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

TL;DR

This paper introduces new upper bounds on the dimension of simple games, improving previous bounds by linking the dimension to the size of binary covering codes, thus advancing understanding of simple game complexity.

Contribution

The paper presents two new upper bounds on the dimension of simple games, notably connecting it to binary covering codes, which improves upon the existing Taylor/Zwicker bound.

Findings

Bound on dimension improved from ${n race n/2}$ to covering code size

Constructive method for bounding dimension using binary covering codes

Progress towards closing the dimensionality gap for 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.