American Political Science Review Vol. 93, No. 4
Structure of the Game
1. The leader in nation A chooses between war and negotiations. If she selects war, then she also
chooses how hard to fight, gA E [0, 1]. 2. If war occurs, then the leader of B, having observed
A’s effort level, chooses how hard to fight, gB E [0, 1].
3. Nature determines the outcome of the war. 4. In each nation, the members of the winning coali-
tion, having observed the international outcome and the level of private goods allocated to them, decide whether to retain their leader or defect to a domes- tic political rival, thereby removing the current leader from office.
We assume that only A can start a war. This assump- tion is benign. The question of whether B wants to initiate is answered by simply flipping the labels A and B. Our central question is how political institutions affect crisis behavior.
Settling Crises by War
We model war as a costly lottery in which each player’s expected utility from the war depends on the probabil- ity that its side will win or lose and the utilities associated with each possible outcome. In this section we develop our notions regarding the probability of victory (and defeat) and the attendant utilities.
The values of victory and defeat are normalized to one and zero, respectively. In addition, players pay a per-capita cost, k, associated with the war’s destruction and the risks of fighting. Therefore, the utility of victory equals 1 – k, and the utility for defeat is -k.
Many factors shape the outcome of a war. Observ- able military capabilities certainly play an important part, as do short-term shifts in government priorities by putting more national resources behind a war effort (Kim and Morrow 1992; Organski and Kugler 1980; Powell 1996a, 1996b). ~The probability of victory is presumed to increase as the total military advantage dedicated to the war effort by one side grows relative to the other side. Therefore, the victor is likely to be the nation with the most total military capabilities dedi- cated to the war.
We consider two types of military capabilities: the observable military balance, M, and the proportion of additional national resources committed to the war
effort, gi. The subscript will refer to nation A or B, as appropriate. The military balance, which takes values between 0 and 1, represents the ratio of observable military assets of the two sides. Additional resources dedicated to the war by either country are drawn from the Ri resources each leader has at her disposal. By choosing to devote the proportion gi of Ri to the war effort, she generates an additional gir in military assets, where r represents the exchange rate between re- sources and military capability.
The probability that A wins in a war, denoted PA = PA (gA, gR, M), is increasing in military balance, M, and A’s effort, go, and is decreasing in B’s effort, gB. The probability B wins is PB = 1 – PA. A general method
of modeling this process is to treatpA as the probability that a random variable a, with distribution 4>(E), is less
than a function of the variables h (gA, gB, M), a common example of which is the probit model, where
4>(E) is the standard normal distribution, and h(gA, g9B M) = M – + gAr – r.7
Negotiations
When nations enter negotiations, we assume they have expectations about the likely outcome of the bargain- ing process. In particular we assume the expected
rewards for A and B from a negotiated settlement are
X and 1 – X, respectively. We might suppose this deal, X, reflects the relative military balance, M, but it need not. Other factors can also be influential. As Fearon (1995) points out in his bargaining model, the indivis- ibility of the policy space, the sustainability of deals in
the future, and the role of current negotiations in altering the future military balance all influence which deals can be achieved through negotiations.8
The payoff to A from a negotiated settlement is X, which is enjoyed by all citizens in A. That is, this payoff is a public good. The citizens in B enjoy the benefit 1 –
X from negotiation.
UA(negotiations) = X; UB(negotiations) = 1 – X.
Table 1 summarizes the policy payoffs associated with each possible outcome of the international dis- pute. We let z represent the generic outcome. Every- one in A receives the policy payoff VA(z) associated with the outcome z, and all members of polity B receive the payoff VB(z).
TABLE 1. The Policy Payoffs Associated with International Outcomes
Outcome VA(Z) VB(Z) Negotiations X 1 – X A wins the war 1 – k -k A loses the war -k 1 – k
7 Other common examples used in international relations include Bueno de Mesquita, Morrow, and Zorick (1997), who set
?()={1 if :- 0
and h(M, mA, mB, gA, gB) = mA + 2(M – 1) – mB + gAr – gBr, where mA and mB are intangible military utilities, and the standard ratio of forces model (see, e.g., Bueno de Mesquita 1981), which sets
0 if&OF i
?(? = if ?E C[0, 1] 1 if,> 1
and h(M, gA, gB) = (M + rgA)/(l + rgA + rgB). Hirshleifer (1989) examines the implications of variation in these functions. 8 Nations’ private information about their capability, which may take the form of intangible military assets of morale and leadership (Reiter and Stam 1998a, 1998b; see also Morrow 1989), also influ- ences negotiations.
