Conway games, algebraically and coalgebraically

TL;DR

This paper extends Conway's game theory to non-terminating, hypergames using coalgebraic methods, providing a new framework for analyzing infinite and non-wellfounded games.

Contribution

It introduces a coalgebraic approach to hypergames, generalizing Conway's theory to include non-wellfounded games and developing a compositional semantics for impartial hypergames.

Findings

Extended Conway's theory to hypergames with coalgebraic methods

Developed a generalized Grundy-Sprague function for hypergames

Discussed equivalences and future directions in hypergame theory

Abstract

Using coalgebraic methods, we extend Conway's theory of games to possibly non-terminating, i.e. non-wellfounded games (hypergames). We take the view that a play which goes on forever is a draw, and hence rather than focussing on winning strategies, we focus on non-losing strategies. Hypergames are a fruitful metaphor for non-terminating processes, Conway's sum being similar to shuffling. We develop a theory of hypergames, which extends in a non-trivial way Conway's theory; in particular, we generalize Conway's results on game determinacy and characterization of strategies. Hypergames have a rather interesting theory, already in the case of impartial hypergames, for which we give a compositional semantics, in terms of a generalized Grundy-Sprague function and a system of generalized Nim games. Equivalences and congruences on games and hypergames are discussed. We indicate a number of intriguing directions for future work. We briefly compare hypergames with other notions of games used in computer science.