The second chapter of Potential Game Theory: Theories and Applications is now complete, and I am rather pleased with how it came together. Let me share what lies at its heart.
The central question of Chapter 2 is deceptively simple: what does it mean for a group of self-interested decision-makers to all be, in some deep sense, following the same playbook? In a so-called exact potential game, the answer is precise: every player’s incentive to change their action is perfectly mirrored by a single shared function—the potential. If you can improve your payoff by switching strategies, the potential increases by exactly the same amount. Everyone is, quite literally, climbing the same hill.
This is a powerful idea. It means that the entire strategic complexity of such a game—involving potentially many players with quite different payoff structures—can be summarised in a single real-valued function. The equilibrium analysis, the learning dynamics, the welfare properties: all of these are anchored in one object. The Cournot oligopoly, in which firms compete by choosing output levels, turns out to be a canonical example, and it serves as a recurring illustration throughout the chapter.
Chapter 2 develops several characterisations of exact potential games. The most elegant, due to Monderer and Shapley (1996), applies to games with smooth, differentiable payoff functions. There, a game has an exact potential if and only if a certain symmetry condition holds in the cross-effects between players: the way one player’s marginal payoff responds to a rival’s action must match how that rival’s marginal payoff responds in return. A beautiful constraint—and a surprisingly practical one, as it allows us to rule out entire classes of games quite efficiently. Rent-seeking contests, for instance, fail this criterion in a rather decisive fashion.
A second characterisation, due to Ui (2000), reveals an unexpected bridge to cooperative game theory. Every exact potential game corresponds, in a precise sense, to a cooperative game whose Shapley value—the canonical fair-division concept in cooperative settings—yields exactly the payoff functions in the strategic game. The two branches of game theory, long treated as rather separate enterprises, turn out to be more intimately connected than they first appear.
The chapter concludes with a question that I find particularly compelling: what do we make of games that are not exact potential games, but are close to being one? Building on a decomposition by Jiang and colleagues, Bichler et al. (2025) introduce a scalar measure of potentialness that quantifies, for any finite game, the fraction of its strategic structure that behaves like a potential game. The remainder is harmonic—capturing cyclical, never-settling incentives. This measure turns out to have real predictive power for learning dynamics.