# An Extension of the Normal Play Convention to $N$-player Combinatorial Games

## TL;DR

This paper extends the normal play convention to multi-player combinatorial games, analyzing outcomes and structures, especially for three-player Nim, and showing many two-player theorems have natural analogues.

## Contribution

It introduces a new play convention for multi-player games and explores its theoretical implications, including detailed analysis of three-player Nim.

## Key findings

- Many two-player game theorems have natural analogues in the multi-player setting.
- Detailed analysis of outcomes in three-player Nim.
- Characterization of the structure of three-player Nim outcomes.

## Abstract

We examine short combinatorial games for three or more players under a new play convention in which a player who cannot move on their turn is the unique loser. We show that many theorems of impartial and partizan two-player games under normal play have natural analogues in this setting. For impartial games with three players, we investigate the possible outcomes of a sum in detail, and determine the outcomes and structure of three-player Nim.