A Theory of Democratic Peace and Power Shift Ilhan Sezer1 Abstract We study a model of war in which governments pave the way for wars by creating uncertainty about their military powers. In this model countries’ resource levels, military technologies and victory values are common knowledge. Governments simultaneously decide how much to invest in military power. The outcome of the war is deterministic in that the country with the stronger army wins and receives its victory value. We characterize the unique equilibrium when the war is between two democracies or between two autocracies. Our model delivers results on the intensity of wars in terms of their origin in the societal and the international system level. In the societal level, countries have smaller armies, they fight less severe wars, and their citizens are better off in a democratic world than in an autocratic one, because democracies, unlike autocracies, are highly attentive to the costs of wars. In the systemic level, these results follow from an increase in aggression inequality, because the non-aggressive country becomes less likely to win and invests less in warfare when there is a decline in its military technology or an advancement in the aggressive country’s military technology. Moreover, wars completely disappear when the aggressive country has absolute superiority in military technology. As an extension, we consider the case when countries’ victory values are private information, and get results in line with the complete information case. JEL classification: D44; D74; H56 Keywords: War; Auction; Uncertainty; Democratic peace; Power shift
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This paper is based on my dissertation, submitted to Princeton University, Department of Economics. I am grateful to my advisor Faruk Gul for his guidance and continuous help. I would like to thank Sylvain Chassang, Dilip Abreu, Zafer Akin and a number of seminar participants for helpful comments and encouragement. Mailing address: 351 Lemonick Ct, Unit 102, Princeton, NJ. Fax: +1 (609) 258-6419. Telephone number: +1 (609) 933-5470. E-mail address:
[email protected].
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1
Introduction
The outcome of wars is uncertain from the perspective of the involved countries. In standard models, the outcome is uncertain because of luck on the battlefield. In this paper we study an alternative model of wars in which countries are uncertain about the outcome because they lack information about their opponent. In our model, there are two countries characterized by their initial resource levels, military technologies and victory values. Each country decides how much of their own resources they will allocate to military production. The uncertainty about the military production levels, which is created by governments acting in their own self-interest in the complete information case, causes war. The country with greater military power wins the war and gains its victory value. We characterize monotone continuous equilibrium strategies for the wars between democracies and between autocracies for both complete and incomplete information cases. The equilibrium is unique for all possible values of the parameters in either case. When governments have complete information, governments choose mixed strategies and create uncertainty that paves the way for wars, whereas when they have incomplete information, wars recur due to their exogenous uncertainty about each other. In both cases the following results hold: First, the world of democracies is more peaceful than the world of autocracies in terms of countries having smaller military powers and fighting less severe wars, and all citizens are better off in the democratic world. Second, if one of the countries has absolute superiority in military technology, then investments in warfare and wars will almost disappear. Finally, when governments have complete information, if there is a small advancement in the aggressive country’s military technology or a small decline in the non-aggressive country’s military technology, both countries will have smaller armies, they will fight less severe wars, and their citizens will be better off. The theoretical literature on wars contains two main strands: why wars emerge and how they are fought. Our model primarily contributes to the second strand. The models in this strand have four common features.2 First, war takes place for exogenous reasons. Second, countries allocate their resources to warfare and production. Third, the winner is determined probabilistically, often modeled using a Tullock contest success function.3 Fourth, the winner consumes all production. We keep the first two features. However, we assume that the stronger army wins for sure, and the winner consumes its victory value, which does not have to be equal to all production. Thus, in our model countries are uncertain about the outcome 2 3
See Garfinkel and Skaperdas [8] and Hodler and Yektas [13]. Tullock contest success function was first introduced by Tullock [36].
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of the war not because of luck on the battlefield, but because they lack information about their opponent’s military power. Moreover, since victory does not necessarily have the same value for all countries, our model allows us to examine a wide range of wars.4 Despite the differences in the setup, our model and the standard models of wars have some common results: If the countries have the same military technologies and victory values, the resource poorer country allocates a higher share of its resources to warfare, and fully compensates its disadvantage, that is, both countries win the war with the same probability.5 However, our model allows difference in military technologies and victory values, and, by doing so, addresses power shift. Moreover, our model shows that democracies tend to fight less severe wars with each other than do autocracies and democratic peace has not been addressed in this manner by the standard models. In the literature, democratic peace generally posits that democracies are hesitant to engage in wars with other democracies, and therefore the likelihood of war between two democracies is smaller than the likelihood of war between two countries of which at least one is not a democracy. Its foundations date back to Paine [27] and Kant [17], and it has been revitalized by Doyle [6, 7], which have been followed by an outbreak of studies on democracy.6 However, there are strong criticisms of this view. For instance, Russet [29] says that democracies were too scarce and far apart before the Second World War, hence the absence of wars between democracies was not surprising. Even though his core thesis is that democracies do not fight with one another, he lists several other factors such as alliances, wealth and political stability that have influenced the alleged peace between democracies, and restricts the influence of democracy on peace to the very recent past. Small and Singer [33] find an absence of wars between democratic states with two “marginal exceptions,” but argue that this pattern is statistically insignificant. Spiro [34] shows that the absence of wars between democracies is not statistically significant, except for a brief period during World War I. Gowa [10] shows that the democratic peace before 1945 4
For example, if the reason for war is to acquire a land, a mineral mine or all production, such as in Garfinkel and Skaperdas [8] and Hodler and Yektas [13], the value of victory for a country represents the value of that land, mine or production for that country. However, if the two countries are fighting due to political differences, such as separation versus unification in Bolton and Roland [4], then the value of victory for a country represents the extra benefit that country gets by implementing its own policy instead of its opponent’s policy. 5 This result is known as the Paradox of Power and is one of the most famous results in the theoretical literature on wars. For standard models see Garfinkel and Skaperdas [8] and Hirshleifer [12]. 6 See also de Tocqueville [5].
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is insignificant, and offers an alternate explanation for the following period such that peace is an artifact of the Cold War, and the threat from the communist states forced democracies to ally with one another. Schwartz and Skinner [32] take the criticism further, and argue that neither the historical record nor the theoretical arguments advanced for the purpose provide any support for democratic pacifism, and show that there have been as many wars between democracies as one would expect between any other pair of states.7 Rummel [28] tries to reconcile the democratic peace theory with these criticisms, and argues that a regime’s warlikeness should be measured in terms of the severity of wars instead of the frequency count of wars, and shows that democracies are less warlike in terms of the severity of their wars between 1900 and 1987. As a result of these studies, the recent consensus on the democratic peace is in line with Rummel [28], that is, democracies are not different than other types of regimes in terms of frequency of war, but their wars are less severe. In our model, we follow Kant [17] and argue that democratic governments are accountable to the citizens, they are highly attentive to the costs of wars and bear all burdens of wars, whereas the same cannot be said about autocratic governments, which enjoy the benefits of wars and avoid most of their costs. However, we deviate from Kant’s conclusion that democracies almost never go to war with each other. We show that democracies fight democracies, but wars between democracies tend to be less severe than wars between autocracies. In other words, unlike Kant, we measure a regime’s warlikeness in terms of the severity of wars instead of the frequency count of wars, which gives the same count of one for a country that has suffered only a few casualties and for another country that has lost several million citizens, and show that democracies are less warlike in a world of democracies than are autocracies in a world of autocracies. Hence, our model offers a theoretical background to the validity of the recent consensus on democratic peace, that is, a world of democracies is expected to be more peaceful than a world of autocracies when peace is measured in terms of mildness of wars. The differences between the incentives of governments and citizens have been studied in the literature. For instance, Lake [20] argues that in democracies the society’s cost of controlling the state is relatively low compared that cost in autocracies, so democracies are less likely to fight wars with each other while they are more likely to fight wars with autocratic states. Schultz [31] presents an incomplete information model of crisis bargaining to explore the effect of domestic political competition on the escalation of international crises and comes up with a similar result: the inclusion of a strategic opposition party decreases the ex-ante probability of war by helping to reveal information about a state’s preferences. Hence, he suggests a mechanism 7
However, they include wars between young and dubious democracies as well as very small wars.
