The Nash product is a utility representation of the Pareto ordering

The paper deals with different completions of partial orderings on finite dimensional compact sets and an application to bargaining games. In particular, the Nash product turns out to be a continuous utility representation of the Pareto ordering in the sense of [Peleg, B., Econometrica 38 (1970) 93-96.] and [Sondermann, D., Journal of Economic Theory 23 (1980) 183-188.]. This provides an interesting "straightforward interpretation" that the Nash product according to [Osborne, M.J. and A. Rubinstein, A Course in Game Theory (1994), MIT Press: Cambridge, Massachusetts., p. 303] is lacking. For each payoff allocation admissible in the bargaining problem, it measures the set of admissible allocations Pareto dominated by it. The two sets of resulting maximal elements of the two completions are the Pareto efficient boundary and the Nash bargaining solution, respectively.