Combinatorial Structures of Max-Type Functions characterizing the Optimal Solution of the Equivalence Problem

This paper presents a solution to the so called equivalence problem which was introduced in [2]. The equivalence problem consists of a semismooth system of equations which has the following form: ystem can be formulated in a more simplified representation by ctor ũ can be interpreted as possible control parameters of a time-discrete system, whereas the vector x̃ can be seen as bargaining solution of a cooperative game. In [2] the bargaining solution is identical to the τ-value, which was introduced in [3]. For the special case of three actors the τ-value lies always in the core, if we assume 1-convexity. Under these assumptions, all properties are expressed by the general formulation of (2) where the functions f g and h are of the form ξ(u) = min{u1,…, un} = −max{−u1, …, − un}. By exploiting the combinatorial structure of max-type functions we can show that a solution of (2) exists and may be found via Newton type methods which are treated in [1].