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through which democratic states overcome the informational asymmetries between their governments and their citizens, a central obstacle to negotiation. Jackson and Morelli [15] examine how countries’ incentives to engage in a war depend on the political bias of their leaders, where political bias refers to the discrepancy between the interests of decision makers and citizens. They show that it is possible to avoid a war when there is no political bias, but when there is sufficiently strong bias on the part of one or both countries, war cannot be prevented via transfers, and the probability of winning the war depends on the countries’ levels of wealth. However, in contrast to democratic peace theory, they show that when transfers are possible, at least one country will choose a biased leader as that leads to a strong bargaining position and extraction of transfers. We differ from this literature by modeling war as an auction problem and by showing that this discrepancy makes autocracies more warlike than democracies, even if there exists no informational asymmetry between governments and their citizens or no possible transfers between countries. On the other hand, power shift is a systemic-level explanation of the causes of war.8 Among systemic-level explanations, we see two opposite approaches that have developed through studies on the relationship between power distributions and war. The balance-of-power approach insists on peaceful consequences of power equality, while the power-shift approach claims the opposite, that is, a shift from power equality towards power inequality produces a more peaceful world. Balance-of-power approach is concerned with explaining national strategies, the formation of blocking coalitions, the avoidance of hegemony, and the stability of the system. Balance-ofpower supporters agree that some form of equilibrium of military capabilities, bipolarity9 or multipolarity10 , increases the stability of the system, which is generally defined as the relative absence of major wars, and that movements toward unipolarity are destabilizing because they trigger a hegemonic war to restore equilibrium. Even though they rely heavily on polarity as a key explanatory variable, they do so with little supporting evidence. For instance, Sabrosky [30] shows that bipolarity is no less war-prone than multipolarity, that wars occur under a variety of structural conditions, and that polarity is not a primary causal factor in the outbreak of war. As an alternative to balance-of-power approach, power-shift approach is a form of hegemonic theory that emphasizes the peaceful consequences of a hierarchical sys8
Ever since Waltz [38] classified the causes of wars in terms of their origins in the individual level, the societal level and the international-system level, international relations theorists have agreed on using it for the levels of analysis. 9 See Waltz [39] and Mearsheimer [23] 10 See Morgenthau [25] and Gulick [11]
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tem.11 Many of the theoretical analyses of power-shift approach focus on transitions between a hegemonic state and a challenger, and study the causes of the rise and fall of hegemons as well as the precise identity of hegemonic war such as who initiated the war, whether the declining hegemonic state did to block the rising challenger while the chance was still available or the challenger did so to bring its benefits from the system into line with its rising military power.12 Analyses of power-shift approach generally include a broader international system with a hierarchy among states. However, some studies of power-shift approach apply to any two states in the system, and require that countries are more warlike when there is an equality of power and less warlike when one state has a preponderance of power over the other. Kugler and Lemke [19] shows the widespread support for peaceful consequences of power preponderance of a state over another in the empirical literature. Hence, even though there are a number of strong adherents to both approaches, the recent empirical studies tend to favor power-shift approach over balance-of-power approach, and in this paper we present an auction type model which offers a theoretical background for power-shift approach. In our model, the war game between two governments is like an all pay auction. When we look at the relationship between power distributions and war, our model offers a theoretical explanation for the power-shift approach through the following observation on auctions. In an all pay auctions, an increase in the strong bidder’s valuation or a decrease in the weak bidder’s valuation causes a decrease in the weak bidder’s probability of victory, and this leads to a decrease in the weak bidder’s bid. The strong bidder also bids less aggressively due to the decrease in the weak bidder’s bid. Therefore, bids will be higher when bidders’ valuations are equal than when one bidder’s valuation is preponderance over the other’s.13 The same mechanism works in our model when we replace bidders’ bids with governments’ effective bids and bidders’ valuations with governments’ levels of aggression.14 In our analysis, we offer a new interpretation of power-shift approach, and show that our model offers a theoretical background that supports both our and classical interpretations of this approach. Powerful countries do not intervene in the conflicts between other countries (or powers) as long as the conflicts is not important to them. For instance, France intervened in Libya during the Arab Spring in 2011, but chose not to intervene in 11
See Organski and Kugler [26], Gilpin [9], and Thompson [35]. For a detailed survey see Levy [21] and Vasquez [37]. 13 See Baye, Kovenock and de Vries [3] and Amann and Leininger [1]. 14 Governments’ effective bids are equal to governments’ bids amplified by their military technology, and governments’ levels of aggression are their victory values amplified by their military technology. 12
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Bosnia and Herzegovina two decades ago; or the U.S. intervened in Korea, Vietnam, Iraq et cetera, but failed to intervene in Rwanda and Syria. These show that countries do not intervene in conflicts just because they are powerful, but they choose to become a part of the conflict if they have a motive for intervening. Hence, we think that countries’ levels of aggression is a better measure of their motivation than their power levels, and we offer a different interpretation of power-shift approach which depends on countries’ aggression levels. According to our interpretation, power-shift approach states that a shift towards higher aggression inequality, instead of higher power inequality, produces a more peaceful world. In line with our understanding of power-shift approach, we show that when governments have complete information, investment in warfare will decrease and loss from wars will decline with an advancement in the more aggressive government’s military technology or a decline in the less aggressive government’s military technology, that is, a step towards higher aggression inequality will also be a step towards a more peaceful world. Moreover, in line with both our and classical interpretation of power-shift approach, independent of whether governments have complete or incomplete information, we show that investments in warfare and wars will completely disappear with the absolute superiority of one of the countries in military technology. As we model war as an auction, our study also relates to the literature on auction theory. Wars between autocracies share some features with all-pay auctions. Resources allocated to warfare cannot be used to produce consumption goods, hence, as in all-pay auctions, all bids need to be paid. However, there is a small difference such that the winner is the government with the highest effective bid in our war game, whereas the winner is the bidder with the highest bid in an all-pay auction. Baye, Kovenock and de Vries [3] fully characterize equilibrium for all-pay auctions with complete information, and we adapt their characterization for wars between autocracies that have complete information. On the other hand, Amann and Leininger [1] prove the existence and uniqueness of Bayesian equilibrium for a class of generally asymmetric all-pay auctions with incomplete information, which resembles our model when the war is between autocracies that have incomplete information. However, when the war is between democracies, our model deviates from the literature on all-pay auctions in that the payoff structures are quite different, because governments’ payoffs depend on each others’ bids. In the auction literature, Hodler and Yektas [13] is the closest to our model. They model wars as asymmetric auctions with incomplete information. In their model, the outcome of the war is uncertain since countries’ realized resource levels are private information. Each country allocates its resources to production and warfare, and the country with greater military power wins and consumes all goods that have been produced in the two countries. They 7
show that war takes place due to incomplete information, and both comparative and absolute advantages matter. There are major differences between their model and ours. In our model: (i) countries’ victory values, which play a similar role to the “resource levels” of Hodler and Yektas [13], can be both private information and common knowledge, whereas in Hodler and Yektas [13] resource levels are common knowledge, (ii) the winner gains only its own value of the war, while in Hodler and Yektas [13], the winner consumes all production, and (iii) both governments pay their bids and the costs of war, unlike Hodler and Yektas [13] in which the loser pays nothing. Moreover, our results are different in that our model has a say about power-shift approach and democratic peace. The remaining of the paper is organized as follows. Section 2 lays out the basic model. We analyze the complete information case in Section 3. As an extension, we study the incomplete information case in Section 4. Section 5 concludes. All the proofs are relegated to the Appendix.
