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Free Riding, Network Effects, and Burden Sharing in Defense Cooperation Networks

Published online by Cambridge University Press:  11 January 2023

Brandon J. Kinne*
Department of Political Science, University of California, Davis, USA
Stephanie N. Kang
Department of Political Science, University of California, Davis, USA
*Corresponding author. Email:


How do states distribute the burdens of collective defense? This paper develops a network theory of burden sharing. We focus on bilateral defense cooperation agreements (DCAs), which promote cooperation in a variety of defense, military, and security issue areas. Using a computational model, we show that DCA partners’ defense spending depends on the network structure of their agreements. In bilateral terms, DCAs increase defense spending by committing states to defense activities and allowing partners to reciprocally punish free riding. However, as a state's local network of defense partnerships grows more densely connected, with many transitive “friend of a friend” relations, DCAs have the countervailing effect of reducing defense spending. The more deeply integrated states are in bilateral defense networks, the less they spend on defense. We distinguish two potential mechanisms behind this effect—one based on efficiency improvements, the other on free riding. An empirical analysis using multilevel inferential network models points more to efficiency than to free riding. Defense networks reduce defense spending, and they do so by allowing countries to produce security more efficiently.

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Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The IO Foundation

Problems of burden sharing and defense cooperation have come to the fore of public debate. During his tenure in office, US president Donald Trump frequently took aim at NATO, singling out member states for “not paying their fair share”Footnote 1 and insisting that “the distribution of costs has to be changed.”Footnote 2 Burden sharing encapsulates a fundamental collective-action problem—how best to divide the burden of common defense among partners—that extends across the realm of international security.

This paper develops and tests a network theory of burden sharing. Recent work on networks shows that the structure of international relations substantially influences state behavior.Footnote 3 Classic public-goods models of burden sharing—in which contributions or “spill-ins” from alliance partners affect individual defense effortFootnote 4—implicitly acknowledge this network context but do not consider its implications. Empirical studies of burden sharing often focus on country-level determinants of defense spending, such as economic constraints or political ideology.Footnote 5 Much less is known about how states select defense partners, and how the resulting network structure of those partnerships influences burden sharing.

Our analysis focuses on defense partnerships established through bilateral defense cooperation agreements, or DCAs. These agreements have proliferated dramatically since the early 1990s and now play a central role in the global security environment.Footnote 6 By promoting a broad range of cooperative activities—intelligence sharing, arms trade, training and officer exchanges, peacekeeping operations, and joint military exercises, among othersFootnote 7—DCAs help countries modernize their militaries and pool resources against shared threats. Because these activities require defense outlays, DCAs naturally pose the question of whether formal defense commitments lead to increased defense effort.

Our network approach to burden sharing shows that defense agreements have divergent effects on defense spending. In strictly bilateral terms, DCAs enable detection and informal punishment of free riders. Contrary to the “large numbers” problem of public-goods models, defense spending increases as countries sign more DCAs. However, the network structure of defense agreements exercises a countervailing influence. As states form increasingly dense defense ties, characterized by friend-of-friend relations or “transitive triads” in their local networks of partners, defense agreements in fact reduce spending. The overall impact of defense cooperation on burden sharing thus depends on how defense relations are structured. We use an agent-based model (ABM) of network–behavior coevolution to develop the argument and derive hypotheses, and we test the hypotheses with inferential network models.

This analysis makes three contributions to our understanding of burden sharing. First, we show that forming defense partnerships and determining an appropriate level of defense effort are interdependent processes. Governments do not create defense agreements randomly but instead select those partners that best contribute to mutual defense.Footnote 8 At the same time, these partnerships determine the influence of agreements on burden sharing. Put differently, defense partnerships and individual defense effort coevolve over time, and the influence of those partnerships on burden sharing depends, in part, on how states select partners.

Second, we theoretically separate the bilateral influence and network influence of DCAs, and we empirically assess the independent impact of each. We find that although bilateral DCAs put upward pressure on defense expenditures, as those agreements congeal into dense local networks they instead reduce individual defense effort. This network effect is wholly unapparent from a bilateral perspective. Post-estimation analysis further reveals that the DCA network is a more important determinant of defense spending than virtually all covariates, including regime type, alliance membership, and economic growth.

Finally, we identify two distinct mechanisms that connect network structure to reduced defense effort. On the one hand, dense local networks may promote convergence in defense policies, rendering defense cooperation less costly and more efficient. On the other hand, dense local networks may undermine reciprocity-based punishments and increase the publicness of defense goods, inviting free riding. Scholarship on burden sharing recognizes the importance of both mechanisms.Footnote 9 We incorporate these mechanisms into the ABM, derive testable hypotheses for each, and empirically determine which effect dominates. The results suggest that DCA networks reduce defense effort primarily by generating efficiencies, not by inducing free riding.

Overall, the network perspective shows both that particular structures of cooperation reduce defense effort, and that such reductions are not necessarily suboptimal. Rather, networks may allow states to produce security more efficiently. This finding means that individual contributions to burden sharing should be viewed from a broader network context, as instances of apparent free riding may in fact reflect network efficiencies. More generally, these findings underscore the importance of theorizing and empirically modeling international relations as a global network, where network structure influences behaviors in ways that are unobservable from standard dyadic perspectives.

Defense Cooperation and Burden Sharing

The literature on burden sharing typically focuses on formal alliances.Footnote 10 The economic theory of alliances, first articulated by Olson and Zeckhauser, is perhaps the most widely employed burden-sharing framework.Footnote 11 In this view, alliances produce pure public goods, such as deterrence or reduced militarized conflict. Because these goods are nonexcludable and nonrival, they create incentives for countries to free ride, or enjoy the security benefits of defense cooperation at little or no cost to themselves. Others argue that alliances generate “joint products”—not only pure public goods, but also private and impure public goods. Free riding declines, but does not altogether disappear, when the gains of defense cooperation are more commensurate with individual effort.Footnote 12

While research on alliances has produced important insights on defense cooperation, we know little about burden sharing beyond alliances. Formal alliances exist primarily to deter conflict,Footnote 13 and they focus more on partner reliability in militarized confrontations than on routine cooperation.Footnote 14 DCAs, by contrast, involve no defensive, offensive, neutrality or other conflict-specific commitments, and they explicitly establish “institutional frameworks for routine defense cooperation.”Footnote 15 DCAs encourage such day-to-day defense activities as military exercises, training and officer exchanges, arms trade, peacekeeping, research and development, and sharing of classified information, with a combined focus on interstate security issues and more nontraditional threats like terrorism, maritime security, and nonstate armed groups.

States continue to sign new DCAs at a high rate, and there are now nearly as many dyads with DCAs in place as there are dyads with alliances (Figure 1). These empirical trends, combined with the emphasis of DCAs on routine, substantive defense cooperation—which places an immediate expectation on states to contribute to joint security—present an opportunity to explore burden sharing beyond formal alliances. Further, the institutional features of DCAs mitigate common methodological problems. DCAs tend to be (1) similarly structured, prioritizing a common set of issue areas; (2) virtually always bilateral; and (3) lacking in institutionalized enforcement mechanisms.Footnote 16 These features hold constant the potential confounding influences of institutional design, issue-area variation, multilateral politics, and intergovernmental organizations.

Figure 1. Trends in defense cooperation agreements (DCAs) and alliances, 1990–2010

Most importantly, existing research does not consider how endogenous aspects of defense cooperation affect burden sharing. The choice to sign defense agreements with particular partners is not independent of the subsequent influence of those agreements on defense behavior. Defense scholars typically treat partner selection as exogenous.Footnote 17 Yet, as Digiuseppe and Poast observe with regard to alliance formation, “defense pacts and military spending are co-determinous processes.”Footnote 18 In networks, states strategically select partners to maximize their utility, and this selection process generates distinct network structures.Footnote 19 Scholars of international relations have shown that across issue areas—alliance formation, militarized conflict, status and reputation, trade and human rights, and even joint security productionFootnote 20—network structures influence behavior. Yet, despite the implicit network logic of free-riding arguments,Footnote 21 studies of burden sharing typically focus on exogenous factors like external threats, domestic politics, leader attributes, or financial constraints.Footnote 22 We show that network structures matter both for the selection of defense partners and for the influence of partners on individual defense effort.

