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Internet Economics כלכלת האינטרנט Class 11 – Externalities, cascades and the Braesss paradox. 1

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Reminder: Course duties 2 Work in pairs. – Exceptions (single students) are possible. Presentation and seminar paper. – Same topic – Same partner Submission of the (optional) problem set – individually - not in pairs. – You are expected to do it by yourselves.

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Course duties: choosing a topic 3 Choose a topic: – paper/book-chapter from the list in the course weblog. – Or any other academic paper or part of a book. See references to the literature in the papers from the list. In either case, you need my approval for the topic chosen. Deadline: January 1 st, – I recommend choosing a topic ASAP. –כל הקודם זוכה – This is the deadline for getting an approval. Means that you need to send it before (in case paper is already taken, or not approved for other reasons).

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Course duties: choosing a topic 4 Approval methods: 1. to me (preferred) - 2.Come to my office hours. ( first) 3.Write a comment in the articles page in the blog. (shows others that you have already chosen a certain paper)

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Suggested articles 5 A variety – Some theoretical/mathematical – Some empirical – Some surveys Mathematical depth will be appreciated. – Not mandatory, you can also go in depth in other directions. Papers from related fields may be approved (for example, business, computer science, game theory)

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סוף מעשה במחשבה תחילה 6 Please invest effort in choosing the article. – Read parts of it first. – Look at other papers. – Check if the math level is appropriate for you. Most problems in previous years: students that discovered too late (just before the presentation) that they would like to change a paper.

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ראשי פרקים 7 To encourage you to read the paper, you should submit an outline of the presentation by January 12 th. ½ to 1 page. Font 12. Double spaced. Please send it to the teaching assistant of the course Avi Lichtig. –

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Time constraints 8 We will schedule the presentations during the semester break. Please send your hard time constraints (miluim, ski vacations, (your own) weddings). – To Avi, by Januray 12 th in the same as the outline of the presentation. You can also mention soft constraints (I would like to present before Pesach as Ill have exams afterwards), but we may not be able to fulfill them. After the schedule is prepared, changes are very difficult, very often impossible.

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Summary: your duties for the next couple of weeks 9 The following actions are mandatory for participating in the course: Send me an with the names of students in your team + get my approval for a – By January 1 st. Send Avi an The outline of your presentation Your time constraints for presenting in semester B. By January 12.

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Todays Outline 10 Network effects Positive externalities: Diffusion and cascades Negative externalities: Selfish routing.

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Decisions in a network 11 When making decisions: – We often do not care about the whole population – Mainly care about friends and colleagues. E.g., technological gadgets, political views, clothes, choosing a job,. Etc.

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What affects our decisions? 12 Possible reasons: – Informational effects: Choices of others might indirectly point to something they know. if my computer-geek friend buys a Mac, it is probably better than other computers – Network effects (direct benefit): My actual value from my decisions changes with the number of other persons that choose it. if most of my friends use ICQ, I would be better off using it too Todays topic

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Main questions 13 How new behaviors spread from person to person in a social network. – Opinions, technology, etc. Why a new innovation fails although it has relative advantages over existing alternatives? What about the opposite case, where I tend to choose the opposite choice than my friends?

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Network effects 14 My value from a product x is v i (n x ): depends on the number n x of people that are using it. Positive externalities: – New technologies: Fax, , messenger, which social network to join, Skype. – v i (n x ) increasing with n x. Negative externalities: – Traffic: I am worse off when more people use the same road as I. – Internet service provider: less Internet bandwidth when more people use it. – v i (n x ) decreasing with n x.

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Network effects 15 We will first consider a model with positive externalities.

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Network effects 16 Examples: VHS vs. Beta (80s) Internet Explorer vs. Netscape (90s) Blue ray vs. HD DVD (00s)

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Diffusion of new technology 17 What can go wrong? Homophily is a burden: people interact with people like themselves, and technologies tend to come from outside. – We will formalize this assertion. You will adapt a new technology only when a sufficient proportion of your friends (neighbours in the network) already adapted the technology.

