ora-0175

15.7 Universal equilibrium objects

A game declaration combines player-indexed information categories, action objects, and value functors. Its consistency stage should construct an equilibrium candidate as a universal compatibility object; its observer then measures the failure of unilateral-deviation diagrams to commute. Minimax duality becomes a comparison between two orders of universal construction. Equality requires explicit hypotheses rather than a formal interchange of limits and colimits.

For a bifunctorial payoff \(G:A\times B\to V\), the two orders have the schematic form

\[ \operatorname *{colim}_{a\in A} \operatorname *{lim}_{b\in B}G(a,b) \longrightarrow \operatorname *{lim}_{b\in B} \operatorname *{colim}_{a\in A}G(a,b). \]

The arrow is an interchange comparison, not generally an isomorphism. A minimax theorem supplies hypotheses—convexity, compactness, continuity, and the correct enrichment—under which the observer regards it as exact.

Definition 15.7 Deviation defect

For a candidate profile \(x\), its deviation defect is the player-indexed family of comparison maps from the realized value at \(x\) to the values obtained by admitted unilateral deviations. An equilibrium object is a profile whose deviation defect lies in the observer’s zero or acceptable subobject.

This definition retains localization: a failed equilibrium comes with the player, information restriction, and deviation face that failed. Scalarizing too early destroys precisely the data needed for repair.