MultiLevelMarketing_EC11

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Mechanisms for Multi-Level Marketing

Yuval EmekComputer Engineering and

Networks Laboratory

ETH ZurichZurich, Switzerland

[email protected]

Ron KaridiMicrosoft Israel Innovation Lab

Herzeliya, Israel

[email protected]

Moshe TennenholtzMicrosoft Israel R&D Center

and

TechnionIsrael Institute ofTechnology

Herzeliya, [email protected]

Aviv ZoharMicrosoft Research

Silicon Valley LabMountain View, CA, [email protected]

ABSTRACT

Multi-level marketing is a marketing approach that moti-vates its participants to promote a certain product amongtheir friends. The popularity of this approach increases dueto the accessibility of modern social networks, however, itexisted in one form or the other long before the Internetage began (the infamous Pyramid scheme that dates backat least a century is in fact a special case of multi-level mar-keting). This paper lays foundations for the study of rewardmechanisms in multi-level marketing within social networks.We provide a set of desired properties for such mechanismsand show that they are uniquely satisfied by geometric re-ward mechanisms. The resilience of mechanisms to false-name manipulations is also considered; while geometric re-ward mechanisms fail against such manipulations, we exhibitother mechanisms which are false-name-proof.

Categories and Subject DescriptorsJ.4 [Social and Behavioral Sciences]: Economics

General Terms

Economics, Theory

Keywords

multi-level marketing, pyramid scheme, reward mechanisms

1. INTRODUCTIONSocial networks are everywhere: our e-mail and phone ad-

dress books, our family relatives, and our business connec-

tions, all define either explicit or implicit social networks.

Permission to make digital or hard copies of all or part of this work forpersonal or classroom use is granted without fee provided that copies arenot made or distributed for profit or commercial advantage and that copiesbear this notice and the full citation on the first page. To copy otherwise, torepublish, to post on servers or to redistribute to lists, requires prior specificpermission and/or a fee.

EC11,June 59, 2011, San Jose, California, USA.Copyright 2011 ACM 978-1-4503-0261-6/11/06 ...$10.00.

Social networks have existed long before the Internet, buttheir recent web-based form, as exhibited by companies likeFacebook, Twitter, or LinkedIn, made them more tangible.

In their new manifestation, social networks have become anattractive playground for viral marketing: the dream of anymarketer is that her products will be promoted via word ofmouth (which relies on social networks). In order to makethat dream a reality, various forms of marketing have beenadvocated. The so-called affiliate marketing, direct mar-keting, and multi-level marketing all refer to (overlapping)approaches that facilitate viral marketing. In this paper, weshall adhere to the term multi-level marketing, as it seemsto be the least restrictive one.

The fundamental idea behind multi-level marketing is thatAlice, who already purchased the product, is rewarded forreferrals, i.e., for purchases made by Bob as a result of Alicespromotion. The reward mechanism associated with multi-level marketing may take various forms. In particular, Alicemay be rewarded for both purchases made by Bob and forBobs own referrals in a recursive manner.

The potential to accumulate small rewards from each per-son to a sizable sum is important as it allows advertisersto attract early adopters and trendsetters that are of greatvalue to them. On the downside, the possibility of gath-ering a large sum has also inspired more illicit versions ofmulti-level marketing, namely pyramid schemes. These ille-gal1 mechanisms, essentially based on the notion of indirectreferrals, are not intended to promote a real product, butrather to collect money from the social network, althoughsometimes a product is used in an attempt to cover the na-ture of the pyramid scheme and bypass legal restrictions.In these cases, customers seldom enjoy the actual product

being promoted (when a product is being promoted), butare only participating in the (usually false) hope of gettingrewards from recruiting others.

Needless to say that selecting an appropriate reward mech-anism is inherent to the design of a successful multi-levelmarketing scheme. Interestingly, despite the popularity of

1The current paper does not take legal issues into consider-ation. In particular, our analysis will not make the distinc-tion between legitimate multi-level marketing and pyramidschemes.


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work on information spreading and influence in social net-works (see, e.g., the survey in [11]) the study of reward mech-anism design in that context has been almost completelyneglected. Such study is the main subject of the currentpaper.

Consider for example the following basic coupon drivenscheme. Upon purchase of the product, Alice is givencoupons that she can distribute among her friends. Then,for any purchase made by Bob in which Alices coupon is

used, Alice is rewarded with appropriate rebates on futurepurchases. This scheme and similar ones, are easy to imple-ment and have become quite standard in our daily life. Notethat the coupon driven scheme does not exploit indirect re-ferrals: Alice is not rewarded by purchases made with Bobscoupons or with the coupons of Bobs referrals.

Reward mechanisms that exploit indirect referrals usedto be difficult to implement as they require some centralauthority that keeps track of the referral structure. Infor-mation technology has made this task much easier. Considerfor example the setting in which Alice promotes a productby publishing a link to the sellers web-site in her blog orFacebook page. Bob can buy the product by clicking onthat link; together with the actual product, Bob receives a

link to the sellers web-site that he can also publish in hisblog or Facebook page. The sellers web-site can easily iden-tify Bob as a buyer that followed Alices link.2 This way acomplete record of direct and indirect referrals can clearlybe maintained. This ease of implementation makes rewardmechanisms that take indirect referrals into account evenmore appealing than they have been before.