A Network Theory of Burden Sharing

In Olson's classic model, provision of public goods decreases as group size grows.Footnote 23 Large alliances thus encourage free riding.Footnote 24 This logic provides a baseline expectation for DCAs. As a focal state's number of DCA partners increases, its incentive to free ride on its partners’ efforts increases, and its defense spending declines. This well-known “large numbers problem” relies on three assumptions: (1) there is a distinct group of actors, and the fraction of the group benefit for any given actor shrinks as the group grows; (2) states cannot easily implement the strategic interactions needed to encourage contributions; and (3) organization costs are high.Footnote 25

These assumptions readily apply to multilateral agreements and formal organizations. But DCAs are not multilateral and do not create organizations.Footnote 26 Instead, DCAs are separable bilateral agreements, and they produce defense benefits that, prima facie, resemble club goods, shared among the members of the arrangement.Footnote 27 In practical terms, DCAs provide a way for states to align their defense policies toward shared goals and interests.Footnote 28 This alignment may involve large-scale strategic issues, such as determining which global threats to prioritize and how to respond to those threats.Footnote 29 Or alignment may involve technical considerations like interoperability, logistics and supply, training standards, and information-sharing protocols, among others. When defense policies align, states prioritize similar types of threats and implement similar operational standards in addressing those threats.

Production of collective security goods requires not only that states align their policies but also that they make ongoing contributions to mutual defense. To modernize their militaries, for example, governments must invest in research and development, weapons procurement, training, and innovative military doctrines. Improving access to classified intelligence requires investments in signal and human intelligence-gathering capabilities (such as satellites and spies), as well as implementation of safeguard protocols at the organizational level. Any agreed-upon joint actions—military exercises, peacekeeping operations, counterterrorism operations—necessitate further operation-specific expenditures. Without mutual contributions, the benefits of defense cooperation are minimal.

As bilateral agreements, DCAs include an informal punishment mechanism.Footnote 30 If one partner falters in its obligations, the other withholds its own contribution, and the club good is not produced. The bilateral nature of DCAs enables tit-for-tat strategies that encourage individual defense effort.Footnote 31 Governments are acutely aware of this mechanism. Turkey's DCA with Indonesia, for example, explicitly defines cooperation as “activities based upon reciprocity.”Footnote 32 Implementing reciprocity-based punishments is simply a matter of withholding cooperation in response to perceived lack of effort by partners. Such reciprocity gradually undermined a 1995 DCA between Indonesia and Australia, until the agreement was abrogated in 1999.Footnote 33 Less dramatically, reciprocal punishments leave agreements to languish, such that they fail to produce security goods. In strictly bilateral terms, contrary to the large-numbers problem, this potential for punitive reciprocity should encourage defense spending.

The Network Structure of Defense Cooperation

However, DCAs are not merely bilateral. They also comprise a larger network. A network consists of a set of agents, or “nodes,” connected by a set of ties, or “edges.” Network ties are interdependent; the formation, maintenance, and/or termination of one edge depends on edges elsewhere in the network. In addition to maintaining their ties, nodes engage in various unit-level behaviors, such as allocating expenditures to defense, which may affect—or be affected by—edge formation. This network–behavior setup allows us to theorize the selection of defense partners and the influence of those partners on defense effort from within a single coevolutionary framework.

As the network evolves, distinctive structures emerge. These structures may generate higher-order effects that disrupt the straightforward logic of bilateral cooperation.Footnote 34 Accordingly, we separate the bilateral influence of DCAs from their network influence. We focus on one particular network structure, the transitive triad, also known as a triangle, wherein three nodes are mutually connected by three unique edges, as illustrated in Figure 2.Footnote 35 Triads are the building blocks of networks. They enable social arrangements—mediation, brokerage, coalitions—that are impossible with only two actors.Footnote 36 Triads are also essential to more complex network features, such as hierarchy, clustering, and modularity.Footnote 37 As the most elementary form of network structure, triads provide crucial insight into the effects of structure on behavior.

Figure 2. Two types of triadic structures

Analyzing the topology of the DCA network reveals that DCA ties have grown increasingly transitive, such that bilateral DCA partners often have mutual DCA ties to common third parties (Figure 3). At the same time, the network has grown denser and less “centralized,” or less dominated by a small number of highly active nodes (such as major powers). The key question is how these structural features matter for burden sharing.

Figure 3. DCA network topology at two time points

Selection and Influence at the Network Level

To understand how network structure influences defense effort, we merge network-analytic concepts with insights from the burden-sharing literature. We identify two specific mechanisms—one based on policy convergence and efficiencies, the other on free riding—that connect transitive triads to burden sharing.

Efficiency concerns, or the quantity of security produced relative to spending, are central to burden sharing.Footnote 38 Research on alliances shows that certain defense policies—such as “sharing and standardization schemes” to ensure interoperability of weapons and forcesFootnote 39—can reduce defense outlays by promoting complementarity and mitigating risk.Footnote 40 Building on this insight, we focus on network efficiencies. A network efficiency exists when actors derive utility not only from their individual ties but also from the structure of those ties.Footnote 41 For transitive triads, network efficiency means that the aggregate utility generated by a triangle is greater than the sum of those ties’ individual utilities. Put differently, states benefit from the joint defense products of each DCA in which they are members, and they benefit from the overall connectedness of their DCA partners. By this logic, the structure in Figure 2(b) generates more utility for a given focal node than does the structure in 2(a), even if its number of direct ties is identical in the two cases. Network efficiencies are similar to synergy effects.Footnote 42

Transitive triads can generate efficiencies by facilitating convergence in the defense policies of states, thus reducing the costs of defense cooperation.Footnote 43 This argument prioritizes the role of DCAs in defense policy coordination and builds on the burden-sharing literature's emphasis on policy-driven sources of efficiency. As Pannier and Schmitt observe, “policy convergence is a prerequisite to effective cooperation.”Footnote 44 When governments share strategic goals, they can more readily identify and act on mutual threats. In technical areas, by coordinating policies on interoperability, training, logistics, and other standards, governments can subsequently engage in joint exercises, counterterrorism and peacekeeping operations, arms trade, and investment in research and development. For example, policies that improve interoperability—on such issues as ammunition caliber, aerial refueling technology, and software protocols, for example—allow defense partners to adopt exchangeable force units and hardware, increase compatibility in communications technology, and standardize data exchanges.Footnote 45

Defense policy adjustments are costly and must contend with budgets, bureaucracies, technological limitations, and other barriers.Footnote 46 The wider the policy gap between prospective partners, the greater the costs of adjustment.Footnote 47 In a transitive triad, focal node i's partners j and k are also aligned with one another by virtue of their direct tie, jk. Thus any policy adjustments i makes with regard to one partner are extensible, in part, to others. This effect is similar to regulatory convergence, where governments adopt a modal policy in lieu of multiple, potentially contradictory regulations.Footnote 48 On sharing of classified information, for example, the US maintains different standards with South Korea than it does with Japan, largely due to an absence of direct Japan–Korea standardization. When attempting to cooperate on issues of mutual interest, such as the North Korean threat, this patchwork arrangement “only covers about half the necessary information and creates significant lag in the information flow.”Footnote 49 Standardization between Japan and South Korea would allow the US to implement a common intelligence-sharing policy with regard to both states. More generally, transitive triads reduce the need for discrepant standards and allow states to coordinate policies at lower cost.

