Bayesian inference for weighted voting games

Weighted voting games are ubiquitous models which are used in many fields ranging from economics and political science to bioinformatics and machine learning. They model scenarios where agents with associated a non-negative weight, vote in favour or against a decision. The decision is made if and only if the total weight of those voting in favour of the decision is equal to or greater than a given quota. In this work we consider a pre-electoral setup, so to speak, where the weight configuration has to be learnt from sampling data. Tackling this statistical problem from a Bayesian viewpoint, we place a Dirichlet prior over the weight vector to obtain – by coniugacy – a Dirichlet posterior distri- bution that can then be used to make inference on various solution concepts of interest. In particular we consider the posterior behaviour of the Shapley value, a key index used to quantify the "political" power of each agent, and show how useful some tools from computational and discrete geometry are in this respect. Finally we briefly touch upon an extension of our framework where also the co- operation structure is modeled as a random (weighted) graph with connectivity estimated from (historical) data.