គោលការណ៍នៃការបង្រួបបង្រួម

 The Principle of Convergence

The model shows that a change in war aims (expectations) is necessary for war to end (Fox 1970, 7). War is not so much about twisting arms, but about influencing expectations. How can players systematically and strategically influence each others’ beliefs? The tension between the desire to find a settlement and the desire to give up as little as possible produces inefficiency as opponents learn more about each other.

Learning occurs through two channels: a nonstrategic, nonmanipulable, and involuntary one—the battlefield—and a strategic, manipulable, and voluntary one—the negotiation table. The first is imprecise and noisy, and although players can infer something about the distribution of power, it is not sufficient to ensure convergence in beliefs. The strategic channel is more useful because offers, counteroffers, and rejections are all rational decisions that reveal information about the privately known parameter. However, as noted before, some of the strength of its impact is due to the requirement that players always make counteroffers.

Since negotiations can be so revealing, it is perhaps not surprising that empirically warring parties try to avoid initiating talks until after an armistice.¹⁶ In the cases where explicit bargaining was contemporaneous with fighting, the offers and counteroffers were heavily dependent on dramatic developments of the battlefield but as war progressed, convergence did occur despite lack of decisive engagements (Pillar 1983).

The Principle of Convergence posits that warfare ceases to be useful when it loses its informational content, which occurs when strategic and involuntary revelations make beliefs “irreversible” in the sense that both sides can agree on the relative likelihoods of different outcomes.

While variants of the principle can be derived from the other models, it exhibits slightly different dynamics. In A. Smith and Stam 2001, players only take into account battlefield outcomes and ignore the manipulable source. Their model cannot address the issue of wartime negotiations at all. Filson and Werner solve their model for an artificially restricted case that limits possible fighting to only two battles, and the one-sided bargaining protocol (shared with R. Powell'’s model ignores the possiblity that an informed player may signal strategically. It is, however, important to note that this principle appears quite robust to different specifications, which is usually not the case because bargaining models are quite sensitive to the precise protocol.

J. Smith (1995) states that the main reason for war continuation is belief in victory, and so one of the most important requirements for a cease-fire agreement is a clear military trend. Similar arguments abound, and all share a conclusion that a convergence of expectations about the military attrition trend are a necessary condition for termination of armed conflict (Calahan 1944; Foster and Brewer 1976; Kecskemeti 1970). The principle of convergence is much weaker for it only requires that opponents agree on the relative likelihoods of different outcomes, not on who the eventual winner is going to be.

A Substitute for Victory

Traditionally, the purpose of the military instrument has been to secure victory on the battlefield. As General Douglas MacArthur claimed, “War's very objective is victory—not prolonged indecision. In war there is no substitute for victory.” The thinking is reflected in American strategic doctrines and permeates theories of war that invariably focus on its outcome expressed in strictly military terms: victory, defeat, or stalemate.

The Principle of Convergence suggest that this view is flawed because a crucial result is that belief in defeat (or victory) is not a necessary condition to agree to terminate a war, which lends support to von Clausewitz's ([1832] 1984, 91–92) claim that “not every war need be fought until one side collapses . . . if one side cannot completely disarm the other, the desire for peace on either side will rise and fall with the probability of further successes and the amount of effort these would require.”

The presence of uncertainty allows weaker states to exploit informational asymmetries because they know their opponent has incentives to settle as soon as possible and so will make offers that exceed the ones under complete information. In other words, weak adversaries can actually profit from fighting a little and then settling, even though they know very well that in the long run they will inevitably lose.

Although it is possible for war to end with the complete military defeat of one side, such instances will be rare. This is what we observe empirically as well: Most wars do not terminate with the obliteration of the losing side but are settled long before that. This follows immediately from the informational role of warfare—once opponents learn “enough” about each other, they can find a mutually acceptable settlement. How much “enough” is and how long it takes to learn it depend on prior beliefs, the battlefield performance, and the speed with which information is revealed during the war.

There are substitutes for victory in war, and these are the political settlements that terminate the war. Since it is not necessary to win a war in order to end it, it is not necessary to agree on who the eventual winner will be either.

