, it is plainly required; likewise, flexible response presents no genuine choice of responses without it. Under flexible response, the critical question is which response option Defender would choose in light of Challenger's capability to counterescalate. We consider this question below.

To model aversion to the costs and risks of conflict, we assume that each player prefers to gain the upper hand or, if it must, lose the advantage at the lowest possible level of conflict. Thus, for instance, Challenger prefers payoff DC to ED, and so does Defender. We further as sume that the status quo (CC) is the highest ranked outcome for Defender, and second only to unilateral defection (DC) for Challenger.

Taken together, these assumptions restrict the players' utilities as follows:

Challenger: [c.sub.DC] > [c.sub.CC] > [c.sub.ED]

> [c.sub.DD] > [[c.sub.EE] and [c.sub.DE]]

Defender: [d.sub.CC] > [d.sub.DE] > [d.sub.DD]

> [d.sub.DC] > [[d.sub.EE] and [d.sub.ED]]

where "[is greater than]" means "is greater than." For now, we leave unspecified whether Challenger prefers EE or DE and whether Defender prefers EE or ED. Thus we make no fixed assumption about whether either player prefers ultimately to capitulate or to fight. This preference, we hold, depends on the stakes, the anticipated costs of war, and other factors. Our model, in fact, allows for two types of players: Hard players who prefer all-out war to capitulation at the final opportunity; and Soft players with the opposite preference.

Each player in our model is presumed to know its own and the other's utilities, except that the payoffs to Challenger and Defender at outcome EE ([C.sub.EE] and [D.sub.EE]) are binary random variables--denoted by upper-case letters--with known distributions; a player knows the realized value of only its own variable. More specifically, it is common knowledge that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This postulate affords each player probabilistic knowledge of the other's preference between capitulating and fighting at the highest level of conflict. In other words, in our model Defender {Challenger} will be seen by Challenger {Defender} to prefer EE to ED {DE} with probability 1-[p.sub.Def] {1- [p.sub.Ch]}. [p.sub.Def] {1 - [P.sub.Def]} and [p.sub.Ch] {1 - [p.sub.Ch]} are the probabilities that Defender and Challenger are perceived to be Hard {Soft}. Since these preferences determine whether the players can escalate rationally to the highest level of conflict, these probabilities can be taken to measure the credibilities of their threats to do so. The higher [p.sub.Ch] and [p.sub.Def], the more credible Challenger and Defender, respectively and conversely.(17)

II. ESCALATION AND INCOMPLETE INFORMATION

What are the effects of uncertainty on the escalation process when a defender's threat to respond in kind is inherently credible? What is the connection between the players' credibilities and the stability of the status quo when a defender adopts a flexible response deployment policy? How credible must each player's endgame threat be to deter escalation or retaliation? Under what conditions might a substrategic war be waged?

We address these questions with the Asymmetric Escalation Game (Figure 1), using backward induction to analyze Defender's choice at the last node (4). Node 4 is reached when Challenger upsets the status quo by choosing D, Defender responds in kind by also choosing D, and Challenger then escalates by choosing E. Defender's choice at Node 4 is easy to analyze, since Defender always knows its own preference between ED and EE and has no reason to conceal this preference. A Hard Defender, preferring EE to ED, always escalates, while a Soft Defender, with the opposite preference, concedes.

The same is true of Challenger's choice at the node reached (3b) after Challenger selects D and Defender escalates instead of responding in kind. If Challenger is Hard and prefers EE to DE, it will always counterescalate; if it is Soft, it will yield.

Because Challenger's and Defender's behavior at these nodes is strictly determined by their types, the only strategic decisions that require analysis are Challenger's Node 1 choice of C or D, its choice of c or E at Node 3a, and Defender's Node 2 choice of C, D, or E. Unlike the nodes already analyzed,,these decisions can depend on the decision maker's beliefs about the opponent. We denote the probabilities of these choices as follows:

[x.sub.H] = probability that a Hard Challenger initiates at Node 1

[x.sub.S] = probability that a Soft Challenger initiates at Node 1

[W.sub.H] = probability that a Hard Challenger escalates at Node 3a

[W.sub.S] = probability that a Soft Challenger escalates at Node 3a

[y.sub.H] = probability that a Hard Defender responds in kind at Node 2

[y.sub.S] = probability that a Soft Defender responds in kind at Node 2

[z.sub.H] = probability that a Hard Defender escalates at Node 2

[z.sub.S] = probability that a Soft Defender escalates at Node 2

In a game of complete information ([p.sub.Ch] and [p.sub.Def] equal to either 0 or 1), the (Nash) Equilibria of the two-stage escalation model can be described by a vector of probabilities describing action choices ([x.sub.H], [x.sub.S], [W.sub.H], [W.sub.S]; [y.sub.H], [y.sub.S], [z.sub.H], [z.sub.S]). In a game of incomplete information, the natural extension of a Nash Equilibrium is a Perfect Bayesian Equilibrium. A Perfect Bayesian Equilibrium consists of a complete plan of action for each player (that is, a strategy) plus a set of beliefs for each player about the opponent's type (in our case, Hardness or Softness) such that at each stage a player acts to maximize its expected utility given its beliefs about its opponent's type, and it updates its beliefs about its opponent's type rationally, in terms of actions it has observed.(18)

Table 1 summarizes the action choices associated with each Perfect Bayesian Equilibrium of the game of Figure 1. Here we simply provide an informal characterization; the equilibria are more completely specified in the appendix. To facilitate the subsequent discussion, these equilibria are grouped into four major categories: Deterrence Equilibrium, No-Response Equilibrium (NRE), No-Limited-Response Equilibrium (NLRE), and Limited-Response Equilibrium (LRE).

TABLE 1 PERFECT BAYESIAN EQUIBRIA(a)
                    Challenger
                    Initiates
               [x.sub.H]   [x.sub.S]
Deterrence
Equilibrium        0           0

No-Response
Equilibrium        1           1

Form 1
No-Limited-

Response
Equilibrium        1           1

Form II
No-Limited
Response
Equilibrium        1          (.)

Form III
No-Limited
Response
Equilibrium        1          (.)

No-First-Use
Equilibrium        1           1

Warfighting
Deterrence
Equilibrium        1           (.)

                    Challenger
                    Escalates
               [w.sub.H]   [w.sub.s]
Deterrence
Equilibrium        -           -

No-Response
Equilibrium        -           -

Form 1
No-Limited-

Response
Equilibrium        -           -

Form II
No-Limited
Response
Equilibrium        -           -

Form III
No-Limited
Response
Equilibrium        -           -

No-First-Use
Equilibrium       (.)          0

Warfighting
Deterrence
Equilibrium       (.)          0

                    Defender
                    Responds in Kind
               [y.sub.H]   [y.sub.S]
Deterrence
Equilibrium        -           -

No-Response
Equilibrium        0           0

Form 1
No-Limited-

Response
Equilibrium        0           0

Form II
No-Limited
Response
Equilibrium        0           0

Form III
No-Limited
Response
Equilibrium        0           0

No-First-Use
Equilibrium        1          (.)

Warfighting
Deterrence
Equilibrium       (.)         (.)

                    Defender
                    Escalates
               [z.sub.H]   [z.sub.S]
Deterrence
Equilibrium        -           -

No-Response
Equilibrium        0           O

Form 1
No-Limited-

Response
Equilibrium        1           0

Form II
No-Limited
Response
Equilibrium        1          (.)

Form III
No-Limited
Response
Equilibrium       (.)          0

No-First-Use
Equilibrium        0           0

Warfighting
Deterrence
Equilibrium       (.)          0




(a) "(.) = fixed value between 0 and 1; "--" = value not fixed, although some restrictions may apply.

DETERRENCE EQUILIBRIUM

A Deterrence Equilibrium of the two-stage escalation game is an equilibrium in which there is no initiation (that is, [x.sub.H] = [x.sub.S] = 0). Under a Deterrence Equilibrium, a Challenger--whether Hard or Soft-never defects and the status quo is never disturbed.

In the appendix the conditions under which a Deterrence Equilibrium (0, 0, [W.sub.H], [W.sub.S]; [y.sub.H], [y.sub.S], [z.sub.H], [z.sub.S]) can exist are discussed. The Deterrence Equilibrium does not depend upon initial beliefs about the credibility of either Defender's or Challenger's threat to wage all-out conflict. Rather, a Deterrence Equilibrium will exist either if Defender plans to escalate at Node 2 with a high probability (whatever its type and whatever its perception of Challenger's type) or if Challenger (whatever its type or perception of Defender's type) plans to escalate at Node 3 with a small probability and Defender plans to respond in kind at Node 2 with a high probability. In other words, a Deterrence Equilibrium is possible if [z.sub.H] and [z.sub.S] are large enough, or if [y.sub.H] and [y.sub.S] are large enough and [W.sub.H] and [W.sub.S] are small enough.

Because the Deterrence Equilibrium is independent of any (initial) beliefs that either the Challenger or the Defender might have about the other's type, some might consider the beliefs associated with it difficult to justify. Note also that these beliefs concern events that are "off the equilibrium path"; that is, the beliefs are never tested when a Deterrence Equilibrium is in play.

The (likely) contents of the beliefs associated with the Deterrence Equilibrium are consistent with the view of deterrence that Bundy calls existential deterrence, namely, the position that the very existence of nuclear weapons and the fear they almost surely instill in decision makers assures a stable international system and the continuation of what Gaddis calls "the Long Peace."(19) For example, it is consistent with the existence of a Deterrence Equilibrium for Challenger to decide not to defect (at Node 1), either because it believes that Defender will likely escalate (at Node 2), or because, expecting Defender to respond in kind (at Node 2) and fearing escalation at Node 4, it intends not to escalate at Node 3a.

It is noteworthy that these two belief sets correspond roughly to the two distinct deployment strategies Daalder associates with existential deterrence: pure deterrence and conventional deterrence.(20)

Pure deterrence is independent of any specific deployment strategy and indeed is inconsistent with the very notion of flexible response as it is commonly understood.(21) Pure deterrence theorists deny the existence of clear thresholds in war. In this view, any overt conflict makes escalation to the highest rung of the ladder almost inevitable (that is, [z.sub.H] and [z.sub.S] are high). It is precisely the inevitability of escalation that serves to deter any potential challenger. Consequently, the careful calibration of conventional and tactical nuclear forces to deter an attack is futile.

