The simple Nash demand game is analysed in an evolutionary context. The evolutionarily stable strategies (ESS) are characterised in the cases where players have symmetric and asymmetric roles. Introducing stochastic effects we show conditions under which the set of Nash equilibria converges to the Nash bargaining solution as the noise becomes negligible. Two evolutionary models are given for which these conditions are satisfied. In each case above the simulated limit outcome of the replicator selection dynamics is given for a range of parameter values. Both approaches give evolutionary support for the Nash bargaining solution.