The effect of decision weights in bargaining problems.
The present paper investigates bargaining problems, under the welfare-like approach, where the bargainer's preferences are represented as a rank-dependent utility (functional). M. E. Yaari (1986) provided an intuitive definition of risk aversion, and showed that, under expected utility, one decision-maker's more risk averse than another if and only if the utility function of the former is a strictly increasing concave transformation of that of the latter. In the rank-dependent utility model, in contrast to the expected utility one, not only the utility function but also a probability weighting function determines the risk attitude of a decision-maker. In the present paper, two risk aversion factors are distinguished: the utility risk aversion, associated with the utility function as in the expected utility model, and the probability weighting function as in the rank-dependent utility model. Focusing on the probability risk aversion, the authors try to answer the classical question in bargaining problems: is it advantageous or disadvantageous to bargain with a more risk-averse person? After necessary preliminary definitions (Section 2), Section 3 investigates the impact of both utility and probability risk aversion for the bargaining solution developed by Y. Kalai and M. Smorodinsky (1975). The main result of this section is that, for a large class of bargaining problems (whose solutions exhibit a weak monotonicity property), it is advantageous to have a less probability risk averse, and a more utility risk averse opponent. Apparently surprising, this result is nevertheless not counterintuitive since the increased probability risk aversion of the opponent could be bad for a player, because such an opponent might insist on larger probabilities for good alternatives and thereby reduce the other player's utility. The obtained approach may also have strategic implications if it is assumed that bargainers can be dishonest about their true preferences, and pretend to evaluate lotteries by means of a different utility or probability weighting function. Section 4 of the paper presents the axiomatization of Kalai and Smorodinsky bargaining solution in the rank-dependent utility setting. These results can be generalised to \(n\)-person bargaining problems with the restriction that the feasible sets are made comprehensive, the bargaining solutions exhibiting certain monotonicity properties. An important remark is that the obtained solutions do not include the (classical) Mash bargaining solution. This is because not only those feasible sets are not necessarily convex (entailing non-well-defined solutions), but even if Nash solution is well-defined it does not behave regularly: under rank-dependent, a player can gain or lose with respect to the Nash bargaining solution when bargaining with a more probabilistically risk-averse opponent.
scientific article; zbMATH DE number 1957020
scientific article