In many real-life situations, we need to bargain. What is the best bargaining strategy? If you are already in a negotiating process, your previous offer was a, the seller's last offer was A > a, what next offer a' should you make? A usual commonsense recommendation is to "split the difference", i.e., to offer a' = (a + A) / 2, or, more generally, to offer a linear combination a' = k * A + (1 - k) * a (for some parameter k from the interval (0,1)).
The bargaining problem falls under the scope of the theory of cooperative games. In cooperative games, there are many reasonable solution concepts. Some of these solution concepts -- like Nash's bargaining solution that recommends maximizing the product of utility gains -- lead to offers that linearly depend on a and A; other concepts lead to non-linear dependence. From the practical viewpoint, it is desirable to come up with a recommendation that would not depend on a specific selection of the solution concept -- and on specific difficult-to-verify assumptions about the utility function etc.
In this paper, we deliver such a recommendation: specifically, we show that under reasonable assumption, we should always select an offer that linearly depends on a and A.