1. The Axiomatic Approach and the Internet Moshe Tennenholtz Work carried out at: Microsoft Research and Technion

2. Acknowledgements Research initiated in a UAI paper on the axiomatic approach to reputation systems. Initial work on Ranking Systems and Trust systems started in joint work with Alon Altman. Work on trust-based recommendation systems is with MSR-NE. Work on propagating distrust with MSR (Christian Borgs, Jennifer Chayes, Adam Kalai) Work with Ola Rozenfeld on consistent recommendation systems. Work on selection systems with Noga Alon, Felix Fischer and Ariel Procaccia. Work on social distribution / viral marketing carried out at MS Israel jointly with its innovation labs. Many thanks to the MSR, ATL, ASG teams at Microsoft ILDC.

3. 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 Viral marketing (e.g. affiliate marketing) Appear everywhere (search, reputation mechanisms, social networks). No systematic rigorous mathematical evaluation.

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

5. The “Big Data” Paradigm Online social systems are typically associated with large amount of data collected online, which leads to the ``big data'' approach: data mining, followed by suggestions, modifications and A/B testing. However, the abundance of available data does not suffice to come up with the right design for online systems in the first place or to explore the overwhelming number of possibilities. Available data is typically generated by the use of a particular system, and mining the data generated by users while interacting with one system does not provide a tool for exploring the overwhelming large design space, nor a way to deal with the user’s reaction to the change.

6. From Major Internet Applications To 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 and cooperative game theory. Mathematical Social Choice / cooperative game theory offer the axiomatic approach that can be adapted to the context of ranking/trust/recommendation/viral marketing systems in order to overcome the challenge.

7. Social Choice The classical social choice setting is comprised of: –A set of agents –A set of alternatives –A preference relation for each agent over the set of alternatives. A social welfare function is a mapping between the agents' individual preferences into a social ranking over the alternatives. The goal: produce “good” social welfare functions.

8. Social Choice - Example

9. The Axiomatic Approach We try to find basic properties (axioms) satisfied by social choice systems. Axioms are typically some “minimal requirements”, e.g. “if all voters prefer alternative A to alternative B then the society should prefer A to B”. Encompasses two distinct approaches: –The normative approach, in which we study sets of axioms that should be satisfied by a system; and –The descriptive approach, in which we devise a set of axioms that are uniquely satisfied by a given system

10. The Axiomatic Approach: A Celebrated Result Arrow's(1963) impossibility theorem is one of the most important results of the normative approach in Social Choice, and the basis for one of the most celebrated Nobel prizes in economics. It provides a set of natural requirements showing that only dictatorship (having a fixed voter ranking determine the society ranking) can satisfy them, which resulted in a huge number of papers suggesting weaker requirements and studying resulting systems.

11. Ranking Systems Systems in which agents’ ranks (votes) for each other are aggregated into a social ranking. Examples: Page Ranking Reputation System

12. Ranking Systems The design of ranking systems can be seen as a variation of the social choice problem where the agents and alternatives coincide.

13. Graph Ranking Systems Each agent may only make binary votes: only specify some subset of the agents as “good”. Preferences of all the agents may be represented as a graph, where the agents are the vertices and the votes are the edges. Applies e.g. for ranking WWW pages.

14. Ranking systems

15. The Axiomatic Approach Revisited The axioms studied in the huge literature on mathematical social choice do not make sense in ranking systems and our other applications due to the fact the set of voters and the set of alternatives coincide (and additional reasons). This requires to re-visit this basic theory, looking at new axioms, leading to novel conceptual and technical contributions.

16. Transitive Effects The rank of your voters should affect your own.

17. Transitivity An agent voted by two strong agents should be ranked higher than one voted by strong+weak

18. Transitivity An agent voted by two agents should be ranked lower than an agent who is voted by three agents two of which are ranked similarly to the voters of the former agent:

19. Strong Transitivity Formally, a ranking system F satisfies strong transitivity if for every two vertices x,y where F ranks x's predecessor set P(x) (strictly) weaker than P(y), then F ranks x (strictly) weaker than y. We define a predecessor set P(x) as being weaker than P(y) as the existence of a 1-1 mapping between P(x) and P(y) where every vertex in P(x) is mapped to a stronger or equal vertex in P(y) and at least one of the comparisons is strict, or the mapping is not onto.

