Inaugural Article

Networks of military alliances, wars, and international trade

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  1. Contributed by Matthew O. Jackson, Oct 30, 2022 (sent for review August 28, 2015; reviewed by Rachel Kranton and Alex Teytelboym)

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Significance

The incidence of interstate wars has dropped dramatically over fourth dimension: The number of wars per pair of countries per year from 1950 to 2000 was roughly a 10th equally high every bit it was from 1820 to 1949. This meaning decrease in the frequency of wars correlates with a substantial increase in the number of armed services alliances per country and the stability of those alliances. We prove that i possible explanation of this is an accompanying expansion of international trade. Increased merchandise decreases countries' incentives to attack each other and increases their incentives to defend each other, leading to a stable and peaceful network of armed services and trade alliances that is consequent with observed information.

Abstruse

We investigate the role of networks of alliances in preventing (multilateral) interstate wars. Nosotros first prove that, in the absence of international merchandise, no network of alliances is peaceful and stable. Nosotros so show that international merchandise induces peaceful and stable networks: Trade increases the density of alliances so that countries are less vulnerable to attack and also reduces countries' incentives to attack an ally. We present historical data on wars and trade showing that the dramatic drop in interstate wars since 1950 is paralleled by a densification and stabilization of trading relationships and alliances. Based on the model we also examine some specific relationships, finding that countries with high levels of trade with their allies are less likely to be involved in wars with any other countries (including allies and nonallies), and that an increment in trade between ii countries correlates with a lower adventure that they will get to war with each other.

  • war
  • networks
  • international trade
  • conflict

Wars are caused past undefended wealth.

Ernest Hemingway (repeated by Douglas MacArthur in lobbying to fortify the Philippines in the 1930s) (1)

There is only one thing worse than fighting with allies, and that is fighting without them.

Winston Churchill, Apr 1, 1945 (ii)

The enormous costs of war make it imperative to sympathise the weather under which wars can exist prevented. Although much is known about bilateral conflicts, at that place is no formal theory of how networks of multilateral international relationships foster and deter interstate wars. We introduce a model of networks of armed forces alliances and international trade that can serve as a foundation for study of international alliance structure and conflict.

The idea of arranging multiple alliances to ensure world peace found perhaps information technology almost famous proponent in Otto von Bismarck and his conventionalities that the European states could exist allied in ways that would maintain a peaceful balance of power (eastward.thou., see ref. 3). The alliances that emerged were briefly stable following the unification and expansion of Germany that took place up through the early 1870s but were ultimately unable to prevent World War I. Indeed, many world conflicts involve multiple countries allied together in defensive and offensive groups, from the shifting alliances of the Peloponnesian and Corinthian wars of ancient Greece to the Centrality and Allies of World State of war Ii, and then studying the textile of alliances is necessary for understanding international (in)stability.

Between 1823 and 2003, 40% of wars with more 1,000 casualties involved more than two countries, and many of the well-nigh subversive (e.g., the World Wars, Korean War, Vietnam, First and Second Congo Wars, etc.) involved multilateral conflicts.* Most importantly, this is actually a network problem. Multilateral wars never involved cliques (fully allied coalitions of more than than two countries) confronting cliques. Out of the 23 wars between 1823 and 2003 that authorize equally having at least one side with three or more than countries, none of them involved a clique versus a clique. Thus, an approach of modeling networks of alliances, rather than coalitions (in which countries are partitioned into allied groups), is warranted. Also, a network arroyo meshes well with patterns of international trade, which are critical in determining which countries accept incentives to attack or defend which others.

The history of interstate alliances between the early 1800s and the present tin can be broken into two periods. The early period (pre-1950) involved relatively sparse, very dynamic, and unstable networks, with more than 30% of alliances turning over every v years, and many wars. The later on flow (post-1950) involves networks that are more than four times as dumbo, and with high stability— with but 5% of alliances turning over every five years (see Data on Trade and Wars). These differences also correspond to the dramatic drop in wars seen in Fig. i likewise as a parallel increase in the density of trading partnerships.

To gain insight into the relationship between networks of alliances and the incidence of wars, we model the incentives of countries to set on each other, to form alliances, and to trade with each other. Nosotros first ascertain a concept of networks that are stable against wars from a military indicate of view, when trade is ignored. We testify that there are no (nonempty) networks of alliances that are stable, where stability requires that no country exist vulnerable to defeat accounting for offensive and defensive alliances, and no country wants to add or driblet alliances. This instability is suggestive of the shifting alliances and recurring wars of the 19th and outset half of the 20th centuries. Wars, even so, have greatly subsided in parallel with the huge increment of trade, trends that nosotros discuss in detail below. We thus enrich the base model to include international trade, introducing a concept of a network of alliances being war-and-trade-stable, which allows countries to class alliances for either economic or war machine considerations. This enables the existence of networks of alliances that are stable against disharmonize and the add-on or deletion of alliances. Trade helps in two ways: First, it provides economic motivations to maintain alliances, and the resulting denser network has a deterrent effect; second, it reduces the incentives of a land to attack trading partners. This reduces the potential set of conflicts and allows for a rich family of stable networks that can exhibit structures like to modernistic trade and alliance networks. In examining the information, we find that an increase in the number of a country's allies correlates with a lower frequency with which it is attacked, and increasing trade between two countries correlates with a lower frequency of conflict between them.