On War Initiation

The Principle of Convergence also has startling implications for the initiation of war. If it is not necessary to win a war in order to profit from it, it is not necessary to believe in victory to start one. The traditional theoretical puzzle is framed as one about divergent optimistic expectations about the outcome of war, and the literature has sought ways to account for such discrepancies in rationalist terms (Fearon 1995).

My results suggest that this places unnecessary demands on optimism: It is quite rational to start a war a state expects to lose as long as its opponent believes that this state is stronger because this induces the opponent to offer better terms. It is not necessary for both sides to be optimistic about the military outcome.

This further implies that asymmetric conflict may not be as puzzling as once thought. It does not require suicidal or desperate rulers for a weak state to challenge a strong one, although it is necessary for the strong one to believe its adversary is not as weak as it actually is. Filson and Werner find that war can begin when the initiator underestimates the strength of the defender, who then fights to demonstrate that it does not have to concede as much. This also happens in the present model, but the additional rationale for a weak state fighting an opponent because the opponent may overestimate its strength cannot be derived from their model. This is a consequence of limiting the types of opponents to only two, which deprives moderately strong players of the ability to separate from the weak while pooling with the strong, which is what generates their higher payoff in the present model.

Duration of Total and Restricted Wars

Scholars often make a conceptual distinction between restricted and total war (Kecskemeti 1958; Manwaring 1987; Wagner 2000).¹⁷ The difference between the two is mainly in the way they end or are expected to end. The former type is seen as achieving limited goals with qualified military engagements. Restricted wars usually end with a settlement and both sides retain fighting capacity. The latter type is seen as seeking the complete destruction of the military capability of the defeated state.

The Principle of Convergence challenges the distinction between wars fought “solely for the purpose of revealing information and wars fought to disarm the adversary” (Wagner 2000). In the model war can end with the complete military defeat of one state before a bargain is struck, but the outcome arises from the same mechanism that produces the political settlements. Moreover, it is very unlikely that such an outcome will occur because, all things equal, war results in relatively quick disclosure of information, and so is wont to lose its informational content without significant delay. Extreme uncertainty can produce delays that are long enough for parties to fail to reach an agreement. However, no one starts out intending to wage an absolute war, it just may turn out that way, as it happened in the Second World War.

The ceteris paribus qualification is important. Innovations can significantly alter the prewar capabilities while fighting lasts. Depending on the identity of the successful innovator, this may result in further delays— a state that has become stronger demands more and rejects terms—or dramatic shortening—a weak state finds it no longer can extract concessions from a now significantly stronger opponent.

The Shadow of the Future

States must value the future sufficiently for war to occur because only when the settlement is important enough to the players do they have incentives to bear costly delays to secure better outcomes. When players discount the future too much, only semiseparating and pooling equilibria exist instead. This implies that more stable governments will be harder to settle with, which also helps explain why it is often the case that the parties who begin the war are not the same ones that finish it, especially on the losing side (Calahan 1944).

In international relations theory, the shadow of the future is usually thought to have a benign effect because it makes cooperative behavior possible in equilibrium. However, in a situation where uncooperative behavior in the short term may secure better benefits in the long run, the shadow of the future has the exact opposite effect. The more patient players are, the more incentives they have to delay agreement and fight.

CONCLUSION

If warfare is purely instrumental, then its role as an information transmitter is paramount. War results not from bargaining failure but from incentives to determine the type of opponent and obtain a better negotiated settlement. In a costly, but noisy, fighting environment such information takes time to accrue, and only when players learn sufficiently about their prospects in war will they agree to a settlement that reflects these expectations. The Principle of Convergence posits that warfare ceases to be useful when it loses its informational content.

These results have significant implications for the theory of war because they show the standard puzzle about both sides being optimistic about victory to be irrelevant. States do not have to believe in victory to engage in war, and neither do they have to believe in defeat to end it. As long as both sides want to settle as quickly as practicable, weaker states can benefit from uncertainty and obtain a deal through fighting that they would not have been able to obtain under complete information.

More attention must be paid to the different sources of information in war. The battlefield is a noisy but involuntary source, while the bargaining table is a precise but manipulable one. There are other sources of information between the two extremes, like public opinion and intelligence. It will be worth investigating how

More attention must be paid to the different sources of information in war. The battlefield is a noisy but involuntary source, while the bargaining table is a precise but manipulable one. There are other sources of information between the two extremes, like public opinion and intelligence. It will be worth investigating how these can be utilized and how states interpret information acquired during war.

