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.

Abstract

One of the more visible means by which the Internet has disrupted traditional activity is the manner in which advertising is sold. Offline, the price for advertising is typically set by negotiation or posted price. Online, much advertising is sold via auction. Most prominently, Web search engines like Google and Yahoo! auction space next to search results, a practice known as sponsored search. This chapter describes the auctions used and how the theory developed in earlier chapters of this book can shed light on their properties. We close with a brief discussion of unresolved issues associated with the sale of advertising on the Internet.

Introduction

Web search engines like Google and Yahoo! monetize their service by auctioning off advertising space next to their standard algorithmic search results. For example, Apple or Best Buy may bid to appear among the advertisements – usually located above or to the right of the algorithmic results – whenever users search for “ipod.” These sponsored results are displayed in a format similar to algorithmic results: as a list of items each containing a title, a text description, and a hyperlink to the advertiser's Web page. We call each position in the list a slot. Generally, advertisements that appear in a higher ranked slot (higher on the page) garner more attention and more clicks from users. Thus, all else being equal, merchants generally prefer higher ranked slots to lower ranked slots.

Abstract

We study the integration of game theoretic and computational considerations. In particular, we study, the design of computationally efficient and incentive compatible mechanisms, for several different problem domains. Issues like the dimensionality of the domain, and the goal of the algorithm designer, are examined by providing a technical discussion on four results: (i) approximation mechanisms, for single-dimensional scheduling, where truthfulness reduces to a simple monotonicity condition; (ii) randomness as a tool to resolve the computational vs. incentives clash for Combinatorial Auctions, a central multidimensional domain where this clash is notable; (iii) the impossibilities of deterministic dominant-strategy implementability in multidimensional domains; and (iv) alternative solution concepts that fit worst-case analysis, and aim to resolve the above impossibilities.

Introduction

Algorithms in computer science, and Mechanisms in game theory, are very close in nature. Both disciplines aim to implement desirable properties, drawn from “real-life” needs and limitations, but the resulting two sets of properties are completely different. A natural need is then to merge them – to simultaneously exhibit “good” game theoretic properties as well as “good” computational properties. The growing importance of the Internet as a platform for computational interactions only strengthens the motivation for this.

However, this integration task poses many difficult challenges. The two disciplines clash and contradict in several different ways, and new understandings must be obtained to achieve this hybridization. The classic Mechanism Design literature is rich and contains many technical solutions when incentive issues are the key goal.

Abstract

Despite impossibility results on general domains, there are some classes of situations in which there exist interesting dominant-strategy mechanisms. While some of these situations (and the resulting mechanisms) involve the transfer of money, we examine some that do not. Specifically, we analyze problems where agents have single-peaked preferences over a one-dimensional “public” policy space; and problems where agents must match with each other.

Introduction

The Gibbard–Satterthwaite Theorem (Theorem 9. 8) is a Procrustean bed that is escaped only by relaxing its assumptions. In conjunction with the Revelation Principle (Proposition 9. 25), it states that on the general domain of preferences, only dictatorial rules can be implemented in dominant strategies (if the range contains at least three alternatives). In this chapter we escape Procrustes by examining dominant strategy implementation on restricted domains of preferences.

In most applications it is clearly unreasonable to assume that agents' preferences are completely unrestricted, as was assumed in the voting context of Section 9.2.4. For instance, in situations involving the allocation of goods, including money, one can safely assume that each agent prefers to receive more money (or other goods). As can be seen in the following chapters, the ability for agents to make monetary transfers allows for a rich class of strategy-proof rules.

Nevertheless there are many important environments where money cannot be used as a medium of compensation. This constraint can arise from ethical and/or institutional considerations: many political decisions must be made without monetary transfers; organ donations can be arranged by “trade” involving multiple needy patients and their relatives, yet monetary compensation is illegal.

Abstract

We give an introduction to the design of mechanisms for profit maximization with a focus on single parameter settings.

Introduction

In previous chapters, we have studied the design of truthful mechanisms that implement social choice functions, such as social welfare maximization. Another fundamental objective, and the focus of this chapter, is the design of mechanisms in which the goal of the mechanism designer is profit maximization. In economics, this topic is referred to as optimal mechanism design.

Our focus will be on the design of profit-maximizing auctions in settings in which an auctioneer is selling (respectively, buying) a set of goods/services. Formally, there are n agents, each of whom desires some particular service. We assume that agents are single-parameter; i.e., agent i's valuation for receiving service is vi and their valuation for no service is normalized to zero. A mechanism takes as input sealed bids from the agents, where agent i's bid bi represents his valuation vi, and computes an outcome consisting of an allocation x = (x1, …, xn) and prices p = (p1, …, pn). Setting xi = 1 represents agent i being allocated service whereas xi = 0 is for no service, and pi is the amount agent i is required to pay the auctioneer. We assume that agents have quasi-linear utility expressed by ui = vixi – pi.

