  Most work in mechanism design assumes that buyers are risk neutral; some
considers risk aversion arising due to a non-linear utility for money. Yet
behavioral studies have established that real agents exhibit risk attitudes
which cannot be captured by any expected utility model. We initiate the study
of revenue-optimal mechanisms under buyer behavioral models beyond expected
utility theory. We adopt a model from prospect theory which arose to explain
these discrepancies and incorporates agents under-weighting uncertain outcomes.
In our model, an event occurring with probability $x < 1$ is worth strictly
less to the agent than $x$ times the value of the event when it occurs with
certainty.
  In contrast to the risk-neutral setting, the optimal mechanism may be
randomized and appears challenging to find, even for a single buyer and a
single item for sale. Nevertheless, we give a characterization of the optimal
mechanism which enables positive approximation results. In particular, we show
that under a reasonable bounded-risk-aversion assumption, posted pricing
obtains a constant approximation. Notably, this result is "risk-robust" in that
it does not depend on the details of the buyer's risk attitude. Finally, we
examine a dynamic setting in which the buyer is uncertain about his future
value. In contrast to positive results for a risk-neutral buyer, we show that
the buyer's risk aversion may prevent the seller from approximating the optimal
revenue in a risk-robust manner.
