Addendum: The Shapley value of a perfectly competitive market may not exist

Abstract

A counter-example for the existence of the Shapley value of non-differentiable perfectly competitive Walrasian (i.e., pure exchange) economies is given. The model used is that of a non-atomic continuum of traders. The appropriate — and most powerful — value in the non-differentiable case, introduced by Mertens (1988) is considered; the existence and unicity of this value for monetary (i.e., transferable utility) markets was established by Mertens (1989), without any differentiability assumption. Moreover, we show in fact that, for any concave utility representation of this economy, the corresponding side-payment game has an asymptotic value, and that this value necessarily involves non-zero transfers. The non-existence of the value in our example, far from being exceptional, appears for an open set (with strictly positive measure) of initial allocations.

This work is an abridged version of CORE Discussion Paper 9037, that is part of the author’s Ph.D. thesis done under the supervision of Professor J.F. Mertens. The author wishes to thank him for helpful discussions on the subject.