Abstract

This chapter presents motivation and definitions for quantifying the inefficiency of equilibria in noncooperative games. We illustrate the basic concepts in four fundamental network models, which are studied in depth in subsequent chapters. We also discuss how measures of the inefficiency of equilibria can guide mechanism and network design.

Introduction

The Inefficiency of Equilibria

The previous two parts of this book provided numerous examples demonstrating that the outcome of rational behavior by self-interested players can be inferior to a centrally designed outcome. This part of the book is devoted to the question: by how much?

To begin, recall the Prisoner's Dilemma (Example 1.1). Both players suffer a cost of 4 in the unique Nash equilibrium of this game, while both could incur a cost of 2 by coordinating. There are several ways to formalize the fact that the Nash equilibrium in the Prisoner's Dilemma is inefficient. A qualitative observation is that the equilibrium is strictly Pareto inefficient, in the sense that there is another outcome in which all of the players achieve a smaller cost. This qualitative perspective is particularly appropriate in applications where the “cost” or “payoff” to a player is an abstract quantity that only expresses the player's preferences between different outcomes. However, payoffs and costs have concrete interpretations in many applications, such as money or the delay incurred in a network.

Abstract

In this chapter, we study two types of pricing mechanisms: one where the goal of the pricing scheme is to achieve some socially beneficial objective for the network and the other where prices are set by multiple competing service providers to maximize their revenues. For both cases, we present an overview of the mathematical models involved and the relevant optimization and game-theoretic techniques needed to study these models. We study the impact of different degrees of strategic interactions among users and between users and service providers on the network performance. We also relate our models and solutions to practical resource allocation mechanisms used in communication networks such as congestion control, routing, and scheduling. We conclude the chapter with a brief introduction to other game-theoretic topics in emerging networks.

This chapter studies the problem of decentralized resource allocation among competing users in communication networks. The growth in the scale of communication networks and the newly emerging interactions between administrative domains and end users with different needs and quality of service requirements necessitate new approaches to the modeling and control of communication networks that recognize the difficulty of formulating and implementing centralized control protocols for resource allocation. The current research in this area has developed a range of such approaches. Central to most of these approaches is the modeling of end users and sometimes also of service providers as self-interested agents that make decentralized and selfish decisions.

Abstract

We explain algorithms for computing Nash equilibria of two-player games given in strategic form or extensive form. The strategic form is a table that lists the players' strategies and resulting payoffs. The “best response” condition states that in equilibrium, all pure strategies in the support of a mixed strategy must get maximal, and hence equal, payoff. The resulting equations and inequalities define polytopes, whose “completely labeled” vertex pairs are the Nash equilibria of the game. The Lemke–Howson algorithm follows a path of edges of the polytope pair that leads to one equilibrium. Extensive games are game trees, with information sets that model imperfect information of the players. Strategies in an extensive game are combinations of moves, so the strategic form has exponential size. In contrast, the linear-sized sequence form of the extensive game describes sequences of moves and how to randomize between them.

Introduction

A basic model in noncooperative game theory is the strategic form that defines a game by a set of strategies for each player and a payoff to each player for any strategy profile (which is a combination of strategies, one for each player). The central solution concept for such games is the Nash equilibrium, a strategy profile where each strategy is a best response to the fixed strategies of the other players. In general, equilibria exist only in mixed (randomized) strategies, with probabilities that fulfill certain equations and inequalities.

Abstract

Suppose that a set of weighted tasks shall be assigned to a set of machines with possibly different speeds such that the load is distributed evenly among the machines. In computer science, this problem is traditionally treated as an optimization problem. One of the classical objectives is to minimize the makespan, i.e., the maximum load over all machines. Here we study a natural game theoretic variant of this problem: We assume that the tasks are managed by selfish agents, i.e., each task has an agent that aims at placing the task on the machine with smallest load. We study the Nash equilibria of this game and compare them with optimal solutions with respect to the makespan. The ratio between the worst-case makespan in a Nash equilibrium and the optimal makespan is called the price of anarchy. In this chapter, we study the price of anarchy for such load balancing games in four different variants, and we investigate the complexity of computing equilibria.

Introduction

The problem of load balancing is fundamental to networks and distributed systems. Whenever a set of tasks should be executed on a set of resources, one needs to balance the load among the resources in order to exploit the available resources efficiently. Often also fairness aspects have to be taken into account. Load balancing has been studied extensively and in many variants.