Measuring Claim Initiation

The dependent variable captures the initiation of a territorial claim by one state against another. We use data from the Issue Correlates of War (ICOW) Project, which covers territorial claims from 1816 to 2001.Footnote ²⁹ We exclude offshore island claims as well as claims over non-homeland or dependent territory (for example, colonial possessions).Footnote ³⁰ For each directed dyad-year, the dependent variable equals 1 in the year that the challenger initiates a claim against the target, 0 in any year with no claim initiation, and missing in the event that a claim is ongoing in a given year.Footnote ³¹ Claim onsets are relatively rare, occurring in approximately 1 percent of directed dyad-years.

Identifying Potential Claims

To determine whether states are vulnerable to a breakdown of the territorial status quo, we construct several measures of potential claims. A potential claim exists when a state (the potential challenger) perceives something of value on the territory controlled by a neighbor (the potential target) that has not yet been the subject of an actual claim. By identifying potential claims in the system, we can derive, for each state, two measures that capture vulnerability to the activation of potential claims: (1) a state is vulnerable to potential claims when it is the potential target of one or more neighbors, and (2) a state is more vulnerable to territorial instability in its neighborhood the more potential claims exist in its vicinity.

Identifying potential claims is not straightforward, since what makes a territory valuable can vary with historical, demographic, and geographic conditions.Footnote ³² Historical and demographic conditions gave rise to different dynamics in different regions. For example, in the European and African contexts, definitions of homeland were often connected to ethnic identity, but this consideration is largely absent in Latin America, where homeland claims were generally rooted in (unclear) colonial borders. Moreover, if states seek territorial expansion for the sake of power and profit, they could, in principle, have incentives to make claims on all of their neighbors, simply waiting for the right conditions to arise. To deal with this, we capture states’ latent incentives to make claims in several different ways.

Shared borders. Our first measure of a potential claim is the presence of a shared border. This indicator assumes that every border presents both an opportunity for expansion and a source of vulnerability that increases with the number of neighboring states.Footnote ³³ A nice feature of this measure is that it is easy to observe and available for the entire time period of our data, and the number of countries that a state borders is plausibly exogenous to local dispute initiation. A significant downside is that research on the origins of territorial disputes suggests that not all borders are equally likely to be contested, suggesting that we need more refined indicators of potential claims.

Transborder ethnic kin. Prior work emphasizes that territorial claims are particularly likely when politically powerful groups in one state have kin residing in neighboring states.Footnote ³⁴ We use data from the Transborder Ethnic Kin (TEK) Dataset and the Ethnic Power Relations (EPR) data sets to capture this condition.Footnote ³⁵ We say that there is a potential claim based on TEK if a group holding political power in one state also resides in the neighbor, where political power means being at least a senior partner in government.Footnote ³⁶

Although this variable has the advantage of capturing an important motive for territorial disputes, the TEK data are available starting in only 1946, which limits the temporal span of tests based on these data. For Europe, however, we can construct a similar indicator using the Historical Ethnic Group (HEG) data, which go back to 1886.Footnote ³⁷ Tests performed on European subsamples use the HEG data.

Lost territory. The third proxy for potential claims is an indicator for whether a state previously lost territory to another. Territorial claims are often based on previous or historical ownership of a piece of territory.Footnote ³⁸ In some cases, lost territory is associated with the partitioning of ethnic kin who lived there. In other cases, historical control may be associated with sites that are important for national or religious reasons. Moreover, political leaders can often mobilize support around the perceived injustice of the loss or the urge to restore an imagined “golden age.”Footnote ³⁹

To code lost territories, we use the COW Territorial Change Dataset (TCD, Version 6), which compiles global territorial transfers since 1816.Footnote ⁴⁰ Due to limitations in these data, however, we supplement them with information from other sources. The process for coding territorial losses is described at greater length in Appendix A. The result is a variable indicating, for each directed dyad-year, whether or not the potential target controlled territory that previously belonged to the potential challenger. This variable exists for the entire time span of the data, and not only for the post-1945 period.

A concern, however, is that losses are endogenous to territorial disputes: they are often the result of a previous claim. Losses could resolve rather than generate potential claims. Since there is no a priori way to identify losses that leave both sides satisfied and those that generate new grievance, we address this concern by coding only losses that arose before or at the birth of the dyad, generally because the territory was transferred during colonial rule or one state in the dyad seceded from the other. We also restrict the sample to observations up to the first dispute in the dyad, as these inherited losses are unlikely to affect subsequent disputes.Footnote ⁴¹

Loss or TEK. Since territorial disputes might be motivated by either (or both) a historical loss or transborder kin, we also create an indicator that captures the presence of either source of potential claim. Although this combination captures different motives for potential claims, this variable inherits the constrained time frame of the TEK data and the endogeneity concern with the loss-based indicator, restricting tests to the post-1945 period (post-1886 for Europe) and observations that precede the first claim in a dyad. It is most useful, then, for newly decolonized regions.

