Time Lotteries and Stochastic Impatience

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We study preferences over lotteries in which both the prize and the payment date are uncertain. We call time lotteries those in which the prize is fixed and only the date in which it will be received is uncertain. The standard model of risk and time preferences, Expected Discounted Utility, implies that individuals must be risk seeking over time lotteries (RSTL). In contrast, we show experimentally that almost all subjects violate this property. Our main contributions are theoretical. We first establish an impossibility result: within a very broad model, that includes many known forms of non Expected Utility and time discounting, it is impossible to accommodate even a single violation of RSTL without violating a property we termed Stochastic Impatience, a risky counterpart of standard Impatience. We then offer two positive results. First, if one is willing to forego Stochastic Impatience, violations of RSTL can be accommodated with a simple generalization of Expected Discounted Utility, obtained from imposing solely the behavioral postulates of Discounted Utility and Expected Utility. Second, if one instead wishes to maintain Stochastic Impatience, then RSTL can be accommodated by a novel type of relaxation of Independence.

Joint work with Patrick DeJarnette, Daniel Gottlieb , Pietro Ortoleva.