Are indirect referrals really that important? We believeso. To demonstrate their significance, suppose that Bob is arock music authority and that following Alices promotion,he downloads a new rock song. If Bob recommends this songin his blog, and consequently many other users downloadthis song, then Alice certainly played a major role in thepromotion process, even if she only had a few direct referrals.A reward mechanism that depends only on direct referrals

is therefore bound to miss the bigger picture.

The referrals tree model.There are many possible ways to take the social network

that forms the basis of the referral process into account.In principle, one may wish to consider the times at whichpromoting messages were sent from one user to another, toconsider referrals that were not followed up by a purchase ofthe product being promoted, or even to consider the sociallinks along which a referral was notmade. However, all ofthis information may not be available to the original seller. 3

We therefore take the straightforward approach of lookingonly at the structure of successful referrals. For each buyer,we mark only a single referrer for introducing the product

to her (in reality, this would typically be specified at thetime of purchase). The induced structure of referrals formsa collection of directed trees, each rooted at a node that

2This can be implemented by associating Alices link witha unique identifier. In the current paper we abstract awaythis technical issue.3In some social networks such as Facebook there is oftenmore explicit knowledge of social connections, but generalreferral systems do not necessarily have all the informationabout the underlying social structure and may not be ableto track messages in the network.

corresponds to some buyer that has purchased the productdirectly from the seller. We shall refer to this tree collectionas the referrals forest, denotedT, and to the rooted trees inTas the referrals trees. We find the assumption that T canbe maintained by the seller sufficiently weak.

It should be clarified that the referrals forest correspondsto a single multi-level marketing campaign (typically associ-ated with a single product). Moreover, social network usersthat did not purchase the product are not represented in

T even if some of their friends attempted at promoting theproduct to them. For ease of presentation, we assume thatT is fully known when the rewards are to be distributed,although all the mechanisms explored in this paper are alsosuited for incremental payments performed online. It willalso be convenient to identify the buyers with their corre-sponding nodes in T, denoting the reward of (the buyercorresponding to) node u under the referrals forest T byRT(u).

Constraints on the reward mechanism.The reward mechanism is essentially a function that maps

the referrals forest Tto the non-negative real rewards of itsnodes. However, not every such function should b e consid-

ered; specifically, we impose three constraints on the rewardmechanisms. The first one is the subtree constraint: RT(u)is uniquely determined by Tu, namely, by the subtree ofTrooted at u. This is sensible, as each user u can really becredited only for bringing in users she promoted the productto, either directly (the children ofuin T) or indirectly (lowerlevel descendants ofu). Moreover, a dependence of RT(u)on the position ofu withinT (rather than on Tu only) mayresult in an undesirable behavior on behalf of u: in somecases u is better off delaying the purchase of the productafter receiving a referral in hope for a better offer, i.e., fora referral that would place u in a better position within T.

One of the consequences of the subtree constraint is thatthere is no point in dealing with the referrals forest T infull, but rather focus on trees which are rooted at the nodes

whose reward we are trying to calculate. In other words, thereward mechanism is completely specified by the functionR(T) that maps the rooted tree T to the non-negative realreward of its root (which may be an internal node withinthe whole referrals forest).

The second constraint that we impose on the reward mech-anism is thebudget constraint: the seller is willing to spendat most a certain fraction 1 of her total income on re-warding her buyers for referrals. Given that the price of theproduct is, this means that the total sum of rewards givento all nodes is at most |T |. We assume without loss ofgenerality that and are scaled so that = 1. Thus,

uTR(Tu) |T | .

The third constraint is the unbounded reward constraint:there is no limit to the rewards one can potentially receiveeven under the assumption that each user has a limited circleof friends in the underlying social network (imposing a lim-ited number of direct referrals). Formally, the unboundedreward constraint dictates that there exists some positiveinteger d (a property of the reward mechanism) such thatfor every real R, there exists some tree Tof maximum de-gree d (i.e., every node has at most d children) such thatR(T) R. In particular, this constraint implies that the


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reward mechanisms we consider must take indirect referralsinto account.

Our results.We begin our exploration of reward mechanisms with a

well known family of mechanisms, namely, geometric rewardmechanisms. Under these mechanisms, the contribution ofa node to the rewards of its ancestors in the referrals treedecreases exponentially (by a fixed factor) with the distancefrom these ancestors. Three desired properties of geometricreward mechanisms are listed: additivity, child-dependence,and depth-level-dependence. We show that these three prop-erties fully characterize the family of geometric mechanismsin the sense that any mechanism that satisfies all three prop-erties must be a geometric mechanism. (This may explainwhy pyramid schemes typically rely on geometric rewardmechanisms or some close variant of them.) We go on toshow that none of the properties is redundant: if any one ofthe three is left out, alternative reward mechanisms can befound that possess the remaining three.

We then look at one more important property of rewardmechanisms, namely, resilience to false-name manipulations(a.k.a. Sybil attacks). The geometric mechanism family

turns out to be susceptible to manipulations by users thatcan create false identities. In fact, we show that mecha-nisms that are resilient to false-name manipulations cannotguarantee a user some constant fraction of the reward ofeven its least influential child. Moreover, it turns out thateven if one replaces the child-dependence and depth-level-dependence properties with the much weaker monotonicityproperty, resilience to false-name manipulations is still im-possible. On the positive side, we present and analyze tworeward mechanisms that maintain resilience to false-namemanipulations; the two mechanisms differ in their level ofresilience and ease of implementation.

