Public Choice 68: 117-135, 1991. Kluwer Academic ... · Figure 1. The threat game. rence: averting conflict through the threat of retaliation, which, if carried out, could be costly

Public Choice 68: 117-135, 1991. © 1991 Kluwer Academic Publishers. Printed in the Netherlands.

R a t i o n a l retal iat ion:

S u p e r p o w e r response to crisis, 1 9 4 8 - 1 9 7 9 "

PATRICK JAMES Department o f Political Science, McGill University, Montreal, Quebec, Canada H3A 2T7

Submitted 26 July 1988; accepted 18 July 1989.

Abstract. Using a modified version of Chicken, referred to as the Threat Game, Brams and Kilgour (1987a; 1987b) have developed a theoretical model of rational retaliation in superpower crises. They derive explicit thresholds for sufficient retaliation against provocations to deter them in the first place. When the retaliating move meets or is more coercive than a threshold value, it is rational for the adversaries to prefer to desist from conflict, whereas a response that is too cooperative will not provide the same incentive.

Data on superpower crises from 1948 to 1979 will be used to test a modified version of the model developed by Brains and Kilgour. A description of their model, culminating in some revisions, is the point of departure. Selection of cases from the International Crisis Behavior (ICB) Project data set, operationalization of the model and testing will follow. Finally, implications of the generally positive findings are discussed.

1. Introduction

Since every international conflict holds some risk of escalation - perhaps even resulting in superpower confrontation at the nuclear level - it is worthwhile

to explore more effective means of crisis management. The objective of this study is to use data on superpower crises to test a modified version of the model of rational retaliation developed by Brams and Kilgour (1987a; 1987b). A description of their model, culminating in some revisions, is the first stage of the investigation. Selection of cases and operationalization of the model comprise the second phase and testing is the third. The final stage concerns the implications of the findings.

Of course, formal models of international conflict other than that of Brams and Kilgour could be applied to the superpower rivalry and then tested. A salient - and more general - model, also based on economic theory, is developed by Hirshleifer (1988) in the special issue of Synthese devoted to formal analysis of international relations. Although his rigorous model of continuing

* I am grateful to Steven J. Brams, Michael Brecher, Arlene Broadhurst, John A. Galbraith, Frank Harvey, Marc Kilgour, Hudson Meadwell, John Oneal, Joe A. Oppenheimer, Blema Stein- berg and an anonymous referee for helpful commentaries and to Maria Marcheschi for word processing services.

118

conflict could be tested, none of its variants is designed specifically to deal with crisis-oriented, superpower nuclear rivalry. Although Hirshleifer's (1988: 221) Threat-and-Promise solutions most clearly resemble the one that will be adapted from Brams and Kilgour, only a single round of play is assumed: " the con- troller will have prespecified a reaction function, a set of threats and promises, that will govern his last-word responses to the subordinate's choice". In sum, the model derived by Brams and Kilgour has more direct application to superpower crises, whereas that of Hirshleifer (and others in the special issue of Syn- these) could be used more profitably to analyze arms races, terrorism and other

continuing conflicts.

2. The threat game and rational retaliation

According to Brams (1985), the game of "Chicken" is well-suited as a model of nuclear deterrence between the superpowers.1 With two modifications, as Brams and Kilgour (1987a, 1987b) have demonstrated, Chicken can be trans- formed into a superior model of bargaining at the superpower level: (i) the players can make quantitative choices of levels of cooperation (C) or noncooperation (C'), not just qualitative choices of C or C'; and (ii) once these initial choices, which can be interpreted as levels of nonpreemption (versus preemption) are made, the less preemptive player (i.e., who chose the lower level of preemption initially, if there was one) can retaliate by choosing a different - and presumably higher - level of noncooperation subsequently (Brams and Kilgour, 1987a). This version of Chicken is known as the "Threat Game" . The two changes effectively respond to some of the most telling criticisms of game

matrices (Morgan, 1984). Specifically, the players are no longer assumed to have just two options;

there is a continuum of cooperation and conflict. Simultaneous choices that

conclude play also are not entailed by the Threat Game, because the second modification allows for altered tactics for at least one of the antagonists. This also appears to be more consistent with the reality of superpower bargaining

in crises. There are two ways in which the game can terminate. If the initial levels of

C or C' are the same, play is terminated; when they are not equivalent, the game ends after the more cooperative player has retaliated. This is intended to replicate the process of escalation in a bilateral crisis (Brams and Kilgour,

1987a: 835). Brams and Kilgour "telescope" escalation into a "single retaliatory counter-

move" by the less preemptive player (1987a: 835). Further rounds of escalation could be included, but " the present simple sequence captures well both the process that might trigger further escalation and the core meaning of deter-

119

) Preplay position (1,1 ~ ,

YO

q ( x 0 )

' r4, c 2 )

( r 2, c S

) Initial position

Final position

( r 1, e I ) ( 1 / 2 ) ( 0 , 0 )

Column

Key :

( r i , cj ) = (payoff to row, payoff to column )

r 4 , c 4 = best; r 3 , c 3 = next best; t ~ , c 2 = next worst; r 1 , c I = w o r s t

YO ' Xo = initial strategy choices of row and column, respectively

q ( x 0 ) = subsequent strategy choice of row (more cooperative player initially)

0,1)

2 / 3 )

( 1 / 3 )

Row

Source: This figure has been adapted from Brams and Kilgour (1987a: 836)

F i g u r e 1. T h e t h r e a t g a m e .

rence: averting conflict through the threat of retaliation, which, if carried out, could be costly to both players".

