RICHARDSON'S PUZZLE

In 1948, the English physicist and meteorologist Lewis F. Richardson published a landmark paper entitled “Variation of the Frequency of Fatal Quarrels with Magnitude” (Richardson 1948). Richardson divided domestic and international cases of violence between 1820 and 1945 into logarithmic categories μ = 3,4,5,6, and 7 corresponding to casualties measured in powers of 10. Based on his own updated compilation of conflict statistics, Richardson (1960) recorded 188, 63, 24, five, and two events that matched each category, respectively, the latter two being the two world wars. His calculations revealed that the frequency of each size category follows a simple multiplicative law: for each 10-fold increase in severity, the frequency decreased by somewhat less than a factor of three.

To investigate whether these findings hold up in the light of more recent quantitative evidence, I use data from the Correlates of War (COW) Project (Geller and Singer 1998) while restricting the focus to interstate wars. Instead of relying on direct frequency counts for each order of magnitude as did Richardson, my calculations center on the cumulative relative frequencies of war sizes N(S > s), where S is the random variable of war sizes. This quantity can be used as an estimate of the probability P(S > s) that there are wars of greater severity than s. Thus, whereas for small wars the likelihood of larger conflicts occurring has to be close to one, this probability approaches zero for very large events because it is very unlikely that there will be any larger calamities.

In formal terms, it can be postulated that the cumulative probability scales as a power law:

where C is a positive constant and D is a negative number.

with the intercept logC and the slope coefficient D.

As can be readily seen in Figure 1, the linear fit is strikingly good (R² = 0.985), confirming that the distribution follows a power law. The data points in the lower right-hand corner represent the world wars. The vast majority of all other wars reach considerably lower levels of severity, though without straying very far from the estimated line. The slope estimate (−0.41) implies that a 10-fold increase in war severity decreases the probability of war by a factor of 2.6 (=1/10^−0.41).

This regularity appears to be robust. It can be shown that these findings generalize beyond the two last centuries covered by the COW data. Similar calculations applied to Levy's (1983) compilation of European great power wars from 1495 to 1965 yield a similarly straight line in a log–log diagram with an R² of 0.99, though with a steeper slope (−0.57 instead of −0.41).²

Given these strong results, it may seem surprising that so few scholars have attempted to account for what seems to be an important empirical law. In fact, the situation is not very different from the economists' failure to explain the power law governing the distribution of city sizes, also known as Zipf's law. Using urban population data from many countries, researchers have established that the rank of city size typically correlates astonishingly well with city size.³

Richardson's law remains an equally acute embarrassment. Although the law has been known for a long time, the vast majority of researchers have paid scant attention to it. For example, Geller and Singer (1998) make no mention of it in their comprehensive survey of quantitative peace research dating back several decades (see also Midlarsky 1989 and Vasquez 1993). Those scholars who have focused explicitly on the relationship between war severity and frequency have observed an inverse correlation but have typically not framed their findings in terms of power laws (e.g., Gilpin 1981, 216; Levy and Morgan 1984). To my knowledge there are extremely few studies that address Richardson's law directly.⁴

Given the discrepancy between the strong empirical findings and the almost-complete absence of theoretical foundations on which to rely to account for them, we are confronted with a classical puzzle. This scholarly lacuna becomes all the more puzzling because of the notorious scarcity of robust empirical laws in political science or international relations. Despite decades of concerted efforts to find regularities, why have so few scholars followed in the footsteps of Richardson, who, after all, is considered to be one of the pioneers of the systematic analysis of conflict? Postponing consideration of this question to the concluding section, I instead turn to a literature that appears to have more promise in accounting for the regularity.

