power bent on economic warfare against the Conflict Escalation, Arab world. Washington, however, sent ineffective signals that increased Saddam’s Signaling and Screening suspicion of the United States’ intention3. As a Compatibility or Competition? result, the United States failed to reassure Kazumichi Uchida Saddam and encouraged him to invade Kuwait. In this sense, the war was caused by

their failure of reassurance. hat is the cause of war under Today, states have developed a conditions of asymmetric variety of techniques to overcome asymmetric information? Some argue that information. Some might declare their war is caused by states’ failure in W determination by taking costly actions such as deterrence of preemptive attacks, while others arms buildups, while some might test their argue that it is caused by states’ fear of other neighbors’ limits by making proposals. The states’ intention. By focusing on how states most important point of those techniques is exchange their resolves to resort to military to interchange their resolution with each other forces designed to prevent preemptive attacks, in a proper manner. This paper seeks to this study explores the causes of war under clarify under what condition states can conditions of asymmetric information. transmit their resolves effectively and address States need to deter its neighbors the problem, and under what condition they from aggressive actions; however, they also fail in transmission and war is likely. need to relieve them at the same time. If a This article proceeds as follows. state puts too much stress on deterrence, it First, I review previous studies and their may frighten the neighbors, which encourages deficiencies, and propose ways of correcting them to attack preemptively. Thus, it is these defects. Second, I present a basic model difficult for all states to declare their that makes it possible to consider the intentions in a proper manner under compatibility of both signaling and screening. conditions of asymmetric information. Based on this model, I clarify several findings The in 1990-91 provides a regarding when war is more likely and when it good example of the central issues. Before is less likely. invading Kuwait, Iraqi President Saddam

Hussein made a variety of demands of Kuwait.1 However, neither Kuwait nor the THEORETICAL BACKGROUND United States ever tried to determine the Prevention of the Preemptive Attacks concession point at which Saddam would In existing literatures on the causes have been satisfied by making increasingly of war, many have argued that the main cause generous offers.2 As a result, they failed to of war is the gap between distribution of deter Saddam’s invasion. In this sense, the war power and distribution of benefit, meaning was caused by their failure of deterrence. that the greater the disparity between distribution of power and distribution of In the meantime, Washington also 4 failed to reassure Saddam through the benefit, the more likely war will occur. If a signaling of benign intentions. In those days, Saddam regarded America as an imperialist 3 Ibid, 161-65. 4 Robert Powell, In the Shadow of Power: States 1 Janice Gross Stein, “Deterrence and and Strategies in International Politics Compellence in the Gulf, 1990-91: – A Failed or (Princeton: Princeton University Press, 1999), 6; Impossible Task?,” International Security, 17 Suzanne Werner, “The Precarious Nature of (1992): 147-7. Peace: Resolving the Issues, Enforcing the 2 Ibid, 159-60. Settlement, and Renegotiation the Terms,” JPI: Fall 2015 Issue 21 state feels that it is not getting its fair share of concessions that states make to other states is benefits, it has an incentive to change the the decisive cause of war. If states succeed in status quo, and that includes use of force. making concessions to other states properly, The relevant diplomatic policies for they can settle disputes. In other words, the the prevention of such calamities combine failure of reassurance could cause a war. Deterrence and Reassurance.5 Deterrence here is defined as the use of threats to dissuade an Asymmetric information opponent from undertaking damaging actions Then, why do states fail to take those in the future.6 The origin of bargaining theory measures against preemptive attacks？ The in international relations can be found in main reason is that every state has an James Fearon’s Rationalist Explanations for incentive to misrepresent its resolve under War.7 Assuming that a state’s resolve is based conditions of asymmetric information.11 in asymmetric information, Fearon has Hence, every state questions the credibility of attempted to elaborate the causes of war by other states’ discourses, and needs a way to observing whether it can send proper signals make their messages credible in order to reach to deter other states.8 Thus, war could be agreements with other states and avoid regarded as the failure of deterrence. unnecessary wars becomes paramount. There In contrast, Reassurance is defined as are two potential means of making messages the communication of benign and credible: Signaling and Screening. Signaling is unaggressive intentions by the defender to the defined as taking action to make threats to use 9 prospective challenger. Hence, deterrence is force credible.12 For example, if a state leader designed to use aggressive intentions to decides to expand troop numbers or to dissuade the opponent from preemptive declare publicly that the state will resist if attack, while reassurance relies on the attacked, it can declare its resolution more demonstration of benign intentions. Robert strongly to other states.13 Powell’s In the Shadow of Power developed the Screening is defined as the theory by starting with the rigid assumption identification of how much degree an that the gap between the distribution of opponent is resolved through the use of benefits and the distribution of power causes increasingly generous proposals that facilitate 10 war. Powell thus argued that the failure of settling early with the weakly resolved types.14 For example, if a state suggests a rigorous