20. Strong Transitivity Doesn't always apply

21. Strong Transitivity too Strong? My Homepage Seminar Homepage = Many Pages Assume: =

22. More about Transitivity Weak Transitivity –The idea: Only match predecessors with equal out- degree. –We assume nothing about predecessors of different out-degrees. –Otherwise, same as Strong Transitivity. PageRank satisfies Weak Transitivity but not Strong Transitivity. Strong Transitivity can be satisfied by a nontrivial Ranking System [Tennenholtz 2004]

23. Consistency Consider the statement: “An agent with votes from two weak agents should be ranked the same as one with a vote from one strong agent”. Such statement should hold / not hold in our ranking in a consistent way.

24. Consistency We would like such comparisons to be consistent. That is, in every profile such as the one described in the previous slide we should decide >/</= consistently.

25. Impossibility Theorem: There exists no general Ranking System that satisfies Weak Transitivity and Consistency. An interesting note: the impossibility reveals graphs for which “special treatment” has been provided in existing page ranking systems.

26. Impossibility Proof – Part 1 c a a bd a b c is weakest Assume b ≤ a a < b – Contradiction! → A vertex with two equal predecessors is stronger than one with one weaker and one stronger predecessor.

27. Impossibility Proof – Part 2 c a a bd a b c is strongest Assume a ≤ b b < a – Contradiction! → A vertex with two equal predecessors is weaker than one with one weaker and one stronger predecessor. → Contradiction to part 1. QED

28. Stronger Impossibility Results Our impossibility result exists even in very limited domains: –Small graphs (4 agents are enough with Strong Transitivity). –Strongly connected graphs –Bipartite (buyer/seller) graphs. –Single vote per agent

29. Positive Results Weak Transitivity is satisfied by Google's PageRank. Strong Transitivity can be satisfied by a nontrivial ranking system [Tennenholtz 2004] If we weaken our notion of transitivity in another way,we find a nontrivial ranking system satisfying consistency + strong quasi-transitivity.

30. Strong Quasi-Transitivity We define the notion of strong quasi-transitivity as requiring only non-strict comparisons, unless there is a 1-1 mapping from each vertex to a strictly stronger vertex.

31. Strong Quasi-Transitivity Here purple is preferable to green:

32. Positive Result Theorem: There exists a ranking system satisfying consistency and strong Quasi- Transitivity. The system is the recursive-indegree ranking system: This is a non-trivial system that is based on ranking first based on in-degree, and propagating this in a way which captures strong quasi transitivity, taking special care of cycles.

33. Incentives An agent may choose to cheat and not report its real preferences, in order to improve its position. A system is strongly incentive compatible if such manipulation is not beneficial.

34. Incentives We have classified several types of incentive compatible ranking systems, under a wide range of axioms. –This classification has shown that full incentive compatibility is impossible for any practical purpose. We have also quantified the incentive compatibility of known ranking systems, and suggested useful new ranking systems that are almost incentive compatible. PageRank for example is highly manipulable while some of the systems we have shown have much better resistance.

35. Intermediate Conclusions: Ranking systems Basic impossibility results reveal limitations Positive results by weakening requirements New systems derived from the axiomatic study Quantifying Incentive Compatibility

36. Trust Systems: Personalized Ranking Systems The “client” of the ranking system may also be a participant. A trust/personalized ranking system is like a general ranking system, except: –Additional parameter: the source, i.e. the agent under whose perspective we're ranking –The input graph now is viewed as a “web of trust” in which agents are ranked from the perspective of a particular agent based on trust relationships.

37. Examples of PRSs/Trust Systems s b ac d e f Personalized PageRank (d=0.2) d, s, f, b, a, c, e (d=0.5) s, b, a, d, f, c, e Distance s, a=b, c=d, e=f α -Rank s, b, a, d, c, f, e

38. Properties of PRSs/Trust Systems A PRS satisfies self-confidence if the source s is ranked stronger than all other vertices. The following properties from general ranking systems could be adapted to PRSs. –Strong/Quasi/Weak transitivity –Consistency –Strong Incentive Compatibility

39. Classification of PRSs Consistency Incentive Compatibility Strong Quasi- Transitivity α -Rank Strong Transitivity Personalized PageRank distance ? ? ? ? ??

40. Strong Count Systems Assume the source is the strongest x will be ranked higher than y when the strongest predecessor of x is stronger than the strongest predecessor of y. If the strongest predecessors of x and y are of equal power, then the ranking of x and y is determined by a fixed non-decreasing function from the number of strongest predecessors to the integers.

41. Classification Theorem Theorem: F is a PRS that satisfies self confidence, strong quasi transitivity, consistency and strong incentive compatibility, if and only if F is a strong count system. Consistency Incentive Compat. Str. Quasi- Transitivity Strong ? ? ? ? ??

