Academia.eduAcademia.edu

Outline

Title
Abstract
Introduction
Particular Cases
References
All Topics
Computer Science
Theoretical Computer Science

Centroids of the core of exact capacities: a comparative study

Profile image of Enrique MirandaEnrique Miranda

Annals of Operations Research

https://doi.org/10.1007/S10479-022-05097-1
visibility

…

description

41 pages

link

1 file

Abstract

Capacities are a common tool in decision making. Each capacity determines a core, which is a polytope formed by additive measures. The problem of eliciting a single probability from the core is interesting in a number of fields: in coalitional game theory for selecting a fair way of splitting the wealth between the players, in the transferable belief model from evidence theory or for transforming a second order into a first order model. In this paper, we study this problem when the goal is to determine the centroid of the core of a capacity, and we compare four approaches: the Shapley value, the average of the extreme points, the incenter with respect to the total variation distance and the limit of a procedure of uniform contraction. We show that these four centroids do not coincide in general, we give some sufficient conditions for their equality, and we analyse their axiomatic properties. We also discuss how to define a notion of centrality measure indicating the degree of centra...

Related papers

On the core of linear production games
Guillermo Owen

Mathematical Programming, 1975

We show that, in cooperative production games, when the production functions are not concave, the core may well be empty. However, as the number of players increases (subject to some regularity conditions) the relative deficit obtained by using concavified functions decreases to zero. Furthermore, differentiability of the functions will cause the absolute deficit to go to zero.

View PDFchevron_right
Analyzing cooperative game theory solutions: core and Shapley value in cartesian product of two sets
Mohammed El Kamli

Frontiers in applied mathematics and statistics, 2024

The core and the Shapley value stand out as the most renowned solutions for addressing sharing problems in cooperative game theory. These concepts are widely acknowledged for their e ectiveness in tackling negotiation, resource allocation, and power dynamics. The present study aims to extend various notions of cooperative games from the standard set N to a new class of cooperative games defined on the cartesian product N × N ′ (utilizing the specific coalition A * B). This extension encompasses fundamental concepts such as rationality, core, and Shapley value. The findings presented in this study demonstrate that the core concept as a solution yields a set of imputations without favoring any specific point within the set, in contrast to the Shapley value, which o ers a singular solution. Moreover, the results confirm that the Shapley value satisfies the conditions defining the core of a game. Through both theoretical analysis and numerical findings, employing a practical example, it becomes evident that the Shapley value o ers a more distinct solution to the sharing problem compared with the core solution.

View PDFchevron_right
The Shapley value for capacities and games on set systems
Fabien Lange

2006

We propose a generalization of capacities which encompass in a large extent the class of Choquet’s capacities. Then, we define the class of probabilistic values over these capacities, which are values satisfying classical axioms, the well-known Shapley value being one. Lastly, we propose a value on these capacities by borrowing ideas from electric networks theory.

View PDFchevron_right
Linear measures of inequality for cooperative games
Ezra Einy

Journal of Economic Theory, 1991

We provide a self-contained axiomatization of linear inequality measures for coalitional games. Adding one axiom to those of Weymark (Math. Sot. Sci. 1 (1981) 4099430), we prove that all such measures must be generalized Gini functions of the Shapley value. Also, we generalize the foregoing result to the spaces pNA(p) of non-atomic games. Journal of Economic Literature Classification Numbers: 024. 026.

View PDFchevron_right
On approximate cores of non-convex economies
Shlomo Weber

Economics Letters, 1984

In this note we investigate, for non-convex finite economies, the relationship between the existence of approximate core allocations and the size of an economy.

View PDFchevron_right
The equity core and the Lorenz-maximal allocations in the equal division core
Francesc Llerena Garrés

RePEc: Research Papers in Economics, 2013

In this note, we give a geometrical decomposition of the equity core as a finite union of polyhedrons. As a consequence, we characterize the non-emptiness of the equity core (Selten in Decision theory and social ethics: issues in social choice, Reidel, Dordrecht, 289-301, 1978) and provide a method, easy to implement, for computing the Lorenz-maximal allocations in the equal division core (Dutta and Ray in Games Econ Behav 3:403-422, 1991).

View PDFchevron_right
Diversification, convex preferences and non-empty core in the Choquet expected utility model
Jean-marc Tallony, Alain Chateauneuf

Economic Theory, 2002

We s h o w, in the Choquet expected utility model, that preference for diversication, that is, convex preferences, is equivalent to a concave utility index and a convex capacity. W e then introduce a weaker notion of diversication, namely \sure diversication." W e show that this implies that the core of the capacity i s non-empty. The converse holds under concavity of the utility index. This property i s s h o wn to be equivalent to the notion of c o m onotone diversication ; notion that we i ntroduce in the paper. Finally, in the expected utility model, all these notions of diversication are equivalent and are represented by the concavity of the utility i ndex. . z We thank R.A. Dana for useful discussions, as well as I. Gilboa and P. W akker for comments on a previous draft.

