Measuring Voting Power in Convex Policy Spaces

Abstract

Classical power index analysis considers the individual's ability to influence the aggregated group decision by changing its own vote, where all decisions and votes are assumed to be binary. In many practical applications we have more options than either "yes" or "no". Here we generalize three important power indices to continuous convex policy spaces. This allows the analysis of a collection of economic problems like, e.g., tax rates or spending that otherwise would not be covered in binary models.

Axiomatizations for the Shapley-Shubik Power Index for Games With Several Levels of Approval in the Input and Output

“…This line of reasoning can be used to motivate a definition of a Shapley-Shubik index for (j, k) simple games as defined in [5], c.f. [12]. Suppose that voters successively and independently each choose a level of approval in J with equal probability.…”

“…The later games were called simple aggregation functions in [13], linking to the literature on aggregation functions [9], and interval simple games in [14]. A Shapley-Shubik like index for those games was motivated and introduced in [12], an axiomatization is given in [14].…”

“…[1] for some current research directions. We think that the variants from [5], for (j, k)-simple games, and from [12], for interval simple games, form one consistent way to generalize the Shapley-Shubik index for simple games. Here we mainly focus on an axiomatic justification, see our main result in Theorem 5.1.…”

An Axiomatization of the Shapley-Shubik Index for Interval Decisions

“…So, we want to study [0, 1] n → [0, 1]-functions v with v(0) = 0, v(1) = 1, and v(x) ≤ v(y) for all x, y ∈ [0, 1] n with x ≤ y. In [20] the author called those objects continuous simple games since, for simplicity, v was assumed to be continuous. To go in line with the above naming we call them interval simple games here 5 .…”

“…Mimicking the properties of a simple game we speak of interval simple games for real-valued decisions in [0,1]. A generalization of the Shapley-Shubik index to that context was proposed in [20]. Here we give an axiomatic justification for that power index.…”