Constructive comparison in bidding combinatorial games

TL;DR

This paper introduces an algorithmic method for comparing bidding combinatorial games, extending classical game theory results and exploring properties of various number classes within this new framework.

Contribution

It generalizes standard game comparison techniques to bidding games and demonstrates their implications for classical number properties and game structures.

Findings

Algorithmic comparison method for bidding games

Generalization of classical results to bidding game context

Identification of properties of integers, dyadics, and numbers in this setting

Abstract

A class of discrete Bidding Combinatorial Games that generalize alternating normal play was introduced by Kant, Larsson, Rai, and Upasany (2022). The major questions concerning optimal outcomes were resolved. By generalizing standard game comparison techniques from alternating normal play, we propose an algorithmic play-solution to the problem of game comparison for bidding games. We demonstrate some consequences of this result that generalize classical results in alternating play (from Winning Ways 1982 and On Numbers and Games 1976). In particular, integers, dyadics and numbers have many nice properties, such as group structures, but on the other hand the game * is non-invertible. We state a couple of thrilling conjectures and open problems for readers to dive into this promising path of bidding combinatorial games.