As noted previously, the definition of potential claims excludes those that have already been activated and, potentially, resolved. Thus for example, if a state has TEK in a neighbor but has already made a territorial claim on that neighbor, the potential claim has been activated. Pandora’s box logic implies that the deterrent effect, if any, should come from potential claims that have not yet given rise to an actual claim. Thus when using the TEK- and/or loss-based indicators, we identify a potential claim only if there has been no previous claim in the directed dyad during the spell of time in which the TEK or loss existed.Footnote ⁴² However, when we measure potential claims with the number of shared borders, we include all of a state’s borders in order to preserve the strict exogeneity of that measure.

Later, and in Appendix B, we present evidence showing that these indicators of potential claims do a good job of predicting actual claims in most regions. The presence of TEK and/or lost territory in the target state is strongly and positively correlated with claim initiation, nearly doubling the risk (or tripling, in the case of the combined indicator). This evidence gives us some confidence that states that host TEK and/or lost territory are vulnerable to a potential claim. These variables fare poorly, however, in the Americas, where ethnic politics were less salient and boundary disputes were often rooted in ambiguities in colonial-era borders.Footnote ⁴³ Because this region has a very different historical experience, we will develop an alternative indicator for potential claims, using data compiled by Lee to identify areas that had elevated risk of claims due to changes in administrative borders or unclear jurisdiction prior to independence.Footnote ⁴⁴ We defer the treatment of the region to Appendix E and, in what follows, omit the Americas from tests that use the TEK and/or loss indicators.

Measuring Vulnerability to Territorial Revisionism

Once we have identified potential claims, we generate several measures of vulnerability to a breakdown of territorial order. For each potential challenger $i$ facing potential target $j$ in year $t$ , let ${K_t}$ denote the set of third-party states that have a potential claim on $i$ . Our first measure of vulnerability is simply the count of states in ${K_t}$ . So, for example, when using shared borders as an indicator of potential claims, this variable equals the number of states bordering $i$ (excluding $j$ ).

Because vulnerability depends on both the incentives and capabilities of third parties to claim territory, we also construct measures of vulnerability that weight potential claims by the relative power of each potential claimant. In particular, for each state $k \in {K_t}$ , we calculate its capabilities relative to $i$ ’s as ${p_{kit}} = {{Ca{p_{kt}}} \over {Ca{p_{kt}} + Ca{p_{it}}}}$ , where $Cap$ is the COW composite index of capabilities.Footnote ⁴⁵ We then sum the ${p_{kit}}$ across all states in ${K_t}$ .

We measure vulnerability to the negative externality from instability in the region as the number of potential claims in a state’s neighborhood, where the neighborhood around state $i$ is the set of dyads $kl$ such that both $k$ and $l$ border on $i$ . In principle, the relevant neighborhood might be larger than this, but we suspect that these dyads would have the greatest effect on $i$ ’s decisions.

Model Specification

In each model, the key independent variable is a measure of vulnerability to potential territorial claims. This measure captures the deterrent effect of potential claims by third parties. But we also control for and estimate the direct effect of potential claims on claim initiation: if state $i$ has a potential claim on state $j$ then $i$ should have greater incentive to claim $j$ ’s territory, independent of the potential claims it must contend with from third parties. There could also be a reciprocal effect of potential claims by the target $j$ against the challenger $i$ . It is not uncommon for states to engage in reciprocal disputes, each claiming part of the other, and we control for the presence of these “reverse” potential claims as well. We also control for the active claims that third parties have already initiated against a challenger state.

All models control for time dynamics arising from the fact that territorial claims tend to arise early in the life of a dyad, often soon after independence, and become much less likely over time. To capture this flexibly, we include a cubic polynomial of the age of the border.Footnote ⁴⁶ All models also control for several variables that have been shown to effect the incidence of territorial claims:

• The challenger’s share of dyadic capabilities to capture the direct effects of relative power on claim initiation. For this measure, as with the $p$ terms in the weighted measures, we used the COW national capabilities scores.Footnote ⁴⁷

• The challenger’s level of industrialization, proxied by its energy consumption per capita,Footnote ⁴⁸ to capture the argument by Markowitz, Fariss, and McMahon that states with more industrialized economies have fewer incentives to engage in territorial conquest.Footnote ⁴⁹

• Indicators for regime type based on the Polity and V-Dem data sets.Footnote ⁵⁰ We define a country as democratic if its liberal democracy score is at least 0.5 (on a 0 to 1 scale) as defined by V-Dem; where V-Dem scores are unavailable, we impute them using Polity data. Models include controls for both challenger and target values of each of these variables as well as for whether both members of a dyad are democratic.

