Generated Sufficiency for Equilibrium and Game Theory: A General Directional Mapping Approach

First-order conditions identify candidate equilibria, but only curvature conditions showthat a candidate is a best response for every agent. Classical sufficiency asks payoffsto be concave in the strategy variable. Agents, however, reason in their own workingvariables, such as logarithms of quantities, batches or scaled units, in which payoffsneed not be concave. This paper introduces generated sufficiency: a condition statedin each agent’s perceived units, which we call perceived diminishing returns, underwhich a profile where no agent perceives a marginal gain is a Nash equilibrium. Threepropositions describe the results: (1) perceived diminishing returns and first-orderconditions with shadow prices are sufficient for equilibrium; (2) a strict version makesthe loss from any deviation grow with the perceived distance and each best responseunique; (3) the condition holds whenever payoffs are concave after a smooth relabelingof strategies, even if they are not concave in the working variable. Examples includea capacity-constrained Cournot economy and a game in log-quantities. The resultsapply Hanson-type sufficiency to games in a scale-free formulation, and their limits arestated before the conclusion.