Mechanism for Multi-Level Marketing

1. 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 and Microsoft ILDC Innovation Labs

2. The setting Sell an information good over a social network information good = Seller’s dream: Invest %s of income in rewarding “good” buyers =credited for bringing many other buyers can make a profit from their recommendations

3. 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 clickingu’s link • new links are generated • The purchasing site keeps track of:“who clicked whose link?”

4. The purchasing tree T • The induced purchasing tree: • root = purchasing site (or the seller) • node v is a child of node u if user vbought the product by clicking user u’s link • u should be credited for bringing her descendents in T v u

5. 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

6. 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 Tdoesn’t affect r(u) • r(u) is uniquely determined by Tu

7. 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

8. 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.

9. 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

10. The characterization

11. 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’srewards u

12. Summing contributions • [SC] sequence {ck}k≥1 s.t. each new buyer contributesck to its kth ancestor in T, k = 1, 2, …reward = sum (over all descendents) of these contributions • LEMMA: SC  DLD + ADD u

13. SC  DLD + ADD New buyer contributes ck to its kth ancestor • Direction  trivial • f(d) = reward of a node with dklevel 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

14. 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+) – ckwhere 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)

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

16. f(d) = c1 d1 +  + ck dk + g’(d) Fix ck= g(1k) and g’(d) = g(1k d+) – ck , where d+ = {di}i>k

17. 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) = ia(r(vi) + b)

18. 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 T1 T2

19. SC + CD  {ck}k≥1 T’1 T’2 T1 T2 CD: r(T’1) = r(T’2)

20. 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

21. 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

22. Split: from one node to… T1 T2 T3 T4 T5 T6 T7 T8

23. several replica T4 T5 T6 T7 T8 T1c T2 T3

24. Split and reward Reward = 8x0.5=4

25. Reward Gain in a Split Reward = 8x0.5+8x0.25-1.5 =4.5

26. 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

27. 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.

28. 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.

29. 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.

30. 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).

31. 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.

32. Ranking, Trust, and Recommendation Systems What is the right model / a good system?

33. 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.

34. Incentives in recommendation systems –an axiomatic approach Sell an information good over a social network information good = Seller’s dream: Invest %s of income in rewarding “good” buyers =credited for bringing many other buyers can make a profit from their recommendations

35. 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).