Avoider-Enforcer game

There, the positions are all the edges of the complete graph on 6 vertices.

Players take turns to shade a line in their color, and lose when they form a full triangle of their own color: the losing sets are all the triangles.

For example, consider the biased version of the games, in which the first player takes p elements each turn and the second player takes q elements each turn (in the standard version p=1 and q=1).

Maker-Breaker games are bias-monotonic: taking more elements is always an advantage.

Avoider-Enforcer games are not bias-monotonic: taking more elements is not always a disadvantage.

For example, consider a very simple Avoider-Enforcer game where the losing sets are {w,x} and {y,z}.

There is a monotone variant of the (p:q) Avoider-Enforcer game-rules, in which Avoider has to pick at least p elements each turn and Enforcer has to pick at least q elements each turn; this variant is bias-monotonic.

Suppose Avoider needs to avoid taking at least a fraction t of the elements in any winning-set (i.e., take at most 1-t of the elements in any set), and Enforcer needs to prevent this, i.e., Enforcer needs to take less than a fraction t of the elements in some winning-set.

Biased positional game#A winning condition for Avoider