Published online by Cambridge University Press: 13 June 2011
The relation between stability and the distribution of power is an important and long-debated problem in international relations theory. The balance-of-powcr school argues that an even distribution of power is more stable, while the preponderance-of-power school argues that a preponderance of power is more stable. Empirical efforts to estimate this relation have yielded contradictory results. This essay examines the relation between stability and the distribution of power in an infinite-horizon game-theoretic model in which two states are bargaining about revising the international status quo. The states make offers and counteroffers until they reach a mutually acceptable settlement or until one of them becomes so pessimistic about the prospects of reaching an agreement that it uses force to impose a new settlement. The equilibrium of the game contradicts the expectations of both schools and offers an explanation for the conflicting empirical estimates. In the model stability is greatest when the status quo distribution of benefits reflects the expected distribution of benefits that the use of force would impose.
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3 Bueno de Mesquita and Lalman (fn. 2, 1992), 204–5,190.
4 Fearon, James, “War, Relative Power, and Private Information” (Paper presented at the annual meeting of the International Studies Association, Atlanta, March 31-April 4, 1992), 20Google Scholar.
6 Although the potential use offeree is not the source of the coercive pressure, some suggest that international regimes are more likely to break down when the underlying distribution of power differs from the distribution of benefits the regime confers. See Keohane, Robert and Nye, Joseph, Power and Interdependence (Boston: Little Brown, 1977), 139Google Scholar; and Krasner, Stephen, “Regimes and the Limits of Realism,” in Krasner, , ed., International Regimes (Ithaca, N.Y.: Cornell University Press, 1983), 357–58Google Scholar
7 Fearon (fn. 4).
8 Bueno de Mesquita and Lalman (fn. 2,1992), 190, 204–5.
9 In a more general model the two states might be bargaining about altering the international status quo where the set of feasible outcomes is an n-dimensional policy space. Each point in this space represents a different international order and a different distribution of benefits for the two states. The formal analysis developed below applies equally well to this more general formulation.
10 Organski (fn. 1), 294.
11 Wolfers (fn. 1), 120. See also Claude (fn. 1), 62.
13 Rubinstein (fn. 12) showed that this game has a unique subgame-perfect equilibrium.
14 More formally, U1 is assumed to be twice differentiate with U1> 0 and U1≤0. S1 is risk neutral if U1=0 and risk averse if U1 < 0.
15 There are no offensive or defensive advantages in the present formulation, so it makes no difference whether S1, or S2 attacks.
16 See Levy (fn. 1), 231–32; Mansfield (fn. 2, 1994), 17–18; and Wright (fn. 1), 755.
17 For efforts to examine the relation between stability and the distribution of power in a setting in which there are more than two states, see Niou, Emerson and Ordeshook, Peter, “Stability in Anarchic Internatonal Systems” American Journal of Political Science 84 (December 1990)Google Scholar; Wagner (fn. 1); and idem, “The Theory of Games and the Balance of Power”, World Politics 38 (July 1986)Google Scholar.
18 Wright (fn. 1), 755.
19 Mearsheimer (fn. 1), 18; emphasis added.
22 For efforts in this direction, see Fearon (fn. 4); Kim, Woosang and Morrow, James, “When Do Power Transitions Lead to War?” American journal ofPolitical Science 36 (November 1992)CrossRefGoogle Scholar; and Powell, , “Appeasement as a Game of Timing” (Manuscript, Department of Political Science, University of California, Berkeley, July 1995).Google Scholar
22 Blainey (fn. 1).
23 Fearon (fn. 4) first makes and develops this point.
24 Wagner (fn. 1).
25 As will be seen, at most only one state can be dissatisfied.
26 This follows from the assumptions that the states are risk neutral or risk averse, that they agree on the distribution of power, and that fighting is costly. Because the states are risk neutral or risk averse, the utility functions are concave, which implies U1 (q)≤ q and U2 (1 –q) ≤ 1– q. If both states are dissatisfied, then p – c1 < U1 (q) ≤q and 1 – p – c2 < U2 (1 –q)≤ 1–q Adding these inequalities gives 1 –1 - c2 < 1 or 0 < c1 + c2. But fighying is costly, so c1 ≤ 0 and c2 ≤ 0. Thus, c1 + c2 cannot be less than zero, and this contradiction implies that at least one state must be satisfied.
27 The complete-information game is a straightforward modification of Rubinstein's (fn. 12) model with an outside option. For an analysis of the unique subgame-perfect equilibrium of this game, see Osborne, Martin and Rubinstein, Ariel, Bargaining and Markets (New York: Academic Press, 1990), 54–58.Google Scholar
28 If S2 preferred fighting to offering, x then 1−p − c2 < U2(1-x) where p−c1 = U1(x) Adding these relations and recalling that U2 U2 are concave leave the contradiction 0 < c1 + c2.
29 These cumulative distributions functions are assumed to have continuous densities that are positive over the intervals (c1, c1) and (c2, c2). These distributions are also common knowledge.