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The Basic Model
Two countries, country A and country B, engage in a war to acquire the spoils of victory. Each country i ∈ {A, B} is characterized by three parameters, its initial resource level Ri ∈ R+ , its military technology λi ∈ R+ , and its victory value vi ∈ R+ , that is, country i ∈ {A, B} has Ri units of resources, it can produce λi units of military goods by using one unit of resources, and it gains a benefit of vi units of resources by winning the war. Each country has a government. Governments simultaneously decide how much of their own resources they will allocate to military production. Countries’ initial resource levels are not a binding constraint on their military production. After the production of military goods, countries engage in a war. The outcome of the war is deterministic and the country with greater military power wins the war. More formally, for i ∈ {A, B}, let bi denote the amount of resources allocated for military production by government i. Then, country i ∈ {A, B} will have a military power of λi bi . We call bi government i0 s bid and λi bi government i0 s effective bid. Effective bids determine the winner of the war, such that the country with the higher effective bid wins the war and in case of equality both countries win with equal probabilities. War is costly, because countries produce military goods and also because war causes losses, such as wounded civilians and soldiers, destroyed or damaged buildings and other financial costs. We assume that countries have to spend one unit of their military power to destroy one unit of their opponent’s military power, and they do not give up fighting as long as they have military power, that is, the war is over whenever 8
the military power of at least one of the countries is completely destroyed. As a result of this, the size of the smaller army determines both countries’ losses during the war. We measure all losses in terms of resources. Therefore, more formally, given bi , bj and λj bj ≤ λi bi , country i and country j spend bi and bj units of resources to produce λi bi λ and λj bj units of military goods, respectively, and suffer losses of λji bj and bj units of resources during the war, respectively.15 The size of the war is the total war-related 1 losses of the involved countries, that is bA + bB + min{λA bA , λB bB } λA + λ1B . In each country, there is a representative citizen, citizen A and citizen B. These representative citizens are risk-neutral and they enjoy the resources in their countries after the war. If country i wins the war, that is, if λj bj < λi bi , then it uses bi units of λ resources for military production, suffers a loss of λji bj units of resources during the war, and acquires vi units of resources by winning the war; hence, citizen i enjoys λ Ri + vi − bi − λji bj units of resources. Following the same reasoning, if both countries win the war with equal probabilities, that is, if λj bj = λi bi , then citizen i enjoys Ri + vi − 2bi or Ri − 2bi units of resources with equal probabilities. Similarly, if country i loses the war, that is, if λj bj > λi bi , then citizen i enjoys only Ri − 2bi units of resources. Governments are one of two types: autocratic or democratic. Autocratic governments suffer the costs of military production, however, unlike their representative citizens, they do not suffer any losses during the war. Hence, it follows from the above discussion that when country i is an autocracy, government i’s utility is: for λi bi < λj bj Ri − bi , vi uARi (bi , bj ) = Ri + 2 − bi , for λi bi = λj bj (1) Ri + vi − bi , for λi bi > λj bj . Democratic governments, unlike their autocratic counterparts, are accountable to their citizens and bear the burdens of wars just like their representative citizens. Hence, a democratically elected government i’s utility is the same as citizen i’s utility, and it is: min{λi bi , λj bj } uDRi (bi , bj ) = uARi (bi , bj ) − λi if λi bi < λj bj Ri − 2bi , = Ri + v2i − 2bi , (2) if λi bi = λj bj λj b j Ri + vi − bi − λi , if λi bi > λj bj . 15
Hence, the country that is more advanced in military technology suffers less in terms of resources during the war.
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In this paper, we examine wars between two democracies and between two autocracies and compare a world of democracies with a world of autocracies. We make this comparison with two different information setups: complete and incomplete. The parameters that characterize country A and country B are common knowledge in the complete information case, whereas countries’ victory values, that is, vA and vB , are private information in the incomplete information case.
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Complete Information
In this section, we analyze the complete information case. Hence, all parameters of country A and country B are common knowledge. For i ∈ {A, B}, government i can produce upto λi vi units of military goods by using the spoils of victory.16 We call λi vi government i0 s level of aggression. We say that government i becomes more aggressive when λi vi increases, and similarly becomes less aggressive when λi vi decreases. Without loss of generality, we assume that government A is weakly more aggressive than government B, that is, λA vA ≥ λB vB , and call governments A and B the aggressive and non-aggressive governments, respectively. We define aggression inequality as the ratio of the aggressive government’s level of aggression over the non-aggressive government’s level of aggression, and denote it with: s=
λA vA . λB vB
(3)
Hence, governments’ aggression levels are alike when aggression inequality is low, whereas government A is much more aggressive than its opponent when aggression inequality is high. First, we look at wars between two autocratic governments. The war game between two autocracies is similar to all-pay auctions with complete information and private valuation, and differs only in that the winner is the government with the highest effective bid.17 The equilibrium will require mixed strategies. We denote governments’ bidding strategies by the distribution functions FA and FB , where Fi (x) is the probability that the amount of resources allocated for military production by government i is less than x units. 16 Government i has no incentive to produce more than λi vi units of military goods, since it is better off with no military production than with a military production of more than λi vi units, even though having no military production leads to a sure failure in the war. 17 Notice that an all-pay auction is a particular case of our war game with λA = λB = 1.
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Proposition 3.1. When both countries are autocracies, there is a unique Nash equilibrium. In this equilibrium, governments’ bidding strategies are: h v i x A FA (x) = s · for x ∈ 0, , (4) vA s 1 1 x FB (x) = 1 − + · for x ∈ [0, vB ]. (5) s s vB War does not produce anything of use to citizens other than its determination of who will acquire the spoils of victory, and is costly both in terms of military production and losses. Hence, war always causes inefficiency. Autocratic governments have the option of choosing pure strategies and avoiding uncertainty, which causes war. However, Proposition 3.1 shows us that they choose mixed strategies, create uncertainty about their military powers, and cause war. In the unique equilibrium, government A’s effective bid has first-order stochastic dominance over government B’s effective bid. Hence, government A, which is more aggressive than its opponent, bids more aggressively and is more likely to win the war.18 Second, we look at wars between democracies, which, unlike autocracies, bear all the burdens of war. As in the case of autocratic governments, the equilibrium requires mixed strategies, and we denote governments’ bidding strategies by the distribution functions of the resources they allocate for military production, which we denote by FA and FB . Proposition 3.2. When both countries are democracies, there is a unique Nash 18
The probabilities of victory for the countries are: Z
vA s
P (country A wins) =
FB
0
Z
vA s
= 0
=
λA x λB
fA (x)dx ! λ x 1 λAB − vB s 1+ dx s vB vA
2s − 1 , and 2s
P (country B wins) = 1 − P (country A wins) = Since s > 12 , it follows that P (country A wins) > P (country B wins).
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(6)
1 . 2s
(7)
equilibrium. In this equilibrium, governments’ bidding strategies are: vA ln 2 −s· vx FA (x) = 2 − 2 · e A for x ∈ 0, s 1
FB (x) = 2 − 2 s · e
− 1s · vx
B
for x ∈ [0, vB ln 2].
(8) (9)
Proposition 3.2 shows that when there is a war between two democracies with complete information, both democratic governments, just like their autocratic counterparts, choose mixed strategies for military spending, create uncertainty about their military powers, and cause war. From Proposition 3.1 and Proposition 3.2, we see that our model offers an individual-level explanation of wars in that the informational asymmetry created endogenously by governments causes war. Moreover, the aggressive government chooses to lose the war with a positive probability, since its mixed bidding strategy results in a higher payoff due to its lower cost of military production when compared with the cost of other bidding strategies that makes victory sure. In the equilibrium, the highest effective bids of both governments are equal because their payoffs decrease in the resources allocated to warfare, and none of them will bid more than the amount necessary for a sure victory. However, the aggressive government’s effective bid has first-order stochastic dominance over the non-aggressive government’s effective bid, and it is more likely to win the war and to acquire the spoils of victory.19 In the standard models, such as Garfinkel and Skaperdas [8], both countries win the war with equal probabilities regardless of their initial resource distribution, which is known as the Paradox of Power. However, unlike standard models, in our model unequal probabilities of victory are plausible. The reason for this is the following. In the standard models, the winner claims all resources, which is the same for both governments, and both countries have the same military technologies, whereas in our 19
In the equilibrium, the probabilities of victory for the countries are: Z P (country A wins) =
vA ln 2 s
FB
λA x λB
fA (x)dx λ x ! Z vAsln 2 vB ln 2− A λB 2s − vsx svB = 2−e e A dx v A 0 s+1 s =2− 1 − 2 s , and s+1 s s+1 P (country B wins) = 2 s − 1 − 1. s+1 0
It can be easily shown that P (country A wins) >
1 2
12
> P (country B wins) for s > 1.