American Journal of Political Science, 43 (1999): 929. 11 Fearon, “Rationalist Explanations for War,” 5 Richard Lebow, “Deterrence and Reassurance: International Organization, 379- 414. Recent Lessons from the Cold War,” Global Dialogue, 3 studies have tended to emphasize the (2001): 128. Commitment Problem as an explanatory cause 6 Alexander George, Forceful Persuasion: of war. However, the analysis of that factor Coercive Diplomacy as an Alternative to War requires a firm basis in Fearon’s first explanatory (Washington, DC: United States Institute of variable, Asymmetric Information, which I thus Peace Press, 1991), 5. focus on as the cause of war in this study. 7 James D. Fearon, “Rationalist Explanations for 12 Fearon, “Signaling Foreign Policy Interests: War,” International Organization 49 (3), 379-414. Tying Hands Versus Sinking Costs,” 69. 8 James D. Fearon, “Signaling Foreign Policy 13 James D. Fearon, “Domestic Political Interests: Tying Hands versus Sinking Costs,” Audiences and the Escalation of International Journal of Conflict Resolution 41 (1), 68-90. Disputes,” American Political Science Review, 9 Lebow, “Deterrence and Reassurance: 88 (1994), 577- 92. Lessons from the Cold War,” 432. 14 Scott Wolford, Dan Reiter, and Clifford 10 Powell, In the Shadow of Power: States and Carrubba, “Information, Commitment, and War,” Strategies in International Politics, 86 – 104. Journal of Conflict Resolution, 55 (2011), 567. JPI: Fall 2015 Issue 22 proposal on some issue to an opponent and states, in order to prevent preemptive attacks the opponent accepts it quickly, it can be from weaker actors. assumed that an opponent is not resolved so The next section will examine how much to fight over the issue. and under what conditions states’ Therefore, in this section, two main expectations about both the distribution of points could be drawn: First, states fail either power and the resolve of states converge. And in deterrence or reassurance due to this convergence is necessary in order to asymmetric information. Second, states prevent preemptive attacks. overcome asymmetric information by means of signaling and screening. THEORETICAL ANALYSIS Assume that two states in Table 1. Diplomatic Policy and its Techniques international system argue over strategic

resources such as oil, coal, or territory. The In previous studies, many scholars total amount of resources is normalized as 1. Signaling Screening In addition, let us assume that state 1 is a Deterrence Fearon (1997) × defending state, while state 2 is a challenging Reassurance × Powell (1999) state. have looked into conditions and success factors for either signaling or screening Let �! be the expected utility of state 1, let p separately, but not together. However, wars (0 ≤ � < 1) be the probability of winning of have occurred because these two techniques state 1, let �! be the war value of state 1, and have not been compatible.15 This study let c be the cost of war. To put it simply, I assumes that it is important to examine the assume that if they win, they gain everything, 17 compatibility of signaling and screening and to while if they lose, they lose everything. find the conditions under which we can �! = �! ∙ � ∙ 1 + 1 − � ∙ 0 − � achieve both at the same time, and thus clarify = � ∙ �! − � the cause of war. In the same fashion, we can express the utility of state 2 as, Convergence of States’ Beliefs �! = � ∙ �! − � （where � + � = 1）. How can previous research be Note that p also stands for the modified in order to address the problem? As balance of power in this international system mentioned, signaling and screening need to be or how resources are shared among states. examined in combination in order to analyze This is because the distribution of resources the causes of war. In addition to Powell’s, among states is determined by the probability Slantchev argues that even though there is a of winning wars. great disparity in capability among states, a weaker state may threaten to provoke a war in order to induce a stronger state to offer better terms.16 Therefore, expectations need to converge not only with respect to the distribution of power, but also the resolve of

15 Branislav L. Slantchev, “The Principle of 17 This model is consistent with those of Fearon Convergence in Wartime Negotiations,” (1997) and Wagner (2000) in the sense that it American Political Science Review, 97 (2003), takes war to be a costly lottery. Harrison 626. Wagner, “Bargaining and War,” American 16 Ibid. 621-32. Journal of Political Science, 44 (2000), 470. JPI: Fall 2015 Issue 23 Figure 1. Conflict Resolution and Escalation and amount to peaceful resolution. The lower bound of this range is a signaling point, while q q its upper bound is a screening point, and they stand respectively for the minimum and RESOLUTION maximum bounds necessary for convergence of belief. From here forward, I deduce those two points, which must be compatible to achieve their diplomatic purposes in this Reaction Function of State 1 model.

