On the Effects of Priors in Weighted Voting Games
نویسندگان
چکیده
We study the Shapley value in weighted voting games. The Shapley value has been used as an index for measuring the power of individual agents in decision-making bodies or political organizations, in which decisions are made by a voting process. Previous studies assume that agent weights (corresponding to the size of a caucus or a political party) are fixed; we analyze new domains in which the weights are generated according to a certain probability distribution, modeling, for example, the effect of elections. We study how the parameters of the weight generating process affect the power disparity in the resulting game, by showing how different prior beliefs regarding the weight distribution affect the expected Shapley values of the agents. We examine several natural weight generation processes: the binomial distribution; the Balls and Bins model, with uniform as well as exponentially-decaying probabilities; and i.i.d. weights drawn from a known distribution. In particular, we draw a novel connection between the case of i.i.d. weights and renewal theory. We also analyze weights that admit a super-increasing sequence, answering several open questions pertaining to the Shapley value in such games.
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