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⁴⁴ The model was constructed using NetLogo. Additional details and the code are available from the authors.

⁴⁵ The model assumes that interactions among allied actors produce policy outcomes at the alliance's effective ideal point. In two-member alliances, this point is identical to the policy outcome that would have arisen in the absence of an alliance. In larger alliances, however, the outcome with the alliance differs from the outcome without it, providing an additional potential basis to accept or reject an alliance proposal.

⁴⁶ An appendix detailing the pseudocode, as well as other details of the model, is available from the authors.

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⁴⁹ Note that accepting an alliance does not preclude an actor from subsequently taking a turn as alliance proposer, although any alliance that that actor proposes must be approved by their ally before it can come into effect.

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⁵¹ For actors with no alliance partners, k is fixed to 1, as alliance inefficiencies obviously are not a pertinent factor.

⁵² Because interactions with allies produce outcomes at the alliance effective ideal point, i gains the same utility from every interaction with an ally. We can thus simply multiply that utility by the number of allies, rather than having to sum over each ally.

⁵³ This point illustrates the tensions inherent in the use of an agent-based model. Relative to a game-theoretic model, actors in our model are not particularly forward-looking. For example, they do not consider the possibility that agreeing to an alliance proposal now might allow the new alliance partner to veto a more attractive alliance proposal at a later date. We would only be able to capture this sort of foresight by dramatically limiting the number of actors in the resulting game-theoretic model, a much less appealing compromise given the objectives of this study.

⁵⁴ A detailed set of representative results at different parameter values appears in the online appendix.

⁵⁵ For simplicity, the model contains democracies and autocracies, but does not include ‘anocracies’. The empirical world includes states that are neither fully democratic nor autocratic, adding to the challenge of connecting theory with evidence. As we will see, however, this distinction does not appear to critically affect tests of the theory.

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⁵⁷ Plotting only jointly democratic alliance dyads reveals a similar pattern of rising propensity to ally over time (available from the authors). Our interest here is in the relative shift in democratic preference for democratic partners.

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⁷⁰ This variable is an admittedly imperfect measure of common threat. An alternative approach would be to use the number of MIDs against similar actors, but this raises difficult questions about an appropriate time lag between disputes in different dyads consistent with the existence of a common threat. We leave this issue for future research.

⁷¹ The substitution of democracies for autocracies is not one-for-one, as many states fall into the middle (‘anocracy’) category. The point is not deductive, but simply an empirical one that democracies are ‘crowding out’ autocracies.

⁷² This problem may occur in other contexts as well. It is common to extend datasets in international relations temporally. Any differences between older analyses using shorter time-series and more recent studies including this new data could either be the result of more or better data or temporal changes in underlying causal relationships.