Reputation Systems: An Axiomatic Approach

1. Reputation Systems: An Axiomatic Approach Moshe Tennenholtz Technion—Israel Institute of Technology

2. The Internet: A Computational Social System The Internet allows several remarkably powerful capabilities: • Powerful search capabilities based on page ranking technology. • Reputation-based commerce adopting users ranking technology. The above are based on viewing the Internet as a computational social system, where computers/people/organizations provide input about one another. As a result, the theory of social choice may provide essential tools for understanding and improving upon these technologies.

3. Social Choice: Voters and Alternatives Alice Yahoo Bob M’Soft Chris Amazon

4. The Internet: Voters and Alternatives Coincide Yahoo Positive Reputation Systems: An important page is a page that important pages link to it. Amazon M’soft

5. The Basic Setup • G=(V,E) – a (positive) reputation system setting V – agents E  V2 -- set of positive feedbacks/links R(v)={u  V: (u,v)  E} – the supporters (support set) of v The social ranking S takes a graph G, and returns a ranking (total pre-order) S(G):V  {1,2,……,|V|} of its nodes.

6. Requirements • Classical social choice attempts to identify good social rules for the aggregation of individual preferences into a social preference, by introducing a set of postulates/axioms/requirements. • Classical requirements of the theory of social choice such as the independent of irrelevant alternative make no sense in our setting: the social ranking of agents based on individual rankings will change when new alternatives are added, since these alternatives are agents that may link to the previously existing alternatives/agents. • Google’s PageRank is a particular approach to aggregating individual preferences into a social preference in this setting!

7. Positive Reputation Systems: The importance relation Low rank = 5 R(David) is more important than R(Chris) Low rank = 5 Alice Bob Jon High rank=100 David Chris Low rank=5 Jeff

8. Positive Reputation Systems: The importance relation rank = 5 R(Chris) is more important than R(David) rank = 5 Alice Bob rank=2 Jon rank=5 Jane David Chris rank=5 Jeff

9. Positive Reputation Systems: The importance relation Given a reputation system setting G=(V,E), and a social ranking S(G), R(vi) is more important than R(vj) if there is a 1-1 mapping f: R(vj) R(vi) such that for every vR(vj) there exist f(v)  R(vi) such that v  f(v) and either f is not onto or there exist v  R(vj) such that v < f(v).

10. Positive Reputation Systems: Transitivity • Transitivity [T]: Given a positive reputation system setting G=(V,E) and a social ranking S(G), then for everyvi ,vj  V, if R(vi ) is more important thanR(vj)then vi >vj .

11. Positive Reputation Systems: Transitivity Low rank = 5 David should be ranked higher than Chris since his support is stronger Low rank = 5 Alice Bob M’Soft High rank=100 David Chris High rank=100 ? Amazon ?

12. Positive Reputation Systems: Beyond Transitivity Low rank = 5 Chris should not be ranked higher than David (but may be ranked similarly) since no one in Chris support is as strong as someone in David support. Low rank = 5 Alice Bob M’Soft High rank=100 David Chris ? ?

13. Positive Reputation Systems: Weak Monotonicity • Weak Monotonicity [M]: Given a positive reputation systems setting G=(V,E) and a social ranking S(G), then for everyvi ,vj  V, we have that if R(vi) is not more important thanR(vj )but vi >vj then it must be the case that there exist v1  R(vi) and v2  R(vj)such that v1 > v2.

14. Generality • Generality [G]: A positive reputation systems S should associate with any reputation system setting G a social ranking S(G).

15. An Impossibility Result • Theorem: there is no social reputation rule that satisfies G,T,M.

16. A Possibility Result • Theorem: We can satisfy any pair of the postulates G,T,M.

17. General Transitive Ranking • Iteration 0 – rank the nodes according to their in-degree. • Iteration I+1 refines the ranking of iteration I: A). Choose a node v, such that R(v) > R(t) and there is no s such that R(S) > R(v) [according to the rankings in iteration I, where s,v,t refer to nodes of the same rank in that iteration]. B). Refine the ranking, so that the nodes of rank of v in I will be partitioned into two: v and all nodes in its (previous) rank who have a support of the same power and the rest of nodes (including t) in its (previous) rank.

18. General Transitive Ranking 3(I) Kim Bob 2(I) 3(I) Jon Alice Mark 3 (I) 5(I) 5(I) Jeff Helen Jane 4(I) 6 (I)

19. General Transitive Ranking Notice that there always will be the case that the second lowest agent in the support of Alice (Jane) which is ranked higher than than the two lowest agents in the support of Bob (while the supports are of equal size), so we won’t get into cycles. 3 (I+1) Kim Bob 2(I+1) 3(I+1) Jon Alice Mark 3.5 (I+1) 5(I+1) 5(I+1) Jeff Helen Jane 4(I+1) 6 (I+1)

20. General Transitive Ranking Bob Chris 1 1 1 0 2 Jane Alice David

21. General Transitive Ranking Bob Chris 1 1 1.5 0 2 Jane Alice David

22. General Transitive Ranking Bob Chris 1.4 1 1.5 0 2 Jane Alice David

23. General Transitive Ranking Bob Chris 1.4 1.3 1.5 0 2 Jane Alice David

24. General Transitive Ranking Bob Chris 2 1 3 0 4 Jane Alice David

25. Negative Reputation Systems Alice Chris provides negative feedback about Bob, and Bob provides negative feedback about Alice. Bob Chris Ranking agents based on such information is the basis of reputation based commerce!