Proposition 1 ! Reaction Function of Let � = � − 2� + 2� (where α is a real ! ! ! � State 2 �∗ ∗ number). In the equilibrium, if � > �, the conflict is resolved, while if � < �, the conflict is escalated. p Proof: See Appendix. �∗ ESCALATION This proposition has several implications. First, � stands for the signaling point, meaning the minimum amount by which state 1 should demonstrate its resolve through Reaction Function of State 1 activities such as increasing or deploying troops in order to make its message credible. If state 1 demonstrates resolve to less than this degree, other states will think that state 1 Reaction Function of State 2 is not resolved and will be tempted to take advantage of it. Therefore, if state 1 seeks to make its signal credible, it should go at least to the degree �, which leads to conflict p resolution. This is because its belief in p �∗ converges with the equilibrium point, and no other states take action. As shown in Figure 1 above, I define the resolution of conflict as the convergence Proposition 2 of beliefs about the probability of winning a ! ! ! ! ! Let � = − �! + �! + 2�!�! + war, while I define the escalation of conflict as !!!!! ! ! the divergence of beliefs in that regard. The �! − � ∙ �! . reason why we can define conflict as shown In the equilibrium, if � < �, the conflict is above is because if their beliefs converge, they � > � have no incentive to change the status quo; if resolved, while if , the conflict is they diverge, they do have an incentive to escalated. ! change the status quo. The same rule is If we let A= � + � , we can get � = A + ! ! ! applicable to states’ resolve, according to the !!!! !!! ! ! ! (where � is a real number). explanation by Slantchev. ! There is some latitude in states’ Proof: See Appendix actions within which their beliefs converge JPI: Fall 2015 Issue 24 This proposition also has several ! ! �! − 2�! + 2� increases and � ≡ A + � ! ! implications. First, stands for the screening !!!! !!! ! ! ! decreases, it becomes more point, meaning the maximum degree by which ! state 1 should demonstrate its resolve in the difficult to converge expectations. I call this effort to make its message credible. The area the danger zone. I represent these two areas reason for regarding the maximum point as graphically in Figure 2 below. However, I the screening point is that it is necessary for a omit two variables, α and β, from the state to determine the maximum amount it calculation for simplicity. can gain in searching for an opponent’s resolve. If it crosses the limit, other states will Figure 2. The Security Zone and the Danger get cautious and take some action against it. If Zone state 1 wants to make its discourses credible, �! ! it should go at most to this point because here �! = �! beliefs in p converge. 1 � = � Security Zone ! 2 ! Proposition 3 In the equilibrium, if � < � < �, the conflict 1 is resolved; otherwise, it is escalated. 4

Proof: From the two propositions above, we can �! deduce that this proposition is inevitable. 0 1

2 The most striking aspect of this proposition is that just by putting p at any point within this range, state 1 can make its Danger Zone messages credible and conflicts among states will be resolved. This is because any point within this range is Nash Equilibria: everyone Finally, let us consider a case in which � goes will think that state 1 has no incentive to ahead of � and there is no room for states to misrepresent its resolve within this range. take any measures to prevent war. More Now we can think about three cases. ! ! formally, if � − 2� + 2� > A + In the first case, the range between � and � ! ! ! ! !!!! !!! ! expands. In this case, it becomes easier for ! ! , that is, if ! states to put p within this range and to resolve ! ! !! !! !! ! ! !! !! !! ! � < ! ! ! , ! ! ! < � , conflicts. More formally, if � ≡ � − ! ! ! ! ! ! ! the expectations of states never converge. 2� + 2� decreases and � ≡ A + ! ! This indicates that if the resolve of state 1 is !!!! !!! ! ! ! increases, it becomes easier to either too much or too little, they have no ! choice but to go to war. converge expectations. I call this area the Based on this analysis, we can security zone. deduce several findings. The main finding is The second case we will consider is that war is more likely if the resolve of a one in which the range between � and � challenging state exceeds half that of a shrinks. In this case, it becomes more difficult defending state, while war is less likely if the for states to put p within this range and resolve of a challenging state falls short of half ! ! resolve conflicts. More formally, if � ≡ that of a defending state. Note that : ! !