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An Institutional Explanation of the Democratic Peace December 1999
Reselection
Following the international dispute, the leaders in each nation face reselection. The members of the selector- ate evaluate the payoff they received under the incum- bent. They compare this with what they expect to receive if they depose the incumbent and choose a domestic challenger. Deposition is not simply a matter of concluding that the leader did a poor job during the dispute. Rather, it is a question of whether the mem- bers of W believe they will be better off under alterna- tive leadership.
Incumbents are deposed when they can no longer convince W members of the selectorate to support them. If the package of benefits an incumbent offers to her supporters is better than the rewards any chal- lenger can credibly offer, then the incumbent can find W members of the selectorate who will retain her in office. If, however, the incumbent fails to provide benefits to the winning coalition in excess of what a challenger can credibly promise to provide, then the incumbent can no longer garner enough support to form a winning coalition. At this point, supporters defect, and the incumbent is ousted.
Defection is risky. There is no shortage of challeng- ers and prospective defectors. Consequently, a member of the current winning coalition cannot be certain of being essential to the successor government. That is, by defecting, members of the winning coalition place their private goods benefits at risk. We characterize that risk as increasing as the pool of available members for a winning coalition increases and as decreasing as the necessary size of the winning coalition increases, so that the risk increases as S increases and decreases as W increases. For simplicity’s sake, we treat the proba- bility of being a member of the successor winning coalition in A as WAISA. Then, the expected utility from private goods in a new coalition if one defects to a challenger is simply (RAIWA) (WAISA) = RAISA.
Like the incumbent,Athe challenger proposes a mix of public and private goods allocations. Of course, the selectorate does not know what the challenger can actually deliver, whereas they have observed the per- formance of the incumbent. The selectorate must infer the ability of the challenger (Riker 1996). We denote the competence, strength, or ability of the challenger in A as CA and use comparable notation for the challenger in B.9 When the incumbent is making choices about whether to fight or negotiate and how to allocate
9 In autocracies and other nondemocratic regimes, campaigns may not be public but instead take such forms as secret discussions about a coup, assassination, or other means to overthrow the incumbent. Success depends on attracting enough supporters to depose the sitting government and replace it with one led by the challenger. Because of the risks (captured by our notion that the probability of being essential to a successor government is equal to WIS), challeng- ers in autocracies generally operate in secret. It would be a mistake to believe that challengers do not always exist. Even in the most autocratic regime there are always people eager to take over the role of leadership. One obvious indication is that following the sudden death of an incumbent there is always someone willing to step in and assume leadership. Frequently, many people compete for the oppor- tunity, although the closest supporters of the leader are the obvious candidates.
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resources, she is uncertain of the qualities of a prospec- tive domestic rival. We represent the distribution of possible challengers with the cumulative density func-
tion FA(X), where FA(x) = Pr(CA ? x). For technical convenience, we assume the distribution of challengers is exponential: FA(x) = 1 – e-x
What can a challenger credibly offer the selectorate? He cannot make credible promises regarding how he will perform during a dispute or on other policy questions. Knowing this, the selectorate’s members focus on the reservation value they expect if they choose a new leader. We assume that the reservation value for picking the challenger is CA + RAISA.
Incumbents can anticipate what they must give sup- porters to defeat challengers. They simply must pro- vide more utility for their coalition members than that offered by the challenger, or
RA (1 A) – + RA + VA (Z),
where Vi represents the performance of the leader on all policy dimensions other than the international dispute. This utility term is quite intuitive. VA (z) is the utility supporters derive from the outcome of the policy
of the leader in the international dispute.10 (1 – gA) is the proportion of resources reserved for distribution as private goods to the winning coalition after spending
gA on the war effort. Of course, if there is no war, then gA = 0. The total pool of resources (RA) is diminished by whatever portion goes to the war effort, if any. What remains is distributed evenly to the members of the winning coalition so that each member receives (1 –
gA) RAIWA. Of note is that members of the winning coalition are assumed to be certain to receive their share of private goods if they stay with the incumbent, but they only receive such goods probabilistically if they defect. In addition, neither the incumbent nor the challenger can promise to distribute any resources that are destroyed or lost by the state during a war. This proves important later in understanding why autocrats do not make the same allocation decision as demo- cratic leaders.
