Analysis in social networks with usage of modified Raiffa solution for~cooperative games

Ondřej Černík ; Radim Valenčík ; Petr Wawrosz

Contributions to game theory and management, Tome 8 (2015), pp. 8-20

Résumé

We present our approach based on Nash bargaining problem for $n$-player definition as a set $B$ settled pairs $(S, d)$. The elements $B$ of are called instance (examples) of the problem $B$, elements $S$ are called variants or vector of utility, point d is called the point of disagreement, or status quo. From the point of view that we develop it is interesting Raiffa's solution that was proposed in the early 1950's. Raiffa (1957) suggested dynamic procedures for the cooperative bargaining in which the set $S$ of possible alternatives is kept unchanged while the disagreement point $d$ gradually changes. He considers two variants of such process — a discrete one and the continuous one. Discrete Raiffa's solution is the limit of so called dictated revenues. Diskin A., Koppel M., Samet D. (2011) have provided an axiomatization of a family of generalized Raiffa's discrete solutions. The solution concept which is composed of two solution functions. One solution function specifies an interim agreement and the other specifies the terminal agreement. The solution that we suggest and that we called von Neumann–Morgenstern modified discrete Raiffa's solution for $n = 3$. Our approach modifies Raiffa solution as a value of compensation, that implies from affinity of the player Xr to the player Xs based on the assumption that they will be in the same $(n-1)$ members of coalition. Comparing the results of the original game with the game extended of player affinities brings valuable results if analysing various types of social networks. Particularly when examining relations based on investing in social status and when analysing the structures based on mutual covering of violations of the generally accepted principles.

Détail

Keywords: Nash bargaining problem for $n$-player, Raiffa solution, three-person game, social network, coalition affinity, social networks based on mutual covering violate the generally accepted principles, von Neumann–Morgernstern stable set.