Voting games with abstention: a probabilistic characterization of power and a special case of Penrose's Limit Theorem

In general, analyses of voting power are performed through the notion of a simple voting game (SVG) in which every voter can choose between two options: 'yes' or 'no'. Felsenthal and Machover (1997) introduced the concept of ternary voting games (TVGs) which recognizes abstention alongside. They derive appropriate generalizations of the Shapley-Shubik and Banzhaf indices in TVGs. Braham and Steffen (2002) argued that the decision-making structure of a TVG may not be justified. They propose a sequential structure in which voters first decide between participation and abstention and then between yes or no. The purpose of this paper is twofold. First, it compares the two approaches and shows how the probabilistic interpretation of power provides a unifying characterization of analogues of the Banzhaf (Bz) measure. Second, using the probabilistic approach we shall prove a special case of Penrose's Limit Theorem (PLT). This theorem deals with an asymptotic property in weighted voting games with an increasing number of voters. It says that under certain conditions the ratio between the voting power of any two voters (according to various measures of voting power) approaches the ratio between their weights. We show that PLT holds in TVGs for analogues of Bz measures, irrespective of the particular nature of abstention.