Computing optimal contracts in combinatorial agencies

We study an economic setting in which a principal motivates a team of strategic agents to exert costly effort toward the success of a joint project. The action of each agent is hidden and affects the outcome of the agent's individual task in a stochastic manner. A Boolean technology function maps the outcomes of the individual tasks to the project's outcome. The principal induces a Nash equilibrium on the agents' actions through payments which are conditioned on the project's outcome and the main challenge is that of determining the Nash equilibrium that maximizes the principal's net utility, namely, the optimal contract. Babaioff, Feldman and Nisan study a basic combinatorial agency model for this setting, and provide a full analysis of the AND technology. Here, we concentrate mainly on OR technologies that, surprisingly, turn out to be much more complex. We provide a complete analysis of the computational complexity of the optimal contract problem in OR technologies which resolves an open question and disproves a conjecture raised by Babaioff et al. While the AND case admits a polynomial time algorithm, we show that computing the optimal contract in an OR technology is NP-hard. On the positive side, we devise an FPTAS for OR technologies. We also study series-parallel (SP) technologies, which are constructed inductively from AND and OR technologies. We establish a scheme that given any SP technology, provides a (1+ε)-approximation for all but an ε̂-fraction of the relevant instances in time polynomial in the size of the technology and in the reciprocals of ε and ε̂.