Power Diagrams

This chapter considers the case when the attributes are d-dimensional vectors, the matching surplus is the scalar product, the distribution of workers' attributes is continuous, and the distribution of the firms is discrete. It discusses the geometry of the optimal assignment of workers to firms and relates it to the important notion of power diagrams in computational geometry. It shows that the optimal assignment map is the gradient of a piecewise affine convex function, and the equilibrium prices of the firms are shown to be the solution to a finite-dimensional convex minimization problem. The chapter discusses how to perform this computation in practice.