JPI: Fall 2015 Issue 25 equates to 2: 1, which represents the Just before he decided to invade proportion of defending states and Kuwait, he assumed that the outcome of the challenging ones. Iran- had propelled Iraq into the In addition, war is less likely only leadership of the Arab world in the post-Cold when the resolve of both states is positive; War era. This outcome imposed upon Iraq the and the resolve of defending states should be duty to deter both Iranian hordes and Israeli neither too high nor too low; paradoxically, if adventurism, which spurred Iraq to defending states are too resolved, they are reconstruct its economy and to expand its significantly damaged as well. industrial-technological infrastructure. This could only be achieved by means of its oil TESTS OF THE THEORY revenues.20 The theory above will be tested At the end of May 1990, during an according to two criteria.18 First, what Arab summit meeting convened in Baghdad, predictions can be inferred from the above Saddam denounced the Arabs of the Gulf theory？ Second, how much history does this who were keeping the price of oil artificially theory explain? low by producing beyond their OPEC quotas - to such an extent that the price in certain Predictions and Tests instances had plummeted to $7 per barrel, The theory’s predictions derived although the agreed-upon price was $18 per from its primary hypothesis. Namely, that war barrel-, and thereby engaging in economic 21 is more likely if the resolve of challenging sabotage of Iraq. In a memorandum dated states exceeds half that of defending states, July 15, addressed to the Secretary-General of while war is less likely if the resolve of the United Nations, Iraqi Foreign Minister challenging states falls short of half that of Tariq Aziz explicitly named Kuwait and the defending states. The following two United Arab Emirates as the two “culprits” in 22 predictions based on this theory are tested in a overproduction. To make matters worse, case study of the Gulf War of 1990-91. The during the hostilities with Iran, Kuwait was outcomes vary sharply across time, creating a drawing more than its allotment from the good setting for multiple within–case shared northern Rumaila oil field, whose comparisons tests that contrast different southern tip straddled the border inland from 23 periods within the same case.19 the Gulf. Defending states succeed in On July 16, at Saddam’s request, prevention of preemptive attacks if the Aziz sent a memorandum to Kuwait resolve of challenging states’ does not exceed demanding $2.4 billion in compensation for the proportion. oil that Kuwait had pumped from the Defending states fail in prevention of preemptive attacks if the resolve of 20 Walid Khalidi, “Iraq vs. Kuwait: Claims and challenging states’ exceeds the proportion. Counterclaims” in The Gulf War Reader, ed. Micah Sifry and Christopher Cerf (Ithaca, NY: Test 1: Failure in Prevention TimesBooks /Random House, 1991), 60-61. 21 Stein, “Deterrence and Compellence in the Gulf, 1990-91 –A Failed or Impossible Task?,” 18 Stephen Van Evera, Guide to Methods for 147-179; Khalidi, “Iraq vs. Kuwait”, 63. Students of Political Science (Ithaca, NY: 22 Bishara A. Bahbah, “The Crisis in the Gulf: – Cornell University Press, 1997); Stephen Van Why Iraq Invaded Kuwait” in Phyllis Bennis and Evera, “Offense, Defense, and the Causes of Michel Moushabeck, eds., Beyond The Storm: – War,” International Security, 22 (1998): 22. A Gulf Crisis Reader (New York: Olive Branch 19 Van Evera, Guide to Methods for Students of Press, 1991), 52-3; Khalidi, “Iraq vs. Kuwait,” 63. Political Science, 58-63. 23 Khalidi, “Iraq vs. Kuwait,” 63. JPI: Fall 2015 Issue 26 disputed Rumaila oil field; $12 billion in Glaspie. Her assumption was that the Iraqi compensation for the depressed oil prices tanks were meant to intimidate Kuwait at the brought about by Kuwait’s overproduction; negotiating table, and that they would forgiveness of Iraq’s war debt of $10 billion; probably succeed in doing so. She found it and a lease on the strategic island of Bubiyan tolerable.28 that controlled access to Iraq’s only port, We can assume from the Umm Qasr.24 Irrespective of the mediation appearances that the resolve of Iraq was efforts of King Fahd of Saudi Arabia, intensified day by day, while the United States President Hosni Mubarak of Egypt, and King was not resolved at all, which made war more Hussein of Jordan, on July 24, two Iraqi and more likely. Saddam also thought that the armored divisions moved from their bases to United States would not wage