We discuss below how the model fits with, and sheds light on, other information and theories, such as the advent of nuclear weapons, the Common cold State of war, and the increase in the number of democracies.

Our analysis fits firmly into the "rationalist" tradition based on cost and benefit analyses, with roots in ref. five.† To our knowledge, there are no previous models of conflict that game-theoretically model networks of alliances between multiple agents/countries based on costs and benefits of wars.‡ In that location are previous models of coalitions in conflict settings (e.one thousand., come across ref. 12 for a survey). Hither, network structures add several things to the picture show. Our model examines group conflict (east.g., ref. xiii) but enriches information technology to acknowledge network structures of alliances and of international trade. This allows united states to admit patterns that are consequent with the networks of alliances that are actually observed, which are far from beingness partitions, moving the models toward matching observed patterns of wars, merchandise, and alliances. Moreover, as nosotros see below both empirically and theoretically, the patterns and number of partners of a given state matter beyond group membership or overall levels of interaction. Finally, our model illuminates the relationships between international trade, stable network structures, and peace, something not actualization in the previous literature—because the previous literature that involves international merchandise and conflict generally revolves effectually bilateral reasoning or focuses on instability and armament (e.g., ref. 14) and does not accost the questions that we accost hither. The circuitous relationship between trade and disharmonize is also the subject of an active empirical literature (e.g., refs. fifteen⇓ ⇓ ⇓–19). The complex correlations between conflict and merchandise are illuminated by a model that provides structure with which to interpret some of the observations.

Data on Trade and Wars

Trends in Military Alliance Networks.

Marked differences occur betwixt the military alliances we run across in the Alliance Treaty Obligation and Provisions Project data at different points in fourth dimension.§ At that place are two major changes that we see comparing the period before to that after 1950 (see as well SI Appendix). The first major modify is that big turnover in alliances, which constantly shift in the period from 1816 to 1950, drops substantially. Specifically, consider an alliance between two countries that is nowadays in year t, and summate the frequency with which those two countries are still allied in year t + 5 . Doing this for each yr from 1816 to 1950, we find the frequency to exist 0.695. When doing this for each yr from 1950 to 2003 the frequency becomes 0.949. Thus, there is an nigh one-3rd hazard that whatever given alliance disappears in the next 5 years in the pre-WWII catamenia, and so only a five% chance that any given alliance at any given time volition disappear within the adjacent v years in the post-WWII period. The 2nd major modify is that the network of alliances greatly densifies. Between 1816 and 1950 a country had on average 2.525 alliances (SD 3.809). During the period of 1951–2003 this grows by a factor of more than four to 10.523 (SD of 1.918). Thus, there are significantly more alliances per country in the mail service-1950 than the pre-1950 period.

To summarize, in the pre-WWII period countries take simply a couple of alliances on boilerplate and those alliances rapidly changed; in dissimilarity, in the mail service-WWII period countries grade on average more than ten alliances and alter much more rarely.

Trends of Wars and Conflicts.

Another trend is that the number of wars per country has decreased dramatically post-WWII, even though the number of countries has increased—so in that location are many more pairs of countries that could get to war. For example, the average number of wars per pair of countries per year from 1820 to 1949 was 0.00059, whereas from 1950 to 2000 it was 0.00006, roughly a 10th of what it was in the previous period. We run across this in Fig. 1 (SI Appendix shows that this is robust beyond a variety of methods of measurement). Also, with the exceptions of the anomalous Falklands War, and the Korean and Vietnam Wars—which had Cold War considerations with major protagonists on both sides, including nations outside of Korea and Vietnam—the 24 other Militarized Interstate Disputes (MID) 5s since 1950 involved lesser-developed (lower-merchandise) countries as the major protagonist on at least ane (and often both) sides of the dispute. Moreover, major trading partners at the time do not appear on opposite sides of the dispute.

Trade and Wars.

International trade has had two major periods of growth over the terminal two centuries. The first was from the 1870s through 1913, during which world merchandise exports as a percentage of gross domestic product (GDP) grew from 5% to about 12%, disrupted by the First World State of war, and then picking upwardly once more after the Second World State of war, during which exports grew from vii% in 1950 to over 25% by 2012. Putting this trend together with that of wars, we see that the subtract in wars is mirrored by an increase in trade (see SI Appendix for more background on trade and war).

Moreover, trade has get less full-bodied: In the late 19th century almost of the world'southward trade was concentrated amidst a pocket-size portion of countries, and not all of them traded extensively with each other. Trade is more balanced, and in our model that enables disharmonize-costless networks, and such trade has emerged both in scale and scope by and large in the by six decades. This answers an important puzzle from the data: Export levels in 1913 are like to those in 1973, and notwithstanding the drib in wars occurs from the 1950s onward, and non in the early on 20th century. In this regard, our model beneath is helpful in suggesting a different measure out to apply than export levels. In our assay, conflict ends upwardly related non only to the level of trade, just the patterns of merchandise partnerships. The numbers of trading partners paint a clearer picture between trade and wars, and a different pattern than the overall level of trade, which turns out to be concentrated amidst a smaller number of countries before WWI. In particular, equally we see in Tabular array 1, the number of trading partners per country is higher in 1950 than in 1913, and the number roughly doubles by 1973, depending on which threshold 1 uses to define "partner".# These numbers do not imply causation because there are many confounding variables that make it hard to test whatever theory straight. Notwithstanding, this motivates developing a model that predicts which factors matter and how they relate to each other.

