Iterated Kalai–Smorodinsky–Nash compromise

The results of Rachmilevitch (2014) are independent from the previous characterization results since KSNR neither implies nor is implied by the set of standard axioms, involving IEUR, SY, and WPO.

This axiom of Kalai (1977) requires that for any two 0-normalized bargaining problems S and \(S'\) with \(S'\supseteq S\), the solution on \(S'\) can be obtained by first calculating the solution on S and then taking it to be the starting point for the distribution of the utilities in \(S'\) to calculate the solution on this normalized set.

Various forms of decomposability axioms were earlier used by Salonen (1988), Rachmilevitch (2012), Saglam (2014), and Trockel (2014, 2015), among others.

Given two vectors x and y in \({\mathbb {R}}^2_+\), \(x > y\) means \(x_i > y_i\) for \(i=1,2\) and \(x \ge y\) means \(x_i \ge y_i\) for \(i=1,2\).

Any game \((S,d) \in \Sigma ^2\) is said to be a hyperplane game if the Pareto frontier of the set \(IR(S,d)=\{x\in S|x\ge d\}\) is a line segment in \(\mathfrak {R}^2_+\).

Thomson (1983) showed that in the n-person case the Kalai–Smorodinsky solution turns out to be the only solution that possesses Anonymity (AN), Continuity (CONT), IEUR, Monotonicity With Respect to Changes in the Number of Agents (MON), and WPO. Of these axioms, IEUR and WPO are generalizations of the former conditions used by Kalai and Smorodinsky (1975) for the two-person case. The axiom AN is a strengthening of SY, and it requires that not only the names of the agents in a given group do not matter but also that any other group of agents of the same size would reach the same bargaining outcome. CONT implores that a small change in the bargaining set causes only a small change in the bargaining outcome. Finally, MON says that if the expansion of a group of agents requires a sacrifice to support the entrants, then every incumbent must contribute. The Nash solution satisfies AN, CONT, IEUR, and WPO; but it does not satisfy MON as directly shown by Thomson and Lensberg (1989, pp. 41–42).