296 M. Hoffman et al.
Temptation to Defect
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X Fig. 3. A single stage of the Envelope Game
is arepeated game with two players. In each round, player 1 receives a sealed enve- lope, 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 cooper- ate 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.” This equilibrium emerges in agent-based simulations of evolution and learning processes.’ 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 tempta- tions. Not looking permits cooperative equilibria in the face of large temptations.
The Envelope Game is meant to capture the essential features of many interest- ing aspects of our morality, as described next.
Authentic Altruism. Many have asked whether “[doing good is] always and exclu- sively motivated by the prospect of some benefit for ourselves, however subtle” (Batson, 2014), for example, the conscious anticipation of feeling good (Andreoni,
? Technically, the conditions under which we expect players to avoid looking and attend to looking are c,>a/(l—w)>cp+c(1—p) and bp+d(1—p)<0), where c, and c, 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.
>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 temp- tation 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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