As transitive triads persist, they push defense policies further into alignment via third-party influences. When two states align their policies with a common third party, they indirectly align those same policies with one another. This convergence, in turn, reduces the operational costs of defense activities. For example, the defense policies that Japan and Australia individually adopted in their respective relations with the US ultimately lowered barriers to direct Japan–Australia cooperation on arms procurement, joint ballistic missile defense, intelligence sharing, and force complementarity.Footnote 50 Similarly, the post-World War II hub-and-spoke system of bilateral ties in East Asia, defined by policy coordination with the US, facilitated coordination between the spokes themselves, yielding a “web of security relations” and frequent joint military activities.Footnote 51 This indirect convergence effect mirrors a general homophily effect, observed in many social networks, where third-party ties increase belief similarity.Footnote 52 Such third-party influences promote efficiency by lowering policy barriers to defense activities.

Finally, transitive triads reduce the operational costs of trilateral or larger plurilateral actions, such as peacekeeping, military exercises, and counterterror operations. Effective trilateralism requires extensive “spoke-to-spoke” interaction, where each leg of the triangle effectively coordinates with the others.Footnote 53 From i's perspective, the more closely aligned j and k are, the more easily all three states can leverage their respective bilateral ties for trilateral purposes. For example, Turkey has long encouraged greater policy coordination between Azerbaijan and Georgia—its two key defense partners in the Caucasus—in the hopes of promoting trilateral activities, improving regional security, and avoiding the expense of holding separate bilateral exercises, trainings, and exchanges.Footnote 54 This policy alignment within transitive triads lowers operational costs in similar fashion to “minilateral” strategies.Footnote 55

The allure of reduced adjustment and operational costs directly affects partner selection. Countries have an incentive to select partners in a way that yields transitive triads and captures network efficiencies. As they do so, dense local networks emerge, where a given node's defense partners are also partnered among themselves (Figure 4). This structure maximizes triangle-based efficiencies. While an accumulation of bilateral ties—as in a sparse local network—increases a given node's defense obligations, a dense local network lowers the costs of those obligations. Ceteris paribus, the lower the costs of security production, the weaker the demand for defense outlays.Footnote 56 Precisely this outcome motivates states to form triangles in the first place. In terms of influence, then, the focal node's defense spending should decrease as the density of its local network increases.

Figure 4. Sparse versus dense local networks

Importantly, efficiency-driven reductions in defense effort are the result of lowered costs, not free riding. However, dense local networks may also incentivize free riding, in two ways. First, as with network effects more generally,Footnote 57 the utility generated by efficiencies increases with the number of participants. As states converge in their security strategies, a growing number of partners stands to benefit from the actions of a few vigilant actors. Policy convergence increases the odds that the security-minded efforts of some states will address threats of importance to others, thus increasing the publicness of defense goods. By demarcating groups of like-minded states—defense partners in dense local networks—and expanding the range of beneficiaries, network efficiencies resurrect a key assumption behind the large-numbers problem: that a distinct group exists, and larger groups reduce the fractional benefit for any given actor.Footnote 58

Second, dense local networks may endanger the reciprocity mechanism that deters free riding at the bilateral level. As political economists have long arguedFootnote 59—and as Farrell and Newman show with regard to economic networks and security issuesFootnote 60—policy interdependence reduces flexibility. In a dense local network, j's attempts to punish a free-riding i are more likely to negatively impact j's other partners—who expect to benefit from production of defense goods—regardless of whether those third parties engage in free riding themselves. If dense local networks indeed reduce states’ ability to deter free riding via strategic interactions, they effectively reinstate the additional two assumptions behind the large-numbers problem—that is, states lack strategic mechanisms to encourage contributions, and they face organizational costs.Footnote 61 Opportunistic states, recognizing the limited ability of their partners to inflict punishments, can more easily free ride on those partners’ efforts. Notably, free riding in this case arises not from multilateral treaties or international organizations but from network structure.

Both efficiency and free riding reduce defense spending, but for starkly different reasons. Separating these mechanisms is necessary to illuminate the overall security benefits of defense cooperation. A key distinction is that efficiencies are a general effect of network structure as such,Footnote 62 while free riding is conditional on the defense effort of one's partners.Footnote 63 That is, efficiencies emerge when defense partnerships coalesce into transitive triads. By contrast, the incentive to free ride on the efforts of other states depends on the effort expended by those states. We exploit this distinction to separate the effects of network efficiencies from triangle-induced free riding.

A Computational Model of Network–Behavior Coevolution

To derive testable hypotheses, we must (1) explicitly model interdependence between selection and influence; (2) separate the bilateral and network influences of DCAs; and (3) distinguish the effects of network efficiencies from those of free riding. Because network relations and individual behaviors are mutually endogenous, models that rely on analytic solutions, such as game-theoretic and decision-theoretic models, are generally not feasible. Instead, we build an ABM using a network–behavior coevolution approach, which is designed to assess how network relations and individual behaviors mutually influence one another.Footnote 64 Similar ABMs have been used to study selection–influence dynamics across a range of networks and behaviors, including the international system.Footnote 65 A methodological benefit of this approach is that it is readily extensible to empirical analysis, as discussed later. This subsection explains the key elements of the ABM. The online supplement gives a thorough presentation.

Consider a finite set of agents, N = {1, …, n}, with ties g ∈ {0, 1}, in an n × n matrix, representing bilateral DCAs. The n × 1 matrix r ∈ {1, …, M} defines an individual behavior, scaled across M ordinal categories, representing defense effort. Agents adjust their ties or behaviors when given an opportunity to do so according to separate rate functions, such that g and r coevolve in continuous time. A focal agent i may create a new network tie, terminate an existing one, or make no change at all. Let g + denote the network that exists after i has been given an opportunity to adjust its ties. Similarly, let r + denote the matrix of behaviors that results from i having an opportunity to change its behavior. When agents adjust their ties or behavior, they maximize their utility with respect to two objective functions: $f_i^{{\rm net}} ( {g, \;g^ + , \;r} ) $ for the network and $f_i^{{\rm beh}} ( {g, \;r, \;r^ + } ) $ for behavior.

The network choice probabilities for i are defined as

(1)$$P( {g^ + } ) = \displaystyle{{\exp ( {\,f_i^{{\rm net}} ( {g, \;g^ + , \;r} ) } ) } \over {\mathop \sum \limits_{k = 1}^n \exp ( {\,f_i^{{\rm net}} ( {g, \;g^{ + k}, \;r} ) } ) }}$$

where the sum in the denominator refers to all possible g +k states of the network, or the options available to i for toggling its network ties.Footnote 66 Similarly, the choice probabilities for behavior change are defined as

(2)$$P( {r^ + } ) = \displaystyle{{\exp ( {\,f_i^{{\rm beh}} ( {g, \;r, \;r^ + } ) } ) } \over {\mathop \sum \limits_{k = 1}^M \exp ( {\,f_i^{{\rm beh}} ( {g, \;r, \;r^{ + k}} ) } ) }}$$

where the sum in the denominator refers to possible r +k levels of the behavior.

The ABM implements these objective functions as linear combinations of effects:

(3)$$f_i^{{\rm net}} ( {g, \;g^ + , \;r} ) = \mathop \sum \limits_h \beta _h^{{\rm net}} s_h^{{\rm net}} ( {i, \;g, \;g^ + , \;r} ) $$
(4)$$f_i^{{\rm beh}} ( {g, \;r, \;r^ + } ) = \mathop \sum \limits_h \beta _h^{{\rm beh}} s_h^{{\rm beh}} ( {i, \;g, \;r, \;r^ + } ) , \;$$

where the statistics s h must be specified on the basis of theory and may include endogenous features of the network g, various aspects of behavior r, or exogenous covariates. The β h parameters are weights that determine the extent to which agents attempt to achieve a network–behavioral state that yields large values for the corresponding s h statistics. The ABM captures network–behavior coevolution by including behavior terms in the network equation and network terms in the behavior equation.