View PDFchevron_right
Axiomatic characterizations of the core and the Shapley value of the broadcasting game
Juan D. Moreno-Ternero

International journal of game theory, 2024

We study the cooperative game associated with a broadcasting problem (the allocation of revenues raised from the collective sale of broadcasting rights for a sports tournament). We show that the set of core allocations can be characterized with three axioms: additivity, null team and monotonicity. We also show that the Shapley value can be characterized with additivity, equal treatment of equals and core selection.

View PDFchevron_right
On the coincidence of the core and the bargaining sets
Carles Rafels

2012

View PDFchevron_right
On extensions of the core and the anticore of transferable utility games
Hans Peters, Peter Sudhölter

International Journal of Game Theory, 2014

We consider several related set extensions of the core and the anticore of games with transferable utility. An efficient allocation is undominated if it cannot be improved, in a specific way, by sidepayments changing the allocation or the game. The set of all such allocations is called the undominated set, and we show that it consists of finitely many polytopes with a core-like structure. One of these polytopes is the L1-center, consisting of all efficient allocations that minimize the sum of the absolute values of the excesses. The excess Pareto optimal set contains the allocations that are Pareto optimal in the set obtained by ordering the sums of the absolute values of the excesses of coalitions and the absolute values of the excesses of their complements. The L1-center is contained in the excess Pareto optimal set, which in turn is contained in the undominated set. For three-person games all these sets coincide. These three sets also coincide with the core for balanced games and with the anticore for antibalanced games. We study properties of these sets and provide characterizations in terms of balanced collections of coalitions. We also propose a singlevalued selection from the excess Pareto optimal set, the min-prenucleolus, which is defined as the prenucleolus of the minimum of a game and its dual.

View PDFchevron_right

References (48)

