We analyze bargaining situations where the agents' payoffs from disagreement depend on who among them breaks down the negotiations. We model such problems as a superset of the standard domain of Nash [Nash, J.F., 1950. The bargaining problem. Econometrica 18, 155-162]. On our extended domain, we analyze the implications of two central properties which, on the Nash domain, are known to be incompatible: strong monotonicity [Kalai, E., 1977. Proportional solutions to bargaining situations: Interpersonal utility comparisons. Econometrica 45, 1623-1630] and scale invariance [Nash, J.F., 1950. The bargaining problem. Econometrica 18, 155-162]. We first show that a class of monotone path rules uniquely satisfy strong monotonicity, scale invariance, weak Pareto optimality, and "continuity". We also show that dropping scale invariance from this list characterizes the whole class of monotone path rules. We then introduce a symmetric monotone path rule that we call the Cardinal Egalitarian rule and show that it is weakly Pareto optimal, strongly monotonic, scale invariant, symmetric and that it is the only rule to satisfy these properties on a class of two-agent problems.
