Efficient Prior-Free Mechanisms for No-Regret Agents

We study a repeated Principal Agent problem between a long lived Principal and Agent pair in a prior free setting. In our setting, the sequence of realized states of nature may be adversarially chosen, the Agent is non-myopic, and the Principal aims for a strong form of policy regret. Following Camara, Hartline, and Johnson, we model the Agent's long-run behavior with behavioral assumptions that relax the common prior assumption (for example, that the Agent has no swap regret). Within this framework, we revisit the mechanism proposed by Camara et al., which informally uses calibrated forecasts of the unknown states of nature in place of a common prior. We give two main improvements. First, we give a mechanism that has an exponentially improved dependence (in terms of both running time and regret bounds) on the number of distinct states of nature. To do this, we show that our mechanism does not require truly calibrated forecasts, but rather forecasts that are unbiased subject to only a polynomially sized collection of events -- which can be produced with polynomial overhead. Second, in several important special cases -- including the focal linear contracting setting -- we show how to remove strong ``Alignment'' assumptions (which informally require that near-ties are always broken in favor of the Principal) by specifically deploying ``stable'' policies that do not have any near ties that are payoff relevant to the Principal. Taken together, our new mechanism makes the compelling framework proposed by Camara et al. much more powerful, now able to be realized over polynomially sized state spaces, and while requiring only mild assumptions on Agent behavior.