Related work.The general idea of diffusion of opinions and conventions

in societies has for long been a topic of study in the socialsciences [15, 9] and got attention by game theorists (e.g.,[20]) and AI researchers (e.g., [16]) among others, quite awhile ago. The effects of the social structure on emergentbehavior and norms has also been studied, e.g., in [14, 17].

The more explicit algorithmic questions that arise whenone considers an endeavor such as viral marketing have beenposed more recently. The original question of how to select agood set of influential users has appeared in a seminal paperby Domingo & Richardson [8], and has later been explored(with various related models) in a series of papers, e.g. [7]and many other following works [10, 3, 5]. These generallyassume that the spread of information in the social networkoccurs through some contagion model (i.e., that a user is

more likely to be infected with an idea if more of his neigh-bors are) in which users do not explicitly exert effort. Therigorous study of incentive design for facilitating diffusion orproduct adoption was typically left without proper explicittreatment, and all works we are aware of in the context ofviral marketing do not try to influence the amount of effortexerted to spread the information further.

The issue of incentives in social networks has however re-ceived attention in other particular contexts. For example,fair distribution of costs/gains of members in a network, us-ing standard power indices from cooperative game theory,

such as the Shapley and Banzhaf values has been a subjectof study (see e.g., [13, 4, 2]). However, these do not providea general rigorous study of reward mechanisms for socialdistribution. Kleinberg & Raghavan [12] consider a settingthat is perhaps the most similar in spirit to our own, inwhich they elicit effort from agents that forward queries ina social network. Unlike our setting, they allow each agentthat receives the query and forwards it to offer its own re-ward for a successful answer. The final rewards are only

allocated along the path to the agent that gave the answeras each agent along the way receives the reward for passingthe answer back along the path. A similar reward mecha-nism (that was more structured) was used by the team fromMIT that won the DARPA network challenge [1].

Finally, our work also deals with the issue of Sybil attacksthat have appeared in many other contexts such as reputa-tion mechanisms [6], combinatorial auctions [18], and socialchoice [19].

2. PRELIMINARIESUnless stated otherwise, all trees addressed in this paper

are assumed to be finite and directed from the (unique) root

towards the leaves. A typical tree will be denoted by T;its root is denoted by r. We use the standard (directed)tree notions of parent, child, leaf, descendant, and ancestorin their natural sense; the parent of a (non-root) node uin T is denoted pT(u). The degree of node u T, denoteddegT(u), is just the number of childrenuhas inT. Given twonodesu, v Tsuch that u is an ancestor ofv, we define thedistancefromutov, denotedT(u, v), as the number of hops(i.e., edges) along the unique path inT leading fromu to v;the distance fromu to itself is defined to be T(u, u) = 0. IfT(u, v) =k >0, then we refer tou as thek

th ancestorofv.We denote the subtree ofT rooted at u byTu. The heightofT, denoted h(T), is defined to be the maximum distancefromr to any leaf in T; theheightofu in T, denoted hT(u),is simply h(Tu). The depth ofu in T, denoted depT(u), is

the distanceT(r, u) from the root ofT tou. When the treeTis clear from the context, we may omit the subscripts andsimply write p(u), deg(u),(u, v), h(u), and dep(u).

A reward mechanism is a function that maps a non-negative real reward R(T) to every finite rooted treeT. Wethink ofTas a (subtree of a) referrals tree and of R(T) as thereward of the rootr ofT. (Recall that defining the reward inthat manner is made possible due to the subtree constraintrequiring that the reward of a node in the referrals forestdepends only on its subtree.) The profit of r is actuallyR(T) as r paid 1 for purchasing the product whenshe joined the referrals forest. By the budget constraint, itis assumed that

uTR(Tu) |T |. The unbounded reward

constraint guarantees the existence of some positive integer

dsuch that for every real R, there exists some treeTof max-imum degree d such that R(T) R. The notation RM()is used when it is important to emphasize that the functionR() is associated with the reward mechanism M.

3. THE GEOMETRIC MECHANISMIn this section we focus on the following family of reward

mechanisms, referred to as geometric mechanisms. Giventwo constants 0 < a < 1 and b > 0 such that b+ 1 1/a,the reward from a referral tree Tunder the (a, b)-geometric


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mechanismis defined to be

R(T) =uT

adep(u) b .

The constraints on a and b ensure that the amount con-tributed by each node to the reward of its ancestors willnot exceed 1. This simple mechanism is very p opular withpyramid schemes; as we will show soon, this is no coinci-dence. Let us begin by defining and discussing three basic

properties of reward mechanisms:Additivity (ADD): We define the operation on trees suchthat ifT1, T2 are trees, then T1 T2 is the tree formed bycontracting (or merging) the roots ofT1 and T2. ADDis thenstated as follows: R(T) + R(T) = R(T T).This property suggests that if two disjoint trees are mergedat the root, then the reward of the root is exactly the sumof the rewards of the two original trees. Generally speak-ing this property implies that the reward to each node canbe independently attributed to the subtrees rooted at itschildren.Child Dependence (CD): The reward of the root isuniquely determined by the rewards of its children. Thisproperty ensures that the actual computation of the rewards

can be performed locally. In fact, we shall consider a weakercondition for this property: If the root of T has a singlechildu, then R(T) is uniquely determined by R(Tu). Thisis captured by a function : R0 R0 (a property of themechanism) so that R(T) = (R(Tu)).Depth Level Dependence (DLD): R(T) is uniquely deter-mined by the number of nodes on each depth level in T.We denote by dk the number of nodes ofTat depth levelk > 0, and the infinite vector containing these numbers forall depth levels by d = (d1, . . . , dh, 0, 0, . . . ), where h is theheight of the tree. Let D be the set of all such vectors, i.e.,the set of all infinite vectors over Z0with a strictly positiveprefix followed by a countably infinite suffix of zeros. ThenDLD implies that there exists some function f :D R0 (aproperty of the mechanism) such that R(T) = f(d).

