Cooperative grey games and the grey Shapley value

This contribution is located in the common area of operational research and economics, with a close relation and joint future potential with optimization: game theory. We focus on collaborative game theory under uncertainty. This study is on a new class of cooperative games where the set of players is finite and the coalition values are interval grey numbers. An interesting solution concept, the grey Shapley value, is introduced and characterized with the properties of additivity, efficiency, symmetry and dummy player, which are straightforward generalizations of the corresponding properties in the classical cooperative game theory. The paper ends with a conclusion and an outlook to future studies.

[1]  L. Shapley.  Cores of convex games , 1971 .

[2]  Daniel Granot,et al.  Cooperative Games in Stochastic Characteristic Function Form , 1977 .

[3]  Ramon E. Moore.  Methods and applications of interval analysis , 1979, SIAM studies in applied mathematics.

[4]  Deng Ju-Long,et al.  Control problems of grey systems , 1982 .

[5]  H. Young.  Monotonic solutions of cooperative games , 1985 .

[6]  L. Shapley.  A Value for n-person Games , 1988 .

[7]  Hubertus Th. Jongen,et al.  Semi-infinite optimization: Structure and stability of the feasible set , 1990 .

[8]  Hubertus Th. Jongen,et al.  Nonlinear optimization: Characterization of structural stability , 1991, J. Glob. Optim..

[9]  Hubertus Th. Jongen,et al.  One-Parametric Semi-Infinite Optimization: On the Stability of the Feasible Set , 1994, SIAM J. Optim..

[10]  Anastasia Telesetsky,et al.  The Kyoto Protocol , 1999 .

[11]  Anja De Waegenaere,et al.  Cooperative games with stochastic payoffs , 1999, Eur. J. Oper. Res..

[12]  G. W. Weber,et al.  AN ALGORITHMIC APPROACH BY LINEAR PROGRAMMING PROBLEMS IN GENERALIZED SEMI-INFINITE OPTIMIZATION ¤ , 2000 .

[13]  Gerhard-Wilhelm Weber.  Impulses from Results of Inverse Problems to Discrete Tomography , 2001, Electron. Notes Discret. Math..

[14]  Klaus Meer,et al.  Some aspects of studying an optimization or decision problem in different computational models , 2002, Eur. J. Oper. Res..

[15]  S. Pickl.  An Iterative Solution to the Nonlinear Time‐Discrete TEM Model ‐ The Occurrence of Chaos and a Control Theoretic Algorithmic Approach , 2002 .

[16]  Elliot Mendelson.  Introducing Game Theory and its Applications , 2004 .

[17]  Desheng Dash Wu,et al.  The method of grey related analysis to multiple attribute decision making problems with interval numbers , 2005, Math. Comput. Model..

[18]  Yi Lin,et al.  Grey Information - Theory and Practical Applications , 2005, Advanced Information and Knowledge Processing.

[19]  P. Taylan,et al.  NEW APPROACHES TO REGRESSION IN FINANCIAL MATHEMATICS BY ADDITIVE MODELS , 2007 .

[20]  S. Z. Alparslan-Gök,et al.  SEMI-INFINITE AND CONIC OPTIMIZATION IN MODERN HUMA N, LIFE AND FINANCIAL SCIENCES UNDER UNCERTAINTY , 2007 .

[21]  G.-W. Weber,et al.  Optimization of gene-environment networks in the presence of errors and uncertainty with Chebychev approximation , 2008 .

[22]  Mohamed Elnourani.  COGNITIVE RADIO AND GAME THEORY : OVERVIEW AND SIMULATION , 2008 .

[23]  Hans Peters,et al.  Game Theory: A Multi-Leveled Approach , 2008 .

[24]  Rodica Branzei,et al.  Convex Interval Games , 2009, Adv. Decis. Sci..

[25]  Gerhard-Wilhelm Weber,et al.  FINANCIAL REGRESSION AND ORGANIZATION , 2009 .

[26]  Gerhard-Wilhelm Weber,et al.  On optimization, dynamics and uncertainty: A tutorial for gene-environment networks , 2009, Discret. Appl. Math..

[27]  Erik Kropat,et al.  Regression Analysis for Clusters in Gene-Environment Networks based on Ellipsoidal Calculus and Optimization , 2009 .

[28]  Erik Kropat,et al.  OPTIMIZATION APPLIED ON REGULATORY AND ECO-FINANCE NETWORKS - SURVEY AND NEW DEVELOPMENTS - , 2010 .

[29]  ilkay Ulusoy,et al.  Inverse solution for parameter estimation of a dynamic volume spline based forehead skin model , 2010 .

[30]  Rodica Branzei,et al.  Cooperative interval games: a survey , 2010, Central Eur. J. Oper. Res..

[31]  Rodica Branzei,et al.  The interval Shapley value: an axiomatization , 2010, Central Eur. J. Oper. Res..

[32]  Rodica Branzei,et al.  Cooperative games under interval uncertainty: on the convexity of the interval undominated cores , 2011, Central Eur. J. Oper. Res..

[33]  G. Weber,et al.  RCMARS: Robustification of CMARS with different scenarios under polyhedral uncertainty set , 2011 .

[34]  Erik Kropat,et al.  Dynamical Gene-Environment Networks Under Ellipsoidal Uncertainty: Set-Theoretic Regression Analysis Based on Ellipsoidal OR , 2011 .

[35]  R. Branzei,et al.  Set-valued solution concepts using interval-type payoffs for interval games , 2011 .

[36]  Stef Tijs,et al.  Fuzzy interval cooperative games , 2011, Fuzzy Sets Syst..

[37]  G. Weber,et al.  CMARS: a new contribution to nonparametric regression with multivariate adaptive regression splines supported by continuous optimization , 2012 .

[38]  Yu-Hsien Liao,et al.  Alternative Axiomatizations of the Shapley Value under Interval Uncertainty , 2012, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[40]  S. Z. A. Gök,et al.  On the interval Shapley value , 2014 .

[41]  Erik Kropat,et al.  Spline regression models for complex multi-modal regulatory networks , 2014, Optim. Methods Softw..