1. Internet Economics כלכלת 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 x is vi(nx): depends on the number nx of people that are using it.
Positive externalities:
New technologies: Fax, , 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. A B
(a,a)
(0,0)
(b,b)

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 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

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, …
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. Diffusion 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 A B B B B A B B B B

16. Example
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%

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 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.)

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. Previously we saw a threshold q=b/(a+b)
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. Previously we saw a threshold q=b/(a+b)
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 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)

38. Previously we saw a threshold q=b/(a+b)
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’s neighbours 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
Cooperate
Defect
-1, -1
-5, 0
0, -5
-3,-3
Cooperate
Defect
4, 4
0, 5
5, 0
2,2
Efficient outcome: efficiency=4+4=8
(Worst) Nashe Equilibrium: efficiency=2+2=4
Price of anarchy: 1/2

46. Example 1  “Price of anarchy”: 3/4 S T
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. S T
C(x)=x
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

47. Example 2
c(x)=x
c(x)=1 S T
c(x)=1
c(x)=x
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

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

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

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.

51. Summary
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…)