30 Although the outcome is unique, there are multiple equilibria because different off-the-equihbrium-path beliefs will support this equilibrium. The fact that there is a unique outcome is surprising. Typically in bargaining games in which an informed bargainer (i.e., a bargainer with private information) can make offers, there is a multiplicity of equilibrium outcomes. For an excellent introduction to bargaining models, see Fudenberg, Drew and Tirole, Jean, Game Theory (Cambridge: MIT Press, 1991)Google Scholar. A good survey is also found in Kennan, John and Wilson, Robert, “Bargaining with Private Information”, Journal ofEconomic Literature 31 (March 1993).Google Scholar
31 Robert Powell, “Bargaining in the Shadow of Power”, Games and Economic Behavior (forthcoming)
32 To see that this offer will be accepted immediately, recall that ẋ is designed to ensure that the dissatisfied state cannot do better than accepting ẋ by fighting instead. In equilibrium, the dissatisfied state will never reject ẋ in order to fight. And since S2 will never offer more than ẋ the dissatisfied state can gain nothing by holding out for a better offer. Indeed, the dissatisfied state will lose by not reaping the benefits from a favorable shift in the status quo from q to x as soon as possible. Thus, the dissatisfied state will accept a counteroffer of ẋ immediately.
33 To sec that payoff is strictly less that W1(0), note that the definition of S2 implies that S is indifferent to accepting ŷ now, which leaves S 2 with 1 − ŷ forever, and settling on ẋ in the next period, which leaves S2 with 1 − ẋ forever. Thus U 2(1 − ŷ)/(1 − δ) = U2(1 − q) + δU2(1 − ẋ)/(1 − δ).But S2 preface the status quo share of 1 − q to what it will have after seeling on either 1 − ŷ or 1 ẋ. That is, 1 − q > 1 − ŷ. Given 1 − q > 1 − ŷ, then the previous equality implies 1 − ŷ > 1 − ẋ or, equivalently, ẋ > ŷ. finaly, the fact that ẋ > ŷ > q leaves W1(0) = (p − ŷ1)/(1 − δ) = U1(ẋ)/(1 − δ) > U1(q) + δ U1(ŷ)/1 − δ).
34 The analysis of the case in which x = q is completely analogous
35 The expression for P(x) was derived on the basis of the assumption that x > q. If x ≤ q, s2's expected payoff to offering x is P(q). Accordingly, s2 is indifferent to offering any x ≤ q and it will suffice to maximize P over the range x ≥ q.
36 s1 rejects x if p − c1 > U1(x) or p − U1(x) > c1. Given that 1 is distributed according to the cumulative distribution function F1, the probability that c1 is less than p − U1(x) is F1(p − U1(x)).
37 Although cumbersome to show, the probability that the bargaining will end in war is essentially the same regardless of whether the satisfied state or the potentially dissatisfied state makes the initial offer. See Powell (fn. 31) for the details.
38 The specific value of 1/4 at which the function levels off is a result of the particular assumption that the states' costs are uniformly distributed between the bounds of zero and one. Different assumptions about these bounds would cause the function to level off at a different value.
39 Bueno de Mesquita and Lalman (fn. 2,1992).
41 Ibid., 190. In a second version of their game, Bueno de Mesquita and Lalman let the state's demands be determined as part of their equilibrium strategies. But this game has complete information. The combination of endogenous demands and complete information means that the probability of war is zero in their model—as well as in the model analyzed above—regardless of the distribution of power. Bueno de Mesquita and Lalman do not examine the case in which there are both endogenous demands and incomplete information.
42 Bueno de Mesquita and Lalman (fn. 2,1992), 190.
43 Strictly speaking, the probability of war increases as ∣p − q∣ increases only as long as ∣p − q∣ < 1/2. After this, the probability of war levels off.
44 Fearon (fn. 4), 20.
46 Siverson andTennefoss (fn. 2).
47 Kim (fn. 2,1992); and Moul (fn. 2).
48 Singer, Brerner, and Stuckey (fn. 2).
49 Maoz (fn. 2); and Bueno de Mesquita and Lalman (fn. 2,1992).
50 Unfortunately, the theoretical results derived here, while indicating the importance of controlling for the status quo, do not identify a means of doing so. The basic problem is to find a way of measuring a state's utility for the status quo. Bueno de Mesquita and Lalman's (fn. 2, 1992) measure of this utility may offer a start in this direction, but their current formulation is inadequate. If the states are risk neutral as in the example above, then Bueno de Mesquita and Lalman's measure reduces to assuming that the status quo distribution is constant in all cases. In terms of the example above, they are in effect assuming that q always equals 1/2. (To obtain q = 1/2, assume the state is risk neutral by taking rA = 1 in equation A1.3 in Bueno de Mesquita and Lalman (fn. 2, 1992), 294, and normalize the state's utility to be one if it obtains everything it demands and zero if the other state obtains everything it demands by setting UA(δA) = 1 and UA(δB) = 0.) Controlling for the status quo, however, requires the status quo to be treated as a variable across cases, and it is unclear how to do this with their measure as it is currently formulated.
51 To see this, suppose that the distribution of power closely mirrors the status quo distribution of territory in that ∣p − q∣ is less than the minimium of the lowest possible costs of fighting, i.e., the minimum of c1 and c2 Then, p − q ≤ c1, and q − p ≤ c2. These inequalities and the conavity of U1 and U2 imply p − c1 ≤ q ≤ U1(q), and 1 − q − p − c2 ≤ 1 − q ≤ U2(1 − q). Thus, both S 1 and S 2 are satisfied, and the probability of war is zero.
52 The case of risk-acceptant states introduces nonconvexities and technical difficulties, and the analysis of this case remains a task for future work.
53 More formally, suppose that the distributions of cost F1 and F2 were also a function ofp and that the mean and variance of F1 decrease in p while the mean and variance of F2 decrease in 1 − p.
54 Even if F1 and F2 depend on p the argument in footnote 26 goes through with only minor modification.
55 The model's answer is only partial because the shape of the function relating the probability of war to the distribution of power, while always zero when p = q, is likely to depend on the states' utility functions and on the functional forms of the probability distributions representing their beliefs.