Competitive Equilibrium with Indivisibilities

This paper studies an exchange economy with a finite number of agents in which each agent is initially endowed with a finite number of (personalized) indivisible commodities. We observe that the core equivalence theorem may not hold for this economy when the coalitional form game is generated in the standard manner. We provide an alternative definition of the coalitional form game to resolve this problem so that the balancedness of the new defined game provides a useful necessary and sufficient condition for the existence of competitive equilibrium for the original economy. We also observe that the nice strategy proof property of the minimum competitive price mechanism in the assignment problem and the Vickrey auction model does not carry over to the above economy. We show that examples of exchange economies exist for which no competitive price mechanism is individually (coalitionally) strategy proof.