Ordering of Games

This is a foundational concept of CGT.

In terms of the game ``0``, one can proceed by defining what it means to have ``G >= 0``, for any game ``G``, namely that Left can win ``G`` if Right starts. More generally, the outcome class of ``G`` says who wins given who starts.

One would expect ``G >= H`` to mean that ``G - H >= 0`` (where ``G - H`` is the so-called difference game), and it does indeed make sense to take that as the definition.

Games are only partially ordered, since two games may not be comparable at all with respect to the order. If a game is a first player win, it is said to be “fuzzy”, that is, neither greater than nor , nor equal to nor less than ``0``. If games ``G`` and ``H`` are such that ``G - H`` is fuzzy, we can't use the order relation to compare ``G`` with ``H``.

Significance of the ordering

If a player has two options, ``H_0`` and ``H_1`` in a game ``G``, with ``H_0 >= H_1``, they can always do at least as well by playing ``H_0`` rather than ``H_1``, which is said to be dominated, and may be eliminated from ``G`` without affecting its result.

If, however ``H_0 ∥ H_1`` (``H_0`` is fuzzy against ``H_1``), neither option may be eliminated, because each is more favourable than the other under some circumstances.