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Tabular array 1.

Average number of trading partners per state

A Model of Networks of Alliances

Countries and Networks.

Due north = { one , … , northward } is a fix of countries.

Countries are linked through alliances, represented past a network of alliances, g ⊂ N 2 , with the interpretation that if i j ∈ g countries i and j are allies. So, for case, the network yard = { 12,23,45 } on a set of countries North = { ane,ii,3,4,five } represents situations where country 2 is allied to both i and 3, and iv is allied with 5, and where no other alliances are nowadays. Alliances correspond channels through which countries tin can coordinate military deportment, either offensively or defensively. The presence of alliances does not require countries to come to each other's aid, because that volition accept to be incentive-compatible, as embodied in our definitions below. The operative part of the assumption is that countries either need to have an alliance or add one to coordinate their military activities. North i ( yard ) = { j : i j ∈ g } are the allies of i.

For a given alliance i j ∉ 1000 , let grand + i j denote the network obtained by adding the brotherhood i j to m, and for i j ∈ thousand , allow yard − i j denote the network obtained by deleting the alliance i j from k. In a slight abuse of annotation, let one thousand − i denote the network obtained by deleting all alliances of the form i k , m ∈ N , from g.

Military Strengths and Wars.

Each group of countries C ⊂ Northward has a collective military strength K ( C ) . Let Thou i announce M ( { i } ) . A prominent case is with M ( C ) = ∑ i ∈ C Yard i .||

If there is a war between sets of countries C and C ′ , with C existence the aggressor, then C "wins" if M ( C ) > γ ( C , C ′ ) 1000 ( C ′ ) . The parameter γ ( C , C ′ ) > 0 is the defensive reward (of C ′ over C if γ ( C , C ′ ) > 1 ) or offensive advantage (of C over C ′ if γ ( C , C ′ ) < one ) advantage in the war.

The dependence of the parameter γ ( C , C ′ ) on the groups of countries in question allows the model to incorporate diverse geographic and technological considerations (e.g., land and sea accessibility betwixt countries, nuclear versus conventional capabilities, troop locations, etc.).**

We maintain weak monotonicity conditions on the functions:††

  • M ( C ″ ) ≥ K ( C ) whenever C ⊂ C ″ : bigger groups of countries in terms of set up inclusion are at least every bit strong as smaller groups.

  • γ ( C ″ , C ‴ ) ≤ γ ( C , C ′ ) whenever C ⊂ C ″ and C ‴ ⊂ C ′ : adding countries to the attacking grouping and/or subtracting them from the defending grouping does not increase the defensive advantage.

Vulnerable Countries and Networks.

A land i is said to be vulnerable at a network g if in that location exists j and C ⊂ Due north j ( m ) ∪ { j } such that j ∈ C , i ∉ C and M ( C ) > γ ( C , C ′ ) M ( C ′ ) , where C ′ = i ∪ ( N i ( one thousand ) ∩ C c ) and C c is the complement of C. In this instance, country j is a potential aggressor at a network m.‡‡

Vulnerability is illustrated in Fig. ii, Left. Annotation that state five cannot join countries 2, 3, and iv in attacking country one because information technology is non allied with any of them, and countries 2 and 3 can attack i despite existence allies with 1.

Fig. 2.

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Fig. 2.

(Left) Land ii, and its allies 3 and 4, attack ane, which is defended past 5. Country i is vulnerable if Yard ( { 2 ; 3 ; 4 } ) > γ ( { 2 ; 3 ; 4 } , { 1 ; 5 } ) M ( { 1 ; 5 } ) . (Right) A band network that is not war-stable for any parameter values.

For a group of countries to set on they must coordinate via some state j, so the target land i can be defended by its neighbors.§§

Vulnerability just defines the engineering of war. If country i was vulnerable to set on past 2, 3, and 4, then 5 might accept no involvement in actually defending 1, which might too requite upwardly with minimal resistance. Vulnerability identifies when information technology is that there is an instability and some country could exist attacked successfully.

Incentives and War-Stable Networks.

We now introduce the concept of war stability that accounts for countries' incentives to conquer other countries and to add or delete alliances. We first nowadays the definition that does not include trade. The motivation for attacking another land comes from the economical spoils.¶¶ Netted from this are expected damages and other costs of war. The expected net gain from winning a state of war is so represented as E i k ( g , C ) , which are the full economic gains that accrue to land k if country i is conquered by a coalition C with yard ∈ C when the network is g and i is defended by the coalition C ′ = { i } ∪ ( N i ( k ) ∩ C c ) .## For example, these include natural resources or other potential spoils of war.

Finally, there are costs to alliances. The price of country i's having an brotherhood with land j is some c i j ( grand ) > 0 . This could include costs of opening diplomatic, military and advice channels, coordinating military operations or intelligence, or other related costs. We generally have costs of alliances to be minor relative to the potential spoils of winning a war, because otherwise the analysis is degenerate.*** The costs are also sufficiently small-scale that whatever country i is willing to pay c i j ( g + i j ) to add an alliance with j, provided that the improver of the alliance would change i from being vulnerable to not vulnerable.

Ascertain a network 1000 to be state of war-stable if three conditions are met:

  • S1 no country is vulnerable at g;

  • S2 no two countries both benefit past calculation an alliance to thousand; and

  • S3 no land has an incentive to delete whatever of its alliances.

