Consider the domain of economic environments E whose typical element is ξ = (U
2, Ω, ω
*), where ui
are Neumann-Morgenstern utility functions, Ω is a set of lotteries on a fixed finite set of alternatives, and ω* ∈ Ω. A mechanism f associates to each ξ a lottery f(ξ) in Ω. Formulate the natural version of Nash’s axioms, from his bargaining solution, for mechanisms on this domain. (e.g., IIA says that if ξ′ = (U
2, Δ, ω′), Δ ⊂ Ω, and f ∈ Δ then f(ξ′) = f(ξ).) It is shown that the Nash axioms (Pareto, symmetry, IIA, invariance w.r.t. cardinal transformations of the utility functions) hardly restrict the behavior of the mechanism at all. In particular, for any integer M, choose M environments ξi, i = 1, … , M, and choose a Pareto optimal lottery ωi
, restricted only so that no axiom is directly contradicted by these choices. Then there is a mechanism f for which f(ξi
) = ωi
, which satisfies all the axioms, and is continuous on E.