"The question of existence of a Nash equilibrium is one of the most important questions in game theory. This thesis aims to advance our understanding of the question in two broad directions: 1) by providing weaker sets of sufficient conditions; and 2) by introducing novel techniques, which allow for straightforward proofs and results that give new economic insights. Discontinuous games: This project considers the existence problem in games in which strategy sets are compact and convex, but preferences of the players are represented by numerical functions that may not be continuous. One way to show existence of equilibrium in such games is to apply the 'better reply security' logic, introduced by Reny (1999), combined with some form of quasiconcavity of utility functions. As long as players have securing strategies that are robust to other players' small deviations, one can show that the game has an equilibrium. The novelty of the techniques used relies in combining three main ideas: 1) a local continuous selection of each player's strict upper contour set; 2) the idea of activating different players locally; and 3) a weak notion of convexity of preferences. Bayesian games: Games of incomplete information have been shown to apply to a huge variety of economic, political and other social interactions. Still, the question of existence of equilibria in such games has been largely dealt with on a case-by-case basis. That is, given a particular game with incomplete information, one has to find the equilibrium to prove that it exists. The reason is that the usual tools to prove existence (continuous and quasiconcave payoffs defined on convex and compact strategy sets) do not apply to the general framework of games with incomplete information. This project investigates sufficient conditions for the existence of pure-strategy Nash equilibrium in games of incomplete information. The assumption of nonatomicity of the distribution of types allows for an analysis that does not require convexity assumptions on action spaces and/or payoff functions"--Pages iv-v.

This dissertation consists of three essays on evolutionary game theory and its applications. The first essay considers mechanism design in the evolutionary game-theoretic framework. The second essay studies equilibrium selection of coordination games by using an evolutionary game-theoretic concept. The third essay formulates a multi-regional economic growth model as an evolutionary game and characterizes the stability of its equilibria under an evolutionary dynamic. The summaries of each essay are provided below. In the first essay, I consider an implementation problem in a class of congestion games with players that have heterogeneous costs of taking actions. One application is to traffic congestion with drivers having heterogeneous time costs. The planner would like to design a price scheme under which the economy converges to an epsilon-optimum from any initial state when he does not have full knowledge of the cost functions, and he can observe only the aggregate strategy distribution. Although the planner would like to internalize the externalities, the informational constraints compel him to estimate their values. Using the optimality and equilibrium conditions, I construct a practical estimation procedure that yields the true values of externalities in the long-run. Moreover, I show that our scheme makes the epsilon-optimum globally stable under the best response dynamic if the externalities among players taking the same action are sufficiently large relative to those among players taking different actions. In the second essay, I study the long-run outcomes of noisy asynchronous repeated games with players that are heterogeneous in in terms of their patience. The players repeatedly play a 2-by-2 coordination game with random pair-wise matching. The games are noisy because the players may make mistakes when choosing their actions and are asynchronous because only one player can move in each period. I characterize the long-run outcomes of Markov perfect equilibrium that are robust to the mistakes and show that if there is a sufficiently patient player, the efficient state can be the unique robust outcome even if it is risk-dominated. Because I need heterogeneity for the result, I argue that it enables the most patient player in effect to be the leader. In the third essay, I consider a microfounded urban growth model with two regions and a mass of mobile workers to study interactions among growth, agglomeration, and urban congestion. Unlike previous research in the urban growth literature, I formulate the model as a one-shot game and take an evolutionary game-theoretic approach for stability analysis. My approach enables us to analyze the stability of nonstationary equilibria in which populations of each region are not constant over time. I show that if both the expenditure share for housing and inter-regional transport cost are small, a stable stationary equilibrium does not exist. Moreover, in such a case, I show that there can exist a stable nonstationary equilibrium in which mobile workers agglomerate in one region at first but some of them migrate to the other region later. I argue that such a nonstationary location pattern is related to return migration.

This state-of-the-art collection of papers on the theory of Cournotian competition focuses on two main subjects: oligopolistic Cournot competition and contests. The contributors present various applications of the Cournotian Equilibrium Theory, addressing topics such as equilibrium existence and uniqueness, equilibrium structure, dynamic processes, coalitional behavior and welfare. Special emphasis is placed on the aggregative nature of the games that are relevant to such theory. This contributed volume was written to celebrate the 80th birthday of Prof. Koji Okuguchi, a pioneer in oligopoly theory.