  1. Abellán, J., & Moral, S. (2003). Maximum entropy for credal sets. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 11, 587-597.
  2. Angilella, S., Bottero, M., Corrente, S., Ferretti, Greco, S., & Lami, I. M. (2016). Non additive robust ordinal regression for urban and territorial planning: An application for sitting in urban waste landfill. Annals of Operations Research, 245, 427-456.
  3. Augustin, T., Coolen, F., de Cooman, G., & Troffaes, M. (Eds). (2014). Introduction to Imprecise Probabilities. Wiley Series in Probability and Statistics. Wiley.
  4. Bader, U., Gelander, T., & Monod, N. (2012). A fixed point theorem for L 1 spaces. Inventiones mathematicae, 189, 143-148.
  5. Banzhaf, J. F. (1965). Weighted voting does not work: A mathematical analysis. Rutgers Law Review, 19, 317-343.
  6. Baroni, P., & Vicig, P. (2005). An uncertainty interchange format with imprecise probabilities. International Journal of Approximate Reasoning, 40, 147-180.
  7. Cascos, I. (2009). Data depth: Multivariate statistics and geometry. In W. S. Kendall & I. Molchanov (Eds.), New perspectives in stochastic geometry. Oxford Scholarship Online.
  8. Choquet, G. (1953). Theory of capacities. Annales de l'Institut Fourier, 5, 131-295.
  9. Corsato, C., Pelessoni, R., & Vicig, P. (2019). Nearly-linear uncertainty measures. International Journal of Approximate Reasoning, 114, 1-28.
  10. de Campos, L. M., Huete, J. F., & Moral, S. (1994). Probability intervals: A tool for uncertain reasoning. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 2, 167-196.
  11. Destercke, S. (2017). On the median in imprecise ordinal problems. Annals of Operations Research, 256, 375-392.
  12. Destercke, S., Montes, I., & Miranda, E. (2022). Processing multiple distortion models: A comparative study. International Journal of Approximate Reasoning, 145(C), 91-120.
  13. Dubois, D., & Prade, H. (1980). Fuzzy sets and systems. Theory and applications. Academic Press.
  14. Elbassioni, K., & Tiway, H. R. (2012). Complexity of approximating the vertex centroid of a polyhedron. Theoretical Computer Science, 421, 56-61.
  15. Gilboa, I., & Schmeidler, D. (1989). Maxmin expected utility with a non-unique prior. Journal of Mathematical Economics, 18, 141-153.
  16. Grabisch, M. (2013). The core of games on ordered structures and graphs. Annals of Operations Research, 204, 33-64.
  17. Grabisch, M. (2016). Set functions, games and capacities in decision making. Springer.
  18. Huntley, N., & Troffaes, M. (2012). Normal form backward induction for decision trees with coherent lower previsions. Annals of Operations Research, 195, 111-134.
  19. Jaffray, J. (1995). On the maximum-entropy probability which is consistent with a convex capacity. Interna- tional Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 3(1), 27-33.
  20. Keith, A., & Ahner, D. (2021). A survey of decision making and optimization under uncertainty. Annals of Operations Research, 300, 319-353.
  21. Klibanoff, P., Marinacci, M., & Mukerji, S. (2005). A smooth model of decision making under ambiguity. Econometrica, 73(6), 1849-1892.
  22. Klir, G. J., & Parviz, B. (1992). Probability-possibility transformations: A comparison. International Journal of General Systems, 21, 291-310.
  23. Kumar, I. E., Venkatasubramanian, S., Scheidegger, C., & Friedler, S. A. (2020). Problems with Shapley- value-based explanations as feature importance measures. arXiv:2002.11097.
  24. Levi, I. (1980). The enterprise of knowledge. MIT Press.
  25. Levin, D. A., Peres, Y., & Wilmer, E. L. (2009). Markov chains and mixing times. American Mathematical Society.
  26. Lundberg, S., & Lee, S. I. (2017). A unified approach to interpreting model predictions. arXiv:1705.07874.
  27. Miranda, E., Couso, I., & Gil, P. (2003). Extreme points of the credal sets generated by 2-alternating capacities. International Journal of Approximate Reasoning, 33(1), 95-115.
  28. Miranda, E., & Destercke, S. (2015). Extreme points of the credal sets generated by comparative probabilities. Journal of Mathematical Psychology, 64(65), 44-57.
  29. Miranda, E., & Montes, I. (2018). Shapley and Banzhaf values as probability transformations. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 26(6), 917-947.
  30. Miranda, E., & Montes, I. (2021). Centroids of credal sets: a comparative study. In J. Vejnarová, & N. Wilson (Eds.), Symbolic and quantitative approaches to reasoning with uncertainty. ECSQARU 2021, Volume 12897 of Lecture Notes in Artificial Intelligence (pp. 427-441). Springer.
  31. Montes, I., & Destercke, S. (2017). Extreme points of p-boxes and belief functions. Annals of Mathematics and Artificial Intelligence, 81(3), 405-428.
  32. Montes, I., Miranda, E., & Destercke, S. (2019). Pari-mutuel probabilities as an model. Information Sciences, 481, 550-573.
  33. Montes, I., Miranda, E., & Destercke, S. (2020a). Unifying neighbourhood and distortion models: Part I-New results on old models. International Journal of General Systems, 49(6), 602-635.
  34. Montes, I., Miranda, E., & Destercke, S. (2020b). Unifying neighbourhood and distortion models: Part II-New models and synthesis. International Journal of General Systems, 49(6), 636-674.
  35. Neyman, A. (2002). Values of games with infinitely many players. Handbook of Game Theory with Economic Applications, 3, 2121-2167.
  36. Pelessoni, R., Vicig, P., & Zaffalon, M. (2010). Inference and risk measurement with the pari-mutuel model. International Journal of Approximate Reasoning, 51, 1145-1158.
  37. Sarin, R., & Wakker, P. (1992). A simple axiomatization of nonadditive expected utility. Econometrica, 60(6), 1255-1272.
  38. Shafer, G. (1976). A mathematical theory of evidence. Princeton University Press.
  39. Shapley, L. S. (1953). A value for n-person game. Annals of Mathematics Studies, 28, 307-317.
  40. Shapley, L. S. (1971). Cores of convex games. International Journal of Game Theory, 1, 11-26.
  41. Smets, P. (2005). Decision making in the TBM: The necessity of the pignistic transformation. International Journal of Approximate Reasoning, 38, 133-147.
  42. Smets, P., & Kennes, R. (1994). The transferable belief model. Artificial Intelligence, 66(2), 191-234.
  43. Troffaes, M. C. M. (2007). Decision making under uncertainty using imprecise probabilities. International Journal of Approximate Reasoning, 45(1), 17-29.
  44. Tukey, J. (1975). Mathematics and the picturing of data. In Proceedings of the international congress of mathematicians (Vol. 2, pp. 523-531).
  45. Walley, P. (1981). Coherent lower (and upper) probabilities. Statistics research report.
  46. Walley, P. (1991). Statistical reasoning with imprecise probabilities. Chapman and Hall.
  47. Wallner, A. (2007). Extreme points of coherent probabilities in finite spaces. International Journal of Approx- imate Reasoning, 44(3), 339-357.
  48. Weber, R. J. (1988). Probabilistic values for games. In A. E. Roth (Ed.), The Shapley value. Essays in honour of L.S. Shapley (pp. 101-119). Cambridge University Press.