To facilitate coefficient interpretation, all estimates are obtained using linear probability models and the dependent variable is multiplied by 100, so the effects can be interpreted as percentage point changes in the probability of claim initiation. Models do not include cross-section fixed effects since most variables of interest vary little with time. In all specifications, we use cluster-robust standard errors for dyadic data as proposed in Aronow, Samii, and Assenova.Footnote ⁵¹

Results

We present the results at two levels of analysis. First, data aggregated at the regional level allow an initial assessment of whether the regional variation in dispute frequency might be explained by regional variation in our measures of vulnerability. Then, we turn to regression analysis of the directed-dyadic data to determine whether vulnerability to challenges from third parties is associated with a lower likelihood of claim initiation.

Regional patterns. Do regions with more vulnerable states experience fewer claims? Figure 1 documents the variation in the dependent variable, showing the proportion of dyadic borders that were ever disputed in each region.Footnote ⁵² For Europe, we also estimate effects in pre- and post-World War II samples, splitting the time period after the 1947 peace settlement that resolved most outstanding claims from that conflict. African borders are the least likely to have experienced a dispute, consistent with conventional wisdom. Europe also had a low rate of territorial disputes, but this is driven by the very low dispute rate after 1947. At the other end of the scale, the Americas are the most dispute-prone region, with over 80 percent of borders experiencing at least one claim, followed by the Middle East and Northern Africa (MENA) and then Asia.

Figure 1. Claim frequency by region

Figure 2 summarizes the mean vulnerability score across directed dyads in each region. We focus on three measures of vulnerability: the number of third-party states, the number of third parties with a potential claim based on TEK or loss, and the number of dyads with a potential claim based on TEK or loss in the challenger’s neighborhood. For the first two, we show the counts weighted by relative capabilities.

Figure 2. Mean levels of vulnerability by region

Comparing these figures reveals some observations that are consistent with the expectation that high vulnerability is associated with low dispute rates, but also some that are not. Africa generally scores high on these measures, consistent with its low rate of territorial revisionism. The Americas have the lowest rate of vulnerability measured by shared borders, consistent with its high rate of disputes. Post-1947 Europe also has high levels of vulnerability, but those levels are essentially unchanged from the pre-1947 period. Thus the marked drop in European dispute rates after the World War II settlement cannot be explained by a change in vulnerability to territorial threats alone. The MENA region, which has the second highest rate of disputes, also scores high in terms of vulnerability to third parties with potential claims.

Directed-dyad level patterns. Figure 3 displays the results of a series of regressions on the directed-dyad data using different indicators of potential claims and different regional subsamples.Footnote ⁵³ For each indicator, the figure reports three effects: the direct effect of a potential claim by the challenger against the target (gray circles), the estimated deterrent effect using the raw count of third parties with a potential claim on the challenger (black squares), and the estimated deterrent effect using the weighted measure that takes the relative capabilities of third parties into account (black diamonds).Footnote ⁵⁴ If a given indicator is a good predictor of claim initiation, then the coefficient in gray will be positive; if vulnerability to claims by third parties has a deterrent effect, then coefficients shown in black will be negative.

Figure 3. Effect of vulnerability to potential claims on claim initiation

In the full sample, there is evidence consistent with this pattern. The indicators of potential claims all have a positive direct effect, confirming that they are valid predictors of claim risk, and the estimated deterrent effects are generally negative, particularly when using the weighted measures.Footnote ⁵⁵ The evidence that vulnerability deters claims is strongest when using the weighted count of third-party neighbors and the weighted indicator that captures potential third-party claims due to TEK or historical loss.

The figure suggests, however, that the pooled results mask significant regional heterogeneity. The pattern of results is most consistent in the sub-Saharan Africa subsample. Every measure of potential claims has positive direct effects and negative deterrent effects. The estimates suggest that each additional neighbor is associated with a 0.0016 decrease in the probability of a challenge, or around 25 percent of the base rate of claim initiation in the region (0.006). The coefficient on the weighted TEK indicator means that adding a third-party claimant with the same capabilities as the challenger (that is, $p = 0.5$ ) is associated with a 0.0021 reduction in the probability of a challenge, or one-third of the base rate. These results are robust to the exclusion of any individual African country.