war due to their positions on Kuwait’s border.25 Vietnam complex and the low tolerance of the While the crisis was intensified, American public to U.S. causalities.29 Early in Washington’s signals to Iraq were ambiguous the morning on August 2, two Iraqi armored and contradictory.26 On July 25, Saddam asked divisions spearheaded the attack against to meet with Ambassador April Glaspie Kuwait. within the hour, in response to the U. S. joint military exercises with the United Arab Emirates the previous day. Glaspie assured Test 2: Success in Prevention Saddam that the United States should express At the first full no opinion on the Kuwait issue, and that the Council (NSC) meeting after the Iraqi issue was not associated with them. She then invasion of Kuwait on August 2, 1991, asked Saddam about the intention of Iraqi Schwarzkopf outlined two military options for troop movements to the border with Kuwait. the U.S. response: (1) punitive air strikes by He informed her that he had just asked carrier-based aircraft, and (2) a detailed President Mubarak to assure Kuwait that they existing contingency plan, Operation Plan would do nothing until they held meetings. (OP) 90-1002, which called for deploying He added that if there were any hope in more than four divisions and three aircraft negotiations, nothing would happen, while if carriers, for a total of 100,000 to 200,000 U.S. they were unable to find a solution, it would troops. The key divide in this August 3 be natural for Iraq not to accept death.27 This meeting was between National Security statement was designed to satisfy Ambassador Advisor Brent Scowcroft and Secretary of Defense Richard Cheney, who favored an

24 expansive response, and Chairman of the Stein, “Deterrence and Compellence in the Joint Chiefs of Staff General , Gulf, 1990-91 –A Failed or Impossible Task?,” who remained more cautious about 147-179. 25 “Kuwait: How The West Blundered”, in The committing U.S. military power and Gulf War Reader, ed. Micah Sifry and advocated the more limited goal of defending Christopher Cerf (Ithaca, NY: TimesBooks Saudi Arabia from further Iraqi aggression. In /Random House, 1991), 104; Stein, “Deterrence Powell’s view, which became known as the and Compellence in the Gulf, 1990-91 –A Failed “Powell Doctrine,” the military was to be or Impossible Task?,” 147-179. 26 used as a last resort, and only then in an Stein, “Deterrence and Compellence in the overwhelming fashion. By this time, however, Gulf, 1990-91 –A Failed or Impossible Task?,” President Bush had already begun to lean 147-179. 27 “Kuwait: How The West Blundered”, 104-5; toward OP 90-1002, the “” option Stein, “Deterrence and Compellence in the Gulf, 1990-91 –A Failed or Impossible Task?,” 147- 28 “Kuwait: How The West Blundered,” 105. 179. 29 Bahbah, “The Crisis in the Gulf,” 54. JPI: Fall 2015 Issue 27 over the more limited goal of offensive attack by Saddam into Saudi Arabia “.”30 less likely.36 The official Saudi “invitation” for In those days, when air power was U.S. military intervention came on August 7.31 not as well recognized, it was ground troops The next day, August 8, a Defense more than airstrikes what showed the Intelligence Agency officer informed Prince governments’ resolve. Saddam had moved his Bandar of what had happened during the eight divisions, an estimated 100,000 men, to Kuwait invasion and how Saddam was the Saudi border while the United States went massing the same elite force of eight divisions public with the concrete number of troops on the Saudi border.32 who would be sent to defend Saudi Arabia From then on, the U.S. was and then put the plan into practice. What is strengthening their signals to deter Iraq from important in this study is the fact that when invading Saudi Arabia, but at an incremental Saddam recognized that the United States put rate in order to reassure Iraq rather than twice as many troops as his on the Saudi encourage a preemptive attack. The first signal border, he gave up invading Saudi Arabia. was sent on August 17 by U.S. officials This corroborates the prediction that including Defense Secretary Cheney. They defending states succeed in both deterrence intentionally leaked word to the Associated and reassurance when their resolve exceeds Press that the U.S. was preparing for a “long twice that of challenging states’. commitment” in the Gulf.33 They also leaked the real number of troops up to 250,000 to CONCLUSION the press.34 The next signal was the This article argues that war is more Pentagon’s exponentially inflated troop likely when the resolve