Figure 1 displays the Threat Game, which is played on the unit square. 2 The players, Row and Column, select preemption levels Yo and x o, respectively. These choices range from 0 (maximum) to 1 (minimum) preemption. This simultaneous selection by the players results in (x o, Yo), referred to as the initial position. The preplay position is (1,1), prior to noncooperation by the players.

Movement from the preplay to the initial position means noncooperation at certain levels, resulting in (x o, Yo) as the initial position, with x o and Yo mea- sured along the horizontal (x) and vertical (y) axes, respectively. If Yo > Xo, Row is the more cooperative player and retaliates according to the retaliation function q(Xo), making the final position of the game (x o, q(Xo) ). If Yo < Xo,

120

then Column retaliates with P(Yo), resulting in (P(Yo), Yo) as the final position. The retaliation function in each case depends upon the level of noncooperation selected by the adversary, and the relative magnitudes of Yo and x o determine which player will retaliate.

Each corner of the unit square corresponds to one of the " p u r e " states of Chicken, with r i = Row's payoff and cj = Column's payoff. The subscripts provide the rankings, with 4 as the best and 1 being worst. The points along the x and y axes cover all possible payoff combinations.

Payoffs at the final position are weighted averages of the payoffs at the corners of the game board. (The direction of the horizontal axis in Figure 1 is reversed, in order to make the corners correspond to the cells of the Chicken matrix.) The payoff functions are bilinear, meaning linear in each coordinate. For example, the final position of the game would correspond to x = x o, y = q(Xo) or x = P(Yo), Y = Yo, with Row and Column, respectively, as the initially more cooperative player. In calculating payoffs for Row and Column at any point (x,y), referred to as PR(x,y) and Pc(x,y) respectively, the payoffs at each corner are "weighted by the product of the distances, parallel to the axes, from (x,y) to the opposite corner" (Brams and Kilgour, 1987a: 837):

PR(x,y) = xyr 3 + x(1--y)r 4 + ( 1 - x ) y r 2 + ( 1 - x ) ( l - - y ) r 1 (1)

Pc(x,y) = xyc 3 + y ( 1 - x ) c 4 + ( 1 - y ) x c 2 + ( 1 - x ) ( 1 - y ) c 1 (2)

A numerical example will be provided later for Equations 1, 2 and 5 (which follows). Although a different weighting system would alter the proportions of the resulting payoffs, it would not alter " the basic nature" of the Nash equilibria within the game. 3

One equilibrium that results from the Threat Game is the deterrence equilibrium (DE). The initial strategies and retaliation functions which

produce payoffs (r3,c 3) are as follows:

Yo = 1, q(Xo) _- ql(Xo) (3)

x o = 1, P(Yo) -< Pl(Yo) (4)

where

C 3 - - XoC 2

ql(x°) = 1 - x o ( 1 - c a + c 2 ) ' O - < x o < 1, and (5)

r 3 - Yor2 Pl(Yo) = 1 - Yo(1 - r 3 +r2) ' 0 _ Yo < 1. (6)

121

At DE (or Yo--Xo--1) neither player preempts. Equations 3 and 4 create a Nash equilibrium because

any level of preemption by each player's opponent will be at least as costly (after retaliation) as no preemption. Hence, the players do at least as well, and generally better (if the inequalities are strict), by choosing no preemption initially ([Yo = Xo] = 1).

Equations 5 and 6 give the minimal levels of threatened retaliation needed to render (1,1) stable, yielding payoffs of PR(1,1) = r 3 and Pc(l,1) = c 3. If the superpowers seek to restore the nonpreemption position at (1,1), then the retaliation functions ql(Xo) and Pl(Yo) determine the threshold for rational response by the less coercive player. 4

Before exploring crisis-related implications, it is essential to deal with Zagare's (1985: 160) analysis of Chicken, Prisoners' Dilemma (PD) and deterrence:

Chicken may very well encapsulate the problem of mutual deterrence, especially in the nuclear age, wherein deterrent threats are inherently incredible. But if, as the theory of deterrence suggests, stable mutual deterrence requires that both players possess a credible threat, then Prisoners' Dilemma represents the solution to the problem of stable mutual deterrence.

Zagare argues that the nonmyopic equilibrium, a solution concept for PD developed by Brams and Wittman (1981), is consistent with the record of MAD. Thus it could be inferred that PD, not Chicken, should be used to generate a model of rational retaliation. However, Zagare (1985: 164) also has acknowledged that, when neither player prefers to punish the other's departure from the preplay position, "the game is Chicken". It will be argued at a later point that US-Soviet crises do indeed possess that property; the payoffs corresponding to combinations of pure strategies contain an outcome of mutual catastrophe, represented by nuclear war. By contrast, in PD, it is preferable to punish the adversary's departure from cooperation, in order to escape the "sucker's" payoff.