(10) (11)
model, the winner claims the spoils of victory, which may be different for governments, and countries may have different military technologies. Yet if both governments have the same level of aggression, that is, the amount of military goods they can produce using the spoils of victory, then our model will lead to the same result as the standard models, and both countries will win the war with equal probabilities. Next, we compare the wars between two democracies with the wars between two autocracies, by keeping all parameters identical in both Proposition 3.1 and Proposition 3.2, and show that democracies are more peaceful as stated in the following corollary. Corollary 3.1. In the world of two democracies, countries have smaller military powers, the size of the war between them is smaller, and their representative citizens are better off than they would be in the world of two autocracies.20 Democratic governments, unlike autocratic governments, suffer all the burdens of the war just as their citizens do. As a result of this, they are expected to invest less in military production and build smaller armies than their autocratic counterparts. Wars between smaller armies will be less severe, therefore, if all parameters other than regime types are the same, the war between democracies is expected to be less severe than the war between autocracies. Finally, all representative citizens are expected to be better off in a world of democracies than they would be in a world of autocracies because of the following reasoning. Consider a change in countries’ regimes from autocracy to democracy. Then, the aggressive government’s probability of victory will decrease.21 However, the expected sizes of armies and wars will also decrease as a result of this regime change. The benefit to representative citizen of the aggressive government from building smaller armies and fighting less severe wars exceeds the cost of losing the spoils of victory with a higher probability, and he will be better off. The non-aggressive government’s representative citizen will also be better off because after the regime change his country will win the war with a higher probability, which will add further to the benefit received from building smaller armies and fighting less severe wars. Corollary 3.1 considers the effects of a regime change from democracy to autocracy. In the following, we determine the effects of a change in one of the countries’ military technology, and show that our model offers a a theoretical explanation for the power-shift approach. 20
Governments choose mixed strategies, therefore we compare the expected values of the corresponding parameters. 21 It follows from the comparison of 6 and 10.
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Corollary 3.2. Assume both countries have the same regime. Then, if there is an advancement in the aggressive country’s military technology or a decline in the nonaggressive country’s military technology, both countries will have smaller armies, the size of the war between them will be smaller, and both representative citizens will be better off. Moreover, wars will completely disappear when the aggressive country has absolute superiority in military technology, that is, when λA /λB → ∞. This corollary is in line with our interpretation of the power-shift approach, since a step towards higher aggression inequality is also a step towards a more peaceful world. The intuition is the following. When aggression inequality increases due to an advancement in country A’s military technology or a decline in country B’s military technology, government B realizes that it is unlikely to be victorious and invests less in warfare. In response to this, government A also decreases its investment in warfare. Therefore, both countries will have smaller armies, and the war between them will be less severe. Moreover, this change in military technology is in favor of government A, and government A uses this advantage to increase its probability of victory. Citizen A benefits from this increase in aggression inequality, because his country’s probability of victory increases while the costs of military production and war decrease. When we look at the representative citizen of the non-aggressive government, he is hurt by the decrease in the winning probability of his country. However, his benefit, due to the decrease in the costs of military production and war, (weakly) exceeds his loss and he is (weakly) better off. On the other hand, Corollary 3.2 offers an explanation for the validity of the classical interpretation of power-shift approach. Country A’s absolute superiority in military technology leads to absolute power inequality, where country i’s power is measured by the size of its army or by the magnitude of military power it can produce using its resources, and this power inequality induced by the absolute superiority of one of the countries in military technology paves the way for absolute peace. The intuition is as follows. If country A’s military technology is overwhelmingly more advanced than its opponent’s, government B will choose to produce almost no military goods since it will have almost no hope to win the war. As a result of this, government A will win almost surely by investing almost no resources in warfare and wars will almost completely disappear. The countries’ victory values, vA and vB , have been common knowledge up to this point. In the following section, we will investigate the war game in which countries’ victory values are private information.
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4
Incomplete Information
In this section, we analyze the incomplete information case. The countries’ victory values, vA and vB , are independently and randomly drawn from the intervals [0, v¯A ] and [0, v¯B ], respectively, where v¯i ∈ R+ for i ∈ {A, B}. The realizations of vA and vB are private information, whereas their distributions are common knowledge. Country i ∈ {A, B} can produce up to λi v¯i units of military goods by using the spoils of victory. As in the previous section, we call λi v¯i government i0 s level of aggression, and we say that government i becomes more aggressive when λi v¯i increases, and it becomes less aggressive when λi v¯i decreases. Without loss of generality, we assume government A is weakly more aggressive than government B, that is, λA v¯A ≥ λB v¯B . We call governments A and B the aggressive and non-aggressive governments, respectively. We denote the ratio of government A’s level of aggression over government B’s level of aggression with: λA v¯A , (12) s¯ = λB v¯B and call it aggression inequality. First, we investigate the war game between two autocracies. Governments have incomplete information about each other’s victory values, hence it is a Bayesian game. Following Hodler and Yektas [13], we look for Bayesian Nash equilibria in monotone continuous strategies that are differentiable with non-zero derivatives everywhere except at the boundary points. Proposition 4.1. When both countries are autocracies, there is a unique Bayesian Nash equilibrium. In this equilibrium, governments’ bidding strategies are: 1/¯s vA 1 · · vA for vA ∈ [0, v¯A ], and (13) bA (vA ) = s¯ + 1 v¯A s¯ vB s¯ · · vB for vB ∈ [0, v¯B ]. (14) bB (vB ) = s¯ + 1 v¯B Proposition 4.1 shows that when autocratic governments have incomplete information, they choose pure strategies in equilibria and do not create uncertainty. However, there is exogenous uncertainty, that is, governments’ uncertainty about each other’s victory values, and it causes war. In the equilibrium, government A, which is more aggressive, bids more aggressively, produces more military goods, and is more likely to win the war.22 These results are parallel to those obtained in the previous section. 22
We get that country A has greater military power by comparing (64) with (63). On the other
15
On the other hand, unlike the complete information case, the size of the war does not necessarily decrease with an increase in aggression inequality. However, as stated in the following corollary, if country A’s military technology is overwhelmingly more advanced than country B’s military technology, wars will almost completely disappear. Corollary 4.1. Assume both countries are autocracies. Then, wars will completely disappear when country A has absolute superiority in military technology, that is, when λA /λB → ∞. The intuition for this corollary is as follows. When country A has overwhelmingly advanced military technology, government B chooses to spend almost no resources on military production since it will almost surely lose the war. In response to this, government A also chooses to invest almost no resources in warfare, however wins almost surely due to its superiority in military technology. As a result of the lack of investment in warfare, wars will fade away. In other words, absolute power inequality as well as absolute aggression inequality, induced by one of the two countries’ absolute superiority in military technology, paves the way for peace.23 Hence, our model offers a theoretical background to the validity of the power-shift approach for both complete and incomplete information cases. Next, we look at the war game between two democracies. In this case, a general closed form solution cannot be found. Hence, we study a particular case in which both countries have the same victory value and military technology, that is, v¯A = v¯B and λA = λB . In this Bayesian game, we look for the Bayesian Nash equilibria hand, country A wins the war if and only if: 1 λA s¯ + 1
vA v¯A
s¯+1 s ¯ v¯A > λB
s¯ s¯ + 1
vB v¯B
s¯+1 v¯B ⇔
vB vA 1/¯s . < v¯B v¯A
Since vA and vB are independent and uniformly distributed on [0, v¯A ] and [0, v¯B ], the probabilities of victory for the countries in the equilibrium are: Z
v ¯A
P (country A wins) = 0
P (country B wins) =
vA v¯A
1 . s¯ + 1
1/¯s
1 s¯ dvA = , and v¯A s¯ + 1
(15) (16)
It follows that P (country A wins) > P (country B wins). 23 The disappearance of wars due to country A’s absolute superiority in military technology makes citizen A best off, more so than any other possible state of military technologies. However, the same cannot be said for citizen B, who is better off when government B is as aggressive as government A.
