Nash-implementation of the no-envy solution on symmetric domains of economies

We show that a simple game form, which resembles the “Divide-and-Choose” procedure, Nash-implements the no-envy solution on domains of economies where the set of feasible allocations is symmetric (an allocation obtained from a feasible allocation by interchanging the bundles of any two agents is also feasible) and preferences are complete (each agent can compare any two bundles). Our result extends a result by Thomson (2005) and it is applicable to a wide class of models including the classical model of fair allocation, the unidimensional single-peaked model, cake division model, and allocation of indivisible objects with monetary transfers. We show that, even when the preferences exhibit consumption externalities, an extension of the no-envy solution is Nash-implementable on general domains of economies.