Coordination in a Social Network Bertil Hatt ∗ ´ EconomiX , FranceTelecom† March 31, 2007 Marginal costs can be subtracted from a, intrinsic utility of the product can reduce c: however, our idea is to focus on social network externalities. Nonlinear modeling is possible, but produce unnecessary tedious results. An agent would subscribe if his expected utility is positive, hence if he knows enough subscribers:

We consider a service useful if used by relatives too, like instant messaging or document standard. Social ties are modeled along a “large interaction network”. Communication about the service is decisive in its adoption. Competing against a service with an installed based is made more difficult, whether offering a better service with more features, or by being cheaper or easier to use. For simultaneous entry, we discuss the ambiguous role of out-switching costs.

i∈S

1

Utility function of a social service

⇔

ui ≥ 0

⇔

dSi ≥ δ =

c . a

(2)

It is therefore reasonable not to subscribe to such a service, even if it is fine, if anticipating it will have no success among one’s relatives; a no-subscribing convention is a sub-optimal Nash equilibrium. This is inspired from Arthur (1989), but in a local fashion. If two relatives have agreed to subscribe, though they have mutual interest in keeping their word, one can fail to its promise if he discovers other relatives are not subscribing. Coordination is key, beyond one’s direct acquaintances.

We consider a large population; each agent i has relatives j, and the set of ties draws a graph, or social network. Such a simplifying vision enlightens many new properties (Albert and Barab´ asi 2001, Newman 2003). The number of i’s relatives is his degree di . We consider a service S that presents:

• a fixed subscription or entry cost c, which can be monetary or attention (possibly monetized This model suits communication services, like though ads), it can be filling forms or under- phone, fax, e-mail, conferencing tools, instant messtanding efforts, both reduced by a good design; saging (IM) and more specifically social network ser• a utility a for each of his relatives subscribed too, vices (SNS) — postal mail does not require subscripdSi in total; this can be format or features, struc- tion, though. On the other hand, chat rooms, forums and virtual words1 have similar communication featural management — hence the total utility: tures, and a similar viral diffusion, but their point is more to meet new people, not existing relatives; some ui = a dSi − c (1) ties established on-line offer a user-base, with similar ∗ Universit´ e Paris-X Nanterre, K 104 – 200 avenue de la positive externalities and “stickiness”. R´ epublique, 92 001 Nanterre C´ edex, France † FranceTelecom R&D TECH/LEU, ´ F 225 – 38-40 avenue du G´ en´ eral-Leclerc, 92 794 Issy-les-Moulineaux C´ edex 9, France

1 We consider here both massively multi-players role-playing on-line games (MMORPG) and digital universe, similar to SecondLifeTM.

1

φ(·) is the fixed
point function: φ(d) = h(d|d) = d 1 − G(d|d) ; hence, if i has d relatives, φ(d) is the expected number of i’s relatives with more then d relatives; and

Our result could apply to many services whose utility is non exclusively based on relatives’ choice, but with decisive social network effects: fashion and cultural goods, but also all standard-driven technology, like software, come to mind. Complex goods demanding expert assistance can be preferred when help is provided on a gift-exchange basis. Other non-digital services can be understood with this model: caf´es and bars in Europe are mostly for enjoyable conversation; conferences are preferred when attended with like-minded experts; night-clubs are increasingly appreciated when joining a large group.

2

¯ ¯ φ(·) is the upper limit of the fixed point: φ(d) = mink≥d φ(k); people with d relatives or more ex¯ pect to have φ(d) relatives or more with a degree higher then d. φ¯ and φ are different if φ is decreasing— for instance, if many of the more connected individuals are mostly acquainted to lonely people. Social network studies have established that, on the contrary, sociable people tend to know sociable people, and lonely people befriend similarly isolated person: “degree homophily” or “assortativity”.

