Series of combinatorial games

TL;DR

This paper introduces a new framework for summing sequences of combinatorial games, unifying classical sums for real numbers and ordinal sums, and exploring novel variants like string limits and transfinite sums.

Contribution

It defines a general sum for sequences of combinatorial games that encompasses classical and ordinal sums and discusses innovative variants like string limits and Dadaist sums.

Findings

Unified sum definition for combinatorial games and real/ordinal sequences

Introduction of string limits and transfinite sums for game sequences

Discussion of loopfree variants with transfinite runs

Abstract

We present a definition for the sum of a sequence of combinatorial games. This sum coincides with the classical sum in the case of a converging sequence of real numbers and with the infinitary natural sum in the case of a sequence of ordinal numbers. We briefly discuss other possibilities, such as the string limit, some "magical" variants of Hackenbush, as well as "Dadaist" infinite sums, which allow transfinite runs, while still being loopfree.