SCALING AND SELF-ORGANIZED CRITICALITY

Natural scientists have been studying power laws in various settings for more than a decade. Usually organized under the notion of self-organized criticality (SOC), the pioneering contributions of Per Bak and others have evolved into a burgeoning literature that covers topics as diverse as earthquakes, biological extinction events, epidemics, forest fires, traffic jams, city growth, market fluctuations, firm sizes, and, indeed, wars (for popular introductions, see Bak 1996 and Buchanan 2000). Alternatively, physicists refer to the key properties of these systems under the heading of “scale invariance” (Stanley et al. 2000).

Self-organized criticality is the umbrella term that connotes slowly driven threshold systems that exhibit a series of meta-stable equilibria interrupted by disturbances with sizes scaling as power laws (Jensen 1998, 126; Turcotte 1999, 1380). In this context, thresholds generate nonlinearities that allow tension to build up. As the name of the phenomenon indicates, there must be elements of both self-organization and criticality. Physicists have known for a long time that, if constantly fine-tuned, complex systems, such as magnets, sometimes reach a critical state between order and chaos (Buchanan 2000, chap. 6; Jensen 1998, 2–3). What is unique about SOC systems, however, is that they do not have to be carefully tuned to stay at the critical point where they generate the scale-free output responsible for the power laws.⁵

Using a sandpile as a master metaphor, Bak (1996, chap. 3) constructed a simple computer model that produces this type of regularity (see Bak and Chen 1991 and Bak, Tang, and Wiesenfeld 1987). If grains of sand trickle down slowly on the pile, power law-distributed avalanches will be triggered from time to time. This example illustrates the abstract idea of SOC: A steady, linear input generates tensions inside a system that in turn lead to nonlinear and delayed output ranging from small events to huge ones.

Whereas macro-level distributions emerge as stable features of scale-free systems, at the micro level, such systems exhibit a strong degree of path dependence (Arthur 1994; Pierson 2000). To use the sandpile as an illustration, it matters exactly where and when the grains land. This means that point prediction often turns out to be futile, as exemplified by earthquakes. This does not mean, however, that no regularities exist. In particular, it is important to distinguish complex self-organized systems of the SOC kind from mere chaos, which also generates unpredictable behavior (Axelrod and Cohen 1999, xv; Bak 1996, 29–31; Buchanan 2000, 14–15).

All this is interesting, but do these insights really generalize to interstate warfare? Though useful as a diagnostic, the mere presence of power laws does not guarantee that the underlying process is characterized by SOC. Like any other class of explanations, such accounts ultimately hinge on the theoretical and empirical plausibility of the relevant causal mechanisms. This is precisely the weakness afflicting the few attempts that have so far been made to explain why wars are power law distributed. Recently, Turcotte (1999, 1418–1420) has observed that Richardson's result resembles a model of forest fires (see also Roberts and Turcotte 1998). Computational models of such phenomena are known to produce slope coefficients not unlike the one in Figure 1. If forest fires start when lightning ignites sparks that spread from tree to tree, Turcotte (1999, 1419) suggests, “ a war must begin in a manner similar to the ignition of a forest. One country may invade another country, or a prominent politician may be assassinated. The war will then spread over the contiguous region of metastable countries” (see also Buchanan 2000, 189).

Though suggestive, this analogy cannot serve as an explanation in its own right, because at the level of mechanisms, there are simply too many differences between forests and state systems. Nevertheless, Turcotte's conjecture points in the right direction. The key to any explanation of war sizes depends on how wars spread, and we therefore need to explore what is known or suspected about this topic.

EXPLAINING THE SCOPE OF WARFARE

Accounting for the size of wars is equivalent to explaining how conflicts spread. Rather than treating large wars, such as the world wars, as qualitatively distinct events requiring separate explanations (e.g., Midlarsky 1990), it is preferable to advance a unified theory that explains all wars regardless of their size (e.g., Kim and Morrow 1992). Apart from the inherent desirability of more general explanations, the stress on SOC encourages us to search for scale-invariant explanations.

