Internet Economics כלכלת האינטרנט Class 11 – Externalities, cascades and the Braess’s paradox.

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

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.

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

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?

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, 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.

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

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

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.

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)

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

A diffusion model (cont.)

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.

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”

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

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%

Example 1

Example 1 Two early adopters of the technology A

Example 1

Example 1 A full cascade!

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

Example 2 Again, two early adopters

Example 2

Example 2

Example 2 Dynamic process stops: a partial cascade

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.

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.)

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

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.

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

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

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.

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

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)

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.

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.

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)

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.

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.

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

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.

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

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…)

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.

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

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

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

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!? !!!!

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.

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.

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…)