A Fraudulent Expert and Short-Lived Customers

A market where short-lived customers interact with a long-lived expert is considered. An expert privately observes whether or not a particular treatment is necessary for his customers and has an incentive to recommend the treatment even if it is unnecessary. Customers imperfectly observe the expert's past actions. Truthful reporting at all times yields the expert his best equilibrium payoff when the expert is known to be opportunist (i.e., rational in the usual sense). If the customers believe that the expert might be an honest type, who always reports truthfully, then the expert can build his reputation for honesty, so then he defrauds his customers to achieve a higher payoff. In equilibrium, deception for a long period of time is extremely unlikely. However, it is a probability one event (in some of the equilibrium) when the expert's customer is also a long-lived agent.