A Mathematical Analysis of the Potential Function Method of Cooperative Relationships

• In Event: 425 - Regular Session. Mathematical Sociology

Abstract

This paper analyzes the emergence of cooperative relationships under competitions. Many times, competitions are “zero-sum”; i.e., if he or she gets a reward, then I will less likely get a reward. Hence, cooperation under zero-sum does not seem to be possible. But, by using potential game theory, this paper argues that, contrary to our intuition, cooperative relationships can emerge under certain zero-sum situations. Moreover, our analysis suggests competitions do encourage the emergence of some local cooperative networks. On the other hand, if there is no competition, contrary to our intuition, cooperative relationships do not likely emerge because one’s own utility function and the positive externalities of the network cannot establish a stable equilibrium. Even if a cooperative relationship emerges, it is unstable and does not last for the long run. Hence, this paper suggests competitions are necessary ingredients for the emergence of stable cooperative relationships.
In addition, this paper proposes a further extension of potential game approach by integrating the Exponential Random Graph Model. By doing this, we may be able to develop a reliable predictive model of the emergence of certain cooperative networks.

Author

• Robert Hideo Mamada, University of California-Irvine