By contrast, advocates of a conventional deterrence deployment policy aim to avoid the instability associated with the "stability-instability paradox," namely, the increased potential for substrategic conflict implied by a strategic stalemate.(22) In this view, when strategic forces are balanced and hence mutually deterred (for example, when [w.sub.H] and [w.sub.S], and [z.sub.H] and [z.sub.S], are low), they are unavailable for deterring lower-level conflicts. Thus, advocates of conventional deterrence hold that

deterrence is enhanced by the prospects of a conventional defense capable

of denying the adversary the achievement of his objectives. The

conventional strategy therefore emphasizes a conventional response to

attack in the hope that nuclear escalation can be avoided. Extended

deterrence persists, however, by deploying some nuclear weapons in

Europe to pose the existential risk that war could escalate to all-out

nuclear war, thus coupling the stability of the mutual

U.S.-Soviet deterrence relationship to Europe.(23)

The existence of the Deterrence Equilibrium in our model verifies the connection between the premise and the conclusion of those deterrence theorists who view the postwar world order as unusually stable and robust;(24) it does not verify the premise, however. Nevertheless, the model indicates that an unconditionally stable deterrence relationship is possible, provided relevant decision makers somehow acquire either of the sets of beliefs associated with this equilibrium. In fact, such beliefs lead to an especially benevolent self-fulfilling prophecy.

NO-RESPONSE EQUILIBRIUM (NRE)

A No-Response Equilibrium is an equilibrium in which there is some possibility of initiation ([x.sub.H] + [x.sub.S] [is greater than] 0) but there is never any response ([y.sub.H] = [y.sub.S] = [z.sub.H] = [z.sub.S] = 0). As shown in the appendix, at any No-Response Equilibrium, Challenger always defects (that is, [x.sub.H] = [x.sub.S] = 1) and maintains a sufficiently credible threat to escalate should Defender resist by responding in kind (that is, [w.sub.H] and [w.sub.H] are large). Under this equilibrium, therefore, the status quo is never stable but (limited and unlimited) conflict and escalation are nonetheless precluded. Challenger always gets its way and the outcome is always DC. When a No-Response Equilibrium exists, Challenger attacks with impunity, as perhaps the Soviets did in Hungary in 1956, in Czechoslovakia in 1968, or in Afghanistan in 1979, and as the United States did in Grenada in 1983, in Libya in 1986, or in Panama in 1989.

Knowing the conditions under which a No-Response Equilibrium might exist provides additional insight into the dynamics of the Asymmetric Escalation Game. To understand these conditions, as well as those associated with the remaining equilibria of this two-stage escalation game, consider Figure 2. The horizontal axis of Figure 2 measures [P.sub.Ch], the a priori probability that Challenger's threat to escalate at the highest level is credible. Similarly, [p.sub.Def], the a priori probability that Defender is Hard and prefers all-out war to capitulation, is measured along the vertical axis. Several constants, which are defined and discussed in the appendix, are also indicated on these axes. These constants are thresholds that separate the various equilibria.

[Figure 2 ILLUSTRATION OMITTED]

The No-Response Equilibrium can be found only in the eastern region of Figure 2, where [p.sub.Ch] is large. Equilibria of this type exist, therefore, if and only if Challenger's credibility is high enough to deter Defender from escalating. (As well, Challenger's commitment to counterescalate [at Node 3a] must be strong enough to dissuade even a Hard Defender from responding in kind.) Because Defender is, in effect, completely deterred when a No-Response Equilibrium is in play, while Challenger is entirely undeterred, it should not be surprising that the possibility of a No-Response Equilibrium depends on Challenger's credibility being high but is unrelated to Defender's credibility.

The threshold value of Challenger credibility ([d.sub.2]) associated with the existence of a No-Response Equilibrium is defined in the appendix and discussed more fully elsewhere.(25) For now we simply note that [d.sub.2] moves to the right, thereby reducing the region where the No-Response Equilibrium exists, as Defender's payoff from immediate escalation increases or as its payoff from immediate capitulation decreases. As one might expect, the more value Defender attaches to unilateral escalation, or the lower its value for immediate capitulation, the higher must be Challenger's credibility to dissuade Defender from resisting.

The threshold [d.sub.2] also rises as the value a Hard Defender places on the all-out conflict outcome increases. In other words, as central war becomes less onerous to Defender, the higher must be Challenger's a priori credibility to induce a No-Response Equilibrium. By increasing the cost of conflict, then, Challenger makes it more likely that a No-Response Equilibrium exists, so that even a Hard Defender will capitulate. But if Defender is already Soft, such increases are devoid of strategic implications. In general, whenever Challenger can threaten an all-out war with even moderate credibility, a Soft Defender will not rationally resist.

NO-LIMITED-RESPONSE EQUILIBRIUM (NLRE)

A No-Limited-Response Equilibrium is an equilibrium in which there is some possibility of Challenger initiating (that is, [x.sub.H] + [x.sub.S] [is greater than] 0), some possibility of Defender escalating (that is, [z.sub.H] + [z.sub.H] [is greater than] 0), but no possibility that Defender will respond in kind when Challenger initiates a confrontation (that is, [y.sub.H] = [y.sub.S] = 0). For a No-Limited-Response Equilibrium to exist, Challenger's commitment (at Node 3a) to escalate first (that is, its values of [w.sub.H] and [w.sub.S]) must be large enough to deter any type of Defender from responding in kind (at Node 2) should Challenger dispute the status quo.

In the appendix it is shown that there are three distinct forms of No-Limited-Response Equilibria. When a No-Limited-Response Equilibrium is in play, Hard Challengers always defect (that is, [x.sub.H] = 1) and there is always some possibility that Soft Challengers will defect as well (that is, [x.sub.S] [is greater than] 0).(26) Since Defenders either escalate or do not respond at all, nothing like a limited conflict can emerge under any of the No-Limited-Response Equilibria.

FORM I NO-LIMITED-RESPONSE EQUILIBRIUM

Like the No-Response Equilibrium, the Form I No-Limited-Response Equilibrium involves certain initiation by Challenger (regardless of its type) and certain capitulation by a Soft Defender. When a Form I equilibrium is in play, Hard Defenders escalate with certainty (that is, [z.sub.H] = 1)- Soft Challengers then suffer a humiliating defeat, capitulating after Defender's harsh reaction, while Hard Challengers set in motion a process that culminates in all-out conflict.

As Figure 2 indicates, a Form I NLRE exists at intermediate levels of Challenger credibility, and at lower levels of Defender credibility. Unlike the No-Response Equilibrium, the Form I NLRE depends not only on Challenger's credibility, but also on Defender's. (See appendix for details.) Form I NLRE occur when the a priori probability that Challenger is Hard is high enough to deter a Soft Defender from retaliating, yet not so high that a Hard Defender capitulates (that is, [d.sub.1] [is less than] [p.sub.Ch] [is less than] [d.sub.2]). At the same time, Defender's credibility must be low enough that even a Soft Challenger would not hesitate before testing the water (that is, [p.sub.Def] [is less than or equal to] [c.sub.1]).(27)

FORMS II AND III NO-LIMITED-RESPONSE EQUILIBRIA

Like all equilibria of the Asymmetric Escalation Game other than the Deterrence Equilibrium, the two remaining No-Limited-Response Equilibria (Forms II and III) involve certain initiation by a Hard Challenger. Unlike the No-Response Equilibrium and the Form I NLRE, however, the Form II and Form III NLRE are associated with probabilistic (as opposed to certain) initiation by a Soft Challenger.

Under a Form II equilibrium, Hard Defenders always resist and Soft Defenders sometimes resist. But at a Form III equilibrium, only Hard Defenders resist, and only probabilistically at that. In each region, then, a Challenger unwilling to wage war might rationally precipitate a crisis; but only in region II might a reluctant Defender rationally call a Challenger's bluff.

Form II NLRE are to be found near the origin in Figure 2, at the lowest levels of Challenger and Defender credibility. It should therefore not be surprising that rational behavior in this region involves the possibility of bluffing by both players. While all-out wars can transpire anywhere save for the region of the No-Response Equilibrium, it is only in the area of Form II NLRE that a Soft Defender can find itself involved in a war it would prefer to avoid, for only in this region is it rational for a Soft Defender sometimes to defy Challenger by escalating first. In the unlikely event that Challenger happens to be Hard, the unthinkable will occur.

Hard Challengers always initiate when a Form II NLRE is in play, while Soft Challengers initiate with a certain probability. This probability increases steadily from 0 to 1 as one moves from left ([p.sub.Ch] = 0) to right ([p.sub.Ch] = [d.sub.1]) across the region. In other words, as Challenger's credibility rises, so does its tendency to test Defender's resolve, even when Challenger is in fact Soft. Likewise, a Soft Defender's optimal response strategy is to resist probabilistically. This probability, however, decreases steadily as one moves upward in the region, approaching zero at the upper boundary ([p.sub.Def] = [c.sub.1]). Thus a more credible Soft Defender resists less often under a Form II NLRE.

Like the Form II NLRE, the Form III NLRE also involves certain initiation by a Hard Challenger and probabilistic probing by a Soft Challenger. While this probability increases from 0 to 1 as Challenger's credibility rises from [p.sub.Ch] = 0 to [p.sub.Ch] = [d.sub.2], it increases more slowly than does the corresponding probability when a Form II NLRE is in play. In fact, equally credible Soft Challengers are more circumspect at a Form III NLRE than at a Form II NLRE. This is as it should be, since Defender's credibility is after all higher when a Form III NLRE exists, giving a Soft Challenger more reason to hesitate.

On the other hand, Defender's behavior is somewhat different under a Form III NLRE, relative to the Form II case. Specifically, under a Form III NLRE a Soft Defender never resists, and a Hard Defender resists only probabilistically. As with Soft Defenders in Form II, a more credible Hard Defender is less likely to resist under a Form III NLRE. The decrease in this tendency is a logical consequence of a Soft Challenger's willingness to make its initiation decision randomly at this equilibrium.