42. Relaxing the Axioms All axioms are required for the previous result. There is an artificial system that satisfies all axioms excluding self confidence, and an artificial system that satisfies all axioms excluding strong quasi-transitivity. The Path Count PRS ranks vertices based on distance, and break ties based on the number of shortest directed paths each vertex has from the source; it satisfies all axioms excluding consistency. The recursive in-degree ranking system can be adapted to the PRSs/trust systems setting by giving the source vertex a maximal value; it satisfies all axioms excluding incentive compatibility.

43. Consistency Incentive Compatibility Strong Quasi- Transitivity α -Rank Strong Transitivity Personalized PageRank Weak Maximum Transitivity Full Classification rec. indegree path count * Artificial Ranking Systems * * * * strong ?? ? ?

44. Intermediate Conclusions Trust/personalized ranking systems: Full characterization of the systems that are transitive, consistent, and incentive compatible. New trust systems offered. Impossibility results are reversed.

45. Trust-Based Recommendation Systems (MSR Collaboration) : Jennifer Chayes Christian Borgs Reid Anderson Uri Feige Abie Flaxman Adam Kalai Vahab Mirrokni Moshe Tennenholtz

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

47. Old-Fashioned Model I want a recommendation about an item, e.g., –Professor –Product –Service –Restaurant –… I ask my trusted friends –Some have a priori opinions (first-hand experience) –Others ask their friends, and so on I form my own opinion based on feedback, which I may pass on to others as a recommendation

48. A Model “Trust graph” –Node set N, one node per agent –Edge multiset E N 2 Edge from u to v means “u trusts v” Multiple parallel edges indicate more trust Votes: disjoint V +, V – N –V + is set of agents that like the item –V – is set of agents that dislike the item Rec. system (software) assigns {–,0,+} rec. R s (N,E,V +,V – ) to each nonvoter + – + + 0 +

49. Axiomatization #1: Few Axioms Characterizing The Random Walk system 1. Symmetry 2. Positive response 3. Scale invariance (edge repl.) 4. Independence of Irrelevant Stuff 5. Consensus nodes Trust Propogation 6. Trust Propogation – 0 + + 1 – + s + – ? u v  THM: Axioms 1-6 are satisfied uniquely by random walk system.

50. Random Walk System Input: voting network, source (nonvoter) s. –Consider hypothetical random walk: Start at s Follow random edges Stop when you reach a voter –Let p s = Pr[walk stops at + voter] –Let q s = Pr[walk stops at – voter] Output rec. for s = + if p s > q s 0 if p s = q s – if p s < q s + – + + 0 + + – + 0 –

51. Axiomatization #2 No Groupthink Axiom If a set S of nonvoters are all + rec’s, then a majority of the edges from S to N \ S are to + voters or + rec’s If a set S of nonvoters are all – or 0 rec’s, then it cannot be that a majority of the edges from S to N \ S are to + voters or + rec’s + + + (and symmetric – conditions) – – – THM : The “No groupthink” axiom on undirected graphs uniquely implies the min-cut system

52. Min-Cut-System (Undirected graphs only) Def: A +cut is a subset of edges that, when removed, leaves no path between –/+ voters Def: A min+cut is a cut of minimal size The rec for node s is: + if in every min+cut s is connected to a + voter, – if in every min+cut s is connected to a – voter, 0 otherwise + – + + 0 +

53. Intermediate Conclusions Trust-Based Recommendation Systems Simple model of trust-based rec systems –Simplify matters by rating one item (at a time) –Generalizes to real-valued weights, votes & rec’s Two axiomatizations leading to unique systems –Random walk system for directed graphs –Min-cut system for undirected graphs Some additional work: Nice systems/axioms in the multi-valued/continuous case. Dealing with distrust as well.

54. Affiliate/Viral Marketing Work by Microsoft Research, Herzlyia & ILDC Innovations Labs Slides based on: Mechanisms for multi-level marketing Y. Emek, R. Karidi, MT, A. Zohar

55. 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:

56. Possible implementation: identified links Upon buying the product, user u obtains a link to 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?”

57. 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 u v

58. 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 buyer receive? –Depends on the purchasing tree Looking for a rewarding mechanism: –easy to implement –robust –buyers have an incentive to promote a good product

59. What’s a reasonable mechanism? Sub-tree constraint: Reward depends on descendents; not on position in T Otherwise may wait for a “better offer” (to have a better position in T) instead of buying the product right away (failure of e.g. Shapley value). 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.