16
in monotone continuous strategies that are differentiable with non-zero derivatives everywhere except at the boundary points. Proposition 4.2. Assume both countries are democracies, v¯A = v¯B = v¯, and λA = λB . Then, there is a unique Bayesian Nash equilibrium. In this equilibrium, governments’ bidding strategies are: 2¯ v b(v) = 2¯ v ln − v for v ∈ [0, v¯]. (17) 2¯ v−v Proposition 4.2 shows that when democratic governments have incomplete information, they choose pure strategies and do not create uncertainty as their autocratic counterparts. Hence, the war is caused only by exogenous uncertainty, that is, governments’ uncertainty about each other’s victory values. Moreover, governments’ bids increase in their victory values, that is, b0 (v) > 0, because they will be willing to invest more for a higher expected return. Using Proposition 4.1 and Proposition 4.2, we compare the world of two democracies with the world of two autocracies, and obtain the following corollary. Corollary 4.2. Assume v¯A = v¯B and λA = λB . Then, the countries have smaller military powers, the size of the war between them is smaller, and their representative citizens are better off in the world of two democracies than they would be in the world of two autocracies.24 This corollary shows that Corollary 3.1 can be extended to the incomplete information case at least for the particular case of v¯A = v¯B and λA = λB . Democratic governments, who, unlike their autocratic counterparts, suffer all the burdens of the war, invest less in military production, build smaller armies and fight less severe wars. Hence, in line with the democratic peace theory, regime changes from democracy to autocracy aggravate wars. Moreover, representative citizens are always better off in the democratic world, due to the decreases in the costs of building armies and fighting wars.91p;
5
Conclusion
We have presented a game-theoretic model of war in which two countries with complete information about each other fight. We have characterized its unique equilibrium when the war is between two democracies or between two autocracies. We 24
As in Corollary 3.1, we compare the expected values of the corresponding parameters.
17
have obtained three significant results which offer individual-level, societal-level and systemic-level analyses, respectively. First, governments’ lack of information about each other, which is created endogenously by governments’ mixed strategy choices, paves the way for the war. Second, in line with the democratic peace theory, a world of two democracies is more peaceful than a world of two autocracies in terms of governments investing less in military, countries fighting less severe wars, and citizens being better off. Third, even a small shift towards higher aggression inequality will produce a more peaceful world in the same terms as are indicated above, in line with our interpretation of power-shift approach. Moreover, wars will completely disappear when the aggressive country has absolute superiority in military technology, which is in line with both our and classical interpretation. As an extension, we have considered the incomplete information case and obtained similar results: Governments’ lack of information, which exist exogenously, paves the way for the war; a democratic world is more peaceful than an autocratic one in the same terms as are indicated above; and wars will completely disappear with absolute aggressionpower inequality.
Appendix Proof of Proposition 3.1 Let us define two new countries A0 and B 0 such that given the bid of other government, bj 0 , government i0 tries to maximize: for bi0 < bj 0 −bi0 vi0 (18) ui0 (bi0 , bj 0 ) = − bi0 for bi0 = bj 0 2 vi0 − bi0 for bi0 > bj 0 where vi0 = λi vi . It can be easily seen (bA0 , bB 0 ) is an equilibrium of this game if b b and only if λAA0 , λBB0 is an equilibrium of our original war game. Hence, we will characterize the equilibria of this new game to complete the proof. Notice that this new game is the original all-pay auction with complete information which has been solved by Baye, Kovenock and de Vries [3], and we repeat their solution.25 First, we assume that s > 1, that is, vA0 > vB 0 . For i0 ∈ {A, B}, let si0 and si0 be the lower and upper bounds of government i0 ’s equilibrium bid distribution function, that is, Fi0 . Let αi0 (x) denote the size of mass at point x in government 25 The proof for all-pay auctions under complete information when the valuations are homogeneous or heterogenous can be seen in Baye, Kovenock and de Vries [3] .
18
i0 ’s distribution function,26 and u∗i0 be government i0 ’s equilibrium payoff. We first obtain the supports of the mixed strategies in the equilibrium. Step 1: Let us show that sA0 = sB 0 = 0. Government i0 can guarantee at least a non-negative payoff by bidding bi0 = 0, so all bids greater than vi0 are ruled out. Thus, vi0 ≥ si0 ≥ si0 ≥ 0. Given sA0 = sB 0 , αA0 (sA0 ) > 0 and αB 0 (sB 0 ) > 0, we see that both government A0 and government B 0 has an incentive to raise their lower bound by a small amount. Therefore, it follows that: sA0 = sB 0 ⇒ there exists h ∈ {A, B} s.t. αh (sh ) = 0.
(19)
Let uj 0 (bj 0 , Fi0 ) denote government j 0 ’s payoff. If si0 ≥ sj 0 and αi0 (sj 0 ) = 0, then uj 0 (sj 0 , Fi0 ) = −sj 0 < 0 = uj (0, Fi0 ) for sj 0 > 0. We will get the same result when we replace sj 0 with any bj 0 ∈ (0, si0 ). Thus, it follows that: si0 ≥ sj 0 and αi0 (sj 0 ) = 0 ⇒ sj 0 = 0 and Fj 0 (0) = lim Fj 0 (x). x↑si0
(20)
Moreover, it can be easily seen that: si0 ≥ sj 0 and αi0 (si0 ) = 0 ⇒ sj 0 = 0 and Fj 0 (0) = Fj 0 (si0 ).
(21)
Now, let us show that sA0 = sB 0 . Assume not. Then, there exists i0 s.t. si0 > sj 0 . If αi0 (si0 ) = 0, then from (21) we get Fj 0 (0) = Fj 0 (si0 ), and as a result of this ui0 (si0 , Fj 0 ) < ui0 (x, Fj 0 ) for x very small but greater than 0, which is a contradiction. On the other hand, if αi0 (si0 ) > 0, then αj 0 (si0 ) = 0, since otherwise government i0 has an incentive to raise his lower bound by a small amount. In this case, bidding a positive amount lower than si0 causes only loss for government j 0 , we get that Fj 0 (0) = Fj 0 (si0 ). As a result of this, ui0 (si0 , Fj 0 ) < ui0 (x, Fj 0 ) for x very small but greater than 0, which is a contradiction. Thus, we conclude that sA0 = sB 0 . Combining this result with (19) and (20), it follows that: sA0 = sB 0 = 0.
(22)
Step 2: We will show that sA0 = sB 0 = vB 0 and u∗B 0 = 0. Let us denote s = max{sA0 , sB 0 }. Government B 0 will never bid above vB 0 , since 0 strictly dominates any bid above vB 0 . Therefore, government A0 has no incentive to bid above vB 0 , and it follows that s ≤ vB 0 . Moreover, government A0 ’s equilibrium payoff, that is, u∗A0 , cannot be less than vA0 − vB 0 > 0, because sB 0 ≤ vB 0 and government A0 wins the election for sure with any bid greater than sB 0 . 26
αi0 (x) is greater than zero if and only if x is a masspoint.