Assortativity and adoption

Instead of giving detailed information on the social ties, let us consider several distributions, or network statistics:

℘

f (·) the distribution of degree2 in the population: f (d) is the ratio of agents with degree d;

If we assume risk-neutral agents who know about:

F (·) the cumulative distribution of degree in the population: F (d) is the ratio of agents with a degree lower then d;

• the assortativity, measured by the cross distribution φ¯

g(·|·) the cross-distribution, i. e. if we consider all the relatives of agents with degree d, g(k|d) is the share of those with a degree k;3

then an agent of degree d who anticipates that a ser¯ vice is good enough, i. e. with δ ≤ φ(d), is worthy: all agents with a higher degree will be interested in subscribing too, sustaining the equilibrium. However, this is dependent on

• their degree d, and

G(·|·) is the cumulative distribution, i. e. if we consider the relatives of all the agents with degree d, G(k|d) is the ratio of those with a degree lower then k; therefore,

• the Common Knowledge (Dupuy 1989) of φ¯ and • the d-convention4 that all agents with a degree ¯ d ≥ d, φ(d) ≥ δ, subscribe.

h(·|·) is the degree-weithged distribution: around an agent with d relatives, there are h(k|d) = d 1 − This result allow us to use formal institutional frame
G(k|d) of them with a degree k or higher; works to model viral marketing. A maximal convention would be that all agents ap2 In the literature, the degree distribution is often assumed ply, who anticipate that they can be part of a subto be a power- law or Zipfian; it is sometimes following a log¯ ≥ δ}. A maxnormal curve. We are working on a distinct paper that explores scription equilibrium: d = min{φ(·) those skewed distributions. imal estimate for the adoption of a communication 3 Two properties can be deducted straight away: service is therefore 1 − F (d), provided it is advertised P ∀d ∈ N∗ ,

g(k|d)

=

1

(3)

∀d, k ∈ N∗2 , f (k) k g(d|k)

=

f (d) d g(k|d)

(4)

k

4 We shall insist that no adoption, or a “lower” subscription ¯ convention — let d verify φ(d) > δ, and all agents with a degree higher then d subscribe — are both sustainable equilibrium.

2

well enough for everyone to agree on such a convention. Such a threshold might not exist, if the maximum of φ¯ is too low i. e. lim+∞ φ < δ, then no interconnected core can support an initial adoption. A service such that δ := ac ≥ max φ¯ is too expensive, unclear, or with a low an added value per subscribed contact or insufficient features. That adoption is based on expected utility, and is not a Nash-equilibrium ex post for two reasons:

With such a model, in the second period, a disappointed user program it to chose between (a dSi − c) and (−k). Considering the convention is to subscribe if one’s degree is higher then d, dSi is the number of i’s relative with a degree higher then that. To model that, we define the following density: b( · , · , · ) the cross-density: if we consider all the agents i of degree d, then b(`, k, d) is the ratio of those i that have ` relatives with a degree k or higher;

• some agents considering adoption prior to communicating with their friends might have B( · , · , · ) the cumulative cross-density: B(`, k, d) is over-estimated their interest — namely here, the is the ratio of those i that have ` or less their relatives’ degree; relatives with a degree k or higher;

This is an even more accurate description of the social network, detailing how regular is assortativity.5 Let there be a d-convention, B(δ, d, d) is the ratio of user with a degree d that would be disappointed by the service S: there have less then δ relative with a degree higher then d, that is less then enough relative that would subscribe to justify they own subscription. 0 If δ 0 = c−k a , B(δ , d, d) is the number of user of degree d, disappointed enough to unsubscribe by themselves. B(δ 0, δ 0, d) is the number of user of degree d, that will unsubscribe because their relatives were disappointed, a second-order reasoning. As a convention cannot be irrational, d > δ; furthermore δ = ac > c−k = δ 0 : an agent cannot una subscribe if he has interest in subscribing; therefore, d > δ 0 : a convention is not compatible with unsubscription. Hence, B(δ 0, d, d) > B(δ 0, δ 0, d): the secondorder criteria is more demanding. —— he expected a positive utility for subscribing, prior to knowing more then the maintained subscription threshold δ 0 , the conventional subscription threshold d and b, i. e. he anticipated a greater gain from taking the chance to have enough relatives subscribing and a possible lost from un-subscribing, then from doing nothing:

