Voters' power in voting games with abstention: Influence relation and ordinal equivalence of power theories

The influence relation was introduced by Isbell [Isbell, J.R., 1958. A class of simple games. Duke Math. J. 25, 423-439] to qualitatively compare the a priori influence of voters in a simple game, which by construction allows only "yes" and "no" votes. We extend this relation to voting games with abstention (VGAs), in which abstention is permitted as an intermediate option between a "yes" and a "no" vote. Unlike in simple games, this relation is not a preorder in VGAs in general. It is not complete either, but we characterize VGAs for which it is complete, and show that it is a preorder whenever it is complete. We also compare the influence relation with recent generalizations to VGAs of the Shapley-Shubik and Banzhaf-Coleman power indices [Felsenthal, D.S., Machover, M., 1997. Ternary voting games. Int. J. Game Theory 26, 335-351; Freixas, J., 2005a. The Shapley-Shubik power index for games with several levels of approval in the input and output. Dec. Support Systems 39, 185-195; Freixas, J., 2005b. The Banzhaf index for games with several levels of approval in the input and output. Ann. Operations Res. 137, 45-66]. For weakly equitable VGAs, the influence relation is a subset of the preorderings defined by these two power theories. We characterize VGAs for which the three relations are equivalent.

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