# Dominated Options

__Keywords__: Theory

In combinatorial game theory making a 'move' is always conceived of as choosing between options (also called followers).

If a player has a *dominated option*, that means the same as a move without any possible motivation. In practical terms, 'why would you ever play that way?'

This comes down to the ordering of games: if G >= H then under all possible circumstances one of the players (Left) will choose G rather than H, whenever offered the choice; and the other (Right) naturally will choose H rather than G.

The games may be numbers, in which case we are saying Left always prefers higher numbers, Right always lower numbers.

Therefore playing in a dominated option corresponds to what Go players would call 'taking an unconditional loss', measured against best play.

If two of the options are equal as games, then each makes the other a dominated option. In practical terms this means that you can remove one or the other from consideration.

Tom: Another explanation.

A game G is a set {GL1,GL2,...} of left options (to which left may move) and a set of {GR1,GR2,...} of right options (to which right may move). Each of these options are games in their own rights. If GL1 >= GL2 (informally: left prefers GL1) then GL2 is said to be a dominated option. Similarly, if GR1 >= GR2 then GR1 is a dominated option.

This definition is useful because it has been proved that dominated options can be deleted without changing the value of the game. This accords with common sense, you would never make a move that, even locally, has a superior alternative. It follows that the game would have the same winner if the inferior move were not available.