  We study a single-agent contracting environment where the agent has
misspecified beliefs about the outcome distributions for each chosen action.
First, we show that for a myopic Bayesian learning agent with only two possible
actions, the empirical frequency of the chosen actions converges to a Berk-Nash
equilibrium. However, through a constructed example, we illustrate that this
convergence in action frequencies fails when the agent has three or more
actions. Furthermore, with multiple actions, even computing an
$\varepsilon$-Berk-Nash equilibrium requires at least quasi-polynomial time
under the Exponential Time Hypothesis (ETH) for the PPAD-class. This finding
poses a significant challenge to the existence of simple learning dynamics that
converge in action frequencies. Motivated by this challenge, we focus on the
contract design problems for an agent with misspecified beliefs and two
possible actions. We show that the revenue-optimal contract, under a Berk-Nash
equilibrium, can be computed in polynomial time. Perhaps surprisingly, we show
that even a minor degree of misspecification can result in a significant
reduction in optimal revenue.