Related papers

Weighted values and the core
D Monderer

International Journal of Game Theory, 1992

View PDFchevron_right
From set-valued solutions to single-valued solutions: the centroid and the core-center. Working Paper 03-09
Estela Sánchez

In the framework of cooperative game theory there are several solution concepts that give rise to different ways of dividing the worth of the grand coalition v(N) among the players. We work on the set of transferable utility games, shortly, TU games. Although solution concepts admit different classifications, to our purpose, we divide them into two groups: set-valued solutions and single-valued solutions. Roughly speaking, set-valued solutions provide a subset of the set of all assignments, this subset is chosen according to some specific criteria, which depend on the kind of solution we are aiming for. Examples of this type are the stable sets (von Neumann and Morgenstern, 1944), the core (Gillies, 1953), the kernel (Davis and Maschler, 1965), the bargaining sets (Aumann and Maschler, 1964; among others), etc. On the other hand, one can establish some desirable properties or axioms that determine a unique solution for each game, and that is known as a single-valued solution. The Shapley value (Shapley, 1953) and the nucleolus (Schmeidler, 1969) are solutions of this type. Each solution concept has its interpretation and attends to specific principles (fairness, stability...) and all of them enrich the setting of cooperative game theory. Besides, there are many papers that research on relations between single-valued solutions and set-valued solutions. To mention some of the results, when the core is non-empty, the nucleolus selects an imputation in it, and for the class of convex games, the Shapley value is in the core. Our work establishes a direct connection between a set-valued solution and a particular singled-valued solution obtained from it. * We thank Ignacio García-Jurado for valuable comments and conversations. The authors received finantial support from the Ministerio de Ciencia y Tecnología and the FEDER through grant BEC2002-04102-C02-02.

View PDFchevron_right
Diversification, Convex Preferences and Non-Empty Core
Alain Chateauneuf

Econometric Society World Congress 2000 Contributed Papers, 1998

We s h o w, in the Choquet expected utility model, that preference for diversication, that is, convex preferences, is equivalent to a concave utility index and a convex capacity. W e then introduce a weaker notion of diversication, namely \sure diversication." W e show that this implies that the core of the capacity i s non-empty. The converse holds under concavity of the utility index. This property i s s h o wn to be equivalent to the notion of c o m onotone diversication ; notion that we i ntroduce in the paper. Finally, in the expected utility model, all these notions of diversication are equivalent and are represented by the concavity of the utility i ndex.

View PDFchevron_right
Dempster-Shafer Updating and Additivity of the Core (Applications of Discrete Convex Analysis to Game Theory and Mathematical Economics)
Atsushi Kajii

We discuss an interpretation of the Dempster-Shafer updating rule for belief functions in terms of unanimity games, utilizing the additivity of the core operation. This interpretation leads us to a simple proof for the equivalence of Dempster-Shafer updating rule and the maximum likelihood updating rule.

View PDFchevron_right
Cores of Cooperative Games in Information Theory
Mokshay Madiman

EURASIP Journal on Wireless Communications and Networking, 2008

Cores of cooperative games are ubiquitous in information theory and arise most frequently in the characterization of fundamental limits in various scenarios involving multiple users. Examples include classical settings in network information theory such as Slepian-Wolf source coding and multiple access channels, classical settings in statistics such as robust hypothesis testing, and new settings at the intersection of networking and statistics such as distributed estimation problems for sensor networks. Cooperative game theory allows one to understand aspects of all these problems from a fresh and unifying perspective that treats users as players in a game, sometimes leading to new insights. At the heart of these analyses are fundamental dualities that have been long studied in the context of cooperative games; for information theoretic purposes, these are dualities between information inequalities on the one hand and properties of rate, capacity, or other resource allocation regions on the other.

View PDFchevron_right

Related topics

  • Mathematics
  • Theory Of Computation

    • Cited by

    Inner approximations of coherent lower probabilities and their application to decision making problems
    Andres Presa

    Annals of Operations Research

    We consider a decision making problem under imprecision, where the probabilistic information is given in terms of a set of probability measures, and where finding the optimal alternative(s) may be difficult. To ease the computation, we propose to transform the initial model into another one that (1) belongs to some subclass with better mathematical properties, such as supermodularity or complete monotonicity; (2) is at least as informative as the original model, while being as close as possible to it. We show that the problem can be approached in terms of linear or quadratic programming and that it can be connected with the one of determining the incenter of a credal set. Finally, we compare the solutions of a decision making problem with the initial and the transformed models and illustrate how our approach can be applied in a decision making problem under severe uncertainty.

    View PDFchevron_right
    580 California St., Suite 400
    San Francisco, CA, 94104
    © 2025 Academia. All rights reserved