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.

Abstract

The objective of cooperative game theory is to study ways to enforce and sustain cooperation among agents willing to cooperate. A central question in this field is how the benefits (or costs) of a joint effort can be divided among participants, taking into account individual and group incentives, as well as various fairness properties.

In this chapter, we define basic concepts and review some of the classical results in the cooperative game theory literature. Our focus is on games that are based on combinatorial optimization problems such as facility location. We define the notion of cost sharing, and explore various incentive and fairness properties cost-sharing methods are often expected to satisfy. We show how cost-sharing methods satisfying a certain property termed cross-monotonicity can be used to design mechanisms that are robust against collusion, and study the algorithmic question of designing cross-monotonic cost-sharing schemes for combinatorial optimization games. Interestingly, this problem is closely related to linear-programming-based techniques developed in the field of approximation algorithms. We explore this connection, and explain a general method for designing cross-monotonic cost-sharing schemes, as well as a technique for proving impossibility bounds on such schemes. We will also discuss an axiomatic approach to characterize two widely applicable solution concepts: the Shapley value for cooperative games, and the Nash bargaining solution for a more restricted framework for surplus sharing.

Abstract

The flow of information or influence through a large social network can be thought of as unfolding with the dynamics of an epidemic: as individuals become aware of new ideas, technologies, fads, rumors, or gossip, they have the potential to pass them on to their friends and colleagues, causing the resulting behavior to cascade through the network.

We consider a collection of probabilistic and game-theoretic models for such phenomena proposed in the mathematical social sciences, as well as recent algorithmic work on the problem by computer scientists. Building on this, we discuss the implications of cascading behavior in a number of online settings, including word-of-mouth effects (also known as “viral marketing”) in the success of new products, and the influence of social networks in the growth of online communities.

Introduction

The process by which new ideas and new behaviors spread through a population has long been a fundamental question in the social sciences. New religious beliefs or political movements; shifts in society that lead to greater tolerance or greater polarization; the adoption of new technological, medical, or agricultural innovations; the sudden success of a new product; the rise to prominence of a celebrity or political candidate; the emergence of bubbles in financial markets and their subsequent implosion – these phenomena all share some important qualitative properties.

Abstract

Many interesting and important new applications of game theory have been discovered over the past 7 years in the context of research into the economics of information security. Many systems fail not ultimately for technical reasons but because incentives are wrong. For example, the people who guard a system often are not the people who suffer the full costs of failure, and as a result they make less effort than would be socially optimal. Some aspects of information security are public goods, like clean air or water; externalities often decide which security products succeed in the marketplace; and some information risks are not insurable because they are correlated in ways that cause insurance markets to fail.

Deeper applications of game-theoretic ideas can be found in the games of incomplete information that occur when critical information, such as about software quality or defender efforts, is hidden from some principals. An interesting application lies in the analysis of distributed system architectures; it took several years of experimentation for designers of peer-to-peer systems to understand incentive issues that we can now analyze reasonably well. Evolutionary game theory has recently allowed us to tie together a number of ideas from network analysis and elsewhere to explain why basing peer-to-peer systems on rings is a bad idea, and why revolutionaries use cells instead. The economics of distributed systems looks like being a very fruitful field of research.

Abstract

Large computer networks such as the Internet are built, operated, and used by a large number of diverse and competitive entities. In light of these competing forces, it is surprising how efficient these networks are. An exciting challenge in the area of algorithmic game theory is to understand the success of these networks in game theoretic terms: what principles of interaction lead selfish participants to form such efficient networks?

In this chapter we present a number of network formation games. We focus on simple games that have been analyzed in terms of the efficiency loss that results from selfishness. We also highlight a fundamental technique used in analyzing inefficiency in many games: the potential function method.

Introduction

The design and operation of many large computer networks, such as the Internet, are carried out by a large number of independent service providers (Autonomous Systems), all of whom seek to selfishly optimize the quality and cost of their own operation. Game theory provides a natural framework for modeling such selfish interests and the networks they generate. These models in turn facilitate a quantitative study of the trade-off between efficiency and stability in network formation. In this chapter, we consider a range of simple network formation games that model distinct ways in which selfish agents might create and evaluate networks.

Abstract

Most discussions of algorithmic mechanism design (AMD) presume the existence of a trusted center that implements the required economic mechanisms. This chapter focuses on mechanism-design problems that are inherently distributed, i.e., those in which such a trusted center cannot be used. Such problems require that the AMD paradigm be generalized to distributed algorithmic mechanism design (DAMD).

We begin this chapter by exploring the reasons that DAMD is needed and why it requires different notions of economic equilibrium and computational complexity than centralized AMD. We then consider two DAMD problems, namely distributed VCG computation and multicast cost sharing, that illustrate the concepts of ex-post Nash equilibrium and network complexity, respectively.

