Essays in Game Theory

Abstract

My dissertation deals with two topics: collusion in auctions and bargaining; two chapters are devoted to each. Regarding collusion, my goal is to address the following questions (a) what can the cartel members achieve via collusion, and (b) what are the implications of this activity on social efficiency. The answer to both questions depends on various factors, such as the ability to organize side transfers, the availability of communication and commitment devices, whether future interactions between the colluding parties are expected, and more. I therefore address these questions within two main models, each emphasizing different issues; both, however, share the same mathematical structure: a 2-person IPV setting with a continuum of types. In Chapter 1 I consider the case of a one-shot second-price auction which is preceded by two rounds of bribing, where each bidder can try to bribe his opponent to drop out of the auction. The main result of this chapter is a characterization of the efficient equilibria in monotonic and continuous strategies. Additionally, a first-price auction which is preceded by one round of bribing (i.e., a "take-it- or-leave-it" game) is analyzed; it is shown that under the restriction to monotonic and continuous bribing functions, every equilibrium of this game is a bribe-free equilibrium. In Chapter 2 I dispense with the assumption that the players can communicate, exchange money, and deliver (and keep) binding promises; instead, I consider in a repeated game setting. Here, the only instruments for collusion are future payoffs and threats. The analysis of this chapter is more general than that of Chapter 1, in the sense that it applies to any standard auction format. Here, the main contribution is the construction of an incentive-compatible collusive scheme, endogenous bid rotation, which delivers payoffs greater than simple bid rotation payoffs. In the special case where the auction format is second-price, the scheme has the following additional properties: (1) it is supported in equilibrium independently of the discount factor, and (2) it is supported in equilibrium even if each bidder's participation decision in each auction is not known to his opponent. Finally, in a model with two types, the analog of this scheme delivers first-best collusion. Regarding bargaining, my focus is on the role of outside options. Intuitively, one would expect a bargainer's payoff to increase in his outside option. Does this monotonic relation have further implications? In Chapters 3 and 4 I show that indeed it has. Formally, I introduce a new axiom into Nash's (1950) axiomatic bargaining model, disagreement point monotonicity; it requires that a player's solution payoff be a strictly increasing function of his disagreement payoff. In Chapter 3 I show that on the domain of strictly comprehensive n-person problems, this axiom, together with Nash's IIA and other standard axioms, uniquely characterizes the family of proportional solutions (they were first characterized by Kalai (1977), by means of different monotonicity axioms). In Chapter 4 I show that on the domain of 2-person problems, the Kalai-Smorodinsky bargaining solution (due to Kalai and Smorodinsky (1975)) is characterized by disagreement point monotonicity, scale-invariance, and other standard axioms. It is worth noting that from the three major bargaining solutions in the literature—the Nash solution, the Kalai-Smorodinsky solution, and the proportional solution—only the Nash solution is a scale-invariant and IIA-satisfying solution; however, it fails to satisfy disagreement point monotonicity. Each of the other two solutions satisfies exactly one of IIA or scale-invariance, and both satisfy disagreement point monotonicity.