Essays on bargaining games

This dissertation is centered around bargaining games. I study both the cooperative and the non-cooperative approaches to the bargaining games. The two chapters of my dissertation deal with each of these approaches. ❧ Coming back to the bargaining problem; in the first chapter of this dissertation, I study two related but separate questions on a cooperative bargaining problem. I will give a brief historical background to explain the first question I tackle. Nash (1950) characterized a solution for bargaining problems in terms of four simple axioms. Later research showed that when the disagreement point in the analysis is replaced with an infeasible ideal point, there is no solution that can be characterized with the axioms that are counterparts to the ones used in Nash's approach. One way to deal with this problem is suggested by Conley, McLean, and Wilkie (forthcoming). Instead of using independence from affine transformations, their characterization of the Nash solution employs a property that guarantees proportional gains from an endogenous disagreement point. Then, the same property is used to characterize another solution when the endogenous disagreement point is replaced by an infeasible ideal point. That is, the corresponding property guarantees proportional losses from this ideal point. I extend their characterization to weighted bargaining games by a simple modification of their proportional loss property. ❧ In the second part of the paper, I tweak Nash's original proof to characterize solutions that are independent under common scale transformations. This weakening of the independence from affine transformations is required since we know that there is no solution that satisfies it along with other nice properties under the ideal point approach. Then, instead of showing that the Nash solution can be obtained in the entire domain from the solution of a symmetric set, I show that any solution that satisfies independence under common scale transformations can be obtained from the solutions of a certain class of hyperplanes. ❧ In the second chapter of this dissertation, I look at the bargaining problem from the non-cooperative side. I provide results that extend and complement Yildiz (2003). Yildiz studies bargaining games without a common prior; that is, games where players have subjective beliefs about their probabilities of getting selected at each period. This departs from the original alternating offers games where such selection procedures are strict. Assuming no learning and transferable utility, he shows that as long as both players are optimistic for long enough, they will reach an agreement immediately. Intuitively, players will reach an agreement immediately if both of them are highly optimistic, since their expectations about rents in future periods are so low that they settle immediately. Another key result from his work is that players will also reach an immediate agreement if the drop in optimism from one period to the next is within certain bounds. This second result does not require high optimism. ❧ By relaxing the transferable utility assumption, I study the effects of risk aversion on bargaining games without common priors. I find that it takes a higher level of optimism (as compared with transferable utility) to reach an immediate agreement as the agents get more risk averse. Intuitively, risk aversion means higher rents in future periods, which will make immediate agreement a less attractive option. This is in line with Yildiz's findings. Also, I show that the bound required for immediate agreement as mentioned in the above paragraph is strictly narrower than the transferable utility case. In fact, I am currently trying to show that immediate agreement is only reached at constant levels of optimism as the agents get more and more risk averse.