• symmetrically, agents with a degree lower then d can still know more then δ subscribers or similarly interested candidates: such situations can happen with clustering and be resolve through local communication. A similar result is possible with a non-linear utility function: considering ui = a(dSi ) − c, then a service will not be adopted if ¯ > c. a(max φ)

3

(5)

Un-subscription

If an agent over-estimated his relatives’ willingness to use the service, he can unsubscribe at a cost k < c. The model now unfolds on two periods: Each agents knows its degree d, the cross- distribution density g; convention is maximal adoption: all agents subscribe if there is an equilibrium that makes rational. 1. A service S appears, at cost c and with a uniform utility per known user a. Agents consider to subscribe if they have a degree larger then d.

P(dSi ≥ δ 0 )(a dSi − c) − P(dSi < δ 0 )k > 0;

2. Once subscribed, all agents can see who adopted the service, and can unsubscribe at a cost k if their are better off without S.

5 Among the few obvious properties: b(`, k, d) = 0 if ` > d and b(`, 1, d) = 1(` = d). P We can also average it` to reach the ´ cross cumulative density: 1≤`≤d ` b(`, k, d) = d 1 − G(k, d) .

3

4

more accurately, i considers his subscribed relatives j that are not planing to unsubscribe:

Successive entry

An incumbent has had time to offer a service S1 , and holds a stable user-base; and entrant offers a service S2 : we make two technological assumptions:

P(dSi ≥ δ 0 ) = P(#{j|dSj ≥ δ 0 } ≥ δ 0 ) = b(δ 0 , δ 0 , di )

• services are not perfectly compatible: we introduce a deteriorated cross-experience a1−2 < a2 , hence u2 = a2 d2i + a1−2 d1−2 − c2 (6) i

—— All conventional subscription threshold d have to verify: X (a d − c)b(d, δ 0 , d) > k B(δ 0 , δ 0 , d).

where d1−2 is the number of contacts of i using i S1 , but not S2 ; we also assume that the crossexperience a1−2 < a1 between services is symmetric:

d≥δ 0

We are considering d, the minimum of such threshold, which allows the maximum number of subscribers.

u1 = a1 d1i + a1−2 d2−1 − c1 ; i

(7)

℘ • subscribing to both services (“multi-homing”) is possible but one relationship prefers to use just one medium, and the utility from one contact can only be a1 or a2 or a1&2 = a1 + a2 − a1−2 :

d is therefore decreasing in k: a low un-subscription cost both deters less users from trying the service, and encourages some with little reason to believe they know interested people to give it a try.

Nonetheless, it would be preferable to include, u1&2 = a1 d1i + a2 d2i − a1−2 d1&2 − c1 − c2 . (8) i in such a dynamic subscription approach, an subscription opportunity in the second period too, for marginal users who then noticed S was relevant for We consider that unsubscribing costs k1 refrain users them. Two arguments might park such a reasonable from canceling their use of S1 before they have an established base of relatives using S2 . suggestion: Can S2 compete against the user-base, by quality, • S being a communication technology, agents use features and richer format, or by price and usability? it to see who else has subscribed. However, communications outside S would support the assumptions of Common Knowledge and conven- 4.1 Quality competition tion, and would ease S directory constraints. We shall consider in this first subsection that the entrant S2 offers a better quality of service, more fea• We are interested in disappointments in the sectures, or richer formats: a2 > a1 . With a cost too ond period, and their anticipation; a second wave low, it is Common Knowledge that a general switch of adoption will come too late to prevent it. to S2 would be profitable to everyone: if not, a lack Without this unsubscribing option, and a possible of coordination would once again be at stake; as we latter adoption, marginal users might wait, and could consider reasonable conventions, we will assume next not trigger a sufficient “critical mass” if the network that some users have no interest in a general switch: is dis-assortative; in an assortative network, most relc2 + k1 c1 evant services can spread though the core first. < δ20 = (9) δ1 = a1 a2 We shall now introduce competition: firstly succesa2 i. e. k1 > c1 − c2 . (10) sive, and then simultaneous. a 1