Implicit in this definition (in S1) is that if some country were vulnerable, and so a group that could defeat the country and its remaining allies would have an incentive to attack and defeat the country—so that would exist unstable. Also, implicit in the definition is that if the state and its allies could exist successful in fending off an attack, and then they would practice so then things would exist stable. For now, we simply assume that winning a state of war (or successfully aiding an ally in defense) is desired and losing a war is not. When we model trade, we are more explicit near gains and losses.

Given our assumptions on the relative sizes of the spoils of war and the costs of alliances, and that alliances are costly, S2 is equivalent to S2 ∀ j k ∉ g , no country is vulnerable at chiliad + j m , and S3 is equivalent to S3 ∀ j k ∈ g , both j and k are vulnerable at g − j g .

This definition is similar to that of pairwise stability (26) in that we consider changes in the network one alliance at a time, and both additions or deletions—requiring two countries to benefit to form an alliance, just only one land to do good to break an alliance. Given that there is already a large literature on possible variations on definitions of network formation, we focus on this basic definition here.†††

Nonexistence of State of war-Stable Networks.

For the example of two countries, it is direct to check that the only possible stable network is the empty network and it is war-stable only if each land has a sufficient defensive advantage. Thus, nosotros consider the more interesting instance with northward ≥ 3 .

Before presenting the results on lack of existence of war-stable networks, let us illustrate some of the main insights.

Consider a network like that pictured in Fig. two, Correct, the ring network with five countries. In social club for 1 not to be vulnerable under the improver of the link 53, it must be that γ ( { two,3,four,v } , { ane } ) Thou i ≥ M ( { ii,3,4,5 } ) (because it must not be vulnerable to 3 and its allies 2, 4, and five). However, this implies that 1 is non vulnerable in the original network if it deletes an alliance regardless of the attacking coalition, and so this contradicts war stability.

The following theorem shows that these arguments extend.

Theorem i.

Let n ≥ 3 . There are no nonempty state of war-stable networks. The empty network is war-stable if and only if M ( { j , k } ) ≤ γ ( { j , k } , { i } ) Thou i for all distinct i , j , yard .

War-stable networks merely exist in the extreme example in which the defensive parameter is so high that the weakest country can withstand an attack by the ii strongest countries in the globe, in which case the empty network is stable. Outside of that case, in that location are no war-stable networks. The intuition behind this theorem is similar to that of the example: Outside of the extreme case, requiring that a country not be vulnerable, nor vulnerable to the add-on of whatever alliances, implies that a country has extraneous alliances.

The nonexistence of war-stable networks extends to other definitions of vulnerability, with some interesting variations, as nosotros discuss in the SI Appendix.

Nuclear Weapons.

An obvious difference post WWII is the presence of nuclear weapons, leading to dramatic changes in the technology of war. Although rarely used, nuclear weapons change the technology.‡‡‡ Nonetheless, we emphasize that their beingness lonely does not lead to stability: Our model (when attacking countries can exist joined by their neighbors) allows for completely arbitrary asymmetries in military strengths and in offensive/defensive advantages. There is no way for countries to ally themselves, as a function of their strengths and nuclear capabilities, to produce a stable and nonempty network. The simply fashion in which nuclear weapons could stabilize things would be for all countries to have them and for the empty network to ensue. This is clearly non the case.

Therefore, one needs to add together merchandise, or some other consideration, to the model to explain why nosotros see denser networks that are stable and why many nonnuclear countries also live in relative peace. Thus, we plow to the analysis of the stability when at that place are substantial trade considerations.

War and Merchandise-Stable Networks.

We at present generalize the model to include payoffs that accrue to countries equally a office of the network equally a event of trade.

Country i gets a payoff or utility from the network g given by u i ( g ) , representing the economical benefits from the trade as a role of the network 1000.

A link now represents both a potential trading relationship and potential to coordinate military activities. The assumption is that if two countries trade (significantly) with each other, then they can come to each other's help in the event of a military disharmonize—so the armed services aspect of an alliance does non demand to be made explicit if there is merchandise. The two assumptions that nosotros are using are thus: (i) Having an "alliance" involves costs, nonetheless tiny, that must be commencement by some benefits either via merchandise or war; and (ii) without having any relationship, countries are not able to coordinate either in attacking or in defending. Without (two) "alliances" have no real meaning. Alliances can exist fairly inexpensive merely however serve a purpose of making clear who could defend whom in various situations.§§§

The presumption that substantial trade allows for potential military coordination captures the thought that both the interests and channels of communication needed for military coordination are present, regardless of whether there is then an explicit armed services alliance. An example is the US aid to Kuwait in the Persian Gulf War.

Vulnerability and Stability with Trade.

We now introduce a concept of vulnerability based on the incentives of countries to assail others when bookkeeping for the benefits and costs, including merchandise consequences, associated with conquering a land.

