Simultaneous battles and sequential battles in bargaining models of war

Nakao, Keisuke Article Simultaneous battles and sequential battles in bargaining models of war Peace Economics, Peace Science and Public Policy (PEPS) Provided in Cooperation with: De Gruyter Brill Suggested Citation: Nakao, Keisuke (2025) : Simultaneous battles and sequential battles in bargaining models of war, Peace Economics, Peace Science and Public Policy (PEPS), ISSN 1554-8597, De Gruyter, Berlin, Vol. 31, Iss. 1, pp. 1-20, https://doi.org/10.1515/peps-2024-0033 This Version is available at: https://hdl.handle.net/10419/333336 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Keisuke Nakao* Simultaneous Battles and Sequential Battles in Bargaining Models of War https://doi.org/10.1515/peps-2024-0033 Received July 30, 2024; accepted October 18, 2024; published online November 18, 2024 Abstract: A war consists of multiple battles, yet the theoretical literature of International Relations has given little attention to how battles relate within a war. By integrating two ultimatum games with private information played by an Aggressor and a Defender, we develop bargaining models of war with two structures: (i) parallel war, where battles occur simultaneously in two domains, as the Aggressor can access both directly; and (ii) series war, where a battle in one domain (e.g., sea) precedes a battle in another (e.g., land), as the Aggressor must first control the former domain to instigate conflict in the latter. In a theoretical comparison between parallel and series wars, we demonstrate that although series war imposes structural disadvantages on the Aggressor, series war is more likely to break out than parallel war under broad circumstances. If prewar bargaining of series war fails, the Aggressor may infer that future bargaining is also likely to fail, leading him to take a greater risk of war when issuing an ultimatum. Such dynamics are absent in parallel war. We also discuss further developments in theories of war (182 words). Keywords: bargaining model; likelihood of war; ultimatum game JEL Classification: D74; F51; F52 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; I thank Hiroshi Uno, Yasutomo Murasawa, and two anonymous reviewers for valuable comments. *Corresponding author: Keisuke Nakao, College of Business and Economics, University of Hawaii at Hilo, 200 W. Kawili St., Hilo, HI 96720, USA, E-mail: [email protected]. https://orcid.org/0000-00019109-2542 Peace Econ. Peace Sci. Pub. Pol. 2025; 31(1): 1–20 Open Access. © 2024 the author(s), published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 International License. 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 counterterrorism (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. 1Alongside recent developments in Blotto models, there has also been some dynamic models where players allocate resources across a sequence of fights (Rinott, Scarsini, and Yu 2012; Sela and Erez 2013). 2The land and sea are mere metaphors for the two domains (or battlefields). They can be replaced with the cyber and physical spaces in some instances, or conventional and nuclear theaters in others. 3Throughout the article, we assign the masculine pronouns to the Aggressor and the feminine pronouns to the Defender. 4Parallel war and series war are named after the corresponding circuits in an electrical network. 