We also see some evidence of deterrent effects in Asia, though neither as strong nor consistent. The indicators of potential claims have positive direct effects, but they are only significantly different from zero at the 10 percent level, and deterrent effects are evident only when using the weighted measures. This latter result is due to China, which, on the basis of raw counts, appears to be highly vulnerable to third-party claims due to its large number of neighbors. Since China initiated disputes against almost all of its neighbors, including it in the sample with the unweighted measure can obscure the average deterrent effect elsewhere in the region. Once we weight potential third-party claims to take into account relative power, however, China appears much less vulnerable. The most robust effect in Asia, in terms of both significance and sensitivity to exclusion of individual countries, is that of shared borders: an additional neighbor with capabilities equal to those of the challenger reduces the predicted probability of a challenge by 0.0045, or about 25 percent the base rate of claim initiations.

By contrast, there is no evidence of deterrent third-party effects in either MENA or Europe, even in the post-1947 time period that is associated with a stabilization of borders. In MENA, the only indicator of potential claims that consistently predicts actual claims is the combined indicator; however, there is no evidence that vulnerability to third-party claims using this indicator had a deterrent effect.Footnote ⁵⁶

In Europe, there is no evidence of any deterrent effects associated with vulnerability to third-party claims; if anything, estimated effects of third-party claims are positive. This is true even in the post-1947 period, which generally had a low level of territorial conflict. This suggests that the relative stability of this period did not emerge from within-region variation in vulnerability to territorial claims. Instead, the post-1947 results are what would be expected if the region as a whole experienced systemic pressure that muted the effect of local conditions on individual states—that is, pressure from the two superpowers during the Cold War to keep a lid on historic grievances and ethnic tensions.

In Appendix E, we show that vulnerability was not associated with mutual restraint in South America, where territorial claims were driven less by TEK and historical losses than by ambiguities in colonial-era boundaries.

Finally, we ask whether states are deterred from initiating claims by their vulnerability to negative externalities from potential disputes nearby. To do so, we add to the previous models the indicator for the number of potential claims in neighboring dyads, defined as dyads in which both states border on the potential challenger.

Figure 4 reports the estimated effects of potential claims in the neighborhood using the different indicators and regional subsamples.Footnote ⁵⁷ As the figure shows, most models show a positive association between this variable and the probability of a challenge. This effect is largely driven by MENA and pre-1947 Europe.Footnote ⁵⁸ These results cast doubt on the idea that states are deterred from making challenges by the risk of setting off instability among their neighbors.

Figure 4. Effect of potential claims in the neighborhood on claim initiation

Collectively, this descriptive evidence presents a puzzle. Despite the intuitive appeal of Pandora’s box logic, vulnerability to territorial revisionism has an inconsistent relationship with claim-making behavior both at the regional and directed-dyadic levels. We observe regions where vulnerability was associated with a reduced risk of dispute initiation (Africa and Asia) as well as regions in which widespread vulnerability was associated with widespread conflict (MENA, South America, pre-1947 Europe). And in Europe, we see a major decrease in territorial disputes after World War II despite little change in the (high) levels of vulnerability. What might explain these patterns?

A Two-State Model of Territorial Conflict

A two-state model of territorial conflict begins by depicting the two states’ capitals as the endpoints of a line that represents the territory between them. Any point on the line represents a possible border dividing the territory into shares possessed by each state. In particular, for any dyad consisting of states $i$ and $j$ , a border ${x_{ij}} \in \left[ {0,1} \right]$ divides the territory into shares ${x_{ij}}$ and ${x_{ji}} = 1 - {x_{ij}}$ .

Taking inspiration from Caselli, Morelli, and Rohner,Footnote ⁶⁴ we assume that the value of the territory varies along its length as a function of demographic, historical, cultural, or economic endowments at any given location. Formally, let ${\nu_{ij}}\left( {{x_{ij}}} \right)$ denote the value that $i$ gets from the territory located between its capital and a border at ${x_{ij}}$ . We assume that this function is continuous, non-negative, and non-decreasing.Footnote ⁶⁵ Thus each state prefers more territory to less, but the marginal value of additional territory can vary.

The game is infinitely repeated. Let ${q_{ij}} = 1 - {q_{ji}}$ denote the status quo border at the start of the game. This is the border the states are “born” with and reflects some unmodeled history that determined the distribution of territory at the dyad’s independence. Thereafter, let ${q_{ijt}}$ denote the distribution of territory at the end of each period $t \ge 1$ .

In each period, the states decide simultaneously whether or not to contest their border. For simplicity, we black box the process of a border dispute and assume that its outcome can be described by two sets of parameters. First, let ${p_{ij}} = 1 - {p_{ji}}$ denote the division of territory after a dispute. Second, let ${c_{ij}}$ and ${c_{ji}}$ denote the expected costs imposed on both states as a result of the dispute. For mathematical convenience, we assume that the costs effectively reduce the value of the territory for all time, rather than being paid one time in the period of the contest.

