In Chapter 8, we considered a first extended application of game-theoretic ideas in our analysis of traffic flow through a network. Here we consider a second major application – the behavior of buyers and sellers in an auction.

An auction is a kind of economic activity that has been brought into many people's everyday lives by the Internet, through sites such as eBay. But auctions also have a long history that spans many different domains. For example, the U.S. government uses auctions to sell Treasury bills and timber and oil leases; Christie's and Sotheby's use them to sell art; and Morrell & Company and the Chicago Wine Company use them to sell wine.

Auctions will also play an important and recurring role in this book, since the simplified form of buyer–seller interaction they embody is closely related to more complex forms of economic interaction as well. In particular, in the next part of the book, when we discuss markets in which multiple buyers and sellers are connected by an underlying network structure, we'll use ideas initially developed in this chapter for understanding simpler auction formats. Similarly, in Chapter 15, we'll study a more complex kind of auction in the context of a Web search application, analyzing the ways in which search companies like Google, Yahoo!, and Microsoft use an auction format to sell advertising rights for keywords.

Types of Auctions

In this chapter we focus on different simple types of auctions and how they promote different kinds of behavior among bidders. We consider the case of a seller auctioning one item to a set of buyers.

In Chapter 3 we considered some of the typical structures that characterize social networks, and some of the typical processes that affect the formation of links in the network. Our discussion in Chapter 3 focused primarily on the network as an object of study in itself, relatively independent of the broader world in which it exists.

However, the contexts in which a social network is embedded will generally have significant effects on its structure. Each individual in a social network has a distinctive set of personal characteristics, and similarities and compatibilities between two people's characteristics can strongly influence whether a link forms between them. Each individual also engages in a set of behaviors and activities that can shape the formation of links within the network. These considerations suggest what we mean by a network's surrounding contexts: factors that exist outside the nodes and edges of a networks, but which nonetheless affect how the network's structure evolves.

In this chapter we consider how such effects operate and what they imply about the structure of social networks. Among other observations, we will find that the surrounding contexts affecting a network's formation can, to some extent, be viewed in network terms. By expanding the network to represent the contexts together with the individuals, we will see in fact that several different processes of network formation can be described in a common framework.

Homophily

One of the most basic notions governing the structure of social networks is homophily – the principle that we tend to be similar to our friends. Typically, your friends don't look like a random sample of the underlying population.

Popularity as a Network Phenomenon

For the past two chapters, we have been studying situations in which a person's behavior or decisions depend on the choices made by other people — either because the person's rewards are dependent on what other people do or because the choices of other people convey information that is useful in the decision-making process. We've seen that these types of coupled decisions, where behavior is correlated across a population, can lead to outcomes very different from what we find in cases where individuals make independent decisions.

Here we apply this network approach to analyze the general notion of popularity. Popularity is a phenomenon characterized by extreme imbalances: while almost everyone goes through life known only to people in their immediate social circles, a few people achieve wider visibility, and a very, very few attain global name recognition. Analogous things could be said of books, movies, or almost anything that commands an audience. How can we quantify these imbalances? Why do they arise? Are they somehow intrinsic to the whole idea of popularity?

We will see that some basic models of network behavior can provide significant insight into these questions. To begin the discussion, we focus on the Web as a concrete domain in which it is possible to measure popularity very accurately. While it may be difficult to estimate the number of people worldwide who have heard of famous individuals such as Barack Obama or Bill Gates, it is easy to take a snapshot of the full Web and simply count the number of links to high-profile Web sites such as Google, Amazon, or Wikipedia.

In our analysis of economic transactions on networks, particularly the model in Chapter 11, we considered how a node's position in a network affects its power in the market. In some cases, we were able to come up with precise predictions about prices and power, but in others the analysis left open a range of possibilities. For example, in the case of perfect competition between traders, we could conclude that the traders would make no profit, but it was not possible to say whether the resulting situation would favor particular buyers or sellers – different divisions of the available surplus were possible. This is an instance of a broader phenomenon that we discussed earlier, in Chapter 6: when there are multiple equilibria, some of which favor one player and some of which favor another, we may need to look for additional sources of information to predict how things will turn out.

In this chapter, we formulate a perspective on power in networks that can help us further refine our predictions for the outcomes of different participants. This perspective arises dominantly from research in sociology, and it addresses not just economic transactions but also a range of social interactions more generally that are mediated by networks. We will develop a set of formal principles that aim to capture some subtle distinctions in how a node's network position affects its power. The goal will be to create a succinct mathematical framework enabling predictions of which nodes have power, and how much power they have, for arbitrary networks.

Price Setting in Markets

In Chapter 10 we developed an analysis of trade and prices on a bipartite graph consisting of buyers, sellers, and the edges connecting them. Most importantly, we showed that market-clearing prices exist and that trade at these prices results in maximal total valuation among the buyers and sellers, and we found a procedure that allowed us to construct market-clearing prices. This analysis shows in a striking way how prices have the power to direct the allocation of goods in a desirable way. What it doesn't do is provide a clear picture of where prices in real markets tend to come from. That is, who sets the prices in real markets, and why do they choose the particular prices they do?

Auctions, which we discussed in Chapter 9, provide a concrete example of price determination in a controlled setting. In our discussion of auctions, we found that if a seller with a single object runs a second-price sealed-bid auction – or equivalently an ascending-bid auction – then buyers bid their true values for the seller's object. In that discussion, the buyers were choosing prices (via their bids) in a procedure selected by the seller. We could also consider a procurement auction in which the roles of buyers and sellers are reversed, and a single buyer is interested in purchasing an object from one of several sellers. Here, our auction results imply that if the buyer runs a second-price sealed-bid auction (buying from the lowest bidder at the second-lowest price), or equivalently a descending-offer auction, then the sellers will offer to sell at their true costs.