To summarize, the model of rational retaliation within the Threat Game entails more than a single round of play; that is, it deals with nonmyopic stability. Thus the current model is intended as an analogue to Zagare's PD-based solution to the game of nuclear deterrence.

Brams and Kilgour have not derived and tested propositions, although they do discuss the impact of varying the corner payoffs on the process of escalation. To take the analysis further, hypotheses focusing on rational choice and crisis outcomes will be formulated and tested. Certain aspects of the Threat Game, however, first will require revision.

122

One problem is that the path of crisis bargaining is not incorporated in the original model, as Brams and Kilgour (1987a: 848-849) have acknowledged. The fundamental problem with that approach is that Yo < 1 and x o < 1 may not emerge at about the same time in actual crises. In other words, what if the initial preemptive move of the more cooperative player always approximates full cooperation? If so, that would entail movement from (1,1) to (Xo,1), as opposed to a position (Xo,Yo) on the interior of the unit square. Thus an initial position, (Xo,1), would be reached by movement along the upper horizontal axis. When Row retaliates, the final position (x o, q(Xo) ) is reached. This process, according to Brains and Kilgour, constitutes a special case of the model, although the calculations focusing on the retaliation threshold would not be affected.

Symmetry is implicit in the Threat Game formulated by Brams and Kilgour. However, it is not clear that the superpowers, which would play the game in reality, attach equal values to the four pure outcomes. In other words, sym-

metric ordering of the alternatives would be preserved - as Brams and Kilgour require - but cardinal symmetry might not hold true. The most salient poten- tial difference concerns the preplay position, (r 3, c3). Perhaps the leader and challenger, to use the terminology of Organski and Kugler (1980), will view that situation differently. The US might be expected to see a smaller difference between c 3 and c 4 and a greater gap from c 3 to c 2 because it finds the status quo relatively more appealing. In other words, it would seek to preserve the status quo and avert losses. By contrast, as challenger for global leadership, the USSR would not be inclined toward a risk-averse outlook regarding the pure states of the game. Thus the eventual measurement of payoffs should allow for both identical (i.e., r i = ci) and symmetric but unequal (r i ~ci) values. 5

Even if the game is assumed to have cardinal symmetry, the identity of the defender could affect the predictive performance of the model. One superpower might conform more than the other to expectations generated by the retaliation function. Different roles could affect responses; the leading state might be more inclined to react vigorously to any threat from the challenger. It would be of some practical value to discover which (if either) of the con- tenders has followed the dictates of the model more consistently.

The meaning of threat is a further question arising about the sequence of play. A " th rea t " , in the narrow sense, is not an action but an assertion of intent to commit some act contingent upon the adversary's subsequent behavior. Crisis bargaining, however, frequently entails tangible, coercive acts as well as statements regarding intentions. To deal with this variation, a scale of coercion must grant more credibility, all other things being equal, to an action rather than a threat. More specifically, violence should be accorded more weight than non-violent military actions, which in turn would stand above threatening statements. In sum, "actions speak louder than words".

123

Having revised the model of superpower response, the three propositions may now be presented. With no loss of generality, each has Column as the initiator (or more preemptive player) and Row as the defender (or more cooperative

player), i.e., Yo > Xo" One proposition deals with the nature of Row's response:

Proposition 1: Row's reaction to a preemptive move by Column will meet or exceed the threshold represented by q(x o) < ql(Xo).

Given the unattractive nature of mutual non-cooperation, the players are assumed to prefer a return to the preplay position. The threshold noted in the proposition must be matched or passed in order to create an incentive for Column to pursue de-escalation.

Two propositions deal with the impact of the defender's choice on goal achievement specifically and, in more general terms, satisfaction with the outcome of the crisis:

Proposition 2: If q(Xo) _.< ql(Xo), it is more likely that the defender will achieve at least some of its specific goals in a crisis.

Proposition 3: If q(Xo) < ql(Xo), it is more likely that the defender will be satisfied with the outcome of the crisis.

Specific policy objectives and overall satisfaction for the defender depend upon selection of a sufficiently coercive response. The value for ql(Xo) sets the standard because (1,1) is the eventual target for the defender. The latter also seeks to preserve (and even advance) its own interests, not necessarily achieved at (1,1), a possibly Pareto-inferior point. Consequently, ql(Xo) is the relevant criterion in the appraisal of superpower response. By contrast, if q(Xo) > ql(Xo), there is no incentive for Column to return to the preplay position. Thus achievement of immediate goals and overall satisfaction for the defender are less probable when (x o, q(Xo) ) (with q(Xo) > ql(Xo)), rather than (1,1) is more appealing to the initiator.

Several measurements must be derived prior to testing the present model. An operational definition of crisis is required, in order to permit selection of relevant cases. The payoffs and levels of preemption which are central to the Threat Game require explicit measurement. Finally, the outcome variables must be rendered operational.

