Equality of Games

In combinatorial game theory, equality of games is a defined concept. It certainly doesn’t coincide with the idea of having the same game (what you could call an identical copy).

Firstly one has an ordering of games, such that ``G >= H`` is a relation defined to hold just when the difference game ``G - H >= 0``.

Then two games ``G`` and ``H`` are by definition equal when ``G >= H`` and ``H >= G``.

This is an obvious definition to make, from a mathematical point of view. One also wants to be able to compute with this idea. The theory of canonical forms of games is designed to do that.

Charles Matthews

Two equal games ``G`` and ``H`` have the basic property that in a sum of games, one can be replaced by the other, without changing the overall outcome, i.e. who can win depending on who starts.