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A Course in Networks and Markets A Course in Networks and Markets Rafael Pass Cornell Tech Last updated: September 26, 2017 c 2018 Rafael Pass All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission by the author. ISBN: 10 9 8 7 6 5 4 3 2 1 First pre-edition: February 2017 Introduction In this course, using tools from game theory and graph theory, we explore how network structures and network effects play a role in economic and information markets. Let us start by providing an overview through a few examples. Markets with Network Effects: iPhone vs. Android Consider a mar- ket with two competing mobile phone brands, where the buyers are connected through a social network (see Figure 0.1). Each phone has some intrinsic value to a buyer, but the actual value of phone is affected by how many of the buyer's friends (i.e., the nodes connected to them in the social network) have the same phone|this is referred to a network effect: For instance, even if I prefer iPhones in isolation (i.e., they have a higher intrinsic value to me), I may prefer to get an Android phone (or even switch to one, if I currently have an iPhone), if enough of my friends have an Android. Some questions that naturally arise are: • Will there eventually be a stable solution, where everyone is happy with their phone, or will people keep switching phones? • If we arrive at a stable solution, what will it look like? (For instance, will everyone eventually have the same phone, or can we get a market for both?) A A A A I I A I Figure 0.1: An example of a small social network for the Android/iPhone game. Nodes correspond to buyers. We draw an edge between nodes that are friends. i • If we want to market iPhones, to which “influential" individuals should we offer discounts in order to most efficiently take over the market? • How should we set the price of a phone to best market it? (Perhaps start low and increase the price as more people buy it?) We will study models that allow us to answer these questions. Note that this type of modeling is not only useful to study the spread of products, but can also be used to reason about the spread of (e.g., political) news or other information (or disinformation) in a social networks. For instance, I may post a news article on my Facebook wall if many of my friends do it. The Role of Beliefs. In the example above, to market a phone, it may suf- fice that enough people believe that their friends will buy the phone. If people believe that their friends will buy the phone (accomplished e.g., by advertis- ing), their perceived value of the phone will increase, and they will be more likely to buy it|we get a \self-fulfilling prophecy". As we shall see, in some situations, it may even be enough that there exist people who believe there exist people who believe (etc.) that enough people will buy the phone for this effect to happen|that is, so-called higher-level beliefs can have a large impact. We will study models for discussing and analyzing such higher-level beliefs| perhaps surprisingly, networks will prove useful also for modeling higher-level beliefs. We shall next use these models to shed light on the emergence of bubbles and crashes in economic markets. More generally, we will discuss how crowds process information and how and why the following phenomena can occur: • The wisdom of crowds: In some situations, the aggregate behavior of a group can give a significantly better estimate of the \truth" than any one individual (e.g., prediction teams outperforming single analysts in elections). • The foolishness of crowds: In other situations, \misinformation" can be circulated through a social network in \information cascades" (e.g., the spread of urban legends/\fake news" through a social network). Matching Markets, Auctions and Voting. Let us finally consider a quite different type of market. Assume we have three people A; B; C and three houses called H1;H2;H3. The people may have some constraints on what houses are acceptable to them; we can depict the situation using a graph as shown in Figure 0.2. Can we find a \matching" (i.e., pairing) between people and houses that respects these acceptability constraints? In this simple Figure 0.2: The \acceptability" graph in a matching problem. We draw a edge between a person and a house if the person finds the house acceptable. example, it is easy to see that A can be matched with H2, B with H1, and C with H3; we will study algorithms for solving this problem, and more generally, understanding when a matching where everyone gets matched exists. Consider, now, a variant of this problem where everyone finds all houses acceptable, but everyone prefers H1 to H2, and H2 to H3. How should we now assign houses to people? Note that no matter how we assign houses to people, 2 people will be unhappy with their house (in the sense that they would have preferred a different house)! The key for overcoming this problem is to assign prices to the three houses. This gives rise to the following questions: • Can we set prices for these three houses so that everyone can be matched with their most preferred house (taking into account the price of the house)? Indeed, we will show that such, so-called, \market-clearing prices" are guaranteed to exist (and the hope is that the market will converge on these prices over time). • Can we design a mechanism that incentivizes people to truthfully report how much each house is worth to them, so that we can assign houses to people in a way that maximizes the total \happiness" of all the peo- ple? Indeed, we shall study the Vickrey-Clark-Groves (VCG) auction mechanism that enables doing this. We next note that the methods we use to provide answers to the above ques- tions form the basis for the auction mechanisms used in sponsored search, where advertisers bid on \slots" for sponsored results in Internet search queries (and need to pay to get their advertisement shown)|in this context, the goal is to find a matching between advertisers and slots. We will also consider the \standard" (non-sponsored) web search problem: think of it as matching webpages with \slots" in the search ranking, but the difference with the sponsored search problem is that now there are no payments. We will discuss the \relevance" algorithms used by search engines (e.g., Google's PageRank algorithm) to determine how (non-paying) pages returned by a search should be ordered. Here, the network structure of the Internet will be the central factor for computing a relevance score. The basic idea behind these methods is to implement a voting mechanism whereby other pages \vote" for each page's relevance by linking to it. Finally, we will discuss voting schemes (e.g., for presidential elections) more generally, and investigate why such schemes typically are susceptible to \strategic voting", where voters are incentivized to not truthfully report their actual preferences (for instance, if your favorite candidate in the US presidential election is a third-party candidate, you may be inclined to vote for your second choice). Outline of the course. The course is divided into four main parts. • Part 1: Games and Graphs. In Part 1, we first introduce basic concepts from game theory (the study of how rational agents, trying to maximize their utility, interact) and graph theory (the study of graphs, mathemati- cal constructs used to model networks of interconnected nodes). We then use concepts from both to analyze \networked coordination games" on social networks|such games provide a framework for analyzing situa- tions similar to the Android/iPhone game discussed above. • Part 2: Markets on Networks. In Part 2, we begin by introducing some more advanced algorithms for exploring graphs, and then use these al- gorithms to explore various different types of markets on networks (in- cluding e.g., the above-discussed market for matching houses to people). • Part 3: Mechanisms for Networks. In Part 3, we discuss mechanisms for taming the above-mentioned auctions, web search, voting, and matching, problems. • Part 4: The Role of Beliefs. Finally, in Part 4, we discuss various ways of modeling players' beliefs and knowledge, and explore how players' beliefs (and the above-mentioned higher-level beliefs) play a role in auctions and markets. Comparison with Easley-Kleinberg. The topics covered here, as well as the whole premise of using a combination of game-theory and graph-theory to study markets, is heavily inspired by Easley and Kleinberg's (EK) beau- tiful book \Networks, Crowds and Markets" [EK10]. However, whereas our selection of topics closely follows EK, our treatment of many (but not all) of the topics is somewhat different. In particular, our goal is to provide a for- mal treatment, with full proofs, of the simplest models exhibiting the above- described phenomena, while only making assumptions about the rationality of agents. As such, we are also covering fewer topics than EK: in particular, we are simply assuming that the network structure (e.g., the social-network in the first example) is exoneously given|we do not consider how this network is formed, and do not discuss properties on it. There is a number of beau- tiful models and results regarding the structure of social networks (e.g., the Barabasi-Albert preferential attachment model [BA99], Watts-Strogatz small worlds model [WS98], and Kleinberg's decentralized search model [Kle00]), which are discussed in depth in EK.
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