A Special Case of Penrose’s Limit Theorem When Abstention is Allowed

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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 [Felsenthal, D.S. and Machover, M. (1997), International Journal of Game Theory 26, 335–351.] 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 [Braham, M. and Steffen, F. (2002), in Holler, et al. (eds.), Power and Fairness, Jahrbuch fur Neue Politische Okonomie 20, Mohr Siebeck, pp. 333–348.] 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 two-fold. First, we compare the two approaches and show 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.