This paper addresses Monte Carlo algorithms for calculating the Shapley-Shubik power index in weighted majority games. First, we analyze a naive Monte Carlo algorithm and discuss the required number of samples. We then propose an efficient Monte Carlo algorithm and show that our algorithm reduces the required number of samples as compared to the naive algorithm.
We investigate a class of weighted voting games for which weights are randomly distributed over the standard probability simplex. We provide close-formed formulae for the expectation and density of the distribution of weight of the $k$-th largest player under the uniform distribution. We analyze the average voting power of the $k$-th largest player and its dependence on the quota, obtaining analytical and numerical results for small values of $n$ and a general theorem about the functional form of the relation between the average Penrose--Banzhaf power index and the quota for the uniform measure on the simplex. We also analyze the power of a collectivity to act (Coleman efficiency index) of random weighted voting games, obtaining analytical upper bounds therefor.
In this paper, we propose a pseudo polynomial size LP formulation for finding a payoff vector in the least core of a weighted voting game. The numbers of variables and constraints in our formulation are both bounded by $mbox{O}(n W_+)$, where $n$ is the number of players and $W_+$ is the total sum of (integer) voting weights. When we employ our formulation, a commercial LP solver calculates a payoff vector in the least core of practical weighted voting games in a few seconds. We also extend our approach to vector weighted voting games.
The attribution problem, that is the problem of attributing a models prediction to its base features, is well-studied. We extend the notion of attribution to also apply to feature interactions. The Shapley value is a commonly used method to attribute a models prediction to its base features. We propose a generalization of the Shapley value called Shapley-Taylor index that attributes the models prediction to interactions of subsets of features up to some size k. The method is analogous to how the truncated Taylor Series decomposes the function value at a certain point using its derivatives at a different point. In fact, we show that the Shapley Taylor index is equal to the Taylor Series of the multilinear extension of the set-theoretic behavior of the model. We axiomatize this method using the standard Shapley axioms -- linearity, dummy, symmetry and efficiency -- and an additional axiom that we call the interaction distribution axiom. This new axiom explicitly characterizes how interactions are distributed for a class of functions that model pure interaction. We contrast the Shapley-Taylor index against the previously proposed Shapley Interaction index (cf. [9]) from the cooperative game theory literature. We also apply the Shapley Taylor index to three models and identify interesting qualitative insights.
In the last decade, sequential Monte-Carlo methods (SMC) emerged as a key tool in computational statistics. These algorithms approximate a sequence of distributions by a sequence of weighted empirical measures associated to a weighted population of particles. These particles and weights are generated recursively according to elementary transformations: mutation and selection. Examples of applications include the sequential Monte-Carlo techniques to solve optimal non-linear filtering problems in state-space models, molecular simulation, genetic optimization, etc. Despite many theoretical advances the asymptotic property of these approximations remains of course a question of central interest. In this paper, we analyze sequential Monte Carlo methods from an asymptotic perspective, that is, we establish law of large numbers and invariance principle as the number of particles gets large. We introduce the concepts of weighted sample consistency and asymptotic normality, and derive conditions under which the mutation and the selection procedure used in the sequential Monte-Carlo build-up preserve these properties. To illustrate our findings, we analyze SMC algorithms to approximate the filtering distribution in state-space models. We show how our techniques allow to relax restrictive technical conditions used in previously reported works and provide grounds to analyze more sophisticated sequential sampling strategies.
We address the approximation of functionals depending on a system of particles, described by stochastic differential equations (SDEs), in the mean-field limit when the number of particles approaches infinity. This problem is equivalent to estimating the weak solution of the limiting McKean-Vlasov SDE. To that end, our approach uses systems with finite numbers of particles and a time-stepping scheme. In this case, there are two discretization parameters: the number of time steps and the number of particles. Based on these two parameters, we consider different variants of the Monte Carlo and Multilevel Monte Carlo (MLMC) methods and show that, in the best case, the optimal work complexity of MLMC, to estimate the functional in one typical setting with an error tolerance of $mathrm{TOL}$, is $mathcal Oleft({mathrm{TOL}^{-3}}right)$ when using the partitioning estimator and the Milstein time-stepping scheme. We also consider a method that uses the recent Multi-index Monte Carlo method and show an improved work complexity in the same typical setting of $mathcal Oleft(mathrm{TOL}^{-2}log(mathrm{TOL}^{-1})^2right)$. Our numerical experiments are carried out on the so-called Kuramoto model, a system of coupled oscillators.
Yuto Ushioda
,Masato Tanaka
,Tomomi Matsui
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(2021)
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"Monte Carlo Methods for Calculating Shapley-Shubik Power Index in Weighted Majority Games"
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Tomomi Matsui
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