  We study the price of anarchy of mechanisms in the presence of risk-averse
agents. Previous work has focused on agents with quasilinear utilities,
possibly with a budget. Our model subsumes this as a special case but also
captures that agents might be less sensitive to payments than in the
risk-neutral model. We show that many positive price-of-anarchy results proved
in the smoothness framework continue to hold in the more general risk-averse
setting. A sufficient condition is that agents can never end up with negative
quasilinear utility after playing an undominated strategy. This is true, e.g.,
for first-price and second-price auctions. For all-pay auctions, similar
results do not hold: We show that there are Bayes-Nash equilibria with
arbitrarily bad social welfare compared to the optimum.