19
Let us show that government A0 does not have a masspoint at 0. Assume not. Then, government A0 has a masspoint at 0. From (19) and (22), we know that at least one of the governments does not have a masspoint at 0, and if it is not government A0 , then it is government B 0 . Given government B 0 does not have a masspoint at 0, it follows that government A0 will get a payoff 0 - which is less than vA0 − vB 0 - by bidding sA0 = 0, a contradiction. Since u∗A0 > 0, government A0 should outbid government B 0 with a probability bounded away from zero. This is possible only when government B 0 has a masspoint at 0, because sA0 = 0. Thus, 0 is a masspoint for government B 0 but not for government A0 , and we get that u∗B 0 = 0. If s < vB 0 , then government B can get more than 0 by bidding a little above s, which is a contradiction. Hence, s = vB 0 . Since none of the governments have an incentive to bid more than the highest bid by the other government in the equilibrium (which follows from step 3, that is, there exists no masspoints on (0, vB 0 ]), we conclude that sA0 = sB 0 = vB 0 . Step 3: Let us show that there are no point masses on the interval (0, vB 0 ]. Suppose that Fi0 has a masspoint at xi0 ∈ (0, vB 0 ]. If xi0 < vj 0 , then government j 0 has an incentive to transfer mass from (xi0 − , xi0 ] to xi0 + δ for a small and a very small δ. If xi0 = vj 0 , then government j 0 has an incentive to transfer mass from (vj 0 − , vj 0 ] to 0 for a small . Thus, there would be a neighborhood below xi0 in which government j 0 would put no mass. However, then it is not an equilibrium strategy for player i0 to put mass at xi0 , which is a contradiction. Step 4: Let us show that, for x, y ∈ [0, vB 0 ] and i0 ∈ {A, B}, if x > y, then Fi0 (x) − Fi0 (y) > 0. Assume that there exists x, y ∈ (0, vB 0 ] such that x > y and Fi0 (x) − Fi0 (y) = 0. x, y ∈ (0, vB 0 ] cannot be masspoints, which follows from step 3, and putting a mass on (y, x] causes only loss for government j 0 . Therefore, Fj 0 (x) − Fj 0 (y) = 0. Denote x¯ = sup{z ∈ [x, vB 0 ] : FA0 (z) − FA0 (y) = 0 or FB 0 (z) − FB 0 (y) = 0}. Then FA0 (¯ x) − FA0 (y) = 0 and FB 0 (¯ x) − FB 0 (y) = 0. If x¯ < vB 0 , government A (and government B) has an incentive to transfer mass from (¯ x, x¯ + ) for small to y, which is a 0 0 contradiction. If x¯ = vB , then y ≥ sA , which is a contradiction, because we already have vB 0 > y and sA0 = vB 0 . On the other hand, the case for y = 0 is straight forward, because there exists y 0 ∈ (0, x) and it directly follows that Fi0 (x) > Fi0 (y 0 ) ≥ Fi0 (0). Step 5: For i0 ∈ {A, B}, Fi0 is increasing at all points ∈ [0, vB 0 ]. Therefore, the payoff government i0 gets by bidding any xi0 ∈ [0, vB 0 ] is constant and equal to u∗i0 , where u∗A0 = vA0 − vB 0 and u∗B 0 = 0. As a result of this, for all x ∈ [0, vB 0 ], we have: (vA0 − x)FB 0 (x) − x(1 − FB 0 (x)) = u∗A0 ⇔ FB 0 (x) = 20
vA0 − vB 0 + x , and vA0
(23)
x . (24) vB 0 Rearranging these distribution functions, we obtain that the unique Bayesian Nash equilibrium of the war game is given by (4) and (5). The same proof can easily be adapted to the case s = 1. (vB 0 − x)FA0 (x) − x(1 − FA0 (x)) = u∗B 0 ⇔ FA0 (x) =
Proof of Proposition 3.2 Let us define two new governments A0 and B 0 such that given the bid of other government, bj 0 , government i0 tries to maximize: for bi0 < bj 0 −2bi0 vi0 ui0 (bi0 , bj 0 ) = (25) − 2bi0 for bi0 = bj 0 2 vi0 − bi0 − bj 0 for bi0 > bj 0 where vi0 = λi vi . It can be easily seen (bA0 , bB 0 ) is an equilibrium of this game if b b and only if λAA0 , λBB0 is an equilibrium of our original war game. Hence, we will characterize the equilibria of this new game to complete the proof. First, we assume that s > 1, that is, vA0 > vB 0 . Let si0 , si0 , Fi0 , αi0 , and u∗i0 be same as in the proof of Proposition 3.1. We begin with obtaining the supports of the mixed strategies in the equilibrium. Step 1: sA0 = sB 0 = 0. It follows from the proof of step 1 in the proof of Proposition 3.1 with a small difference, which is that uj 0 (sj 0 , Fi0 ) = −2sj 0 . Step 2: Let us show that sA0 = sB 0 = s ≤ vB 0 and there are no masspoints on the interval (0, s]. There are no masspoints on the interval (0, vB 0 ], which is shown same as step 3 in the proof of Proposition 3.1. Government B 0 has no incentive to bid more than vB 0 , and since vB 0 is not a masspoint for government B 0 , government A0 has no incentive to bid more than vB 0 . On the other hand, since there exists no masspoints on (0, vB 0 ], none of governments has an incentive to bid more than the highest bid of the other government. Thus, sA0 = sB 0 = s ≤ vB 0 , and there exists no masspoints on (0, s]. Step 3: For x, y ∈ [0, s] and i0 ∈ {A, B}, if x > y, then Fi0 (x) − Fi0 (y) > 0. The proof of this step follows from the proof of step 4 of the proof of Proposition 3.1, when we replace vB 0 with s. Step 4: For i0 ∈ {A, B}, Fi0 is increasing at all points in the interval [0, s]. Therefore, the payoff government i0 gets by bidding any xi0 ∈ [0, s] is constant and equal to u∗i0 . As a result of this, for all x ∈ [0, s], the payoff government A0 gets by bidding x is: Z x Z s ∗ uA0 = (vA0 − x − y)fB 0 (y)dy + (−2x)fB 0 (y)dy 0
x
21
Z
x
= vA0 FB 0 (x) − x(2 − FB 0 (x)) −
yfB 0 (y)dy
(26)
0
and the payoff government B 0 gets by bidding x is: Z s Z x ∗ 0 0 (vB − x − y)fA (y)dy + uB 0 = (−2x)fA0 (y)dy 0
x
Z
x
= vB 0 FA0 (x) − x(2 − FA0 (x)) −
yfA0 (y)dy
(27)
0
Taking the derivative of (26) with respect to x we get: 0 = vA0 fB 0 (x) − 2 + FB 0 (x) + xfB 0 (x) − xfB 0 (x) ⇔ FB 0 (x) = 2 − vA0 fB 0 (x)
(28)
Since FB 0 is differentiable, it follows that fB 0 is also differentiable. Then, taking the derivative of (28) with respect to x, we get: fB 0 (x) = −vA0 fB0 0 (x) ⇒ fB 0 (x) = M e−x/vA0 and FB 0 (x) = 2 − M vA0 e−x/vA0 where M is a constant. Since FB 0 (s) = 1, we obtain M =
es/vA0 v A0
and:
s−x
FB 0 (x) = 2 − e vA0
(29)
Taking the derivative of (27) with respect to x we get: 0 = vB 0 fA0 (x) − 2 + FA0 (x) + xfA0 (x) − xfA0 (x) ⇔ FA0 (x) = 2 − vB 0 fA0 (x)
(30)
Since FA0 is differentiable, it follows that fA0 is differentiable. Taking the derivative of (30) with respect to x we get: fA0 (x) = −vB 0 fA0 0 (x) ⇒ fA0 (x) = N e−x/vB0 and FA0 (x) = 2 − N vB 0 e−x/vB0 where N is a constant. We know that FA0 (s) = 1. Therefore, N =
es/vB 0 vB 0
and:
s−x
FA0 (x) = 2 − e vB0
(31)
From step 1 and (19), it follows that there exists i0 such that αi0 (0) = 0, that is, Fi0 (0) = 0. Since vA0 > vB 0 , we have FB 0 (0) > FA0 (0). Hence, FA0 (0) = 0, and, from (31), it follows that: s = vB 0 ln 2. (32) 22
Plugging (32) in (29) and (31), we obtain: −x
FA0 (x) = 2 − 2e vB0 and FB 0 (x) = 2 − e
vB 0 ln 2−x vA0
on x ∈ [0, vB 0 ln 2].
(33)
Rearranging these distribution functions, we conclude that governments’ bidding strategies in the unique Bayesian Nash equilibrium of the original war game are given by (8) and (9). The same proof can easily be adapted to the case s = 1.27 Proof of Corollary 3.1 In a world of autocracies, using governments’ bidding strategies stated in Proposition 3.1, we get that country A’s expected military power is: Z M PA =
vA s
λA xfA (x)dx = 0
λB vB , 2
country B’s expected military power is: Z vB λB vB λB xfB (x)dx = M PB = , 2s 0 the size of war is: Z vA Z vB s WS = xfA (x)dx + xfB (x)dx 0
(34)
(35)
0 vA s
Z vB 1 1 + min{λA x, λB y}fB (y)fA (x)dydx + λA λB 0 0 # Z vA "Z λA x Z vB s λB vA vB 1 1 1 = + + λB ydy + λA xdy dx + λA x 2s 2s λA λB v v B A 0 0 λB 1 1 5 = + 2 vA + vB , (36) 2s 3s 6s the expected utility of citizen A is: Z vA Z vB s uDRA (x, y)fB (y)dy fA (x)dx EUA =RA + 0 0 ! Z vA Z λA x Z vB s λB 1 λB y s−1 s =RA + vA − x − dy − 2xdy + (vA − x) dx λA x svB λA s v A 0 0 λB 1 1 (37) =RA + vA 1 − − 2 , s 3s
Z
27
There is a small difference, in that at the end of step 4, we obtain FB 0 (0) = FA0 (0) = 0 which follows from vA0 = vB 0 . We use it to find upper bound of the governments’ bids.