Although most of the literature focuses on the causes of war, some researchers have attempted to account for how wars expand in time and space (Siverson and Starr 1991). Most of these efforts center on diffusion through borders and alliances. Territorial contiguity is perhaps the most obvious factor enabling conflict to spread (Vasquez 1993, 237–40). Evidence indicates that states exposed to “warring border nations” are more likely to engage in conflict than those that are not so exposed (Siverson and Starr 1991, chap. 3). Geopolitical adjacency in itself says little about how warfare expands, however. The main logic pertains to how geopolitical instability changes strategic calculations by altering contextual conditions: “An ongoing war, no matter what its initial cause, is likely to change the existing political world of those contiguous to the belligerents, creating new opportunities, as well as threats” (Vasquez 1993, 239; see also Wesley 1962). Henceforth, I will refer to this mechanism as context activation.

The consensus among researchers is that alignment also serves as a primary conduit of conflict by entangling states in conflictual clusters (see Vasquez 1993, 234–37). In fact, the impact of “warring alliance partners” appears to be stronger than that of warring border nations (Siverson and Starr 1991). Despite the obvious importance of alliances, however, I consider only contiguity. Because of its simplicity, the alliance-free scenario serves as a useful baseline for further investigations. Drawing on Vasquez' reference to strategic context, I assume that military victory resulting in conquest changes the balance-of-power calculations of the affected states. The conqueror typically grows stronger while the weaker side loses power. This opens up new opportunities for conquest, sometimes prompting a chain reaction that will stop only when deterrence or infrastructural constraints dampen the process (e.g., Gilpin 1981; Liberman 1996).

What could turn the balance of power into such a period of instability? The list of sources of change is long, but I highlight one crucial class of mechanisms relating to environmental factors. Gilpin (1981, chap. 2) asserts that change tends to be driven by innovations in terms of technology and infrastructure. Such cases of technological change may facilitate both resource extraction and power projection. In Gilpin's words, “technological improvements in transportation may greatly enhance the distance and area over which a state can exercise effective military power and political influence” (57).

As Gilpin (1981, 60) points out, technological change often gives a particular state an advantage that can translate into territorial expansion. Even so, “international political history reveals that in many instances a relative advantage in military technique has been short-lived. The permanence of military advantage is a function both of the scale and complexity of the innovation on which it is based and of the prerequisites for its adoption by other societies.” Under the pressure of geopolitical competition, new military or logistical techniques typically travel quickly from country to country until the entire system has adopted the more effective solution. It is especially in such a window of opportunity that conquest takes place.

Going back to the sandpile metaphor, it is instructive to liken the process of technological change to the stream of sand falling on the pile. As innovations continue to be introduced, there is a trend toward formation of larger political entities thanks to the economies of scale. Context activation, in turn, provokes “avalanches” as contingent chains of war decisions. If the SOC conjecture is correct, the wars erupting as a consequence of this geopolitical process should conform with a power law.

MODELING GEOPOLITICS: GEOSIM

How can we move from models of sandpiles and forest fires to more explicit formalizations of war diffusion? Because the power law in Figure 1 stretches over two centuries, it is necessary to factor in Braudel's longue durée of history. But such a perspective raises the explanatory bar considerably, because this requires a view of states as territorial entities with dynamically fluctuating borders rather than as fixed billiard balls (Cederman 1997, 2002; Cederman and Daase 2003). Levy's data, focusing on great power wars in Europe, for example, coincide with massive rewriting of the geopolitical map of Europe. In early modern Europe, there were as many as 500 independent geopolitical units in Europe, a number that decreased to some 20 by the end of Levy's sample period (Tilly 1975, 24; cf. also Cusack and Stoll 1990, chap. 1).

