A Pre-Kernel Characterization and Orthogonal Projection

Abstract

We have established that a quadratic and convex function can be attained from each payoff equivalence class. After that, we proved that the objective function, from which a pre-kernel element can be pursued, is composed of a finite collection of quadratic and convex functions. In addition, from each payoff equivalence class a linear transformation can be derived that maps payoff vectors into the space of unbalanced excess configurations. The resultant column vectors of the linear mapping constitutes a spanning system of a vector space of balanced excesses. Similar to payoff vectors, any vector of unbalanced excesses is mapped by an orthogonal projection on a m-dimensional flat of balanced excesses, whereas m ≤ n. Moreover, each payoff set determines the dimension and location of a particularly balanced excess flat in the vector space of unbalanced excesses. Since, a spanning system or basis of a flat is not unique, we can derive a set of transition matrices where each transition matrix constitutes a change of basis. This basis change has a natural interpretation, which transforms a bargaining situation into another equivalent bargaining situation. It is established that the transition matrices belong to the positive general linear group ${\text{GL}}^{+}(m; \mathbb{R})$. As a consequence, a group action can be identified on the set of all ordered bases of a flat of balanced excesses. Any induced payoff equivalence class of a TU game can be associated with a specific basis or bargaining situation. Finally, a first pre-kernel result with regard to the orthogonal projection method is given.