Computing Optimal Contracts in Series-Parallel Heterogeneous 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 taken by each agent is hidden and affects the (binary) outcome of the agent's individual task in a stochastic manner. A Boolean function, called technology, maps the individual tasks' outcomes to the outcome of the whole project. The principal induces a Nash equilibrium on the agents' actions through payments that are conditioned on the project's outcome (rather than the agents' actual actions) and the main challenge is that of determining the Nash equilibrium that maximizes the principal's net utility, referred to as the optimal contract. Babaioff, Feldman and Nisan suggest and 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 and on series-parallel (SP) technologies, which are constructed inductively from their building blocks -- the AND and OR technologies. 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. In particular, we show that while the AND case admits a polynomial time algorithm, computing the optimal contract in an OR technology is NP-hard. On the positive side, we devise an FPTAS for the OR case and establish a scheme that given any SP technology, provides a (1 + ?) -approximation for all but an $\hat{\epsilon}$-fraction of the relevant instances (for which a failure message is output) in time polynomial in the size of the technology and in the reciprocals of ? and $\hat{\epsilon}$.