of challenging states projections needed to “defend” Saudi Arabia exceeds twice that of defending states. With against an Iraqi attack. On August 22, the respect to the causes of war, states cannot president called up an initial 40,000 members achieve both diplomatic policies, namely, of military reserve units, and by September 6 deterrence and reassurance, at the same time the number of troops in Saudi Arabia had unless they combine the techniques of reached 100,000. On September 15, U.S. signaling and screening. officials announced they had established a Based on the assumption that war credible defense and were now building up can be represented as the divergence of state their offensive capability.35 Finally, on beliefs about the probability of winning and September 21, U.S. intelligence claimed that state resolve, I establish the minimum and Saddam‘s forces were digging in, moving into maximum points necessary for their beliefs to even more defensive positions. This made an converge, which can be defined as signaling and screening points. I then assume that if 30 Jonathan Monten and Andrew Bennett, those ranges expand, it becomes easy for “Models of Crisis Decision Making and the 1990- states to determine the points that are 91 Gulf War,” International Security, 19 necessary for the settling of disputes, and vice (2010):498-503. versa. This enables me to present the 31 Steve Niva, “The Battle is Joined,” in Beyond hypotheses mentioned above. The Storm: – A Gulf Crisis Reader, ed. Michel I deduce several predictions from those Moushabeck and Phyllis Bennis (New York: hypotheses, which are tested with the case of Olive Branch Press, 1991), 57. 32 Bob Woodward, The Commanders (New the Gulf War of 1990-91, which is rich in York: Simon and Schuster, 1991), 278. both political success and failure. I then 33 Niva, “The Battle is Joined,” 59. 34 Woodward, The Commanders, 297. 35 Niva, “The Battle is Joined,” 59. 36 Woodward, The Commanders, 297. JPI: Fall 2015 Issue 28 demonstrate that these predictions correlated equilibrium is that the condition of conflict with what happened in the Gulf. resolution is � < 1. This study leaves many themes of Because the reaction function of state 1 is !! remaining research. The first topic we need to � = �, we can get ! = � − � = 0. tackle is how to deal with the influence of !" ! state growth, which is categorized as a Therefore, � = �! − � ∙ � + � = � ∙ ! ! commitment problem. In order to complete ! � − � + � (where β is a real number). the research, we need to consider a dynamic ! version of this model by introducing new Since we assume �! = � ∙ �! − �, we can get ! variables such as the growth rates of states. � = � − �. Moreover, since � + � = 1, we ! ! Some previous research has already tackled ! 37 can get 1 = � − � …① . this problem with distinguished implications. ! ! What we need to do next is to combine these On the other hand, because the reaction achievements with the findings of this function of state 2 is � = � ∙ � + �, we can !! research in order to improve the ways in get ! = �� + � − � = 0 . which states can achieve their diplomatic !" ! Therefore, � = − �! + �� + � � + � = purposes, even when there is a great disparity ! ! in growth rates among states. ! q − � + �� + � + �(where ! APPENDIX γ is a real number). Since we assume �! = � ∙ �! − �, we can get ! Proof of Proposition 1 �! = − � + �� + � . Consider the Cournot Equilibrium. ! Now, since � + � = 1, we can get 1 = The reason why I consider the Cournot Equilibrium is because in equilibrium both 2� + 1 � − 2�! + 2�…② . actors’ beliefs about the probability of Because ① and ② should be compatible, ! winning war are supposed to converge. � − � = 2� + 1 � − 2� + 2� . ! ! ! Thus, I start with both actors’ reaction Therefore we can get functions so that their beliefs will converge as −�! + 2�! − 2� 1 a result of their profit maximizations. � = 2� + 4 . Assume that state 1 is a defender, When � ≥ 0, the condition of � < 1 is and its reaction function is � = �, while state −� + 2� − 2� ! ! + 1 < 1 2 is a challenger, and its reaction function is 2� 4 . ! � = � ∙ � + � (where α is a real number). Therefore, we can get � > −� + 2� − ! ! ! In this case, the condition of equilibrium at 2� . point t is �! = � ∙ �!!! + �.In addition, in When � < 0, the condition of � < 1 is equilibrium, �∗ = � ∙ �∗ + � . − −� + 2� − 2� ! ! − 1 < 1. If we assume that ∆�! ≡ �! − �∗, we can get 2� 4 ∆� = �∆� = ⋯ = �!∆� . (where � is a ! ! !!! ! Therefore, we can get � > �! − 2�! + real number). The important point in this ! 2� . Q.E.D.

