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Summary: Discrepancy games
Noga Alon
Michael Krivelevich
Joel Spencer
Tibor Szab´o §
June 15, 2005
Abstract
We investigate a game played on a hypergraph H = (V, E) by two players, Bal-
ancer 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
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.
1 Introduction
In the classical theory of Maker/Breaker positional games a hypergraph H = (V, E) is
given and the players, Maker and Breaker, take turns in occupying a previously unoccupied
element of the "board" V . The goal of Breaker is to prevent Maker from fully occupying an
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