Abstract

In this chapter we examine the representational and algorithmic aspects of a class of graph-theoretic models for multiplayer games. Known broadly as graphical games, these models specify restrictions on the direct payoff influences among the player population. In addition to a number of nice computational properties, these models have close connections to well-studied graphical models for probabilistic inference in machine learning and statistics.

Introduction

Representing multiplayer games with large player populations in the normal form is undesirable for both practical and conceptual reasons. On the practical side, the number of parameters that must be specified grows exponentially with the size of the population. On the conceptual side, the normal form may fail to capture structure that is present in the strategic interaction, and which can aid understanding of the game and computation of its equilibria. For this reason, there have been many proposals for parametric multiplayer game representations that are more succinct than the normal form, and attempt to model naturally arising structural properties. Examples include congestion and potential games and related models (Monderer and Shapley, 1996; Rosenthal, 1973).

Graphical games are a representation of multiplayer games meant to capture and exploit locality or sparsity of direct influences. They are most appropriate for large population games in which the payoffs of each player are determined by the actions of only a small subpopulation. As such, they form a natural counterpart to earlier parametric models.

Abstract

This chapter studies the inefficiency of equilibria in noncooperative routing games, in which self-interested players route traffic through a congested network. Our goals are threefold: to introduce the most important models and examples of routing games; to survey optimal bounds on the price of anarchy in these models; and to develop proof techniques that are useful for bounding the inefficiency of equilibria in a range of applications.

Introduction

A majority of the current literature on the inefficiency of equilibria concerns routing games. One reason for this popularity is that routing games shed light on an important practical problem: how to route traffic in a large communication network, such as the Internet, that has no central authority. The routing games studied in this chapter are relevant for networks with “source routing,” in which each end user chooses a full route for its traffic, and also for networks in which traffic is routed in a distributed, congestion-sensitive manner. Section 18.6 contains further details on these applications.

This chapter focuses on two different models of routing games, although the inefficiency of equilibria has been successfully quantified in a range of others (see Section 18.6). The first model, nonatomic selfish routing, is a natural generalization of Pigou's example (Example 17.1) to more complex networks. The modifier “nonatomic” refers to the assumption that there are a very large number of players, each controlling a negligible fraction of the overall traffic.

As the Second World War was coming to its end, John von Neumann, arguably the foremost mathematician of that time, was busy initiating two intellectual currents that would shape the rest of the twentieth century: game theory and algorithms. In 1944 (16 years after the minmax theorem) he published, with Oscar Morgenstern, his Games and Economic Behavior, thus founding not only game theory but also utility theory and microeconomics. Two years later he wrote his draft report on the EDVAC, inaugurating the era of the digital computer and its software and its algorithms. Von Neumann wrote in 1952 the first paper in which a polynomial algorithm was hailed as a meaningful advance. And, he was the recipient, shortly before his early death four years later, of Gödel's letter in which the P vs. NP question was first discussed.

Could von Neumann have anticipated that his twin creations would converge half a century later? He was certainly far ahead of his contemporaries in his conception of computation as something dynamic, ubiquitous, and enmeshed in society, almost organic – witness his self-reproducing automata, his fault-tolerant network design, and his prediction that computing technology will advance in lock-step with the economy (for which he had already postulated exponential growth in his 1937 Vienna Colloquium paper).

Abstract

Combinatorial polynomial time algorithms are presented for finding equilibrium prices and allocations for the linear utilities case of the Fisher and Arrow–Debreu models using the primal-dual schema and an auction-based approach, respectively. An intersting feature of the first algorithm is that it finds an optimal solution to a nonlinear convex program, the Eisenberg-Gale program.

Resource allocation markets in Kelly's model are also discussed and a strongly polynomial combinatorial algorithm is presented for one of them.

Introduction

Thinkers and philosophers have pondered over the notions of markets and money through the ages. The credit for initiating formal mathematical modeling and study of these notions is generally attributed to nineteenth-century economist Leon Walras (1874). The fact that Western economies are capitalistic had a lot to do with the over-whelming importance given to this study within mathematical economics – essentially, our most critical decision-making is relegated to pricing mechanisms. They largely determine the relative prices of goods and services, ensure that the economy is efficient, in that goods and services are made available to entities that produce items that are most in demand, and ensure a stable operation of the economy.

A central tenet in pricing mechanisms is that prices be such that demand equals supply; that is, the economy should operate at equilibrium. It is not surprising therefore that perhaps the most celebrated theorem within general equilibrium theory, the Arrow–Debreu Theorem, establishes precisely the existence of such prices under a very general model of the economy.

Abstract

In combinatorial auctions, a large number of items are auctioned concurrently and bidders are allowed to express preferences on bundles of items. This is preferable to selling each item separately when there are dependencies between the different items. This problem has direct applications, may be viewed as a general abstraction of complex resource allocation, and is the paradigmatic problem on the interface of economics and computer science. We give a brief survey of this field, concentrating on theoretical treatment.

