Variable temptations and black mark reputations

Reputations often guide sequential decisions to trust and to reward trust. We consider situations where each player is randomly matched with a partner in every period. One player – the truster – decides whether to trust. If trusted, the other player – the temptee – has a temptation to betray. The strength of temptation, private information to the temptee, varies across encounters. Betrayals are recorded as publicly known black marks. First, we identify equilibria when players only condition on the number of a temptee's black marks. Second, we show that conditioning on the number of interactions as well as on the number of black marks does not prolong trust. Third, we examine stochastic variations where black marks may be forgotten. Perhaps surprisingly, such variations do not improve outcomes. Fourth, when players condition on more general summary statistics of a temptee's past, we study equilibria where trust is suspended temporarily.