Mechanism for Multi-Level Marketing

Yuval Emek (ETH) Ron Karidi (Microsoft)

Moshe Tennenholtz (Microsoft & Technion) Aviv Zohar (Microsoft)

Collaboration of Microsoft Israel Basic Research Group

andMicrosoft ILDC Innovation Labs

The setting

Sell an information good over a social network

information good =

Invest %s of income in rewarding “good” buyers =credited for bringing many other buyers

can make a profit from their recommendations

Seller’s dream:

Possible implementation:identified links

• Upon buying the product, user u obtains a link to the product’s section in the purchasing site– this link encodes the id of u– can be installed in u’s page, blog or sent by e-mail, sms

• Other users can buy the product by clicking u’s link– new links are generated

• The purchasing site keeps track of:“who clicked whose link?”

The purchasing tree T• The induced purchasing tree:– root = purchasing site (or the seller)– node v is a child of node u if user v bought the product by

clicking user u’s link– u should be credited for bringing her descendents in T

uv

Rewarding mechanism

• Suppose that the price of the product is $2 and that the seller is willing to spend ≤ $1 out of it on rewards– n buyers at most $n spent on rewards

• The focus of our research project:how much reward should each user receive?– depends on the purchasing tree

• Looking for a rewarding mechanism:– fair– easy to implement– buyers have an incentive to promote a good product

Failed attempt

• Split the $1 equally among all ancestors of the new buyer in T (a.k.a. Shapley value)

• CON: user prefers to have small depth in T (root = depth 0)– share (future) rewards with less nodes– may wait for a “better offer” instead of buying the product

right away• Any reasonable rewarding mechanism should satisfy:– the position of u in T doesn’t affect r(u)– r(u) is uniquely determined by Tu

Axiomatic approach

• Identify a minimal set of requirements that a rewarding mechanism should satisfy (a.k.a. spec)

• Characterize the collection of rewarding mechanisms that satisfy these requirements

• Saves on running many experiments

What’s a reasonable mechanism?• Sub-tree constraint: Reward depends on descendents;

not on position in T• Budget constraint: the seller is willing to spend at

most a certain fraction π ≤ 1 of her budget on referrals. Given the price of product is ф , we normalize so that πф =1.

• Unbounded reward constraint: there is no limit to the rewards one can potentially receive even under the assumption that each user has a limited circle of friends.

An axiomatic approach• [DLD] Depth Level Dependence: Really matters: #descendants on each depth level– capturing the magnitude of credit

r(u) is uniquely determined by #descendents of u on each depth level

• [ADD] Additivity: r(T) + r(T’) = r(T T’)• [CD] Child Dependence: r(u) can be calculated from

children’s rewards– easy to implement

u

The characterization

Goal

• Characterize the collection of mechanisms that satisfy:• [DLD] r(u) is uniquely determined by #descendents of

u on each depth level• [ADD] r(T) + r(T’) = r(T T’)• [CD] r(u) can be calculated from children’s rewards

u

Summing contributions

• [SC] sequence {ck}k≥1 s.t. each new buyer contributes ck to its kth ancestor in T, k = 1, 2, …reward = sum (over all descendents) of these contributions

• LEMMA: SC DLD + ADD

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SC DLD + ADD

• Direction trivial• f(d) = reward of a node with dk level k descendants

– promised by DLD• Direction established by showing that sequence

{ck}k≥1 s.t. f(d) = ck dk

• Prove by induction on k thatf(d) = c1 d1 + + ck dk + g(d)where g(d) doesn’t depend on d1, …, dk and is additive

• Base case: guaranteed by ADD

New buyer contributes ck to its kth ancestor

SC DLD + ADD

• Assume that f(d) = c1 d1 + + ck-1 dk-1 + g(d)where g(d) doesn’t depend on d1, …, dk-1 and is additive

• Show that f(d) = c1 d1 + + ck dk + g’(d)where g’(d) doesn’t depend on d1, …, dk and is additive

• Fix ck = g(1k) and g’(d) = g(1k d+) – ck where d+ = {di}i>k

• ck is a constant and g’(d) doesn’t depend on d1, …, dk

• Remains to show:– g’ is additive– f(d) = c1 d1 + + ck dk + g’(d)

g’ is additive

Fix ck = g(1k) and g’(d) = g(1k d+) – ck , where d+ = {di}i>k

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Geometric progression• We showed that SC DLD + ADD• SC is characterized by a sequence {ck}k≥1