Taken together, the strategic options associated with the No-Response Equilibrium and the three No-Limited-Response Equilibria correspond roughly to the policy prescription Daalder terms escalatory deterrence. Proponents of escalatory deterrence hold that the threat of deliberate escalation, rather than any denial capability associated with the deployment of a potent conventional force, is the most efficacious way of deterring aggression. According to Daalder, "The escalatory-deterrence strategy, in recognizing the reality of certain thresholds in war, seeks to deter by posing the threat of unacceptable damage through the potential use of nuclear escalation. It thus extends the deterrent threat provided by mutual assured destruction to Europe by threatening to enlarge a conflict in Europe to general nuclear war."(28)

Like a pure deterrence deployment policy, then, escalatory deterrence relies on an opponent's fear of escalation to deter aggression. Advocates of pure deterrence contend that this fear is inherent in any contentious nuclear relationship. Thus, deterrence in Europe can be enhanced simply by coupling European security with American security. In this view, nuclear weapons should be deployed in a way that "ensures that escalation to general nuclear war is inherent in the very use of nuclear weapons."(29) By contrast, proponents of escalatory deterrence recommend a more deliberate tactical response to a challenge. Specifically, they counsel deemphasizing a conventional response and the early, if not the first, use of nuclear weapons in a confrontation.

The reason we associate the No-Response Equilibrium and the three No-Limited-Response Equilibria with an escalatory deterrence deployment policy is that none of these equilibria requires that Defender's threat to respond in kind to a challenge be credible. Moreover, unlike the Deterrence Equilibrium, which we link with pure deterrence, each of the escalatory equilibria requires that Defender prefer to escalate rather than accede to Challenger's demands for a change in the status quo (that is, Defender prefers outcome DE over outcome DC).(30)

Even a cursory examination of the behavioral characteristics of these equilibria reveals the limits of escalatory deterrence. Relying primarily on a threat to escalate is insufficient to deter Challenger if Challenger is Hard or if the game is played in the region of the No-Response Equilibrium or the Form I NLRE. Only when a Form II or III NLRE is in play does escalatory deterrence offer the possibility of a stable status quo. But even here the promise of stability is weak; in each of these two regions, deterrence works only when Challenger is Soft, and only probabilistically at that. Defender's best hope is to project high enough credibility ([p.sub.Def] [is greater than] [c.sub.1]) that with a less credible Challenger ([p.sub.Ch] [is less than] [d.sub.2]), a Form III NLRE evolves. Deterrence of a Soft Challenger is more likely in that region than anywhere else in Figure 2.

LIMITED-RESPONSE EQUILIBRIA (LRE)

As a group, the escalatory equilibria share one additional important characteristic: none admits the possibility of Defender responding in kind to a challenge (that is, [y.sub.H] + [y.sub.S] = 0). This means that after initiation Defender either escalates or does not respond at all. Clearly, when a No-Response Equilibrium or one of the No-Limited-Response Equilibria is in play, limited conflict (associated with the choice of D by both players) is rationally precluded.

Significantly, these four equilibria and the (pure) Deterrence Equilibrium are the only possible equilibria when Defender's first-stage threat is known to lack credibility, that is, when Defender prefers outright capitulation (outcome DC) to limited conflict (outcome DD).(31) In the Asymmetric Escalation Game, however, we assume that Defender prefers DD to DC and that this preference is known, making Defender's first-stage deterrent threat credible. Thus, the remaining equilibria of the Asymmetric Escalation Game arise as a consequence of this critical assumption about Defender's credibility. These additional patterns of behavior emerge when Defender's response options become unfettered.

Putting this differently, when Defender's only credible response is to escalate, substrategic conflict is not rationally possible. It is for this reason that, in another context, we associate these five equilibria with the policy of massive retaliation.(32) By contrast, the remaining Perfect Bayesian Equilibria of the Asymmetric Escalation Game admit the possibility of a nonescalatory (that is, D) response by Defender; hence, we call them Limited-Response Equilibria and associate them with deployment policies, such as, flexible response, that provide Defender with a credible, nonescalatory alternative for resisting a challenge.

It is important to stress, however, that for these equilibria to exist, Defender's threat to respond in kind must be credible; that is, Defender must prefer outcome DD to outcome DC. Since this preference is precisely what flexible response was designed to achieve, our model allows us to assess the immediate strategic consequences of this approach to extended deterrence.(33) The existence conditions associated with the Limited-Response Equilibria are particularly interesting, since they speak directly to the possibility of a constrained conflict or a limited war and to the viability of two additional extended deterrence postures, no-first-use and warfighting deterrence.

First proposed publicly by Robert McNamara in 1982, a no-first-use declaratory policy implies commitment to a nonnuclear defense against a nonnuclear attack, or, in terms of the model, to a response in kind.(34) Like the conventional deterrence policy associated with the Deterrence Equilibrium, no-first-use relies on nonescalatory responses to deter aggression. Nonetheless, there are important differences between these two extended deterrence policy stances. One subtle difference involves the point at which NATO would use nuclear weapons. Supporters of conventional deterrence, like former Secretary of Defense James Schlesinger, recommend that nuclear weapons be used "as late as possible" but "as early as necessary."(35) Under no-first-use, by contrast, nuclear weapons would be used only in response to a nuclear attack.

Another salient difference is rooted in the relationship between conventional and strategic options under no-first-use. According to Daalder, "The assumption [of conventional deterrence advocates] that escalation cannot be controlled does provide a coupling mechanism, if only an existential one."(36) By contrast, under a no-first-use deployment, these response options are decoupled. Thus proponents of no-first-use suggest a sufficiently capable conventional defense to deny an adversary victory in a limited, nonnuclear war. In their view, there is no role for tactical nuclear weapons; indeed, one of the benefits of a no-first-use deployment policy is that only survivable, second-strike nuclear weapons need be deployed. Or as Stromseth puts it, should this policy be implemented, "the ultimate reliance on nuclear weapons to shore up a failing conventional defense would be eliminated, and conventional forces would no longer function as a 'delayed trip-wire' for nuclear war.(37)

Warfighting deterrence, like no-first-use, does not deny the need for a potent conventional capability but rather stresses the need for a range of local options, including escalatory options, in order to deter aggression. A warfighting capability, then, does not rely solely on a credible substrategic response, as does no-first-use, or a capable strategic response, as does escalatory deterrence, or even an existential link between them, as do pure and conventional deterrence policies. Rather, this deployment policy depends upon maintaining credible response options at both the substrategic and the strategic level.

The key to warfighting deterrence, then, is escalation dominance, coupled with the ability to deny an adversary an advantage at any level of attack. According to Daalder, the strategy "relies on NATO's ability to dominate the escalation process up to the highest level of violence."(38) Thus, warfighting deterrence seeks to avoid war by denying an adversary an advantage at any level of outright conflict, even if it means escalating first.

LIMITED-RESPONSE EQUILIBRIA: GENERAL CHARACTERISTICS

At a Limited-Response Equilibrium there is some possibility of a challenge and a nonescalatory response (that is, [x.sub.H] + [x.sub.S] [is greater than] 0 and [y.sub.H] + [y.sub.S] [is greater than] 0). As we indicate below, it is possible but unlikely for the status quo to remain stable under this equilibrium. Moreover, the genuine possibility of limited conflict does not rule out higher levels--unilateral escalation and all-out war. Nonetheless, constrained conflict can occur only under a Limited-Response Equilibrium, which is therefore a necessary, although not a sufficient, condition for limited war.

Interestingly, certain probing and bluffing activity is less likely at a Limited-Response Equilibrium than elsewhere. To be sure, Soft Challengers may rationally initiate conflict, and Soft Defenders may rationally respond in kind to a challenge. Nevertheless, Soft Challengers and Soft Defenders never escalate first (that is, [w.sub.S] = [z.sub.S] = 0) at a Limited-Response Equilibrium. Thus all-out conflict can occur only when both players are Hard. War is impossible unless both sides want it, unlike a Form II No-Limited Response Equilibrium, where even a Soft Defender may rationally escalate.

On the other hand, Hard Challengers always intend to escalate first under a Limited-Response Equilibrium, but only probabilistically (that is, 0 [is less than] [w.sub.H] [is less than] 1). As might be expected, therefore, limited war deployment policies like no-first-use or warfighting deterrence always entail the risk of deliberate escalation. In other words, no limited war deployment policy can eliminate the possibility of rational escalation.

Challenger's behavior under a Limited-Response Equilibrium is the reverse of the bluffing activity associated with Forms II and III No-Limited-Response Equilibria. Under the latter equilibria, Hard Challengers always defect and Soft Challengers defect probabilistically. Soft Challengers may therefore act as if they are Hard. Under a Limited-Response Equilibrium, Soft Challengers never escalate and Hard Challengers may choose not to. In other words, Challengers may act Soft even when they are Hard!

Challenger's probabilistic intention to escalate first if Hard is, in a sense, a protective mechanism. Since Soft Challengers never escalate, Defender would never capitulate unless there was some probability of things getting out of hand. Conversely, were this probability a certainty, Defender would never respond in kind (see below). This intention thus benefits Challenger, especially when it does not have to act on it.

It is easy to understand why there is no Limited-Response Equilibrium under which Challenger escalates first for certain (that is, where [W.sub.H] = 1). If so, there would be no possibility of a limited conflict; Hard Defenders would escalate in response to initiation, while Soft Defenders would capitulate.

In fact, Soft Defenders respond probabilistically at any Limited-Response Equilibrium, and Hard Defenders always respond, either in kind or by escalating (that is, [Y.sub.H] + [Z.sub.H = 1). This means that when a Limited-Response Equilibrium coexists with a No-Response Equilibrium (see below) and Defender is Hard, it should be easy to determine which equilibrium is in play.

In sum, at any Limited-Response Equilibrium, the possibility of a challenge and a nonescalatory response is real; all-out conflict is possible only if both players prefer it to capitulation and only Hard players unilaterally escalate. Finally, Hard Defenders always respond in some way to a challenge under a Limited-Response Equilibrium. Nonetheless, the two forms of Limited-Response Equilibrium do have some salient distinguishing features, which we describe next.

