People increasingly compete against AI agents, and organisations decide what to tell them about these opponents. We distinguish an opponent effect, specific to the system faced, from an information effect, a property of the disclosure. In a preregistered, deception-free experiment (N = 1,395), we crossed what participants were told about their opponent with whom they faced in a live all-pay auction in which escalation arises from the incentives. Holding the opponent fixed, being told a human was or might be the opponent raised the median price by 6.7% of the prize value, and being told an optimising machine might be lowered it by 8.8%. Facing an AI agent lowered prices and made it rarer that both players finished with positive earnings. Exploratory analyses indicate that the labels changed how often people raised again while already ahead, not how often they responded when behind. Describing an AI competitor is not behaviourally neutral.
In many applications of cooperative game theory -- from corporate governance and cartel formation to parliamentary voting -- not all winning coalitions are feasible. Ideological distances, institutional constraints, or pre-electoral agreements may render certain coalitions implausible. Classical power indices ignore this and weight all winning coalitions equally. We introduce cohesion structures to quantify coalition feasibility and axiomatically characterize two families of cohesion-sensitive power indices, represented as expected marginal contributions under Luce-type distributions. In the Banzhaf branch, coalition weights are a power transformation of cohesion; in the Shapley branch, additional axioms separate size from cohesion, recovering the classical size weights with cohesion acting within each size class. All results have been mechanically verified in Lean 4 with Mathlib. We illustrate the framework on the German Bundestag and the French Assemblée Nationale, where cordon sanitaire and double cordon scenarios produce sharp, interpretable power shifts.