Balancing rationality and social influence: Alpha-rational Nash equilibrium in games with herding

TL;DR

This paper introduces the concept of alpha-Rational NE in mean-field games with both rational and herding-irrational players, revealing that irrational behaviour can sometimes lead to higher utilities for all players and alter classical equilibria.

Contribution

It proposes a novel equilibrium concept for mixed rational and herding-irrational players and analyzes its implications, showing benefits for rational players and potential advantages of irrationality.

Findings

Rational players can achieve higher utility leveraging herding behaviour.

New equilibria emerge, and some classical NEs are eliminated due to herding players.

Irrational players can attain utilities close to social optimality.

Abstract

The classical game theory models rational players and proposes Nash equilibrium (NE) as the solution. However, real-world scenarios rarely feature rational players; instead, players make inconsistent and irrational decisions. Often, irrational players exhibit herding behaviour by simply following the majority. In this paper, we consider the mean-field game with $\alpha$-fraction of rational players and the rest being herding-irrational players. For such a game, we introduce a novel concept of equilibrium named $\alpha$-Rational NE (in short, $\alpha$-RNE). The $\alpha$-RNEs and their implications are extensively analyzed in the game with two actions. Due to herding-irrational players, new equilibria may arise, and some classical NEs may be deleted. The rational players are not harmed but benefit from the presence of irrational players. Notably, we demonstrate through examples that rational players leverage upon the herding behaviour of irrational players and may attain higher utility (under $\alpha$-RNE) than social optimal utility (in the classical setting). Interestingly, the irrational players may also benefit by not being rational. We observe that irrational players do not lose compared to some classical NEs for participation and bandwidth sharing games. More importantly, in bandwidth sharing game, irrational players receive utility that approaches the social optimal utility. Such examples indicate that it may sometimes be `rational' to be irrational.