Bayesian Network Games

This thesis builds from the realization that Bayesian Nash equilibria are the natural definition of optimal behavior in a network of distributed autonomous agents. Game equilibria are often behavior models of competing rational agents that take actions that are strategic reactions to the predicted actions of other players. In autonomous systems however, equilibria are used as models of optimal behavior for a different reason: Agents are forced to play strategically against inherent uncertainty. While it may be that agents have conflicting intentions, more often than not, their goals are aligned. However, barring unreasonable accuracy of environmental information and unjustifiable levels of coordination, they still can\u27t be sure of what the actions of other agents will be. Agents have to focus their strategic reasoning on what they believe the information available to other agents is, how they think other agents will respond to this hypothetical information, and choose what they deem to be their best response to these uncertain estimates. If agents model the behavior of each other as equally strategic, the optimal response of the network as a whole is a Bayesian Nash equilibrium. We say that the agents are playing a Bayesian network game when they repeatedly act according to a stage Bayesian Nash equilibrium and receive information from their neighbors in the network. The first part of the thesis is concerned with the development and analysis of algorithms that agents can use to compute their equilibrium actions in a game of incomplete information with repeated interactions over a network. In this regard, the burden of computing a Bayesian Nash equilibrium in repeated games is, in general, overwhelming. This thesis shows that actions are computable in the particular case when the local information that agents receive follows a Gaussian distribution and the game\u27s payoff is represented by a utility function that is quadratic in the actions of all agents and an unknown parameter. This solution comes in the form of the Quadratic Network Game filter that agents can run locally, i.e., without access to all private signals, to compute their equilibrium actions. For the more generic payoff case of Bayesian potential games, i.e., payoffs represented by a potential function that depends on population actions and an unknown state of the world, distributed versions of fictitious play that converge to Nash equilibrium with identical beliefs on the state are derived. This algorithm highlights the fact that in order to determine optimal actions there are two problems that have to be solved: (i) Construction of a belief on the state of the world and the actions of other agents. (ii) Determination of optimal responses to the acquired beliefs. In the case of symmetric and strictly supermodular games, i.e., games with coordination incentives, the thesis also derives qualitative properties of Bayesian network games played in the time limit. In particular, we ask whether agents that play and observe equilibrium actions are able to coordinate on an action and learn about others\u27 behavior from only observing peers\u27 actions. The analysis described here shows that agents eventually coordinate on a consensus action. The second part of this thesis considers the application of the algorithms developed in the first part to the analysis of energy markets. Consumer demand profiles and fluctuating renewable power generation are two main sources of uncertainty in matching demand and supply in an energy market. We propose a model of the electricity market that captures the uncertainties on both, the operator and the user side. The system operator (SO) implements a temporal linear pricing strategy that depends on real-time demand and renewable generation in the considered period combining Real-Time Pricing with Time-of-Use Pricing. The announced pricing strategy sets up a noncooperative game of incomplete information among the users with heterogeneous but correlated consumption preferences. An explicit characterization of the optimal user behavior using the Bayesian Nash equilibrium solution concept is derived. This explicit characterization allows the SO to derive pricing policies that influence demand to serve practical objectives such as minimizing peak-to-average ratio or attaining a desired rate of return. Numerical experiments show that the pricing policies yield close to optimal welfare values while improving these practical objectives. We then analyze the sensitivity of the proposed pricing schemes to user behavior and information exchange models. Selfish, altruistic and welfare maximizing user behavior models are considered. Furthermore, information exchange models in which users only have private information, communicate or receive broadcasted information are considered. For each pair of behavior and information exchange models, rational price anticipating consumption strategy is characterized. In all of the information exchange models, equilibrium actions can be computed using the Quadratic Network Game filter. Further experiments reveal that communication model is beneficial for the expected aggregate payoff while it does not affect the expected net revenue of the system operator. Moreover, additional information to the users reduces the variance of total consumption among runs, increasing the accuracy of demand predictions

Robust Comparative Statics in Large Dynamic Economies

We consider infinite horizon economies populated by a continuum of agents who are subject to idiosyncratic shocks. This framework contains models of saving and capital accumulation with incomplete markets in the spirit of works by Bewley, Aiyagari, and Huggett, and models of entry, exit and industry dynamics in the spirit of Hopenhayn's work as special cases. Robust and easy-to-apply comparative statics results are established with respect to exogenous parameters as well as various kinds of changes in the Markov processes governing the law of motion of the idiosyncratic shocks. These results complement the existing literature which uses simulations and numerical analysis to study this class of models and are illustrated using a number of examples

Social Networks in Determining Employment and Wages: Patterns, Dynamics, and Inequality