Although these assumptions resemble those made in bargaining models that treat war as a costly lottery, we adopt a more general interpretation of a “contest” here. The decision to challenge a border causes the game to enter an unmodeled subgame that generates the new border through a variety of possible mechanisms, some peaceful, some not. All we assume is that there are costs, in expectation, associated with territorial disputes, which can include costs of fighting but which may also include costs of military preparation, disruptions in economic relations, diplomatic costs, costs associated with ill-will from the target state, and so on.Footnote ⁶⁶ When states annex neighbors’ territory, they must also pay costs associated with governance and with fortification against repossession. Each state effectively has an option in each period to pay this cost to shift the border.

If a border is contested in period $t$ , a new distribution of territory is created at ${q_{ijt}} = {p_{ij}}$ . Otherwise, the previous status quo is maintained, or ${q_{ijt}} = {q_{ij,t - 1}}$ . The states’ payoff at the end of each period equals the value of the territory they possess, that is, ${\nu_{ij}}\left( {{q_{ijt}}} \right)$ , minus any costs they have incurred from being involved in a contest, either as initiator or target. The states discount future payoffs by a common discount factor $\delta \in \left[ {0,1} \right)$ .

Several results follow from these assumptions. First, once the border has been contested, the new status quo will reflect the division that can be obtained via a contest, ${p_{ij}}$ . That parameter remains constant over time, so there can be no benefit to either state from contesting the border a second time. Thus once contested, the border will remain at ${p_{ij}}$ forever.Footnote ⁶⁷

Given this observation, in any period $t$ , the current expected value to state $i$ of contesting its border with $j$ is ${{{\nu_{ij}}\left( {{p_{ij}}} \right) - {c_{ij}}} \over {1 - \delta }}$ . Its current expected value from maintaining the status quo inherited from the previous period is ${{{\nu_{ij}}\left( {{q_{ij,t - 1}}} \right)} \over {1 - \delta }}$ . Thus state $i$ can profitably contest the status quo border with state $j$ if

(1) \begin{align}{\nu_{ij}}\left( {{p_{ij}}} \right) - {\nu_{ij}}\left( {{q_{ij,t - l}}} \right) \gt {c_{ij}} \end{align}

—in other words, if the value of additional territory acquired via a contest exceeds the costs. This expression depends on both $i$ ’s bargaining power, ${p_{ij}}$ , and the value of the territory that sits on the other side of the status quo border (including, for example, whether it is home to ethnic kin).

Following Powell,Footnote ⁶⁸ we will say that, if condition 1 holds, then state $i$ is “dissatisfied” with its border with $j$ . We can also say that $i$ has a potential challenge on $j$ or that $j$ is a potential target or is vulnerable to a claim. Notice that a necessary condition for $i$ to be dissatisfied with $j$ is that ${p_{ij}} \gt {q_{ij,t - 1}}$ . Because ${p_{ji}} = 1 - {p_{ij}}$ and ${q_{ji,t - 1}} = 1 - {q_{ij,t - 1}}$ , it is not possible for both states to be dissatisfied simultaneously.

It follows that, in the two-state version of the game, the unique subgame perfect equilibrium is for a dissatisfied state, if one exists, to contest its border in the first period. Lacking any external factors, there is no basis for restraint. Territorial stability is only possible if neither state is dissatisfied.

The Multistate Model

We now construct a multistate generalization drawing on concepts and notations from graph theory. The system of states is represented by a network whose nodes correspond to the states’ capitals. The edges of this network represent the territory between each state that shares a border, whose length we normalize to 1. Formally, let $N = \left\{ {1, \ldots ,n} \right\}$ denote a set of states with $n \gt 2$ , and let $b$ denote a $n \times n$ matrix of borders such that ${b_{ij}} = 1$ if states $i$ and $j$ share a border. Let ${B_i}\left( {N,b} \right) = \left\{ {j:{b_{ij}} = 1} \right\}$ denote the set of states that share a border with state $i$ . A border consists of a division of each edge into territory belonging to each state. As before, assume that, at the outset of each period $t$ , state $i$ ’s share of its territory with each state $j \in {B_i}\left( {N,b} \right)$ is given by ${q_{ij,t - 1}}$ . This game also inherits the assumptions and notation from the two-state version regarding how territory is valued and how the outcomes of contests are determined.

To build intuition, we will refer at times to the simple network illustrated in Figure 5, which shows three states, $A$ , $B$ , and $C$ , whose capitals are depicted as the vertices of the triangle; the territories between them are edges, with the status quo distributions labeled.