2K. Nakao 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 bL> 0 and bS> 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 iwith probability pL i>0, and the sea battle won by iwith probability pS i>0 such that pd A+pd 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,Aincurs cost cd A.A’s expected payofffrom fighting in dcan be defined as: πd A≡pd Abd−cd A>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, Ais uncertain about D’s resolve (“type”)λ, but Aonly knows that λfollows the uniform distribution on 0,Λ [] Simultaneous Battles and Sequential Battles 3 (λ∼U0,Λ [] ). D’s costs of fighting are determined by λ:cd D|λ≡λkd, where kd> 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: πd D|λ≡pd Dbd−λkd. Thus, Dwith a lower λis 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 Assumption 1: The Aggressor is so uncertain over the Defender’s type λthat: 6 (i)Λ>cL A+cS A kL+kS(1) (ii)Λ> 2kS+pS A 2kL [] cL A kL+cS A kS+pS A 2kL.(2) 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,Aplaces an ultimatum θLS ∈0,bLS [] to D, where bLS ≡bL+bS.D’s response to θLS is denoted as σLS λ.IfDaccepts θLS, the game ends with payoffs bLS −θLS,θLS () .IfDrejects θLS, the sea and land battles are fought simultaneously between Aand D. The expected payoffs from fighting the war can be shown as πS A+πL A,πS D|λ+πL D|λ () . The extensive form of parallel war appears in Figure 1. 5While the next two sections focus on equilibria where the probability of war is positive, other equilibria and their conditions are discussed in the Appendix. 6Another way to interpret Assumption 1 is that the costs cL Aand cS Aare so small that the Aggressor is willing to fight battles with positive probabilities. 4K. Nakao 3.2 Equilibrium The equilibrium can be derived backward by finding the sequentially-rational strategy at every information set. Any type λof Daccepts the ultimatum θLS if and only if it is larger than or equal to her expected payofffrom fighting the two battles (θLS ≥πL D|λ+πS D|λ). Let λ LS θLS () be the threshold of λ, with which Dis indifferent between accepting θLS and fighting. For λ=λ LS θLS () , θLS =πL D|λ+πS D|λ =pL DbL+pS DbS−kL+kS [] λ LS θLS () . Anticipating D’s response above, Aseeks the balance between a compromise in peace and the risk of war so as to maximize his continuation payoffby choosing θLS: ΠLS AθLS () ≡1−Pr BatLS ()[] bLS −θLS [] +Pr BatLS () πL A+πS A [] , where Pr BatLS () is the probability that parallel war breaks out: Pr BatLS () ≡Pr(λ<λ LS θLS ()) = λ LS θLS () Λ. The payoff-maximizing θLS can be derived from the first-order condition of ΠLS AθLS () . The second-order condition is guaranteed by the non-decreasing hazard rate of the uniform distribution of λ(cf. Fudenberg and Tirole 1991: 267). The equilibrium can be summarized as follows: Proposition 1: In the bargaining model of parallel war,there is a unique perfect Bayesian equilibrium θLS*,σLS* λ () such that θLS*=pL DbL+pS DbS−kL+kS [] Λ−cL A+cS A [] 2 σLS* λ= accept for λ≥λ LS θLS () fight for λ<λ LS θLS () , ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ Figure 1: Extensive form of parallel war. Simultaneous Battles and Sequential Battles 5 where λ LS θLS () is the threshold of λthat determines whether D accepts θLS or fights: λ LS θLS () =pL DbL+pS DbS−θLS kL+kS. Moreover, because the threshold of λin the equilibrium is: λ LS θLS* () =pL DbL+pS DbS−θLS* kL+kS [] Λ = pL DbL+pS DbS−pL DbL+pS DbS−kL+kS [] Λ−cL A+cS A [] 2 () kL+kS [] Λ =1 2Λ−cL A+cS A kL+kS [] ,(3) The probability that parallel war breaks out can be shown as: Pr BatLS* () = λ LS θLS* () Λ =1 21−cL A+cS A kL+kS [] Λ ⎡ ⎢ ⎣⎤⎥⎦,(4) which is positive by Inequality (1). 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.Athen issues an ultimatum θS∈0,bLS [] to D.D’s response to θSis denoted as σS λ.IfDaccepts θS, the game ends with payoffsbLS −θS,θS () .IfDrejects θS, the sea battle is fought. Based on D’s decision, Aupdates its belief about λ.IfDwins the sea battle, Dsecures its interests both in the sea and in the land, whereas Anot only fails in the sea but also abandons its invasion to the land, so that the game ends with payoffs−cS A,bLS −cS D|λ () . If Awins the sea battle, Agains bSand further demands θL∈0,bL [] , to which 6K. Nakao Dresponds with σL λ.IfDaccepts θL, the game ends with payoffs bLS −cS A−θL,−cS D|λ+θL () .IfDrejects θL, the land battle is fought, resulting in payoffs bLS −cS A+πL A,−cS D|λ+πL D|λ () . The extensive form of series war is shown in Figure 2. 