Apt, K. R. (2013). Potential Games. Lecture slides, CWI Amsterdam and University of Amsterdam.
Key takeaways
From the reading, NOTE-1427.
- Definitions (slides 6, 10, 12, 14). Exact potential: every unilateral payoff change equals the potential’s change. Ordinal: the payoff rises if and only if the potential does. Generalized ordinal: a rise in payoff implies a rise in potential. FIP: every improvement path is finite.
- The theorem (slide 15). “Every finite game has a generalized ordinal potential iff it has the FIP”, credited to “Monderer and Shapley ‘96”. The forward direction is proved; the converse is a sketch that counts, for each profile, the improvement sequences ending there.
- Ordinal is stronger (slide 13). A two-by-two game has the FIP but no ordinal potential, because one player’s indifference forces two profiles to equal potential that a chain of strict improvements orders.
- Congestion games (slides 17–26) are potential games (Rosenthal 1973), so they have pure equilibria.
Standing in the record
Filed on 2026-10-30 for THEORY-896. It confirms, second-hand, that the class matched to the FIP is the generalized ordinal one. Biggar and Shames’s weaker restatement, which THEORY-896 records, says only that ordinal potential games have the FIP; the slides’ two-by-two game shows that the converse fails for ordinal potentials, so an equivalence can hold only for the generalized class. Its sketch agrees with the record’s proof. It cannot confirm the lemma’s wording, number or page in Monderer and Shapley.