Products of Representations Characterize the Products of Dispersions and the Consistency of Beliefs

A "dispersion" specifies the relative probability between any two elements of a finite domain. It thereby partitions the domain into equivalence classes separated by infinite relative probability. The paper's novelty is to numerically represent not only the order of the equivalence classes, but also the "magnitude" of the gaps between them. The paper's main theorem is that the many products of two dispersions are characterized algebraically by varying the magnitudes of the gaps between each factor's equivalence classes. An immediate corollary is that the many beliefs consistent with two strategies are characterized by varying each player's "steadiness" in avoiding various zero-probability optionsconsistent beliefs, relative probability

Dynamic Programming for Pure-Strategy Subgame Perfection in an Arbitrary Game

This paper uses value functions to characterize the pure-strategy subgame-perfect equilibria of an arbitrary, possibly infinite-horizon game. It specifies the game's extensive form as a pentaform (Streufert 2023p, arXiv:2107.10801v4), which is a set of quintuples formalizing the abstract relationships between nodes, actions, players, and situations (situations generalize information sets). Because a pentaform is a set, this paper can explicitly partition the game form into piece forms, each of which starts at a (Selten) subroot and contains all subsequent nodes except those that follow a subsequent subroot. Then the set of subroots becomes the domain of a value function, and the piece-form partition becomes the framework for a value recursion which generalizes the Bellman equation from dynamic programming. The main results connect the value recursion with the subgame-perfect equilibria of the original game, under the assumptions of upper- and lower-convergence. Finally, a corollary characterizes subgame perfection as the absence of an improving one-piece deviation.Comment: 56 pages, 7 figures. Version 3 provides better non-convergent examples in Section 2.

2020-4 The Category of Node-and-Choice Extensive-Form Games

This paper develops the category NCG. Its objects are node-and-choice games, which include essentially all extensive-form games. Its morphisms allow arbitrary transformations of a game\u27s nodes, choices, and players, as well as monotonic transformations of the utility functions of the game\u27s players. Among the morphisms are subgame inclusions. Several characterizations and numerous properties of the isomorphisms are derived. Also, the game-theoretic concepts of no-absentmindedness, perfect-information, and (pure-strategy) Nash-equilibrium are shown to be isomorphically invariant. Finally, full subcategories are defined for choice-sequence games and choice-set games, and relationships among these two subcategories and NCG itself are expressed and derived via isomorphic inclusions and equivalences