With Irem Portakal, organization of the mini-symposium Algebraic Game Theory
at 2026 General Meeting of European Women in Mathematics (EWM).
The EMW General Meeting will take place at the Mathematics Institute of the University of Warwick 31/8 - 4/9 2026
List of speakers:
Fabrizio Germano, Universitat Pompeu Fabra Barcelona, Spain
Title: Understanding Human Behavior via Similarity: A Geometric and Behavioral Rules-Based Approach to Games
Abstract: We develop an analytical and experimental framework for studying similarity in a complete domain of one-shot 2x2 games. We define similarity geometrically through a neighborhood structure on games and continuity of behavior, allowing direct comparison of the partitions induced by theoretical solution concepts, behavioral rules, and observed play. In a large-scale experiment, subjects play every game in the domain without feedback between decisions. We find that empirically inferred similarity classes differ substantially from those implied by Nash equilibrium and dominance reasoning. Instead, they align closely with the partition induced by a variant of level-k reasoning, with systematic deviations arising when fairness and efficiency considerations conflict with strategic reasoning. At the individual level, subjects are classified according to primary and secondary behavioral rules corresponding either to a variant of level-k reasoning (with 0 ≤ k ≤ 5), max-max or a fairness- and efficiency-based heuristic. The framework provides a unified basis for comparing theoretical and empirical classifications of strategic environments.
Linda Hoyer, RWTH Aachen University, Germany
Title: On the dimensions of correlated equilibrium polytopes of generic games.
Abstract: Assume that we have two car drivers that meet at an intersection. They both want to get to their destination as quickly as possible, yet if they both started driving at the same time, they’d risk a car crash. This situation is solved by introducing a third, neutral party (a traffic light) that suggests who goes first. When no driver benefits from deviating from the suggestion, the ”game” has reached a so called correlated equilibrium. For a given game G, the set of all correlated equilibria is a polytope P, the correlated equilibrium polytope. It is a subset of the probability simplex and contains all Nash equilibria of G. We show that if P is not full-dimensional, then there exists a subgame whose correlated equilibrium polytope is affinely isomorphic to that of the original game. This is joint work with Jan Draisma and Irem Portakal.
Sahar Jahani, London School of Economics and Political Science, United Kingdom
Title: Equilibrium Numbers of Small Multiplayer Games
Abstract: A Nash equilibrium of a noncooperative game defines a mixed (randomized) strategy for each player that is optimal against the fixed strategies of the other players. It is characterized by constraints (equations and inequalities) on the mixed-strategy probabilities for the players' expected payoffs and a complementarity condition. The constraints are linear for two players but involve polynomials for more than two players. We show that the simplest multiplayer game of three players with two actions each has generically at most nine equilibria. The proof, using the small size of the game, relies on properties of hyperbolas. Another tool is the index of an equilibrium, an orientation that implies that the number of equilibria is odd. We discuss a more general conjecture by Hertling and Vujic about equilibrium numbers for two-action games with many players. Joint work with Prof. Bernhard von Stengel.
Elke Neuhaus, Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany
Title: Nash discriminants
Abstract: The totally mixed Nash equilibria (tmNE) can be defined via equations cutting out an algebraic scheme. If in the game, no player has an unfair surplus of strategies, the expected number of tmNE is nonzero and finite. An unexpected number of tmNE can be characterised in this scheme via the Nash discriminant variety. In order to understand this variety better, we find the unique component of maximal dimension, which is the A-discriminant variety of some polytope depending on the format.
Lucas Pahl, University of Sheffield, United Kingdom
Title: Index and Robustness of Mixed Equilibria: An Algebraic Approach
Abstract: We develop an algebraic method for computing the index of isolated completely mixed Nash equilibria in finite games. The method associates the local system of payoff-indifference equations with a finite-dimensional local algebra and computes its topological degree through the signature of an associated bilinear form. Unlike standard index calculations, the procedure does not require perturbing the game or enumerating the nearby regular equilibria created by a perturbation.
We use this method to establish that an isolated completely mixed equilibrium can have any integer as its index. For a class of mildly singular equilibria, which we call monogenic, the possible indices reduce to (0), (+1), and (-1). Within this class, payoff robustness is equivalent to having a nonzero index. We also exhibit singular equilibria with nonzero index and discuss extensions to boundary equilibria, nondegenerate equilibrium components, and isolated equilibrium outcomes of extensive-form games. The analysis clarifies both the possibilities and the limitations of the nonzero-index criterion and identifies the existence of a payoff-robust, completely mixed equilibrium of index zero as a central remaining question.
Javier Arranz Sendra, CUNEF Universidad Madrid, Spain
Title: Conditional Independence equilibria and Spohn CI varieties.
Abstract: We further develop the algebraic--geometric foundations of conditional independence (CI) equilibria, a refinement of Nash and dependency equilibria that integrates conditional independence relations from graphical models into strategic reasoning. The set of CI equilibria, defined as the intersection of the open probability simplex with the Spohn CI variety, contain the set of totally mixed Nash equilibria and are contained within the set of totally mixed dependency equilibria. Extending earlier work on binary games, we analyze the structure of the associated Spohn CI varieties for generic games of arbitrary format. We compute the dimension of the Spohn CI variety for generic games. We show that when non-empty, the set of totally mixed CI equilibria forms a smooth manifold for generic games. For cluster graphical models, we introduce the class of Nash CI varieties, prove their irreducibility, and describe their defining equations, degrees, and conditions for the existence of totally mixed CI equilibria for generic games.