This property essentially means that the credit for a referraldepends solely on how direct (or better said, indirect) thisreferral is.

Theorem 1. A reward mechanism satisfiesDLD, ADD, andCD if and only if it is a geometric mechanism.

To prove the theorem, we will need the definition of thefollowing additional property:Summing Contributions (SC): There exists a sequence{ck}k1 of non-negative reals such that

R(T) =uT

cdep(u) =k=1

#nodes at depth level k ck .

That is,SC

implies that each node in the tree T contributessome independent amount to the root, and that amount de-pends only on its depth. The following lemma reveals theconnection between this property and the ones we have al-ready defined.

Lemma 2. A reward mechanism satisfies SC if and onlyif it satisfies DLDand ADD.

Proof. We start with some useful notation. Let d>k

denotes the infinite vector d after the first k elements havebeen removed, i.e., d>k = (dk+1, dk+2, . . . ). We denote by

m d the infinite vector that starts with the element mand continues with the elements of the infinite vector d, i.e.,m d = (m, d1, d2, . . . ). Finally, we denote the all-zeros

infinite vector by0 = (0, 0, . . . ).The direction in which SC implies DLD and ADD is trivial.

We shall establish the converse direction using an inductiveargument. Specifically, we prove by induction on k that foreveryk 0, there exist non-negative reals c1, . . . , ck and anadditive function4 gk : D R0 such that

R(T) =k

i=1

ci di+gk(d>k) .

The base of the induction, for k = 0, is satisfied by settingg0(d) = f(d), where f is the function promised by DLD. Theadditivity ofg0 is then guaranteed by ADD.

For the inductive step, we assume that there exist somenon-negative reals c1, . . . , ck1 and an additive functiongk1 : D R0 such that

R(T) =k1i=1

ci di+gk1(d>k1) .

The assertion for k is established by setting

ck = gk1(1 0) ; and

gk(d) = gk1(1 d) ck .

To verify that the functiongk is indeed additive, we employthe additivity ofgk1, observing that

gk(d+e) = gk1(1 (d+e)) ck

= gk1(2 (d+e)) gk1(1 0) ck

= gk1(1 d) +gk1(1 e) 2 ck

= gk(d) +gk(e) .

The proof of the inductive step can now be completed dueto the inductive hypothesis since

f(d)

=k1i=1

ci di+gk1(d>k1)

=k1i=1

ci di+gk1(dk d>k)

=k1i=1

ci di+gk1(1 d>k) +gk1((dk 1) 0)

=k1i=1

ci di+gk(d>k) +ck+ (dk 1) ck

=k

i=1 ci di+gk(d>k) .The lemma follows by taking k = h(T) as every additive

functiong must satisfyg(0) = 0 (by definition, g(0) = g(0 +0) = 2 g(0)).

Theorem 1 is established by showing that the contributionvaluesck form a geometric progression.4In this context a functiong : D R0is said to be additiveifg(d + d) = g(d) + g(d), where the summation ind + d iscoordinate-wise.


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Lemma 3. A reward mechanism satisfiesSCandCDif andonly if it is a geometric mechanism.

Proof. It is trivial to show that a geometric mechanismsatisfies both properties, so we focus on the converse direc-tion. Let us restrict our attention to a specific class of trees:For n > 1 and m > 0, we denote by T(n, m) the tree con-sisting ofn+m nodes organized as a path of length n 1emerging from the root with the last node in this path hav-

ingm children, all of which are leaves. Refer to Figure 1 forillustration.Recall that SC implies the existence of constants c1, c2, . . .

that determine the contribution of nodes at each depth levelto the reward of the root. We first argue that ck must bestrictly positive for every k 1. To that end, suppose thatck = 0 for some k

1 and consider the trees T(k, m)and T(k, m) for some m, m > 0, m = m. SC impliesthat R(T(k, m)) = R(T(k, m)) since ck = 0. By CD, weconclude that the same holds for T(k + 1, m) and T(k +1, m), namely, R(T(k +1, m)) = R(T(k+ 1, m)), becauseboth the root ofT(k + 1, m) and that ofT(k + 1, m) havea single child whose reward is R(T(k, m)) = R(T(k, m)).This implies that ck+1 must also be 0 and by induction,that ck = 0 for every k k

. But this contradicts the

unbounded reward constraint: if ck = 0 for every k k,then no tree T of maximum degree d can provide a rewardgreater than 2 dk

.So, assume hereafter that ck > 0 for every k 1. In

attempt to simplify the analysis, we shall impose anotherassumption on the contribution values ck. Specifically, weassume that eachck is rational, so that ck = xk/yk for somepositive integers xk and yk. We later on outline how thisassumption can be lifted.

