Game theory is the mathematical study of interaction among independent, self-interested agents. It has been applied to disciplines as diverse as economics (historically, its main area of application), political science, biology, psychology, linguistics—and computer science. In this chapter we will concentrate on what has become the dominant branch of game theory, called noncooperative game theory, and specifically on normal-form games, a canonical representation in this discipline.

As an aside, the name “noncooperative game theory” could be misleading, since it may suggest that the theory applies exclusively to situations in which the interests of different agents conflict. This is not the case, although it is fair to say that the theory is most interesting in such situations. By the same token, in Chapter 12 we will see that coalitional game theory (also known as cooperative game theory) does not apply only in situations in which the interests of the agents align with each other. The essential difference between the two branches is that in noncooperative game theory the basic modeling unit is the individual (including his beliefs, preferences, and possible actions) while in coalitional game theory the basic modeling unit is the group. We will return to that later in Chapter 12, but for now let us proceed with the individualistic approach.

Self-interested agents

What does it mean to say that agents are self-interested? It does not necessarily mean that they want to cause harm to each other, or even that they care only about themselves.

In this chapter we consider the problem of allocating (discrete) resources among selfish agents in a multiagent system. Auctions—an interesting and important application of mechanism design—turn out to provide a general solution to this problem. We describe various different flavors of auctions, including single-good, multiunit, and combinatorial auctions. In each case, we survey some of the key theoretical, practical, and computational insights from the literature.

The auction setting is important for two reasons. First, auctions are widely used in real life, in consumer, corporate, as well as government settings. Millions of people use auctions daily on Internet consumer Web sites to trade goods. More complex types of auctions have been used by governments around the world to sell important public resources such as access to electromagnetic spectrum. Indeed, all financial markets constitute a type of auction (one of the family of so-called double auctions). Auctions are also often used in computational settings, to efficiently allocate bandwidth and processing power to applications and users.

The second—and more fundamental—reason to care about auctions is that they provide a general theoretical framework for understanding resource allocation problems among self-interested agents. Formally speaking, an auction is any protocol that allows agents to indicate their interest in one or more resources and that uses these indications of interest to determine both an allocation of resources and a set of payments by the agents.

The capacity to learn is a key facet of intelligent behavior, and it is no surprise that much attention has been devoted to the subject in the various disciplines that study intelligence and rationality. We will concentrate on techniques drawn primarily from two such disciplines—artificial intelligence and game theory—although those in turn borrow from a variety of disciplines, including control theory, statistics, psychology and biology, to name a few. We start with an informal discussion of the various subtle aspects of learning in multiagent systems and then discuss representative theories in this area.

Why the subject of “learning” is complex

The subject matter of this chapter is fraught with subtleties, and so we begin with an informal discussion of the area. We address three issues—the interaction between learning and teaching, the settings in which learning takes place and what constitutes learning in those settings, and the yardsticks by which to measure this or that theory of learning in multiagent systems.

The interaction between learning and teaching

Most work in artificial intelligence concerns the learning performed by an individual agent. In that setting the goal is to design an agent that learns to function successfully in an environment that is unknown and potentially also changes as the agent is learning. A broad range of techniques have been developed, and learning rules have become quite sophisticated.

This chapter discusses some of the basic properties of the integers, including the notions of divisibility and primality, unique factorization into primes, greatest common divisors, and least common multiples.

Divisibility and primality

A central concept in number theory is divisibility.

Consider the integers ℤ = {…,−2,−1, 0, 1, 2, …}. For a, b ∈ ℤ, we say that adividesb if az = b for some z ∈ ℤ. If a divides b, we write a | b, and we may say that a is a divisor of b, or that b is a multiple of a, or that b is divisible bya. If a does not divide b, then we write.

We first state some simple facts about divisibility:

Theorem 1.1. For all a, b, c ∈ ℤ, we have

1. (i) a | a, 1 | a, and a | 0;

2. (ii) 0 | a if and only if a = 0;

3. (iii) a | b if and only if − a | b if and only if a | −b;

4. (iv) a | b and a | c implies a | (b + c);

5. (v) a | b and b | c implies a | c.

Proof. These properties can be easily derived from the definition of divisibility, using elementary algebraic properties of the integers. For example, a | a because we can write a · 1 = a; 1 · a because we can write 1 · a = a; a | 0 because we can write a · 0 = 0.

This chapter introduces the basic properties of congruences modulo n, along with the related notion of residue classes modulo n. Other items discussed include the Chinese remainder theorem, Euler's phi function, Euler's theorem, Fermat's little theorem, quadratic residues, and finally, summations over divisors.

