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formalizing an equilibrium notion for conflict.

We may now formalize an equilibrium notion for conflict. This is a collection of attack and success probabilities, a*ðyÞ and p*ðyÞ, one such pair for every victim income y, such that a* is determined by the optimal decisions of the population of potential attackers, given p*, while p* is determined by the optimal decisions of potential victims, given a*. A simple single-crossing argument, which we record in the Appendix, assures us that the protection function is decreasing, while the attack function is increasing. Their unique intersection determines the equilibrium for every y. Observation 1. For every y, the protection function generates success probabilities p that weakly decrease in a, while the attack function generates attack probabilities a that strictly increase in p. There is a unique equilibrium. The Appendix contains a proof. Panel A of figure 1 summarizes an equilibrium for a given victim income. The upward-sloping line is the attack function that generates a as a function of p. The downwardsloping line is the protection function. Either function may have jumps, but we can use indifferences ðand the assumption of a large population of potential attackersÞ to fill in these jumps so that the resulting graph is closed. These jumps will actually arise in our later specification of two kinds of protection technologies. The two lines intersect once, telling us that there is a unique second-stage equilibrium, as in observation 1. In what follows, we are interested in conflict outcomes, specifically, whether or not they are “successful” from the point of view of the aggressor. With a large population of potential attackers, this is equivalent to studying the overall probability of attack