The ABM defines a micro-level, actor-oriented process,Footnote 67 which approximates the decision-theoretic approaches common in the study of burden sharing.Footnote 68 The outcomes of interest are macro-level features of the network and behavior, which are not themselves explicitly modeled but are emergent properties.Footnote 69 We derive hypotheses by observing how the ABM specification affects these macro-level outcomes. The key outcome in this case is mean defense effort. The ABM outcomes are statistical approximations of stable equilibria in which agents cannot further improve their utility.Footnote 70 See the online supplement for equilibrium analysis.

We calibrate the ABM using observed empirical data on the DCA network and defense spending for the year 2000.Footnote 71 The initial g matrix is the observed DCA network, where g ij = 1 indicates a DCA in force. The initial r matrix is an eleven-point scale (M = 11) of country-level defense effort, derived from defense spending as a percentage of GDP. By assuming that individual behavior takes on ordinal values, we can represent network–behavior coevolution in a common statistical framework—specifically, a continuous-time Markov chain with a discrete outcome space.Footnote 72 A continuous behavior metric would entail a virtually infinite number of choices (see equation (2)), which is computationally infeasible. In network–behavior models, discretization is widely used and well established.Footnote 73 We discretize defense-spending data at absolute 1 percent increments from zero to 10 percent, with a residual category for spending above 10 percent; that is, we specify cut points at [0, 0.01, …, 0.1, 1].Footnote 74

DCA Degree and Defense Spending

We first examine the $f_i^{{\rm beh}} ( {g, \;r, \;r^ + } ) $ behavior function in isolation. We specify the standard public-goods model described by Sandler and Hartley, where optimal defense effort for country i is a function of the price of defense goods, national income, level of threat, and spill-ins from defense partners.Footnote 75 When the relationship between spill-ins and defense effort is the focus, the additional terms in the model can be held constant, which allows derivation of reaction functions that indicate i's optimal response to the efforts of its partner j.Footnote 76 We model this response as

(5)$$f_i^{{\rm beh}} ( {g, \;r, \;r^ + } ) = \alpha ^{{\rm beh}}r_i + \pi c_i + \gamma d_i, \;$$

where {α beh, π, γ} are the $\beta _h^{{\rm beh}} $ parameters of equation (4), given unique designations for clarity. r i is agent i's current level of defense effort. c i is the ith observation of an n × 1 random variable with an exponential distribution that represents exogenous, unit-specific demands for defense spending, such as variations in national income and/or exposure to security threats. d i is i's number of partners in the DCA network, or “nodal degree,” which reflects anticipated defense contributions from one's partners and thus corresponds to spill-in in the public-goods framework.Footnote 77 Table 1 lists all the component variables of the ABM, with formal definitions. See the online supplement for parameter profiles.

Table 1. Summary of terms in the agent-based model

The γ parameter determines the effect of i's DCAs on its utility. The claim that free riding increases with group size assumes that γ is negative.Footnote 78 As defense partners increase in number, the incentive to spend on defense declines. Accordingly, we first set γ at incrementally decreasing values from zero. Because public-goods models do not consider network structure, at this point all parameters in the network equation are zero; that is, states select DCA partners at random.Footnote 79 We then simulate the coevolution of the DCA network and defense spending. Figure 5(a) illustrates equilibrium defense spending behavior as γ decreases. These equilibria mirror the familiar response curve of a decision-theoretic public-goods model.Footnote 80 The greater the anticipated spill-ins from defense partnerships, the less effort i expends on defense.

Figure 5. Equilibrium outcomes in the network–behavior agent-based model

By contrast, if bilateral agreements enable detection and punishment, then i cannot accrue the benefits of defense partnerships without making contributions. In this case, γ is positive, reflecting the utility i derives from spending on defense and ensuring continued cooperation from its partners. Figure 5(b) illustrates equilibrium defense spending as γ increases from zero. Contrary to the expectations of free-riding models, increases in γ sharply increase defense effort—despite the pressure of exogenous influences and the nonzero costs of defense. Two clarifications are in order. First, because γ parameterizes the relationship between nodal degree and individual behavior, the effect illustrated in Figure 5 depends not only on γ but also on each agent's respective nodal degree. That is, as a country's number of ties increases, so does the impact of γ on defense effort. Second, because this result is agnostic about network structure and the underlying process of network formation, it is strictly limited to the bilateral effects of DCAs. Overall, under the assumption that bilateralism facilitates reciprocal punishments, and as a counter to public-goods expectations, the model yields the following testable hypothesis:

H1: As a state's number of DCA partners increases, its defense spending increases.

Influence of Transitive Triads

To incorporate the influence of network structure on defense spending, we must first specify the components of the network objective function, $f_i^{{\rm net}} ( {g, \;g^ + , \;r} ) $. Drawing on empirical work on DCA networks,Footnote 81 we specify

(6)$$f_i^{{\rm net}} ( {g, \;g^ + , \;r} ) = \alpha ^{{\rm net}}g_{i\bullet } + \phi c_j + \xi w_{ij} + \zeta r_j + \tau a_{ij} + \delta b_{ij}$$

where {α net, ϕ, ξ, ζ, τ, δ} are the $\beta _h^{{\rm net}} $ parameters of equation (3). The $\alpha ^{{\rm net}}g_{i\bullet }$ term models i's baseline tendency to form ties. As in equation (5), c j is an n × 1 random variable, which in this case reflects exogenous monadic attributes of j that influence partner selection. w ij is a random variable in the form of an n × n matrix to account for exogenous dyadic influences. The ζr j term models the tendency for i to select DCA partners that spend highly on defense. The endogenous term τa ij accounts for the tendency of high-degree nodes to sign DCAs with other high-degree nodes.Footnote 82

The final term, δb ij, is the most critical. b ij is the sum of triangles in i's network ties. The parameter δ determines the utility i derives from selecting defense partners in a way that yields triangles. If δ is positive, agents prefer to form dense local networks; if negative, agents prefer intransitive triads and sparse local networks. We set δ at a positive value to reflect the attraction of network efficiencies, consistent with existing empirical work.Footnote 83

With equation (6) in place, we update equation (5) to include network structure:

(7)$$f_i^{{\rm beh}} ( {g, \;r, \;r^ + } ) = \alpha ^{{\rm beh}}r_i + \pi c_i + \gamma d_i + \psi q_i$$

where q i is a count of triangles in i's local network. The ψ parameter determines the additional utility that triangles generate for i. If triangles are irrelevant, then ψ = 0. If triangles generate network efficiencies, then ψ is negative.

The ABM uses both equations (6) and (7) to simulate the coevolution of the DCA network and defense spending. We vary $\psi $ from negative to positive values while holding γ at a constant positive value, consistent with H1. This specification captures the countervailing effect of network structure even as, in purely bilateral terms, DCAs exert consistent upward pressure on defense effort. Figure 6(a) illustrates equilibrium defense-spending behavior. Positive values of $\psi $—which might obtain if, say, dense local networks improve detection and punishment—generate small increases in defense spending. By contrast, negative values of $\psi $ result in correspondingly larger reductions in defense effort, consistent with the influence of network efficiencies. This result obtains even though the positive bilateral effect of DCAs, reflected in $\gamma $, remains unchanged.