A number of testable hypotheses can be derived from the model. The skimming property of the equilibrium implies that as war progresses, the outcome becomes less advantageous for the worse informed party. Since it takes optimism to select oneself into war, it is likely that this party will be the initiator.

Information acquired during the war outweighs pre-war information derived from capabilities and economic resources. If the war begins with a series of early victories against a strong opponent, the screening process slows down because the proposer fails to reduce its demands fast enough to induce its opponent to quit. A series of initial defeats accelerates the screening process and will lead to shorter wars. Thus, we can address the question of how initial performance in war influences its duration.

The military position achieved immediately prior to negotiating has a strong effect on the bargaining outcome in the sense that a state may prefer to delay agreement following a victory when it will prefer to settle following defeat.

Since behavior at bargaining table has such important consequences for the termination of war, states will try to manipulate the prospect of negotiations by refusing to come to the table.

Because it is the strong that are most hurt by the informational asymmetries, powerful states will seek to reveal their strength, and moderately strong or weak ones will oppose such transparency.

The Weinberger–Powell Doctrine places rather stringent restrictions on the use of force because it views warfare through the prism of military victory. This is counterproductive because it may lead to failure to engage in situations where a determined, yet limited, application of force can yield satisfactory results. The emphasis on the use of overwhelming force, on the other hand, seems well placed, both because it discourages adversaries from costly delays and because it enhances the deterrent posture as long as the use of force is credibly contemplated.

Insofar as the new National Security Strategy of the Bush administration signifies increased willingness to use force, it should provide for better deterrence against traditional adversaries. The strategy implicitly seeks to shorten the time horizon of the opponent (regime change for state actors, preemptive strike against non-state enemies), a distinctly novel approach in contrast to (sometimes limited) détente with status quo powers like the Soviet Union during the Cold War. Since long time horizons are a major reason for engaging in conflict, this shift should produce desirable effects by making adversaries less willing to engage in costly contests with the United States.

APPENDIX

In any stationary no-delay MPE, player 1 must offer player 2 at least what that player expects to obtain by rejecting a proposal. Since in this equilibrium player 2's offer is immediately accepted (or else the game ends), player 1's offer must satisfy,

for 1 < k < N − 1,

π − x₁* = (1 − δ)b₂ + δ[py₂* + (1 − p)π],

π − xₖ* = (1 − δ)b₂ + δ[pyₖ₊₁* + (1 − p)yₖ₋₁*], (1)

π − xₙ₋₁* = (1 − δ)b₂ + δ(1 − p)yₙ₋₂*.

The corresponding equations for player 2's offers are

π − y₁* = (1 − δ)b₁ + δpx₂*,

π − yₖ* = (1 − δ)b₁ + δ[pxₖ₊₁* + (1 − p)xₖ₋₁*],

π − yₙ₋₁* = (1 − δ)b₁ + δ[pπ + (1 − p)xₙ₋₂*].

This defines a system of 2(N − 1) simultaneous equations. Fix some arbitrary N ≥ 2 and let n = N − 1. Label the equilibrium offers such that x₁*, x₂*, ..., xₙ* correspond to z₁, z₂, ..., zₙ, and y₁*, y₂*, ..., yₙ* correspond to zₙ₊₁, zₙ₊₂, ..., zₙ₊ₙ. Construct the 2n × 2n matrix A of coefficients in the usual way and let w > 0 be the corresponding RHS vector. The following lemma establishes that there is a unique solution to the system of equations.

Lemma 3. There exists a unique z* = A⁻¹w.

Proof. Since w ≠ 0, it is sufficient to establish that A⁻¹ exists. A can be partitioned into four square submatrices:

A = ( I M )

( M I )

where I is the identity matrix of size n and M is an n × n matrix whose diagonal elements are 0, immediate lower off-diagonal elements are δ(1 − p), immediate upper off-diagonal elements are δp, and everything else is 0. Like M, each element of M² is nonnegative, and the sum of entries in each column is less than one. By Theorem 8.13 in Simon and Blume 1994, 175, this implies that (I − M²)⁻¹ exists. But since det A = det(I − M²), it follows that det A ≠ 0 as well. 