In this final part of the book, we build on the principles developed thus far to consider the design of institutions and how different institutions can produce different forms of aggregate behavior. By an institution, we mean something very general — any set of rules, conventions, or mechanisms that synthesizes individual behavior across a population into an overall outcome. In the next three chapters, we will focus on three fundamental classes of institutions: markets, voting, and property rights.

We begin by discussing markets, and specifically their role in aggregating and conveying information across a population. Each individual participant in the market arrives with certain beliefs and expectations — about the value of assets or products, and about the likelihood of events that may affect these values. The markets we study will be structured so as to combine this set of beliefs into an overall outcome — generally in the form of market prices — that represents a kind of synthesis of the underlying information.

This is part of a broad issue we have seen several times so far: the fact that individuals' expectations affect their behavior. For example, we saw this in Chapter 8 with Braess's Paradox, where the optimal route depends on which routes others are expected to choose; in Chapter 16 on information cascades, where people draw inferences about the unknown desirability of alternatives (restaurants or fashions) from the behavior of others; and in Chapter 17 on network effects, where the unknown value of a product (a fax machine or a social networking site) depends on how many others are also expected to use the product.

At the beginning of Chapter 16, we discussed two fundamentally different reasons why individuals might imitate the behavior of others. One reason was based on informational effects: since the behavior of other people conveys information about what they know, observing this behavior and copying it (even against the evidence of one's own private information) can sometimes be a rational decision. This was our focus in Chapter 16. The other reason was based on direct-benefit effects, also called network effects: for some kinds of decisions, you incur an explicit benefit when you align your behavior with the behavior of others. This is what we will consider in this chapter.

A natural setting where network effects arise is in the adoption of technologies for which interaction or compatibility with others is important. For example, when the fax machine was first introduced as a product, its value to a potential consumer depended on how many others were also using the same technology. The value of a social networking or media-sharing site exhibits the same properties: it's valuable to the extent that other people are using it as well. Similarly, a computer operating system can be more useful if many other people are using it: even if the primary purpose of the operating system itself is not to interact with others, an operating system with more users will tend to have a larger amount of software written for it and will use file formats (e.g., for documents, images, and movies) that more people can easily read.

We have now seen a number of ways of thinking both about network structure and about the behavior of agents as they interact with each other. A few of our examples have brought these together directly – such as the issue of traffic in a network, including Braess's Paradox – and in the next few chapters we explore this convergence of network structure and strategic interaction more fully, and in a range of different settings.

First, we think about markets as a prime example of network-structured interaction between many agents. When we consider markets creating opportunities for interaction among buyers and sellers, there is an implicit network that encodes the access between these buyers and sellers. In fact, there are a number of ways of using networks to model interactions among market participants, and we will discuss several of these models. Later, in Chapter 12 on network exchange theory, we will discuss how market-style interactions become a metaphor for the broad notion of social exchange, in which the social dynamics within a group can be modeled by the power imbalances of the interactions within the group's social network.

Bipartite Graphs and Perfect Matchings

Matching markets form the first class of models we consider, as the focus of the current chapter. Matching markets have a long history of study in economics, operations research, and other areas because they embody, in a very clean and stylized way, a number of basic principles: the way in which people may have different preferences for different kinds of goods, the way in which prices can decentralize the allocation of goods to people, and the way in which such prices can in fact lead to allocations that are socially optimal.

Among the initial examples in our discussion of game theory in Chapter 6, we noted that traveling through a transportation network, or sending packets through the Internet, involves fundamentally game-theoretic reasoning: rather than simply choosing a route in isolation, individuals must evaluate routes in the presence of the congestion resulting from the decisions made by themselves and everyone else. In this chapter, we develop models for network traffic using the game-theoretic ideas developed thus far. In the process, we will discover a rather unexpected result – known as Braess's Paradox [76] – which shows that adding capacity to a network can sometimes actually slow down the traffic.

Traffic at Equilibrium

Let's begin by developing a model of a transportation network and how it responds to traffic congestion; with this model in place, we can then introduce the game-theoretic aspects of the problem.

We represent a transportation network by a directed graph: we consider the edges to be highways, and the nodes to be exits where you can get on or off a particular highway. There are two particular nodes, which we call A and B, and we assume everyone wants to drive from A to B. For example, we can imagine that A is an exit in the suburbs, B is an exit downtown, and we're looking at a large collection of morning commuters. Finally, each edge has a designated travel time that depends on the amount of traffic it contains.

In our discussion of networks thus far, we have generally viewed the relationships contained in these networks as having positive connotations – links have typically indicated such things as friendship, collaboration, sharing of information, or membership in a group. The terminology of online social networks reflects a largely similar view, through its emphasis on the connections one forms with friends, fans, followers, and so forth. But in most network settings, there are also negative effects at work. Some relations are friendly, but others are antagonistic or hostile; interactions between people or groups are regularly beset by controversy, disagreement, and sometimes outright conflict. How should we reason about the mix of positive and negative relationships that take place within a network?

Here we describe a rich part of social network theory that involves taking a network and annotating its links (i.e., its edges) with positive and negative signs. Positive links represent friendship while negative links represent antagonism, and an important problem in the study of social networks is to understand the tension between these two forces. The notion of structural balance that we discuss in this chapter is one of the basic frameworks for doing this.

In addition to introducing some of the basics of structural balance, our discussion in this chapter serves a second, methodological purpose: it illustrates a nice connection between local and global network properties. A recurring issue in the analysis of networked systems is the way in which local effects – phenomena involving only a few nodes at a time – can have global consequences that are observable at the level of the network as a whole.