Twin relationships in Parsimonious Games: some results

TL;DR

This paper explores twin relationships in Parsimonious games, revealing that twin games share the same minimal winning quota and possess a unique balanced lottery determined by their weights.

Contribution

It provides new insights into twin relationships in Parsimonious games, including properties of their quotas and lotteries based on transposition of incidence matrices.

Findings

Twin games have the same minimal winning quota.

Each Parsimonious game has a unique balanced lottery.

Lottery probabilities are given by the twin game's weights.

Abstract

In a vintage paper concerning Parsimonious games, a subset of constant sum homogeneous weighted majority games, Isbell introduced a twin relationship based on transposition properties of the incidence matrices upon minimal winning coalitions of such games. A careful investigation of such properties allowed the discovery of some results on twin games presented in this paper. In detail we show that a) twin games have the same minimal winning quota and b) each Parsimonious game admits a unique balanced lottery on minimal winning coalitions, whose probabilities are given by the individual weights of its twin game.