We develop a model where agents obtain information about job opportunities through an explicitly modeled network of social contacts. We show that an improvement in the employment status of either an agent's direct or indirect contacts leads to an increase in the agent's employment probability and expected wages, in the sense of first order stochastic dominance. A similar effect results from an increase in the network contacts of an agent. In terms of dynamics and patterns, we show that employment is positively correlated across time and agents, and the same is true for wages. Moreover, unemployment exhibits persistence in the sense of duration dependence: the probability of obtaining a job decreases in the length of time that an agent has been unemployed. Finally, we examine inequality between two groups. If staying in the labor market is costly (in opportunity costs, education costs, or skills maintenance) and one group starts with a worse employment status or a smaller network, then that group's drop-out rate will be higher and their employment prospects and wages will be persistently below that of the other group.Networks, Labor Markets, Employment, Unemployment, Wages, Wage Inequality, Drop-Out Rates, Duration Dependence

Inference And Learning: Computational Difficulty And Efficiency

In this thesis, we mainly investigate two collections of problems: statistical network inference and model selection in regression. The common feature shared by these two types of problems is that they typically exhibit an interesting phenomenon in terms of computational difficulty and efficiency. For statistical network inference, our goal is to infer the network structure based on a noisy observation of the network. Statistically, we model the network as generated from the structural information with the presence of noise, for example, planted submatrix model (for bipartite weighted graph), stochastic block model, and Watts-Strogatz model. As the relative amount of ``signal-to-noise\u27\u27 varies, the problems exhibit different stages of computational difficulty. On the theoretical side, we investigate these stages through characterizing the transition thresholds on the ``signal-to-noise\u27\u27 ratio, for the aforementioned models. On the methodological side, we provide new computationally efficient procedures to reconstruct the network structure for each model. For model selection in regression, our goal is to learn a ``good\u27\u27 model based on a certain model class from the observed data sequences (feature and response pairs), when the model can be misspecified. More concretely, we study two model selection problems: to learn from general classes of functions based on i.i.d. data with minimal assumptions, and to select from the sparse linear model class based on possibly adversarially chosen data in a sequential fashion. We develop new theoretical and algorithmic tools beyond empirical risk minimization to study these problems from a learning theory point of view

Dynamic Pricing and Inventory Management: Theory and Applications

We develop the models and methods to study the impact of some emerging trends in technology, marketplace, and society upon the pricing and inventory policy of a firm. We focus on the situation where the firm is in a dynamic, uncertain, and (possibly) competitive market environment. The market trends of particular interest to us are: (a) social networks, (b) sustainability concerns, and (c) customer behaviors. The two main running questions this dissertation aims to address are: (a) How these emerging market trends would influence the operations decisions and profitability of a firm; and (b) What pricing and inventory strategies a firm could use to leverage these trends. We also develop an effective comparative statics analysis method to address these two questions under different market trends. Overall, our results suggest that the current market trends of social networks, sustainability concerns, and customer behaviors have significant and interesting impact upon the operations policy of a firm, and that the firm could adopt some innovative pricing and inventory strategies to exploit these trends and substantially improve its profit. Our main findings are: (a) Network externalities (the monopoly setting). We find that network externalities prompt a firm to face the tradeoff between generating current profits and inducing future demands when making the price and inventory decisions, so that it should increase the base-stock level, and to decrease [increase] the sales price when the network size is small [large]. Our extensive numerical experiments also demonstrate the effectiveness of the heuristic policies that leverage network externalities by balancing generating current profits and inducing demands in the near future. (Chapter 2.) (b) Network externalities (the dynamic competition setting). In a competitive market with network externalities, the competing firms face the tradeoff between generating current profits and winning future market shares (i.e., the exploitation-induction tradeoff). We characterize the pure strategy Markov perfect equilibrium in both the simultaneous competition and the promotion-first competition. We show that, to balance the exploitation-induction tradeoff, the competing firms should increase promotional efforts, offer price discounts, and improve service levels. The exploitation-induction tradeoff could be a new driving force for the fat-cat effect (i.e., the equilibrium promotional efforts are higher under the promotion-first competition than those under the simultaneous competition). (Chapter 3.) (d) Trade-in remanufacturing. We show that, with the adoption of the very commonly used trade-in remanufacturing program, the firm may enjoy a higher profit with strategic customers than with myopic customers. Moreover, trade-in remanufacturing creates a tension between firm profitability and environmental sustainability with strategic customers, but benefits both the firm and the environment with myopic customers. We also find that, with either strategic or myopic customers, the socially optimal outcome can be achieved by using a simple linear subsidy and tax scheme. The commonly used government policy to subsidize for remanufacturing alone, however, does not induce the social optimum in general. (Chapter 4.) (d) Scarcity effect of inventory. We show that the scarcity effect drives both optimal prices and order-up-to levels down, whereas increased operational flexibilities (e.g., the inventory disposal and inventory withholding opportunities) mitigate the demand loss caused by high excess inventory and increase the optimal order-up-to levels and sales prices. Our extensive numerical studies also demonstrate that dynamic pricing leads to a much more significant profit improvement with the scarcity effect of inventory than without. (Chapter 5.) (e) Comparative statics analysis method. We develop a comparative statics method to study a general joint pricing and inventory management model with multiple demand segments, multiple suppliers, and stochastically evolving market conditions. Our new method makes componentwise comparisons between the focal decision variables under different parameter values, so it is capable of performing comparative statics analysis in a model where part of the decision variables are non-monotone, and it is well scalable. Hence, our new method is promising for comparative statics analysis in other operations management models. (Chapter 6.

