Recently, behaviors that seem to function as punishment or apology have bee
n reported among non-human primates as well as humans. Such behaviors appea
r to play an important role in maintaining cooperation between individuals.
Therefore, the evolution of these behaviors should be examined from the vi
ewpoint of the evolution of cooperation. The iterated prisoner's dilemma (I
PD) game is generally considered to be a standard model for the evolution o
f cooperation. In the present study, strategies accompanied by punishment-l
ike attacks or apology-like behavior were introduced into the common IPD si
mulation. Punishment and apology were represented by the P signal and the A
S signal given immediately after defection. A strategy with the P and AS si
gnals, named the pPAS strategy, was proved to be an evolutionarily stable s
trategy under certain conditions. Numerical simulations were carried out ac
cording to different assigned values of the costs of punishment and apology
. The simulations showed that pPAS could dominate the population (1) when t
he cost of giving P is relatively small, (2) when the cost of receiving P i
s relatively large, or (3) when the cost of giving AS is relatively large.
The relative cost of giving AS had the clearest effect on the success of pP
AS. pPAS can dominate the population even when a dominance asymmetry of the
costs between two players was introduced. The present results suggest the
possible evolution of social behaviors like punishment or apology as a mean
s of maintaining cooperation.