We consider MultiCriteria Decision Analysis models which are defined over discrete attributes, taking a finite number of values. We do not assume that the model is monotonically increasing with respect to the attributes values. Our aim is to define an importance index for such general models, considering that they are equivalent to @math -ary games (multichoice games). We show that classical solutions like the Shapley value are not suitable for such models, essentially because of the efficiency axiom which does not make sense in this context. We propose an importance index which is a kind of average variation of the model along the attributes. We give an axiomatic characterization of it.
Grabisch and Lange @cite_13 did not use unanimity games, but took an axiomatic approach to define a Shapley value in a more general context for games over lattices. They define the Shapley value for multichoice game as follows, @math
@cite_13
2127813674 2167073044 2436401953 2128449113
Coalitional games allow subsets (coalitions) of players to cooperate to receive a collective payoff. This payoff is then distributed “fairly” among the members of that coalition according to some division scheme. Various solution concepts have been proposed as reasonable schemes for generating fair allocations. The Shapley value is one classic solution concept: player i’s share is precisely equal to i’s expected marginal contribution if the players join the coalition one at a time, in a uniformly random order. In this paper, we consider the class of supermodular games (sometimes called convex games), and give a fully polynomial-time randomized approximation scheme (FPRAS) to compute the Shapley value to within a (1 ±e) factor in monotone supermodular games. We show that this result is tight in several senses: no deterministic algorithm can approximate Shapley value as well, no randomized algorithm can do better, and both monotonicity and supermodularity are required for the existence of an efficient (1 ±e)-approximation algorithm. We also argue that, relative to supermodularity, monotonicity is a mild assumption, and we discuss how to transform supermodular games to be monotonic. We present a new approach to representing coalitional games based on rules that describe the marginal contributions of the agents. This representation scheme captures characteristics of the interactions among the agents in a natural and concise manner. We also develop efficient algorithms for two of the most important solution concepts, the Shapley value and the core, under this representation. The Shapley value can be computed in time linear in the size of the input. The emptiness of the core can be determined in time exponential only in the treewidth of a graphical interpretation of our representation. The Shapley value is a key solution concept for coalitional games in general and voting games in particular. Its main advantage is that it provides a unique and fair solution, but its main drawback is the complexity of computing it (e.g., for voting games this complexity is #p-complete). However, given the importance of the Shapley value and voting games, a number of approximation methods have been developed to overcome this complexity. Among these, Owen's multi-linear extension method is the most time efficient, being linear in the number of players. Now, in addition to speed, the other key criterion for an approximation algorithm is its approximation error. On this dimension, the multi-linear extension method is less impressive. Against this background, this paper presents a new approximation algorithm, based on randomization, for computing the Shapley value of voting games. This method has time complexity linear in the number of players, but has an approximation error that is, on average, lower than Owen's. In addition to this comparative study, we empirically evaluate the error for our method and show how the different parameters of the voting game affect it. Specifically, we show the following effects. First, as the number of players in a voting game increases, the average percentage error decreases. Second, as the quota increases, the average percentage error decreases. Third, the error is different for players with different weights; players with weight closer to the mean weight have a lower error than those with weight further away. We then extend our approximation to the more general k-majority voting games and show that, for n players, the method has time complexity O(k^2n) and the upper bound on its approximation error is O(k^2 n). Preface 1. Introduction to the Shapley value Alvin E. Roth Part I. Ancestral Papers: 2. A value for n-person games Lloyd S. Shapley 3. A method for evaluating the distribution of power in a committee system Lloyd S. Shapley and Martin Shubik Part II. Reformulation and Generalizations: 4. The expected utility of playing a game Alvin E. Roth 5. The Shapley-Shubik and Banzhaf power indices as probabilities Philip D. Straffin, Jr. 6. Weighted Shapley values Ehud Kalai and Dov Samet 7. Probabilistic values for games Robert James Weber 8. Combinatorial representations of the Shapley value based on average relative payoffs Uriel G. Rothblum 9. The potential of the Shapley value Sergiu Hart and Andreu Mas-Colell 10. Multilinear extensions of games Guillermo Owen Part III. Coalitions: 11. Coalitional value Mordecai Kurz 12. Endogenous formation of links between players and the coalitions: an application of the Shapley value Robert J. Aumann and Roger B. Myerson Part IV. Large Games: 13. Values of large finite games Myrna Holtz Wooders and William R. Zame 14. Payoffs in nonatomic economics: an axiomatic approach Pradeep Dubley and Abraham Neyman 15. Values of smooth nonatomic games: the method of multilinear approximation Dov Monderer and Abraham Neyman 16. Nondifferentiable TU markets: the value Jean-Francois Mertens Part V. Cost Allocation and Fair Division: 17. Individual contribution and just compensation H. P. Young 18. The Aumann-Shapley prices: a survey Yair Tauman Part VI. NTU Games: 19. Utility comparison and the theory of games Lloyd S. Shapley 20. Paths leading to the Nash set Michael Maschler, Guillermo Owen and Bezalel Peleg.