Abstract
We consider legislative voting rules that govern collective approval or disapproval of a bill or a motion, and that allow abstention (or absence) as a “middle option” distinct from a yes or no vote. In contrast with Peter Fishburn’s work on representative systems, or RSs, we do not treat collective approval and disapproval symmetrically; a voting rule may have a built-in bias against passing motions, for example. In this asymmetric case, the additional assumption that a rule is anonymous (all votes count equally) still allows for a significant variety of rules, a number of which are used by real voting bodies (see Freixas & Zwicker (2003)). We provide three characterizations of weighted voting in this context, and discuss potential applications. In real legislative voting bodies an abstention or absence often does have an effect different from a voter’s yes or no vote. Yet since the publication of Theory of Games and Economic Behavior von Neumann & Morgenstern (1949) the standard mathematical model for a legislative voting system has been the simple game, which by virtue of its structure treats any non-yes vote as a no. Peter Fishburns 1973 work seems to be the earliest to have taken abstention seriously, but others followed: Rubenstein (1980); Bolger (1986, 1993a,b); Felsenthal & Machover (1997, 1998); Amer et al. (1998); Freixas & Zwicker (2003); Corte-Real & Pereira (2004); Dougherty & Edward (2004); Bilbao, Fernandez, Jimenez, & Lopez (2005a,b). Distinguishing features of a RS, as defined in Fishburn (1973), include:
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