Let us compare the reward that is given to the root nodein two specific T(n, m) trees (recall that a T(n, m) tree hasone node at each level 0 to n 1, andm nodes at level n):

R(T(k 1, xk yk1+ 1))

=k2i=1

ci+ (xk yk1+ 1) ck1

=k1i=1

ci+xk1 xk

=k1i=1

ci+xk1 yk ck

= R(T(k, xk1 yk)). (1)

Now, observe that CD implies that if R(T(n, m)) =R(T(n, m)), then R(T(n+ 1, m)) = R(T(n + 1, m)). Byapplying this observation to Equation 1, we conclude that

R(T(k, xk yk1+ 1)) = R(T(k+ 1, xk1 yk)).

SC then implies that

k1i=1

ci+ (xk yk1+ 1) ck =k

i=1

ci+ (xk1 yk) ck+1 ,

hencexk yk1 ck = (xk1 yk) ck+1. It follows that

(ck)2 =ck1 ck+1 , (2)

which implies a geometric progression (ck is the geometricmean ofck1 andck+1).

Figure 1: T(n, m) for n= 4 and m= 10.

Recall that our proof thus far only works if all cks arerational numbers. We can extend the proof to irrationalnumbers if we add a requirement on the continuity of thefunction that determines the rewards of a parent from thereward of its children. If the cks are not rational, it ispossible to approximate them as closely as one wishes withrational numbersxk/yk and with the extra assumption, thederivation above results in an equation similar to Equation 2which is modified with terms that represent the error in theapproximation ofck. As this error can be made arbitrarily

small, Equation 2 holds even for irrational values.

Another appealing property of geometric mechanisms isthat the contribution of descendants to their ancestor de-creases with distance. This reflects the fact that the ancestorgets less credit for more distant indirect referrals.

It is important to point out that each of the three proper-ties we used to characterize the family of geometric mecha-nisms is needed, i.e., if we remove one of the three propertiesthen there exists another mechanism (outside the geometricfamily) for which the remaining three hold. This is estab-lished in the full version of this paper.


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4. SYBIL ATTACKSOur goal in this section is to develop reward mechanisms

which are not vulnerable to forging identities on behalf of theusers. Let us begin by introducing the notions of a splitandalocal split. Consider some treeTand some nodev T andletu1, . . . , uk be the children ofv in T. Intuitively speaking,a split of v refers to a scenario in which v presents itselfas several nodes a.k.a. replicas thus modifying the(sub)treeTv that determines its reward (possibly turning itinto several trees), while keepingTu1 , . . . , T uk intact. A localsplit refers to the special case of a split in which u1, . . . , ukare forced to share the same parent in the resulting tree.

Formally, we say that the tree collection {T1, . . . ,Tm} canbe obtained fromTv by a split ofv if(1) for every 1 i k , there exists a single 1 j (i) m

such that ui Tj(i); and(2)Tj(i)ui =Tui for every 1 i k.The nodes inT1 Tm Tj(1)u1 Tj(k)uk are referred to as the replicas of v under that split. Bydefinition, ui must be a (direct) child of some replica of v

for every 1 i k as otherwise, at least one of the subtreesrooted atu1, . . . , uk must have been changed, thus violatingcondition (2). Refer to Figure 2 for an illustration of a split.The split is called local ifu1, . . . , uk are all children of thesame replica ofv .

The semantic of the aforementioned split is as follows. Theuser corresponding tov forges some new identities that cor-

respond to its replicas inT1, . . . ,Tm. Some of these replicaspurchase the product via referrals from pT(v) they form

the roots of the treesT1, . . . ,Tm; the rest of the replicaspurchase the product via the referrals of other replicas thatalready purchased it. The referrals to u1, . . . , uk are thenmade from the appropriate replicas, so that eventually Tv is

replaced by the treesT1, . . . ,T

m, the roots of which are all

children ofpT(v).It is assumed that ui does not distinguish between theoriginal v and its new replicas. Therefore the new referral(made from a replica of v rather than from v itself) looksto ui like a referral from v and she relies on it to purchasethe product just like she did with the referral from v inthe original scenario. Under local splits, this assumption islifted as all children ofv purchase the product from the samereplica that may have the identity ofv herself.

Whats in it for v ? Clearly, v has to invest #replicasin introducing the new replicas (purchasing new copies ofthe product). However, she now collects the rewards fromall her replicas, which sums up to

m

i=1

replica u ofv in Ti

R(Tiu) .Thus, the profit5 ofv changes from R(Tv) to

mi=1

replica u ofv in Ti

R(Tiu) #replicas ;5Here, we assume that a user has no usage in more than onecopy of the product. This is a valid assumption, e.g., in thecontext of information goods.

the split is called profitable for v if this change is positive.This leads to the definition of the following two propertiesof reward mechanisms.Split Proof (SP): A reward mechanism satisfies SP if it doesnot admit a profitable split for any node v in any tree T.Local Split Proof (LSP): A reward mechanism satisfiesLSPif it does not admit a profitable local split for any nodev in any tree T.