Leaders want to remain in office. At the time leaders choose their actions, including whether to wage war and how to allocate resources, they cannot be certain of the quality of prospective political rivals. Yet, given the distribution of possible challengers, the incumbent can assess how outcomes influence her prospects for reselection. Leader A remains in office as long as she offers her winning coalition at least as much as a potential challenger credibly promises. Hence, the probability the incumbent remains in office given out- comezandeffortgAisPr((l – gA)RAIWA + VA(z) +
10 We assume here that all members of society benefit equally from international outcomes. As earlier work shows (Bueno de Mesquita et al. 1997), however, foreign policy objectives are a function of institutional arrangements. Countries with small winning coalitions typically fight for resources that can be targeted as private goods to the regime’s supporters. By contrast, countries with large winning coalitions fight over policies that have larger public goods compo- nents.
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American Political Science Review Vol. 93, No. 4
[WA 2 CA ? RAISA) = Pr((1 – gA) RAIWA- RAISA + VA(z) + I-A 2 CA) = F(VA(z) + (1 – gA) RAIWA – RAISA + PAA)
All else equal, leaders who perform well on other
issues (large Vi) and those with large selectorates (large S) find it easiest to remain in office. When the selec- torate is large, challengers can draw their political backing from a large pool of potential supporters. This means supporters of the current incumbent cannot be certain of also being included in future coalitions. Hence, members of the current winning coalition only probabilisitically receive private goods under future leaders. By contrast, current members of the winning coalition receive private goods with certainty from the incumbent. This forms the basis of the leader’s incum- bency advantage.
The size of the leader’s incumbency advantage de- pends upon the configuration of the polity’s political institutions. The smaller the selectorate, the greater are the private goods benefits members of the current winning coalition can expect from any challenger and, therefore, the greater the private benefits the incum- bent must provide in order to remain in power. Simi- larly, as the size of the required winning coalition decreases, the number of people with whom private benefits must be shared decreases, which makes the value of the benefits to each member that much greater. Therefore, as the winning coalition becomes larger, the incumbency advantage diminishes because the value of the private benefits to individual members of the winning coalition gets smaller. When the win- ning coalition is small and the selectorate is large, supporters of the incumbent jeopardize their welfare if they defect to a political rival of the incumbent, since they face a high risk of being cut off from private benefits under the new leader. The risk of being excluded from the private payoffs of future coalitions grows as the size of the selectorate increases and as the size of the winning coalition decreases, so that the risk is greatest in autocracies and smallest in democracies.
The incumbent is advantaged in her ability to supply private goods because current members of the coalition are sure of receiving them. Given her advantage in private goods, the incumbent survives so long as she does not do such a poor job on public policy that she is judged grossly incompetent as compared to the chal- lenger. That is, as long as CA is not especially large or p. strongly negative, the incumbent has the upper hand. What constitutes sufficient policy incompetence by the leader to warrant deposition, however, depends on the structure of the polity. If the leader has a huge advantage over the challenger in her ability to supply private goods, then she can survive disastrous policy outcomes. Although leaders from systems with large winning coalitions, such as democracies, have some advantage in the supply of private goods, the magni- tude of this advantage is small, so these leaders cannot tolerate policy failure as well as can autocrats or even monarchs.”1
11 Bueno de Mesquita and Siverson (1995) and King, Tomz, and Wittenberg (1998) find empirical support for these predictions.
The preceding discussion suggests the incentives of leaders. We now specify those formally by denoting the payoffs to leaders under each contingency they face. In particular, if the outcome of the international dispute is
z and the leader’s effort level is gA, and assuming the leader also gets the payoffs received by any other member of the winning coalition, then the leader’s payoff is:
/ ~~~RA UA(Z, gA) = VA(Z) + PFAK VA(Z) + (1 -gA) WA
RA XRA + L S ) + ( 1go)
where T is the leader’s utility for remaining in office, and the other terms refer to the payoff from member- ship in the winning coalition.12 FA(), recall, is the distribution of challenger types; hence, FA (VA (z) +
(1 – gA) RAIWA + p. – RAISA) is the probability that the leader of A retains power given the international
outcome z and effort gA. During a dispute, the incumbent leader in A decides
whether to fight and, if so, how hard to try. If attacked, the leader in B chooses an allocation of resources to dedicate to the war effort. Following the end of the dispute, members of the winning coalition decide whether to remain loyal or to defect, thereby retaining or bringing down the incumbent leadership. This is the sequence of play in the game. With the utilities all specified, we can now turn to solving the game.
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