26. Negative Reputation Systems Alice Chris complains about Bob, and Bob Complains about Alice. Bob Chris Given a reputation system setting G=(V,E), and a social ranking S(G), R(vi) is more reliable than R(vj) if there is a 1-1 mapping such f: R(vj) R(vi) that for every v  R(vj) there exist f(v)  R(vi) such that v  f(v) and either f is not onto or there exist v  R(vj) such that v < f(v).

27. Negative Reputation Systems: Transitivity • B-Transitivity [BT]: Given a negative reputation systems setting G=(V,E) and a social ranking S(G), then for everyvi ,vj  V, if R(vi ) is more reliable thanR(vj )then vi <vj .

28. Negative Reputation System: Transitivity Alice No one complains about Chris, who should be ranked the highest. This means that Bob should be ranked the lowest. Alice will be ranked in between Bob and Chris. Bob Chris Chris > Alice > Bob

29. Negative Reputation Systems: Weak Monotonicity • B-Weak-Monotonicity [BM]: Given a negative reputation systems setting G=(V,E) and a social ranking S(G), then for everyvi ,vj  V, we have that if R(vi ) is not more reliable thanR(vj )but vi <vj then it must be the case that there exist v1  R(vi ) and v2  R(vj )such that v1 > v2.

30. Negative Reputation Systems: Weak Monotonicity Chris should not be ranked higher than David (but may be ranked similarly) since no complain about David is by someone as reliable as at least one of the agents who complain about Chris. High rank = 100 High rank = 100 Alice Bob Low rank=5 Jon Judith Low rank=5 David Chris ? ?

31. An Impossibility Result – Negative Reputation Systems • Theorem: there is no social reputation rule that satisfies G,BT,BM.

32. A Possibility Result – Negative Reputation Systems • Theorem: We can satisfy any pair of the postulates G,BT,BM.

33. Reputation Systems with both negative and positive feedbacks • Two types of edges/links – good and bad. • Rb(v) – the agents who provide negative feedback on v. • Rg(v) – the agents who provide positive feedback on v. • R(V) – the agents that link/point to V. R(vi) is socially stronger than R(vj) if Rb(vi) is less reliable or as reliable as Rb(vj), and Rg(vi) is more important or as important as Rg(vj), with at least one strict inequality.

34. Reputation Systems with both negative and positive feedbacks • Tc ---for everyvi ,vj  V, if R(vi ) is socially stronger thanR(vj ) then vi >vj . • Mc ---for everyvi ,vj  V, we have that if R(vi ) is not socially stronger thanR(vj )but vi >vj then it must be the case that there exist v1  Rg(vi ) and v2  Rg(vj )such that v1 > v2 or that there exist v3  Rb(vi ) and v4 Rb(vj )such that v3 < v4

35. Reputation Systems with both negative and positive feedbacks • Theorem: there is no social reputation rule that satisfies G,Tc,Mc. • Theorem: We can satisfy any pair of the postulates G,Tc,Mc.

36. Relaxing the axioms: strongly connected systems • One issue brought by practitioners is that it may be useful to restrict our attention to strongly connected graphs, where there is a directed path between any pair of nodes. • We refer to the related axiom as WG (“weak generality”).

37. Relaxing the axioms: very weak monotonicity • The complain against weak monotonicity is that vi might be preferable (in e.g. positive reputation systems) to vj although transitivity do not hold and there is no one that links to vi who is preferable to someone who links to vj , since the number of agents that link to vi is much larger than the number of agents that link to vj. • One (strong) relaxation is very-weak monotonicity (VWM): Given a positive reputation systems setting G=(V,E) and a social ranking S(G), then for every vi ,vj  V, where |R(vi)| |R(vj)|+1 we have that if R(vi ) is not more important thanR(vj )but vi >vj then it must be the case that there exist v1  R(vi ) and v2  R(vj )such that v1 > v2.

38. Relaxing the axioms • Theorem: There is no social reputation rule that satisfies WG,T,VWM. • Similar results can be obtained for negative reputation systems.

39. Further work • The approach presented is a normative one, but a complementary study deals with a descriptive approach, where sound and complete axiomatization is provided to known reputations systems. • In a pending paper Altman and Tennenholtz provide such (ordinal, graph-theoretic) representation to Google’s PageRank. • Other parts of study refer to agent incentives, and to the uniqueness of the ranking procedure.

40. Conclusion • We introduced an axiomatic study of reputation systems, adopting a social choice setting where the set of voters and the set of alternatives coincide. • We provided impossibility and possibility results for a variety of settings, including both positive and negative reputation systems.