The geometric mechanisms presented in Section 3 do not

satisfy SP. In fact, they do not even satisfy LSP (see Sec-tion 4.1). A simple mechanism that do satisfy SPis the sin-gle levelmechanism defined by fixing RMsl(T) = deg(r)for some constant 1, however, this mechanism does notadhere to the unbounded reward constraint.

In Section 4.1 we establish two negative results regardingthe design of split-proof mechanisms. On the positive side,we devise two mechanisms which are resilient to false-namemanipulations. The first one, presented in Section 4, satisfiesthe stronger property SP, however, it is complicated to im-plement and not very appealing. The second one, presentedin Section 4.3, satisfies only LSP, but it is more natural andmuch easier to implement.

4.1 Negative ResultsLet us now exhibit two negative results (proofs are de-

ferred to the full version of the paper) regarding the designof split-proof mechanisms. The first result shows that thereward guaranteed to a node in a split-proof mechanism can-not be a constant fraction of even its least influential child.

Lemma 4. A reward mechanism that satisfiesLSPcannotguarantee a node some fraction0< 1 of the reward ofits least rewarded child.

Next, we show that even a family of reward mechanismsmuch wider than geometric mechanisms is still not split-proof. This requires the introduction of another propertywhich is clearly satisfied by every geometric mechanism, yet,

cannot replace any of the characterizing properties listed inTheorem 1.Monotonicity (MONO): If the tree Tcan be obtained fromthe tree T by removing some leaf, then R(T)< R(T).

Lemma 5. A reward mechanism that satisfies MONO andADDcannot satisfy SP.

4.2 A Split-Proof MechanismIn this section we present a split-proof reward mechanism,

denoted Msplit. Informally, the mechanism Msplitis definedin two stages: in the first stage, we define a simple basemechanism, denoted Mbase; Msplit is then defined with re-spect to the maximum profit a node can make under Mbasefrom splits.

Mechanism Mbase.The base mechanism Mbase is defined by setting

RMbase(T) to be the maximumh Z0such thatTexhibitsas a subtree, a perfect binary tree6 B rooted at r whoseheight is h. In that case we say that B realizes RMbase(T)(see Figure 3). If there are several perfect binary trees that

6A perfect binary tree is a rooted tree in which all leaves areat the same distance from the root and all non-leaves haveexactly two children.


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v

2

4

5

6

7

1

3

8

(a)Tv

1

2

3

4

6

5

7

8

(b)T1 andT2Figure 2: The tree collection { T1,T2} can be obtained from Tv by a split of v. The white circles depict thechildren ofv in T(2(a)) and their positions inT1 andT2 after the split (2(b)). The gray circles in 2(b) depictthe replicas ofv under that split.

Figure 3: The tree Tand the perfect binary tree thatrealizes RMbase(T) (nodes depicted by gray circles).

can realize RMbase(T), then it will be convenient to take thefirst one in a lexicographic order based on a breadth-first-search traversal and consider it as theperfect binary treethat realizes RMbase(T).

A node u T is said to be visible in T if it belongs tothe perfect binary tree that realizes RMbase(T); otherwise,u is said to be invisible in T. This definition is extendedas follows: given some ancestor v of u in T, u is said tobe visible (respectively, invisible) to v ifu is visible (resp.,invisible) in Tv. Note that ifu is invisible to v, then it isalso invisible to pT(v) (assuming of course that v =r). Thecontrary is not necessarily true: u may be visible to v butinvisible to pT(v). By definition, for every nodeu T andfor every j Z1, it holds that u admits either 2

j or 0visible depth-j descendants.

The mechanismMbase can be redefined by setting

RMbase(T) =

visibleuT

2T(u,r) .

This alternative view of mechanismMbasecalls for the defi-nition ofcontributions: a node u T contributes2k to thereward of its k th ancestorv , k 1, ifu is visible to v; oth-erwise, u does not contribute anything to the reward ofv.Let CMbase(u, v) denote the contribution ofu to the reward

of v under Mbase. The reward of a node can now be cal-

culated by summing the contributions that its descendantsmake to it. This implies that Mbase satisfies the budgetconstraint: the total contribution made by a node u Tto all its ancestors is bounded from above by the geometric

sum(u,r)

j=1 2j < 1, hence, by changing the summation,

we conclude that

vTRMbase(Tv)< |T|.

Mechanism Msplit.The mechanism Msplit is defined by setting RMsplit(T)

so that it reflects the maximum profit that r can get underMbase from splits. More formally, let Sbe the collection ofall tree collections that can be obtained from Tby a split ofr. ThenMsplit is defined by setting

RMsplit(T)

= supT={ T1...,Tm}S

mi=1

repl. v ofr in Ti

RMbase(Tiv)

T1 Tm |T| .

To avoid cumbersome notation, we shall denote (T) mi=1

replica v ofr in TiRMbase(

Tiv) and T T1 Tmso that RMsplit(T) = sup T S{(

T)(| T | |T|)}. Referto Figure 4 for illustration.

