Setting nash versus Kalai-Smorodinsky bargaining approach: Computing the continuous-time controllable Markov game

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 continuoustime, controllable and ergodicMarkov 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.