Risk aversion in the Nash bargaining problem with risky outcomes and risky disagreement points.

*(English)*Zbl 0747.90116Summary: The Nash bargaining problem has attracted a lot of attention in the last decade. In a deterministic framework, R. Kihlstrom, A. Roth and D. Schmeidler [in: Game theory and mathematical economics, Proc. Semin., Bonn/Hagen 1980, 65-71 (1981; Zbl 0481.90098)] have analyzed the effect of risk aversion on several solution concepts, particularly on the Nash solution, and found that increasing risk aversion is disadvantageous to the player who becomes more risk averse. A. Roth and U. Rothblum [Econometrica 50, 639-647 (1982; Zbl 0478.90090)] have dealt with the same problem in a framework that contains risky outcomes as well, but where the disagreement outcome is still deterministic. They have shown that the deterministic results no longer hold. Specifically, if the disagreement outcome is preferred to a deterministic outcome that is part of the Nash solution (i.e., that can occur with positive probability in the potential agreement), then increase in risk aversion may be advantageous to the player who becomes more risk averse.

In our opinion the analysis of the Nash bargaining problem with risky outcomes must include the cases where the disagreement outcome itself is risky as well. In numerous bargaining situations we observe risky disagreement outcomes (e.g., the threat “going to court” usually contains some risk). The purpose of this note is to generalize the R-R results to models with risky disagreement outcome as well. It turns out that their result will depend, in some cases, upon the “degree of change” in risk aversion. Particularly, when the potential agreement (which is random) has an outcome which makes player 2 worse off compared to the disagreement, and if the disagreement does not dominate that outcome (i.e., its support contains a worse outcome), then when player 2 becomes “sufficiently” more risk averse player 1 becomes better off. Thus we obtain that increase in risk aversion may be disadvantageous to player 2 even if the risky disagreement is preferred to some outcome of the (risky) agreement. This is in contrast to the result of R-R where the disagreement is certain. However, when one of the outcomes in the potential agreement is dominated by the disagreement and the disagreement is dominated by the other agreements’ outcome, then increase in risk aversion is advantageous to player 2.

In our opinion the analysis of the Nash bargaining problem with risky outcomes must include the cases where the disagreement outcome itself is risky as well. In numerous bargaining situations we observe risky disagreement outcomes (e.g., the threat “going to court” usually contains some risk). The purpose of this note is to generalize the R-R results to models with risky disagreement outcome as well. It turns out that their result will depend, in some cases, upon the “degree of change” in risk aversion. Particularly, when the potential agreement (which is random) has an outcome which makes player 2 worse off compared to the disagreement, and if the disagreement does not dominate that outcome (i.e., its support contains a worse outcome), then when player 2 becomes “sufficiently” more risk averse player 1 becomes better off. Thus we obtain that increase in risk aversion may be disadvantageous to player 2 even if the risky disagreement is preferred to some outcome of the (risky) agreement. This is in contrast to the result of R-R where the disagreement is certain. However, when one of the outcomes in the potential agreement is dominated by the disagreement and the disagreement is dominated by the other agreements’ outcome, then increase in risk aversion is advantageous to player 2.

##### MSC:

91A12 | Cooperative games |