Introduction

A large part of computer science as well as a large part of economics may be viewed as addressing the “allocation problem”: how should we allocate “resources” among the different possible uses of these resources. An auction of a single item may be viewed as a simple abstraction of this question: we have a single indivisible resource, and two (or more) players desire using it – who should get it? Being such a simple and general abstraction explains the pivotal role of simple auctions in mechanism design theory.

From a similar point of view, “combinatorial auctions” abstract this issue when multiple resources are involved: how do I allocate a collection of interrelated resources? In general, the “interrelations” of the different resources may be combinatorially complex, and thus handling them requires effective handling of this complexity. It should thus come as no surprise that the field of “combinatorial auctions” – the subject of this chapter – is gaining a central place in the interface between computer science and economics.

Abstract

We give an introduction to the micro-economic field of Mechanism Design slightly biased toward a computer-scientist's point of view.

Introduction

Mechanism Design is a subfield of economic theory that is rather unique within economics in having an engineering perspective. It is interested in designing economic mechanisms, just like computer scientists are interested in designing algorithms, protocols, or systems. It is best to view the goals of the designed mechanisms in the very abstract terms of social choice. A social choice is simply an aggregation of the preferences of the different participants toward a single joint decision. Mechanism Design attempts implementing desired social choices in a strategic setting – assuming that the different members of society each act rationally in a game theoretic sense. Such strategic design is necessary since usually the preferences of the participants are private.

This high-level abstraction of aggregation of preferences may be seen as a common generalization of a multitude of scenarios in economics as well as in other social settings such as political science. Here are some basic classic examples:

• Elections: In political elections each voter has his own preferences between the different candidates, and the outcome of the elections is a single social choice.

• Markets: Classical economic theory usually assumes the existence and functioning of a “perfect market.” In reality, of course, we have only interactions between people, governed by some protocols. Each participant in such an interaction has his own preferences, but the outcome is a single social choice: the reallocation of goods and money.

• […]

Abstract

Computing a Nash equilibrium, given a game in normal form, is a fundamental problem for Algorithmic Game Theory. The problem is essentially combinatorial, and in the case of two players it can be solved by a pivoting technique called the Lemke–Howson algorithm, which however is exponential in the worst case. We outline the recent proof that finding a Nash equilibrium is complete for the complexity class PPAD, even in the case of two players; this is evidence that the problem is intractable. We also introduce several variants of succinctly representable games, a genre important in terms of both applications and computational considerations, and discuss algorithms for correlated equilibria, a more relaxed equilibrium concept.

Introduction

Nash's theorem – stating that every finite game has a mixed Nash equilibrium (Nash, 1951) – is a very reassuring fact: Any game can, in principle, reach a quiescent state, one in which no player has an incentive to change his or her behavior. One question arises immediately: Can this state be reached in practice? Is there an efficient algorithm for finding the equilibrium that is guaranteed to exist? This is the question explored in this chapter.

But why should we be interested in the issue of computational complexity in connection to Nash equilibria? After all, a Nash equilibrium is above all a conceptual tool, a prediction about rational strategic behavior by agents in situations of conflict – a context that is completely devoid of computation.

Abstract

In this chapter, we study the allocation of a single infinitely divisible resource among multiple competing users. While we aim for efficient allocation of the resource, the task is complicated by the fact that users' utility functions are typically unknown to the resource manager. We study the design of resource allocation mechanisms that are approximately efficient (i.e., have a low price of anarchy), with low communication requirements (i.e., the strategy spaces of users are low dimensional).

Our main results concern the proportional allocation mechanism, for which a tight bound on the price of anarchy can be provided. We also show that in a wide range of market mechanisms that use a single market-clearing price, the proportional allocation mechanism minimizes the price of anarchy. Finally, we relax the assumption of a single market-clearing price, and show that by extending the class of Vickrey–Clarke–Groves mechanisms all Nash equilibria can be guaranteed to be fully efficient.

Introduction

This chapter deals with a canonical resource allocation problem. Suppose that a finite number of users compete to acquire a share of an infinitely divisible resource of fixed capacity. How should the resource be shared among the users? We will frame this problem as an economic problem: we assume that each user has a utility function that is increasing in the amount of the resource received, and then design a mechanism to maximize aggregate utility.

This book covers an area that straddles two fields, algorithms and game theory, and has applications in several others, including networking and artificial intelligence. Its text is pitched at a beginning graduate student in computer science – we hope that this makes the book accessible to readers across a wide range of areas.

We started this project with the belief that the time was ripe for a book that clearly develops some of the central ideas and results of algorithmic game theory – a book that can be used as a textbook for the variety of courses that were already being offered at many universities. We felt that the only way to produce a book of such breadth in a reasonable amount of time was to invite many experts from this area to contribute chapters to a comprehensive volume on the topic.

