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⁵⁴ A more detailed description of parameters can be found in the Appendix on the Journal website.

⁵⁵ To prevent the unlikely event of an agent randomly drawing zero capability, we round the random draw of each agent's initial capability to the next highest positive integer.

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⁶⁵ In other words, ι determines the range of possible values of β s for an agent, indicating uncertainty over the other agent's capability (κ). Rather than assuming that an agent calculates an expected value of β based on an underlying probability distribution, an agent's β is determined through a random draw from the distribution (${{ u}_v} $). In other words, we do not impose a method of calculation that assumes all agents arrive at β in a similar manner.

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⁷² In the two-agent case, the strongest agent makes an offer that it always accepts. Assuming that χ = 1, if the weaker agent adds 1 to β every time the stronger agent accepts its own offer, then it may quickly begin to lower its belief about the capability of the stronger agent. This is similar to the weaker agent assuming that when the stronger agent makes an offer to itself that seems high, it is an indication that the stronger agent is stronger than the weaker agent believes. It could be the case, however, that the weaker agent believes the stronger agent is the one that is overestimating its capability, or that the stronger agent is bluffing (see Fearon, ‘Rationalist Explanations for War’). To reflect the different conclusions that can be drawn about the stronger agent, the weaker agent alternates between adding 1 to α and 1 to β at the end of each negotiation phase.

⁷³ By using the term ‘experiment’, we are not suggesting that the model in any way provides an empirical ‘test’ similar to a laboratory or field experiment. Rather, we begin with a baseline condition similar to a control group and then incrementally change parameters in such a way that we can compare each successive condition against the baseline. While this is not the same as a randomized experiment, because we can compare against a baseline or control condition, using the term ‘experiment’ is useful.

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⁷⁶ Although there are five graphs with five points on the x-axis, the midpoint across all of the graphs represents the same experiment in which all parameters are set at their midpoints. It follows that each of the additional four points on the x-axis represents a new experiment (where all other key parameters are held constant at their midpoints), for a total of 21 experiments. (Given that the mid-point of each figure represents the same experiment, the total number of experiments amounts to 21 rather than 25.)

⁷⁷ Fearon, ‘Rationalist Explanations for War’; Gartzke, ‘War is in the Error Term’.

⁷⁸ Slantchev, ‘The Principle of Convergence in Wartime Negotiations’.

⁷⁹ See Slantchev, ‘The Principle of Convergence in Wartime Negotiations’.

⁸⁰ The small number of victories in certain parameter combinations means that the duration findings are sensitive to a small number of simulations. In subsequent discussion, we exercise caution when interpreting findings on the duration of victories, especially when the number of victories is small.

⁸¹ This limit of ten is chosen to simplify the number of agents, such that the analysis of the simulation and resulting plots will be easier to interpret. In the sensitivity analyses reported in the Appendix, we change the maximum number to 15 and 21 and find that the results are qualitatively similar to those reported in the article.

⁸² Uppsala Conflict Database, 2008.

⁸³ Uppsala Conflict Database, 2008.

⁸⁴ Like most evolutionary models, we allow mutation with a very small probability. We include mutation to allow the possibility that the strongest agents do not immediately defeat all other agents. Like many wars, as certain combatants become stronger, they can also lose their ability to adapt and can suffer setbacks. In the model, mutation alters one agent's beliefs, typically one of the strongest, about another agent such that the belief becomes less accurate on average. A mutation parameter μ represents the probability that a randomly selected agent will undergo mutation. A new random draw is made (u_μ∈[0,1]), and if it is less than the probability of an agent mutating (i.e., if μ < s), then the selected agent (assume m) undergoes mutation in which one of agent m's β's is changed. After randomly determining the other agent, a target, another random draw (u_t∈[0,1]) is used to change the mutating agent's β in regards to agent target such that

$$ {{\beta }_{m,target}}\, = \,{{\beta }_{m,target}}\, + \,(({{\beta }_{m,target}}\,\times \,\psi )\,\times \,(1{\rm{ - }}(2\,\times \,{{u}_t}))), $$

where ψ represents the maximum possible percentage change in agent m's β parameter in regards to the target agent.

⁸⁵ In the sensitivity analysis, we consider the case in which the new agents’ beliefs (i.e., how much they deviate from the true capability ratios) are derived from the parent's beliefs. When doing so, the results are broadly similar as the probability of fragmentation increases, but less conclusive when one considers the effect of increasing the deviation of beliefs from the parent's. See Figures 21 and 22 of the Appendix.

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⁸⁹ One alternative would be to allow both coalitions to fight battles with each other, but this does not match empirical cases very well. In the Democratic Republic of the Congo, for example, fighting often took place on multiple fronts with limited engagement of different groups. Furthermore, this alternative requires strong assumptions about actors’ abilities to update beliefs in extremely complicated circumstances. We nonetheless modelled this possibility and report the results in the Appendix (Figures 15 and 16). Overall, the results are qualitatively similar to those that we find here.

⁹⁰ We constructed an alternative model in which all agents can observe battles (but not necessarily participate) and still update their beliefs about the battling agents. The results of the model presented in Figures 4 and 5 of this article are similar to the results in this alternative specification (see Appendix, Figures 7 and 8).

⁹¹ In the Appendix, we consider how the other baseline results (belief deviation, negotiation updating and battle updating) change, once fragmentation is introduced.

⁹² Each of the new parameters is set at a midpoint, and then varied using a high, medium-high, medium, medium-low, or low value (see Table 2), resulting in 25 experimental runs. Each of these experiments consisted of 1,000 runs of the simulation, and the results of 25 experiments appear in Figure 4, and 25 more experiments appear in Figure 5. We also ran a different set of experiments for each of the baseline runs where each of the baseline runs varied a single baseline parameter, sweeping each of the parameter ranges, for a final total of 1,134 experiments. Given the large number of experiments conducted, it would be impossible to do justice to each. An Appendix is available on the Journal website and details the results for most of the experiments. We also consider a number of additional sensitivity analyses, such as coalition shifting, coalition battling, increasing the maximum number of agents, among others, which we report in the Appendix.

⁹³ We do this for a number of reasons. First, it seems unlikely that a war will end with a single member of each coalition agreeing to negotiated terms, without regard for the number and relative strength of other agents. The conflict in Darfur continues despite the agreement of the government and one of the SLM factions in the May 2006 peace agreement, for example. Secondly, as one might expect, settlements occur relatively quickly under settlement rules requiring the strongest of each coalition and a majority of each coalition to agree for settlement to take place, because fewer agents must agree to a settlement under these settlement rules.

⁹⁴ Cunningham, ‘Veto Players and Civil War Duration’.

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¹⁰¹ Ward, ‘A Behavioural Model of Bargaining’.