  We propose a consistent analytic approach to the efficiency of collisional
Penrose process in the vicinity of a maximally rotating Kerr black hole. We
focus on a collision with arbitrarily high center-of-mass energy, which occurs
if either of the colliding particles has its angular momentum fine-tuned to the
critical value to enter the horizon. We show that if the fine-tuned particle is
ingoing on the collision, the upper limit of the efficiency is
$(2+\sqrt{3})(2-\sqrt{2})\simeq 2.186$, while if the fine-tuned particle is
bounced back before the collision, the upper limit is $(2+\sqrt{3})^{2}\simeq
13.93$. Despite earlier claims, the former can be attained for inverse Compton
scattering if the fine-tuned particle is massive and starts at rest at
infinity, while the latter can be attained for various particle reactions, such
as inverse Compton scattering and pair annihilation, if the fine-tuned particle
is either massless or highly relativistic at infinity. We discuss the
difference between the present and earlier analyses.