A state i is vulnerable despite trade in a network g to a country j and coalition C ⊂ N j ( g ) ∪ { j } if j ∈ C , i ∉ C and

  • M ( C ) > γ ( C , C ′ ) 1000 ( C ′ ) , where C ′ = { i } ∪ ( North i ( g ) ∩ C c ) (i.east., C could conquer i), and

  • u yard ( g − i ) + E i k ( g , C ) ≥ u g ( g ) ∀ k ∈ C with some strict inequality: Every k ∈ C would benefit from conquering i, factoring in economic gains of acquisition and gains or losses in subsequent payoffs from the network.¶¶¶

The second particular is new to this definition of vulnerability and incorporates ii aspects of economic incentives of countries to attack each other: E i 1000 ( g , C ) represents the potential net benefits that k enjoys from conquering i as role of the coalition C in a network 1000. If a country is poor in natural resources, and much of its economy is built upon nontransferable or hard-to-extract man capital, information technology would tend to take a lower E i k and would be less attractive. u one thousand ( g − i ) accounts for the fact that as i is conquered then the network of trade volition adjust. If k is a trading partner of i, and so k could lose via the emptying of i, with u m ( g − i ) < u k ( 1000 ) .### Note that this effect can work both ways: It might likewise be that a country k benefits from the elimination of some country i.

With this framework, we now define a stability notion corresponding to war stability but calculation the economic considerations. Our definition of state of war and trade stability incorporates two incentives for calculation or deleting alliances. Beginning, countries might add or maintain an alliance considering of its military value in either preventing a war or in instigating one, just equally with state of war stability. This is similar to what we considered before, except that countries now consider the economic spoils and consequences of state of war in deciding whether to take office in an attack. 2d, countries add or maintain alliances for the economic benefits in terms of trade.

Consider the incentives for countries to add together an alliance and attack some other land. Starting from a network grand, some alliance j thou ∉ g is war-beneficial if at that place exists some C ∈ Northward j ( m + j m ) ∪ { j } with j ∈ C , k ∈ C , and i ∉ C such that i is vulnerable despite trade to C at g + j k and

  • u j ( g + j m − i ) + E i j ( g , C ) ≥ u j ( g ) , so, j would benefit from forming the link and attacking, and

  • u yard ( 1000 + j k − i ) + E i thousand ( m , C ) ≥ u 1000 ( g ) , similarly for k, with one of these inequalities holding strictly.

A network grand is state of war-and-merchandise-stable if three conditions are met:

  • ES1 no land is vulnerable despite trade at g;

  • ES2 ∀ j thou ∉ g : if u j ( 1000 + j k ) > u i ( g ) then u k ( g + j k ) < u grand ( 1000 ) , and also j one thousand is non war-beneficial

  • ES3 ∀ j m ∈ m either u j ( g − j k ) ≤ u j ( g ) or j is vulnerable despite trade at k − j k , and similarly for k.

So, a network of alliances is war-and-trade-stable if no state is vulnerable despite trade, if no ii countries tin add an alliance and both turn a profit either through economic or state of war means, and either economic or war considerations foreclose any country from severing any of its alliances.****

We say that a network grand is strongly war-and-trade-stable if it is war-and-merchandise-stable for whatever (nonnegative) specification of the Due east i j s.

Existence of War-and-Trade-Stable Networks.

Nosotros begin our assay of war-and-trade-stable networks by identifying a condition that is sufficient for war-and-trade stability.

Proposition 1.

Suppose that thou is pairwise stable with respect to u.†††† If no country is vulnerable despite trade at yard or g + j 1000 for whatsoever j k ∉ g , and then g is war-and-trade-stable. Moreover, if no country is vulnerable at g or g + j yard for whatever j thousand ∉ k , then g is strongly state of war-and-trade-stable.

The proof of the proposition is straightforward and thus omitted.

At that place are many examples of networks that are war-and-merchandise-stable but not state of war-stable, because any of the nonempty ones below will satisfy that benchmark. The following theorem outlines classes of war-and-trade-stable networks, showing that economic considerations restore full general existence results.

For the following effective results, we specialize to the example of symmetric countries (then the u i ( ⋅ ) , E i j ( ⋅ ) , and M i s are independent of i and j) and where γ(⋅) is a constant, just it should be clear that similar results extend to the asymmetric case under more complicated conditions.

We also consider a canonical example in which u i ( one thousand ) = f ( d i ( grand ) ) − c ⋅ d i ( g ) , where d i ( g ) is the degree of i and f is concave, nondecreasing, and such that there exists some d ≤ north − 1 such that f ( d ) < c ⋅ d . This is a unproblematic model of gains from merchandise and costs of having trading relationships, abstracting from heterogeneity in appurtenances and trading partners and interdependencies in trading relationships beyond diminishing returns—over again it should be clear that like results concur for richer models. Let d ∗ maximize f ( d ) − c ⋅ d among nonnegative integers.

In improver, permit E i j ( g , C ) = East ( d i ( thousand ) ) / | C | , then that each country'south economic spoils from a war depend only on that state'due south degree, and then are divided equally among the attacking countries.

Theorem 2.

Consider the symmetric model with d * ≥ 2 .

  • Any d * -regular network (i.e., such that each country has d * alliances) for which no ii countries have more than k < d * − 1 allies in common is strongly war-and-trade-stable network if γ ≥ ( d * + 1 ) / ( d * − k − 1 ) .

  • If Due east ( d * ) ≤ ii [ f ( d * ) − f ( d * − 1 ) − c ] , then any d * -regular network (in any configuration, including combinations of cliques) is a war-and-trade-stable network if γ ≥ ( d * + 1 ) / ( d * − 1 ) .