Figure 5. A three-state model of territorial conflict

To capture the pattern of potential claims in the network, let ${g_t}$ denote, for each period $t$ , an $n \times n$ matrix of borders such that ${g_{ijt}} = 1$ if state $i$ is dissatisfied with its border with $j$ in period $t$ . Note that while $b$ describes an undirected network (since $j$ borders $i$ if $i$ borders $j$ ), the network described by each ${g_t}$ is directed, since only one state in a dyad can be dissatisfied. Let ${C_{it}}\left( {N,{g_t}} \right) = \left\{ {j:{g_{ijt}} = 1} \right\}$ denote the set of neighbors for which state $i$ is a potential challenger in period $t$ , and let ${T_{it}}\left( {N,{g_t}} \right) = \left\{ {j:{g_{jit}} = 1} \right\}$ denote the set of neighbors for which state $i$ is a potential target.

As a final preliminary, notice that, in the single-period version of this game, the Nash equilibrium is for every dissatisfied state to press its potential claim(s). Because each such state strictly benefits from making a challenge, there is no incentive to engage in restraint. This means that, in the iterated version of the game, there always exists an equilibrium in which every border that can be profitably challenged is challenged.

As before, territorial stability will arise trivially if no states are dissatisfied. The interesting question is whether stability can also arise as an equilibrium in the presence of one or more dissatisfied states. Notice that, because contests redistribute territory, an equilibrium that preserves the status quo is Pareto optimal. This is true even if the initial endowments are widely regarded as unfair and leave all or most states dissatisfied.Footnote ⁶⁹

A Pandora’s Box Equilibrium

We seek an equilibrium that captures Pandora’s box logic: each state refrains from acting on its territorial ambitions for fear that doing so will trigger a cascade of further claims. Formally, this equilibrium takes the following form:

1. 1. In each period $t$ , no state contests its borders as long as no state has contested its borders in a previous period.

2. 2. If at least one state contests its borders in period $t$ , then all dissatisfied states contest their borders in period $t + 1$ .

We can think of the equilibrium as instantiating a norm of mutual restraint that is upheld by the expectation that everything unravels if the norm is violated. Much like the “grim trigger” strategy in the repeated Prisoners’ Dilemma, the equilibrium strategies include a history-dependent punishment regime that kicks in after the first deviation; however, the punishment entails only one period of mass defection, after which new, stable borders form.

Under what conditions does this equilibrium exist? First, note that behavior off the equilibrium path (2) follows immediately from the fact that every state contesting the borders with which it is dissatisfied is the Nash equilibrium of the single-shot game. Once every state expects restraint to unravel, the expectation is self-fulfilling. Under the assumption that every potential claim will be made, there is no longer any benefit from restraint.

On the equilibrium path, the distribution of territory never changes, so ${q_{ijt}} = {q_{ij}}$ for all $t$ . Thus for every state $i$ , the current discounted value of not contesting any borders—under the assumption that others will similarly refrain—is

\begin{align}E{U_i}\left( {\sim Challenge} \right) = {1 \over {1 - \delta }}\mathop \sum \limits_{j \in {B_i}\left( {N,b} \right)} {\nu_{ij}}\left( {{q_{ij}}} \right) \end{align}

What is the payoff from deviating from the equilibrium? Since any challenge by $i$ will trigger all potential claims against it, the most profitable deviation is to make all potential challenges at once. The expected value of this deviation isFootnote ⁷⁰

The terms outside the brackets capture the value of $i$ ’s challenges in the current period plus the value of the status quo on borders that $i$ will not challenge. The terms inside the brackets capture the future stream of payoffs starting in the next period, when all of the potential challenges on $i$ are triggered. It follows that state $i$ will not deviate if

(2) \begin{align}\delta \ge \delta _i^{\rm{*}} = {{\mathop \sum_{j \in ^{C_i}\left( {N,g} \right)} \left[ {{\nu_{ij}}\left( {{p_{ij}}} \right) - {\nu_{ij}}\left( {{q_{ij}}} \right) - {c_{ij}}} \right]} \over {\mathop \sum \nolimits_{k \in {T_i}\left( {N,g} \right)} \left[ {{\nu_{ik}}\left( {{q_{ik}}} \right) - {\nu_{ik}}\left( {{p_{ik}}} \right) - {c_{ik}}} \right]}} \end{align}

This expression has a natural interpretation. The numerator is the sum of all net gains from profitable challenges, and the denominator is the sum of all losses due to challenges by dissatisfied neighbors. The threshold becomes smaller, and thus the condition is easier to meet, the less $i$ expects to gain from revisionism and the more it has to lose from the equilibrium unraveling.

