Sponsored Search, Market Equilibria, and the Hungarian Method

TL;DR

This paper extends the Hungarian Method to compute market equilibria in sponsored search markets with budgets, demonstrating its limitations and conditions for incentive compatibility.

Contribution

It introduces a modified Hungarian Method for budgeted settings and analyzes incentive compatibility constraints in such markets.

Findings

Modified Hungarian Method finds feasible, envy-free, bidder optimal outcomes with budgets.

No universally incentive-compatible mechanism exists for budgeted settings.

For general position inputs, the proposed mechanism is incentive compatible.

Abstract

Matching markets play a prominent role in economic theory. A prime example of such a market is the sponsored search market. Here, as in other markets of that kind, market equilibria correspond to feasible, envy free, and bidder optimal outcomes. For settings without budgets such an outcome always exists and can be computed in polynomial-time by the so-called Hungarian Method. Moreover, every mechanism that computes such an outcome is incentive compatible. We show that the Hungarian Method can be modified so that it finds a feasible, envy free, and bidder optimal outcome for settings with budgets. We also show that in settings with budgets no mechanism that computes such an outcome can be incentive compatible for all inputs. For inputs in general position, however, the presented mechanism---as any other mechanism that computes such an outcome for settings with budgets---is incentive compatible.