It therefore seems hopeless to trace macro patterns of warfare without problematizing the very boundaries of states. Fortunately, a family of models exists that does precisely that. Bremer and Mihalka (1977) introduced an imaginative framework of this type featuring conquest in a hexagonal grid, later extended and further explored by Cusack and Stoll (1990). Building on the same principles, the current model, which was originally coded in Pascal (Cederman 1997) and is here implemented in the Java-based toolkit Repast (see http://repast.sourceforge.net), differs in several respects from its predecessors (see also Cederman 2001a). These models are all agent-based. Agent-based modeling is a computational methodology that allows scientists to create, analyze, and experiment with, artificial worlds populated by agents that interact in non-trivial ways and that constitute their own environment (see Axelrod 1997; Epstein and Axtell 1996).

Most importantly, due to its sequential activation of actors interacting in pairs that hardwires the activation regime, Bremer and Mihalka's framework is not well suited for studying the scope of conflicts. In contrast, the quasi-parallel execution of the model presented here allows conflict to spread and diffuse, potentially over long periods of time. Moreover, in the Bremer–Mihalka configuration, combat outcomes concern entire countries at a time, whereas in the present formalization, they affect single provinces at the local level (see Cederman 1997, 82–83). Without this more fine-grained rendering of conflicts, it is difficult to measure the size of wars accurately.

The standard initial configuration consists of a 50 × 50 square lattice populated by about 200 composite, state-like agents interacting locally. Because of the boundary-transforming influence of conquest, interactions among states take place in a dynamic network rather than directly in the lattice. In each time period, the actors allocate resources to each of their fronts and then choose whether or not to fight with their territorial neighbors. Half of each state's resources is allocated evenly to its fronts and the remaining half goes to a pool of fungible resources distributed in proportion to the neighbors' power. This scheme assures that military action on one front dilutes the remaining resources available for mobilization, and this dilution in turn creates a strong strategic interdependence that ultimately affects other states' decision-making. The Appendix describes this and all the other computational rules in greater detail.

For the time being, let us assume that all states use the same “grim-trigger” strategy in their relations. Normally, they reciprocate their neighbors' actions. Should one of the adjacent actors attack them, they respond in kind without relenting until the battle is won by either side or ends with a draw. Unprovoked attacks can happen as soon as a state finds itself in a sufficiently superior situation vis-á-vis a neighbor. Set at a ratio of three to one, a probabilistic threshold defines the decision criterion for such attacks.

Context activation is implemented as an increased alertness to geopolitical changes in case of conflict in a state’s immediate neighborhood. Due to the difficulties of planning an attack, actors challenge the status quo with a probability per period as low as 0.01. If fighting involves neighboring states, however, the contextual activation mechanism prompts the state to enter alert status, during which unprovoked attacks are attempted in every period. This mechanism of contextual activation captures the shift from general to immediate deterrence in crises (Huth 1988, chap. 2).⁶

When the local capability balance tips decisively in favor of the stronger party, conquest results, implying that the victor absorbs the targeted unit. This is how composite actors form. If the target was already a part of another multiprovince state, the latter loses its province. Successful campaigns against the capital of composite actors lead to their complete collapse.⁷

Territorial expansion has important consequences for the states' overall resource levels. After conquest, the capitals of conquered territories are able to “tax” the incorporated provinces. As shown in Figure 2, the extraction rate depends on the loss-of-strength gradient, which approaches one for the capital province but falls quickly as the distance from the center increases (Boulding 1963; Gilpin 1981, 115; cf. Cederman 1997, 129). Far away, the rate flattens out around 10% (again, see Appendix for details). This function also governs power projection for deterrence and combat. Given this formalization of logistical constraints, technological change is modeled by shifting the threshold to the right, a process that allows the capital to extract more resources and project them farther away from the center. In the simulating runs reported in this paper, the transformation follows a linear process in time.

Together these rules have four consequences. First, the number of states decreases as the power-seeking states absorb their victims. Second, as a consequence of conquest, the surviving actors increase in territorial size. Third, decentralized competition creates emergent boundaries around the composite actors. Fourth, once both sides of a border reach a point at which no one is ready to launch an attack, a local equilibrium materializes. If all borders are characterized by such balances, a global equilibrium emerges. Yet such an equilibrium is likely to be temporary because decision-making always involves an element of chance, and, in addition, technological change affects the geopolitical environment of the states.

