9. Bargaining Games or How to Negotiate

The term “bargaining game” describes a scenario in which participants may reach a win-win agreement. There is a conflict of interest here over whether an agreement should be reached or whether no agreement should be reached without the consent of each player. Amazingly, negotiating protocols, duopoly market games, business transactions, arbitration, and other key scenarios have all used bargaining and its game-theoretic solutions. The underpinning for all of these research applications is equilibrium computation. This chapter explores the theory behind bargaining games and offers a way for solving the game-theoretic models of bargaining put out by Nash and Kalai-Smorodinsky, which suggest a beautiful axiomatic solution to the problem based on several fairness standards. We consider a class of continuous-time, controllable, and ergodic Markov games. The Nash bargaining solution is first introduced and axiomatized. The Kalai-Smorodinsky approach, which adds the monotonicity postulate to the Nash’s model, is then presented. We recommend using a bargaining solver to resolve the issue, which is carried out iteratively using a set of nonlinear equations represented by the Lagrange principle and the Tikhonov regularization approach to guarantee convergence to a particular 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. This chapter’s key finding illustrates how equilibrium calculations work in bargaining games. We specifically discuss the convergence analysis and rate of convergence of the suggested technique. A numerical example contrasting the Nash and Kalai-Smorodinsky bargaining solution problem is used to illustrate the value of the considered method.