3. Selection of cases and operationalization of the model

Superpower crises comprise the subset of conflicts in world politics to which the Threat Game is directly relevant. To identify these cases, the micro-level

124

or foreign policy crisis defined by the International Crisis Behavior (ICB) Project (Brecher and Wilkenfeld et al., 1988: 3) is an appropriate description to follow: 6

A foreign policy crisis, that is, a crisis viewed from the perspective of an individual state, is a situation with three necessary and sufficient conditions deriving from a change in a state's external or internal environment. All three are perceptions held by the highest level decision-makers of the actor concerned: a threat to basic values, along with the awareness of a finite time for response to the external value threat, and a high probability o f involvement in military hostilities.

The trigger of a foreign policy crisis, which creates these perceptions, corresponds to some level of noncooperation by one of the adversaries. It constitutes the opening move, that is, a departure f rom the preplay position (1,1).

Column has been designated arbitrarily as the first player to commit a noncooperative act, thus triggering a foreign policy crisis for Row and moving the players from (1,1) to (x o, 1), the initial position. This description of the first move constitutes a special case of the theoretical treatment of crisis initiation and retaliation presented by Brains and Kilgour (1987a: 835). They asserted

that the initial strategy choices of Row and Column, Yo and x o respectively, are "simultaneous and determine a point on the unit square". However, in practical terms, the trigger and response moves cannot be treated as simultaneous choices. It is not uncommon in foreign policy crises for the principal response of a target state to occur several days - and sometimes much later than that - after a noncooperative act. 7 Therefore, Yo = 1 by definition, with the "initial posit ion" being (x o, 1).

Since it is the more cooperative player initially, Row then can retaliate at some level q(Xo). This action corresponds to the major response variable in the ICB data set. Thus the final position of the game is (Xo, q(Xo)), as in the case of the Brams and Kilgour model, although the path toward it is through

(x o,1). Termination of play in the Threat Game corresponds to removal of the nega-

tively valued psychological conditions which define a foreign policy crisis. For example, the US experienced a foreign policy crisis on 13 August 1961 because of the initial construction of the Berlin Wall. This action threatened a basic value held by the US: the viability of West Berlin. The Wall also created perceptions of finite time for response and heightened risk of war. The crisis ended for the US with the Soviet 22nd Party Congress, when Khrushchev "pro- nounced the Four Powers' Statute as invalid, signalling Soviet intention to keep the city divided. After that, no further dramatic developments occurred with respect to Berlin" (Brecher and Wilkenfeld et al., 1988: 253). Although

125

Khrushchev's assertion left the Wall standing, it also indicated that the USSR was at least temporarily satisfied with the situation. The US no longer had reason to perceive a crisis over West Berlin, so the players effectively had returned to the preplay position, (1,1), despite an obvious and seemingly permanent physical change in the terrain of Europe.

Table 1 displays the 27 foreign policy crises that are regarded as appropriate for testing the game-theoretic model of superpower rivalry and rational retaliation proposed by Brams and Kilgour. There are several reasons behind the selection of these cases from the 627 available in the ICB data set of foreign policy crises from 1929 to 1979 (Wilkenfeld and Brecher et al., 1988). 8

First, it is appropriate to include only cases that involve the USSR and the

US as adversaries. In order to test the analysis of Brams and Kilgour, one superpower must be threatened directly (or indirectly) by the other. This criterion eliminates intra-bloc crises such as Hungary in 1956, where one superpower threatens its own client state, not the more immediate interests of the rival.

Second, cases in which the client state of one superpower threatens the client state of another also are excluded. In these crises it can be virtually impossible

to attribute responsibility. Third, it is essential to focus on the period in which the superpowers had

some chance of reaching (rl,cl), the worst outcome on the unit square. This result, viewed in practical terms, would amount to a nuclear exchange. The United States achieved nuclear status before the end of World War II, while the USSR demonstrated its nuclear capacity in 1949 by testing a weapon suc- cessfuly. Of course, the USSR could not have used its nascent nuclear capabilities effectively against the US at that stage. But there are at least two reasons to believe that the US had to take account of Soviet nuclear forces by the year in question: One is that the impact of even one crude nuclear weapon would be viewed as a catastrophe by a target state, much more damaging than, for example, defeat in a battle involving conventional forces alone. Put differently, the psychological impact of Hiroshima and Nagasaki on the postwar world should not be underestimated simply because contemporary nuclear weapons are potentially far more destructive. Another reason for the US to have taken account of Soviet capabilities is that, although it is unlikely that the USSR

could have delivered a single nuclear weapon to the US' soil in 1949, even a small probability would be sufficient to make full scale war the least preferred option for the US.

Having considered the relevant arguments about the time frame, foreign policy crises in progress at the outset of 1949 will be included in the analysis because, at that time, the USSR had reached the threshold of nuclear status. While any choice for a transition point would be arbitrary, the inclusion of crises in motion by 1949 means that all cases in which (r 1,cl) could occur will

126

I o~

0 r~

2

0

x ~

x ~

0

\ ~ \ x < < \ \ \ \ \ \ < \ < \ \ \ \ x \ ~ 7 \

127

appear in the analysis. At the same time, foreign policy crises which terminated prior to the Soviet nuclear threshold year of 1949 are excluded. The 27 cases

listed in Table 1 include crises initiated as late as 1979, the current terminal point of the ICB data set.