AN ILLUSTRATIVE RUN

The experimental design features two main phases. In the initial stage until time period 500, the initial 200 states are allowed to compete without technological change. Figure 3 shows a sample system at this point. The lines indicate the states' territorial borders; the dots, their capitals. Because of some cases of state collapse, the number of states has actually gone up to 205.

After the initial phase, technological change is triggered and increases linearly for the rest of the simulation until time period 10,500. At the same time, the war counting mechanism is invoked. The task of operationalizing war size involves two problems. First, spatiotemporal conflict clusters have to be identified as wars. Once they have been identified, their severity needs to be measured. Empirical studies usually operationalize severity as the cumulative number of combat casualties among military personnel (Geller and Singer 1998; Levy 1983). To capture this dimension, the algorithm accumulates the total battle damage incurred by all parties to a conflict cluster. The battle damage amounts to 10% of the resources allocated to any particular front (see Appendix).

The question of identification implies a more difficult computational problem. In real historical cases, subject matter experts bring their intuition to bear in determining the boundaries of conflict clusters. Although determining what constitutes a “case” is not always straightforward (Ragin and Becker 1992), wars tend to be reasonably well delimited (but see Levy 1983, chap. 3). In an agent-based model, in contrast, this task poses considerable problems because of the lack of historical intuition. The model therefore includes a spatiotemporal cluster-finding algorithm that distinguishes between active and inactive states. Active states are those that are currently fighting or fought in the last 20 time periods. The latter rule introduces a “war shadow” that blurs the measurement so that brief lulls in combat do not obscure the larger picture. A cluster in a specific period is defined as any group of adjacent fighting states as long as conflictual interaction binds them together. This allows for the merger of separate conflicts into larger wars. The temporal connection is made by letting states that are still fighting in subsequent periods retain their cluster identification. Once no more active state belongs to the conflict cluster, it is defined as a completed war and its accumulated conflict count is reported.

Figure 4 illustrates the sample system at time period 3,326. The three highlighted areas correspond to conflict clusters that remained local. Whereas most conflicts involve two or three actors, some engulf large parts of the system.

The technological diffusion process starts as soon as the initial period is over, as indicated by the states with capitals marked as rings rather than filled squares in Figure 4. These states have experienced at least one shift of their loss-of-strength gradient. This process has dramatic consequences over the course of the simulation. Figure 5 depicts the final configuration of the sample run in time period 10,500. At this stage, only 35 states remain in the system, some of which have increased their territory considerably. Smaller states manage to survive because, as a rule, they have fewer fronts to defend and are in some cases partly protected by the boundary of the grid.

Having explored the qualitative behavior of the model, I can address the question of whether the model is capable of generating robust power laws.

REPLICATIONS

Let us start by exploring the output of the illustrative run. Based on the same type of calculations as in the empirical case, Figure 6 plots the cumulative frequency against corresponding war sizes. It is clear that the model is capable of producing power laws with an impressive linear fit. In fact, the R² of 0.991 surpasses the fit of the empirical distribution reported in Figure 1.⁸

The choice of the sample system is not accidental. In fact, it represents a larger set of systematic replications with respect to linear fit. More precisely, the illustrative run corresponds to the median R² value of a pool of 15 artificial histories, which were generated by varying the random seed that governs all stochastic events, including the initial configuration. Each replication lasted from time zero all the way to time period 10,500. As reported in line 1 in Table 1, regression analysis of these series yields R² values ranging from 0.975 to 0.996, with a median of 0.991, corresponding to the sample run. The table also reveals that although the linear fit and the size range of this run are typical, its slope is far below the median of −0.55. As a complement, Figures 7 and 8 summarize the replication findings graphically. The former histogram confirms that the linear fit of all the runs falls around the median at 0.991. The slope distribution, shown in Figure 8, is somewhat more scattered, but it is not hard to discern a smoother distribution describing the general behavior of the system.

