M.Sc Thesis
M.Sc Student Shamash Elisheva Risk Aversion and Bargaining Games Department of Industrial Engineering and Management Mr. Uriel Rothblum (Deceased)

Abstract

Nash (1950) introduces two-person bargaining games as pairs (S,d), where S  is the set of feasible expected utility payoffs to individuals (players), and d is the joint utility payoff to the players if no agreement is reached. He defined four reasonable requirements from a function that maps such games into , called a solution, and proved that there is a unique solution that satisfies those requirements. Throughout the years, more solutions have been proposed. Among these solutions are the Kalai-Smorodinsky solution (Kalai and Smorodinsky (1975)), the Perles-Maschler solution  (Perles and Maschler (1981)), the Dictatorial solution, the Raiffa solution  (Livne (1989)), the Equal Area solution  (Anbarci and Bigelow (1994)), the Egalitarian solution (Kalai (1977)), the Utilitarian solution (Thomson (1981)) and the Yu solution (Yu (1973)) .

Roth and Rothblum consider the influences of risk aversion on the Nash bargaining solution on 2-person-lottery bargaining games, where outcomes can be lotteries. Extending their analysis, we find that in the Kalai-Smorodinsky, Raiffa solution Equal Area solutions, if a player becomes more risk averse, then there exist 3 subsets of  games:

• Subset I: the player's opponent's utility must increase.

• Subset II: the player's opponent's utility must decrease.

• Subset III: It is impossible to determine whether the player's opponent's utility increases or decreases, without knowing the specific utility transformation which increased the player's risk aversion.

For Dictatorial solution, we show that if a player becomes more risk averse, then his opponent's utility decreases.

We expand the analysis to deterministic bargaining games.  In these games, when a player becomes more risk averse, the convex set of outcomes may lose its convexity. Therefore, we expand the Kalai-Smorodinsky and Nash solutions to bargaining games where S is not necessarily convex using Pareto-convex bargaining games (games where the Pareto surface is not necessarily "convex"). We find that in the extensions of the Nash and Kalai-Smorodinsky solutions to Pareto-convex bargaining games, if a player's becomes more risk averse, then his opponents utility increases. We extend the Mariotti (1998) extension of the Nash solution to compact bargaining games (games where the set S is compact).  Nagahisa and Tanaka (2002) also extend the Kalai-Smorodinsky solution to compact bargaining games. We analyze the influence of player 2 becoming more risk averse on these extensions. We find that in these extensions, the set of outcomes player 1 may receive as a result of player 2 becoming more risk averse, can only improve.