Fourth, and last among the selection criteria, there are some instances in

which one superpower simultaneously threatens both the other superpower and a client state. For example, the US announced a blockade of all "offensive military equipment en route to C u b a " on 22 October 1962, triggering a crisis

for the USSR and Cuba (Brecher and Wilkenfeld et al., 1988: 290-291). It is

appropriate to treat such an incident as one crisis, directly involving the superpowers.

Having selected the cases appropriate for testing, payoffs within the Threat Game require operationalization. To simplify matters, only Column's payoffs

(cj, j = 1 . . . . . 4) will be presented, because the values for Row are identical in this game. Two measurement schemes will be presented; each entails different assumptions about the function representing the decline in payoffs f rom r 4 to

r 1 •

One approach is to treat the decline as uniform and equal for both actors,

with c 4 = 1, c 3 = 2/3, c 2 = 1/3 and c 1 = 0. Payof f c 4 would refer to unilater- al concessions made by the adversary, with c 2 being the reverse. Neither victory nor defeat would entail a nuclear strike; instead, the utility values correspond to gains or losses over the issue at hand, military, economic or others.

The status quo, c 3, is an intermediate value reflecting the situation at the outset of the crisis. Of course, c 1 represents nuclear destruction.

It could be argued that every national leader attaches a very high value to the status quo, c a, perhaps close to 1.0. However, crises which involve the superpowers are not that uncommon, indicating that the USSR and the US some-

times are willing to risk war in order to obtain concessions. The most obvious approach therefore is to use linear increments as a first approximation to the true values.

Another measurement option is to assume that the payoff intervals decline in magnitude differently for the US and the USSR. In other words, for the US, as indicated earlier, the relative value of the preplay payof f would be greater.

A straightforward way of representing this difference would be to expand the

gap between c 3 and c2, thus achieving the intended impact on Equation 5, with the US as the Column player. Appropriate values are c 4' = 1, c 3' = 0.750, c 2' = 0.250 and c 1' = 0. The difference between the intermediate payoffs for the two sets of values is 0.333 and 0.500, respectively. Each of the two measurement schemes will play a role at the stage of testing.

These measurements might be criticized on more general grounds: Why should the payoffs for both superpowers be treated as invariant across cases? An alternative approach would be to estimate the payoffs associated with the

128

Table 2. Intensity of superpower bargaining techniques

Technique Intensity Technique Intensity

i. nuclear strike 0.000 11. multiple including 0.611 non-violent military

2. full scale 0.072 12. external change, 0.661 conventional war political and military

3. indirect full scale 0.141 13. external change, non- 0.709 conventional war violent military

4. serious clashes 0.208 14. economic act 0.755 5. indirect serious 0.272 15. political act 0.800

clashes 6. minor clashes 0.334 16. internal verbal 0.843

challenge to regime 7. indirect minor 0.393 17. verbal act 0.884

clashes 8. multiple, including 0.451 18. other 0.924

violent military 9. internal physical 0.506 19. no action 0.963

challenge to regime 10. non-violent military 0.560 20. cooperation 1.000

pure outcomes for each crisis on an individual basis. While that might lead to

more accurate assessment o f relative utilities in some cases, it also would have

certain disadvantages. For example, if the model is to be useful for prediction,

utility values should not be estimated f rom each side's behavior in a crisis. In-

tensity o f preference among various outcomes therefore must be specified a

priori for each case, a more difficult task. 9 At this point it is appropr ia te to develop the measurements for x o, the act

o f noncoopera t ion that initiates a foreign policy crisis, and q(Xo), the action taken by the superpower on the defensive. The relevant variables f rom the ICB

data set are the crisis trigger and major response. The trigger o f a foreign policy

crisis creates, for the decisional elite o f the target state, a perception o f threat

to basic values, finite time for response and the likelihood of involvement in military hostilities. This constitutes a precise measurement o f the initial non-

cooperat ive act, x o. The major response is the act which best represents q(Xo),

the protec tor ' s reaction to its new situation. Table 2 displays a con t inuum of coercive acts, ranging f rom a nuclear strike

th rough full coopera t ion with the adversary. 1° The points on the scale correspond to the theoretically possible triggers and major responses in a given

superpower crisis. The scale points which appear are based on the ICB coding for major response and trigger, a l though there has been some adjustment o f

the categories for present purposes, n Some explanation of the scale values is in order. The polar points o f nuclear

129

strike = 0.000 and cooperation = 1.000 are self-explanatory. (These are the

only scale values explicitly designated by Brams and Kilgour.) The increments

between 0 and 1 are not evenly spaced. Each difference between scale points is considered to be lower in magnitude than the one preceding. In other words,

the difference between a nuclear strike and full scale conventional war out- weighs the gap between the latter and indirect full scale conventional war, and so on. The rationale is that more cooperative behavior leads to diminishing returns at the interstate level. Put differently, the decline across the categories f rom cooperation to nuclear war occurs at an accelerating pace.

Several functions could be used to generate scale values with the above-noted

property. A straightforward approach is to t ransform the intervals between the twenty scale points in the following manner:

0.693(sk) S k' = 2 - e (7)

where

s k' = t ransformed scale point (k = 1 . . . . . 20).

s k = linear scale point.

Each Sg would have the value ( k - 1)/19, so s 1 = 0, s 2 = 1/19 . . . . . s20 = 1. The t ransformed s k' then would have the desired exponential rate of decline.

