On Covering Codes and Upper Bounds for the Dimension of Simple Games

Consider a situation with n agents or players where some of the players form a coalition with a certain collective objective. Simple games are used to model systems that can decide whether coalitions are successful (winning) or not (losing). A simple game can be viewed as a monotone boolean function. The dimension of a simple game is the smallest positive integer d such that the simple game can be expressed as the intersection of d threshold functions where each threshold function uses a threshold and n weights. Taylor and Zwicker have shown that d is bounded from above by the number of maximal losing coalitions. We present two new upper bounds both containing the Taylor/Zwicker-bound as a special case. The Taylor/Zwicker-bound imply an upper bound of (