Gain and loss of esteem, direct reciprocity and Heider balance

TL;DR

This paper introduces a model for social relations considering gain and loss of esteem, revealing that asymmetry leads to stable jammed states, preventing the attainment of Heider balance and resulting in distinct social phases.

Contribution

It extends existing social relation models by incorporating asymmetry and demonstrates the emergence of jammed states and phase distinctions.

Findings

Most stable solutions are jammed states due to asymmetry.

Three phases identified: jammed, balanced with two hostile groups, and paradise.

Asymmetry prevents Heider balance in the majority of cases.

Abstract

The effect of gain and loss of esteem is introduced into the equations of time evolution of social relations, hostile or friendly, in a group of actors. The equations allow for asymmetric relations. We prove that in the presence of this asymmetry, the majority of stable solutions are jammed states, i.e. the Heider balance is not attained there. A phase diagram is constructed with three phases: the jammed phase, the balanced phase with two mutually hostile groups, and the phase of so-called paradise, where all relations are friendly.