Simultaneous Battles and Sequential Battles in Bargaining Models of War

1 Introduction

Formal theorists in the mainstream of International Relations have portrayed war as a dynamic of two belligerents fighting lengthy battles. This approach to modeling war is found in bargaining models (Fearon 2004, 2007]; Leventoğlu and Slantchev 2007; Powell 2004a, 2004b, 2012]; Slantchev 2003a; Wagner 2000; Wolford, Reiter, and Carrubba 2011) as well as attrition models (Langlois and Langlois 2009, 2012]; Nakao 2022) and random-walk models (Fey and Ramsay 2011; Slantchev 2003b; Smith 1998; Smith and Stam 2003, 2004]). These models typically presume that as a war proceeds, a series of battles evolve along the dimension of time.

In contrast, theorists in another stream – perhaps less influential on International Relations but more aligned with Economics – have treated war as a clash of military forces on multiple battlefields. This approach has been adopted mainly by Blotto models of commanders deploying troops across battlefields (Borel 1953; Golman and Page 2009; Roberson 2006) and their siblings in the literature on counter-terrorism (Bier, Oliveros, and Samuelson 2007; Powell 2007a, 2007b, 2009]). While the theories in the mainstream (henceforth, dynamic models) have focused on the time dimension, those in the latter stream (spatial models) have placed more emphasis on the geographic dimension of war.^[1]

In reality, wars encompass both the time and geographic dimensions – a war can last for months or years and be waged across multiple battlefields, suggesting that existing models present only incomplete pictures of war. However, if war is modeled along both the dimensions, the merits of parsimony would be seriously undermined. In this article, we attempt a theoretical comparison between the dynamic and spatial models in the context of bargaining. More concretely, by combining two ultimatum games with private information in two structurally contrasting ways, we develop and compare two models of war that correspond to the two modeling approaches mentioned above.

One of our models depicts what we label as “parallel war,” where upon a prewar bargaining failure, battles are fought simultaneously in two domains (land and sea).^[2] This model presumes that because an Aggressor and a Defender are geographically contiguous (e.g., France vs. Germany), the Aggressor can directly access both the domains.^[3] The other model is of “series war,” where a battle in one domain precedes a battle in the other domain.^[4] The latter model postulates that because the Aggressor and the Defender are geographically distant (e.g., the U.S. vs. Japan), the Aggressor must win one domain (sea) in order to provoke a conflict in the other (land). To make a comparison of parallel and series wars possible, we ensure the models share the same parameter values and differ only in the sequence of battles.

The comparison produced an unexpected result. At first glance, parallel war looks more likely than series war, because the Aggressor’s entry into the land battle of series war is conditional on his victory at sea. This conditionality can impede the Aggressor’s invasion to the land. However, our theoretical analysis suggests that series war is more likely to break out than parallel war under a wide range of circumstances. In the prewar bargaining of series war, the Aggressor’s ultimatum matters not only for the outbreak of war but also for the condition to restore peace. If a generous ultimatum is presented by the Aggressor but is rejected by the Defender, the Aggressor would infer that the Defender is so resolved that his future offer to end the war is also likely to be rejected. This inference induces the Aggressor to place a tougher ultimatum at a greater risk of war. In contrast, parallel war lacks such dynamic incentives.

The rest of the article proceeds as follows. After configuring the common setup of the two models, we present and analyze the model of parallel war first and the model of series war later. Subsequently, the two models are compared. The last section discusses further developments in theories of war. The Appendix outlines the equilibrium conditions in both the models.

2 Common Setup

To address how the outcome of bargaining and the likelihood of war are influenced by the structural relations across battles in a war, we develop two game-theoretic models of war – one is of parallel war, and the other of series war. To make a comparison between the two forms of war possible, these models must be identical except for the structural relations. Hence, they are assumed to share the following common setup.

There are an Aggressor (A) and a Defender (D) in conflict ( i ∈ A , D ). They have interests b ^L > 0 and b ^S > 0 at stake in two domains/battlefields – land and sea ( d ∈ L , S ) , respectively. Upon a war’s breakout, the land battle is won by i with probability p i L > 0 , and the sea battle won by i with probability p i S > 0 such that p A d + p D d = 1 for each d ∈ L , S . The battle outcomes across domains are independent from each other.

These battles inflict costs on the belligerents as they fight. In fighting a battle in domain d, A incurs cost c A d . A’s expected payoff from fighting in d can be defined as: π A d ≡ p A d b d − c A d > 0 for d ∈ L , S .