The archetypal example of a DAMD challenge is interdomain routing, which we treat in detail. We show that, under certain realistic and general assumptions, one can achieve incentive compatibility in a collusion-proof ex-post Nash equilibrium without payments, simply by executing the Border Gateway Protocol (BGP), which is the standard for interdomain routing in today's Internet.

Introduction

To motivate the material in this chapter, we begin with a review of why game theory is relevant to computer science. As noted in the Preface to this book, computer science has traditionally assumed the existence of a central planner who dictates the algorithms used by computational nodes. While most nodes are assumed to be obedient, some nodes may malfunction or be subverted by attackers; such byzantine nodes may act arbitrarily.

Abstract

We consider some classical games and show how they can arise in the context of the Internet. We also introduce some of the basic solution concepts of game theory for studying such games, and some computational issues that arise for these concepts.

Games, Old and New

The Foreword talks about the usefulness of game theory in situations arising on the Internet. We start the present chapter by giving some classical games and showing how they can arise in the context of the Internet. At first, we appeal to the reader's intuitive notion of a “game”; this notion is formally defined in Section 1.2. For a more in-depth discussion of game theory we refer the readers to books on game theory such as Fudenberg and Tirole (1991), Mas-Colell, Whinston, and Green (1995), or Osborne and Rubinstein (1994).

The Prisoner's Dilemma

Game theory aims to model situations in which multiple participants interact or affect each other's outcomes. We start by describing what is perhaps the most well-known and well-studied game.

Example 1.1 (Prisoners' dilemma) Two prisoners are on trial for a crime and each one faces a choice of confessing to the crime or remaining silent. If they both remain silent, the authorities will not be able to prove charges against them and they will both serve a short prison term, say 2 years, for minor offenses.

Abstract

Peer-to-peer (p2p) systems support many diverse applications, ranging from file-sharing and distributed computation to overlay routing in support of anonymity, resiliency, and scalable multimedia streaming. Yet, they all share the same basic premise of voluntary resource contribution by the participating peers. Thus, the proper design of incentives is essential to induce cooperative behavior by the peers. With the increasing prevalence of p2p systems, we have not only concrete evidence of strategic behavior in large-scale distributed systems but also a live laboratory to validate potential solutions with real user populations. In this chapter we consider theoretical and practical incentive mechanisms, based on reputation, barter, and currency, to facilitate peer cooperation, as well as mechanisms based on contracts to overcome the problem of hidden actions.

Introduction

The public release of Napster in June 1999 and Gnutella in March 2000 introduced the world to the disruptive power of peer-to-peer (p2p) networking. Tens of millions of individuals spread across the world could now self-organize and collaborate in the dissemination and sharing of music and other content, legal or otherwise. Yet, within 6 months of its public release, and long before individual users are threatened by copyright infringement lawsuits, the Gnutella network saw two thirds of its users free-riding, i.e., downloading files from the network without uploading any in return.

Given the large-scale, high-turnover, and relative anonymity of the p2p file-sharing networks, most p2p transactions are one-shot interactions between strangers that will never meet again in the future.

Abstract

Many situations involve repeatedly making decisions in an uncertain environment: for instance, deciding what route to drive to work each day, or repeated play of a game against an opponent with an unknown strategy. In this chapter we describe learning algorithms with strong guarantees for settings of this type, along with connections to game-theoretic equilibria when all players in a system are simultaneously adapting in such a manner.

We begin by presenting algorithms for repeated play of a matrix game with the guarantee that against any opponent, they will perform nearly as well as the best fixed action in hindsight (also called the problem of combining expert advice or minimizing external regret). In a zero-sum game, such algorithms are guaranteed to approach or exceed the minimax value of the game, and even provide a simple proof of the minimax theorem. We then turn to algorithms that minimize an even stronger form of regret, known as internal or swap regret. We present a general reduction showing how to convert any algorithm for minimizing external regret to one that minimizes this stronger form of regret as well. Internal regret is important because when all players in a game minimize this stronger type of regret, the empirical distribution of play is known to converge to correlated equilibrium.

Abstract

The Cryptographic and Game Theory worlds seem to have an intersection in that they both deal with an interaction between mutually distrustful parties which has some end result. In the cryptographic setting the multiparty interaction takes the shape of a set of parties communicating for the purpose of evaluating a function on their inputs, where each party receives at the end some output of the computation. In the game theoretic setting, parties interact in a game that guarantees some payoff for the participants according to the joint actions of all the parties, while the parties wish to maximize their own payoff. In the past few years the relationship between these two areas has been investigated with the hope of having cross fertilization and synergy. In this chapter we describe the two areas, the similarities and differences, and some of the new results stemming from their interaction.

The first and second section will describe the cryptographic and the game theory settings (respectively). In the third section we contrast the two settings, and in the last sections we detail some of the existing results.

Cryptographic Notions and Settings

Cryptography is a vast subject requiring its own book. Therefore, in the following we will give only a high-level overview of the problem of Multi-Party Computation (MPC), ignoring most of the lower-level details and concentrating only on aspects relevant to Game Theory.