When Does Nash Averaging Ignore a Duplicate? The Exact Scope of Redundancy Invariance

Nash averaging reads pairwise results as a zero-sum game and rates each agent by its expected result against an equilibrium. When there is more than one it takes the maximum-entropy equilibrium, the most spread-out one. The method was proposed with the property that an exact copy changes no rating. We show that the property can fail, and that the published proof covers the set of equilibria, not the maximum-entropy equilibrium. A copy tilts the entropy in proportion to the copied agent's probability. We prove that the maximum-entropy equilibrium stays in place exactly when no step inside the equilibrium set gains more from this tilt than it loses in entropy. When it moves, only agents that no equilibrium plays can be re-rated, while the rest tie at the top. A copy never moves a unique equilibrium, including those of the Atari score table and chatbot-arena leaderboard we check. Coarse scores leave more than one equilibrium in forty of the forty-four football seasons studied, often through exact draws. In seventeen of them, adding a copy of one team reorders two others. More than one equilibrium lets a rating jump at an exact draw. Any rule that never jumps is wrong somewhere nearby by at least half that rating's range; a vanishing relaxation of the equilibrium conditions costs at most the whole range and loses invariance to copies. A published Nash average should say which rule picked its equilibrium and whether others exist.