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A diffusion model 18 People have to possible choices: A or B – Facebook or mySpace, PC or Mac, right-wing or left-wing If two people are friends, they have an incentive to make the same choices. – Their payoff is actually higher… Consider the following case: – If both choose A, they gain a. – If both choose B, they gain b. – If choose different options, gain 0. AB A(a,a)(0,0) B (b,b)

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A diffusion model (cont.) 19 So some of my friends choose A, some choose B. What should I do to maximize my payoff? Notations: – A fraction p of my friends choose A – A fraction (1-p) choose B. If I have d neighbours, then: – pd choose A – (1-p)d choose B. With more than 2 agents: My payoff increases by a with every friend of mine that choose A. Increases by b for friends that choose B. Example: If I have 20 friends, and p=0.2: pd=4 choose A (1-p)d=16 choose B Payoff from A: 4a Payoff from B: 16b

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A diffusion model (cont.) 20

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A diffusion model (cont.) 21 Therefore: – Choosing A gain me pda – Choosing B will gain me (1-p)db A would be a better choice then B if: pda > (1-p)db that is, (rearranging the terms) p > b/(a+b) Meaning: If at least a b/(a+b) fraction of my friends choose A, I will also choose A. Does it make sense? When a is large, I will adopt the new technology even when just a few of my friends are using it.

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A diffusion model (cont.) 22 This starts a dynamic model: – At each period, each agent make a choice given the choices of his friends. – After everyone update their choices, everyone update the choices again, – And again, –…–… What is an equilibrium? – Obvious equilibria: everyone chooses A. everyone chooses B. – Possible: equilibria where only part of the population chooses A. complete cascade

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Diffusion 23 Question: Suppose that everyone is initially choosing B – Then, a set of early adopters choose A – Everyone behaves according to the model from previous slides. When the dynamic choice process will create a complete cascade? – If not, what caused the spread of A to stop? Answer will depend, of course, on: – Network structures – The parameters a,b – Choice of early adopters B B B B B B B B B B B B B A A A

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Example 24 Let a=3 b=2 We saw that player will choose A if at least b/(a+b) fraction of his neighbours adopt A. Here, threshold is 2/(3+2)=40%

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Example 1 25

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Example 1 26 Two early adopters of the technology A

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Example 1 27

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Example 1 28 A full cascade!

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Example 2 29 Lets look at a different, larger network

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Example 2 30 Again, two early adopters

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Example 2 31

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Example 2 32

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Example 2 33 Dynamic process stops: a partial cascade

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Partial diffusion 34 Partial diffusion happens in real life? – Different dominant political views between adjacent communities. – Different social-networking sites are dominated by different age groups and lifestyles. – Certain industries heavily use Apple Macintosh computers despite the general prevalence of Windows.

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Partial diffusion: can be fixed? 35 If A is a firm developing technology A, what can it do to dominate the market? – If possible, raise the quality of the technology A a bit. For example, if a=4 instead of a=3, then all nodes will eventually switch to A. (threshold will be lower) Making the innovation slightly better, can have huge implications. – Otherwise, carefully choose a small number of key users and convince them to switch to A. This have a cost of course, for example, giving products for free or invest in heavy marketing. (viral marketing) How to choose the key nodes? (Example in the next slide.)

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Example 2 36 For example: Convincing nodes 13 to move to technology A will restart the diffusion process.

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Cascades and Clusters 37 Why did the cascade stop? Intuition: the spread of a new technology can stop when facing a densely-connected community in the network.

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Cascades and Clusters 38 What is a densely-connected community? If you belong to one, many of your friends also belong. Definition: a cluster of density p is a set of nodes such that each node has at least a p-fraction of her friends in the cluster. h A 2/3 cluster

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Cascades and Clusters 39 What is a densely-connected community? If you belong to one, many of your friends also belong. Definition: a cluster of density p is a set of nodes such that each node has at least a p-fraction of her friends in the cluster. A 2/3 cluster

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Cascades and Clusters 40 What is a densely-connected community? If you belong to one, many of your friends also belong. Definition: a cluster of density p is a set of nodes such that each node has at least a p-fraction of her friends in the cluster. Note: not every two nodes in a cluster have much in common – For example: The whole network is always a p-cluster for every p. Union of any p-clusters is a p-cluster.

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Cascades and Clusters 41 In this network, two 2/3-clusters that the new technology didnt break into. Coincidence?

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Cascades and Clusters 42 It turns out the clusters are the main obstacles for cascades. Theorem: Consider: a set of initial adopters of A, all other nodes have a threshold q (to adopt A). Then: 1. if the other nodes contain a cluster with greater density than 1-q, then there will be no complete cascade. 2. Moreover, if the initial adopters did not cause a cascade, the other nodes must contain a cluster with a density greater than 1-q. Previously we saw a threshold q=b/(a+b)

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Cascades and Clusters 43 In our example, q=0.4 cannot break into p-clusters where p>0.6 Indeed: two clusters with p=2/3 remain with B.

