Faheem H. Gilani

Is a decision maker more or less persistent when she does not know whether a project, if viable, pays large lump sums rarely or small ones often? I study an exponential bandit in which a viable project has an unknown arrival rate but a fixed expected flow payoff. A decision maker should continue while her belief that the project is viable exceeds the known-rate cutoff at the current mean rate conditional on viability. Relative to a known-rate benchmark with the same mean, a lack of breakthroughs erodes this belief more slowly, since it suggests that a viable project pays rarely but handsomely (hope), but raises the cutoff by lowering the conditional mean rate (discouragement). Discouragement dominates early and hope later: absent a breakthrough, this rate–size uncertainty makes her give up sooner on less promising projects and later on more promising ones. Whenever experimentation is worthwhile, it also makes her worse off, more likely to abandon a viable project, and more optimistic about viability when she gives up. With several players, the unique symmetric Markov perfect equilibrium has them experiment fully while sufficiently optimistic and then mix at an explicit, declining intensity that keeps each indifferent between experimenting and free riding; beliefs converge to the single-agent boundary without reaching it.

Two sophisticated, present-biased agents must each complete a task by a deadline, and completing on the same date costs less. An agent may complete early to avoid a partner whose company would tempt her to delay. For any horizon, avoidance requires at least three periods and is impossible under full naiveté. If the partner seeks her company, they play hide-and-seek. With three periods this happens on an open set of parameters, where completion dates are random, company can hurt both agents, and an agent’s cost can rise with her partner’s self-control. Observable, irreversible starts lead to joint completion.

We study strategic experimentation on a resource that, if productive, depletes when harvested and grows deterministically otherwise. Harvest opportunities arrive according to a poisson process whose rate is proportional to total experimentation intensity and whose value depends on the level of the resource at the time of arrival. In a symmetric Markov perfect equilibrium, when players are sufficiently confident that the resource is productive they exploit it at levels where a social planner would have them wait. When they are unsure, they stop experimenting at a higher belief than the planner would, whatever the initial resource level. In between, they exploit the resource to the point where it pays the same as an outside safe option, dissipating its rent as in an open-access fishery. Sustaining this requires Filippov solutions.