Complexity Theory, Game Theory, and Economics: The Barbados Lectures

This document collects the lecture notes from my mini-course "Complexity Theory, Game Theory, and Economics," taught at the Bellairs Research Institute of McGill University, Holetown, Barbados, February 19--23, 2017, as the 29th McGill Invitational Workshop on Computational Complexity. The goal of this mini-course is twofold: (i) to explain how complexity theory has helped illuminate several barriers in economics and game theory; and (ii) to illustrate how game-theoretic questions have led to new and interesting complexity theory, including recent several breakthroughs. It consists of two five-lecture sequences: the Solar Lectures, focusing on the communication and computational complexity of computing equilibria; and the Lunar Lectures, focusing on applications of complexity theory in game theory and economics. No background in game theory is assumed.Comment: Revised v2 from December 2019 corrects some errors in and adds some recent citations to v1 Revised v3 corrects a few typos in v

Dynamical selection of Nash equilibria using Experience Weighted Attraction Learning: emergence of heterogeneous mixed equilibria

We study the distribution of strategies in a large game that models how agents choose among different double auction markets. We classify the possible mean field Nash equilibria, which include potentially segregated states where an agent population can split into subpopulations adopting different strategies. As the game is aggregative, the actual equilibrium strategy distributions remain undetermined, however. We therefore compare with the results of Experience-Weighted Attraction (EWA) learning, which at long times leads to Nash equilibria in the appropriate limits of large intensity of choice, low noise (long agent memory) and perfect imputation of missing scores (fictitious play). The learning dynamics breaks the indeterminacy of the Nash equilibria. Non-trivially, depending on how the relevant limits are taken, more than one type of equilibrium can be selected. These include the standard homogeneous mixed and heterogeneous pure states, but also \emph{heterogeneous mixed} states where different agents play different strategies that are not all pure. The analysis of the EWA learning involves Fokker-Planck modeling combined with large deviation methods. The theoretical results are confirmed by multi-agent simulations.Comment: 35 pages, 16 figure

An Investigation Report on Auction Mechanism Design

Auctions are markets with strict regulations governing the information available to traders in the market and the possible actions they can take. Since well designed auctions achieve desirable economic outcomes, they have been widely used in solving real-world optimization problems, and in structuring stock or futures exchanges. Auctions also provide a very valuable testing-ground for economic theory, and they play an important role in computer-based control systems. Auction mechanism design aims to manipulate the rules of an auction in order to achieve specific goals. Economists traditionally use mathematical methods, mainly game theory, to analyze auctions and design new auction forms. However, due to the high complexity of auctions, the mathematical models are typically simplified to obtain results, and this makes it difficult to apply results derived from such models to market environments in the real world. As a result, researchers are turning to empirical approaches. This report aims to survey the theoretical and empirical approaches to designing auction mechanisms and trading strategies with more weights on empirical ones, and build the foundation for further research in the field

Computing Bayes Nash Equilibrium Strategies in Auction Games via Simultaneous Online Dual Averaging

Auctions are modeled as Bayesian games with continuous type and action spaces. Computing equilibria in auction games is computationally hard in general and no exact solution theory is known. We introduce algorithms computing distributional strategies on a discretized version of the game via online convex optimization. One advantage of distributional strategies is that we do not have to make any assumptions on the shape of the bid function. Besides, the expected utility of agents is linear in the strategies. It follows that if our regularized optimization algorithms converge to a pure strategy, then they converge to an approximate equilibrium of the discretized game with high precision. Importantly, we show that the equilibrium of the discretized game approximates an equilibrium in the continuous game. In a wide variety of auction games, we provide empirical evidence that the method approximates the analytical (pure) Bayes Nash equilibrium closely. This speed and precision is remarkable, because in many finite games learning dynamics do not converge or are even chaotic. In standard models where agents are symmetric, we find equilibrium in seconds. The method allows for interdependent valuations and different types of utility functions and provides a foundation for broadly applicable equilibrium solvers that can push the boundaries of equilibrium analysis in auction markets and beyond

