Abstract
We consider an extension of a noncooperative game problem where players have joint binding constraints. We suggest a shares allocation approach, which replaces the initial problem with a sequence of Nash equilibrium problems together with an upper level set-valued variational inequality as master problem. This transformation maintains the monotonicity properties of the underlying mappings. We also show that the regularization yields a decomposable penalty method, which removes complex functions in constraints within the custom noncooperative game framework and provides the single-valued master problem with strengthened monotonicity of its cost mapping.
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Owen, G.: Game Theory, 3rd edn. Academic Press, San Diego (1995)
Nash, J.: Non-cooperative games. Ann. Math. 54, 286–295 (1951)
Debreu, G.: A social equilibrium existence theorem. Proc. Nat. Acad. Sci. USA 38, 886–893 (1952)
Rosen, J.B.: Existence and uniqueness of equilibrium points for concave N-person games. Econometrica 33, 520–533 (1965)
Zukhovitskii, S.I., Polyak, R.A., Primak, M.E.: Two methods of search for equilibrium points of n-person concave games. Soviet Mathem. Dokl. 10, 279–282 (1969)
Zukhovitskii, S.I., Polyak, R.A., Primak, M.E.: Concave many person games. Ekon. Matem. Metody 7, 888–900 (1971). (in Russian)
Ichiishi, T.: Game Theory for Economic Analysis. Academic Press, New York (1983)
Okuguchi, K., Szidarovszky, F.: The Theory of Oligopoly with Multi-Product Firms. Springer-Verlag, Berlin (1990)
Krawczyk, J.B., Uryasev, S.: Relaxation algorithms to find Nash equilibria with economic applications. Environ. Model Assess 5, 63–73 (2000)
Facchinei, F., Kanzow, C.: Generalized Nash equilibrium problems. 4OR 5, 173–210 (2007)
Harker, P.T.: Generalized Nash games and quasivariational inequalities. Eur. J. Oper. Res. 54, 81–94 (1991)
Contreras, J., Klusch, M., Krawczyk, J.B.: Numerical solutions to Nash-Cournot equilibria in coupled constraint electricity markets. IEEE Trans. Power Syst. 19, 195–206 (2004)
Pang, J.-S., Scutari, G., Facchinei, F., Wang, C.: Distributed power allocation with rate constraints in Gaussian parallel interference channels. IEEE Trans. Inform. Theory 54, 3471–3489 (2008)
Nikaido, H., Isoda, K.: Note on noncooperative convex games. Pacific J. Mathem. 5, 807–815 (1955)
On the penalty method for constrained variational inequalities (New York, 1981)
Muu, L.D., Oettli, W.: A Lagrangian penalty function method for monotone variational inequalities. Num. Funct. Anal. Optim. 10, 1003–1017 (1989)
Konnov, I.V., Pinyagina, O.V.: D-gap functions for a class of equilibrium problems in Banach spaces. Comp. Meth. Appl. Math. 3, 274–286 (2003)
Facchinei, F., Kanzow, C.: Penalty methods for the solution of generalized Nash equilibrium problems. SIAM J. Optim. 20, 2228–2253 (2010)
Konnov, I.V.: On penalty methods for non monotone equilibrium problems. J. Glob. Optim. 59, 131–138 (2014a)
Facchinei, F., Lampariello, L.: Partial penalization for the solution of generalized Nash equilibrium problems. J. Glob. Optim. 50, 39–57 (2011)
Konnov, I.V.: Application of splitting methods to a class of equilibrium problems. J. Nonl. Conv. Anal. 5, 71–83 (2004)
Konnov, I.V.: An inexact splitting method for systems of equilibrium problems. Pacif. J. Optim. 1, 611–624(2005)
Konnov, I.V.: Application of the proximal point method to a system of extended primal-dual equilibrium problems. In: Seeger, A. (ed.) Recent advances in Optimization, Lecture Notes in Economics and Mathematical Systems, vol. 563, pp. 87–102. Springer, New York (2006)
Konnov, I.V.: Combined RelaxationMethods for Variational Inequalities. Springer-Verlag, Berlin (2001)
Belen’kii, V.Z., Volkonskii, V.A. (eds.): Iterative Methods in Game Theory and Programming. Moscow, Nauka (1974). (in Russian)
Mastroeni, G.: Gap functions for equilibrium problems. J. Glob. Optim. 27, 411–426 (2003)
Chadli, O., Konnov, I.V., Yao, J.-C.: Descent methods for equilibrium problems in a Banach space. Comput. Mathem. Appl. 48, 609–616 (2004)
Konnov, I.V., Ali, M.S.S.: Descent methods for monotone equilibrium problems in Banach spaces. J. Comp. Appl. Math. 188, 165–179 (2006)
Konnov, I.V.: Iterative solution methods for mixed equilibrium problems and variational inequalities with non-smooth functions. In: Haugen, I.N., Nilsen, A.S. (eds.) Game Theory: Strategies, Equilibria, and Theorems, Ch 4, pp. 117–160. NOVA, Hauppauge (2008)
Razumikhin, B.S.: Iterative method for the solution and decomposition of linear programming problems. Autom. Remote Control 29, 427–443 (1967)
Umnov, A.E.: The method of penalty functions in problems of large dimension. USSR Comp. Maths. Math. Phys. 15, 32–45 (1975)
Kornai, J., Liptak, T.: Two-level planning. Econometrica 33, 141–169 (1965)
Lasdon, L.S.: Optimization Theory for Large Systems. Macmillan, New York (1970)
Konnov, I.V.: Right-hand side decomposition for variational inequalities. J. Optim. Theory Appl. 160, 221–238 (2014b)
Bruck, R.: On weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space. J. Math. Anal. Appl. 61, 159–164 (1977)
Facchinei, F., Fischer, A., Piccialli, V.: On generalized Nash games and variational inequalities. Oper. Res. Lett. 35, 159–164 (2007)
Kulkarni, A.A., Shanbhag, U.V.: On the variational equilibrium as a refinement of the generalized Nash equilibrium. Automatica 48, 45–55 (2012)
Kulkarni, A.A., Shanbhag, U.V.: Revisiting generalized Nash games and variational inequalities. J. Optim. Theory Appl. 154, 175–186 (2012)
Blum, E., Oettli,W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 127–149 (1994)
Uryas’ev, S.P.: Adaptive Algorithms of Stochastic Optimization and Game Theory. Moscow, Nauka (1990). (in Russian)
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Konnov, I.V. Shares Allocation Methods for Generalized Game Problems with Joint Constraints. Set-Valued Var. Anal 24, 499–516 (2016). https://doi.org/10.1007/s11228-015-0347-2
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DOI: https://doi.org/10.1007/s11228-015-0347-2
Keywords
- Noncooperative games
- Joint constraints
- Generalized equilibrium points
- Shares allocation
- Set-valued variational inequality
- Penalty method