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On the symmetry relation between different characteristic functions for additively separable cooperative games

This study was partially done while E. Gromova was with St. Petersburg State University

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  • We analyze 4 characteristic functions $ V^\alpha $, $ V^\delta $, $ V^\zeta $, and $ V^\eta $, and give a necessary condition for these functions to satisfy the relation $ V^\alpha - V^\delta = V^\zeta - V^\eta $ for all coalitions $ S $. To do so, we define and formally analyze the class of additively separable games. It is shown that many important types of games, both static and dynamic, belong to this class.

    Mathematics Subject Classification: Primary: 91A12;Secondary: 91A06.

    Citation:

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  • [1] E. BacchiegaL. Lambertini and A. Palestini, On the time consistency of equilibria in a class of additively separable differential games, J. Optim. Theory Appl., 145 (2010), 415-427.  doi: 10.1007/s10957-010-9673-6.
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    [3] E. J. Dockner, S. J orgensen, N. V. Long and G. Sorger, Differential Games in Economics and Management Science, Cambridge University Press: Cambridge, 2000. doi: 10.1017/CBO9780511805127.
    [4] W. M. Gorman, Conditions for additive separability, Econometrica: Journal of the Econometric Society, (1968), 605–609.
    [5] E. GromovaE. Marova and D. Gromov, A substitute for the classical Neumann–Morgenstern characteristic function in cooperative differential games, J. Dyn. Games, 7 (2020), 105-122.  doi: 10.3934/jdg.2020007.
    [6] E. Gromova and L. Petrosyan, On a approach to the construction of characteristic function for cooperative differential games, Autom. Remote Control, 78 (2017), 1680-1692.  doi: 10.1134/s0005117917090120.
    [7] T. Kamihigashi and T. Furusawa, Global dynamics in repeated games with additively separable payoffs, Review of Economic Dynamics, 13 (2010), 899-918. 
    [8] E. MarovaE. GromovaP. Barsuk and A. Shagushina, On the effect of the absorption coefficient in a differential game of pollution control, Mathematics, 8 (2020), 961-984. 
    [9] L. Petrosjan and E. Gromova, Two-level cooperation in coalitional differential games, Tr. Inst. Mat. Mekh., 20 (2014), 193-203. 
    [10] L. Petrosjan and G. Zaccour, Time-consistent Shapley value allocation of pollution cost reduction, J. Econom. Dynam. Control, 27 (2003), 381-398.  doi: 10.1016/S0165-1889(01)00053-7.
    [11] L. Petrosyan and N. Zenkevich, Game Theory, 2$^nd$ edition, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2016. doi: 10.1142/9824.
    [12] P. V. Reddy and G. Zaccour, A friendly computable characteristic function, Math. Social Sci., 82 (2016), 18-25.  doi: 10.1016/j.mathsocsci.2016.03.008.
    [13] L. S. Shapley, A value for n-person games, Contributions to the Theory of Games, 28 (1953), 307-317. 
    [14] J. Von Neumann and O. Morgenstern, Game Theory and Economic Behavior, Princeton University Press: Princeton, NJ, USA, 1944.
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