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Reputation Systems: An Axiomatic Approach

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Reputation Systems: An Axiomatic Approach

Moshe Tennenholtz

Technion—Israel Institute of Technology

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.

Alice

Yahoo

Bob

M’Soft

Chris

Amazon

Yahoo

Positive Reputation Systems:

An important page is a page that important pages link to it.

Amazon

M’soft

- 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.

- 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!

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

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

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 vR(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).

- 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 .

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

?

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

?

?

- 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.

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

- Theorem: there is no social reputation rule that satisfies G,T,M.

- Theorem: We can satisfy any pair of the postulates G,T,M.

- 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.

3(I)

Kim

Bob

2(I)

3(I)

Jon

Alice

Mark

3 (I)

5(I)

5(I)

Jeff

Helen

Jane

4(I)

6 (I)

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)

Bob

Chris

1

1

1

0

2

Jane

Alice

David

Bob

Chris

1

1

1.5

0

2

Jane

Alice

David

Bob

Chris

1.4

1

1.5

0

2

Jane

Alice

David

Bob

Chris

1.4

1.3

1.5

0

2

Jane

Alice

David

Bob

Chris

2

1

3

0

4

Jane

Alice

David

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!

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).

- 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 .

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

- 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.

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

?

?

- Theorem: there is no social reputation rule that satisfies G,BT,BM.

- Theorem: We can satisfy any pair of the postulates G,BT,BM.

- 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.

- 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

- Theorem: there is no social reputation rule that satisfies G,Tc,Mc.
- Theorem: We can satisfy any pair of the postulates G,Tc,Mc.

- 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”).

- 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.

- Theorem: There is no social reputation rule that satisfies WG,T,VWM.
- Similar results can be obtained for negative reputation systems.

- 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.

- 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.