Essays in Bargaining in Networks

Abstract

Networks show connections between agents. Among others, friendship, crime, localbuyer–seller, and inter–dealer networks in over–the–counter financial markets are of interest. Time and again, pairs of connected agents have opportunities to profit from cooperation. If they cannot rely on market prices to split the gain, they need to bargain. The outcome of bargaining, including the exact terms of a potential agreement, depends on the network. Moreover, the bargaining outcome can also change the network — for example, two players may not wish to contact one another after a disagreement. The present thesis focuses on the sources of bargaining power in networks, relative stability of networks as well as the prevalence of bargaining failures. Li and Schürhoff (2019) document that two dealers in the municipal bond market are considerably more likely to trade in a given month if they traded in the previous one. This finding has two implications. First, it suggests that an environment with an explicit network structure and regular opportunities for interaction between pairs of connected agents is well–suited to the analysis of decentralized financial markets. Transactions between dealers in decentralized markets involve bilateral bargaining. Second, disagreement between two agents could lower the frequency of future meetings. In the first chapter, I analyze a bargaining game in networks where, if two players fail to reach agreement, the opportunity goes away and their link is eliminated from the network. As a result, they never meet again. The bargaining outcome depends on the surplus from the current meeting, that is, the difference between the total value achieved under agreement and disagreement. Total value is the sum of two terms, the instantaneous payoffs and the players’ continuation values. A player’s continuation value is the expected net present value of his future cost and payoff stream, under the strategy profile in question. In addition to instantaneous payoffs, continuation values also depend on the outcome of bargaining. In case of agreement, the network remains unchanged and both players have the same future opportunities as before. In the event of disagreement, however, the game continues in a smaller network, with fewer future opportunities. The relative bargaining power between two players depends crucially on their change in continuation value from losing the connection. If the players are affected differently, the one with less to lose is in a stronger bargaining position. I consider the solution concept of Markov perfect equilibrium and establish its existence. Under certain conditions, agreement in all bargaining meetings constitutes an equilibrium. In this equilibrium, the network remains intact despite the threat of severance. I prove that agreement in all meetings is an equilibrium if and only if the cost of maintaining a connection is lower than a network specific threshold. Comparison of thresholds across different networks provides insight to their relative stability. I show that star networks are the most stable. The seminal paper of Rubinstein (1982) analyzes a complete and perfect information, non–cooperative bargaining game between two players and shows that there is unique sub– game perfect equilibrium, in which agreement is always reached. This result gave rise to the intuition that bargaining failures are caused by incomplete information. In the second chapter, I examine two complete (and perfect) information bargaining games, in particular, a two–player game and a three–player game where one player is connected to both of the others, who are not connected to each other. In contrast to the game of the first chapter, players can now decide whether to break the connection after disagreement. A disagreement in this framework leads to the loss of the value of cooperation and possibly the connection as well. In other words, players lose out on a profitable opportunity and also potentially limit their future opportunities by not reaching an agreement. Therefore, disagreement constitutes a bargaining failure. As we might expect, the two–player game has a unique sub–game perfect equilibrium where all meetings end in agreement. However, if one of the players has an additional neighbor, multiple equilibria arise, provided that opportunities arrive sufficiently frequently. Some equilibria feature bargaining failures despite complete (and perfect) information. In sub–game perfect equilibria with bargaining failures, agreement and disagreement in a meeting are, in general, followed by different strategy profiles. If, after disagreement, the game continues with a sub–game perfect equilibrium which yields a higher total value for the players, they might decide to forgo the instantaneous value from cooperation. In the third chapter, I revisit the main question of chapter 2, namely whether equilibria with bargaining failures exist. I extend the game to general networks, but consider a different solution concept — that of Markov perfect equilibrium. Markov perfect equilibria are sub–game perfect equilibria in Markov strategies. A Markov strategy is a strategy where, instead of the entire history of the game, players can only condition their actions on the network induced by the history. Whenever bargaining ends in disagreement, unless the players decide to break the connection, the game continues in the same network. Therefore, without credible threats to break the connection, the Markov requirement implies that the same (Markov) strategy profile is played after both agreement and disagreement. I prove that, in any network, there exists a unique Markov perfect equilibrium, in which all bargaining encounters end in agreement.