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 Computer Science , 2012, Abstract: We consider the problem of designing a sparse Gaussian process classifier (SGPC) that generalizes well. Viewing SGPC design as constructing an additive model like in boosting, we present an efficient and effective SGPC design method to perform a stage-wise optimization of a predictive loss function. We introduce new methods for two key components viz., site parameter estimation and basis vector selection in any SGPC design. The proposed adaptive sampling based basis vector selection method aids in achieving improved generalization performance at a reduced computational cost. This method can also be used in conjunction with any other site parameter estimation methods. It has similar computational and storage complexities as the well-known information vector machine and is suitable for large datasets. The hyperparameters can be determined by optimizing a predictive loss function. The experimental results show better generalization performance of the proposed basis vector selection method on several benchmark datasets, particularly for relatively smaller basis vector set sizes or on difficult datasets.
 Computer Science , 2012, Abstract: For $f$ a weighted voting scheme used by $n$ voters to choose between two candidates, the $n$ \emph{Shapley-Shubik Indices} (or {\em Shapley values}) of $f$ provide a measure of how much control each voter can exert over the overall outcome of the vote. Shapley-Shubik indices were introduced by Lloyd Shapley and Martin Shubik in 1954 \cite{SS54} and are widely studied in social choice theory as a measure of the "influence" of voters. The \emph{Inverse Shapley Value Problem} is the problem of designing a weighted voting scheme which (approximately) achieves a desired input vector of values for the Shapley-Shubik indices. Despite much interest in this problem no provably correct and efficient algorithm was known prior to our work. We give the first efficient algorithm with provable performance guarantees for the Inverse Shapley Value Problem. For any constant $\eps > 0$ our algorithm runs in fixed poly$(n)$ time (the degree of the polynomial is independent of $\eps$) and has the following performance guarantee: given as input a vector of desired Shapley values, if any "reasonable" weighted voting scheme (roughly, one in which the threshold is not too skewed) approximately matches the desired vector of values to within some small error, then our algorithm explicitly outputs a weighted voting scheme that achieves this vector of Shapley values to within error $\eps.$ If there is a "reasonable" voting scheme in which all voting weights are integers at most $\poly(n)$ that approximately achieves the desired Shapley values, then our algorithm runs in time $\poly(n)$ and outputs a weighted voting scheme that achieves the target vector of Shapley values to within error $\eps=n^{-1/8}.$
 Mathematics , 2014, Abstract: Following the original interpretation of the Shapley value (Shapley, 1953a) as a priori evaluation of the prospects of a player in a multi-person interaction situation, we propose a group value, which we call the Shapley group value, as a priori evaluation of the prospects of a group of players in a coalitional game when acting as a unit. We study its properties and we give an axiomatic characterization. Relaying on this valuation we analyze the profitability of a group. We motivate our proposal by means of some relevant applications of the Shapley group value, when it is used as an objective function by a decisionmaker who is trying to identify an optimal group of agents in a framework in which agents interact and the attained benefit can be modeled bymeans of a transferable utility game. As an illustrative examplewe analyze the problem of identifying the set of key agents in a terrorist network.
 Mathematics , 2005, DOI: 10.1007/s00285-007-0126-2 Abstract: Every weighted tree corresponds naturally to a cooperative game that we call a "tree game"; it assigns to each subset of leaves the sum of the weights of the minimal subtree spanned by those leaves. In the context of phylogenetic trees, the leaves are species and this assignment captures the diversity present in the coalition of species considered. We consider the Shapley value of tree games and suggest a biological interpretation. We determine the linear transformation M that shows the dependence of the Shapley value on the edge weights of the tree, and we also compute a null space basis of M. Both depend on the "split counts" of the tree. Finally, we characterize the Shapley value on tree games by four axioms, a counterpart to Shapley's original theorem on the larger class of cooperative games.
 Modern Applied Science , 2009, DOI: 10.5539/mas.v2n4p76 Abstract: In this paper we extend the notion of Shapley value to the stochastic cooperative games. We give the definition of marginal vector to the stochastic cooperative games and we define the Shapley value for this game. Furthermore, we discuss the axioms of the Shapley value and give the proofs of these axioms.
 Vladislav Kargin Mathematics , 2003, Abstract: This paper introduces a measure of uncertainty in the determination of the Shapley value, illustrates it with examples, and studies some of its properties. The introduced measure of uncertainty quantifies random variations in a player's marginal contribution during the bargaining process. The measure is symmetric with respect to exchangeable substitutions in the players, equal to zero for dummy player, and convex in the game argument. The measure is illustrated by several examples of abstract games and an example from epidemiology.
 Computer Science , 2014, Abstract: We propose the study of computing the Shapley value for a new class of cooperative games that we call budgeted games, and investigate in particular knapsack budgeted games, a version modeled after the classical knapsack problem. In these games, the "value" of a set $S$ of agents is determined only by a critical subset $T\subseteq S$ of the agents and not the entirety of $S$ due to a budget constraint that limits how large $T$ can be. We show that the Shapley value can be computed in time faster than by the na\"ive exponential time algorithm when there are sufficiently many agents, and also provide an algorithm that approximates the Shapley value within an additive error. For a related budgeted game associated with a greedy heuristic, we show that the Shapley value can be computed in pseudo-polynomial time. Furthermore, we generalize our proof techniques and propose what we term algorithmic representation framework that captures a broad class of cooperative games with the property of efficient computation of the Shapley value. The main idea is that the problem of determining the efficient computation can be reduced to that of finding an alternative representation of the games and an associated algorithm for computing the underlying value function with small time and space complexities in the representation size.
 Irinel Dragan American Journal of Operations Research (AJOR) , 2018, DOI: 10.4236/ajor.2018.86025 Abstract: In a cooperative transferable utilities game, the allocation of the win of the grand coalition is an Egalitarian Allocation, if this win is divided into equal parts among all players. The Inverse Set relative to the Shapley Value of a game is a set of games in which the Shapley Value is the same as the initial one. In the Inverse Set, we determined a family of games for which the Shapley Value is also a coalitional rational value. The Egalitarian Allocation of the game is efficient, so that in the set called the Inverse Set relative to the Shapley Value, the allocation is the same as the initial one, but may not be coalitional rational. In this paper, we shall find out in the same family of the Inverse Set, a subfamily of games with the Egalitarian Allocation is also a coalitional rational value. We show some relationship between the two sets of games, where our values are coalitional rational. Finally, we shall discuss the possibility that our procedure may be used for solving a very similar problem for other efficient values. Numerical examples show the procedure to get solutions for the efficient values.
 M. Pinter Computer Science , 2008, Abstract: We consider Young (1985)'s characterization of the Shapley value, and give a new proof of this axiomatization. Moreover, as applications of the new proof, we show that Young (1985)'s axiomatization of the Shapley value works on various well-known subclasses of TU games.
 Joss Sánchez-Pérez Theoretical Economics Letters (TEL) , 2012, DOI: 10.4236/tel.2012.25100 Abstract: This paper studies the economic inequality as a phenomenon resulting from two variables: The variable of responsibility which depends on the agents’ performance in the economy, and the variable of opportunity which is independent of the individuals’ performance in the economy. Identifying the contribution of the inequality of opportunity in the total inequality as the problem to be measured, we propose a decomposition of the Atkinson inequality index, by using one of the central solution concepts of cooperative game theory, called the Shapley value.
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