Proof of Proposition 1. Consider the following strategies. Player 1 always offers xₖ*, accepts all offers x ≥ xₖ*, and rejects all offers x < xₖ*, where k is the realization of the stochastic process and xₖ* is the kth element of z* from Lemma 3. Player 2's strategy is defined analogously. It is trivial to verify that these strategies are subgame perfect. Since the vector with proposals is unique, there exists at most one stationary no-delay MPE. Agreement is immediate on xₖ₀*. 

Proof of Proposition 2. In the proposed equilibrium, the game in period t = 2 is equivalent to the complete information game with p = pL and starting state k₂. By Proposition 1, this game has a unique stationary no-delay MPE in which player 1 offers and player 2 accepts Vₛ¹(k₂). 

The Two-Period Game

Let k₁ denote the realization of the state variable in period t = 1, and let q₁ʷ = 0, q₁ᵐ > 0, and q₁ˢ = 1 − q₁ᵐ > 0 denote player 1’s prior (i.e., at the beginning of the period, before player 2 makes an offer) belief that its opponent is of type 2w, 2m, and 2s, respectively. The probability that the opponent is weak is zero because players follow equilibrium strategies. Let p₁ = q₁ˢpL + (1 − q₁ˢ)pM be player 1’s ex ante expectation that it will win a fight if it rejects an offer in period 1. Player 2’s unique (by Lemma 2) equilibrium offer that is accepted by player 1 is

y*(q₁ˢ, k₁) = π − (1 − δ)b₁ − δ[p₁Vₛ¹(k₁ + 1) + (1 − p₁)Vₛ¹(k₁ − 1)]. (2)

Since 2ₛ makes an unacceptable offer, it must be the case that

y*(q₁ˢ, k₁) < (1 − δ)b₂ + δ[π − pLVₛ¹(k₁ + 1) − (1 − pL)Vₛ¹(k₁ − 1)],

which, together with condition (2), yields the additional constraint

pM − pL > [ (1 − δ) / (δ(1 − q₁ˢ)) ] [ (π − b₁ − b₂) / (Vₛ¹(k₁ + 1) − Vₛ¹(k₁ − 1)) ] . (3)

Lemma 4. Let δ₁(q₁ˢ) be the smallest discount factor that solves (3) for some q₁ˢ. Whenever δ ≥ δ₁(q₁ˢ), player 2ₛ strictly prefers to make an unacceptable offer in period t = 1 to making an acceptable one.

This lemma implies that the optimal strategy for 2w and 2m in period t =1 is to demand, and receive, y*(q₁ˢ, k₁). Player 2ₛ demands, but does not receive, y > y*(q₁ˢ, k₁) as long as δ ≥ δ₁(q₁ˢ). Player 1 accepts all y ≤ y*(q₁ˢ, k₁), and rejects everything else. Let x₁*(q₁ˢ, k₁) = π − y*(q₁ˢ, k₁) denote player 1’s smallest expected payoff in t = 1.

Beliefs Following a Battle. In equilibrium, an acceptance of the first offer by player 2 signals unambiguously that its type is 2w, and therefore a rejection signals that its type is either 2m or 2s. The prebattle probability that the type is 2s equals qˢ/(qˢ + qᵐ). The post-battle posterior is then

q'₁(I) = (qˢ pL^I (1 − pL)^(1−I)) / (qˢ pL^I (1 − pL)^(1−I) + qᵐ pM^I (1 − pM)^(1−I)), (4)

where I is the battle indicator that equals 1 if player 1 won or 0 if it lost.

The Three-Period Game. Let p₀ = (qᵐpM + qˢpL) / (qᵐ + qˢ) be player 1's expectation that it will win a fight if 2m and 2s reject its offer. Since player 2w accepts player 1's offer, the optimal offer in t = 0 is

x₀* = π − (1 − δ)b₂ − δ[pHy*(q₁ˢ(1), k₀ + 1) + (1 − pH)y*(q₁ˢ(0), k₀ − 1)]. (5)

Player 1's ex ante payoff from a strategy that induces only 2w to accept with probability one is at least

x* = qʷx₀* + (1 − qʷ)[(1 − δ)b₁ + δ(p₀x₁*(q₁ˢ(1), k₀ + 1) + (1 − p₀)x₁*(q₁ˢ(0), k₀ − 1))]. (6)

In equilibrium this must be at least as good as the payoffs from inducing only 2s to separate, or from settling immediately with all three types.