The theorem'due south two existence results reflect dissimilar mechanisms. The first part reflects that if countries have incentives to maintain multiple relationships for trade purposes, then that can result in an alliance network such that no country is vulnerable (even with the addition of new alliances). Stability then requires specific network structures (for instance, but forming cliques where each country has d ∗ allies will not work, considering then all of a country'due south partners can set on the country and win). This office of the result works off of incentives for countries to defend one some other. The 2d result reflects that, with sufficient gains from trade, the potential spoils of a war against a trading partner are outweighed past the lost trade value—and and then countries are never attacked by one of their own allies. This result thus also incorporates a lack of incentives for a country to attack a trade partner, in which example with a sufficient number of alliances a wider range of networks becomes stable. This allows more cliquish structures to be stable.

Some other Look at State of war and Merchandise Data

The model suggests some dimensions that are important in determining peace. For case, as but discussed, a country having more than allies (who are trading partners) and having more merchandise with them would lead the state to be less decumbent to attack. Besides, the country should be less prone to beingness attacked by countries that trade with information technology substantially. Before endmost, we take a brief look at these furnishings in the information, and we concentrate on the period of 1950–2000 for which nosotros have detailed trade and GDP data (from ref. 32).

Tabular array two reports a logistic regression of the probability that 2 countries are at war with each other in a given twelvemonth as a office of the level of trade between the two countries, where the level of trade between the two countries is a measure of the total exchanged (imports plus exports) divided past the maximum of the GDPs of the ii countries as a normalizer. Nosotros limit attending to countries inside one,000 km of each other because virtually other dyads are much less likely to be at war or merchandise with each other. We too consider war to be a MID 5 according to the COW dataset.

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Table 2.

Logistic regressions of dyadic war on dyadic trade

In Tabular array 2 we run across a large and significantly negative human relationship between the trade between two countries and the adventure that they volition become to war.‡‡‡‡ To get an idea of the magnitude of the event, a i SD (0.0087) increase in the normalized dyadic trade decreases the log odds ratio that ii countries are at state of war with each other by a gene of more than 17 (based on the coefficient from column 1)—basically taking the odds ratio to 0.

In the SI Appendix, nosotros show that similar results hold when we just look at new wars (the first year that countries are at state of war), and we also clarify how the probability that a country is at war with another land in a given year depends on the number of trading partners that the land has (i.e., the number of countries with which the country trades at least 0.5% of its GDP), as well as the full trade that the country has with its partners normalized by its GDP, showing that these both significantly subtract incidence of wars. For instance, we find that adding 10 allies (just nether the mean) decreases the odds that a land is at war by over 50%.

Discussion

We provided a model of networks of armed services alliances and the interactions of those with international merchandise. Nosotros showed that regardless of military technologies and asymmetries amongst countries, nonempty stable networks fail to exist unless trade considerations are substantial. Moreover, the network perspective gives us an understanding of how merchandise might prevent conflict, by discouraging countries from turning confronting their allies and encouraging countries to defend their trade partners. Although this points to merchandise as a necessary status for stability, whether it is sufficient for stability depends on size of the costs and benefits of state of war.

In closing, we comment on several other features of international relations that are part of the larger picture show of interstate war. A notable change in alliances during the Cold-War period was from a "multipolar" to a "bipolar" construction, something which has been extensively discussed in the Common cold-War literature (eastward.g., see ref. 12 for references). Although this lasted for part of the postwar flow, and was characterized past a stalemate between the Eastern and Western blocs, such a organisation of two competing cliques of alliances is only war-stable if there are sufficient trade benefits between members of a clique, as shown in our second theorem. Moreover, it is more than of a historical ascertainment than a theory, and it does not account at all for the continued peace that has ensued over the terminal several decades. Thus, this fits well within the scope of the model and does not account for the overall trend in peace.

Some other institutional observation regarding the mail service-WWII calm is that institutions have allowed for coordination of countries onto a peaceful "collective security" equilibrium where any country disrupting international peace is punished by all other countries, and then that war against ane is war against all. However, as shown by ref. 34, this equilibrium is in some sense "weak": It relies heavily upon the assurance that a state tempted to bring together an attacking coalition will refuse and that all countries volition follow through on their punishment commitments, then that far-sighted expectations of off-equilibrium behavior are correct. Given that various modest conflicts since WWII did not precipitate a global response, such doubts of some countries' commitment to follow through on punishments seem reasonable.§§§§ Although collective security does not seem to explicate the lasting peace, it nonetheless does suggest an interesting avenue for extension of our model: taking a repeated games arroyo to networked conflict and trade.

One more than relevant observation regarding changes in patterns of conflict is the so-called democratic peace: Democracies rarely become to state of war with each other.¶¶¶¶ This coupled with a large growth of democracies might be idea to explain the increment in peace. However, one time 1 brings trade back into the picture, it seems that much of the democratic peace may be due to the fact that well-established democracies tend to exist better-developed and trade more. Indeed, studies (38, 39) point that poor democracies are actually significantly more probable to fight each other than other countries, and that paired commonwealth is only significantly correlated with peace when the countries involved have loftier levels of economic development, which is consequent with trade's playing the major role rather than the government structure. Our model abstracts from political considerations, which still could exist meaning, then this suggests another avenue for further extension.####

Nosotros shut by noting some additional directions for further research. One is that both trade and state of war have strong relationships with geography (see, due east.g., ref. 42 as well equally ref. 25, in which the authors discover that from 1945 to 1987 86% of meaning international wars were between neighboring states). Because geography constrains both the opportunities and benefits from trade and war, information technology has ambiguous furnishings on stability. Yet, it plays an of import role in explaining realized networks of trade and alliances and deserves further attending. A 2nd direction concerns expanding the telescopic of the paper. Although we have focused on interstate war, there are analogous forces at piece of work in civil wars in which there are multiple parties involved, besides as other settings in which there may be multiple groups or factions with competing interests and possibilities of gainful alliances (e.yard., firms in an industry with possible research collaborations or product tie-ins, or political parties with possible vote trading). The wide prediction of increased trade interests enabling more cooperative behavior overall would be interesting to explore more generally.