DYNAMIC STRATEGY CHOICE BEHAVIOR UNDER COMPETITIVE ENVIRONMENT: APPLICATION TO ELECTRONIC FREIGHT AUCTION MARKETPLACES

In the electronic market place, the auction is known to be economically efficient, allowing players to maximize their own benefits. Through this mechanism, new spot markets are created, which connect buyers and sellers. In this new spot markets, many problem contexts give rise to competitive decision situations in which players must make repeated decisions along with or in response to competitors' decisions. Auction-based electronic marketplaces for freight service procurement are an example of such environments, and provide the motivating application context for the models presented in this paper. The specific focus is on the decisions of carriers, as bidders for the loads tendered by shippers in spot market situations. This paper is about the learning models used to describe a player's strategy choice behavior using experimental data and explains how that choice arises from the nature of multi-player interactions and their dynamics over multiple bids. Therefore, the principal focus of this paper is how to model a player's dynamic strategy choice behavior under the pressure of competition. A dynamic strategy choice model structure for two type of cognitive learning process is formulated, with alternative specifications corresponding to different levels of cognition capacity. Furthermore, the dynamic strategy choice model structure for mixed learning is developed, which combines both elements of two different learning processes. The model is intended to describe how a player or agent in a non-cooperative game with no perfect information and bounded rationality chooses a bidding strategy. We propose a general dynamic strategy choice model framework using the dynamic mixed logit model structure and estimate the model parametrically, using two sets of experimental data. The paper also presents econometric issues that arise in estimating such models given a time series of auction bids and outcomes, and formulates error structures appropriate to the highly interactive dynamic nature of competitive auction-based marketplaces

Dynamic strategic interactions : analysis and mechanism design

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (p. 225-232).Modern systems, such as engineering systems with autonomous entities, markets, and financial networks, consist of self-interested agents with potentially conflicting objectives. These agents interact in a dynamic manner, modifying their strategies over time to improve their payoffs. The presence of self-interested agents in such systems, necessitates the analysis of the impact of multi-agent decision making on the overall system, and the design of new systems with improved performance guarantees. Motivated by this observation, in the first part of this thesis we focus on fundamental structural properties of games, and exploit them to provide a new framework for analyzing the limiting behavior of strategy update rules in various game-theoretic settings. In the second part, we investigate the design problem of an auctioneer who uses iterative multi-- item auctions for efficient allocation of resources. More specifically, in the first part of the thesis we focus on potential games, a special class of games with desirable equilibrium and dynamic properties, and analyze their preference structure. Exploiting this structure we obtain a decomposition of arbitrary games into three components, which we refer to as the potential, harmonic, and nonstrategic components. Intuitively, the potential component of a game captures interactions that can equivalently be represented as a common interest game, while the harmonic part represents conflicts between the interests of the players. We make this intuition precise by studying the properties of these two components, and establish that indeed they have quite distinct and remarkable characteristics. The decomposition also allows us to approximate a given game with a potential game. We show that the set of approximate equilibria of an arbitrary game can be characterized through the equilibria of a potential game that approximates it. The decomposition provides a valuable tool for the analysis of dynamics in games. Earlier literature established that many natural strategy update rules converge to a Nash equilibrium in potential games. We show that games that are close to a potential game exhibit similar properties. In particular, we focus on three commonly studied discrete-time update rules (better/best response, logit response, and discrete-time fictitious play dynamics), and establish that in near-potential games, the limiting behavior of these update rules can be characterized by an approximate equilibrium set, size of which is proportional to the distance of the original game from a potential game. Since a close potential game to a given game can be systematically found via decomposition, our results suggest a systematic framework for studying the limiting behavior of adaptive dynamics in arbitrary finite strategic form games: the limiting behavior of dynamics in a given game can be characterized by first approximating this game with a potential game, and then analyzing the limiting behavior of dynamics in the potential game. In the second part of the thesis, we change our focus to implementing efficient outcomes in multi-agent settings by using simple mechanisms. In particular, we develop novel efficient iterative auction formats for multi-item environments, where items exhibit value complementarities/substitutabilities. We obtain our results by focusing on a special class of value functions, which we refer to as graphical valuations. These valuations are not fully general, but importantly they capture value complementarity/substitutability in important practical settings, while allowing for a compact representation of the value functions. We start our analysis by first analyzing how the special structure of graphical valuations can be exploited to design simple iterative auction formats. We show that in settings where the underlying value graph is a tree (and satisfies an additional technical condition), a Walrasian equilibrium always exists (even in the presence of value complementarities). Using this result we provide a linear programming formulation of the efficient allocation problem for this class of valuations. Additionally, we demonstrate that a Walrasian equilibrium may not exist, when the underlying value graph is more general. However, we also establish that in this case a more general pricing equilibrium always exists, and provide a stronger linear programming formulation that can be used to identify the efficient allocation for general graphical valuations. We then consider solutions of these linear programming formulations using iterative algorithms. Complementing these iterative algorithms with appropriate payment rules, we obtain iterative auction formats that implement the efficient outcome at an (ex-post perfect) equilibrium. The auction formats we obtain rely on simple pricing rules that, in the most general case, require offering a bidder-specific price for each item, and bidder-specific discounts/markups for pairs of items. Our results in this part of the thesis suggest that when value functions of bidders exhibit some special structure, it is possible to systematically exploit this structure in order to develop simple efficient iterative auction formats.by Utku Ozan Candogan.Ph.D