In contrast, D’s costs of fighting battles depend on her type and are unknown to A. Put more precisely, when prewar bargaining begins, A is uncertain about D’s resolve (“type”) λ, but A only knows that λ follows the uniform distribution on 0 , Λ ( λ ∼ U 0 , Λ ). D’s costs of fighting are determined by λ: c D | λ d ≡ λ k d , where k ^d > 0 can be interpreted as D’s material cost of fighting in d ∈ S , L . D’s payoffs from fighting in d ∈ L , S will be shown as: π D | λ d ≡ p D d b d − λ k d . Thus, D with a lower λ is more resolved to fight battles. To rule out equilibria where no war breaks out, the following restrictions are imposed on the distribution of λ, so that the Aggressor is willing to run the risk of war:^[5]

Both the models assume the simplest possible bargaining protocol at every stage – the Aggressor makes an ultimatum, to which the Defender responds either by accepting it in peace or by rejecting it through fighting. Whenever peace and fighting are payoff-equivalent, the Defender always chooses peace.

3 Parallel War

3.1 Bargaining Model

The game of parallel war begins with Nature choosing D’s type λ. Without knowing the true value of λ, A places an ultimatum θ L S ∈ 0 , b L S to D, where b ^LS ≡ b ^L + b ^S . D’s response to θ ^LS is denoted as σ λ L S . If D accepts θ ^LS , the game ends with payoffs b L S − θ L S , θ L S . If D rejects θ ^LS , the sea and land battles are fought simultaneously between A and D. The expected payoffs from fighting the war can be shown as π A S + π A L , π D | λ S + π D | λ L . The extensive form of parallel war appears in Figure 1.

Figure 1:

Extensive form of parallel war.

4 Series War

4.1 Bargaining Model

In contrast to the simultaneous land and sea battles in parallel war, the sea battle precedes the land battle in series war. That means, the Aggressor must first control the sea to invade and occupy the land at stake.

The game of series war also begins with Nature choosing D’s type λ. A then issues an ultimatum θ S ∈ 0 , b L S to D. D’s response to θ ^S is denoted as σ λ S . If D accepts θ ^S , the game ends with payoffs b L S − θ S , θ S . If D rejects θ ^S , the sea battle is fought. Based on D’s decision, A updates its belief about λ. If D wins the sea battle, D secures its interests both in the sea and in the land, whereas A not only fails in the sea but also abandons its invasion to the land, so that the game ends with payoffs − c A S , b L S − c D | λ S . If A wins the sea battle, A gains b ^S and further demands θ L ∈ 0 , b L , to which D responds with σ λ L . If D accepts θ ^L , the game ends with payoffs b L S − c A S − θ L , − c D | λ S + θ L . If D rejects θ ^L , the land battle is fought, resulting in payoffs b L S − c A S + π A L , − c D | λ S + π D | λ L . The extensive form of series war is shown in Figure 2.

Figure 2:

Extensive form of series war.

5 Comparison

Based on the equilibrium results above, we examine which form of war is more likely than the other and address why this might be the case. At first glance, parallel war appears more likely than series war, because the Aggressor can immediately provoke the land battle in parallel war, unlike in series war where his invasion to the land is contingent upon winning the sea battle. This conditional requirement puts the Aggressor at a disadvantage in prosecuting series war. However, a comparison of the probabilities of parallel war Pr B a t L S * and series war Pr B a t S * (Equations (4) and (16)) predicts the opposite under broad circumstances.

By Equations (3) and (15), the difference between Pr B a t S * and Pr B a t L S * is:

which is positive unless c A S takes a very large value, as only the last term involving c A S is negative. This comparison suggests that series war is more likely than parallel war unless the Aggressor strongly prefers avoiding only the sea battle.

It is evident from Equation (17) that substituting c A L and k ^L with c A S and k ^S , Pr B a t S * − Pr B a t L S * > 0 . Below we offer an intuitive explanation for Proposition 3.

While choosing θ ^L * at the second stage of series war, A seeks a balance between restoring peace at higher costs and risking another battle. On the other hand, while choosing θ ^S * at the first stage, A considers not only the balance between a more generous ultimatum for peace and the risk of war but also how his current offer θ ^S influences future bargaining ( θ L * θ S in Equation (5) and Π A L * θ S in Equation (12)). As A reduces his demand to D by raising θ ^S , λ ̄ S θ S decreases (Equation (14)), indicating that while his ultimatum is more likely to be accepted, only more resolved types of D will reject it and enter the war. Upon the rejection, A updates his belief about D’s type and rebalances the trade-off between restoring peace and fighting the land battle at the second stage, potentially offering an even more generous θ ^L (Recall that as θ ^S increases, λ ̄ S θ S decreases, while θ L * θ S increases). These dynamic incentives matter for A’s decision on the ultimatum θ ^S in series war, but such incentives do not exist in parallel war. Therefore, despite the structural challenges in series war hindering the Aggressor’s land acquisition, series war is predicted to be more likely than parallel war.

