Complexity of Strong Implementability

TL;DR

This paper investigates the computational complexity of determining strong implementability of social choice functions, showing it is decidable in polynomial space and polynomial time for single individuals, using polyhedral theory and linear programming.

Contribution

It establishes the complexity bounds for strong implementability, demonstrating polynomial space decidability and polynomial time solvability for single-agent cases, advancing mechanism design theory.

Findings

Strong implementability can be decided in polynomial space.

Payments for strong implementation can be polynomially bounded.

Single-agent strong implementability is decidable in polynomial time.

Abstract

We consider the question of implementability of a social choice function in a classical setting where the preferences of finitely many selfish individuals with private information have to be aggregated towards a social choice. This is one of the central questions in mechanism design. If the concept of weak implementation is considered, the Revelation Principle states that one can restrict attention to truthful implementations and direct revelation mechanisms, which implies that implementability of a social choice function is easy to check. For the concept of strong implementation, however, the Revelation Principle becomes invalid, and the complexity of deciding whether a given social choice function is strongly implementable has been open so far. In this paper, we show by using methods from polyhedral theory that strong implementability of a social choice function can be decided in polynomial space and that each of the payments needed for strong implementation can always be chosen to be of polynomial encoding length. Moreover, we show that strong implementability of a social choice function involving only a single selfish individual can be decided in polynomial time via linear programming.