NO-FIRST-USE LIMITED-RESPONSE EQUILIBRIUM

Figure 3 superimposes on Figure 2 the location of one manifestation of each form of Limited-Response Equilibrium. (For the details of the other manifestations, see the appendix.) As Figure 3 shows, one form occurs at high levels of Challenger credibility and at low levels of Defender credibility. Under this equilibrium, Defender, whatever its type, never escalates first. Since Defender either responds in kind or not at all, we associate this form with a no-first-use deployment policy.

[Figure 3 ILLUSTRATION OMITTED]

The second form occurs at lower levels of Challenger credibility and at intermediate levels of Defender credibility. This is the only equilibrium of the Asymmetric Escalation Game that includes the possibility that Defender will respond in kind and the possibility that Defender will escalate first. For this reason, we associate this form of Limited-Response Equilibrium with a warfighting deterrence posture.

As Figure 3 indicates, the No-First-Use Equilibrium partially overlaps the areas occupied by the No-Response and the Form I No-Limited-Response Equilibrium. Under certain conditions it may also partially overlap the existence region of the Form III NLRE (see appendix for details). Since the Deterrence Equilibrium occurs throughout the unit square of Figures 2 or 3, the No-First-Use Equilibrium always coexists with it.

Although the No-First-Use Equilibrium includes the possibility of limited conflict, the status quo is never stable. In other words, since Challenger always defects (that is, [X.sub.H] = [X.sub.S] = 1), nothing like deterrence can emerge under a No-First-Use Equilibrium: rational Challengers always initiate.

Fortunately, perhaps, Defender's equilibrium response to initiation never involves escalation (that is, [Z.sub.H] = [Z.sub.S] = 0). In the unlikely event that it is Hard, Defender simply responds in kind (that is, [Y.sub.H] = 1). As detailed in the appendix, Soft Defenders do likewise, but probabilistically. The higher its credibility, the higher the probability that a Soft Defender responds in kind rather than capitulates.

What happens then depends on Challenger's type. If Challenger is Soft, it will stick with its prior choice and a limited conflict ensues (that is, [W.sub.S] = 0). Limited conflict may also occur if Challenger is Hard: when a No-First-Use Equilibrium is in play, a Hard Challenger may also stick with its prior choice at Node 3a (see appendix for details), in which case the conflict remains constrained. However, there is also a probability that a Hard Challenger will escalate; ultimately, the conflict may spiral to the highest level.

In sum, limited conflict is theoretically possible within the existence region of the No-First-Use Equilibrium. Here and only here can a no-first-use policy that relies exclusively on a tactical response to a challenge emerge as an equilibrium choice for Defender. Such a deployment policy does not, however, make the status quo any more stable. The major benefit associated with no-first-use is the relatively small but real possibility, not otherwise present, of limiting conflict and avoiding the stark choice between outright capitulation and certain escalation. This benefit, though, must also be weighed against the possibility of escalation to all-out conflict, also a possibility under a Form I or III Limited-Response Equilibrium but not under a No-Response Equilibrium.

WARFIGHTING LIMITED-RESPONSE EQUILIBRIA

The second major form of Limited-Response Equilibrium is called a Warfighting Equilibrium. It is the only Perfect Bayesian Equilibrium that admits the possibility of either a limited or an escalatory response by Defender; hence its name.

As Figure 3 indicates, the Warfighting Equilibrium occurs at lower levels of Challenger credibility and at intermediate levels of Defender credibility. (Depending on parameter values, the equilibrium may in fact overlap with any of the three No-Limited-Response Equilibria.) Since the underlying conditions associated with the Warfighting Equilibrium differ markedly from those associated with the No-First-Use Equilibrium, it should hardly be surprising that there are significant differences between the two Limited-Response Equilibria.

At the Warfighting Equilibrium, Hard Challengers initiate with certainty and Soft Challengers initiate probabilistically. The probability of an initial probe is therefore lower at a Warfighting Equilibrium than at a No-First-Use Equilibrium. In this case, ceteris paribus, crises should be less frequent and the status quo more stable.

A more significant difference, however, between the No-First-Use Equilibrium and the Warfighting Equilibrium concerns Defender's possible responses. Under a No-First-Use Equilibrium, Defender either responds in kind or does not respond at all. But when a Warfighting Equilibrium is in play, Hard Defenders may also rationally escalate first. (Soft Defenders never escalate first at any Limited-Response Equilibrium.) More specifically, a Hard Defender always responds if challenged (that is, [Y.sub.H] + [Z.sub.H] = 1) and chooses a limited or an escalatory response with prespecified probabilities. (See appendix for details.) Thus Defender utilizes the entire range of response options under a Warfighting Equilibrium.

As with the No-First-Use Equilibrium, limited conflicts are possible when a Warfighting Equilibrium is in play. Clearly, for limited conflicts to occur, Defender must respond in kind when Challenger initiates, and Challenger must subsequently choose not to escalate. Both of these requirements are likely to be met when both players are Soft, but they may also be satisfied when one player is Hard or even if both are Hard. In other words, at a Warfighting Equilibrium, all-out conflicts may be avoided even when the players do not view all-out conflict as the worst possible outcome.

However, this is not to say that limited conflicts are inevitable or even likely at a Warfighting Equilibrium. The entire range of conflict outcomes, including unconstrained conflict, may evolve. As at the No-First-Use Equilibrium, then, the genuine possibility of a limited war carries with it the risk of more extensive conflict.

III. MULTIPLE EQUILIBRIA

As indicated above, the Deterrence Equilibrium coexists with all other equilibria. The No-First-Use Equilibrium may partially overlap the region occupied by the No-Response Equilibrium and by Forms I and III No-Limited-Response Equilibria. Finally, the Warfighting Deterrence Equilibrium may occur simultaneously with all three No-Limited-Response Equilibria.

When two equilibria coexist, it is possible that rational players will find one of them unsustainable. For example, both types of both players may prefer one to the other. As well, equilibrium refinements, which are extensions to the criteria for rationality, sometimes eliminate one of the competing equilibria.(39)

This is never the case with the Deterrence Equilibrium, because it gives Defender its best outcome. All other equilibria, by contrast, always involve the possibility of initiation and therefore at least the risk of a less preferred outcome; thus, both types of Defender always strictly prefer the Deterrence Equilibrium to any of the alternatives. Nevertheless, Challenger would not initiate unless its expected value were at least what it gets at the Deterrence Equilibrium. In fact, a Hard Challenger always prefers the competing equilibrium; a Soft Challenger does also--at the three equilibria where Soft Challengers initiate for certain (No-Response, Form I No-Limited-Response, and No-First-Use). But Soft Challengers are indifferent when it comes to choosing between the Deterrence Equilibrium and the three equilibria where they initiate probabilistically (Forms II and III No-Limited Response and Warfighting). In consequence, these other equilibria simply coexist with the Deterrence Equilibrium--in every instance, there is no rational reason for players to reject either equilibrium in favor of the other.

The status of the No-First-Use Equilibrium is similar, with one important exception. When it coexists with the No-Response Equilibrium, Soft Defenders and both types of Challengers actually prefer the No-Response Equilibrium; only Hard Defenders prefer the No-First-Use Equilibrium. The net result is that either is possible.

However, both types of Challengers and also Hard Defenders strictly prefer the No-First-Use Equilibrium to the Form I NLRE, and Soft Defenders are indifferent. Rational players would therefore never play the Form I NLRE; they would instead choose the No-First-Use Equilibrium in the region of overlap, which must include some, but may include all, of the zone of existence of Form I NLRE.

Finally, under certain conditions, the No-First-Use Equilibrium also overlaps with the Form III NLRE. If so, Soft Challengers strictly prefer the No-First-Use Equilibrium, as do Hard Challengers when [P.sub.Def] is small enough. Likewise, Soft Defenders prefer the Form III NLRE, as do Hard Defenders when [P.sub.Ch] is large enough. This means that neither equilibrium can be eliminated when they coexist.

The situation of the Warfighting Equilibrium is quite different, however. This equilibrium may coexist with any of the three No-Limited-Response Equilibria. In every case, Warfighting is strictly less preferred than the alternative, except that Soft Challengers are indifferent between it and the Form II NLRE, and Soft Defenders between it and the Form I NLRE. It follows that rational players never play the Warfighting Equilibrium but reject it in favor of the appropriate No-Limited-Response alternative.

IV. DISCUSSION

The preceding technical discussion of the equilibria and their characteristics raises a number of difficult empirical and theoretical questions. First, why are substrategic deployments and flexible response policies of limited utility for stabilizing extended deterrence relationships? Second, given the overall tenuous stability of the status quo in the model, how can we account for the "remarkably stable system that emerged in Europe in the late 1940s"?(40) Third, in light of the theoretical improbability of limited conflicts, how can actual instances of limited war be explained? Finally, what does the model suggest about the nature of current and future interstate conflicts now that the cold war is over? In this section, we address each of these questions serially.

There are at least two ways to answer the first question. We begin with the prejudices of the model. On the one hand, by crediting Defender with a perfectly capable and completely credible substrategic threat, and by assuming that there is no particular advantage to escalating first, we have weighted the model in favor of the status quo or limited conflict outcomes.(41) But we have countered that particular prejudice by also presuming that Defender prefers unanswered escalation to limited conflict. This, we acknowledge, is a strong assumption that suggests why substrategic forces may not be of great value to a Defender unwilling to escalate--why, in our model, limited conflicts are rare events.

What, then, is the purpose of this assumption? The most important reason is its presumption by proponents of massive retaliation. Massive retaliation supposes a preference for escalation over limited conflict and capitulation, as long as the challenger does not have the ability to counterescalate.(42) By retaining this assumption we are able to measure directly the strategic implications of substrategic deployments under precisely the conditions that were taken for granted by the first wave of deterrence theorists. These conditions assume the worst not only of Challenger but also of Defender. In the tradition of both classical and structural realism, each state prefers unilateral advantage and is not averse to using force to gain it, unless there is a counterforce capable of preventing aggrandizement by the individual actor.(43) In other words, this assumption best reflects the Hobbesian world feared most by strategic thinkers. To put flexible response to a less severe test would unduly bias our model the other way, by presuming a nonegotistical Defender.

