A Note on Numbers

TL;DR

This paper characterizes when all game positions are numbers, showing that certain properties related to first-move advantage determine whether values are integers, with implications for analyzing game states.

Contribution

It establishes necessary and sufficient conditions for all game positions to be numbers, linking properties to the absence of first-move advantage and integer values.

Findings

Two properties are necessary and sufficient for all positions to be numbers.

Checking these properties can often be done by examining positions on the board.

If the stronger property holds for all positions, then the values are integers.

Abstract

When are all positions of a game numbers? We show that two properties are necessary and sufficient. These properties are consequences of that, in a number, it is not an advantage to be the first player. One of these properties implies the other. However, checking for one or the other, rather than just one, can often be accomplished by only looking at the positions on the `board'. If the stronger property holds for all positions, then the values are integers.