Figure 6. Agent-based model with coevolution and dense local networks

Network–behavior coevolution plays an essential role in generating these outcomes. We thus far have assumed a network formation process in which agents prioritize transitive triads $(\delta > 0)$. Consider an alternative network formation process, known as “preferential attachment,”Footnote 84 where agents prefer ties to high-degree nodes and avoid transitive closure $(\delta < 0)$. This process yields a hub-and-spoke topology, consistent with theories that emphasize hierarchy in the international systemFootnote 85 and the primacy of great power politics.Footnote 86

We simulated the ABM using this alternative parameter profile for network formation (equation (6)) while keeping the behavior profile (equation (7)) unchanged.Footnote 87 This model generates sharply divergent outcomes, illustrated in Figure 6(b). Transitive triads have no discernible effect on defense spending, even though the parameters for equation (7) are identical to those used to produce the results in Figure 6(a). When states have little incentive to form triangles, then the influence of triangles on spending declines to trivial levels, even if, in principle, triangles should reduce defense effort. These results reinforce the crucial insight that network formation and individual behavior are interdependent processes. If we ignore the strategic motivations behind agents’ selection of partners, or if we misunderstand those motivations, we may reach wildly divergent expectations about the influence of network ties on behavior. The model yields the following hypothesis:

H2: As the number of triangles in a state's local DCA network increases, its defense effort decreases.

While these results are consistent with an efficiency mechanism, they may also be driven by free riding. To separate these mechanisms, we capitalize on the distinction noted earlier. Efficiencies are a general feature of network structure, not conditional on nodal attributes.Footnote 88 By contrast, the free-riding incentive depends on spill-ins from the efforts of one's defense partners.Footnote 89 Bilateralism mitigates free riding by enabling reciprocal punishments. Yet, as dense local networks undermine the ability of states to impose punishments, the free-riding incentive reemerges—specifically in the context of a dense local network, where an opportunistic agent can reap the gains of its triangle partners’ efforts while strategically avoiding punishments.

We model $i$'s responsiveness to spill-ins from its triangle partners as

(8)$$f_i^{{\rm beh}} ( {g,r,r^ + } ) = \alpha ^{{\rm beh}}r_i + \pi c_i + \gamma d_i + \psi q_i + \eta z_i,$$

where $z_i$ measures triangles as a function of the defense effort of $i$'s partners, and the parameter $\eta $ determines the strength of the free-riding incentive, or $i$'s prospects for evading bilateral punishments. Considering $\eta $ alongside $\psi $ separates the general effect of network efficiencies from the conditional effect of free riding.

Figure 7 illustrates the plausible parameter space for $\eta $ and $\psi $ as each decreases from zero. Free riding and efficiency are independently capable of reducing defense effort. That is, even if we altogether eliminate free riding $( {\eta = 0} ) $, triangles still push defense effort downward, regardless of partners’ defense efforts, via the general influence of network efficiencies $(\psi < 0)$. Conversely, if we assume that triangles as such generate no utility $( {\psi = 0} ) $ but we allow dense local networks to undermine reciprocity $(\eta < 0)$, an opportunistic agent will reduce its defense effort so long as it obtains spill-ins from its local network partners. These results cleanly separate the causal mechanisms and yield two testable hypotheses:

H3 (Efficiency): Triangles reduce a state's defense effort regardless of the effort of its partners $(\psi < 0,\eta = 0)$.

H4 (Free riding): Triangles reduce a state's defense effort only as the effort of its partners increases $(\psi = 0,\eta < 0)$.

Overall, the ABM provides insights that, to our knowledge, have not been considered in the burden-sharing literature. First, the anticipated effect of defense agreements on defense effort depends on critical assumptions about whether an accumulation of bilateral DCAs raises the same large-$N$ problems that plague collective action in multilateral and organizational frameworks. Second, states engage in a strategic selection–influence dynamic when joining defense agreements, and this dynamic generates distinctive network structures. Third, the network structures that emerge from strategic partner selection influence defense effort in ways that are not apparent from bilateral relations. Finally, these network influences may involve either free riding or efficiency, and these competing mechanisms carry distinct empirical implications.

Figure 7. Effect of efficiency and free riding on defense effort

Research Design and Data

The complexities of the ABM—simultaneous equations across levels of analysis, endogenous influences within the network, and selection–influence dynamics—are intractable in traditional regression-based empirical models. To our knowledge, the only established empirical model capable of modeling such features of network–behavior coevolution is the stochastic actor-oriented model (SAOM).Footnote 90 Fortunately, the network–behavior architecture of the ABM also underlies the SAOM, and moving from the ABM to empirical analysis is straightforward. The SAOM implements a simulated method-of-moments estimator, which uses network–behavior simulations to calculate expected values of included model statistics, compare those expected values to observed values, and derive parameter estimates. In this sense, the SAOM validates the ABM against real-world data.Footnote 91 In empirical studies of international relations, SAOMs have been used to model single networks over time,Footnote 92 network–behavior coevolution,Footnote 93 and dynamics of multiplex networks.Footnote 94

We first update the objective functions to incorporate longitudinal empirical network–behavior data. Let ${\bf g}$ be a $1 \ldots T$ stack of symmetric, binary $n \times n$ matrices, where $T$ is the number of years of data. As in the ABM, $g_{ij,t} = 1$ indicates that a DCA is in force between $i$ and $j$ in year $t$. Let ${\bf r}$ be a $1 \ldots T$ stack of $n \times 1$ matrices. Again mirroring the ABM, each $r_{i,t}$ entry of ${\bf r}$ takes on some ordinal integer value, where larger values indicate greater defense effort.

The empirical implementation of the network objective function in equation (3) can be written as

(9)$$f_i^{\bf g} ( {{\bf g},{\bf r}} ) = \mathop \sum \limits_h \beta _h^{\bf g} s_{ih}^{\bf g} ( {{\bf g},{\bf r}} )$$

and the behavior objective function in equation (4) can be written as

(10)$$f_i^{\bf r} ( {{\bf g},{\bf r}} ) = \mathop \sum \limits_h \beta _h^{\bf r} s_{ih}^{\bf r} ( {{\bf g},{\bf r}} )$$

To ensure that the empirical model aligns with the ABM, we use the same network terms in the SAOM as in the ABM (summarized in Table 1). For clarity, in the SAOM we refer to these terms by their full names. Equation (9) thus includes density, total degree, transitive triads, and defense spendingj, all of which are calculated on empirical DCA data. Similarly, equation (10) includes constant,Footnote 95 dca degree, dca dense triads, and dca triads effort. We incorporate the exogenous covariates of the ABM, c i, c j, and w ij, using multiple monadic and dyadic controls, as discussed later.

The SAOM uses empirical data to calculate observed values for each of the $s_{ih}^{\bf g} $ and $s_{ih}^{\bf r} $ statistics in the respective network and behavior objective functions. To obtain expected values for these statistics, the model simulates network–behavior coevolution using the same choice probabilities as in the ABM, where states adjust ties to maximize the function in equation (9) and adjust defense spending to maximize the function in equation (10). Unlike the ABM, these simulations are fully constrained by the observed data, and the parameter values are estimated rather than determined. A Robbins-Monro Markov-chain Monte Carlo algorithm searches the parameter space and locates vectors of parameter estimates, $\hat{\beta }_h^{\bf g} $ and $\hat{\beta }_h^{\bf r} $, where the expected values of the model statistics, calculated on the simulations, are equal to the observed values. Standard errors are derived using the delta method. Null hypotheses are tested with a standard t-statistic, $t_h = {{{\hat{\beta }}_h} \over {{\rm s}{\rm .e} .( {{\hat{\beta }}_h} ) }}$. See the online supplement for an extensive formal treatment.

We build the DCA network using Kinne's Defense Cooperation Agreement Dataset.Footnote 96 Our analysis employs the “general” category of DCAs, which includes only agreements that institutionalize the full range of countries’ defense cooperative activities, including mutual consultation, training, joint exercises, intelligence sharing, research and development, and arms trade, among others. We obtain similar results if we also include “sector” DCAs (see the online supplement).