On monotone strategy equilibria in simultaneous auctions for complementary goods

We explore existence and properties of equilibrium when N ≥ 2 bidders compete for L ≥ 2 objects via simultaneous but separate auctions. Bidders have private combinatorial valuations over all sets of objects they could win, and objects are complements in the sense that these valuations are supermodular in the set of objects won. We provide a novel partial order on typesunder which best replies are monotone, and demonstrate that Bayesian Nash equilibria which are monotone with respect to this partial order exist on any finite bid lattice. We apply this result to show existence of monotone Bayesian Nash equilibria in continuous bid spaces when a single global bidder competesfor L objects against many local bidders who bid for single objects only. Wethen consider monotone equilibrium with endogenous tiebreaking building onJackson, Simon, Swinkels and Zame (2002), and demonstrate that these existin general. These existence results apply to many auction formats, including first-price, second-price, and all-pay

An aggregation method for large-scale dynamic games

It is a well known fact that many dynamic games are subject to the curse of dimensionality, limiting the ability to use them in the study of real-world problems. I propose a new method to solve complex large-scale dynamic games using aggregation as an approximate solution. I obtain two fundamental characterization results. First, approximations with small within-state variation in the primitives have a smaller maximum error bound. I provide numerical results which compare the exact errors and the bound. Second, I find that for monotone games, order preserving aggregation is a necessary condition of any optimal aggregation. I suggest using quantiles as a straightforward implementation of an order preserving aggregation architecture for industry distributions. I conclude with an illustration, by solving and estimating a stylized dynamic reputation game for the hotel industry. Simulation results show maximal errors between the exact and approximated solutions below 6%, with average errors below 1%.publishersversionpublishe

Essays in Spatial Economics

A central concern in international economics and urban economics is explaining the distributions of economic assets and activity across space. This dissertation contains three essays examining the pattern of specialization across US cities. Chapter 1 investigates the determinants of quality specialization within products. A growing literature suggests that high-income countries export high-quality goods. Two hypotheses may explain such specialization, with different implications for welfare, inequality, and trade policy. Fajgelbaum, Grossman, and Helpman (2011) formalize the Linder (1961) conjecture that home demand determines the pattern of specialization and therefore predict that high-income locations export high-quality products. The factor-proportions model also predicts that skill-abundant, high-income locations export skill-intensive, high-quality products (Schott, 2004). Prior empirical evidence does not separate these explanations. I develop a model that nests both hypotheses and employ microdata on US manufacturing plants' shipments and factor inputs to quantify the two mechanisms' roles in quality specialization across US cities. Home-market demand explains at least as much of the relationship between income and quality as differences in factor usage. In Chapter 2, Donald R. Davis and I develop a theory to jointly explain the distributions of skills, occupations, and industries across cities. Our model incorporates a system of cities, their internal urban structures, and a high-dimensional theory of factor-driven comparative advantage. It predicts that larger cities will be skill-abundant and specialize in skill-intensive activities according to the monotone likelihood ratio property. We test the model using data on 270 US metropolitan areas, 3 to 9 educational categories, 22 occupations, and 21 manufacturing industries. The results provide support for our theory's predictions. Chapter 3 examines whether larger cities are attractive to consumers. Popular and academic discussions celebrate the virtues of large cities for consumption and leisure. But the standard spatial-equilibrium account says that the consumer attractions of larger cities cannot account for their higher nominal wages and more skilled populations. This chapter revisits that conclusion and shows that the consumption motive can play a first-order role in spatial variation in wage distributions when individuals are heterogeneous. I present a general-equilibrium model in which larger cities offer a greater variety of local goods and services, attracting higher-income individuals who value such variety relatively more. Despite the absence of production-related agglomeration economies, the equilibrium outcomes match a series of facts about spatial variation in wage distributions. I present evidence on the spatial choices of retirees, who consume but do not produce, that is consistent with consumption-driven agglomeration