• LEMMA: SC + CD {ck}k≥1 is a geometric progression:– parameters 0 < a < 1, b > 0 s.t. a b ≤ 1 – a– ck = ak b

• We first show that ck > 0 k (details omitted)

• Direction If each node contributes ck = ak b to its kth ancestor,

then the reward of a node u with children v1 … vt is r(u) = i a(r(vi) + b)

SC + CD {ck}k≥1

• Require rational rewards (and contributions)– ck = xk / yk (integers)

• Show that (xk / yk)2 = (xk-1 / yk-1) (xk+1 / yk+1)– ck is the geometric mean of ck-1 and ck+1

– hence {ck}k≥1 is a geometric progression

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Free requirements

• Proved: every rewarding mechanism that satisfiesDLD + ADD + CD is a geometric progression mechanism.

• Getting for free:• credit from a depth k descendent >

credit from a depth k + 1 descendent– natural way to think of the credits

• unbounded rewards potential for every user– even if all degrees are bounded

Strategic split

• Another requirement we would have wanted:• u has no incentive to “split” into several buyers– replace a node with a subtree

• Under a geometric progression mechanism a user with a large reward gains from splitting

• CORROLARY: No rewarding mechanism satisfiesDLD + ADD + CD + SPLIT

Split: from one node to…

T1

T2 T3T4

T5

T6

T7

T8

several replica

T1c

T2 T3T4

T5

T6

T7

T8

Split and reward

Reward = 8x0.5=4

Reward Gain in a Split

Reward = 8x0.5+8x0.25-1.5 =4.5

Split and Weak Split

• In a split a node is replaced by several replica. • Each node in the replica buys the product, potentially from

another node in the replica.• Any buyer who bought from the split node will now buy

from one of its replica. • In a weak split the purchase will be only following the

promotion by the original node.

• Split proof [SP] mechanism – resistant to splits• Weak split proof [WSP} mechanism – resistant to weak splits

Dealing with weak split

• The geometric mechanism can be adapted to be weak split proof using the following idea:

the largest sub-tree of a node is ignored when computing the reward.

• Proof omitted.

Dealing with split

• Base reward mechanism: Given a tree T, originating in X, the reward to X is the maximal height of a perfect binary tree originating from X, embedded in the tree.

• Rsplit: given a tree originating from node X, the reward to X is according to the height of the best split he can do when the reward is according to the base mechanism.

• The above provides an SP mechanism (satisfying also budget constraint and unbounded potential reward, as needed).

• Details omitted/non-trivial proof.

A negative result

• A reward mechanism that satisfies WSP cannot guarantee a node a fraction 0 < α ≤ 1 of the reward of its least rewarded child.

Summary

• Goal: design a rewarding mechanism for social distribution of information goods

• Using an axiomatic approach• The geometric progression family of mechanisms

uniquely satisfies all three requirements• Dealing with splits (Sybil attacks).

Major Internet Applications Call for Help

• Ranking systems (e.g. page ranking systems, global reputation systems).

• Trust systems (e.g. personalized ranking based

on trust relationships). • Recommendation systems

• Appear everywhere (search, reputation mechanisms, social networks).

• No systematic rigorous mathematical evaluation.

Ranking, Trust, and Recommendation Systems

What is the right model / a good system?

From Major Internet ApplicationsTo Extended Social Choice Theory

• Question: • How do we evaluate existing systems, or find novel

good ones?

• Answer: • These settings can be viewed as extensions of classical

social choice. • Mathematical Social Choice offers the

axiomatic approach that can be adapted to the context of ranking/trust/recommendation systems, in order to overcome the challenge.

Incentives in recommendation systems –an axiomatic approach

Sell an information good over a social network

information good =

Invest %s of income in rewarding “good” buyers =credited for bringing many other buyers

can make a profit from their recommendations

Seller’s dream:

Summary

• Goal: design a rewarding mechanism for social distribution of information goods

• Using an axiomatic approach• The geometric progression family of mechanisms

uniquely satisfies all three requirements• Dealing with splits (Sybil attacks).