NTU-Solutions for the Taxation Game

The Taxation Game is a cooperative game played between a set of countries $I$ = {$1,\ldots,n$} admitting the operation of a multinational enterprise (MNE, "the firm" ) within their jurisdiction. The firm, when operating within the territory of a subgroup $S {\subseteq} I$ of countries (a coalition) will generate a profit. Cooperating countries can agree about a share of the profit to be available for imposing taxes according to their rules and specifications, i.e., their tax rate or $tariff$. If the taxation basis, i.e., the profit obtained by the firm, is taken as the (monetary) value of the game, then we obtain a side payment or TU game, represented by a coalitional function $\stackrel{\circ}{\it{v}}$ defined on coalitions. This version is discussed by W. F. Richter. This author suggests that countries should agree on an allocation of the tax basis, i.e., the total profit $v(I)$ generated when the firm is operating in all participating countries. However, the profit obtained by the firm when (hypothetically) operating in in a subgroup (a coalition) $S {\subseteq} I$ of all countries should be taken into account. Consequently, the share of the tax basis alloted to a country should be determined by the Shapley value of the taxation game. The Shapley value "as a tool in theoretical economics" has widely been applied in Game Theory and Equilibrium Theory, but also in applications to Cost Sharing problems. We recall the Tenessee valley project, the determination of airport fees, and many others . By his approach Richter creates an interesting new field of applications. Within this paper we present a modification of this model by introducing the game $\stackrel{*}{\it{v}}$ dual to $\stackrel{\circ}{\it{v}}$. Moreover, we introduce the tariffs of countries, as the incentives of all parties involved (and actually of the firm) are based on their actual tax income depending on the tariffs. Then the resulting game is of NTU character and involves the tariffs. To this NTU game we apply a version of the Shapley NTU value. This way we characterize an agreement of the countries involved regarding the share of profit and the taxes resulting. A particular version of an NTU--game is given by a "bargaining problem" for $n$ countries. For simplicity of the argument we start out with this version and discuss the bargaining solution which are taylored versions of a Shapley version concept. We particularly deal with the Maschler-Perles solution based on superadditivity. Here, we focus on a suitable generalization of the Maschler-Perles solution.