  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.
