Automated Mechanism Design: Complexity Results Stemming from the Single-Agent Setting
The aggregation of conﬂicting preferences is a central problem in multiagent systems. The key difﬁculty is that the agents may report their preferences insincerely. Mechanism design is the art of designing the rules of the game so that the agents are motivated to report their preferences truthfully and a (socially) desirable outcome is chosen. We propose an approach where a mechanism is automatically created for the preference aggregation setting at hand. This has several advantages, but the downside is that the mechanism design optimization problem needs to be solved anew each time. Hence the computational complexity of mechanism design becomes a key issue. In this paper we analyze the single-agent mechanism design problem, whose simplicity allows for elegant and generally applicable results.
We show that designing an optimal deterministic mechanism that does not use payments is N P-complete even if there is only one agent whose type is private information—even when the designer’s objective is social welfare. We show how this hardness result extends to settings with multiple agents with private information. We then show that if the mechanism is allowed to use randomization, the design problem is solvable by linear programming (even for general objectives) and hence in P. This generalizes to any ﬁxed number of agents. We then study settings where side payments are possible and the agents’ preferences are quasilinear. We show that if the designer’s objective is social welfare, an optimal deterministic mechanism is easy to construct; in fact, this mechanism is also ex post optimal. We then show that designing an optimal deterministic mechanism with side payments is N P-complete for general objectives, and this hardness extends to settings with multiple agents. Finally, we show that an optimal randomized mechanism can be designed in polynomial time using linear programming even for general objective functions. This again generalizes to any ﬁxed number of agents.