Box-making game


A box-making game is a biased positional game where two players alternately pick elements from a family of pairwise-disjoint sets. The first player - called BoxMaker - tries to pick all elements of a single box. The second player - called BoxBreaker - tries to pick at least one element of all boxes.
The box game was first presented by Paul Erdős and Václav Chvátal. It was solved later by Hamidoune and Las-Vergnas.

Definition

A box game is defined by:
The first player, BoxMaker, picks p balls. Then the second player, BoxBreaker, breaks q boxes. And so on.
BoxMaker wins if he has managed to pick all balls in at least one box, before BoxBreaker managed to break this box. BoxBreaker wins if he has managed to break all the boxes.

Strategies

In general, the optimal strategy for BoxBreaker is to break the boxes with the smallest number of remaining elements. The optimal strategy for BoxMaker is to try to balance the sizes of all boxes. By simulating these strategies, Hamidoune and Las-Vergnas found a sufficient and necessary condition for each player in the box game.
For the special case where q=1, the each of the following conditions is sufficient: