We agree with the authors that there is a plausible link between moral impurity and breakdown of cooperation. However, such a link does not logically require that immoral behavior negatively impacts self-control. It is enough that immoral behavior signals poor self-control. In fact, it's even enough that it signals impatience. For example, gluttony could harm cooperation even if it does not undermine the glutton's future ability to resist temptations. If potential partners consider gluttony to be a symptom of impatience, they will be reluctant to attempt cooperation.

Below, we articulate our argument with the help of a game-theoretic model. The model combines the theory of discounted repeated games with signaling theory. (For a textbook treatment of these theories, see, e.g., Fudenberg & Tirole, Reference Fudenberg and Tirole1991, Chs. 5 and 8, respectively.) For simplicity, we only consider the role of patience. That is, we refrain from considering lack of self-control, for example in the form of non-exponential discounting. As shown by Obara and Park (Reference Obara and Park2017) and Bernergård (Reference Bernergård2019), exponential and non-exponential discounting serve essentially identical functions with respect to sustaining cooperation in infinitely repeated games.

Two agents, drawn at random from a large population of agents, are involved in the following two-stage interaction. In the first stage, henceforth called the individual stage, each agent faces a choice whether to engage in moderation or gluttony. Moderation yields a utility of 2 today and 2 tomorrow. Gluttony yields utility of 4 today and −1 tomorrow. The patience of agent i is captured by a subjective discount factor δ_i ∈ (0, 1). Thus, the total utility of moderation is 2 + 2δ_i and the total utility of gluttony is 4 − δ_i. The former is larger than the latter if δ_i ≥ 2/3.

In the second stage, henceforth called the group stage, the two agents play an infinitely repeated Prisoners' Dilemma game. Each agent chooses whether to make a sacrifice for the other's benefit or not to do so, that is, to defect. In each round, mutual sacrifice yields a utility of 1 each, mutual defection yields a utility of 0 each, and when one agent defects and the other sacrifices the former gets 2 and the latter gets −1. Suppose for simplicity that agents restrict attention to two strategies at the group stage. One strategy is to always defect. The other strategy is conditional sacrifice, which entails sacrifice in the first round and for as long as the other has sacrificed, and a switch to defection as soon as the opponent defects (either switching forever, as in the “grim trigger strategy” or switching temporarily as in “tit-for tat”). If both play conditional sacrifice, there is cooperation forever, resulting in a utility of 1 + δ_i + δ_i ² + ⋯ = 1/(1 − δ_i) for each of them. If both play always defect, there is mutual defection forever, yielding zero utility. If one plays always defect and the other plays conditional sacrifice, then the defecting player gets 2 and the sacrificing player gets −1, because they will both be defecting from the second round onward. Note that always defect is a best response to always defect, whereas conditional sacrifice is a best response to conditional sacrifice if and only of δ_i ≥ 1/2. Suppose finally that the discount factors in the agent population are uniformly distributed on the interval [1/4, 3/4].

For the solution to the entire game, it matters crucially whether agents observe their partner's choice at the individual stage. If agents do not observe each other's consumption choices, then there is no Nash equilibrium in which conditional sacrifice is played at the group stage. Intuitively, the risk of facing a defecting opponent is enough to deter everyone from attempting to establish cooperation, regardless of their patience.

By contrast, if agents can observe their partner's consumption before deciding their own group-stage strategy, cooperation might get established. Consider the following strategy for player i: (a) At stage 1, choose moderation if and only if δ_i ≥ 1/3; (b) at stage 2 choose conditional sacrifice if and only if both chose moderation at stage 1 and δ_i ≥ 1/2. It is straightforward to show that this strategy profile is a subgame perfect equilibrium of the entire game. In this equilibrium, moderate consumption signals high patience (high δ_i) and a willingness to cooperate. Note that an agent with δ_i between 1/3 and 2/3 would not choose moderation if the game were to end after stage 1 but do so when they can thereby reap the benefit of durable cooperation at stage 2.

Our observation that moral behavior can have signaling value neither contradicts nor detracts from the authors' theory. Presumably, both mechanisms are at play. Only empirical analysis can clarify their absolute and relative importance for explaining puritanical morality. Unobservable puritanical behavior is inconsistent with the signaling theory. On the contrary, deliberately public displays of morality are probably better explained by the signaling theory than by the authors' theory.

Let us end by noting a difference between our signaling argument and those of the prior literature on religiosity as a credible signal (e.g., Iannaccone, Reference Iannaccone1994; Irons, Reference Irons and Nesse2001). The prior literature typically posits that people differ in their commitment to the religious cause. Moreover, it posits that costly displays of devotion are rewarded by other congregation members. Here, we demonstrate that the same kind of prudent behavior might instead be signaling patience, and that the reward might take the form of successful cooperation – possibly also with people outside of the religious community itself.