Having described the coding of crisis bargaining techniques, a numerical

example can be used to illustrate the calculation of payoffs for Row and Column and the threshold value generated by the retaliation function. In Table

1, the case of Syria-Turkey Border places the US in the role of initiator (i.e., Column), with the USSR as protector (i.e., Row). The US made the first move,

x o = 0.88, and the USSR responded with q(Xo) = 0.61. Assuming the first set of corner payoffs for each player (0, 1/3, 2/3, 1), Equations 1 and 2 can be

used to calculate the payoffs to Row and Column, respectively, at (0.88, 0.61):

PR(0.88,0.61) = (0.88)(0.61)(2/3) + (0.88)(1--0.61)(1) + (1-- 0.88)(0.61)(1/3) + (1 -- 0.88)(1 -- 0.61)(0)

= 0.725

Pc(0.88,0.61) = (0.88)(0.61)(2/3) + (0.61)(1-0.88)(1) + ( 1 - 0.61)(0.88)(1/3) + (1 - 0.88)(1 - 0.61)(0). 0.545

(1 ')

(2')

With Row retaliating, Equation 5 is used to generate the threshold value for an acceptable level of response:

130

2/3 - (0.88)(1/3) q1(0"88) = 1 - (0.88)(1 - 2/3 + 1/3)

= 0.903

(5')

Since Row's response is 0.61, the response is sufficient according to the threshold conveyed by Equation 3.

From the perspective of the crisis actor, the results o f a crisis are the relevant

evidence upon which to evaluate its major response. Thus the dependent varia-

bles in the quantitative analysis to follow are goal achievement and overall satisfaction for the defender. The first variable is tr ichotomous, with the

categories being (1) full achievement of the specific goals of the crisis actor, i.e., victory, (2) stalemate or compromise and (3) failure to achieve goals, i.e.,

defeat. Satisfaction with the outcome, the second variable, refers to the overall

reaction of the actor to the situation resulting f rom the crisis. It incorporates goal achievement and the general strategic and tactical position of the actor. This variable is dichotomous, with the categories being (1) satisfied and (2) dissatisfied. 12

4. Analysis of data

Each of the Propositions 1 - 3 will be tested twice. The first set of results is generated under the assumption of cardinal symmetry in the payoffs. In other

words, c 1 = 0, c 2 = 1/3, c 3 = 2/3 and c 4 = 1 for the US and the USSR alike. The second set of results is derived by another scheme of payoffs. The USSR still is assumed to have the above-noted values. However, the US is judged to

favor the status quo more and has the payoff values c 1' = 0, c 2' = 1/4, c 3' = 3/4 and c 4' = 1. The thresholds generated by each scheme of payoffs appear in Table 1 as T1 and T2 respectively.

Proposition 1: In responding to the initial provocation, x o, Row is expected to fulfill the threshold represented by q(x o) _ ql(Xo).

This proposition is supported by the results generated by both sets of payoffs. In the former instance, 19 out o f 27 (70.4%) of the defenders responded below or at the threshold value ql(Xo). Thus in most cases the defender 's response gave the initiator an incentive to prefer a return to the status quo. The difference in mean level of response (q(xo)) compared to the threshold value (ql(xo)) also is noteworthy. Assuming ~ = 0.657 as a sample mean, with N = 27, the 95% confidence interval is f rom 0.558 to 0.756. a3 The value qa - ]~ = 0.796 therefore lies beyond the boundaries of that confidence interval, suggest- ing that the mean response level is lower than the mean threshold value. In

131

other words, defenders generally retaliated with sufficient force to induce rational pursuit of de-escalation by the initiator.

With respect to the results generated by the second set of payoffs, 20 out of 27 (74.1%0) of the defenders responded at or below the threshold value. The combined average value for qla(Xo) and qlw(Xo), corresponding to Soviet and US cases, respectively, also falls beyond the 95% confidence interval. This result indicates that the mean response and threshold values are significantly different.

The seven crises in which the defender failed to respond adequately share an interesting trait: each of these cases occurred from 1961 onward. This result suggests that the relative emphasis on crisis stabilization at (Xo,Yo) - as opposed to efforts toward an immediate return to the preplay position - has in- creased over the years. These milder responses could reflect increasing awareness of the dangers of confrontation. Rather than seeking a direct return to the preplay position, the defender in a given crisis might favor stabilization of the initial position and, under certain conditions, a response above the threshold would be consistent with such an objective.

For each of the seven anomalies, a position (x o, q(Xo) ) can be identified from the data displayed by Table 1. For example, that is (0.21, 0.88) in the Bay of Pigs Crisis. If the objective of the USSR - the defender in that case - is assumed to be stabilization of (0.21, 0.88), as opposed to a return to (1,1), the

threshold of retaliation is calculated as follows: %.21,0.88 (0.21) = 0.88. (The method of calculation is explained in Brams and Kilgour (1987a).) This result means, not surprisingly, that the USSR would have to escalate its response in order to make (0.21, 0.88) stable for the US. The Bay of Pigs and the other anomalies could be studied in greater depth, in order to see whether subsequent actions by the defender suggested pursuit of crisis stability. But even if all of the seven cases had that property, it still would be essential to explain why defenders in other crises of the 1960s and 1970s appeared to prefer an immediate return to the status quo. Further research will be required to account for this variation from 1961 onward.

