Discrepancy Games

We investigate a game played on a hypergraph H = (V,E) by two players, Balancer and Unbalancer. They select one element of the vertex set V alternately until all vertices are selected. Balancer wins if at the end of the game all edges e ∈ E are roughly equally distributed between the two players. We give a polynomial time ∗Research supported in part by a USA-Israel BSF grant, by the Israel Science Foundation, by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University and by the Von Neumann Fund. †Research supported in part by USA-Israel BSF Grant 2002-133 and by grant 64/01 from the Israel Science Foundation. ‡Research supported in part by a USA-Israel BSF Grant. the electronic journal of combinatorics 12 (2005), #R51 1 algorithm for Balancer to win provided the allowed deviation is large enough. In particular, it follows from our result that if H is n-uniform and has m edges, then Balancer can achieve having between n/2−ln(2m)n/2 and n/2+ln(2m)n/2 of his vertices on every edge e of H. We also discuss applications in positional game theory.