On another level we can address, perhaps more intuitively, the restricted utility of credible substrategic threats by referring to the dynamics of the game. Note first that our model reflects the consensus of the wider strategic literature that policies like massive retaliation are not particularly efficacious once a Challenger has a credible counterescalatory threat. Thus, it should not be surprising that when Challenger's threat to retaliate is credible enough, a No-Response Equilibrium exists and the status quo is never an outcome at equilibrium.

But what if the credibility of Challenger's threat to retaliate is also low? Given uncertainty about Defender's response, Challenger can deter Defender from responding in kind by intending to escalate first (at Node 3a). Once Defender is so deterred, Challenger can initiate with impunity. This tendency is accentuated as Defender's credibility to retaliate decreases or as Challenger's credibility increases. As well, the status quo becomes more stable as Challenger's credibility diminishes, ceteris paribus.

How, then, do we account for the persistence of the status quo in postwar Europe? There are a number of ways to explain this fact. One obvious answer is that by the time the Soviet Union had rendered massive retaliation completely obsolete by developing a nuclear capability and the means to deliver it, it had become a status quo power content to exercise control over its own territory and those neighboring states that could provide a buffer against foreign intrusion. This explanation, popular among revisionist historians, of course runs counter to standard realist assumptions and to the logic of deterrence theory.

It is also possible that a deterrence equilibrium was in play and that Soviet leaders came to believe that any attempt to alter the postwar status quo would inevitably lead to all-out nuclear conflict. While this explanation may comfort those who wish to believe that nuclear weapons have forever immunized the world from cataclysmic wars, it is not entirely consistent with Soviet choices to reimpose control in Czechoslovakia in 1968 or Afghanistan in 1979, or the even more provocative decision to try to address a strategic imbalance by shipping medium- and intermediate-range missiles to Cuba in 1962.

A more likely explanation, one that is consistent with the escalation model developed herein, is that the Soviet Union, while motivated to expand, was itself unwilling to fight a costly strategic war to do so and that U.S. leaders knew this. In the terms of the model, the Soviets lacked a credible retaliatory threat.(44) Thus, in situations like Czechoslovakia or Afghanistan, which were of high salience to the Soviets but not to the United States, the Soviets could, in equilibrium, act provocatively; but otherwise they chose not to gamble, behaving like a Soft Challenger under a Form II or Form III No-Limited-Response Equilibrium. Parenthetically we might add that this explanation is not inconsistent with the argument of some strategic thinkers who attribute the absence of a superpower war to luck.(45)

Next, how do we explain real-world examples of limited wars? In our opinion, such events most likely occur outside the parameters of the present model. Recall again that we assume Defender's substrategic threat to be both completely capable and perfectly credible. The capability assumption implies Challenger's preference for the status quo over limited conflict,(46) while the credibility assumption leads Challenger to believe that Defender will never concede unilaterally. Within the confines of the two-stage escalation game, limited conflicts are more probable when either presupposition is relaxed.

Consider first the implications of the capability assumption. It is easy to demonstrate that given complete information and mutually credible strategic-level threats, limited conflict (outcome DD in Figure 1) is an equilibrium if Defender's substrategic threat lacks capability (that is, Challenger actually prefers outcome DD to outcome cc). Under these conditions, Challenger is deterred from escalating at Node 3b and Defender is deterred from escalating at Node 4. Nonetheless, Challenger initiates and Defender rationally responds in kind. Thus, when the capability assumption is relaxed, limited conflicts are much more probable.

Prussia likely operated under constraints like these in 1866. Clearly, Bismarck wanted a war with the Austrians, preferring limited bilateral conflict to an unsatisfactory status quo in which Austria played the dominant role in greater Germany. Given the confidence of the Prussian general staff in a short and decisive military victory, it is unlikely that there was anything that Austria could have done unilaterally to deter Prussia. Nonetheless, against the advice of his generals, Bismarck limited his war aims: the Prussian minister-president feared the involvement of other powers, but especially France. That was a war that Bismarck was as yet unprepared to fight. Consequently, after Koniggratz (Sadowa), he convinced King William I to hold back.

Much the same could be said of the American involvement in Vietnam. The North Vietnamese clearly lacked the capability to deter the United States from aiding the South. Still, the U.S. was careful not to risk a major escalation of this conflict by threatening vital Soviet or Chinese interests.

Now consider the implications of relaxing the assumption of a perfectly credible substrategic threat: once this assumption is dropped, the possibility of an unanticipated limited response is introduced. Such a possibility helps to explain some crises and limited conflicts.(47) In 1948, for instance, Soviet leaders were surprised when the United States and its allies found a way to resist their attempt to block the formation of a pro-Western German state. The resulting crisis over Berlin festered until the Soviets called off their blockade early in 1949. Similarly the Falkland/Malvinas crisis can be traced to the belief of General Galtieri and the Argentine junta that the British would not "go to war for such a problem as these few rocky islands."(48) And UN forces would probably not have crossed the 38th parallel had they correctly gauged Chinese intentions in 1950.

In sum, our pessimism about the possibility of limited war does not apply to asymmetric conflicts in which a strong state uses force to secure limited objectives; nor does it pertain to those less-than-total conflicts that evolve when one state simply misjudges another's willingness to resist. Rather, our conclusions apply most directly to relations among relatively equal powers in which Defender's willingness and capability to engage a Challenger on the substrategic level is common knowledge and, not incidentally, both prefer limited to total war. We believe that these are precisely the conditions against which the efficacy of flexible response strategies like no-first-use and warfighting deterrence ought to be measured. After all, the implied objective of substrategic deployments is to reduce the possibility of strategic wars, not to increase the probability of limited conflicts.

What implications does this have for present-day conflicts? One is that under extreme conditions, restraint in warfare is unlikely. When two determined adversaries square off, chances are that no holds will be barred. The forces of Saddam Hussein, of course, set the oil fields in Kuwait on fire; chemical weapons were used in the war between Iran and Iraq; and Sherman leveled Atlanta. Such behavior is the rule, not the exception. When it occurs, restraint is likely to evolve by accident--when Challenger is unexpectedly confronted by a resolute Defender--or by virtue of a circumspect Challenger that, nonetheless, cannot be deterred from a limited objective.

What is to be done? One strategy is for weak states, like Belgium prior to World War I or the Baltic republics today, to rely on their "own strength and art, for caution against all other[s]."(49) But such states are by definition unlikely to have threats that are sufficiently capable (not to mention credible) to deter a highly motivated Challenger;(50) and, as we have seen, even a strong Defender with a credible threat cannot ensure a small state's integrity. Perhaps the best chance for successful extended deterrence is for a pawn to remain an unattractive prizes;(51) alternatively, a pawn can hope that, in a marginal case, Defender's promised protection is sufficiently credible to dissuade Challenger from attacking. Failing this, only an existential fear of escalation offers even the remotest possibility of a stable status quo.

V. SUMMARY AND CONCLUSIONS

The aim of this essay is to assess the impact of credible substrategic deployments on a wide spectrum of extended deterrence relationships--to ask whether and when flexible response deployment policies can make limited wars possible and total wars less, or more, probable. We know of no other formal work that has asked these questions.

The efficacy of substrategic response options was evaluated using the Asymmetric Escalation Game as a model for extended deterrence. The structure of this game is simple: Challenger must decide whether to contest the status quo; if so, Defender must choose either to capitulate or to respond, and if the latter, whether in kind or in an escalatory fashion. Each player must make its choice without complete information about whether the other prefers fighting an all-out war or backing down in response to escalation. We believe this model to be a rough approximation of the historical relationship of the United States and the Soviet Union after NATO's implementation of a flexible response deployment policy. It applies as well to any venue (for example, Korea) in which a defender seeks to deter substrategic challenges.

All Perfect Bayesian Equilibria of the incomplete information escalation model have been identified, interpreted, and grouped into four mutually exclusive categories: Deterrence, No-Response, No-Limited Response, and Limited-Response Equilibria.

A Deterrence Equilibrium, in which there is no possibility of a challenge, can occur under almost any conditions, provided that the players have an existential fear of escalation. One belief set consistent with the Deterrence Equilibrium requires that Challenger anticipate that any conflict will almost certainly escalate out of control. This view, we argue, is similar to that advanced by spiral theorists and advocates of a pure deterrence deployment policy. Parenthetically, we note that this belief set yields a similar equilibrium even when Defender adopts a policy like massive retaliation and lacks a credible substrategic threat.

A Deterrence Equilibrium might also evolve when Challenger expects Defender to respond in kind to a challenge: because it fears counterescalation, it intends not to escalate itself. This belief set is consistent with a conventional deterrence deployment policy. Advocates of conventional deterrence hold that a strategy of denial, when coupled existentially with the possibility of nuclear escalation, is the most efficacious extended deterrence posture. As one might expect, this belief cannot yield an equilibrium unless Defender possesses a credible substrategic threat. For existential deterrence to emerge, a flexible response deployment policy therefore provides one additional opportunity, beyond that given by massive retaliation.

The second major equilibrium category is the No-Response Equilibrium. The status quo is never stable when a No-Response Equilibrium is in play: Challenger always initiates and Defender always capitulates. For a No-Response Equilibrium to emerge, Challenger must possess a sufficiently credible threat to escalate and counterescalate. In effect, this threat deters Defender from offering any resistance at all.

There are three distinct forms of No-Limited-Response Equilibrium. At any No-Limited-Response Equilibrium there is always the possibility of initiation but no possibility that Defender will respond in kind. In other words, Defender either escalates or does not respond at all.

Like the No-Response Equilibrium, the three No-Limited-Response Equilibria do not depend on any substrategic threat. For this reason, both categories are associated with the escalatory deterrence policy prescription. Clearly, the deliberate threat to cross the nuclear threshold implicit in this deployment policy is not a particularly effective mechanism for extending deterrence, as critics of the Eisenhower administration's massive retaliation policy were quick to point out. The status quo is never stable when either a No-Response Equilibrium or a Form I No-Limited-Response Equilibrium is in play, or when Challenger prefers all-out war to capitulation (that is, is Hard). Only when Challenger's escalatory threat is relatively incredible is it possible for the status quo to persist.

