Categorical Approach to Conflict Resolution:A Corrected Correspondence between Category Theory and the Graph Model for Conflict Resolution

Abstract

This note is a substantially revised version of the author's earlier preprint (version~1, 2023), which proposed a ``Categorical Graph Model for Conflict Resolution'' (C-GMCR). Several of the central definitions and claims of version~1 are incorrect, and this version replaces them. We first observe that the states and one-step moves of a graph model do not form a category; the correct construction is the free category on a quiver whose arrows are moves labelled by the decision maker (DM) who controls them. Controller labels are functorial: they define a functor into the free monoid on the set of DMs, and the legal move sequences underlying coalition reachability are exactly the paths whose label words contain no immediate repetition. Preferences, in contrast, are not functorial. We show that a map representing a DM's preference in a preordered set exists only if the preference is transitive, and that even then it extends to a functor on the path category if and only if every move, regardless of its controller, is weakly improving for the focal DM; in that case general metarationality, symmetric metarationality and sequential stability all collapse to Nash stability for the focal DM. Preferences therefore enter the categorical picture as a selection of a subquiver of improvements, not as a functor. We restate the four standard stability concepts in path language, work them out on the Prisoner's Dilemma, and take a first step towards comparing graph models via induced embeddings: Nash stability is reflected by induced embeddings, whereas general metarationality, symmetric metarationality and sequential stability are neither preserved nor reflected. A section lists each correction to version~1 explicitly.