A Course in Networks and Markets [lecture notes]

244 Pages·2018·7.336 MB·English

by  Rafael Pass

A Course in
Networks and Markets
Rafael Pass
Cornell Tech
Last updated: January 3, 2018
(cid:13)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
To Shira-Perla and Isaak, det finaste jag vet.
Introduction
Inthiscourse, usingtoolsfromgametheoryandgraphtheory, weexplorehow
networkstructuresandnetworkeffectsplayaroleineconomicandinformation
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 as a network effect: For instance, even if
a buyer prefer iPhones in isolation (i.e., they have a higher intrinsic value to
them), they may prefer to get an Android phone (or even switch to one, if
they currently have an iPhone), if enough of their 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
effecttohappen—thatis,so-calledhigher-level beliefs canhavealargeimpact.
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).
MatchingMarkets, AuctionsandVoting. Letusfinallyconsideraquite
different type of market. Assume we have three people A,B,C and three
houses called H ,H ,H . The people may have some constraints on what
1 2 3
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
peopleandhousesthatrespectstheseacceptabilityconstraints? Inthissimple
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 H , B with H , and C
2 1
withH ;wewillstudyalgorithmsforsolvingthisproblem,andmoregenerally,
3
understanding when a matching where everyone gets matched exists.
Consider, now, a variant of this problem where everyone finds all houses
acceptable, buteveryoneprefersH toH , andH toH . Howshouldwenow
1 2 2 3
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)!
Thekeyforovercomingthisproblemistoassignpricestothethreehouses.
This gives rise to the following questions:
• Canwesetpricesforthesethreehousessothateveryonecanbematched
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,
whereadvertisersbidon“slots”forsponsoredresultsinInternetsearchqueries
(and need to pay to get their advertisement displayed)—in this context, the
goal is to find a matching between advertisers and slots.
Wewillalsoconsiderthe“standard”(non-sponsored)websearchproblem:
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
ideabehindthesemethodsistoimplementavoting mechanismwherebyother
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
theirutility,interact)andgraphtheory(thestudyofgraphs,mathemati-
calconstructsusedtomodelnetworksofinterconnectednodes). Wethen
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. InPart3, wediscussmechanismsfor
tamingtheabove-mentionedauctions,websearch,voting,andmatching,
problems.
• Part 4: The Role of Beliefs. Finally, in Part 4, we discuss various ways
of modeling people’s beliefs and knowledge, and explore how people’s
beliefs (and the above-mentioned higher-level beliefs) play a role in auc-
tions 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 assuming that people are “rational agents”,
acting in a way that maximizes some internal“utility” function. 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 of it. There is a number of beautiful models and results
regarding the structure of social networks (e.g., the Barabasi-Albert prefer-
ential attachment model [BA99], Watts-Strogatz small worlds model [WS98],
and Kleinberg’s decentralized search model [Kle00]), which are discussed in
depth in EK. We also do not discuss specific diffusion models (e.g., SIR/SIS
epidemic models) for modeling the spread of diseases in a social network; in-
stead, we focus only on studying diffusion in a game-theoretic setting where
agents rationally decide whether to, for instance, adopt some technology (as
in the first example above).
Finally, we only rarely discuss behavioral or sociological experiments or
observations (whereas EK discusses many intriguing such experiments and
observations)—in a sense, we mostly focus on the mathematical and compu-
tational models. As such, we believe that a reader of these notes should read
EK for the behavioral/sociological context.
Prerequisites. We will assume basic familiarity with probability theory; a
primer on probability theory, which covers all the concepts and results needed
to understand the material in the course, is provided Appendix A. Basic
notions in computing, such as running-time of algorithms, will also be useful
(but the material should be understandable also without it). Finally, we
assume a basic level of mathematical maturity (e.g., comfort with definitions
and proofs).
Intended audience. Most of the material in these notes is appropriate for
a Master’s level, or advanced undergraduate-level, course in Networks and
Markets. We have also included some more advanced material (marked as
such) which could be included in a introductory Ph.D. level course.
Acknowledgements. I am extremely grateful to Andrew Morgan who was
the teaching assistant for CS 5854 in 2016 and 2017; Andrew edited and
typeset my first version of these notes, created all the figures in the notes,
came up with many of the examples in the figures, and found many mistakes
and typos. Andrew also came up with many amazing HW problems! Thank
you so very much!
IamalsoverygratefultothestudentsofCS5854in2016and2017, aswell
asAntonioMarcedoneandThodorisLykouriswhoprovidedusefulfeedbackon
the notes. Finally, I am extremely grateful to Jon Kleinberg, Joseph Halpern
and E´va Tardos for many helpful discussions.
Contents
Contents vii
I Games and Graphs 1
1 Game Theory 3
1.1 The Prisoner’s Dilemma Game . . . . . . . . . . . . . . . . . . 3
1.2 Normal-form games. . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Dominant Strategies . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Iterated Strict Dominance (ISD) . . . . . . . . . . . . . . . . . 6
1.5 Nash Equilibria and Best-Response Dynamics . . . . . . . . . . 8
1.6 A Cautionary Game: The Traveler’s Dilemma . . . . . . . . . . 12
1.7 Mixed-strategy Nash Equilibrium . . . . . . . . . . . . . . . . . 13
2 Graphs and Applications 17
2.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 BFS and Shortest Paths . . . . . . . . . . . . . . . . . . . . . . 21
3 Analyzing Best-Response Dynamics 25
3.1 A Graph Representation of Games . . . . . . . . . . . . . . . . 25
3.2 Characterizing Convergence of BRD . . . . . . . . . . . . . . . 26
3.3 Better-Response Dynamics . . . . . . . . . . . . . . . . . . . . 29
3.4 Games without PNE . . . . . . . . . . . . . . . . . . . . . . . . 30
4 Coordination in Social Networks 33
4.1 Plain Networked Coordination Games . . . . . . . . . . . . . . 33
4.2 Convergence of BRD . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3 Incorporating Intrinsic Values . . . . . . . . . . . . . . . . . . . 36
4.4 The Price of Stability . . . . . . . . . . . . . . . . . . . . . . . 39
vii