The inadequacy of escalatory deterrence policies for maintaining the status quo in Europe explains NATO's migration toward flexible response. Flexible response required that NATO have a capable and credible threat to respond in kind to a substrategic challenge. Accordingly, in the early 1960s NATO's conventional and tactical nuclear forces were augmented. This buildup, it was thought, would allow decision makers to avoid the stark choice between holocaust and humiliation, between suicide and surrender, implicit in escalatory deterrence deployment policies like massive retaliation.

Two additional equilibria arise when Defender's substrategic threat is credible. These equilibria, the Limited-Response Equilibria, capture the additional rational behavioral possibilities that a flexible response deployment policy provides. Significantly, they are the only equilibria of the Asymmetric Escalation Game that involve the possibility of a restrained response to a challenge and, by implication, of a limited war.

The good news is that a limited conflict is theoretically possible under a Limited-Response Equilibrium. The bad news is that deterrence is unlikely when a Limited-Response Equilibrium is in play. Should a No-First-Use Equilibrium evolve, Challenger always initiates. And under a Warfighting Deterrence Equilibrium, there is only a limited chance that the status quo will persist.

Although deterrence is never stable under a No-First-Use Equilibrium, Defender does have certain advantages that might conceivably warrant the deployment stance associated with it. For example, in the region in which they coexist, Defender's expected payoff is greater under a No-First-Use Equilibrium than under a No-Response Equilibrium, provided Defender is Hard. (Soft Defenders prefer the No-Response Equilibrium.) Similarly, Hard Defenders prefer No-First-Use to the Form I No-Limited-Response Equilibrium, and Soft Defenders are indifferent. By contrast, Soft Defenders and, under certain conditions, Hard Defenders as well, actually prefer the Form III No-Limited-Response Equilibrium to the No-First-Use Equilibrium form.

On the other hand, these potential advantages must be weighed against the costs implicit in developing and maintaining a credible substrategic capability and in the realization that they accrue only in the relatively small existence region of the No-First-Use Equilibrium. Thus the probability that a no-first-use policy will be consistent with an equilibrium is relatively low.

It is also important to point out that the No-First-Use Equilibrium exists only when Challenger's credibility to wage all-out war is high and Defender's is low. Under these restricted conditions, Defender is likely Soft. The advantages offered by the No-First-Use Equilibrium to a Soft Defender are minimal. One must question, then, whether on balance the benefits of implementing a no-first-use extended deterrence deployment policy outweigh the costs; escalatory deployment policies, like massive retaliation, are both less expensive and no less effective in securing the status quo.

There is no question, however, about the attractiveness of the Warfighting Equilibrium, which requires Defender to be prepared either to respond in kind to a challenge or to escalate. Such a policy is consistent with equilibrium only at intermediate levels of Defender credibility and at lower levels of Challenger credibility. But a War-fighting Equilibrium is never preferred by Defender (or Challenger) to the No-Limited-Response Equilibrium with which it coexists. There are no conditions, therefore, under which Defender benefits from this deployment policy. In other words, it is almost always the case that a less costly escalatory deterrence deployment policy is preferable to a warfighting deterrence stance (for Defender).

In sum, our model indicates that almost never does a flexible response deployment offer an advantage to a Defender over an escalatory policy; even when it does, the benefits are minimal. This does not mean, however, that escalatory deterrence policies are a particularly efficacious mechanism for extended deterrence. There is no combination of beliefs and player types that guarantees a stable status quo under any of the four Escalatory Deterrence Equilibria. When an Escalatory Equilibrium is in play, Hard Challengers always initiate. There are also some situations in which Soft Challengers initiate with certainty, but no conditions under which a Soft Defender is completely deterred. In any case, the status quo is likely to survive only when Challenger's credibility is very low, when it is almost surely Soft, and when its probability of initiation vanishingly small.

Overall, the strategic position of a Defender limited to an escalatory response to a challenge is not enviable. While Defender's prospects rise along with the credibility of its strategic threat, even perfect credibility may not be sufficient to maintain the status quo. Only when Challenger's escalatory threat is almost totally incredible is the preservation of the status quo likely.(52)

In the absence of an irresolute Challenger, then, perhaps the best one can hope for is that a Deterrence Equilibrium evolves. Such a hope, however, is anathema to strategists and military planners. It depends on no particular deployment policy (other than the existence of strategic nuclear weapons) and is independent of each player's ability to convince the other that it will willfully escalate. Nonetheless, from the perspective of our model, it appears that Jervis was largely correct with respect to extended deterrence relationships when he observed that "a rational strategy for the employment of nuclear weapons is a contradiction in terms."(53) If so, there is little solace to be taken, since an existential explanation of deterrence stability precludes control and manipulation to enhance peace, and it raises questions about both the source and the persistence of the beliefs that sustain the status quo.

APPENDIX

This Appendix contains details of the calculation of the Perfect Bayesian Equilibria of the game of Figure 1, which is described in Section I of the text. The players are Challenger (Ch) and Defender (Def). Refer to Section I for definitions of the payoff parameters

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

the credibility parameters [p.sub.Ch] = [p.sub.C] and [p.sub.Def] = [p.sub.D], and the strategic variables [x.sub.H], [x.sub.S], [y.sub.H], [y.sub.S], [z.sub.H], [z.sub.S], [w.sub.H] and [w.sub.S]. It will be assumed here that 0 [is less than] [p.sub.C] [is less than] 1 and 0 [is less than] [p.sub.D] [is less than] 1. Below, the symbol r will be used to denote Def's conditional probability that Ch is Hard given that Def must take action (that is, that the game reaches Node 2 in Figure 1). Likewise, q will denote Ch's conditional probability that Def is Hard, given that Ch must respond to Def's D response (that is, that the game reaches Node 3a in Figure 1).

It can be shown that values ([x.sub.H], [x.sub.S], [y.sub.H], [y.sub.S], [z.sub.H], [z.sub.S], [w.sub.H], [w.sub.S], r, q) constitute a Perfect Bayesian Equilibrium if and only if the following conditions are met:

(C1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(C2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(C3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(C4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(C5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(C6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(C7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(C8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

DETERRENCE EQUILIBRIA

A Deterrence Equilibrium is any equilibrium with [x.sub.H] = [x.sub.S] = 0 Clearly, only conditions (C1) and (C2) apply. Inspection of these two conditions shows that they hold provided [z.sub.H] and [z.sub.S] are large enough, or when [y.sub.H] and [y.sub.S] are large enough and [w.sub.H] and [w.sub.S] are small enough. For instance, there is a Deterrence Equilibrium ([x.sub.H] = [x.sub.S] = 0) for any values of the parameters when [z.sub.H] = [z.sub.S] = 1. At any Deterrence Equilibrium, the outcome is CC and the prayers' utilities are [c.sub.CC and [d.sub.CC].

NO-RESPONSE EQUILIBRIA

A No-Response Equilibrium (NRE) is any equilibrium with [x.sub.H] + [x.sub.S] [is greater than] 0, but [y.sub.H] = [y.sub.S] = [z.sub.H] = [z.sub.S] = 0 To identify all NRE, first observe from (C1) and (C2) that [y.sub.p] = [z.sub.p] = 0 implies that [x.sub.H] = [x.sub.S] = 1. Then, from (C3), r = [p.sub.C]. The only remaining requirement for an NRE is that all of the coefficients of y and z on the right side of (C4) and (C5) must be non-positive. Because

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

this requirement is equivalent to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is now easy to verify that there is an NRE whenever [p.sub.C] [is greater than or equal to] [d.sub.2] and [w.sub.H] and [w.sub.S] are large enough, as specified above.

NO-LIMITED-RESPONSE EQUILIBRIA

A No-Limited-Response Equilibrium (NLRE) is any equilibrium with [x.sub.H] + [x.sub.S] [is greater than] 0, [z.sub.H] + [z.sub.S] [is greater than] 0, and [y.sub.H] = [y.sub.S] = 0 To identify all NLRE, only conditions (C1) - (C5) need to be considered, provided [y.sub.H] = 0 and [y.sub.S] = 0 are also satisfied.

First we show that, at any NLRE, [x.sub.S] [is greater than] 0 and [x.sub.H] = 1. From (C1) and (C2), note that at any NLRE,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The difference between the right-side coefficients of x is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which proves that if [x.sub.S] [is greater than] 0, then [x.sub.H] = 1. To complete the proof, note that if [x.sub.S] = 0, then [x.sub.H] [is greater than] 0 (for an NLRE) so that r = 1 by (C3). But now the coefficient of z on the right side of (C4) is ([d.sub.DE] - [d.sub.DC]) - ([d.sub.DE] - [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] - [d.sub.DC] [is less than] 0, implying that [z.sub.H] = 0. Using (C5) in a similar way shows that [z.sub.S] = 0, contradicting the definition of an NLRE.

Thus at any NLRE, either [x.sub.H] = [x.sub.S] = 1 or [x.sub.H] = 1 and 0 [is less than] [x.sub.S] [is less than] 1. The first case occurs only when [z.sub.p] [is less than or equal to] [c.sub.DC] - [c.sub.CC]/[c.sub.DC] - [c.sub.DE] = [c.sub.1] and the second when [z.sub.p] = [c.sub.1]. Note that r [is greater than] 0 always.

Subtracting the coefficients of z on the right sides of (C4) and (C5) gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This implies that, at an NLRE (where [y.sub.H] = [y.sub.S] = 0), [z.sub.S] [is greater than] 0 implies [z.sub.H] = 1. Furthermore, [z.sub.H] [is greater than or equal to] 0 at equilibrium iff (if and only if) r [is less than or equal to] [d.sub.2], and [z.sub.S] [is greater than or equal to] 0 at equilibrium

iff r [is less than or equal to] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Evidently, [d.sub.1] [is less than] [d.sub.2], and r [is less than or equal to] [d.sub.2] at any NLRE.