We measure country-level defense effort as defense expenditures divided by GDP.Footnote 97 We discretize this metric into eleven categories of defense effort, identical to the ABM. The online supplement explores discretization in greater depth and shows that the results are robust to numerous alternative approaches, such as increasing or decreasing the number of ordinal categories, taking log transformations, and using alternative metrics of defense effort. Data on expenditures come from the Correlates of War national military indicators data set.Footnote 98 GDP data are from the Penn World Table.Footnote 99

In the network equation of the ABM, the terms w ij and c j reflect exogenous dyadic and monadic influences, respectively. To operationalize these terms, we draw on recent work on DCA networks, which shows that exogenous demand for DCAs is determined largely by geography, shared economic and political interests, membership in formal alliances, and economic resources.Footnote 100 We thus include the following controls:

  • distance: the log-transformed geographic distance between i and j's capital cities.Footnote 101

  • alliance (non-nato): a dummy variable that equals 1 if i and j share membership in any alliance other than NATO.Footnote 102

  • nato: a dummy variable that equals 1 if both i and j are NATO member states.

  • unga ideal points: the distance between the ideal-point estimates of i and j's voting records in the UN General Assembly.Footnote 103

  • trade: the total bilateral trade between i and j, log transformed.Footnote 104

  • democracyj: a dummy variable that equals 1 if the potential DCA partner is a democracy.Footnote 105

  • capabilitiesj: the log-transformed Correlates of War Composite Indicator of National Capability score for j.Footnote 106

The defense-spending equation of the ABM includes the exogenous covariate c i. To operationalize this term, we draw on the empirical literature on defense spending, which emphasizes internal demand-side factors like political ideology, supply-side factors like economic growth, and exogenous factors like militarized conflict, alliance membership, and neighborhood effects. The following controls account for these influences:

  • democracy: a dummy variable that equals 1 if the country is democratic.Footnote 107 Because democracies face domestic challenges in diverting resources from “butter” to “guns,” they may be less willing to spend on defense.Footnote 108

  • gdp growth: the country's annual GDP growth rate.Footnote 109 Ceteris paribus, when governments have more revenue to spend, they spend more on defense.Footnote 110

  • alliances (non-nato): the number of alliances, excluding NATO, in which the country holds membership.Footnote 111 This variable accounts for potential spill-ins in formal alliances.Footnote 112

  • nato member: a dummy variable that equals 1 if the country is a full NATO member state in the current year. NATO's unique institutional structure and expenditure guidelines may influence defense spending differently than other alliances.

  • military regime: a dummy variable that equals 1 if the country's government is a military regime.Footnote 113 This variable accounts for militaristic political ideologies, which may be inclined toward high levels of defense spending.Footnote 114

  • mids: the log-transformed count of the number of militarized interstate disputes in which the country participated in the current year.Footnote 115 Countries that frequently engage in conflict may spend more on defense.Footnote 116

  • spatial lag: the defense spending of the country's geographically contiguous neighbors. High levels of spending in neighboring states may increase spending in the focal state.Footnote 117

The online supplement includes robustness checks with additional control variables, such as bilateral loans, alliance-based spill-ins, and alternative operationalizations of UNGA ideal points.

Results and Discussion

We estimated four models. Figure 8 summarizes the results, with estimates and confidence intervals scaled for legibility. (See the online supplement for unscaled estimates.) Model 1 is a baseline model that examines the coevolution of defense spending and DCA partner selection—via defense spendingj in the DCA equation and dca degree in the spending equation—but does not account for higher-order network effects. The estimates from the DCA equation show partial evidence of strategic selection. For example, the positive estimates for capabilitiesj and democracyj indicate that governments prefer militarily capable and democratic defense partners.Footnote 118 The estimate for defense spendingj is statistically indistinguishable from zero. Governments do not appear to select high-spending partners.Footnote 119 The estimates for the remaining variables in the DCA equation are consistent with prior research.Footnote 120

Figure 8. Stochastic actor-oriented model of DCAs and defense spending

In the defense-spending equation of model 1 (lower panel), the estimate for dca degree is positive and weakly significant (10% level). This result is consistent with a bilateral influence effect and supports H1. Interpretation of SAOM estimates is analogous to multinomial logit. Exponentiating the (unscaled) estimates provides odds ratios. Consider a comparison between two hypothetical countries, i and h, equal on all observed dimensions except i has one more DCA tie than h. The unscaled estimate for dca degree (0.017) indicates that the odds of country i increasing its defense effort by one unit are about 1.7 percent greater than the odds of h increasing its defense effort. While this effect is small, large differences in DCA connectivity accumulate quickly. For example, if i has 10 more DCA ties than h, the odds of i increasing its defense effort are 19 percent greater than h doing so. If i has twenty more DCA ties, the odds are over 40 percent greater. The more DCAs a state signs, the greater the pressure to increase defense effort.

Model 2 introduces the main network selection term, transitive triads, along with the endogenous total degree effect. This model estimates the extent to which countries condition their selection of DCA partners on structural features of the DCA network. The estimates for the network terms are large and highly precise, indicating a substantial network selection effect. Based on the estimate for transitive triads, the odds of a given country i selecting a partner j that shares a third party k in common are about 22 percent greater than the odds of i selecting an otherwise disconnected j, all else equal. This effect increases substantially as i and j's number of shared third-party collaborators increases. If i and j have, say, five DCA partners in common, the odds of i forming a DCA with j are nearly 170 percent greater than i's odds of selecting some disconnected partner. This selection effect reflects the attraction of network efficiencies.

Model 3 introduces the dca dense triads term into the defense spending equation, which indicates the effect of network structure on expenditures. The estimate is negative and highly significant. Substantively, adding one additional triangle lowers a country's odds of increasing its defense effort by about 2 percent. Adding ten triangles lowers those odds by 20 percent; adding twenty lowers them by 35 percent. Consistent with H2, dense local networks put downward pressure on defense effort. At the same time, model 3 shows a larger, more precise estimate for dca degree, which translates into a 9 percent increase in the odds of a one-unit increase in defense spending—substantially larger than the 1.7 percent effect found in model 1. The weakly positive estimate for dca degree in models 1 and 2 obscures the divergent effects of DCAs. Incorporating dca dense triads into the model not only reveals the influence of network structure but also permits a more precise estimate of bilateral influence.

Finally, model 4 incorporates the free-riding term, dca triads effort. Contrary to H4, the estimate is statistically insignificant. And the estimate for dca dense triads remains virtually unchanged. We thus find no empirical evidence that dense local networks encourage states to free ride on the efforts of their defense partners. Rather, triangles as such reduce i's defense effort. This finding supports the more general efficiency mechanism, where network influence is a structural feature of states’ respective local DCA networks and is not conditional on partners’ spending. That said, the results here represent only a first step in assessing efficiency versus free riding; additional analyses may produce more nuanced conclusions. We consider extensions of this analysis in the conclusion.

Overall, the empirical results provide a comprehensive picture of how selection–influence dynamics combine with network effects to yield unexpected outcomes. At a purely bilateral level, states prefer capable defense partners. And when they sign agreements with those partners, ceteris paribus, their defense effort increases. Though contrary to standard public-goods logic, these findings are consistent with the logic of reciprocity-based punishment. Incorporating network-level dynamics, however, complicates the picture. States do not merely select capable partners; they also select partners in a way that yields transitive triads. Defense agreements signed within that context generate conflicting incentives. On the one hand, bilateral agreements pressure states to increase their defense effort and avoid punishments. On the other hand, dense local networks generate network efficiencies that lower the costs of defense production, thus encouraging reductions in spending.

To draw out the substantive implications of these findings, we conducted post-estimation analysis, focusing on the defense-spending equation. Because the free-riding term, dca triads effort, showed no significant effect, we use model 3 for all post-estimation analysis. (The results for model 4 are virtually identical.) The key variables, dca degree and dca dense triads, are functions of one another; an accurate assessment of their respective influence requires that we consider both simultaneously. Further, the influence of transitive triads increases nonlinearly with nodal degree.Footnote 121 Thus, the countervailing effect of network structure on defense spending grows more pronounced as a node's local network densifies.

