Computing the transfer pricing for a multidivisional firm based on cooperative games

This paper suggests a novel cooperative game-theoretic approach to solve the transfer pricing problem for a multidivisional firm considering that each division purchases goods from an upstream division in the supply chain. The formulation of the transfer pricing problem considers costs and taxes in a vertically integrated supply chain. We conceptualize the transfer pricing as a multi-objective problem and present a method for finding the strong Nash equilibrium that maximizes the utility of the entire multidivisional firm. The approach consists on determining a scalar_λ^* and the corresponding strategies_v^*_(λ^*₎ fixing specific bounds on the Pareto front. Bounds correspond to restrictions imposed by each division over the Pareto front that determine the maximum and minimum transfer price legally authorized. For ensuring the existence of a unique strong Nash equilibrium, we employ a penalized regularization method for poly-linear functions. We implement a recurrent procedure for finding the extremal point. For finding the strong Nash equilibrium, we select the Pareto optimal strong strategy_v^*_(λ^*₎ with minimal distance to the utopia point. We show that the regularized functional of the game decreases and converges, proving the existence and uniqueness of strong Nash equilibrium. The usefulness of the method is successfully demonstrated by a numerical example.