Formal verification and synthesis of mechanisms for social choice

Mechanism Design (MD) aims at defining resources allocation protocols that satisfy a predefined set of properties, and Auction Mechanisms are of foremost importance. Core properties of mechanisms, such as strategy-proofness or budget balance, involve: (i) complex strategic concepts such as Nash equilibria, (ii) quantitative aspects such as utilities, and often (iii) imperfect information, with agents' private valuations. We demonstrate that Strategy Logic provides a formal framework fit to model mechanisms and express such properties, and we show that it can be used either to automatically check that a given mechanism satisfies some property (verification), or automatically produce a mechanism that does (synthesis). To do so, we consider a quantitative and variant of Strategy Logic. We first show how to express the implementation of social choice functions. Second, we show how fundamental mechanism properties can be expressed as logical formulas, and thus evaluated by model checking. We then prove that model checking for this particular variant of Strategy Logic can be done in polynomial space. Next, we show how MD can be rephrased as a synthesis problem, where mechanisms are automatically synthesized from a partial or complete logical specification. We solve the automated synthesis of mechanisms in two cases: when the number of actions is bounded, and when agents play in turns. Finally, we provide examples of auction design based for each of these two cases. The benefit of our approach in relation to classical MD is to provide a general framework for addressing a large spectrum of MD problems, which is not tailored to a particular setting or problem.