Figure 9 interprets the parameter estimates for dca degree and dca dense triads across a range of hypothetical degree and triangle values, holding all other variables constant. The left panel shows that the effect of DCAs varies sharply with network context. If a country increases its number of DCAs from zero to twenty-five, and those DCAs involve no transitive triads (that is, the ratio of actual to possible triangles in the country's local network is zero), then the odds of an increase in defense spending grow eight-fold, as shown in the top line. By contrast, if those twenty-five DCAs entail a maximal increase in triangles, as in a dense local network, then the odds of an increase in spending shrink to virtually zero.

Figure 9. Interpretation of degree and triangle effects on defense expenditures

The four panels on the right of Figure 9 illustrate the effect of triangles at fixed degrees. For a country with five DCAs, the odds of an increase in defense spending are greater than 1 for all but the very highest triangle ratios. For a country with ten DCAs, a triangle ratio of 0.4 or greater—indicating that 40 percent or more of the possible triangles in that country's local network in fact exist—pushes the odds of an increase in spending below the 1:1 threshold. And for a country with twenty-five DCAs, a triangle ratio of just 15 percent reduces the odds of an increase in spending from nearly 8:1 to 1:1. The more deeply embedded a country is in the DCA network, the more strongly network structure pushes against bilateral influence.

We also conducted counterfactual analysis using real-world cases (Figure 10). We selected twenty-four high-degree nodes in the 2010 network and split them into two groups: (1) those with high levels of transitivity in their respective local networks, and (2) those with low levels of transitivity. Such countries are especially vulnerable to counterfactual increases or decreases in triangles. Using observed values of all variables, we iteratively simulated the coevolution of the DCA network and defense spending, and we calculated the predicted level of spending for both groups. We then counterfactually reduced the triangle ratio for group 1 and increased the triangle ratio for group 2, and we again simulated coevolution and derived predictions.

Figure 10. Counterfactual analysis of selected countries

Removing transitive triads (top) predicts a sharp upward deviation from current spending levels. For example, using observed values, the model predicts little change in Ukraine's defense spending. However, if Ukraine's local network altogether lacks transitive triads, the model predicts a substantial increase in spending. Put differently, Ukraine's level of defense spending—2.3 percent of GDP in 2010, less than most non-NATO former Soviet republics—is sustained, at least in part, by the structure of its defense relations. By contrast, adding transitive triads (bottom) attenuates that upward pressure—and for some states, pushes defense spending in a negative direction. For example, although Sweden's 2010 expenditures were already quite low (1.4% of GDP), the model predicts that increasing the density of its local network would push spending even lower. We emphasize that this counter-factual exercise does not alter states’ numbers of DCAs; it only manipulates the structure of their respective local networks.

Finally, to compare the influence of DCAs on defense spending to the influence of other components of the model, we assess variable importance.Footnote 122 Figure 11 illustrates the results. In the early 1990s, variations in defense spending were best explained by regime type, NATO membership, and alliances. While these influences continue to matter over time, their importance is gradually eclipsed by DCAs. By 2000, dca degree is the most important variable in the model. By 2009, dca degree and dca dense triads together explain nearly as much variation in defense spending as all other variables combined. (See the online supplement for variable importance by country.)

Figure 11. Relative importance of effects


Burden sharing is an enduring collective-action problem. A network approach highlights aspects of burden sharing that, to our knowledge, have not been given extensive attention. Our main finding—that the effect of defense agreements on burden sharing depends on the network structure of those agreements—raises important questions about the overall provision of security. Our empirical results suggest that efficiencies, not free riding, are responsible for the negative effect of network structure on defense effort. Consequently, spending reductions do not necessarily indicate an underprovision of security. Rather, such reductions should be viewed through the wider lens of network context; they may reflect the efficiency gains of densely connected networks. This insight complements the long-standing observation that efficiencies reduce aggregate spending even as overall provision of security remains optimal.Footnote 123

This research is a first step toward merging network insights with established principles from the study of burden sharing, public goods, and collective action. While our ABM and empirical model implement a theoretically informed way of operationalizing efficiency and free riding, other approaches should be explored. In particular, further research is needed on whether defense networks improve security overall despite reductions in spending. Counterfactual estimates of optimal defense effort are notoriously difficult, even with well-defined institutions like NATO.Footnote 124 Such difficulties are amplified in a bilateral network context. A more promising approach, which we explore in related work,Footnote 125 is to consider whether dense local networks affect participation in bilateral defense actions, such as joint military exercises, arms trade, and peacekeeping operations. Such an analysis requires extensive dyadic data and a carefully specified causal-inference design. If the evidence indicates that defense activities increase among densely connected partners even as aggregate spending declines, states may indeed be producing more security at lower cost.

More broadly, our findings dovetail with recent work showing significant post–Cold War and post-9/11 shifts in how countries cooperate on defense, with a greater emphasis on nontraditional security threats and substantive defense activities.Footnote 126 Network influences are central to these trends.Footnote 127 While scholars have also examined network influences in traditional alliances,Footnote 128 the applicability of such studies to present-day security questions is limited by the relative lack of change in the global alliance network since the early 1990s. In a complex global environment, where interstate threats coexist alongside myriad nontraditional threats, efficient coordination of defense policies is an increasingly essential means of achieving security. Taken as a whole, our results suggest that bilateral agreements, when embedded within dense networks of aligned collaborators, may be a viable strategy for achieving optimal security gains.

Data Availability Statement

Replication files for this article may be found at <>.

Supplementary Material

Supplementary material for this article is available at <>.


This research was supported by the Gerda Henkel Foundation (AZ 08/KF/18), and by the Minerva Research Initiative (W911NF-19-1-0332). The opinions herein are the authors’ own and not those of the Department of Defense or the Army Research Office. For helpful comments, we thank Michael Allen, Timea Balogh, Ryan Brutger, Kai-Yin Lin, Carmela Lutmar, Zeev Maoz, Aila Matanock, Michaela Mattes, Marika Miner, Stephen Quackenbush, Mark Reedy, members of the UC Davis SPINS lab, participants in the UC Berkeley MIRTH workshop, and audiences at the 2021 annual conferences of the American Political Science Association and the International Studies Association.


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2. “A Transcript of Donald Trump's Meeting with the Washington Post Editorial Board,” Washington Post, 21 March 2016.

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7. Kinne Reference Kinne2020, 730.

8. Digiuseppe and Poast Reference Digiuseppe and Poast2018.

9. Olson and Zeckhauser Reference Olson and Zeckhauser1966; Sandler and Hartley Reference Sandler and Hartley2001.

11. Olson and Zeckhauser Reference Olson and Zeckhauser1966.

12. Cornes and Sandler Reference Cornes and Sandler1984.

14. Crescenzi et al. Reference Crescenzi, Kathman, Kleinberg and Wood2012; Leeds Reference Leeds2003; Mattes Reference Mattes2012. According to data from Leeds et al. Reference Leeds, Ritter, Mitchell and Long2002, less than 15 percent of alliances require peacetime contact between member states.

15. Kinne Reference Kinne2018, 803.

17. Lai and Reiter Reference Lai and Reiter2000; Simon and Gartzke Reference Simon and Gartzke1996.

18. Digiuseppe and Poast Reference Digiuseppe and Poast2018, 997.

21. Olson and Zeckhauser Reference Olson and Zeckhauser1966.

22. E.g., DiGiuseppe Reference DiGiuseppe2015; Fuhrmann Reference Fuhrmann2020; Nordhaus, Oneal, and Russett Reference Nordhaus, Oneal and Russett2012; Whitten and Williams Reference Whitten and Williams2011.

