ESSAYS ON EQUILIBRIUM SELECTION AND GAME THEORY

The problem of equilibrium selection has been a relevant and fundamental topic in game theory since the own definition of the Nash equilibrium concept. However, despite its importance, this topic does not appear any longer as a top priority in the game-theoretic research agenda; at least, considering the absence of relevant contributions in the last two decades. In its present state, the existent literature consists of an eclectic collection of methods, each one dependent on a particular methodology, or tailored merely to a specific class of problems. Hence, it becomes almost impossible to organize those approaches under the umbrella of one or two research programs. The current state is partially the product of the initial approach to the equilibrium multiplicity issue; that was the equilibrium refinement research program. Its purpose was the refinement of the equilibrium concept, seeking in the process to produce more reasonable predictions, and to reduce the number of possible solutions in a game. However, the result was a sequence of competing refinements and the absence of a consensual and unanimous concept. Even in extensive-form games, where the sequential equilibrium concept is possibly the most sensible candidate, we can find several variations that depend on the conditions for the consistency of beliefs, none of which being entirely satisfactory. The first approach aiming at the development of rational criteria for the selection of an equilibrium as the solution of a game emerges in the late 1980s, seen as the culmination of the refinement program and led by two of its main contributors, John Harsanyi, and Reinhard Selten. The subsequent interest resulted in some new methods and theories; however, none became utterly dominant. The slowdown that followed was also the product of an emerging paradigm, under which the information provided by the multiplicity of solutions is perhaps more useful than the prediction/prescription of a single one. However, in my opinion, game theory has reached a stage that justifies renewed attention over this topic. In the one hand, the information provided by the multiplicity of solutions does not prevent the selection of a single solution. On the other hand, the dissemination of game theoretical tools in applied research raises the necessity for sensible predictions. Those predictions do not have to correspond to an exact behavior code; instead, they may just have a prescriptive orientation, matching the limitations that real agents face. The absence of such criteria limits the impact of game-theoretical tools and raises questions as to the usefulness of the theory itself. The product of behavioral and experimental economics research also has increased the criticism on game theory, based on the argument that a theory that does not make consistent and recurrent predictions cannot be effectively useful in social sciences. In this dissertation, I treat the selection issue with a prescriptive orientation. Therefore, I propose an equilibrium selection method to static games with complete information and an extension of it to dynamic games with asymmetric information. In the first paper - equilibrium selection in static games - I define criteria of risk and payoff dominance, which I combine into a single measure. That measure - the premium of an equilibrium - represents the risk of an equilibrium to a player, given his perception about risk, and the expected payoff. Such measure helps to rationalize the available experimental evidence by adjusting the importance of each dominance criterion to the selection of an equilibrium according to the characteristics of the game, namely, the distribution of the payoffs across the game outcomes. The solution of a game is an equilibrium that minimizes the premium to the player, that is, which minimizes the risk to a player given their perception of risk and their expected payoff, conditional on the same being true for every opponent. I provide a brief axiomatic characterization of that measure, and show that the solution set is nonempty and that almost all games have a unique solution; therefore, the set of games with multiple solutions have null Lebesgue measure. In the second paper, I extend the method to dynamic games with asymmetric information. Considering the sequential nature of decision making, I show that a solution of an extensive-form game does not necessarily coincide with the solution of its reduced normal-form. I apply the method to the most basic version of Spence signaling game with just two types of worker. I obtain that the solution of such a game depends on the firms’ prior concerning the players’ types. In both dynamic and static games, I show that the method’s solutions respect certain invariance properties. I am then able to identify several directions of research. In one direction, we have the application of the selection method to specific problems in which the multiplicity of solutions play a critical role. Among such problems, I highlight bank-runs, and climate change negotiations. Additionally, I am interested in expanding the selection analysis of Spence game through the inclusion of additional types of worker. Another direction follows a different path, and focus on the identification and further characterization of the epistemic conditions of this selection method, and the comparison with the conditions in other selection methods and/or equilibrium refinement concepts.