Chapter 1: This chapter considers a new class of dynamic, two-player games, where a stage game is continuously repeated but each player can only move at random times that she privately observes. A player's move is an adjustment of her action in the stage game, for example, a duopolist's change of price. Each move is perfectly observed by both players, but a foregone opportunity to move, like a choice to leave one's price unchanged, would not be directly observed by the other player. Some adjustments may be constrained in equilibrium by moral hazard, no matter how patient the players are. For example, a duopolist would not jump up to the monopoly price absent costly incentives. These incentives are provided by strategies that condition on the random waiting times between moves; punishing a player for moving slowly, lest she silently choose not to move. In contrast, if the players are patient enough to maintain the status quo, perhaps the monopoly price, then doing so does not require costly incentives. Deviation from the status quo would be perfectly observed, so punishment need not occur on the equilibrium path. Similarly, moves like jointly optimal price reductions do not require costly incentives. Again, the tempting deviation, to a larger price reduction, would be perfectly observed. This chapter provides a recursive framework for analyzing these games following Abreu, Pearce, and Stacchetti (1990) and the continuous time adaptation of Sannikov (2007). For a class of stage games with monotone public spillovers, like differentiated-product duopoly, I prove that optimal equilibria have three features corresponding to the discussion above: beginning at a "low" position, optimal, upward moves are impeded by moral hazard; beginning at a "high" position, optimal, downward moves are unimpeded by moral hazard; beginning at an intermediate position, optimally maintaining the status quo is similarly unimpeded. Corresponding cooperative dynamics are suggested in the older, non-game-theoretic literature on tacit collusion. Chapter 2: This chapter shows that in finite-horizon games of a certain class, small perturbations of the overall payoff function may yield large changes to unique equilibrium payoffs in periods far from the last. Such perturbations may tie together cooperation across periods in equilibrium, allowing substantial cooperation to accumulate in periods far from the last. Chapter 3: A dynamic choice problem faced by a time-inconsistent individual is typically modeled as a game played by a sequence of her temporal selves, solved by SPNE. It is recognized that this approach yields troublesomely many solutions for infinite-horizon problems, which is often attributed to the existence of implausible equilibria based on self-reward and punishment. This chapter presents a refinement applicable within the special class of strategically constant (SC) problems, which are those where all continuation problems are isomorphic. The refinement requires that each self's strategy be invariant, here that implies history-independence under the isomorphism. I argue that within the class of SC problems, this refinement does little more than rule out self-reward and punishment. The refinement substantially narrows down the set of equilibria in SC problems, but in some cases allows plausible equilibria that are excluded by other refinement approaches. The SC class is limited, but broader than it might seem at first.

In my dissertation, I provide two models of joint contribution games that are relevant to the phenomenon of crowdfunding. I also provide a characterization of Bayes Nash equilibrium. In the first chapter I build a model of crowdfunding. An entrepreneur finances her project with common value via crowdfunding. She chooses a funding mechanism (fixed or flexible), a price, and a funding goal. Under fixed funding money is refunded if the goal is not met; under flexible funding the entrepreneur keeps the money. Backers observe signals about the value and decide whether to contribute or postpone purchase to the retail stage. The optimal crowdfunding campaign is characterized. When the entrepreneur has commitment power, fixed funding generates more revenue than flexible funding. When the entrepreneur has no commitment power, fixed funding serves as a commitment device to eliminate moral hazard In the second chapter I consider a dynamic contribution game under two regimes. The first regime is that all but the last rounds are cheap talk, the other is that in all rounds contribution is sunk. With binary contribution levels and a continuum of types we show that one of the monotone equilibria in the first regime implements the ex-post efficient and ex-post individually rational allocation when the cheap talk period is long enough. In contrast, when commitment is required, no equilibria achieves the same allocation. However, with a continuum of contribution levels, all equilibria of the contribution game with cheap talk will be the same as a one-shot game with no cheap talk, due to severe free riding. In this case, dynamic contribution with commitment provides credibility and can significantly improve efficiency. In the third chapter, coauthored with Ichiro Obara, we prove the following characterization regarding types and Bayes equilibrium actions they play across games: Given any two types in any two countable type spaces, if for all finite games, the two types have the same pure Bayes Nash equilibrium action, then there exists a bijective belief morphism between them. As an application, our result implies that the universal space for Bayes Nash equilibrium that retains non-redundancy does not exist.

Optimal control and differential games continue to attract strong interest from researchers interested in dynamical problems and models in management science. This volume explores the application of these methodologies to new as well as to classical decision problems in management sciences and economics. In Part I, optimal control and dynamical systems approaches are used to analyze problems in areas such as monetary policy, pollution control, relationship marketing, drug control, debt financing, and ethical behavior. In Part II differential games are applied to problems such as oligopolistic competition, common resource management, spillovers in foreign direct investments, marketing channels, incentive strategies, and the computation of Markov perfect Nash equilibria. Optimal Control and Differential Games is an excellent reference for researchers and graduate students covering a wide range of emerging and revisited problems in management science.