Given two agents with von Neumann-Morgenstern utilities who wish to divide n commodities, consider the two-person noncooperative game with strategies consisting of concave, increasing von Neumann-Morgenstern utility functions as well as rules to break ties and whose outcomes are some solution to the bargaining game determined by the strategies used. It is shown that, for a class of bargaining solutions which includes those of Nash and Raiffa, Kalai and Smorodinsky, any constrained equal-income competitive equilibrium allocation for the true utilities is a Nash equilibrium outcome for the noncooperative game.