We now identify necessary conditions for an NLRE with [x.sub.H] = [x.sub.S] = 1, called a Type I NLRE. Note that r = [p.sub.C]. An NLRE with [p.sub.C] = [d.sub.2] requires [z.sub.S] = 0, 0 [is less than or equal to] [z.sub.H] [is less than or equal to] 1, and [z.sub.P] = [p.sub.D] [z.sub.H] [is less than or equal to] [c.sub.1]. The latter inequality can be satisfied for any value of [p.sub.D]. For an NLRE with [d.sub.1] [is less than] [p.sub.C] [is less than] [d.sub.2], [z.sub.S] = 0, [z.sub.H] = 1, and [z.sub.p] = [p.sub.D] [is less than or equal to] [c.sub.1] are required. For an NLRE with [p.sub.C] = [d.sub.1], 0 [is less than or equal to] [Z.sub.S] [is less than or equal to] 1, [z.sub.H] = 1, and [z.sub.p] = [p.sub.D] + (1 - [p.sub.D]) [z.sub.S] [is less than or equal to] [c.sub.1] are necessary. A suitable value of [z.sub.S] can be chosen iff [p.sub.D] [is less than or equal to] [c.sub.1]. Finally, there cannot be an NLRE with [p.sub.C] [is less than] [d.sub.1], for this would imply [z.sub.S] = [z.sub.H] = 1, whence [z.sub.p] = 1 so that [z.sub.p] [is less than or equal to] [c.sub.1] cannot be satisfied.

Now we turn to NLRE with [z.sub.H] = 1, 0 [is less than] [x.sub.S] [is less than] 1. Recall that r = [p.sub.C] / ([p.sub.C] + (1 - [p.sub.C]) [x.sub.S]) [is greater than] [p.sub.C] and [z.sub.p] = [c.sub.1] at such an equilibrium If r = [d.sub.2] then clearly [x.sub.s] = [p.sub.c] (1 - [d.sub.2] /(1 - [p.sub.C]) [d.sub.2] and [p.sub.c] [is less than or equal to] [d.sub.2] is necessary. Also [z.sub.s] = 0 and [z.sub.p.] = [p.sub.D] [z.sub.H] = [c.sub.1] so [z.sub.H] = [c.sub.1] / [p.sub.D] and [p.sub.D] [is greater than or equal to] [c.sub.1] is necessary. If [d.sub.1] [is less than] r [is less than] [d.sub.2], then [z.sub.S] = 0 and [z.sub.H] = 1, so [p.sub.D] = [c.sub.1]. Furthermore [p.sub.C] [is less than] [d.sub.2] is required in order that [x.sub.S] can be chosen to satisfy r [is less than] [d.sub.2]. If r = [d.sub.1], [x.sub.S] = [p.sub.C] (1 - [d.sub.1])/[d.sub.1] (1 - [p.sub.C] and [p.sub.C] [is less than or equal to] [d.sub.1] is necessary. As well, [z.sub.H] = 1 and [z.sub.p] = [p.sub.D] + (1 - [p.sub.D]) [z.sub.S] = [c.sub.1], so [z.sub.S] = ([c.sub.1] - [p.sub.D]) / (1 - [p.sub.D]), end [p.sub.D] [is less than or equal to] [c.sub.1] is necessary. There is no NLRE with r [is less than] [d.sub.1], for in that case [z.sub.H] = [z.sub.S] = 1, and [z.sub.p] = [c.sub.1] is impossible.

Other necessary conditions for NLRE follow from the requirement that [y.sub.H] = [y.sub.S] = 0. Because the equilibria have [z.sub.H] [is greater than] 0, (C4) shows that the condition [y.sub.H] = 0 is equivalent to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which can certainly be satisfied by choosing [w.sub.H] and [w.sub.S] large enough. Furthermore, (C5) shows that [y.sub.S] = 0 holds whenever

[w.sub.r] ([d.sub.DD] - [d.sub.ED]) [is greater than or equal to] [d.sub.DD] - [d.sub.DC]

which again is always true if [w.sub.H] and [w.sub.S] are large enough.

In summary, we have identified three NLRE that exist on sets of positive measure in ([p.sub.C], [p.sub.D]) - space. They are

(I) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(II) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(III) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

All of these equilibria require that [w.sub.H] and [w.sub.S] be sufficiently large, and that the value of r be appropriate, as specified above.

LIMITED-RESPONSE EQUILIBRIA

A Limited-Response Equilibrium (LRE) is any equilibrium with [x.sub.H] + [x.sub.S] [is greater than] 0 and [y.sub.H] + [y.sub.S] [is greater than] 0. Clearly all of conditions (C1) - (C8) need to be satisfied.

First we show that any LRE has [w.sub.S] = 0 and 0 [is less than] [w.sub.H] [is less than] 1. Subtracting the coefficient of w in (C8) from the coefficient in (C7) gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It follows that either q = 0 (in which case [w.sub.H] = [w.sub.S] = 1) or the coefficient for [w.sub.H] strictly exceeds that for [w.sub.S]. In either case, if [w.sub.S] [is greater than] 0, then [w.sub.H] = 1.

To see that there are no LRE with [w.sub.S] = [w.sub.H] = 0, note that this would imply [w.sub.r] = 0, so that the coefficients of y in (C4) and (C5) would be strictly positive, yielding [y.sub.H] + [z.sub.H] = 1 and [y.sub.S] + [z.sub.S = 1. The coefficient of x in (C1) is then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

because [y.sub.p] + [z.sub.p] = 1. It follows that [x.sub.H] = 0, and, by an analogous calculation using (C2), [x.sub.s] = 0. Thus an equilibrium with [w.sub.s] = [w.sub.H] = 0 cannot be an LRE.

Now consider any LRE with [w.sub.H] = 1. Subtracting the coefficient of y from the coefficient of z in (C4) gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is positive unless r = 1. Clearly [y.sub.H] = 0 if r [is less than] 1. If r = 1, direct substitution in (C4) shows [y.sub.H] = [z.sub.H] = 0 at equilibrium. But if [y.sub.H] = 0 at an LRE, then [y.sub.s] [is greater than] 0, q = 0, and [w.sub.s] = 1 all follow. But now [w.sub.r] = 1, and (C5) fails unless [y.sub.s] = 0. This contradiction shows that [w.sub.H] = 1 is impossible at any LRE.

The proof that [w.sub.s] = 0 and 0 [is less than] [w.sub.H] [is less than] 1 at any LRE is now complete. A further requirement, from (C7), is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Observe that 0 [is less than] q [is less than] 1, which in turn implies that [y.sub.H] [is greater than] 0 and [y.sub.s] [is greater than] 0

Comparison of the coefficients of y in (C4) and (C5) shows that if [y.sub.s] [is greater than] 0 at equilibrium, then [y.sub.H] + [z.sub.H] = 1. Furthermore, it must be the case that [y.sub.s] + [z.sub.s] [is less than] 1, because, as proved above, the equalities [y.sub.H] + [z.sub.H] = 1 and [y.sub.s] + [z.sub.s] = 1 are inconsistent with any equilibrium in which [x.sub.H] + [x.sub.s] [is greater than] 0 We now show that, at any LRE, [z.sub.s] = 0

Rewrite conditions (C4) and (C5) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We have already proved that, at any LRE, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We now show that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In order that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is required. Next,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which means that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

After substitution and manipulation, it can be shown that the right side of this inequality is strictly negative, so that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [is less than] 0 as claimed.

Note that, coincidentally, it was demonstrated above that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

at any LRE. In fact, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (necessary for [z.sub.H] [is greater than] 0), and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = 0 (when [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We now identify all LRE with [z.sub.H] = 0. Clearly [y.sub.H] = 1, and [y.sub.s] is determined by the condition q = [c.sub.q]. Application of (C6) gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Evidently, [p.sub.D] [is less than or equal to] [c.sub.q] is necessary in order that [y.sub.s] [is less than or equal to] 1.

It is easy to verify that [z.sub.p] = 0 and that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = 0, so that (C1) and (C2) reduce to

[x.sub.h] = arg max {x [([c.sub.DC]-[c.sub.CC]) - [y.sub.p] ([c.sub.DC] - [c.sub.DD])]

[x.sub.s] = arg max {x [([c.sub.DC]-[c.sub.CC]) - [y.sub.p] ([c.sub.DC] - [c.sub.DD])]

Let [c.sub.x] = ([c.sub.DC] - [c.sub.CC]) - [y.sub.p] ([c.sub.DC] - [c.sub.DD])]. Clearly there is no LRE with [x.sub.H] = [x.sub.s] = 0. There is an LRE with 0 [is less than] [x.sub.H] [is less than] 1 or 0 [is less than] [x.sub.s] [is less than] 1 when [x.sub.s] = 0, but this happens only when

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which occurs only on a set of measure zero in ([P.sub.D], [P.sub.c]) - space.

The remaining possibility for an LRE with [z.sub.H] = 0 is [x.sub.H] = [x.sub.s] = 1, when [C.sub.x] [is greater than or equal to] 0, which occurs exactly when

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note that [P.sub.D] [is less than or equal to] [c.sup.*] implies [P.sub.D] [is less than] [c.sub.q]. Also, the requirement that r [is greater than or equal to] [d.sub.p] can be met only when [P.sub.c] [is greater than or equal to] [d.sub.p]. There is one more necessary condition for

this equilibrium: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It can be verified directly that the preceding necessary conditions are also sufficient. They yield LRE I

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which exists iff [P.sub.c] [is greater than or equal to] [d.sup.*] = max {[d.sub.1], [d.sub.p]} end [P.sub.D] [is less than or equal to] [c.sup.c]. Note that [d.sub.p] [is less than] [d.sub.2], so [d.sub.1] [is less than or equal to] [d.sup.*] [is less than] [d.sub.2]. The parameter [c.sup.*] could be greater than, less than, or equal to the parameter [c.sub.1].

We turn now to the identification of LRE with [z.sub.H] [is greater than] 0. Clearly [z.sub.H] = 1 - [y.sub.H]. The conditions (C1) and (C2) reduce to

(C1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(C2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Because [z.sub.p] [is greater than] 0, the coefficient of x in (C1) exceeds the corresponding coefficient in (C2). The requirement r = [d.sub.p] can therefore be met only by [x.sub.H] = 1 and [x.sub.s] [is greater than] 0, because 0 [is less than] [d.sub.p] [is less than] 1. In fact

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in order that r = [d.sub.p]; it follows that [P.sub.c] [is less than or equal to] [d.sub.p] is necessary for an LRE of this type, to ensure that [x.sub.s] [is less than or equal to] 1.