23
and the expected utility of citizen B is: "Z vA Z vB
#
s
uDRB (y, x)fA (x)dx fB (y)dy
EUB =RB + 0
0
Z
vB
λB y λA
"Z
=RB + 0
0
# Z vA s λA x 1 vB − y − 2ydx dx − dy λB y λB v v A B λ A
vB . =RB − 3s
(38)
Similarly, in a world of democracies, using governments’ bidding strategies stated in Proposition 3.2, we get that country A’s expected military power is: vA ln 2 s
Z M PA =
λA xfA (x)dx = (1 − ln 2)λB vB ,
(39)
0
country B’s expected military power is: Z vB ln 2 1 λB xfB (x)dx = s 2 s − 1 − ln 2 λB vB , M PB =
(40)
0
the size of war is: Z
vA ln 2 s
WS =
Z
vB ln 2
xfB (x)dx
xfA (x)dx + 0
0
Z vA ln 2 Z vB ln 2 s 1 1 + + min{λA x, λB y}fB (y)fA (x)dydx λA λB 0 0 1 vA =(1 − ln 2) + s 2 s − 1 − ln 2 vB s 1+ 1 Z vA ln 2 Z λA x Z vB ln 2 s λB 1 2 s 1 − svy − svy − sx + + λB ye B dy + λA xe B dy e vA dx λA x λA λB vA vB 0 0 λ
B
1 1 s s 1+ 1 1 = 1 − 2s + 2 s vA + 2 s − 1 − 1 vB , 1+s 1+s 1+s
24
(41)
the expected utility of citizen A is: Z
vA ln 2 s
vB ln 2
Z
uDRA (x, y)fB (y)dy fA (x)dx
EUA =RA + 0
0 vA ln 2 s
λA x λB
λB y − svy =RA + e B dy− vA − x − λA 0 0 Z vB ln 2 s 1 1 1 − sx − svy 2s 2xe B dy + (vA − x) 2 − 2 s 2 e vA dx svB λλA x vA B 1 =RA + vA 2 − 2 s . Z
1 2 svB 1 s
Z
and the expected utility of citizen B is: "Z vA ln 2 Z vB ln 2
#
s
EUB =RB +
(42)
uDRB (y, x)fA (x)dx fB (y)dy 0
0 vB ln 2
λB y λA
λA x 2s − vsx e A dx− =RB + vB − y − λB v A 0 0 Z vA ln 2 s 1 2s − vsx 1 − svy 2y e A dx 2 s e B dy = RB λB y v sv A B λ Z
Z
(43)
A
Comparing these, we get the desired results. Proof of Corollary 3.2 It follows from the calculations in the proof of Corollary 3.1. Proof of Proposition 4.1 For i ∈ {A, B}, we define si such that: λi v¯i (44) λA v¯A + λB v¯B We begin with three observations. First, bA (0) = bB (0) = 0. Second, since governments’ bidding strategies are monotone with non-zero derivative, they are strictly increasing functions, and their inverse is also differentiable everywhere but boundary points. Third, λA bA (¯ vA ) = λB bB (¯ vB ), (45) si =
25
because if λi bi (¯ vi ) > λj bj (¯ vj ) for some i ∈ {A, B}, then government i is better of by deviating to lower values than bi (¯ vi ), which is a contradiction. For i ∈ {A, B}, government i bids 0 if and only vi = 0, which follows from the first observation. Now we focus on the bids when victory values are positive. Assume that vA > 0 and vB > 0. Then, we know that the bids are positive. We assume that government B’s bidding strategy, bB , is differentiable with nonzero derivatives everywhere except at the boundary points, and solve for the bid of government A. From the second observation, it follows that bB is a strictly increasing differentiable function whose inverse is also differentiable everywhere but boundary points. On the other hand, country A wins the war if and only if government A bids λB y which is greater than λB bλBA(vB ) , that is, vB < b−1 B (y/k) where k = λA . Hence, given country A’s victory value is vA ∈ [0, v¯A ] and government A bids y, the expected payoff of government A is: y Z v¯B Z b−1 B (k) 1 1 EUARA (y, vA ) = (RA + vA − y) dvB + (RA − y) dvB y −1 v¯B v¯B bB ( k ) 0 vA −1 b (y/k) − y v¯B B −1 dEUARA (y, vA ) b−1 A (y) dbB (y/k) = −1=0 ⇒ −1 vA =bA (y) dy v¯B dy = RA +
⇒
db−1 (y/k) −1 bA (y) B
= v¯B . (46) dy Similarly, given country B’s victory value is vB ∈ [0, v¯B ] and government B bids z, the expected payoff of government B is: Z b−1 Z v¯A A (kz) 1 1 (RB + vB − z) dvA + (RB − z) dvA EUARB (z, vB ) = −1 v¯A v¯A 0 bA (kz) vB −1 b (kz) − z v¯A A −1 dEUARB (z, vB ) b−1 B (z) dbA (kz) ⇒ = −1=0 −1 vB =bB (z) dz v¯A dz db−1 db−1 v¯A −1 A (kz) A (kz) ⇒ b−1 (z) = v ¯ ⇒ b (z) = , A B B dz dkz k and when we replace z with y/k, we get: = RB +
(y) db−1 −1 bB (y/k) A dy
26
=
v¯A . k
(47)
Dividing (47) with (46) side by side we get: b−1 B (y/k) b−1 A (y)
db−1 A (y) dy
db−1 B (y/k) dy
v¯A k
d ln b−1 v¯A d ln b−1 A (y) B (y/k) ⇒ = = v¯B dy k¯ vB dy k¯ vB
−1 v ¯A ⇒ bB (y/k) = K0 [b−1 A (y)]
(48)
where K0 is a constant. Plugging (48) in (46), we get: b−1 A (y)K0
−1 k¯ vB k¯ vB −1 −1 dbA (y) [bA (y)] v¯A = v¯B v¯A dy
k¯ v db−1 v¯A −1 − B A (y) = [bA (y)] v¯A dy kK0 k¯ vB v¯A v ¯A db−1 dy ⇒ [b−1 A (y) = A (y)] kK0 v¯A 1/sA ⇒ sA [b−1 y + constant = A (y)] kK0
⇒
(49)
Since b−1 A (0) = 0, we get constant = 0. Therefore, (49) becomes: b−1 A (y) =
s v¯A y A kK0 sA
(50)
and plugging (50) in (48), we get: −1 bB (y) = K0
v¯A sB y K 0 sA
(51)
From (50) and (51), it follows that: kK0 sA 1/sA x for x ∈ [0, v¯A ], and v¯A K0 sA x 1/sB bB (x) = for x ∈ [0, v¯B ]. v¯A K0
(52)
bA (x) =
(53) −sB /sA
Plugging (52) and (53) in (45), and solving for K0 we get K0 = v¯B v¯A it follows that: x 1/sA bA (x) = sB v¯A for x ∈ [0, v¯A ], and v¯A x 1/sB v¯B for x ∈ [0, v¯B ]. bB (x) = sA v¯B 27
. Therefore, (54) (55)
Notice that governments are bidding less than their victory values, that is, bi (x) < x s¯ 1 for i ∈ {A, B}. We conclude the proof by plugging sA = 1+¯ and sB = 1+¯ in (54) s s and (55). Proof of Corollary 4.1 In a world of autocracies, using governments’ bidding strategies stated in Proposition 4.1, we get that the size of war is: s 1+¯ 1+¯s Z v¯B s ¯ x 1 s¯ x 1 v¯A dx + v¯B dx WS = v¯A v¯A s¯ + 1 v¯B v¯B 0 0 Z vA Z v¯B ( x ) 1s¯ 1+¯s v ¯A 1 1 s¯ y + + λB v¯B dy+ λA λB 1 + s¯ v¯B 0 0 s 1+¯ Z v¯B s ¯ 1 x 1 v¯A dy dx λA 1 1 + s¯ v¯A v¯A v¯B v¯B ( v¯x ) s¯
Z
v¯A
1 s¯ + 1
A
=
