  In a social dilemma, cooperation is collectively optimal, yet individually
each group member prefers to defect. A class of successful strategies of direct
reciprocity were recently found for the iterated prisoner's dilemma and for the
iterated three-person public-goods game: By a successful strategy, we mean that
it constitutes a cooperative Nash equilibrium under implementation error, with
assuring that the long-term payoff never becomes less than the co-players'
regardless of their strategies, when the error rate is small. Although we have
a list of actions prescribed by each successful strategy, the rationale behind
them has not been fully understood for the iterated public-goods game because
the list has hundreds of entries to deal with every relevant history of
previous interactions. In this paper, we propose a method to convert such
history-based representation into an automaton with a minimal number of states.
Our main finding is that a successful strategy for the iterated three-person
public-goods game can be represented as a $10$-state automaton by this method.
In this automaton, each state can be interpreted as the player's internal
judgement of the situation, such as trustworthiness of the co-players and the
need to redeem oneself after defection. This result thus suggests a
comprehensible way to choose an appropriate action at each step towards
cooperation based on a situational judgement, which is mapped from the history
of interactions.
