Game States

We provide an inductive de…nition of game-positions, which generalize the notion of nodes in perfect-information games with no in…nite paths. Game-states are de…ned inductively as well. Essentially, a game-state consists of a speci…cation of the game-states that result from players’actions, and the players’types, specifying their beliefs on game-states. In a dynamic equilibrium each type chooses an action to maximize her expected payo¤ according to her belief, given the actions of other types. Dynamic equilibrium generalizes the notion of Bayes-Nash equilibrium to a dynamic setting in which a common prior or Bayesian updating are not imposed a priori. Various formulations of beliefs enable further generalizations, expressing unawareness or imperfect recall. 1 Short Introduction In the classical de…nition of extensive-form games with imperfect information, the notion of an information set is half-baked. On one hand, it is meant to specify objective circumstances: part of the development of the game this far has been (physically) concealed from the player. On the other hand, it speci…es a subjective state of mind of the player. In the latter dimension, it is restrictive in at least three senses. First, it does not allow for the possibility of mistakes on the part of the player, who always considers as possible the development of the game which actually took place. Second, it speci…es only a qualitative belief –these and those nodes are possible. It leaves the quantitative part (e.g. probabilistic beliefs on the nodes) to be speci…ed as part of the equilibrium notion (e.g. sequential equilibrium). The Economics and Management Department, The Open University of Israel, [email protected].