We first observe that RMsplit() is well defined by show-ing that we may assume without loss of generality that S is

finite. To that end, we argue that it is sufficient to considerT Ssuch that every replica ofr inTis of degree at least2, which means that there are less than degT(r) replicas intotal, and hence there are finitely many different options

forT. Indeed, assume that v is a replica ofr in some treeT T so that deg T(v)< 2 and letT be the tree obtainedfromT by removing v and turning its sole child (in casethat deg T(v) = 1) into a child ofp T(v) or into the new root

ifv is the root ofT. Note that by the definition of splits,T { T} {T} can also be obtained from Tby a split of


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(a)T (b)T1Figure 4: The tree T and a possible split of its root r (into a single treeT1). The gray circles in 4(b) depictthe replicas ofr under that split. If= 1, then this split realizes RMsplit(T) = 2 + 3 + 4 2 = 7.

r. Since deg T(v)< 2, we know that RMbase(Tv) = 0. More-

over, every descendant u ofv inTmust be invisible to theancestors ofv in

T (which are also replicas ofr), hence u

does not contribute anything to the rewards of these ances-

tors under Mbase. The argument is established by recallingthat the total contribution ofv to the rewards of its ances-tors inTis smaller than 1 , which means that it was notworthwhile for r to generate the replica v .

We say that the tree collectionT S realizesRMsplit(T)if RMsplit(T) =(

T) (| T | |T|). If there are severaltree collections that can realize RMsplit(T), then it will beconvenient to choose one (arbitrarily) and consider it as thetree collection that realizes RMsplit(T).

Excessive contributions.It is easy to see that Msplit satisfies the subtree con-

straint and the unbounded reward constraint (Mbase al-ready satisfies the unbounded reward constraint and the

rewards under Msplit dominates those of Mbase). More-over, by definition, Msplit satisfies SP, i.e., a node cannotincrease its profit by splitting (recall that the mechanismtakes every possible split into account). The difficult part isto show that Msplit satisfies the budget constraint, that is,

vTRMsplit(Tv) |T|.The first step towards fulfilling this task is to extend

the notion of contribution to the reward mechanism Msplit.

Consider some nodeu Tand some ancestor v ofu. LetTbe the tree collection obtained from Tv by a split ofv that

realizes RMsplit(Tv) and letTbe the tree inTthat contains

u. By definition, several replicas ofv inTmay be ancestorsofu (in

T); denote these replicas by v1, . . . , v. The contri-

butionofu to v underMsplit, denoted CMsplit(u, v), is then

defined to be

CMsplit(u, v) =

j=1

CMbase(u, vj) . (3)

We argue thatvT

RMsplit(Tv) uT

ancestors v ofu

CMsplit(u, v) . (4)

At first glance, inequality (4) may be surprising as its righthand side does not take into account contributions made by

replicas that do not exist in T. However,M split is definedwith respect to the profit (rather than reward) that a nodecan get under Mbase from splits. The total contribution ofa node to all its ancestors under Mbase is smaller than 1,

hence each replica of node u contributes less than 1 to therewards of the other replicas of u (it does not contributeanything to any other node in T). Inequality (4) followssince is assumed to be larger than 1. Our goal in whatfollows is to show that

uT

ancestors v ofu

CMsplit(u, v) |T| . (5)

In attempt to establish inequality (5), it may seem nat-ural to bound the total contribution that each node makesto its ancestors, showing that this is at most 1, as we didwith Mbase. Unfortunately, this does not work: If the con-tribution CMsplit(u, v) of node u to its ancestor v underMsplit, as defined in equation (3), is composed from Mbase-

contributions to several (more than one) replicas ofv, thenCMsplit(u, v) > CMbase(u, v). Consequently, the total con-tribution that u makes to the rewards of all its ancestorsunderMsplitmay exceed 1. This obstacle requires a carefulaccounting argument that we now turn to describe.

Consider some node u T. Let (u) denote the set of allancestorsv ofu in Tsuch that CMsplit(u, v)> 0. Assumingthat (u)=, let(u) be the nodev (u) that maximizes

T(v, u). Now, consider some nodev (u). LetT be thetree collection obtained fromTv by a split ofv that realizes

RMsplit(Tv), letTbe the tree inT that contains u, and let

v be the highest ancestor ofu inT such that u is visible tov. Note that v must be a replica ofv in

T since v (u).

Moreover, T(v, u) T(v, u) and the contribution ofu to

vs reward under Msplit is CMsplit(u, v) = T(v,u)j=T(v,u)2j.Thus, the parameter T(v, u) plays an important role in thecalculation of CMsplit(u, v) denote it by (u, v), so that

CMsplit(u, v) = 21T(v,u) 2(u,v).

Deficits and surpluses.Recall that CMbase(u, v) = 2

T(v,u) and that such a con-tribution would have resulted in the desired bound of 1 onthe total contribution of node u to all its ancestors. In-formally speaking, if (u, v) > T(v, u), then u exhibits a


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deficit. On the other hand, if(u) = v, thenu does not con-tribute anything to the rewards of the ancestors ofv , whichleaves us with a small surplus. Our argument will rely oncovering the total deficits with the total surpluses.

Formally, given some nodes u T andv (u), define

deficit(u, v) = CMsplit(u, v) CMbase(u, v)

= 2T(v,u) 2(u,v)

and

surplus(u) = 2(u,(u)) .

Using these notions, we havev(u)

CMsplit(u, v)

=

v(u)

21T(u,v) 2(u,v)

T((u),u)k=1

2k +

v(u)

2T(v,u) 2(u,v)

= 1 2T((u),u) +

v(u)deficit(u, v)

= 1 deficit(u, (u)) surplus(u)

+

v(u)

deficit(u, v)

= 1 surplus(u) +

v(u){(u)}

deficit(u, v) .