Equivalence relations

Before discussing congruences, we review the definition and basic properties of equivalence relations.

Let S be a set. A binary relation ∼ on S is called an equivalence relation if it is

reflexive:a ∼ a for all a ∈ S,

symmetric:a ∼ b implies b ∼ a for all a, b ∈ S, and

transitive:a ∼ b and b ∼ c implies a ∼ c for all a, b, c ∈ S.

If ∼ is an equivalence relation on S, then for a ∈ S one defines its equivalence class as the set {x ∈ S : x ∼ a}.

Theorem 2.1. Let ∼ be an equivalence relation on a set S, and for a ∈ S, let [a] denote its equivalence class. Then for all a, b ∈ S, we have:

1. (i) a ∈ [a];

2. (ii) a ∈ [b] implies [a] = [b].

Proof. (i) follows immediately from reflexivity. For (ii), suppose a ∈ [b], so that a ∼ b by definition. We want to show that [a] = [b]. To this end, consider any x ∈ S.

The Internet has become an integral part of our society. It not only supports financial transactions and access to knowledge, but also cultivates relationships and allows people to participate in communities that were hard to find before its existence. From reading the news, to shopping, to making phone calls and watching videos, the Internet has certainly surpassed the expectations of its creators.

This vital organ in today's society comprises a large number of networks, administered by different authorities and spanning the entire globe. In this book, we will take a deep look into what constitutes those entities and how they form the skeleton of the Internet. Going beyond the physical infrastructure, the wires and boxes that make up the building blocks of today's Internet, we will study the procedures that allow a network to make optimal use of those building blocks.

As in traditional telecommunication networks, the Internet constituents rely on a team of network designers and managers, for their design, operation and maintenance. While such functions are well studied within the traditional telecommunication theory literature, the Internet imposes constraints that necessitate a whole new array of methods and techniques for the efficient operation of a network. Departing from the traditional circuit-switched paradigm, every unit of information flows independently across the network, from the source to the destination, and thus is much harder to account for.

Since the late 1990s, the research community has been devising techniques that allow the operators of this new kind of network to allocate and manage their networks' resources optimally.

The Internet is made of many separate routing domains called Autonomous Systems (ASs), each of which runs an IGP such as IS–IS or OSPF. The IGP handles routes to destinations within the AS, but does not calculate routes beyond the AS boundary. Internet Gateway Protocol engineering (or traffic engineering or IGP optimization) is the tuning of local IS–IS or OSPF metrics to improve performance within the AS. Today, IGP engineering is an ad-hoc process where metric tuning is performed by each AS in isolation. That is, each AS optimizes paths within its local network for traffic traversing it without coordinating these changes with neighboring ASs. The primary assumption behind such an assertion is that there is sufficient separation between intra-domain and inter-domain routing.

Beyond the AS boundary, the choice of AS hops is determined by the BGP,; BGP engineering is a less developed and less understood process compared to IGP engineering. In addition to whether there is a physical link between two ASs over which routes and traffic can flow, there are several BGP policies that determine which inter-domain paths are exposed to a neighboring AS. Business peering policies can directly translate into which routes are exported to each AS. After all these policies are applied, the remaining feasible paths are subjected to the “hot-potato” routing policy. Hot-potato routing occurs when there are multiple egress points to reach a destination.

We establish here some terminology, notation, and simple facts that will be used throughout the text.

Logarithms and exponentials

We write log x for the natural logarithm of x, and logbx for the logarithm of x to the base b.

We write ex for the usual exponential function, where e ≈ 2.71828 is the base of the natural logarithm. We may also write exp[x] instead of ex.

Sets and families

We use standard set-theoretic notation: ∅ denotes the empty set; x ∈ A means that x is an element, or member, of the set A; for two sets A, B, A ⊆ B means that A is a subset of B (with A possibly equal to B), and A ⊊ B means that A is a proper subset of B (i.e., A ⊆ B but A ≠ B). Further, A ∪ B denotes the union of A and B, A ∩ B the intersection of A and B, and A \ B the set of all elements of A that are not in B. If A is a set with a finite number of elements, then we write |A| for its size, or cardinality. We use standard notation for describing sets; for example, if we define the set S ≔ {−2,−1, 0, 1, 2}, then {x2 : x ∈ S} = {0, 1, 4} and {x ∈ S : x is even} = {−2, 0, 2}.