Abstract
The bargaining game refers to a situation in which players have the possibility of concluding a mutually beneficial agreement. Here there is a conflict of interests about which agreement to conclude or no-agreement may be imposed on any player without that player’s approval. Remarkably, bargaining and its game-theoretic solutions has been applied in many important contexts like corporate deals, arbitration, duopoly market games, negotiation protocols, etc. Among all these research applications, equilibrium computation serves as a basis. This chapter examines bargaining games from a theoretical perspective and provides a solution method for the game-theoretic models of bargaining presented by Nash and Kalai–Smorodinsky which propose an elegant axiomatic approach to solve the problem depending on different principles of fairness. Our approach is restricted to a class of continuous-time, controllable and ergodic Markov games. We first introduce and axiomatize the Nash bargaining solution. Then, we present the Kalai–Smorodinsky approach that improves the Nash’s model by introducing the monotonicity axiom. For the solution of the problem we suggest a bargaining solver implemented by an iterated procedure of a set of nonlinear equations described by the Lagrange principle and the Tikhonov regularization method to ensure convergence to a unique equilibrium point. Each equation in this solver is an optimization problem for which the necessary condition of a minimum is solved using the projection gradient method. An important result of this chapter shows the equilibrium computation in bargaining games. In particular, we present the analysis of the convergence as well as the rate of convergence of the proposed method. The usefulness of our approach is demonstrated by a numerical example contrasting the Nash and Kalai–Smorodinsky bargaining solution problem.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Abreu, D., Manea, M.: Markov equilibria in a model of bargaining in networks. Games Econ. Behav. 75(1), 1–16 (2012)
Agastya, M.: Adaptive play in multiplayer bargaining situations. Rev. Econ. Stud. 64(3), 411–426 (1997)
Alexander, C.: The Kalai-Smorodinsky bargaining solution in wage negotiations. J. Oper. Res. Soc. 43(8), 779–786 (1992)
Anant, T.C.A., Mukherji, B., Basu, K.: Bargaining without convexity: generalizing the Kalai-Smorodinsky solution. Econ. Lett. 33(2), 115–119 (1990)
Antipin, A.S.: An extraproximal method for solving equilibrium programming problems and games. Comput. Math. Math. Phys. 45(11), 1893–1914 (2005)
Bolt, W., Houba, H.: Strategic bargaining in the variable threat game. Econ. Theor. 11(1), 57–77 (1998)
Cai, H.: Inefficient Markov perfect equilibria in multilateral bargaining. Econ. Theor. 22(3), 583–606 (2003)
Carrillo, L., Escobar, J., Clempner, J.B., Poznyak, A.S.: Solving optimization problems in chemical reactions using continuous-time Markov chains. J. Math. Chem. 54, 1233–1254 (2016)
Clempner, J.B., Poznyak, A.S.: Simple computing of the customer lifetime value: a fixed local-optimal policy approach. J. Syst. Sci. Syst. Eng. 23(4), 439–459 (2014)
Clempner, J.B., Poznyak, A.S.: Stackelberg security games: computing the shortest-path equilibrium. Expert Syst. Appl. 42(8), 3967–3979 (2015)
Clempner, J.B., Poznyak, A.S.: Conforming coalitions in Stackelberg security games: setting max cooperative defenders vs. non-cooperative attackers. Appl. Soft Comput. 47, 1–11 (2016)
Clempner, J.B., Poznyak, A.S.: Multiobjective Markov chains optimization problem with strong pareto frontier: principles of decision making. Expert Syst. Appl. 68, 123–135 (2017)
Clempner, J.B., Poznyak, A.S.: Using the extraproximal method for computing the shortest-path mixed Lyapunov equilibrium in Stackelberg security games. Math. Comput. Simul. (2017). doi:10.1016/j.matcom.2016.12.010
Coles, M.G., Muthoo, A.: Bargaining in a non-stationary environment. J. Econ. Theory 109(1), 70–89 (2003)
Cripps, M.W.: Markov bargaining games. J. Econ. Dyn. Control 22(3), 341–355 (1998)
