Bidding combinatorial games

TL;DR

This paper extends combinatorial game theory by introducing a generalized framework using Richman auctions, characterizing outcome feasibility, and exploring the structure of game forms across various game families.

Contribution

It generalizes classical normal play to infinitely many game families via Richman auctions and characterizes outcome feasibility and game form structures.

Findings

Existence of a game form for each outcome class

Lattice structures of outcome classes

Analogies with alternating play under certain restrictions

Abstract

Combinatorial Game Theory is a branch of mathematics and theoretical computer science that studies sequential 2-player games with perfect information. Normal play is the convention where a player who cannot move loses. Here, we generalize the classical alternating normal play to infinitely many game families, by means of discrete Richman auctions (Develin et al. 2010, Larsson et al. 2021, Lazarus et al. 1996). We generalize the notion of a perfect play outcome, and find an exact characterization of outcome feasibility. As a main result, we prove existence of a game form for each such outcome class; then we describe their lattice structures. By imposing restrictions to the general families, such as impartial and {\em symmetric termination}, we find surprising analogies with alternating play.