Propositions 2 and 3 also are tested with each set of payoffs:

Proposition 2: If q(Xo) _< ql(Xo), it is more likely that the defender will achieve at least some of its specific goals in a crisis.

Proposition 3: If q(x o) _< ql(Xo), it is more likely that the defender will be satisfied with the outcome of the crisis.

For the first set of payoffs, goal achievement is linked to response level. For the 19 cases in which the response level is below or equal to the threshold, goal achievement is as follows: 52.6% success, 42.1% stalemate or compromise,

132

and 5.3°70 failure; the absolute frequencies are i0, 8 and 1 cases, respectively. Among the 8 cases for which the response is above the threshold, the percen-

tages and frequencies are 37.5%/3, 37.5070/3 and 25.0070/2, respectively. The connection involving response level and goal achievement is non-negligible (tau b = 0.20). The results generated by the second payoff scheme are stronger (tau b = 0.29). For the 20 cases with a response below or equal to the threshold, the respective percentages and frequencies are 55.0070/11,40°70/8 and 5.0°/0/1. Among the 7 cases above the threshold, the values are 28.6%/2, 42.9070/3 and 28.607o/2.

With regard to Proposition 3, the results generated by the first payoff function offer solid support (tau b = 0.36). For the cases in which the response level is below or equal to the threshold, actor satisfaction is as follows: 80.0%/16 satisfied and 20.0070/4 dissatisfied, expressed in percentages and case frequencies, respectively. Among cases for which the response is above the threshold, the respective values are 42.9%/3 and 57.1°70/4. This proposition is more strongly confirmed by the results generated by the second set of payoffs (tau b = 0.42). For cases with a response level below or equal to the threshold, the respective percentages and frequencies are 85.0070/17 and 15.007o/3. Among the cases above the threshold, the values are 42.9%/3 and 57.1 °70/4. Although cases with the defender's response above the threshold are divided about evenly in the tables, at least 16 out of 20 (80.0070) of the crises in which q(x o) matched or fell below the threshold resulted in satisfaction for the actor.

Three other observations can be made about the results, collectively speak- ing. The first concerns relative performance by the two sets of payoffs. A second point pertains to the impact of the defender's identity on the predictions of the model, while the third focuses on direct versus extended rivalry.

With respect to the initial issue, the second set of payoffs consistently produced better results than the first. Although the difference should not be exaggerated, when cardinal symmetry is replaced with the assumption that the US values the status quo more than does the USSR, the model appears to be more valid.

A second aspect of the results as a whole concerns the identity of the defender. It had been suggested that the behavior of one superpower might be more consistent with the model than that of the other. But consider, for example, the anomalies from the second phase of testing for Proposition 1. The US plays the role of protector or defender in 2 of those 7 cases, while for the data set as a whole it is the protector in 10 of 27 crises. The respective proportions, 28.6070 and 37.0°70, are not dramatically different. Analogous figures for other stages of testing are similar, in that neither superpower appears to follow the dictates of the model more consistently.

A third point focuses on cases of direct versus extended rivalry. The possibil- ity had been raised that cases in which the target and protector had the same

133

identity (e.g., US as a crisis actor in the Berlin Blockade) would follow a different pattern from those in which a superpower protected a client (e.g., the USSR, with North Vietnam as a crisis actor in Gulf of Tonkin). Once again referring to the second phase of testing from Proposition 1, very little difference emerges between these two types of cases. There are 3 crises among the anomalies in which the identity is the same (42.9%); the proportion overall is 13 of 27 (48.1%). The figures elsewhere in the analysis are quite similar.

5. Conclusion

Theoretical revision, measurement and testing have combined to produce en- couraging results. The ql(Xo) threshold appears to have predictive power with respect to superpower retaliation in crises. The threshold also is able to distin- guish the goal achievement and overall satisfaction levels of defending states. These results are stronger when the US - commonly cast in the role of hegemonic power - is assumed to place a higher value upon preserving the status quo.

Some defenders failed to meet the ql(Xo) threshold. It is especially interesting that the cases of inadequate retaliation occurred from 1961 onward. Perhaps a model with a different set of parameters is required to account for these cases. For example, it could be that, in more recent decades, the corner payoffs have become linked to the specific value attached to a client state.

With respect to the outcome variables, each of the analyses contains deviant cases in need of further explanation. More long-term objectives include the operationalization and testing of other stages in the Threat Game, referring to optimal threats, crisis stability and threat escalation. At this point, it is fair to say that the model of retaliation has received tentative support, thus confirm- ing the view that crises can be understood as rational interactions.

