Internet Economics כלכלת האינטרנט

1. Internet Economicsכלכלת האינטרנט Class 11 – Externalities, cascades and the Braess’s paradox.

2. Today’s Outline • Network effects • Positive externalities: Diffusion and cascades • Negative externalities: Selfish routing.

3. Decisions in a network • 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.

4. What affects our decisions? • 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” Today’s topic

5. Main questions • 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?

6. Network effects • My value from a product xis vi(nx): depends on the number nxof people that are using it. • Positive externalities: • New technologies: Fax, email, messenger, which social network to join, Skype. • vi(nx) increasing with nx. • 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. • vi(nx) decreasing with nx.

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

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

9. Diffusion of new technology • 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.

10. A diffusion model • 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.

11. A diffusion model (cont.) • 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 dneighbours, 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

12. A diffusion model (cont.)

13. A diffusion model (cont.) • 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.

14. A diffusion model (cont.) • 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, • 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”

15. A B Diffusion B B B B A B B B B A B • 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

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

17. Example 1

18. Example 1 Two early adopters of the technology A

19. Example 1

20. Example 1 A full cascade!

21. Example 2 Let’s look at a different, larger network

22. Example 2 Again, two early adopters

23. Example 2

24. Example 2

25. Example 2 Dynamic process stops: a partial cascade

26. Partial diffusion • 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.

27. Partial diffusion: can be fixed? • If A is a firm developing technology A, what can it do to dominate the market? • If possible, raise the quality of the technology Aa 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.)

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

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

30. Cascades and Clusters • 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 nodes has at least a p-fraction of her friends in the cluster. A 2/3 cluster h

31. Cascades and Clusters • 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 nodes has at least a p-fraction of her friends in the cluster. h A 2/3 cluster

32. Cascades and Clusters • 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 nodes 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.

33. Cascades and Clusters In this network, two 2/3-clusters that the new technology didn’t break into. Coincidence?

34. Cascades and Clusters • 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)

35. Cascades and Clusters 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.

36. Cascades and Clusters • 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) Let’s prove this part.

37. Cascades and Clusters • 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 thatv 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)

38. Cascades and Clusters • 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) Let’s prove this part.

39. Cascades and Clusters • 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 w’sneighbours in S is more that 1-q • S is a cluster with density > 1-q.

40. Today’s Outline • Network effects • Positive externalities: Diffusion and cascades  Negative externalities: Selfish routing.

41. Negative externalities • Let’s 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.

42. Example • Many cars try to minimize driving time. • All know the traffic congestion (גלגלצ, PDA’s)

43. Example • 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…)

44. Example • 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.

45. Example • Efficient outcome: efficiency=4+4=8 • (Worst) Nashe Equilibrium: efficiency=2+2=4 • Price of anarchy: 1/2

46. Example 1 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. • 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 S T C(x)=x

47. Example 2 c(x)=x c(x)=1 • In equilibrium: half of the traffic uses upper routehalf uses lower route. • Expected cost: ½*(1/2+1)+1/2*(1+1/2)=1.5 S T c(x)=1 c(x)=x

48. Example 3 Now a new highway was constructed! v c(x)=x c(x)=1 • 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!? S c(x)=0 T W c(x)=1 c(x)=x !!!!

49. Braess’s Paradox Now a new highway was constructed! v c(x)=x c(x)=1 • This example is known as the Braess’s Paradox:sometimes destroying roads can be beneficial for society. S c(x)=0 T W c(x)=1 c(x)=x

50. Selfish routing, the general case • 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.