Continuous Box game

The classical positional game Box was introduced by Chvátal and Erdős in 1978 in their study of the biased connectivity game on the complete graph. Their analysis was subsequently extended by Hamidoune and Las Vergnas. The board of the Box game consists of elements of n disjoint sets (boxes), which might vary in size. The game is played by two players, Maker and Breaker. Maker claims m board elements per move whereas Breaker claims just one. Maker wins this game if and only if he claims all elements of some box by the end of the game. In this paper we introduce the game CBox, a continuous version of the Box game, where the sizes of the boxes need not be integral and in every move Maker puts a nonnegative real weight into each box, such that the weights sum up to the real number m. This new game, while closely related to the original Box game, turns out to be more amenable to analysis – we derive explicit and easy to use criteria for determining the winner in every instance of the game. Consequently, establishing a connection between CBox and Box, we also obtain applicable criteria for the Box game.