Driesen, B., Perea, A., Peters, H.: The Kalai-Smorodinsky bargaining solution with loss aversion. Math. Soc. Sci. 61(1), 58–64 (2011)
Dubra, J.: An asymmetric Kalai-Smorodinsky solution. Econ. Lett. 73(2), 131–136 (2001)
Forgó, F., Szép, J., Szidarovszky, F.: Introduction to the Theory of Games: Concepts, Methods, Applications. Kluwer Academic Publishers, Dordrecht (1999)
Kalai, E.: Solutions to the bargaining problem. Social Goals and Social Organization, pp. 75–105. Cambridge University Press, Cambridge (1985)
Kalai, E., Smorodinsky, M.: Other solutions to Nash’s bargaining problem. Econometrica 43(3), 513–518 (1975)
Kalandrakis, A.: A three-player dynamic majoritarian bargaining game. J. Econ. Theory 116(2), 294–322 (2004)
Kennan, J.: Repeated bargaining with persistent private information. Rev. Econ. Stud. 68, 719–755 (2001)
Köbberling, V., Peters, H.: The effect of decision weights in bargaining problems. J. Econ. Theory 110(1), 154–175 (2003)
Merlo, A., Wilson, C.: A stochastic model of sequential bargaining with complete information. Econometrica 63(2), 371–399 (1995)
Moulin, H.: Implementing the Kalai-Smorodinsky bargaining solution. J. Econ. Theory 33(1), 32–45 (1984)
Muthoo, A.: Bargaining Theory with Applications. Cambridge University Press, Cambridge (2002)
Naidu, S., Hwang, S., Bowles, S.: Evolutiogame bargaining with intentional idiosyncratic play. Econ. Lett. 109(1), 31–33 (2010)
Nash, J.F.: The bargaining problem. Econometrica 18(2), 155–162 (1950)
Nash, J.F.: Two person cooperative games. Econometrica 21, 128–140 (1953)
Osborne, M., Rubinstein, A.: Bargaining and Markets. Academic Press Inc., San Diego (1990)
Peters, H., Tijs, S.: Individually monotonic bargaining solutions for n-person bargaining games. Methods Oper. Res. 51, 377–384 (1984)
Poznyak, A.S.: Advance Mathematical Tools for Automatic Control Engineers. Vol 2 Stochastic Techniques. Elsevier, Amsterdam (2009)
Poznyak, A.S., Najim, K., Gomez-Ramirez, E.: Self-learning Control of Finite Markov Chains. Marcel Dekker, New York (2000)
Raiffa, H.: Arbitration schemes for generalized two-person games. Ann. Math. Stud. 28, 361–387 (1953)
Roth, A.E.: An impossibility result converning n-person bargaining games. Int. J. Game Theory 8(3), 129–132 (1979)
Rubinstein, A., Wolinsky, A.: Equilibrium in a market with sequential bargaining. Econometrica 53(5), 1133–1150 (1985)
Trejo, K.K., Clempner, J.B., Poznyak, A.S.: Computing the Lp-strong Nash equilibrium looking for cooperative stability in multiple agents Markov games. In: 12th International Conference on Electrical Engineering, Computing Science and Automatic Control, pp. 309–314. Mexico City. Mexico (2015)
Trejo, K.K., Clempner, J.B., Poznyak, A.S.: Computing the Stackelberg/Nash equilibria using the extraproximal method: convergence analysis and implementation details for Markov chains games. Int. J. Appl. Math. Comput. Sci. 25(2), 337–351 (2015)
Trejo, K.K., Clempner, J.B., Poznyak, A.S.: A Stackelberg security game with random strategies based on the extraproximal theoretic approach. Eng. Appl. Artif. Intell. 37, 145–153 (2015)
Trejo, K.K., Clempner, J.B., Poznyak, A.S.: Adapting strategies to dynamic environments in controllable Stackelberg security games. In: 55th IEEE Conference on Decision and Control, pp. 5484–5489. Las Vegas, USA (2016)
Trejo, K.K., Clempner, J.B., Poznyak, A.S.: Computing the strong \(L_p\)-Nash equilibrium for Markov chains games: convergence and uniqueness. Appl. Math. Model. 41, 399–418 (2017)
von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1944)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this chapter
Cite this chapter
Trejo, K.K., Clempner, J.B. (2018). Setting Nash Versus Kalai–Smorodinsky Bargaining Approach: Computing the Continuous-Time Controllable Markov Game. In: Clempner, J., Yu, W. (eds) New Perspectives and Applications of Modern Control Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-62464-8_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-62464-8_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-62463-1
Online ISBN: 978-3-319-62464-8
eBook Packages: EngineeringEngineering (R0)