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 Quantitative Biology , 2015, Abstract: The Shapley Value and the Fair Proportion Index of phylogenetic trees have been frequently discussed as prioritization tools in conservation biology. Both indices rank species according to their contribution to total phylogenetic diversity, allowing for a simple conservation criterion. While both indices have their specific advantages and drawbacks, it has recently been shown that both values are closely related. However, as different authors use different definitions of the Shapley Value, the specific degree of relatedness depends on the specific version of the Shapley Value - it ranges from a high correlation index to equality of the indices. In this note, we first give an overview of the different indices. Then we turn our attention to the mere ranking order provided by either of the indices. We show that even though the chance of two rankings being exactly identical (when obtained from different versions of the Shapley Value) decreases with an increasing number of taxa, the distance between the two rankings converges to zero, i.e. the rankings are becoming more and more alike. Moreover, we introduce our software package FairShapley, which was implemented in Perl and with which all calculations have been performed.
 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.
 - , 2015, Abstract: 基于2004—2012年的省际面板数据,利用基尼(Gini)系数测算农业保险的区域差异,并运用Shapley值分解方法对影响区域差异的因素进行分解并测度出各类因素对区域差异的贡献度.结果表明:1)2004—2012年,表征农业保险区域差异的Gini系数均值高达0.64,明显高于公认的警戒水平,但呈现收敛趋势,Gini系数最低的2012年已降至0.47.2)区域差异的50.59%来自地区农村经济发展差异的贡献,地区的社会环境及保险市场差异也是形成区域差异的重要动因,贡献率分别达到26.06%和20.78%,地区固有因素贡献最小仅为2.58%.因此,要将努力缩小地区农村经济发展差距作为破解区域差异过大的主要抓手.与此同时,重视地区间农村教育、科技和社会保障水平的发展差异及地区间保险市场发展差异也应作为缩小区域差异的重要手段.此外,对于发展相对滞后的地区,还要注重发挥后发优势,吸收、借鉴发达地区的经验与教训,主动利用政策倾斜等手段吸引农业保险公司进入并开发地区市场,积极运用培训、宣传手段提升农民的风险与保险意识等.On the basis of the provincial panel data during 2004-2012,the Gini coefficient was used to calculate the regional differences of agricultural insurance,and taking use of the Shapley value decomposition method,the factors affecting the regional differences were decomposed and their contribution degree to the differences was calculated.The results showed from the following two aspects.1) During the years from 2004 to 2012,the mean of the Gini coefficient representing regional differences of agricultural insurance was as high as 0.64,which was absolutely above the accepted alert level,but showing a convergent trend.In 2012,the Gini coefficient reached the lowest level of 0.47.2) 50.59% of the regional differences resulted from regional differences in rural economic development level,and 26.06% from regional differences in social environment and 20.78% from regional differences in insurance market development level.Regional inherent factor contributed the least of 2.58% to the regional differences.Therefore,the task of narrowing economic development level gaps among rural regions should be put in the first place when resolving the problem of big regional differences.Meanwhile,the regional differences in rural education level,science level and social insurance level as well as the insurance market development level should also be taken into account when resolving the problem.Besides,as for those regions lagging behind, making use of the late-mover advantage to draw lessons from the experiences of the developed regions should be considered,the advantages of the preferential policies should be taken to attract agricultural insurance companies to enter and develop regional markets,and methods such as training and advertisement should be used to improve famers' awareness of risk and insurance.
 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.
 Computer Science , 2013, Abstract: This paper concerns the analysis of the Shapley value in matching games. Matching games constitute a fundamental class of cooperative games which help understand and model auctions and assignments. In a matching game, the value of a coalition of vertices is the weight of the maximum size matching in the subgraph induced by the coalition. The Shapley value is one of the most important solution concepts in cooperative game theory. After establishing some general insights, we show that the Shapley value of matching games can be computed in polynomial time for some special cases: graphs with maximum degree two, and graphs that have a small modular decomposition into cliques or cocliques (complete k-partite graphs are a notable special case of this). The latter result extends to various other well-known classes of graph-based cooperative games. We continue by showing that computing the Shapley value of unweighted matching games is #P-complete in general. Finally, a fully polynomial-time randomized approximation scheme (FPRAS) is presented. This FPRAS can be considered the best positive result conceivable, in view of the #P-completeness result.
 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.
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