4.2 Equilibrium The equilibrium of the game of series war can also be derived backward –from bargaining θL,σL λ () to bargaining θS,σS λ () . Bargaining over the Land: The second stage (i.e., bargaining right before the land battle) resembles the game of parallel war but differs from it in twofold: (a) only the land battle is fought; and (b) a fraction of λ∈0,Λ [] is screened out in the first stage (i.e., bargaining before the sea battle). Suppose that those types of Dwith λ≥λ SθS () accept θSin the first stage, and only those with λ<λ SθS () enter the second stage. Any type of Daccepts θLin the second stage if and only if θLis no less than her expected payofffrom fighting the land battle (θL≥πL D|λ). The threshold λ LθL () of λ, which determines whether Daccepts θLor fights, then satisfies that: θL=pL DbL−kLλ LθL () . In response to σL λwith λ LθL () ,Achooses θLto maximize his continuation payofffrom fighting the land battle: ΠL AθL () ≡1−Pr BatL ()[] bL−θL [] +Pr BatL () πL A [] , where Pr BatL () is the probability of the land battle conditional on λ<λ SθS () : Pr BatL () ≡Pr(λ<λ LθL () |λ<λ SθS ()) = λ LθL () λ SθS () . Figure 2: Extensive form of series war. Simultaneous Battles and Sequential Battles 7 A’s optimal θLcan be obtained from the first-order condition of ΠL AθL () . Note that because the probability of the land battle depends on λ SθS () , the optimal θLis a function of θS. Lemma 1: In the second stage of series war following θS,λ SθS ()() ,there is a unique perfect Bayesian equilibrium of the θL*θS () ,σL* λ () ,which satisfies that: θL*θS () =pL DbL−kLλ SθS () −cL A 2(5) σL* λ=accept for λ≥λ LθL () fight for λ<λ LθL () , ⎧ ⎪ ⎨ ⎪ ⎩(6) where λ LθL () is the threshold of λfor D to accept θLor to fight: λ LθL () =pL DbL−θL kL.(7) Moreover, the players’continuation payoffs from the equilibrium of the second stage are: ΠL* AθS () =πL A+ kLλ SθS () +cL A [] 2 4kLλ SθS () (8) ΠL* D|λθS () = θL*θS ()for λ≥λ LθL*θS ()() πL D|λfor λ<λ LθL*θS ()() , ⎧ ⎪ ⎨ ⎪ ⎩(9) for which the threshold of λin the equilibrium λ LθL*θS ()() is: λ LθL*θS ()() =pL DbL−θL*θS () kL = pL DbL−(pL DbL−kLλ SθS () −cL A 2) kL =1 2[λ SθS () −cL A kL],(10) 8K. Nakao 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. Appendix By dropping Assumption 1, the Appendix explores equilibria and their conditions with parameters taking a broader range of values. A Parallel War In the model of parallel war, the conditions for equilibria are rather trivial. If Λ>cL A+cS A kL+kS(Assumption 1-(i)), the equilibrium takes an interior solution to A’smaximization of ΠLS AθLS () –afractionoftypesofDfight, while others do not. In this interior equilibrium, θLS*=pL DbL+pS DbS−kL+kS [] Λ−cL A+cS A [] 2,λ LS θLS* () =1 2Λ−cL A+cS A kL+kS [] >0, and Pr BatLS* () =1 21−cL A+cS A kL+kS [] Λ [] >0 (Proposition 1). If Λ≤cL A+cS A kL+kS, the equilibrium is placed at the corner –all the types of Daccept θLS*, and no war takes place. In the corner equilibrium, θLS*=pL DbL+pS DbS,λ LS θLS* () =0, and Pr BatLS* () =0. B Series War In the model of series war, the form of equilibrium depends on whether the solution to A’s payoffmaximization is interior or corner at each of the first and second stages. B.1 Second Stage At the second stage, rational strategies depend on λ SθS () , or the fraction of types of D entering it. If λ SθS () >cL A kL, the equilibrium is interior: θL*θS () =pL DbL−kLλ SθS () −cL A 