An Algorithm for payoff space in C1 parametric games

Carfì, David and Ricciardello, Angela and Agreste, Santa (2011): An Algorithm for payoff space in C1 parametric games.

Abstract

We present a novel algorithm to determine the payoff-space of certain normal-form C1 parametric games, and - more generally - of continuous families of normal-form C1 games. The algorithm has been implemented by using MATLAB, and it has been applied to several examples. The implementation of the algorithm gives the parametric expressions of the critical zone of any game in the family under consideration both in the bistrategy space and in the payoff space and the graphical representations of the disjoint union (with respect to the parameter set of the parametric game) of the family of all payoff spaces. We have so the parametric representation of the union of all the critical zones. One of the main motivations of our paper is that, in the applications, many normal-form games appear naturally in a parametric fashion; moreover, some efficient models of coopetition are parametric games of the considered type. Specifically, we have realized an algorithm that provides the parametric and graphical representation of the payoff space and of the critical zone of a parametric game in normal-form, supported by a finite family of compact intervals of the real line. Our final goal is to provide a valuable tool to study simply (but completely) normal-form C1-parametric games in two dimensions.

• Item Type: MPRA Paper
• Language: English
• Keywords: two player normal form games; bargaining problems; cooperative games; competitive games; complete study of a normal-form game
• Subjects: C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C71 - Cooperative Games; C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory; C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C72 - Noncooperative Games; C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C73 - Stochastic and Dynamic Games ; Evolutionary Games ; Repeated Games

References

[1] J. Aubin, Optima and Equilibria, second edition Springer Verlag, (1998).

[2] J. Aubin, Mathematical Methods of Game and Economic Theory North- Holland, Amsterdam, (1980).

[3] M. J. Osborne and A. Rubinstein, A Course in Game Theory Academic Press, (2001).

[4] D. Carf`ı, Payoff space of C1 Games, Applied Sciences 11, 35-47 (2009).

[5] D. Carf`ı, Differentiable game complete analysis for tourism firm decisions, in Proceedings of the 2009 International Conference on Tourism and Work- shop on Sustainable Tourism within High Risk Areas of Environmental Crisis, Messina, Italy, 2009.

[6] D. Carf`ı, Optimal boundaries for decisions, Atti Accad. Pelorit. Pericol. Cl. Sci. Fis. Mat. Nat. 86,C1A0801002 (2008).

[7] D. Carf`ı, A. Ricciardello An Algorithm for Payoff Space in C1-Games Atti Accad. Pelorit. Pericol. Cl. Sci. Fis. Mat. Nat. Vol. 88, No. 1, C1A1001003 (2010)

[8] D. Carf`ı, G. Gambarelli, A. Uristani, Balancing interfering elements, pre- print, 2011

[9] D. Carf`ı, G. Patan`e, S. Pellegrino, A Coopetitive approach to Project financ- ing, preprint (2011)

[10] D. Carf`ı, D. Schilir`o, A model of coopetitive game and the Greek crisis, http://arxiv.org/abs/1106.3543

[11] D. Carf`ı, D. Schilir`o, Coopetitive games and transferable utility: an analyt- ical application to the Eurozone countries, pre-print, 2011

[12] D. Carf`ı, D. Schilir`o, Crisis in the Euro area: coopetitive game solutions as new policy tools, TPREF-Theoretical and Practical Re- search in Economic Fields, summer issue pp. 1-30. ISSN 2068 7710, http://www.asers.eu/journals/tpref.html, 2011

[13] D. Carf`ı, D. Schilir`o, A coopetitive model for a global green economy, sent to International Conference “Moving from crisis to sustainability: Emerging issues in the international context”, 2011

[14] G. Owen, Game Theory Academic Press, (2001).

[15] R. B. Myerson, Game Theory Harvard University Press, (1991).