Extreme points of the Harsanyi set and the Weber set

• We apply matrix approach to axiomatize the Harsanyi set by Moebius transformation.
• The Weber set is characterized as the set of specialized Harsanyi payoff vectors.
• We explore the extreme points of two sets by studying the carrier of linear system.
• A recursive algorithm for computing the extreme points of Harsanyi set is proposed.

In this paper, we present firstly a matrix approach, by Moebius transformation, to axiomatize the Harsanyi payoff vectors in the traditional worth system instead of the dividend system. Then by this approach, the Weber set is also characterized as the set of specialized Harsanyi payoff vectors. The study of marginal contribution vectors, the extreme points of the Weber set is pivotal to characterize the Weber set. Recall that an extreme point of a linear system can be recognized by its carriers. A linear system associated to the Weber set is constructed and a second approach to investigate their extreme points is accessed by the concept of carrier. We apply the same technique to study the extreme points of the Harsanyi set. Together with the core-type structure of the Harsanyi set, we present a recursive algorithm for computing the extreme points of the Harsanyi set for any game.