s¯(2¯ s + 7) s¯(7¯ s + 2) v¯A + v¯B 2(¯ s + 1)(¯ s + 2)(2¯ s + 1) 2(¯ s + 1)(¯ s + 2)(2¯ s + 1)
(56)
When country A has absolute superiority in military technology, it can be easily seen that s¯ → ∞. The desired result follows immediately from lims¯→∞ W S = 0. Proof of Proposition 4.2 The three observations in the proof of Proposition 4.1 still hold. First, bA (0) = bB (0) = 0. Second, since governments’ bidding strategies are monotone with non-zero derivative, they are strictly increasing functions, and their inverse is differentiable everywhere but boundary points. Third, λA bA (¯ vA ) = λB bB (¯ vB ), and it becomes bA (¯ v ) = bB (¯ v ) because v¯A = v¯B = v¯ and λA = λB . We denote y ∗ = bA (¯ v ) = bB (¯ v ). Government i bids 0 if and only vi = 0. Assume vA > 0 and vB > 0. Then, we know that the bids are positive. We assume government B’s bidding strategy, bB , is differentiable with non-zero derivatives except at the boundary points, and solve for the bid of government A. From the second observation, we get that bB is a strictly increasing differentiable function and its inverse is also differentiable everywhere but boundary points. Country A wins the war if and only if government A bids y greater than bB (vB ), that is, vB < b−1 B (y). Hence, given country A’s victory value is
28
vA ∈ [0, v¯] and government A bids y ∈ (0, y ∗ ], the expected payoff of government A is: Z v¯ Z b−1 B (y) 1 1 EUDRA (y, vA ) = [RA + vA − y − bB (vB )] dvB + (RA − 2y) dvB v¯ v¯ 0 b−1 B (y) Z b−1 (y) B bB (vB ) vA −1 2¯ v − b−1 B (y) − dvB = RA + bB (y) − y v¯ v¯ v¯ 0 −1 dEUDRA (y, vA ) b−1 b−1 A (y) dbB (y) B (y) ⇒ = − 2 + =0 −1 v =b (y) A A dy v¯ dy v¯ ⇒
db−1 (y) −1 bA (y) B dy
= 2¯ v − b−1 B (y)
(57)
Similarly, from symmetry we get: b−1 B (y)
db−1 A (y) = 2¯ v − b−1 A (y). dy
(58)
−1 ∗ for y ∈ (0, y ∗ ]. First let us show that in equilibrium b−1 A (y) = bB (y) for y ∈ (0, y ). −1 −1 ∗ Assume that there exists x ∈ (0, y ) such that bA (x) > bB (x). Then, dividing equation (57) and (58) side by side and rearranging we get: db−1 B (x) dx db−1 A (x) dx
=
−1 2¯ v − b−1 B (x) bB (x) . −1 2¯ v − b−1 A (x) bA (x) db−1 (x)
db−1 (x)
−1 Since b−1 ¯, it follows that Bdx < Adx . Then, it can be easily B (x) < bA (x) ≤ v −1 seen that bB (z) has to be lower than b−1 A (z) for all z ≥ x. However, this is a −1 ∗ ∗ contradiction, because bB (y ) = v is not lower than b−1 A (y ) = v. We get the same −1 −1 ∗ contradiction when there exists x ∈ (0, y ) such that bA (x) < b−1 B (x). Therefore, bA −1 −1 and b−1 B are the same function, and we will denote it with g. Replacing bA and bB with g in (57), we get:
g dg = 2¯ v−g ⇒ dg = dy dy 2¯ v−g Z g(y) Z y g ⇒ dg = dy for y ∈ [0, y ∗ ] 2¯ v − g g(0) 0 g
(59)
For any y ∈ (0, y ∗ ], we have 0 < g(y) ≤ v¯, and therefore we can find θ ∈ [0, π/2) such that g(y) = 2¯ v (1 − cos θ) (and dg = 2¯ v sin θdθ). We pick yˆ ∈ (0, y ∗ ], and denote 29
ˆ Plugging these back into equation (59), we θ that solves g(ˆ y ) = 2¯ v (1 − cos θ) with θ. get: Z yˆ Z θˆ 2¯ v (1 − cos θ) dy 2¯ v sin θdθ = 2¯ v cos θ 0 0 Z θˆ (tan θ − sin θ)dθ = yˆ ⇒ 2¯ v 0
ˆ − 2¯ ⇒ 2¯ v (cos θˆ − ln cos θ) v (cos 0 − ln cos 0) = yˆ ˆ we get: Since g(ˆ y ) = 2¯ v (1 − cos θ), cos θˆ =
2¯ v − g(ˆ y) . 2¯ v
(60)
(61)
Plugging (61) back into equation (60) and rearranging we get: 2¯ v − g(ˆ y ) = yˆ for all yˆ ∈ (0, y ∗ ] 2¯ v − g(ˆ y)
2¯ v ln
(62)
For any t ∈ (0, v¯], we have bA (t) = bB (t) = g −1 (t) ∈ (0, y ∗ ]. Plugging g −1 (t) instead of yˆ in (62), and remembering bA (0) = bB (0) = 0, we conclude that: 2¯ v − t for all t ∈ [0, v¯]. 2¯ v − t)
bA (t) = bB (t) = g −1 (t) = 2¯ v ln Proof of Corollary 4.2
In a world of autocracies, using governments’ bidding strategies stated in Proposition 4.1, we get that country A’s expected military power is: Z M PA = λA 0
v¯A
1 s¯ + 1
x v¯A
s 1+¯ s ¯
v¯A
1 s¯2 dx = λB v¯B v¯A (¯ s + 1)(2¯ s + 1)
country B’s expected military power is: 1+¯s Z v¯B s¯ x 1 s¯ v¯B dx = λB v¯B M PB = λB s¯ + 1 v¯B v¯B (¯ s + 1)(¯ s + 2) 0
30
(63)
(64)
the expected utility of citizen A is: Z
v¯A
"Z
v¯B
x v ¯A
1
s ¯
s¯+1 s ¯
1 x EUA =RA + x− v¯A − s¯ + 1 v¯A 0 0 # s¯+1 Z v¯B s 1 1 x v¯A dy 2 dx 1 ¯ + 1 v¯A v¯A v¯B v¯B ( v¯x ) s¯ s A s¯(¯ s − 1) s¯ = RA + v¯A + (2¯ s + 1)(¯ s + 1) (¯ s + 1)(2¯ s + 4)
s¯ λB s¯+1
y v¯B
λA
and similarly the expected utility of citizen B is: 1 − s¯ s¯ EUB = RB + v¯B + (¯ s + 1)(¯ s + 2) (¯ s + 1)(4¯ s + 2)
s¯+1
v¯B
! dy−
(65)
(66)
On the other hand, in a world of democracies, using governments’ bidding strategies stated in Proposition 3.2, and assuming v¯A = v¯B = v¯ and λA = λB = λ, we get that country A’s expected military power equals country B’s expected military power, which is: Z v¯ 1 2¯ v − x dx M PA = M PB = λ 2¯ v ln 2¯ v−x v¯ 0 v¯ x=¯ v = λ 2¯ v ln(2¯ v ) − − 2 [(x − 2¯ v ) ln(2¯ v − x) − x]x=0 2 3 = − 2 ln 2 λ¯ v, (67) 2 the size of war is: Z v¯ Z x Z v¯ 2¯ v 1 2¯ v WS = 2¯ v ln − y dy + − x dy 2 dx 2¯ v ln 2¯ v−y 2¯ v−x v¯ 0 x 0 4 = 2 ln 2 − v¯, (68) 3 the expected utility of citizen A is: Z v¯ Z x 2¯ v 2¯ v EUA =RA + x − 2¯ v ln + x − 2¯ v ln + y dy+ 2¯ v−x 2¯ v−y 0 0 Z v¯ 2¯ v 1 v¯ 2x − 4¯ v ln dy 2 dx = RA + , (69) 2¯ v−x v¯ 6 x 31
and similarly the expected utility of citizen B is: v¯ EUB = RB + , 6 The result follows from the comparison of these results and (56) for v¯A = v¯B = v¯ and λA = λB = λ.
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