Scapegoating as Coordination: Vulnerability, Responsibility, and Stochastic Selection

This paper develops a finite crisis game in which claimants competing over a crowd-dependent object choose among continued rivalry, unanimous reform, or coordinated expulsion of one claimant. The crowd-dependent value function captures the Girardian insight that the contested object's value increases with the number who desire it. The model delivers three main theoretical objects and a repeated extension. The stage game has k at least 3 active claimants, each choosing rivalry, reform, or a targeting action against another claimant. Successful targeting requires unanimity among all non-victims, which removes an ad hoc success-threshold parameter. The first main theorem establishes that the canonical target profile against claimant j is a Nash equilibrium if and only if the attacker payoff from expulsion weakly exceeds the rivalry payoff. This is exact and not a tautology of crisis: a negative rivalry payoff does not by itself imply that any targeting is supportable. The existence claim follows as a corollary. When claimants differ in retaliation capacity and causal contribution to the crisis environment, the model separates the vulnerability channel and the responsibility channel. The innocence theorem identifies the exact condition under which targeting tracks vulnerability rather than causal responsibility: the low-retaliation claimant is attacker-optimal for every admissible profile if and only if the retaliation gap weakly exceeds the causal-load gain from removing the more responsible claimant, scaled by the continuation cost sensitivity. A meritocratic blame proposition identifies the opposite region where the causally responsible claimant is strictly preferred even when the vulnerable claimant has lower retaliation capacity, with an explicit minimal numerical counterexample. When protection investment is endogenous through a weakly increasing function of causal contribution, the interaction innocence theorem derives necessary and sufficient conditions for the low-causation claimant to remain attacker-optimal. Under linear load and linear protection the condition collapses to a slope comparison, with three regimes: when the protection slope exceeds the load sensitivity, innocence holds and the lowest-causation claimant is targeted; when it falls short, meritocratic blame holds; at equality, target ranking is independent of causal contribution. The stage game is proved to be an exact potential game in the sense of Monderer and Shapley, with an explicit potential function equal to the payoff gain of each outcome class over rivalry. Under the standard asynchronous one-player-at-a-time log-linear revision protocol, the induced Markov chain has a unique stationary distribution given by a Gibbs measure over the potential, and the stochastically stable action profiles as the noise vanishes are exactly the global maximizers. This supplies a disciplined selection result: reform is selected when its potential exceeds that of every target class, a specific target identity is selected when its payoff uniquely maximizes the potential over reform and rivalry, and ties produce a mixed stochastically stable set. An accessibility theorem proves that successful targeting is one mutation closer to the all-rivalry status quo than reform is, giving exact probability-order results under rare independent mutations. The repeated extension conditions on a sacrifice occurring whenever the active set reaches crisis size and new claimants enter with fixed probability between periods. An exact recurrence formula gives the hitting time distribution as a negative binomial and the mean recurrence time as the crisis-to-current-size gap divided by the entry probability. The unique closed communicating class under positive entry probability is identified as the two-state set around the crisis threshold, with an explicit stationary distribution. A collective double bind is defined precisely by five conditions and verified at the recurrent threshold: three or more claimants trapped in crisis, almost-sure finite-mean recurrence after sacrifice, negative rivalry payoff, crowd-amplified gross desire, and a precise escape prohibition. The conservation theorem proves that pacifying sacrifice reduces the active set temporarily but leaves the crisis generator unchanged: the value function, rivalry cost, reform technology, and entry probability are all unaltered, so the same recurrence law governs every post-sacrifice period. A coalition-breaking penalty for public structural diagnosis is derived endogenously. Announcing a structural frame before the target identity is fixed costs the diagnoser the full sacrifice payoff surplus over rivalry. This converts the cost of structural diagnosis from an exogenous primitive into a derived quantity, and it is zero when diagnosis occurs only after the target outcome is settled. The narrative extension introduces an episodewise replicator update rule for frame competition between personal attribution and structural diagnosis. The log-odds of personal attribution evolves by the difference between an immediate personal payoff and a delayed discounted structural correction, with the delay distributed as the realized inter-crisis recurrence time. Three regimes are identified and proved: certainty trap toward personal attribution when log-odds diverge upward, certainty trap toward structural diagnosis when log-odds diverge downward, and ambiguity trap when log-odds oscillate across zero infinitely often. The stationary replicator result is recovered as a special case. A constructed ambiguity-trap existence proposition proves nonemptiness of the ambiguity regime under an explicit deterministic replacement schedule rather than merely defining it. An application to Bauer, Cahlikova, Chytilova, Roland, and Zelinsky (2023) provides a consistency check. That experiment fixes causal responsibility at zero for every potential scapegoat, which is the degenerate case of the model. The model predicts that target choice is governed solely by vulnerability in that case, consistent with the experimental finding that punishment shifts onto minority targets at higher rates after in-group harm without any systematic difference in the zero-harm baseline. A factorial experimental design is proposed that jointly varies vulnerability and causal responsibility to test the full vulnerability-responsibility boundary.