Nash demand game and the Kalai-Smorodinsky solution

We introduce two new variations on the Nash demand game. In one, as in all past variants of Nash demand games, the Nash bargaining solution is the equilibrium outcome. The other demand game allows for probabilistic continuation in cases of infeasible joint demands. Specifically, with probability (1-p) the game terminates and the players receive their disagreement payoffs; but with probability p the game continues to a second stage in which one of the two incompatible demands is randomly selected and implemented. Surprisingly, the Kalai-Smorodinsky solution is always the outcome of the most robust equilibrium of this game. Moreover, ranking other solution concepts is impossible.