Several results follow directly from this condition. First, expression 2 holds trivially for states that have no profitable challenges, in which case the numerator is 0 (that is, ${C_i}\left( {N,g} \right) = \emptyset $ ). Second, the condition cannot be met for states that have potential challenges but are not a potential target, in which case the denominator is 0 (that is, ${T_i}\left( {N,g} \right) = \emptyset $ ). It follows that, for any state that is a potential challenger, condition 2 can be met only if it is also vulnerable to a challenge by at least one neighbor. Even then the condition may not hold, and there is no guarantee that the threshold is less than one. If the sum of the potential gains for a state are larger than the sum of its potential losses, then the state has no incentive to engage in restraint.

Finally, for this equilibrium to hold, it must be the case that condition 2 holds for every state in the system, or

(3) \begin{align}\delta \ge \mathop {{\rm{max}}}\limits_{i \in N} \left( {\delta _i^{\rm{*}}} \right) \end{align}

Otherwise, at least one state has an incentive to deviate, and the equilibrium cannot be sustained. This condition means that the stability of the system depends on whether the least deterrable state can be deterred by the threat of unraveling. Because states expect a challenge anywhere in the system to lead to universal unraveling, the equilibrium breaks down if any state in the system expects to profit from that outcome.

We can illustrate the nature of this equilibrium by revisiting the three-state version. The foregoing logic suggests that the Pandora’s box equilibrium can hold only if the set of potential claims is as depicted in Figure 6, where arrows indicate the existence and direction of potential claims. This configuration (and the equivalent one in which the claims run clockwise) is the only one in which all potential challengers are also potential targets. In this system, the Pandora’s box equilibrium can hold if all three states satisfy condition 2. One interesting implication is that territorial stability can arise if either none of the states are dissatisfied or all of them are. Starting from the configuration shown in the figure, there would be more territorial conflict if there were one or two fewer dissatisfied states.

Figure 6. A three-state cycle of potential claims

That the stable configuration comprises a cycle of claims is no coincidence. In order for a set of states to all satisfy condition 3, the directed graph $\left( {N,g} \right)$ must contain at least one directed cycle.Footnote ⁷¹ A cycle is necessary to ensure that every dissatisfied state’s behavior is conditioned on the others’. In the system depicted by Figure 6, state A can be deterred from challenging B because that challenge would set off a chain reaction that comes back to haunt it. On the equilibrium path, $B$ ’s potential claim on $C$ deters a challenge against $A$ . A deviation by $A$ would unleash $B$ to make its claim against $C$ , which would in turn have no reason not to contest its border with $A$ .Footnote ⁷² By contrast, if $B$ did not have a potential claim on $C$ , $C$ would have an incentive to challenge $A$ regardless of $A$ ’s behavior; as a result, $A$ would gain nothing from restraint, and the Pandora’s box equilibrium could not hold.Footnote ⁷³

Cross-Regional Variation and “Islands of Stability”

We have thus seen that there exist conditions under which a group of states could coordinate on an equilibrium of mutual restraint. Those conditions are quite restrictive, however, and unlikely to be met as the system grows in size. But if we think of $N$ not as the entire global system but as one neighborhood within the larger system, then cross-regional variation becomes possible.

To motivate this idea, consider the five-state configuration in Figure 7. In this configuration, states $A$ , $B$ , and $C$ are all dissatisfied, as in the three-state version earlier, but state $D$ has a potential challenge on $E$ . As we just saw, the fact that $D$ faces no threat to its own territory means the system as a whole cannot sustain a Pandora’s box equilibrium, since the failure of the $DE$ status quo will cause the rest of the borders to unravel. But it seems plausible that $A$ , $B$ , and $C$ could coordinate on a local equilibrium that would not be disrupted by a challenge elsewhere in the world. Indeed, if conditions 2 and 3 are met for $A$ , $B$ , and $C$ , then there exists an equilibrium in which they make challenges against one another only in response to a challenge within the triad. That is, if they collectively expect that a challenge by $D$ does not trigger conflict among the rest, then they can maintain stability within the triad based on the norm of mutual restraint.

Figure 7. Possible five-state system

Thus we can partition the entire system into neighborhoods $S = \left\{ {{N_1},{N_2}, \ldots ,{N_m}} \right\}$ that contain at least two states and are independent in the sense that none share an edge in common. Then let $M \subseteq S$ denote the set of neighborhoods in which every member satisfies condition 2 in relation to the other members in the neighborhood. The basic insight is that any of the neighborhoods in $M$ could coordinate on the Pandora’s box equilibrium and constitute an “island of stability” that is insulated from conflict elsewhere in the system.

