How should one allocate n indivisible goods to n agents, who have declared cardinal utilities for the goods, in a principled manner? The classic Hylland-Zeckhauser mechanism (1979) accomplishes this using the power of the pricing mechanism: it renders the goods divisible by viewing them as probability shares, seeks an equilibrium allocation and rounds this fractional allocation using the Birkhoff-von Neumann Theorem. As a result, it has several nice properties, including Pareto optimality, envy-freeness and incentive compatibility in the large. However, its intractability makes it unusable in practice, for even n = 10. In this survey talk, I will first describe recent works which gave a proof of intractability, and follow it up with a current counter-proposal of mechanisms based on the Nash bargaining game; however, the jury is still out. In the meantime, there are exciting open problems as well as new impossibility results.

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