Proofs

Proof of Theorem 1.

The conditions on stability can exist recast as requirements on the γ parameter.

Consider a country that has an alliance in a nonempty network, say i, which then has brotherhood to some k. In order for [S3] to be satisfied, it must be that i is vulnerable in g − i k . Thus, at that place is some j and C ⊂ N j ( thousand − i thou ) ∪ j with i ∉ C and Chiliad ( C ) > γ ( C , C ′ ) M ( C ′ ) for every feasible C ′ out of all C ′ ⊂ { i } ∪ Northward i ( k − i one thousand ) ∩ C c that can defend i. Given [S1], it must exist that i was not vulnerable at chiliad, and and then it must exist that thousand ∉ C and in particular that j k ∉ thou . However, if the link j 1000 is added (so that the network chiliad + j k is formed), then C ∪ { grand } can defeat i, considering M ( C ∪ { g } ) ≥ 1000 ( C ) and too γ ( C ∪ { k } , C ′ ) M ( C ′ ) ≤ γ ( C , C ′ ) M ( C ′ ) for whatsoever feasible C ′ ⊂ { i } ∪ N i ( m ) ∩ ( C ∪ { k } ) c that tin defend i, and so K ( C ∪ { k } ) ≥ M ( C ) > γ ( C , C ′ ) One thousand ( C ′ ) ≥ γ ( C ∪ { g } , C ′ ) M ( C ′ ) for whatsoever feasible C ′ ⊂ { i } ∪ N i ( g ) ∩ ( C ∪ { 1000 } ) c that can defend i. This violates [S2] as then j and one thousand benefit from adding the link because i is vulnerable to a coalition containing both j and one thousand, which is a contradiction. This establishes that whatsoever network that is war-stable must be empty. The second sentence of the theorem is obvious.

Proof of Theorem 2.

We apply Proffer ane. Information technology is clear that any network that is d * regular is pairwise stable. Thus, we demand only evidence that no country is vulnerable despite trade and that this remains truthful with the addition of any link. For the outset role of the theorem we likewise demand to show that this is truthful regardless of E ( ⋅ ) for at least some d * -regular networks. For the second part of the theorem, we demand to show this is true under the given assumption on E ( ⋅ ) , simply for any d * regular network.

Offset, notation that no country i is vulnerable to any coalition C that does not include any of its neighbors (even if this comes from the addition of a link non involving whatever neighbors), because under either role of the theorem γ ≥ ( d * + 1 ) / ( d * − 1 ) > ( d * + 2 ) / ( d * + 1 ) . Thus, we need simply verify vulnerability to a coalition that involves at least 1 neighbour, and might possibly involve the add-on of a link.

And so, consider a land i and a coalition C that involves at least one of its neighbors. Nether the showtime function of the theorem, the maximum strength of the coalition (involving adding a link) would be d * + 2 (if lead attacker is not 1 of i's neighbors) and then the defending coalition would involve at to the lowest degree d * − k members, or else the lead attacker is ane of i's neighbors, in which case the forcefulness is at virtually d * + ane and the defence involves at least d ∗ − thou − 1 members. Given that γ ≥ ( d * + 1 ) / ( d * − k − 1 ) , information technology follows that γ ≥ ( d * + 2 ) / ( d * − thousand ) , and then i is not vulnerable in either case.

Under the second function of the theorem, if some neighbors of i in C all the same has only d ∗ links, then considering E ( d ∗ ) / ii ≤ f ( d ∗ ) − f ( d ∗ − 1 ) − c , and the attacking coalition must involve at least 2 countries (given γ and i at a minimum defending itself), then that country would not be willing to follow through with the attack of i. Thus, all of i'due south neighbors in the coalition C must accept formed a new link (and and then have degree d ∗ + 1 ) in society to be willing to follow through with the attack. This implies that the coalition involves at most two of i's neighbors (since at nigh ane link is added), but then because γ ≥ ( d ∗ + 1 ) / ( d ∗ − ane ) ≥ ( d ∗ + ii ) / d ∗ , the attacking coalition cannot defeat i and its remaining neighbors, regardless of whether it involves one or two of i's neighbors.

Acknowledgments

We thank Antonio Cabrales, Matt Elliott, Jim Fearon, Ben Golub, John Ledyard, Massimo Morelli, and the reviewers and various seminar participants for helpful comments. This piece of work was supported by NSF Grants SES-0961481 and SES-1155302 and Grant FA9550-12-1-0411 from AFOSR and DARPA, and ARO MURI Award W911NF-12-i-0509.

Footnotes

  • This contribution is office of the special series of Inaugural Articles by members of the National Academy of Sciences elected in 2015.

  • Author contributions: M.O.J. and S.N. designed research, performed research, contributed new analytic tools, analyzed data, and wrote the newspaper.

  • Reviewers: R.One thousand., Duke University; and A.T., Oxford Academy.

  • The authors declare no conflict of interest.