4

otherwise, S2 holds an instant monopoly position. S2 is better tailored for connected users, so we assume the convention is that all agents with more then δˆ contacts adopt it; then agents with too low interest in S1 — shall it be hubs unconnected to low users in a assortative network, or users with too few contacts to justify the multi- homing. An agent i with d contacts tries S2 with reason if:

  ˆ (a1 − a1,2 ) φ(δ1 |d) − φ(δ|d) − c1

 + ˆ a1 φ(δ1 |d) − φ(δ|d)

ˆ − c1 < −k1 + a1,2 φ(δ|d) ˆ + a1 φ(δ1 |d) < c1 − k(17) (a1,2 − a1 )φ(δ|d) 1 (18) Those shrinking frontiers will meet where agents consider either switching to S2 or to nothing: will subscribe all agent with a degree d so that:  c2 if ac22 < δ1 and S2 goes  a2 0 ¯ beyond S1 user-base, h(d) > δ2 =  c2 +k1→2 −k1→0 otherwise. a2 (19) Some pockets of users can be untouched by the trend, and keep using S1 : a set of agents A is a S1 cluster if ¯ ≥ δ1 . min h

(11)

ˆ − c2 < (a2 − a1,2 ) φ(δ|d) c2 − c1 a1 − a1,2 ˆ φ(δ1 |d) + (12) φ(δ|d) > a −a a −a | 2 {z 1 } | 2 {z 1} α

α > 0 and κ > 0 ˆ φ(δ|d) > α φ(δ1 |d) + κ Hence δˆ = d.e

(16)

κ

(13) (14)

A

(15)

It has not S2 -triggering core if & % & & % &
¯ < δ. ˆ max h As φ( · | · ), δˆ a1 , c1 , a2 , c2 , a1,2 ; nothing surprising A here: the (relatively) better, richer, cheaper or easier to use an alternative, the more likely someone will This cluster will not be converted to S2 if there are no∗ adopt it — the more compatible a challenger, the adoption path, i. e. series of set of agents {An }n ∈ N more pervasive, too. Will S2 overcome S1 ? • increasing: ∀n ∈ N∗ , An ⊂ An+1 ;

¯ > δ; ˆ • converted to S2 : max A1 h • favorable to its adoption: max An ≥ δ20 ; and

δˆ < δ1 ⇔

ˆ δ) ˆ > φ(δ1 |δ) ˆ φ(δ| ˆ δ) ˆ > α φ(δ1 |δ) ˆ +κ φ(δ|

⇔ ⇔ 0 > (α − 1)φ(δ1 |δ1 ) + κ ⇔ (a2 − 2a1 + a1,2 )φ(δ1 |δ1 ) > c2 − c1

&

• finally reaching A: ∃n ∈ N∗ An ⊂ A.

as φ( · | · ) def. of δˆ

Algorithmically, one can try to map local estimates ¯ and then compare them to δ 0 to make a fist of h 2 estimate of the diffusion.

With degree homophily, such a “feature” strategy is more likely to succeed. The installed base of S1 If (α − 1)φ(δ1 |δ1 ) + κ < 0, then S2 has more closes the market by forcing a project to enter only adopters then S1 ˆ instead of more though network cores reactive to δ, ¯ > δ, ˆ a switching convention Therefore, if max h frequent δ2 degree- points. moves a core. This group will reduce the number of here decisive. S1 using contacts for those still preferring it, and This feature-based competition increases users’ utility, but doesn’t necessarily increase user base. • some among the less connected using S1 are not What can we expect from offering lower cost or better interested anymore and unsubscribe. usability? 5

4.2

Price or usability competition

– either stopped using S1 , as their relatives turn to S2 – or kept it, d˜2 = d˜12