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Cascades and Clusters 44 It turns out the clusters are the main obstacles for cascades. Theorem: Consider: a set of initial adopters of A, all other nodes have a threshold q (to adopt A). Then: 1. if the other nodes contain a cluster with greater density than 1-q, then there will be no complete cascade. 2. Moreover, if the initial adopters did not cause a cascade, the other nodes must contain a cluster with a density greater than 1-q. Previously we saw a threshold q=b/(a+b) Lets prove this part.

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Cascades and Clusters 45 Assume that we have a cluster with density of more than 1-q Assume that there is a node v in this cluster that was the first to adopt A We will see that this cannot happen: Assume that v adopted A at time t. Therefore, at time t-1 at least q of his friends chose A Cannot happen, as more than 1-q of his friends are in the cluster (v was the first one to adopt A)

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Cascades and Clusters 46 It turns out the clusters are the main obstacles for cascades. Theorem: Consider: a set of initial adopters of A, all other nodes have a threshold q (to adopt A). Then: 1. if the other nodes contain a cluster with greater density than 1-q, then there will be no complete cascade. 2. Moreover, if the initial adopters did not cause a cascade, the other nodes must contain a cluster with a density greater than 1-q. Previously we saw a threshold q=b/(a+b) Lets prove this part.

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Cascades and Clusters 47 We now prove: not only that clusters are obstacles to cascades, they are the only obstacle! With a partial cascade: there is a cluster in the remaining network with density more than 1-q. Let S be the nodes that use B at the end of the process. A node w in S does not switch to A, therefore less than q of his friends choose A The fraction of his friends that use B is more than 1-q The fraction of ws neighbours in S is more that 1-q S is a cluster with density > 1-q.

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Todays Outline 48 Network effects Positive externalities: Diffusion and cascades Negative externalities: Selfish routing.

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Negative externalities 49 Lets talk now about setting with negative externalities: I am worse off when more users make the same choices as I. Motivation: routing information-packets over the internet. – In the internet, each message is divided to small packets which are delivered via possibly-different routes. In this class, however, we can think about transportation networks.

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Example 50 Many cars try to minimize driving time. All know the traffic congestion ( גלגלצ, PDAs)

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Example 51 Negative externalities: my driving time increases as more drivers take the same route. Nash equilibrium: no driver wants to change his chosen route. Or alternatively: – Equilibrium: for each driver, all routes have the same driving time. (Otherwise the driver will switch to another route…)

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Example 52 Our question: are equilibria efficient? – Would it be better for the society if someone told each driver how to drive??? We would like to compare: – The most efficient outcome (with no incentives) – The worst Nash equilibrium. We will call their ratio: price of anarchy.

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Example 53 Efficient outcome: efficiency=4+4=8 (Worst) Nashe Equilibrium: efficiency=2+2=4 Price of anarchy: 1/2 CooperateDefect Cooperate -1, -1-5, 0 Defect 0, -5-3,-3 CooperateDefect Cooperate 4, 40, 50, 5 Defect 5, 02,22,2

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Example 1 54 Efficient outcome: splitting traffic equally – expected cost: ½*1+1/2*1/2=3/4 The only Nash equilibrium: everyone use lower edge. – Otherwise, if someone chooses upper link, the cost in the lower link is less than 1. – Expected cost: 1*1=1 Price of anarchy: 3/4 C(x)=x C(x)=1 c(x) – the cost (driving time) to users when x users are using this road. Assume that a flow of 1 (million) users use this network. ST

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Example 2 55 In equilibrium: half of the traffic uses upper route half uses lower route. Expected cost: ½*(1/2+1)+1/2*(1+1/2)=1.5 c(x)=x c(x)=1 ST c(x)=x c(x)=1

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Example 3 56 The only equilibrium in this graph: everyone uses the s v w t route. – Expected cost: 1+1=2 Building new highways reduces social welfare!? c(x)=x c(x)=1 ST v W c(x)=x c(x)=1 c(x)=0 Now a new highway was constructed! !!!!

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Braesss Paradox 57 This example is known as the Braesss Paradox: sometimes destroying roads can be beneficial for society. c(x)=x c(x)=1 ST v W c(x)=x c(x)=1 c(x)=0 Now a new highway was constructed!

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Selfish routing, the general case 58 What can we say about the price of anarchy in such networks? We saw a very simple example where it is ¾ Actually, this is the worst possible: Theorem: when the cost functions are linear (c(x)=ax+b), then the price of anarchy in every network is at least ¾.

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Summary 59 Network effects are important in many different aspects of the Internet. Explain many of the phenomena seen in the last couple of decade (and before…)

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