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M. Hoffman et al.

Temptation
to Defect
Low
p
(1)
High
1-p
Look
Don't Look
1
(2)
Cooperate
Defect
1
(3)
Continue
End
2
(4)

Fig. 3 A single stage of the Envelope Game

is a repeated game with two players. In each round, player 1 receives a sealed envelope, which contains a card stating the costs of cooperation (high temptation to defect vs. low temptation to defect). The temptation is assigned randomly and is usually low. Player 1 can choose to look inside the envelope and thus find out the magnitude of the temptation or choose not to look. Then player 1 decides to cooperate or to defect. Subsequently, player 2 can either continue to the next round or end the game. As in the Repeated Prisoner’s Dilemma, the interaction repeats with a given likelihood, and if it does, an envelope is stuffed with a new card and presented to player 1, etc.

In this model, as long as temptations are rare, large, and harmful to player 2, it is a Nash equilibrium for player 1 to “cooperate without looking” in the envelope and for player 2 to continue if and only if player 1 has cooperated and not looked. We refer to this as the cooperate without looking (CWOL) equilibrium.² This equilibrium emerges in agent-based simulations of evolution and learning processes.³ Notice that if player 1 could not avoid looking inside the envelope, or player 2 could not observe whether player 1 looked, there would not be a cooperative equilibrium since player 1 would benefit by deviating to defection in the face of large temptations. Not looking permits cooperative equilibria in the face of large temptations.

The Envelope Game is meant to capture the essential features of many interesting aspects of our morality, as described next.

Authentic Altruism. Many have asked whether “[doing good is] always and exclusively motivated by the prospect of some benefit for ourselves, however subtle” (Batson, 2014), for example, the conscious anticipation of feeling good (Andreoni,

² Technically, the conditions under which we expect players to avoid looking and attend to looking are ch>a/(1-w)>c₁p+ch(1-p) and bp+d(1-p)<0), where ch and c₁ are the magnitudes of the high and low temptations, respectively; p is the likelihood of the low temptation; a/(1-w) is the value of a repeated, cooperative interaction to player 1; and bp+d(1-p) is the expected payoff to player 2 if player 1 only cooperates when the temptation is low.
³ The simulations employ numerical estimation of the replicator dynamics for a limited strategy space: cooperate without looking, cooperate with looking, look and cooperate only when the temptation is low, and always defect for player 1, and end if player 1 looks, end if player 1 defects, and always end for player 2.

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