If player 2 follows the equilibrium strategy, the only way player 1 can induce only 2s to reject the initial offer is to satisfy player 2m, which implies that player 1 can propose at most

x̂₀ = π − (1 − δ)b₂ − δ[pMy*(q₁ˢ(1), k₀ + 1) + (1 − pM)y*(q₁ˢ(0), k₀ − 1)].

If both 2w and 2m accept the initial offer, q₁ˢ = 1, and Lemma 4

has no bite: 2ₛ will not delay but will demand instead V²ₛ(k₁), to which player 1 will agree. Therefore, player 1’s ex ante payoff from inducing both 2w and 2m to accept the initial offer is

x̃ = (1 − qˢ)x̂₀ + qˢ[(1 − δ)b₁ + δ(π − pL V²ₛ(k₀ + 1) − (1 − pL)V²ₛ(k₀ − 1))].

If player 2 follows the equilibrium strategy, the only way player 1 can guarantee that all three types will accept the initial proposal is to satisfy 2ₛ, which implies that player 1 can propose at most

x̂₀ = (1 − δ²)(π − b₂) + δ²[p²L V¹ₛ(k₀ + 2) + 2pL(1 − pL)V¹ₛ(k₀) + (1 − pL)² V¹ₛ(k₀ − 2)].

whose ex ante expected value is x̂ = x̂₀ because the offer is accepted with probability one.

To establish optimality of player 1’s screening strategy, we wish to show that it would not want to deviate by either settling with all three types immediately or separating only 2ₛ for one period. The strategy will be optimal as long as x̃ ≤ x* and x̂ ≤ x*.

After algebraic manipulation, we find that x̂ < x̃ whenever

pM − pL > [qˢ(1 − δ) / δ(1 − qˢ)] [ (π − b₁ − b₂) / (V²ₛ(k₀ − 1) − V²ₛ(k₀ + 1)) ]. (7)

Lemma 5. Let δ₀ be the smallest discount factor that solves (7). Whenever δ ≥ δ₀, player 1 strictly prefers to delay agreement for one period in order to separate 2ₛ from the other types to settling immediately.

Showing that x̂ ≤ x* is a bit involved, but the following result can be established using numerical methods.

Lemma 6. Let δ₂ be the smallest discount factor such that x̂ < x*. Whenever δ ≥ δ₂, player 1 strictly prefers to delay agreement for up to three periods in order to separate each of the three types to settling in any of the prior periods.

With these results, the proof of the proposition is straightforward. It is always the case that δ₁(q₁ˢ(1)) < δ₁(q₁ˢ(0)). Let δ̄ = max {δ₀, δ₁(q₁ˢ(0)), δ₂}, and take any δ ∈ (δ̄, 1). Consider the following strategy for player 1: in t = 0, offer x* from (6); in t = 1, accept any y ≤ y*(q₁ˢ(I₀), k₁) from (2) and reject anything else; in all even periods t ≥ 2, offer V¹ₛ(kₜ); in all odd periods t ≥ 2, accept any y ≤ V²ₛ(kₜ) and reject anything else. In period 1 update to believe that the probability of 2ₛ is q₁ˢ(I₀), as defined in (4), and the probability of 2w is zero. In periods t ≥ 2 update the probability of 2ₛ to 1.

Consider the following type-dependent strategy for player 2. Type 2w accepts any x ≤ x₀* in period t = 0, and follows 2m’s strategy henceforth. Type 2m accepts any x ≤ x̂₀ in period t = 0, offers y*(q₁ˢ(I₀), k₁) in period t = 1, and follows 2s’s strategy henceforth. Type 2s accepts any x ≤ x̂₀ in period t = 0, demands π in period t = 1, and for all t ≥ 2 accepts x ≤ V¹ₛ(kₜ) in even-numbered periods and demands V²ₛ(kₜ) in odd-numbered periods.

Proposition 1 and lemmas 4, 5, and 6 guarantee that the constructed Markov strategies and player 1’s credible beliefs constitute a perfect sequential equilibrium. The construction demonstrates that there is one type of MPSE but it is only generically unique because player 2s can make different nonserious offers in period t = 1. There exists a continuum of equilibria of this type, but they are all payoff-equivalent. 

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