  • ↵*This is based on the Correlates of War (COW) data for which there are data regarding initiators of the war, which nosotros couple with other data for our analysis (Come across SI Appendix for more than information) (4). This does not even include the Napoleonic Wars, because the data begin afterward.

  • ↵†Background tin be found in refs. 6 and seven.

  • ↵‡In that location is a literature that adapts the balance theory of ref. 8 to examine network patterns of enmity (eastward.g., refs. nine⇓–11). The ideas in those works build upon notions of the form that "the enemy of my enemy is my friend" and are complementary to the assay underlying the military and trading alliances considered here.

  • ↵§We apply alliance data reported by the Alliance Treaty Obligation and Provisions Project (atop.rice.edu) (20), including alliances marked every bit containing at to the lowest degree i of a defensive, offensive, or consultation provision. The number of countries in the dataset grows over time, and so we conform on a per-country ground, because otherwise the trends are magnified further. The number of states in 1816 was 23, in 1950 it was 75, and past 2003 it reached 192.

  • ↵If one drops the WWII decade of the 1940s during which most countries were allied in one of two blocs, then this number drops down even farther to ane.722 between 1816 and 1940 (standard departure of 1.366).

  • ↵#GDP numbers per country are mostly not available except for a minor subset of countries prior to 1950s, which is part of why the trade export level data are biased. To try to avoid that bias, we estimate each land's GDP by assigning it a share of globe Gross domestic product equal to its share of world population (see SI Appendix for sources).

  • ↵||Although it would be interesting to endogenize the strengths, for our purposes here nosotros accept these as given. For bilateral models of endogenous military strengths see refs. 21 and 22.

  • ↵**The specification is somewhat redundant at this indicate because one can contain everything into the γ function, just this representation will exist useful when nosotros specialize information technology below.

  • ↵††This modeling of a war outcome based on relative strengths is reminiscent of the approach of ref. 23. 1 could instead work with competition success functions (e.g., every bit in refs. 22 and 24), which would provide for random chances of winning. In our model it would not add much because we are not focused on arming, and so all that matters is whether the expected benefits exceed a threshold of potential costs/losses.

  • ↵‡‡A country tin exist both vulnerable and a potential aggressor at some networks.

  • ↵§§We consider other definitions, presenting assay with other definitions requiring that attacking and/or defending countries grade a clique in SI Appendix. We also nowadays information showing that the neighbor example that nosotros consider in the body of the paper is by far the about relevant example (accounting for more than 3-quarters of wars from 1823 to 2003).

  • ↵¶¶Historically, these accept included land, natural resources, slaves, and access to markets. For important discussions of the spoils of interstate wars see refs. 14 and 25.

  • ↵##We allow for the dependence upon the network g, because once we permit for trade, the economical spoils available will exist a function of the network.

  • ↵***In particular, costs are small plenty so that if there is some 1000 and j k ∉ g such that j is a potential aggressor at k + j chiliad , merely not at yard, with i being vulnerable to being conquered by j, and then c j k ( grand + j yard ) + ∑ south ∈ N g ( j ) [ c j s ( g + j k ) − c j south ( g ) ] ≤ East i j ( m + j m , C ) . Thus, j is e'er willing to add an alliance to some k that would be pivotal in winning a war.

  • ↵†††Come across ref. 27 for an overview of culling network germination definitions.

  • ↵§§§A more complicated model could involve incomplete data about alliances so that making public a nonbinding alliance is useful, equally in ref. 31.

  • ↵‡‡‡There is a large literature on the Cold State of war and a contentious contend on the potential stabilizing or destabilizing bear on of nuclear applied science (e.g., see refs. 28⇓–xxx).

  • ↵¶¶¶It is not essential whether the strict inequality is required for all countries or but some, or must include j, because for a generic E function there will not be equality for whatever countries.

  • ↵###As ref. 18 documents, the economic loss resulting from trade disruption during wars can be of the same society equally more traditional estimates of losses resulting from interstate conflict. This does not even account for the potential loss of trade if a partner is lost altogether. Hither we do not model later recovery of a country or future networks, simply clearly the model extends every bit long as there are some potential short-term gains and losses from war. Also, we abstract from the political conclusion making and how gains from trade and spoils might be distributed inside a country, an of import topic for further research.

  • ↵****Note that if u i ( ⋅ ) is constant for all i, and then war and trade stability reduces to war stability.

  • ↵††††Pairwise stability is the requirement that no two countries weakly benefit from adding a link (at least one strictly), and no single land benefits from deleting some link.

  • ↵‡‡‡‡The event in ref. iii is robust to the inclusion of country fixed effects, clustered standard errors, and controls for the decade, although the coefficient on lagged trade (5) loses significance when clustered at the dyad level (encounter SI Appendix).

  • ↵§§§§One might think of the ascent of international institutions as assuasive larger groups of countries to simultaneously add together alliances, rather than the pairwise addition in our base model below. However, this only decreases the set of potentially state of war-stable networks, once once again indicating that merchandise needs to be incorporated into a model of alliances.

  • ↵¶¶¶¶For early on background see refs. 35⇓–37, and more recent references can be found in ref. vii.

  • ↵####For example, ref. 40 discusses how term limits and electoral accountability bear upon the incentives to go to war, and ref. 41 discusses the difference of the incentives, between politicians and the population that they represent, to go to war.

  • This article contains supporting information online at world wide web.pnas.org/lookup/suppl/doi:10.1073/pnas.1520970112/-/DCSupplemental.