These findings are encouraging. Yet establishing the presence of power laws in a small set of runs is not the same thing as highlighting their causes. Therefore, I now turn to a set of experiments that will help identify the underlying causes of the regularities.

WHAT DRIVES THE RESULTS?

The previous section showed findings based on one specific set of parameters. Table A1 (Appendix) reminds us that there are many knobs to turn. Indeed, the calibration process is rather difficult. Given that war has been a rare event in the Westphalian system, the trick is to construct a system that “simmers” without “boiling over” into perennial warfare. Here I describe a series of experiments that suggest that technological change and contextual activation, rather than other factors, are responsible for the production of power laws.

The easiest way to establish the result relating technological change is to study a set of counterfactual runs with fewer, or no, such transformations. Reflecting a loss-of-strength gradient shifted 10 steps instead of 20, the runs corresponding to line 2 in Table 1 indicate that the linearity becomes somewhat less impressive, with a median R² of 0.980 and a less expansive size range of 3.7. Furthermore, the median slope becomes as steep as −0.62. Once the process of technological change is entirely absent, the scaling behavior disappears altogether, as indicated in line 3. With a representative R² of 0.941 and a range of 1.4, the linear fit falls well short of what could be expected from even imperfect power laws. Experiments with systems lacking contextual activation show that power-law behavior is unlikely without this mechanism. In these runs, the linearity drops to levels even lower than without technological change (see Table 1, line 4).

These experiments confirm that technological change and contextual decisionmaking play crucial roles in generating progressively larger conflicts. However, the findings say little about the general robustness of the scale-free output. It is thus appropriate to investigate the consequences of varying other dimensions of the model. Keeping all other settings identical to the base runs except for the dimension under scrutiny, lines 5 through 11 in Table 1 reveal that the power laws generated in the base runs are not artifacts of a unique parameter combination. Within bounds, these regularities do not seem to be very sensitive to changes in the “war shadow” of the war-counting mechanism (lines 5 and 6), the states' “trigger happiness” (line 7), the fungibility of resource allocation (line 8), the shape of the distance gradient (lines 9 and 10), or the size of the grid (line 11). The last section in the Appendix provides more precise information about these experiments.

Obviously, the point of the sensitivity analysis is not that the power laws hinge exclusively on technological change and the contextual activation mechanism. It is not hard to make the scale-free output vanish by choosing extreme values on any of the dimensions relating to lines 5 to 11 in Table 1, or other parameters for that matter. Because finding this intermediate range of geopolitical consolidation requires considerable parameter adjustment in the current model, the pure case of parameter insensitivity characterizing SOC cannot be said to have been fully satisfied. Yet the qualitative behavior appears to remain for a reasonably large range of values and dimensions. Ultimately, extensive empirical calibration of the these parameters and mechanisms will be required to reach an even firmer conclusion about the causes of Richardson's law.

In general terms, it is clear that scaling behavior depends on both a logistical “brake” slowing down the states' conquering behavior and an “acceleration effect,” in this case represented by the contextual activation mechanism. What self-correcting mechanisms could render the scale-free behavior even less sensitive to parameter variations? It should be noted that I have made a number of simplifying assumptions, which might render the computational power laws less robust than they are in the real world. First, attention must turn to alliances since, as noted above, they have been singled out in the theoretical and empirical literature on war diffusion. More generally, interaction has been restricted to contiguous neighbors. Relaxing that assumption would allow great powers to extend their reach far beyond the neighboring areas. Such a mechanism would help explain how large conflicts spread.

At the structural level, it would be necessary to consider secession and civil wars (see Cederman 2002). These constrains could slow down positive-feedback cycles of imperial expansion through implosion. The exclusive focus on local, contiguous combat assumes away far-reaching interventions by great powers both on land and at sea. Moreover, nationalism affects not only the extractive capacity but also the boundaries of states through national unification and secession.