2, λ LθL*θS ()() =1 2λ SθS () −cL A kL [] >0, and Pr BatL* () =1 2⎡ ⎣1−cL A kLλ SθS () ⎤⎦>0. If λ SθS () ≤cL A kL, the equilibrium appears at the corner, so that no land battle occurs: θL*θS () =pL DbL, λ LθL*θS ()() =0, and Pr BatL* () =0. Simultaneous Battles and Sequential Battles 15 B.2 First Stage At the first stage, A’s objective function depends on whether the second-stage equilibrium is interior or corner: ΠS AθS () ≡1− λ SθS () Λ ⎡ ⎢ ⎣⎤⎥⎦bLS −θS [] + λ SθS () ΛπS A+pS AΠL AθL*θS ()()[] , for which λ SθS () = pS DbS+pS ApL D+pS D [] bL+pS A cL A 2−θS kS+pS A kL 2 if λ SθS () >cL A kL pS DbS+pS D+pS ApL D [] bL−θS kSif λ SθS () ≤cL A kL ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ΠL AθL*θS ()() = πL A+[cL A+kLλ SθS ()] 2 4kLλ SθS () if λ SθS () ≤cL A kL pL AbLif λ SθS () ≤cL A kL. ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Accordingly, the conditions for first-stage equilibria vary with the relative size between λ SθS () and cL A kL. B.2.1 For λ SθS () >cL A kL If the second-stage equilibrium is interior (λ LθL*θS ()() >0byλ SθS () >cL A kL), the firststage equilibrium must also be interior (λ SθS* () >0). By Proposition 2, it is: θS*=pS DbS+pS ApL D+pS D [] bL+pS A cL A 2−kS+pS A kL 2 [] ΛkS+pS A kL 2 [] −cS A 2kS+pS AkL 2 ⎡ ⎢ ⎣⎤⎥⎦ λ SθS* () =1 2Λ+ ΛpS A kL 4 [] −cS A kS+pS AkL 4 ⎡ ⎢ ⎣⎤⎥⎦. 16 K. Nakao The condition that λ SθS* () >cL A kLis translated as Λ> 2kS+pS A 2kL [] cL A kL+cS A kS+pS A 2kL(Assumption 1-(ii)). B.2.2 For λ SθS () ≤cL A kL If the second-stage equilibrium appears at the corner (λ LθL*θS ()() =0by (λ SθS () ≤cL A kL), the first-stage equilibrium can be either interior (λ SθS* () ∈0,cL A kL ()) or at one of the two corners (λ SθS* () =0,cL A kL). The condition for the second-stage corner equilibrium that λ SθS* () ≤cL A kLis equivalent to Λ≤ 2kS+pS A 2kL [] cL A kL+cS A kS+pS A 2kL. In addition, if the firststage equilibrium is interior, θS*=pS DbS+pS ApL D+pS D [] bL−1 2kSΛ−cS A [] , and λ SθS* () =1 2Λ−cS A kS [] ∈0,cL A kL () , which holds if cS A kS<Λ<2cL A kL+cS A kS. If it is at the lower corner, θS*=pS DbS+pS ApL D+pS D [] bL, and λ SθS* () =0, which holds if Λ≤cS A kS. If it is at the upper corner, θS*=pS DbS+pS ApL D+pS D [] bL−kS kLcL A, and λ SθS* () =cL A kL, which holds if Λ≥2cL A kL+cS A kS. B.3 Summary of Equilibrium Conditions To recap, the second-stage equilibrium depends on the relative size between λ SθS* () and cL A kL.Ifλ S θS* () >cL A kL, the second-stage equilibrium is interior (λ LθL*θS* ()() >0), and the first-stage equilibrium must be interior (λ SθS* () >0). If λ SθS* () ≤cL A kL, the second-stage equilibrium is corner (λ LθL*θS* ()() =0), and the first-stage equilibrium depends on whether λ SθS* () is more than 0 and is less than cL A kL. For graphical illustration, Figure A shows A’s objective function at the first stage ΠS AθS () when: (a) Λis so large (Λ= 100) that the second-stage equilibrium is interior (λ LθL*θS* ()() >0, λ SθS* () >cL A kL); and (b) Λis small enough (Λ= 20) that the second-stage equilibrium is corner (λ LθL*θS* ()() =0, λ SθS* () ≤cL A kL), with the following parameter values: bL= 100, bS= 120, cL A=10, cS A=12, kL=1,kS=1,pL A=0.5, pS A=0.4. It can be confirmed that for (a), λ SθS* () =1 2Λ+ΛpS A kL 4 [] −cS A kS+pS A kL 4 [] =49.0909 …>cL A kL=10, and for (b), λ SθS* () =1 2Λ−cS A kS [] =4<cL A kL=10. Simultaneous Battles and Sequential Battles 17 References Bier, Vicki, Santiago Oliveros, and Larry Samuelson. 2007. “Choosing what to Protect: Strategic Defensive Allocation against an Unknown Attacker.”Journal of Public Economic Theory 9 (4): 563–87. Figure A: A’s objective function ΠS AθS () . (a) The equilibrium is interior at both the first and second stages. 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