The Relationship Between Rationality on the Matrix and the Tree∗

The relationship between the matrix and the tree has been the subject of intensive investigation ever since the beginning of game theory. The issue goes back even to Borel and von Neumann. Later, Thompson [11, 1952], followed by Elmes-Reny [4, 1994], uncovered the structural relationship between the matrix and the tree. They showed that, up to the duplication of pure strategies, two games have the same strategic form if and only if they differ by a certain sequence of elementary transformations. There is also the question of the relationship between Nash equilibrium defined on the matrix (perfect, proper, stable equilibria, etc.) and equilibrium defined on the tree (extensive-form perfect, sequential equilibria, etc.). For example, Kohlberg-Mertens [6, 1986] and van Damme [12, 1984] showed that a strategic-form proper equilibrium induces a sequential-equilibrium outcome in any tree with that strategic form. But a very basic question has remained: What is the relationship between dominance in the matrix and dominance in the tree? And, following from this, what is the relationship between iterated dominance in the matrix and iterated dominance in the tree? This note addresses this question. See Gilli [5, 2002] and Shimoji [10, 2004] for related analyses.