Why do allied attackers sometimes concentrate the burden of conflict on a single member and sometimes share it across the group, and what determines whether conflict breaks out at all? This paper studies a two-stage game in which a group of heterogeneous attackers faces a single defender who commits to defense before attacks are chosen, under a contest success function characterized axiomatically by a threshold property: an attack succeeds with positive probability only once effective attack overwhelms effective defense. The game admits a unique pure-strategy subgame-perfect equilibrium in closed form under linear and quadratic effort costs. Cost curvature governs burden sharing: under linear costs only the most efficacious attacker fights and its allies free-ride, while under quadratic costs every attacker participates in proportion to its efficacy, so the war/peace boundary shifts from the leader's individual capability to the group's aggregate capability. The threshold structure delivers a discrete war, peace, and partial-war taxonomy with simple parametric boundaries. A planner comparison quantifies the welfare cost of non-cooperative burden sharing, and war can strictly dominate the armed peace required to deter it, without contradicting the rationalist inefficiency puzzle. Applications include military alliances and patent-litigation coalitions.

When a coalition's members draw attack and defense from a shared budget, who fights? This paper develops the equilibrium theory of a two-stage attack-defense contest between a single defender and a coalition of heterogeneous attackers under joint quadratic effort costs, in which the squared cost applies to each player's total effort. The specification is the natural reduced form for the leading applications, where one national defense budget or firm-wide legal budget governs all of a player's allocations, and it makes participation itself an equilibrium object. Stage-2 behavior is characterized by two cubic equations: a depressed cubic in the defender's total effort and an aggregative cubic for total effective attack. Participation switches by rivals create convex kinks in reduced payoffs that defeat quasi-concavity, so existence is established by construction under a regular-participation condition, and uniqueness holds within the class of strict-participation equilibria. Under right-side symmetry the active set is a top-K set in attack-efficacy order, pinned by an endogenous cutoff. Along a computed sweep the equilibrium climbs a participation staircase, so the path to peace runs through broader participation, with coalition members held exactly at their participation margins on entire parameter intervals. The size of the active set is not monotone in the defender's strength: deterrence of the marginal member and provocation of the aggregate attack pull in opposite directions. Applications include military alliances and patent-litigation coalitions.

Groups look nothing like the wars they fight. This paper asks why, in a threshold contest between a singleton defender and a group of attackers whose prize is member-specific and nonrival, so that nothing is shared and the classical sharing force against alliance formation is absent by construction. Membership is still costly, but its price is not a parameter: a member is a target, and the defender's counterattack is his equilibrium response to the roll. We compute that price, and wherever the defender's fronts do not compete for funds it has the same shape whatever the cost function. Under linear costs the equilibrium group is unique and two-tiered: one member fights, the rest ride, and a candidate rides whenever her stake exceeds her exposure by more than about fifteen percent, a bound independent of every other parameter. Group size and war intensity are then generated by disjoint blocks of primitives, so neither is informative about the other. A candidate too strong to ride but unwilling to lead is excluded outright, and her payoff falls discontinuously in her own capability. Restoring rivalry caps the group at a size independent of the pool, and pricing effort in the sharing rule eliminates the fringe entirely, which is why the classical literature could not find it. Under a shared budget the reading inverts: a passenger drains the defender by obliging him to fund an attack on her, and a twenty-five-fold range of her attack capability leaves the group's success probability unchanged to six decimal places. What she supplies is a front, not an army.

Support for Ukraine since 2022 is distributed in a way the standard theory of alliance burden sharing does not predict: the largest economies carry the smallest shares, and the heaviest burdens fall on small states close to the fighting. This paper shows that the pattern is what a threshold contest model of group conflict predicts, in which each coalition member's effort is governed by its attack efficacy, a composite of effectiveness, stake, and cost, whose main observable component is exposure to the conflict rather than economic size. Calibrating the model to the Kiel Institute Ukraine Support Tracker, I find that the Olson exploitation hypothesis is absent to reversed in this coalition, that an exposure block derived from the model outpredicts economic size in a fair out-of-sample horse race on military burden shares by an order of magnitude, and that participation is front-loaded and stake-driven, with the largest non-contributors being large, unexposed economies. Inverting the model's equilibrium recovers relative efficacies that are highly persistent across the change in coalition leadership. When the United States withdrew in 2025, the remaining coalition scaled up by a factor the model predicts to within about two percent, a rare confrontation of a group contest model with a realized comparative-statics test. The calibration is transparent about what it identifies, relative efficacy on the active set, and what it cannot separate, the underlying primitives.