Gilles' MS-potential framework decomposes every finite game

Game theory's potential games—where a single global function captures players' incentives—have been a powerful lens since Monderer and Shapley (1996) proved they exist precisely when every pair of players' second-order cross-differences agree. Robert P. Gilles's new preprint (arXiv:2608.01967) tackles what happens when that condition breaks. He constructs an 'MS-potential' from the common-interest component of those cross-differences, showing it is unique up to separable payoff terms and exists under a higher-order MS-condition. For exact potential games, it recovers the original potential up to the players' main effects, neatly generalizing the classical theory.

The central contribution is an 'MS-decomposition': every finite game splits into a common-interest MS-potential game and a residual that absorbs all individualistic effects. Gilles proves this decomposition is invariant under relabelling, non-strategic translation, and action duplication—properties the competing CMOP decomposition lacks. Only the least-squares extension falls short, because averaging and centring break invariance. The headline identity: when all players have the same number of actions, an augmented MS-potential equals Candogan et al.'s (2011) potential up to an additive constant; for unequal action counts, the two diverge, and no bound is established. The work gives theorists a universal toolkit for dissecting finite games into strategic and non-strategic components.