Positional games

Discrepancy game

A discrepancy game is a kind of positional game. Like most positional games, it is described by its set of positions/points/elements and a family of sets (- a family of subsets of ). It is played by two players, called Balancer and Unbalancer. Each player in turn picks an element. The goal of Balancer is to ensure that every set in is balanced, i.e., the elements in each set are distributed roughly equally between the players. The goal of Unbalancer is to ensure that at least one set is unbalanced. Formally, the goal of balancer is defined by a vector where n is the number of sets in . Balancer wins if in every set i, the difference between the number of elements taken by Balancer and the number of elements taken by Unbalancer is at most bi. Equivalently, we can think of Balancer as labeling each element with +1 and Unbalancer labeling each element with -1, and Balancer's goal is to ensure the absolute value of the sum of labels in set i is at most bi. The game was introduced by Frieze, Krivelevich, Pikhurko and Szabo, and generalized by Alon, Krivelevich, Spencer and Szabo. (Wikipedia).

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Maker-Breaker game | Positional game | Family of sets