60. An axiomatic approach [DLD] Depth Level Dependence: Really matters: #descendants in each depth level –#descendants uniquely determines r(u) [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

61. The characterization 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

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

63. Geometric progression LEMMA: SC + CD  {c k } k≥1 is a geometric progression: –parameters 0 0 s.t. a  b ≤ 1 – a –c k = a k  b (Darpa Baloon Challenge, Pyramid Schemes, etc.) We’ll show the “flavor” of the  direction

64. SC + CD  Geometric progression Require rational rewards (and contributions) –c k = x k / y k (integers) T2T2 T1T1

65. SC + CD  Geometric progression T1T1 T2T2 CD: r(T’ 1 ) = r(T’ 2 ) T’ 2 T’ 1

66. Strategic Splits (Sybil Attacks) Users may potentially purchase the product via their own links possibly using false identities. This can potentially increase their rewards. A desirable property from mechanisms: resilience to splits

67. Split: from one node to… u T1 T2 T3 T4 T5 T6 T7T7 T8

68. Several replica In our work: two notions of Weaker & Stronger Split T1 T2 T3 T4 T5 T6 T7T7 T8

69. Split and reward (Geometric Progression) Children Parent

70. Reward Gain in a Split Children to parent Children to grandparent Parent to Grandparent Cost to buy the good again Children Parent (Fake) Grandparent

71. u Dealing With Splits: Base mechanism h = height(maximal perfect binary subtree rooted at u) r(u)=h h

72. Split-Proof Mechanism Clearly Split-Proof (mechanism does the work for you) The surprising part: This mechanism does not exceed the budget. A more natural mechanism? Split node u optimally (max reward) Reward u (and its replicas) according to the base mechanism

73. A negative result A reward mechanism that is Weak-Split- Proof cannot guarantee a node a constant fraction 0 < α ≤ 1 of the reward of its least rewarded child.

74. Intermediate Conclusions We designed reward mechanisms for social distribution of information goods. An axiomatic approach is yet again applicable! Characterized geometric progression mechanisms Dealing with splits (Sybil attacks).

75. Conclusion We believe that the axiomatic approach is the way to analyze ranking, trust, and recommendation, and social distribution systems. Novel conceptual contributions: emerge from social choice when the set of agents and the set of alternatives may coincide (and more). Novel technical contributions: a rigorous approach to understanding the properties of existing systems, while offering also new novel systems.

76. Replacing Axioms by Approximating Desired Systems A related/complementary approach is to define the output of a particular system as the desired one, and try to approximate it using an incentive compatible system. We illustrate it using the study of selection systems. Joint work with Noga Alon, Felix Fischer, and Ariel Procaccia.

77. Selection Systems The system selects k nodes (agents, traders, pages). Can be viewed as a ranking system of two levels – selected and non-selected participants.

78. Selection systems (k=2)

79. The In-degree k-Selection System: A natural and popular objective: select a set of k nodes that maximize the sum of in- degrees. Variants are applicable to tweeter popularity on one side and to survivor-like competitions on the other side, but also to internet-based reputation systems.

80. Incentive Compatibility Let G=(V,E) and G'=(V,E') be graphs that differ only in the outgoing edges from vertex v. A selection system is F incentive compatible, if for any such graphs G and G’ and vertex v, where if v is selected, and 0 otherwise.

81. Approximation Ratios A selection has an approximation ratio of l, if the sum of in-degrees in it is at most l times lower than in an optimal selection. We are interested in finding incentive compatible selection systems which are approximately optimal for some small l.

82. A general impossibility result For any k < n, there is no incentive compatible selection systems that give any finite approximation ratio. That is, you can not guarantee to always select in non-empty graphs an agent that someone votes for it.

83. Probabilistic Mechanism A probabilistic mechanism outputs a distribution on the k-tuples selected. The agents care about their expected utilities.

84. The Random m-Partition (m-RP) Mechanism Broadly speaking, the mechanism works as follows: The agents are randomly partitioned into m subsets; we then select the (roughly) k/m agents with largest in-degree from each subset, where only the incoming edges from the other subsets are taken into account.

85. The Random m-Partition (m-RP) Mechanism Theorem : Let For every value of m, m-RP is incentive compatible. Furthermore, 1.2-RP has an approximation ratio of 4. 2. has an approximation ratio of So, when we have to select many candidates (among any number) we get almost 1.

86. Conclusion We believe that the axiomatic approach is the way to analyze ranking, trust, and recommendation systems. Novel conceptual contributions: social choice when the set of agents and the set of alternatives may coincide (and more). Novel technical contributions: a rigorous approach to understanding the properties of existing systems, while offering also new novel systems.