If the number of potential islands is large, so too is the number of possible equilibria in the entire system. Individual islands or groups of islands could coordinate on mutual restraint, while others do not. Which of the possible equilibria are selected depends on the expectations that hold within each island. If the states on an island expect conflict with one another, or if they expect that conflict elsewhere in the system will lead to conflict on the island, then those expectations are self-fulfilling. On the other hand, if the states on a given island believe that stability on the island will unravel only in response to a challenge on the island, then local cooperation can be sustained. The variety of possible equilibria could in principle induce significant variation across regions depending on both the existence of potential islands, if any, in those regions as well as beliefs they hold. For example, states in sub-Saharan Africa might still coordinate on a norm of territorial integrity even if states in the Middle East and North Africa contest their borders with one another. Similarly, vulnerable states in West and Central Africa may have coordinated on territorial restraint even as relatively invulnerable states, such as Somalia—which sought neighboring Somali-inhabited areas while its neighbors had little reason to claim Somali territory—bucked restraint elsewhere in the region.

A Top-Down Pandora’s Box Equilibrium

Finally, as noted at the outset, territorial stability need not arise from coordination among states in a region, but could also be enforced from above by some powerful state(s). The simplest way to incorporate enforcement into the model is to assume that there exists some external actor, $E$ , that can intervene in territorial contests to protect the status quo. In particular, assume that, in the event some state contests a border, $E$ can choose to intervene at some cost $\kappa $ to prevent or reverse any gains.Footnote ⁷⁴

Since enforcement is costly, the enforcer must enjoy some benefit from the status quo. This is particularly likely if E incurs negative externalities from conflict in the system. For example, territorial conflicts might dampen trade, hamper regional integration, create flows of refugees, or introduce instabilities that adversaries might exploit.Footnote ⁷⁵ We can incorporate this consideration by assuming that $E$ incurs some cost due to territorial disputes elsewhere in the system. Let ${W_E}$ denote an $n\! \times\!n$ matrix with elements ${w_{ij}}$ that denote the cost that state $E$ pays in the event of a dispute involving $i$ and $j$ . Then the total negative externality incurred by E in the event of a breakdown of restraint in any neighborhood $m$ in period $t$ is ${\theta _{E,t}} = \mathop \sum \nolimits_{i \in {N_m}} \mathop \sum \nolimits_{j \in {N_m}} {w_{ij}}{g_{ijt}}$ . For ease of notation, let ${\theta _E}$ denote the externality based on the initial configuration of potential claims in the system.

We can now sketch a “top-down” Pandora’s box equilibrium that takes the following form. Within a generic neighborhood ${N_m}$ (which could be the entire system):

1. 1. In each period $t$ , no state contests any borders in the neighborhood as long as (a) no borders in ${N_m}$ were contested in a previous period or (b) every challenger in ${N_m}$ in a previous period has faced intervention by the enforcer.

2. 2. If some state $i \in {N_m}$ contests a border in period $t$ , the enforcer intervenes as long as it has never failed to intervene in any previous period. If the enforcer fails to intervene in any period, then it will not intervene in any subsequent challenges in the neighborhood.

3. 3. If at least one state in the neighborhood contests a border in period $t$ and the enforcer fails to intervene, then all dissatisfied states in ${N_m}$ contest their borders in period $t + 1$ .

In this equilibrium, stability depends on the enforcer keeping a lid on Pandora’s box by preventing changes to the status quo, and everything unravels if the enforcer fails to act. As before, outcomes can vary depending on expectations about the relevant neighborhood. In principle, this top-down equilibrium could hold in one neighborhood, while the bottom-up version holds in another; if so, failure to intervene in the former does not affect outcomes in the latter.Footnote ⁷⁶

What conditions must hold for this equilibrium to exist? Intervention induces restraint in dissatisfied challengers by removing any possibility of gaining territory. The main question is whether the threat of intervention is credible. Formally, if a challenge happens in period $t$ , the cost of unraveling in the next period is $\delta {\theta _{E,t + 1}} = \delta {\theta _E}$ .Footnote ⁷⁷ For enforcement to be sequentially rational this term must exceed the cost $\kappa $ . The key insight is that the benefit from enforcement is increasing in the number of dissatisfied states in the neighborhood, and the weight the enforcer places on the externalities from conflicts that happen there. The worse the consequences of unraveling, the more incentive the enforcer has to take costly action to prevent it.Footnote ⁷⁸

As in the original Pandora’s box equilibrium, stability in this top-down model rests on widespread dissatisfaction. However, in the presence of an enforcer, stability depends on the number and importance of dissatisfied states, not their particular configuration. As long as the enforcer is sufficiently motivated, potential challengers that are not themselves potential targets can be deterred. Hence, where this equilibrium holds, the link between restraint and individual vulnerability is severed.