An Avoider-Enforcer game is a kind of positional game. Like most positional games, it is described by a set of positions/points/elements and a family of subsets, which are called here the losing-sets. It is played by two players, called Avoider and Enforcer, who take turns picking elements until all elements are taken. Avoider wins if he manages to avoid taking a losing set; Enforcer wins if he manages to make Avoider take a losing set. A classic example of such a game is Sim. 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.
The winning condition of an Avoider-Enforcer game is exactly the opposite of the winning condition of the Maker-Breaker game on the same. Thus, the Avoider-Enforcer game is the Misère game variant of the Maker-Breaker game. However, there are counter-intuitive differences between these game-types. For example, consider the biasedversion of the games, in which the first player takes p elements each turn and the second player takes q elements each turn. Maker-Breaker games are bias-monotonic: taking more elements is always an advantage. Formally, if Maker wins the Maker-Breaker game, then he also wins the game and the game. 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 and. Then, Avoider wins the game, Enforcer wins the game and Avoider wins the game. There is a monotone variant of the Avoider-Enforcer game-rules, in which Avoider has to pick at leastp elements each turn and Enforcer has to pick at least q elements each turn; this variant is bias-monotonic.
Similarly to Maker-Breaker games, Avoider-Enforcer games too have fractional generalizations. Suppose Avoider needs to avoid taking at least a fraction t of the elements in any winning-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. Define the constant:.
If, and the total number of elements is even, then Avoider has a winning strategywhenplaying first.