We now consider the chances of a cost, or usability, strategy: c2 < c1 ; similarly, there are agents in the population that have enough contacts, d, to prefer to keep using S1 : a2 d−c2 −k1→2 < a1 d−c1 — otherwise a switching convention would have S2 overthrow S1 . If subscribers to S1 decide to switch, then other subscribers with less contact should be interested too, as S2 is better adapted to less sociable users; hence, all subscribers below a certain threshold would subscribe. Users with a degree just below the adoption threshold δ1 would subscribe too, as their main difference is that they don’t have to pay the switching cost. ¯ −1 (δ1 ) would subAgents with a degree d < d1 = h scribe to S2 provided enough others do; in this case, all non-subscribed agents with a higher degree would be even more interested; therefore, let us consider three cases:

all the agents between the lower threshold and the S1 limit that have to subscribe, and all the agents between the S1 limit and the upper threshold have to switch;
∀d ∈ [d2 ; d1 ], d G(d˜12 |d) − G(d2 |d) > δ2 (22) e e
∀d ∈ [d1 ; d˜12 ], d G(d˜2 |d) − G(d2 |d) > δ20 ;(23) e

either all the agents above that same threshold d˜12 have to stay, or maybe just a core, with degrees above d˜1 maintained its usage: either or

No subscriber to S1 switches: (most probably because k1→2 is very high) the adoption convention is such that there is a threshold d2 , and all agents with a degree in [d2 ; d1 ] switch. The threshold d2 is such that this margin has to be self-sustainable: every one has enough contact in there to be satisfied with S2 :
∀d ∈ [d2 ; d1 ], d G(d1 |d)−G(d2 |d) > δ2 . (20)

∀d > d˜12 ,

d(1 − G(d˜12 |d)) > δ20 (24) ¯ d˜1 ) > δ1 h( (25)

Once again not an always impossible situation, provided k1→2 is low enough to allow it; if feasible, the audience is wider then the first case, but not as large as a monopoly entry: d2 < d2 < d2 < d1 < d˜2 ≤ d˜1 e

provided they can be defined.

S2 fails: no user set anticipate a reasonable switch. The base effect once again is very strong, as the used the initial solution. constraint (20) is much more difficult to abide to If successful, price or usability competition inthen the monopoly rule: creases the user- base, but can reduce the incum
¯ h(d) = d 1 − G(d|d) > δ2 (21) bent’s base if the switching costs are low enough. The incumbent always represents a significant barmostly because G(d1 |d) < 1. The closer G(d1 |d) rier, as he generally holds the core of the social netto 1, i. e. the higher c1 and the lower a1 , the less work. However, scale effects can make this strategy difficult is the market to enter with a price or very profitable, as with a power-law distribution, the usability strategy. midlle-range market is far large then the core. Some subscribers to S1 switch: there are now Significantly better or cheaper services — or rather two or three thresholds to consider: more features or better usability — can replace the • d2 between non-adopter and S2 adopters, incumbent. When the existing service offers effece tive communication externalities, such replacement and goes very fast, spreading on the existing infrastruc• d˜2 between switchers and people who ture:“busts” or “shark fins”. 6

5

Simultaneous competition

is easier to switch from S1 to S2 then the other way round. Risk-averse agents that feel the urge to adopt at least one option, either

Now consider two non-compatible products, offering a similar service — multi-homing is possible, but costly.

5.1

• because their anticipate that immobility will drive to a Paretto-dominated no adoption; or