Moreover, to analyze the effects of uncertainty regarding D’s type λ, when A faces greater uncertainty (with a larger Λ), the probability of parallel war is larger (Equation (4)), the probability of series war is also larger (Equation (16)), but the difference between the two probabilities is smaller (Equation (17)).

Greater uncertainty induces A to take greater risks of war, but this risk-taking incentive is mitigated in the model of series war, because of an additional incentive for A – this uncertainty alters A’s behavior not only in the first stage but also in the second stages of series war ( θ L * θ S * in Equations (5) and (15) and θ ^S * in Equation (13)). When choosing θ ^S , A anticipates the risk of the land battle and thus rebalances the trade-off between restoring peace with a greater compromise θ ^L and the increased likelihood of entering the land battle Pr B a t L . This additional incentive reduces the effect on the probability of series war, making it smaller compared with the effect on the probability of parallel war.

6 Discussion

In this article, we have conducted the first theoretical study to explore the structural relations across battles within a war. In comparing of two bargaining models of war with diametrically contrasting structures, we have demonstrated that the likelihood of war depends on how battles are related to one another within a war. Namely, in broad circumstances, a war is more likely to break out when two battles are fought sequentially rather than simultaneously. If the bargaining process consists of two periods as in the model of series war, the Aggressor must take into account how his ultimatum placed in the current period influences the condition to restore peace in the future period. If the Aggressor’s ultimatum is set to be sufficiently generous but is rejected by the Defender, he would infer that the Defender is so resolved that it is difficult to end the war in future bargaining. This dynamic incentive induces the Aggressor to run a greater risk of war by issuing a tougher ultimatum in prewar bargaining.

We conclude the article by proposing two major paths toward further theoretical developments of war. First, due to different foci, the two streams of theoretical studies of war elaborated in the Introduction have tackled divergent problems with armed conflict – the dynamic models tend to address the postwar division of interests at stake between belligerents, while the spatial models analyze the prewar distribution of military resources within each belligerent. Put simply, the dynamic models study the outputs of war, while the spatial models do the inputs. However, if a theory of national-security policy claims to be compelling, it must illuminate the input-output linkage, or the entire process from the making of a policy (i.e., the deployment of military forces, as in spatial models) toward what the policy will bring about (i.e., the consequences of war, as in bargaining models). To the best of our knowledge, no such theory has been ever made.

Second, we have compared the two simplest possible structures of war. In reality, however, a war can escalate in much more varied and complicated ways. It would be meaningful to delve into how the timings, locations, and conditions of battles are determined in relation to one another within a war. Below, we itemize potentially fruitful directions for modeling war:

1. The form of war can be not only parallel or series, but also complex in the sense that it is a combination of both simultaneous and sequential battles (Table 1). For instance, an aggressor may advance his forces along multiple trails, each containing a series of military engagements en route (e.g., the U.S. leapfrogging strategy in the Pacific War). One of the trails may be further ramified into a few paths of minor engagements.

2. Whereas we can see how battles were waged and linked to each other in past wars, belligerents were often unsure of how their wars would further escalate when they were fighting (Gartzke 1999). In other words, the structural relations of battles in past wars, as we know today, might be mere hindsight. A war could escalate in ways belligerents did not predict or even imagine, as possibly caused by the progress of military technology (such as chemicals and tanks in WWI).

3. As nuanced by “costly lottery” (Reiter 2003; Wagner 2000) or “random walk” (Slantchev 2003b; Smith 1998; Smith and Stam 2003, 2004]), extant models have treated war as a stochastic process, in which as long as belligerents choose to fight, the conclusion of war is an exogenous and probabilistic event. It is true that war is often governed by the so-called “fog,” but developments on the battlefields are also very often up to the choices of belligerents. In particular, when an aggressor mobilizes and deploys his forces, he can exercise the initiative in determining where and when to project his forces against his opponent. Put in another way, war might be better regarded as an endogenous process, as opposed to what the extant theories have presumed.

4. Battles can have symmetric or asymmetric influence on each other. Put more concretely, a battle outcome in a theater can influence the tide of an ongoing battle in another theater (Nakao 2020). Such influence is likely to be within a belligerent’s decision calculus of choosing battlefields.

To summarize, the structural relations of battles in a war could be complex, unpredictable, and endogenous. These elements of war are potentials for – as well as obstacles to – innovating a new theory of armed conflict.