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

Abstract

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 ${n \choose n/2}$. We improve this upper bound significantly by showing constructively that $d$ is bounded from above by the cardinality of any binary covering code with length $n$ and covering radius $1$. This result supplements a recent result where Olsen et al. showed how to construct simple games with dimension $|C|$ for any binary constant weight SECDED code $C$ with length $n$. Our result represents a major step in the attempt to close the dimensionality gap for simple games.