Cooperation is a key to understand social behavior and decision-making in conflict situations in nature or society. The problem of cooperation is formulated as Iterated Prisoner's Dilemma in game theory. To find out pairs of strategies that establish mutual cooperation in equilibrium, several studies used evolutionary computer simulations. Although the payoff matrix of IPD is defined by the two ordinal inequalities, using simulations requires numerical payoff values. Since Axelrod used a numerical payoff matrix, many researchers have adopted the same payoff values with no justification. We guess that there is the common assumption that findings obtained by examining one numerical IPD payoff matrix hold in more general cases. However, this is not trivial, because we have no evidence supporting the assumption. In this article, we verify this assumption by analyzing IPD games of two payoff-maximizing players with memory-1 Markov strategies. To determine equilibria of IPD, we define an extension of Nash equilibrium with the partial derivatives of payoff functions. Our numerical and formal analyses falsify the assumption. We showed the formal evidence for the conditions that the WSLS pair does not become an equilibrium for some IPD games. Contrarily, our results suggest that the TFT pair might be an equilibrium for any IPD games. Falsification of the common assumption requests a new classification of strategy pairs as equilibria in IPD.
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