Differentiated services

• sensitive to “hype” (modeled as a no-adoption Let us consider two services with a1 < a2 and c1 < penalty k0 ); c2 ; apart from the switching cost, the equations are similar to what we have found with a entrant trying will consider an option, and adopt it prior to exchanging information with their peers: switching costs a quality or feature strategy. will be considered — but paradoxically, a lower outswitching cost should initially favor a service: 5.2 In- & out-switches on similar serThis advantage on the initial user-base will now vices be challenged by an easier switch, making the convention-establishment decision central: Two non-compatible services have the same cost or A socially central use makes compatibility a very usability c and the same quality or feature-set a; with needed element; secondary proprietary gizmos will perfectly equivalent services, the competition resolves not challenge a relationship: therefore the importo a population-wide pure coordination game. Intertance of the service to a social tie will decide who personal knowledge is not good enough to have an might switch: accurate opinion on the decisions of each of one’s relative, and explicit communication means proved too • an ostracized minority will refuse to be isolated, clumsy to handle the considerable task of coordinawhatever the cost; tion. • on the opposite, lower switching cost might force An agent considers none- essentials use to switch, even though their are more numerous. • a global distribution of expected adoption: p1 (respectively p2 ) the ratio of agents would should adopt among those with a degree d such that 5.3 With different qualities ¯ h(d) ≥ δ1 (respectively δ2 ); shares; d2i . Of course, if the services are not compatible, and on Agents would subscribe to S1 only if their degree offers a better quality for a higher price, the assord can compensate for the reduced adoption success: tativity (here measured by moments of g) will once ¯ again be a key-element. h(d) ≥ δ 1 p1 . u1i = a d1i + a0 d21 Multi-homing 6 Possible extensions are shared know friend and Let us consider that an agent i anticipates of his 6.1 Changing the utility function relatives will use S1 and d2i will use S2 . Because multi-homing is costly, This model can be extended, with similar results, Without any better argument, the needed ex-ante to a non-linear increasing function of the number coordination will leverage the slightest difference be- of subscribed relatives. A convex function would tween them, to avoid switching costs. strengthen natural monopolies; a concave function To discuss further those, consider their difference would limit the consequence of our model — and long are in the switching costs: k1→2 < k2→1 , namely, it at it is not decreasing. 7

Negative social externalities could modeled from the same aspect, to model activities that are scorned upon, identity facet management or making surprises. We would have to include company creation costs: equilibria would then between multiple isolated services, their cardinal being an balance between social externalities and entrance barriers.

only solution against an incompatible existing service. Reducing out-switching costs can be a better strategy (then focusing on in-switching costs) to fend-off a simultaneous competitor. A trusted brand will have an almost unchallengeable advantage.

Scale and framing effects can be decisive for content-driven communication tools. However, making the reasoning: I’ll use the service Ties can of course be weighted, considering closer rel- to then go to another platform is counter-intuitive, atives as more influential, because we tend to spend and lead users into adopting the service 2, in spite of more time with them. A schedule constraint appears the intended reasonning. then a reasonable addition, especially to reduce the surprising role of the upper-layer of the most connected individuals; this constraint has similar result References as a concave utility function for those constrained, eka and Albert-L´ aszl´ o Barab´ asi, but also induce less skewed, more assortative net- Albert, R´ “Statistical Mechanics of Complex Networks,” work, as they would be less possibles ties to the most June 2001. arXiv. connected individuals. Considering several social network connecting the Arthur, W. Brian, “Competing Technologies, Insame agents differently is also a key aspects in curcreasing Returns, and Lock-In by Historical rent communication services. So far, drawing conEvents,” The Economic Journal, March 1989, clusion from simple models has proven more efficient 99 (384), 161–131. then trying to write tedious parallel systems — but new and interesting results can only be found with Dupuy, Jean-Pierre, “Common knowledge, comthis approach, as the interaction network has proven mon sense,” Theory and decision, 1989, 27 (1-2), so seminal. More importantly, connecting data from 37–62. distinct networks is difficult or prohibited; increasingly sophisticated SNS might provide a source for Newman, Mark E. J., “The Structure and Function of Complex Networks,” March 2003. arXiv. revealing real data here. We also have considered a model with asymmetric social ties, that could represent admiration, expertise or attention issues. Such services can be either publication if considering the out- going links, or aggregators if focusing on in-going ones. Conclusion are that differentiated offers, similar to those considered here, are possible.

6.2

7

Changing the social ties

Conclusion: Trust and lock-in

Both quality and price competition make sense when facing a compatible competitor; differentiation is the 8