On Nash Dynamics of Matching Market Equilibria

In this paper, we study the Nash dynamics of strategic interplays of n buyers in a matching marketsetup by a seller, the market maker. Taking the standard market equilibrium approach, upon receivingsubmitted bid vectors from the buyers, the market maker will decide on a price vector to clear themarket in such a way that each buyer is allocated an item for which he desires the most (a.k.a., a marketequilibrium solution). While such equilibrium outcomes are not unique, the market maker chooses one(max-eq) that optimizes its own objective — revenue maximization. The buyers in turn change bids totheir best interests in order to obtain higher utilities in the next round’s market equilibrium solution.This is an (n+ 1)-person game where buyers place strategic bids to gain the most from the marketmaker’s equilibrium mechanism. The incentives of buyers in deciding their bids and the market maker’schoice of using the max-eq mechanism create a wave of Nash dynamics involved in the market. Wecharacterize Nash equilibria in the dynamics in terms of the relationship between max-eq and min-eq (i.e., minimum revenue equilibrium), and develop convergence results for Nash dynamics from themax-eq policy to a min-eq solution, resulting an outcome equivalent to the truthful VCG mechanism.Our results imply revenue equivalence between max-eq and min-eq, and address the question thatwhy short-term revenue maximization is a poor long run strategy, in a deterministic and dynamic setting.