Proof of Proposition 2 37 Powell, “The Inefficient Use of Power: Costly Consider the Bertrand Equilibrium. Conflict with Complete Information,” American The reason why I consider the Bertrand Political Science Review, 98 (2004): 231-41; Robert Powell, “War as a Commitment Equilibrium is because in equilibrium both Problem,” International Organization, 60 (2006): actors’ beliefs about the war values are 169-203. supposed to converge. JPI: Fall 2015 Issue 29 ! ! ! Thus, I start with both actors’ reaction Therefore, we can get � = ! − ! − � ∙ functions so that their beliefs will converge as ! !! !! ! ! ! a result of their profit maximizations. + � ∙ − � ∙ + 1 . !! !! !! Assume that state 1 is a defender and When � ≥ 0, the condition of � < 1 is its reaction function is �! = �!, while state 2 ! ! ! ! ! ! ! − ! − � ∙ + � ∙ − � ∙ + 1 < 1. is a challenger and its reaction function is ! !! !! !! !! !! � = � ∙ � + � β is a real number) ! !! ! ! (where . Therefore, we can get � > − ∙ �! + ∙ ! !! The equilibrium at point t is �!! = � ∙ ! �! − � − � . �! !!! + �, while in equilibrium, �!∗ = � ∙ ! When � < 0, the condition of � < 1 is �!∗ + � . ! !! !! ! ! ! If we assume that ∆�!! ≡ �!! − �!∗, − − + � ∙ − � ∙ + � ∙ − 1 < ! ! !! !! !! !! !! we can get ∆�!! = �∆�!!!! = ⋯ = � ∆�!" . ! ! ! 1. Therefore, we can get � < − �! + (where � is a real number). The important !!!!! ! ! point in this equilibrium is that the condition �! + 2� � + � − � ∙ � . ! ! ! ! ! ! of conflict resolution is � < 1. Since we need to find the upper bound of p, Because the reaction function of state 1 is we should assume that � < 0. !!! �! = �!, we can get = �! − �! = 0. � < 0 !!! Therefore, we should also assume that ! Therefore, � = − � ! + � ∙ � + � = � ∙ in proposition 1.Q.E.D. ! ! ! ! ! ! ! − � + � + �(γ is a real number ). ! ! ! Since we assume �! = � ∙ �! − �, we can get ! � = − � + � . ! ! ! Because S = � ∙ �! + � ∙ �!, we can get ! S = − � ! + � ∙ � + � ∙ � ・・・① ! ! ! ! ! (I assume that S stands for the supply of public goods provided by those states). On the other hand, the expected utility of state 2 is �! = � ∙ �! + �; hence, we can get !!! = � ∙ �! + � − �! = 0. Therefore,�! = !!! − ! � ! + � ∙ � + � ∙ � + � = ! ! ! ! ! � − � + � ∙ � + � + � (where ! ! ! ! δ is a real number) . Since we assume �! = � ∙ �! − �, we can get ! � = − � + � ∙ � + � . ! ! ! Because � = � ∙ �! + � ∙ �!, we can get S = − ! � ! + � ∙ � + � � + � ∙ � ・・ ! ! ! ! ! ・② Now, both ① and ② should be compatible, ! such that − � ! + � � + � ∙ � = ! ! ! ! ! ! − � ! + � ∙ � + � � + � ∙ � . ! ! ! ! !

JPI: Fall 2015 Issue 30