Notes

1. The matrix which appears below represents the basic game of "Chicken", with standard ordi-

nal payoffs for Row and Column, respectively (i.e., r 4 > r 3 . . . ) ;

The Game of Chicken

Column

Swerve Do not swerve

Row

Swerve

Do not

swerve

r3,c3

r4,c 2

r2,c 4

rl,c I

134

2. The following description of the Threat Game is based on Brams and Kilgour (1987a: 835-843) and (1987b).

3. With regard to weighting the corner payoffs and deriving equilibria, the most salient alternative to the approach taken by Brams and Kilgour would be Euclidean distances. The weight of each pure payoff would be based upon its distance from the point (x,y) within the unit square. However, the desired property of responding to cooperation (i.e., Xo= 1 or Yo = 1) with continued cooperation is not available with a Euclidean weighting scheme for the corner payoffs, because the expression for a deterrence equilibrium would entail division by 0. The somewhat unconventional approach pursued by Brams and Kilgour has the advantages of

mathematical tractability and treatment of the factor weights as "functions of the mixed (i.e., probabilistic) strategies of the players, who can be viewed as choosing strategy mixtures ( x , 1 - x ) and (y,1 - y ) " (Brams and Kilgour, 1988: 108).

4. Brams and Kilgour (1987a) also have identified points of crisis stabilization at (Xo,Yo), the initial position, with and without threat escalation.

5. Of course, the utilities for all but c I could be allowed to vary from one crisis to the next. In testing the revised model, however, the corner payoffs remain constant across cases.

6. This definition, which forms the basis of a 627-case data set from 1929 to 1979, is derived from Brecher (1977: 43-44). (Data through 1985 will be available in the fall of 1989.) Two coders gathered the data for each case and showed an overall reliability of 0.85.

7. There are cases in which the trigger and major response occur in close proximity. However, in the cases that appear in Table 1, the average time lag is over one month.

8. The selection criteria are explained further by James and Harvey (1989a; 1989b). 9. This might suggest the use of data such as that o f Bueno de Mesquita (1981) or James (1988).

However, over the last four decades, the relevant figures for the Soviet-US dyad are quite stable. Thus building in these numbers from one case to the next would greatly complicate the model with only a marginal impact on the results.

10. With respect to triggering acts, it is obvious that neither cooperation nor a lack of action (the final two scale points) can trigger a crisis. However, the Threat Game is assumed to be in progress at all times. Thus each of these non-threatening forms of behavior may be regarded as a choice available to the players at any given moment.

11. The ICB categories and coding procedures are explained in detail by Brecher and Wilkenfeld et al. (1988) and Wilkenfeld and Brecher et al. (1988).

12. The contingency coefficient for specific goal achievement and overall satisfaction is 0.58. This positive, moderate value is appropriate because the two variables should be linked, but not to the extent that one is merely redundant.

13. The confidence interval's boundaries are derived as follows: There are 27 cases in Table 1, with the average level of q(x o) equal to 0.657 and a standard deviation (S.D.) of 0.262. The lower boundary of the confidence interval around the mean of q(Xo) is:

Z.o25 (S.D.) L = q(Xo) (n)l/2

1.96(0.262) = 0.657

5.196 = 0.558

The upper boundary (0.756) is derived by adding the quantity after the mean of the observed values. The average threshold value, 0.796, lies outside the confidence interval.

135

References

Brams, S.J. (1985). Superpower games: Applying game theory to superpower conflict. New Haven: Yale University Press.

Brains, S.J. and Wittman, D. (1981). Nonmyopic equilibria in 2 x 2 games. Conflict Management and Peace Science 6: 39-62.

Brams, S.J. and Kilgour, D.M. (1987a). Threat escalation and crisis stability: A game-theoretic analysis. American Political Science Review 81: 833-850.

Brams, S.J. and Kilgour, D.M. (1987b). Optimal threats. Operations Research 35: 524-536. Brams, S.J. and Kilgour, D.M. (1988). Game theory and nationalsecurity. Oxford and New York:

Basil Blackwell. Brecher, M. (1977). Toward a theory of international crisis behavior: A preliminary report. Inter-

national Studies Quarterly 21: 39-74. Brecher, M. and Wilkenfeld, J., et al. (1988). Crises in the twentieth century, Vol. I: Handbook

o f international crises. Oxford and New York: Pergamon Press. Bueno de Mesquita, B. (1981). The war trap. New Haven: Yale University Press. Hirshleifer, J. (1988). The analytics of continuing conflict. Synthese 76: 201-233. James, P. (1988). Crisis and war. Montreal and Kingston: McGill-Queen's University Press. James, P. and Harvey, F. (1989a). Threat escalation and crisis stability: Superpower cases,

1948-1979. Canadian Journal o f Political Science XXII: 523-545. James, P. and Harvey, F. (1989b). Nuclear deterrence theory: The record of aggregate testing. Un-

published manuscript. Morgan, T.C. (1984). A spatial model of crisis bargaining. International Studies Quarterly 28:

407-426. Organski, A.F.K. and Kugler, J. (1980). The war ledger. Chicago: University of Chicago Press. Wilkenfeld, J. and Brecher, M., et al. (1988). Crises in the twentieth century, Vol. II: Handbook

o f foreign policy crises. Oxford and New York: Pergamon Press. Zagare, F. (1985). Toward a reformulation of the theory of mutual deterrence. International

Studies Quarterly 29: 155-169.

Documents

Public Choice 68: 117-135, 1991. Kluwer Academic ... · Figure 1. The threat game. rence: averting conflict through the threat of retaliation, which, if carried out, could be costly