This book is partitioned into four parts: the first three parts are devoted to core areas, while the fourth covers a range of topics mostly focusing on applications. Chapter 1 serves as a preliminary chapter and it introduces basic game-theoretic definitions that are used throughout the book. The first chapters of Parts II and III provide introductions and preliminaries for the respective parts. The other chapters are largely independent of one another. The authors were requested to focus on a few results highlighting the main issues and techniques, rather than provide comprehensive surveys.

Abstract

This chapter examines the intersection of evolutionary game theory and theoretical computer science. We will show how techniques from each field can be used to answer fundamental questions in the other. In addition, we will analyze a model that arises by combining ideas from both fields. First, we describe the classical model of evolutionary game theory and analyze the computational complexity of its central equilibrium concept. Doing so involves applying techniques from complexity theory to the problem of finding a game-theoretic equilibrium. Second, we show how agents using imitative dynamics, often considered in evolutionary game-theory, converge to an equilibrium in a routing game. This is an instance of an evolutionary game-theoretic concept providing an algorithm for finding an equilibrium. Third, we generalize the classical model of evolutionary game theory to a graph-theoretic setting. Finally, this chapter concludes with directions for future research. Taken as a whole, this chapter describes how the fields of theoretical computer science and evolutionary game theory can inform each other.

Evolutionary Game Theory

Classical evolutionary game theory models organisms in a population interacting and competing for resources. The classical model assumes that the population is infinite. It models interaction by choosing two organisms uniformly at random, who then play a 2-player, symmetric game. The payoffs that these organisms earn represent an increase or a loss in fitness, which either helps or hinders the organisms ability to reproduce.

Abstract

Prediction markets (also known as information markets) are markets established to aggregate knowledge and opinions about the likelihood of future events. This chapter is intended to give an overview of the current research on computational aspects of these markets. We begin with a brief survey of prediction market research, and then give a more detailed description of models and results in three areas: the computational complexity of operating markets for combinatorial events; the design of automated market makers; and the analysis of the computational power and speed of a market as an aggregation tool. We conclude with a discussion of open problems and directions for future research.

Introduction: What Is a Prediction Market?

Consider the following mechanism design problem called the information aggregation problem. Suppose that an individual (“the aggregator”) would like to obtain a prediction about an uncertain variable, say the global average temperature in 2020. A number of individuals (“the informants”) each hold different and nonindependent sets of information bearing on the outcome of the variable. The goal is to design a mechanism that extracts the relevant information from the informants, aggregates the information appropriately, and provides a collective prediction or forecast. The forecast should ideally be equivalent to the omniscient forecast that has direct access to all the information available to all informants.

Abstract

We introduce convex programming techniques to compute market equilibria in general equilibrium models. We show that this approach provides an effective arsenal of tools for several restricted, yet important, classes of markets. We also point out its intrinsic limitations.

Introduction

The market equilibrium problem consists of finding a set of prices and allocations of goods to economic agents such that each agent maximizes her utility, subject to her budget constraints, and the market clears. Since the nineteenth century, economists have introduced models that capture the notion of market equilibrium. In 1874, Walras published the “Elements of Pure Economics,” in which he describes a model for the state of an economic system in terms of demand and supply, and expresses the supply equal demand equilibrium conditions (Walras, 1954). In 1936, Wald gave the first proof of the existence of an equilibrium for the Walrasian system, albeit under severe restrictions (Wald, 1951). In 1954, Nobel laureates Arrow and Debreu proved the existence of an equilibrium under much milder assumptions (Arrow and Debreu, 1954).

The market equilibrium problem can be stated as a fixed point problem, and indeed the proofs of existence of a market equilibrium are based on either Brouwer's or Kakutani's fixed point theorem, depending on the setting (see, e. g., the beautiful monograph (Border, 1985) for a friendly exposition of the main results in this vein).

Abstract

Online mechanisms extend the methods of mechanism design to dynamic environments with multiple agents and private information. Decisions must be made as information about types is revealed online and without knowledge of the future, in the sense of online algorithms. We first consider single-valued preference domains and characterize the space of decision policies that can be truthfully implemented in a dominant strategy equilibrium. Working in a model-free environment, we present truthful auctions for domains with expiring items and limited-supply items. Turning to a more general preference domain, and assuming the existence of a probabilistic model for agent types, we define a dynamic Vickrey–Clarke–Groves mechanism that is efficient and Bayes–Nash incentive compatible. We close with some thoughts about future research directions in this area.

Introduction

The decision problem in many multiagent problem domains is inherently dynamic rather than static. Consider, for instance, the following environments:

• Selling seats on an airplane to buyers arriving over time.

• Allocating computational resources (bandwidth, CPU, etc.) to jobs arriving over time.

• Selling adverts on a search engine to a possibly changing group of buyers and with uncertainty about the future supply of search terms.

• Allocating tasks to a dynamically changing team of agents.

In each of these settings at least one of the following is true: either agents are dynamically arriving or departing, or there is uncertainty about the set of feasible decisions in the future.