The requirement that q = [c.sub.q] implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Along with [z.sub.H]= 1 - [y.sub.H], this equation can be substituted into the coefficient of x in (C2), which must vanish. The resulting equation can be solved for [y.sub.H] to yield

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The only remaining conditions concern 0 [is less than or equal to] [y.sub.H] [is less than or equal to] 1 and 0 [y.sub.s] [is less than or equal to] 1. It is straightforward to show that, if [c.sub.den] [is greater than] 0, then [y.sub.H] [is greater than or equal to] 0 iff [P.sub.D] [is less than or equal to] [c.sub.1], [y.sub.s] [is greater than or equal to] 0 iff

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [y.sub.s] [is less than or equal to] 1 iff

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Likewise, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It can be verified directly that [c.sup.*] [is greater than] [c.sub.s] when [c.sub.den] [is greater than] 0, and [c.sup.*] [is less than] [c.sub.s] when [c.sub.den] [is less than] 0.

It can now be demonstrated that the preceding necessary conditions are also sufficient, yielding LRE II,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which exists iff [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(1) William Kaufmann, "The Requirements of Deterrence," in Kaufmann, ed., Military Policy and National Security (Princeton: Princeton University Press, 1956).

(2) Jane E. Stromseth, The Origins of Flexible Response: NATO's Debate over Strategy in the 1960s (New York: St. Martin's, 1988).

(3) Ivo H. Daalder, The Nature and Practice of Flexible Response: NATO Strategy and Theater Nuclear Forces since 1967 (New York: Columbia University Press, 1991), 2.

(4) Ibid., chap. 2.

(5) The motivation behind this model is discussed at length in Frank C. Zagare, "The Dynamics of Escalation," Information and Decision Technologies 16, no. 3 (1990); idem, "NATO, Rational Escalation and Flexible Response," Journal of Peace Research 4 (November 1992); and Frank C. Zagare and D. Marc Kilgour, "Modeling `Massive Retaliation,'" Conflict Management and Peace Science 13, no. 1 (1993).

(6) General deterrence relationships in which a challenger may contemplate a direct nuclear attack are modeled in D. Marc Kilgour and Frank C. Zagare, "Credibility, Uncertainty, and Deterrence," American Journal of Political Science 35 (May 1991); and Frank C. Zagare and D. Marc Kilgour, "Asymmetric Deterrence," International Studies Quarterly 37 (March 1993).

(7) As R. Harrison Wagner (personal communication, April 24,1992) points out, we assume that the payoffs at outcome EE are the same, regardless of which player escalates first. We accept the point that players in the real world are likely to prefer escalating first. We make this assumption to gain mathematical tractability. Note its implications: Defender's expected payoffs at Node 2 when it chooses E are underrepresented (because it likely prefers the mutual escalation outcome associated with this choice to the mutual escalation outcome associated with the choice of D); it follows that its expected payoff from choosing (D) at Node 2 is overrepresented. The bias of the model, therefore, is toward overreporting the likelihood of a (Limited-Response) equilibrium that involves the possibility that Defender responds in kind to a challenge. As we show below, however, even with this bias, the conditions under which such an equilibrium exists are quite restricted.

(8) In other words, we assume the choice of D is commensurate with Challenger's initiation at Node 1. Thus, at Node 2, Defender may defect by matching in scope and intensity the actions taken by Challenger to contest the status quo, or it may choose unconstrained actions, such as those associated with waging an all-out war, represented by the escalation alternative (E).

(9) Thomas C. Schelling, The Strategy of Conflict (Cambridge: Harvard University Press, 1960).

(10) James D. Fearon's escalation model also has two stages. See Fearon, "Deterrence and the Spiral Model: The Role of Costly Signals in Crisis Bargaining" (Paper presented at the annual meeting of the American Political Science Association, San Francisco, August 30-September 2, 1990). At the first stage, each player is afforded an opportunity to increase its own cost of backing down and, possibly, its credibility; at the second stage, they decide whether or not to fight. Thus, while there are two stages to this model, there is but one mutual conflict outcome. For this reason, we view Fearon's conclusions as extending and complementing our own analysis of one-stage asymmetric deterrence games of incomplete information; see Zagare and Kilgour (fn. 6).

(11) George W. Downs and David Rocke, Tacit Bargaining, Arms Races, and Arms Control (Ann Arbor: University of Michigan Press, 1990), chaps. 1, 4.

(12) See fn. 6.

(13) Barry Nalebuff, "Brinkmanship and Nuclear Deterrence: The Neutrality of Escalation," Conflict Management and Peace Science 9 (Spring 1986); Robert Powell, Nuclear Deterrence Theory: The Search for Credibility (New York: Cambridge University Press, 1990).

(14) Frank C. Zagare, "Rationality and Deterrence," World Politics 42 (January 1990).

(15) We believe the opposite assumption is consistent with a policy like massive retaliation that makes no provision for a credible conventional deterrent. For an analysis of this case, see Zagare and Kilgour (fn. 5). For the public justification of flexible response, see Robert S. McNamera, "Address at the Commencement Exercises" (Ann Arbor: University of Michigan June 16, 1962).

(16) Helmut Schmidt, Defense or Retaliation (New York: Praeger 1962), 211.

(17) Kilgour and Zagare (fn. 6).

(18) Eric Rasmusen, Games and Information (New York: Blackwell, 1989).

(19) McGeorge Bundy, "The Bishops and the Bomb," New York Review of Books (June 16, 1983) John Lewis Gaddis, The Long Peace: Inquiries into the History of the Cold War (New York: Oxford University Press, 1987).

(20) Daalder (fn. 3).

(21) As noted in the text, Daalder (fn. 3) argues that the formal definition of flexible response is deliberately vague, in part to accommodate divergent views of how deterrence operates and how forces should be structured. This is the only sense in which pure deterrence is compatible with flexible response.

(22) Glen H. Snyder, "The Balance of Power and the Balance of Terror," in Paul Seabury, ed., Balance of Power (San Francisco: Chandler, 1965), 199.

(23) Daalder (fn. 3), 52-53.

(24) Bernard Brodie, ed., The Absolute Weapon: Atomic Power and World Order (New York: Harcourt Brace, 1946); Michael D. Intriligator and Dagobert L. Brito, "Can Arms Races Lead to the Outbreak of War?" Journal of Conflict Resolution 28 (March 1984); Robert Jervis, The Illogic of American Nuclear Strategy (Ithaca, N.Y.: Cornell University Press, 1984), Kenneth N. Waltz, "The Spread of Nuclear Weapons: More May Be Better," Adelphi Paper no. 171 (London: International Institute for Strategic Studies, 1981).

(25) Zagare and Kilgour (fn. 5).

(26) As detailed in the appendix, some restrictions on r (i.e., Defender's conditional probability that Challenger is seen to be Hard, given that Challenger initiates) also apply under any No-Limited-Response Equilibrium.

(27) For a discussion of the impact of specific changes in utilities on the relative locations of the threshold values defining the existence regions of all three forms of NLRE (i.e., [d.sub.1], [d.sub.2], and [c.sub.1]), see Zagare and Kilgour (fn. 5).

(28) Daalder (fn. 3), 58.

(29) Ibid., 46.

(30) The Deterrence Equilibrium does not depend on any particular preference relationship. Rather it exists as long as the players have the required beliefs about each other's action choices, whatever their actual preferences happen to be.

(31) Zagare and Kilgour (fn. 5).

(32) Ibid.

(33) For an analysis of the implications of the pawn's value for extended deterrence relationships, see D. Marc Kilgour and Frank C. Zagare, "Uncertainty and the Role of the Pawn in Extended Deterrence," Synthese 100 (September 1994).

(34) McGeorge Bundy, George F. Kennen, Robert S. McNamara, and Gerard Smith, "Nuclear Weapons and the Atlantic Alliance," Foreign Affairs 60 (Spring 1982).

(35) Daalder (fn. 3), 52.

(36) Ibid., 50.

(37) Stromseth (fn. 2), 202.

(38) Daalder (fn. 3), 63

(39) Drew Fudenberg and Jean Tirole, Game Theory (Cambridge: MIT Press, 1991); Robert Gibbons Game Theory for applied Economists (Princeton: Princeton University Press, 1992), and Eric van Damme, Refinements of the Nash Equilibrium Concept (Berlin: Springer-Verlag, 1993).

(40) John J. Mearsheimer, "Back to the Future: Instability in Europe after the Cold War," International Security 15 (Summer 1990).

(41) See fn. 7.

(42) Kaufmann (fn. 1)

(43) Kenneth N. Waltz, Man, the State and War: A Theoretical Analysis (New York: Columbia University Press, 1959), 232.

(44) Of course, a similar argument could be used to explain why the United States never attempted to roll back the Iron Curtain in the 1950s, despite the rhetorical preference of some Republican leaders to do just that.

(45) T. K. Jones and W. Scott Thompson, "Central War and Civil Defense," Orbis 22 (Fall 1978).

(46) Frank C. Zagare, The Dynamics of Deterrence (Chicago: University of Chicago Press, 1987), 34, chap. 4.

(47) D. Marc Kilgour and Frank C. Zagare, "Using Game Theory to Analyze a General Two-Level Escalation Game" (Paper presented at the annual meeting of the American Political Science Association, New York, September 1-4, 1994).

(48) Alexander M. Haig, Caveat: Realism, Reagan and Foreign Policy (New York: Macmillan, 1984).

(49) Thomas Hobbes, Leviathan, ed. C. B. Macpherson (Harmondsworth, Great Britain: Penguin 1968).

(50) Zagare and Kilgour (fn. 6).

(51) Kilgour and Zagare (fn. 33).

(52) Zagare and Kilgour (fn. 5).

(53) Jervis (fn. 24)

FRANK C. ZAGARE and D. MARC KILGOUR, This material is based upon work supported by the National Science Foundation under Grant no. SES-9123219 to Frank C. Zagare. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the National Science Foundation. D. Marc Kilgour gratefully acknowledges the support of the Laurier Centre for Military Strategic and Disarmament Studies, the Laurier Research Professorship, the Natural Sciences and Engineering Research Council of Canada, and the Social Sciences and Humanities Research Council of Canada.
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Date:Apr 1, 1995
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