We establish the fact that Msplit satisfies the budget con-straint by showing that

uT

v(u){(u)}

deficit(u, v) uT

surplus(u) , (6)

thus covering the deficits with the surpluses. This is done

in the following manner.Consider some nodev Tand letTbe the tree collectionobtained from Tv by a split ofv that realizes RMsplit(Tv);

consider some treeT T. Fix some replica v of v inTand let h be the height of the perfect binary tree that re-

alizes RMbase(Tv). Given some 1 h, let Dv, be the

set of descendants u of v inT that satisfy: (1) v is thehighest ancestor of u inT to which u is visible; and (2)(u, v) = T(v, u) = . It is interesting to point out that dif-ferent nodesu in D v, may have different values ofT(v, u),although they all have the same value ofT(v, u) = .

A crucial observation is that for every two nodes u Tand v (u), there exists a unique choice of v and suchthat u Dv,. Let D

+v, be the set of all nodes u Dv,

such that v = (u), i.e., those nodes that do not contributeto the reward of any ancestor ofv . Let Dv, = Dv, D

+v,.

Inequality (6) is established by proving the following lemma.

Lemma 6. The nodes in Dv, satisfyuD

v,

deficit(u, v)

uD+v,

surplus(u).

Proof. Denote D =Dv, andD+ =D+v,. Recall that

v has either 2 or 0 children which are visible to pT(v); thelatter implies that D =, so assume in what follows thatthe former holds and let v1, v2 be the children of v in T

which are visible to pT(v). Let Di =D

Tvi for i = 1, 2.By definition,D =D1 D

2 andD

1 D

2 =. Moreover,

all nodesu in Di have the same value ofT(vi, u) denoteit byki.

Recall that deficit(ui, v) = 2ki1 2 for every node

ui Di . Since|D

i |= 2

ki , we conclude that

uiDideficit(ui, v) = 2

ki

2ki1 2

= 1

2 2ki

and uD

deficit(u, v) = 1 2k1 2k2 .

On the other hand, surplus(u) = 2 for every nodeu D+.Since |D+| = 2 |D1| |D

2| = 2

2k1 2k2 , it followsthat

uD+

surplus(u) = 2

2 2k1 2k2

= 1 2k1 2k2

which establishes the assertion.

4.3 Resilience to Local SplitsThe mechanism presented in Section 4.2 is resilient to

splits, but it is far from being intuitive. Moreover, userswho recruit a large number of buyers may find that they arenot rewarded for many of them (due to limited visibility).We therefore briefly describe another mechanism (in fact, afamily of mechanisms) this time with resilience only tolocal splits that is much simpler than the one we havepreviously shown.

The mechanism M local is similar to the geometric mech-anism (with some choice of parameters 0 < a 0such thatb + 1 1/a), only that here, we ignore the contri-

butions coming from one of the subtrees rooted at rs chil-dren, say, the largest subtree. More formally, let v1, . . . , vkbe the children ofr in Tand assume without loss of gener-ality that|Tv1 | |Tvk |. Then,

RMlocal(T) =

uTv2Tvk

adepT(u) b .

Note that a node with one child obtains no reward.It is easy to see that Mlocalsatisfies the subtree constraint

and the unbounded reward constraint. Moreover, since therewards under M local are dominated by those of a geomet-ric mechanism, it also satisfies the budget constraint. Thefollowing lemma establishes the resilience ofMlocal to localsplits.

Theorem 7. The reward mechanismMlocalsatisfiesLSP.

Proof Proof sketch. Let r be the root of T and letv1, . . . , vk be its children. By definition, in any local splitof r, the nodes v1, . . . , vk are children of the same replicaof r. A straightforward calculation shows that r does notgain anything from replicas that do not have v1, . . . , vk asdescendants, hence it suffices to consider local splits thatreplace T with a tree T obtained by identifying the rootofTwith the (sole) leaf of some directed path P. (In thatcase, the replicas ofr are the nodes ofP.) The design of


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Mlocal implies that the total reward ofr from such a localsplit remains RMlocal(T) which obviously turns the split intoa non-profitable move.

5. CONCLUSIONSWe have presented a theoretical framework for multi-level

marketing mechanisms. Our framework gives host to manypossible reward mechanisms, with varying properties. A full

characterization of the geometric mechanism family, whichis recognized in particular with Pyramid schemes, is estab-lished. While we find that geometric mechanisms are not re-silient to false-name manipulations, we have instead showntwo different mechanisms that can withstand such manipu-lations.

There are many open avenues for future work. First, manydifferent characterizations can be found for the mechanismsthat we presented. It is possible to think of other intuitiveproperties and attempt to substitute them for the ones wehave defined. In addition, the split-proof mechanisms thatwe have shown may be hard to implement (they are hardto explain to users and deny rewards for some of the refer-rals that are made). It will be nice to find mechanisms thatimprove these aspects as well. It will also be interesting to

explore other models for marketing in the context of socialnetworks, including models that take the underlying struc-ture of the network into account, and perhaps other detailssuch as the percentage of successful referrals and the timingof messages. Finally, we believe that much can be learnedfrom experimental evaluation, or even deployment of mecha-nisms that we have suggested. User trials can reveal defectsin these reward schemes, and show how readily human usersare willing to accept them.

6. REFERENCES[1] DARPA network challenge.

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