23. Olson Reference Olson1965, 35.

24. Olson and Zeckhauser Reference Olson and Zeckhauser1966, 268.

25. Kahler Reference Kahler1992, 683.

26. Thompson and Verdier Reference Thompson and Verdier2014.

32. Agreement on Defense Industry Cooperation between the Government of the Republic of Indonesia and the Government of the Republic of Turkey, 29 June 2010, Article III.

33. “Indonesia Revokes Defense Pact with Australia,” Wall Street Journal, 17 September 1999.

35. Holland and Leinhardt Reference Holland and Leinhardt1971.

38. Sandler and Hartley Reference Sandler and Hartley2001, 872.

39. Sandler and Forbes Reference Sandler and Forbes1980, 428.

40. Compare Conybeare Reference Conybeare1994, 415–17.

42. Cranmer, Desmarais, and Kirkland Reference Cranmer, Desmarais and Kirkland2012.

44. Pannier and Schmitt Reference Pannier and Schmitt2014, 271.

47. Drezner Reference Drezner2005, 846.

49. Wicker Reference Wicker2016, 6.

50. Wilkins Reference Wilkins2015, 102–03.

51. Blair and Hanley Reference Blair and Hanley2001, 9–11.

52. Asikainen et al. Reference Asikainen, Iñiguez, Ureña-Carrión, Kaski and Kivelä2020; Kossinets and Watts Reference Kossinets and Watts2009. This effect has also been observed in socioeconomic networks like environmental standards (Loconto Reference Loconto2017) and trade regulations (Corning, Reference Corning2020).

53. Satake Reference Satake2011, 19.

55. Wuthnow Reference Wuthnow2019, 136–37.

56. Sandler and Hartley Reference Sandler and Hartley2001, 872.

57. Katz and Shapiro Reference Katz and Shapiro1994.

58. Kahler Reference Kahler1992, 683.

59. Rodrik Reference Rodrik2000, 182–83.

60. Farrell and Newman Reference Farrell and Newman2019.

61. Kahler Reference Kahler1992, 683.

62. Jackson and Wolinsky Reference Jackson and Wolinsky1996.

63. Olson and Zeckhauser Reference Olson and Zeckhauser1966, 268.

64. See Snijders Reference Snijders2001; Steglich, Snijders, and Pearson Reference Steglich, Snijders and Pearson2010. While there is a growing literature on network games (Jackson and Zenou Reference Jackson, Zenou, Peyton Young and Zamir2015), the bulk of this work addresses fixed networks, with network structure as an exogenous, static influence (Bramoullé and Kranton Reference Bramoullé, Kranton, Bramoullé, Galeotti and Rogers2016). The few existing game-theoretic models of network–behavior coevolution are exploratory and require highly restrictive assumptions about network formation to arrive at closed-form solutions. For example, Canen, Jackson, and Trebbi Reference Canen, Jackson and Trebbi2022 assume that network ties form according to a “random matching protocol,” while Badev Reference Badev2021 assumes a process of chance encounters among a predetermined subset of nodes. Further, network games predict a large number of equilibria (Galeotti et al. Reference Galeotti, Goyal, Jackson, Vega-Redondo and Yariv2010, 219), many of them trivial or unrealistic, and are not amenable to empirical analysis (Badev Reference Badev2021, 1182).

66. Note that because DCA networks are nondirected, target j of i's network tie must confirm the tie in order for it to be created, based on a choice probability similar to that in equation (1). See the online supplement.

67. Snijders and Steglich Reference Snijders and Steglich2015.

68. Butts Reference Butts2017, 48.

69. Pumpuni-Lenss, Blackburn, and Garstenauer Reference Pumpuni-Lenss, Blackburn and Garstenauer2017.

70. De Marchi and Page Reference De Marchi and Page2014, 10–11.

71. The data sources are the same as those used in the empirical analysis. The choice of year is inconsequential.

72. Niezink, Snijders, and van Duijn Reference Niezink, Snijders and van Duijn2019, 296.

73. Niezink Reference Niezink2018, chapter 6.

75. Sandler and Hartley Reference Sandler and Hartley2001, 873.

76. Sandler Reference Sandler1993, 453.

77. Conybeare, Murdoch, and Sandler Reference Conybeare, Murdoch and Sandler1994.

78. Olson Reference Olson1965, 35.

79. Snijders Reference Snijders2001, 373.

80. Sandler Reference Sandler1993, 455.

84. Barabási and Albert Reference Barabási and Albert1999.

87. Note that in this model the quantity a ij in the τa ij term is defined solely in terms of target j's nodal degree, which better represents the process of preferential attachment. See Barabási and Albert Reference Barabási and Albert1999.

88. Jackson and Wolinsky Reference Jackson and Wolinsky1996.

89. Sandler Reference Sandler1993, 451.

90. Snijders Reference Snijders2001; Steglich, Snijders, and Pearson Reference Steglich, Snijders and Pearson2010.

91. Snijders and Steglich Reference Snijders and Steglich2015.

95. We also include the square of Constant in case of nonlinearities.

97. Hartley and Sandler Reference Hartley and Sandler1999, 674.

99. Feenstra, Inklaar, and Timmer Reference Feenstra, Inklaar and Timmer2015.

101. Weidmann, Kuse, and Gleditsch Reference Weidmann, Kuse and Gleditsch2010. Note that for all logged variables, we add 1 before transforming.

103. Bailey, Strezhnev, and Voeten Reference Bailey, Strezhnev and Voeten2017.

104. Barbieri and Keshk Reference Barbieri and Keshk2016; Barbieri, Keshk, and Pollins Reference Barbieri, Keshk and Pollins2009.

105. Boix, Miller, and Rosato Reference Boix, Miller and Rosato2012.

107. Boix, Miller, and Rosato Reference Boix, Miller and Rosato2012.

108. Fordham and Walker Reference Fordham and Walker2005.

109. Feenstra, Inklaar, and Timmer Reference Feenstra, Inklaar and Timmer2015.

110. DiGiuseppe Reference DiGiuseppe2015; Whitten and Williams Reference Whitten and Williams2011.

112. Sandler and Hartley Reference Sandler and Hartley2001, 873.

113. Geddes, Wright, and Frantz Reference Geddes, Wright and Frantz2014.

114. Studies of defense spending often control for liberal/conservative government ideologies (e.g., Whitten and Williams Reference Whitten and Williams2011). Such metrics exclude autocratic regimes, which results in unacceptable levels of missing data.

116. Nordhaus, Oneal, and Russett Reference Nordhaus, Oneal and Russett2012.

117. Yesilyurt and Elhorst Reference Yesilyurt and Elhorst2017.

118. Digiuseppe and Poast Reference Digiuseppe and Poast2018.

119. However, when using absolute defense spending in lieu of the spending/GDP ratio, the estimate for defense spendingj is strongly positive. See the online supplement.

121. For a node with degree d, the maximum number of triangles in its local network is d × (d − 1).

122. Indlekofer and Brandes Reference Indlekofer and Brandes2013.

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Figure 0

Figure 1. Trends in defense cooperation agreements (DCAs) and alliances, 1990–2010

Figure 1

Figure 2. Two types of triadic structures

Figure 2

Figure 3. DCA network topology at two time points

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Figure 4. Sparse versus dense local networks

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Table 1. Summary of terms in the agent-based model

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Figure 5. Equilibrium outcomes in the network–behavior agent-based model

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Figure 6. Agent-based model with coevolution and dense local networks

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Figure 7. Effect of efficiency and free riding on defense effort

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Figure 8. Stochastic actor-oriented model of DCAs and defense spending

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Figure 9. Interpretation of degree and triangle effects on defense expenditures

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Figure 10. Counterfactual analysis of selected countries

Figure 11

Figure 11. Relative importance of effects

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