computational social systems: reputation and non-cooperative computation

Computational Social Systems: Reputation and Non-Cooperative

Computation

Moshe Tennenholtz Technion

Acknowledgements

Many thanks to Dov Monderer for many discussions on Internet reputation systems, and to Yoav Shoham and Rann Smorodinsky for joint work on multi-party computation games.

The part of talk dealing with sequential information elicitation is a joint work with Rann Smorodinsky.

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

Information aggregation and elicitation by brands based on voting technology

The above are based on viewing the Internet as a computational social system, where computers/people/organizations provide input about one another or about product/service features. As a result, the theory of social choice and game-theory may provide essential tools for understanding and improving upon these technologies.

Social Choice: Voters and Alternatives

Alice

Bob

Chris

Yahoo

M’Soft

Amazon

The Internet: Voters and Alternatives Coincide

Yahoo

M’softAmazon

Positive Reputation Systems:An important page is a page that important pages link to it.

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.

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!

Positive Reputation Systems: The importance relation

Jon

Jeff

AliceBob

Chris

David

High rank=100

Low rank=5

Low rank = 5

Low rank = 5 R(David) is more important than R(Chris)


Jon

Jeff

AliceBob

Chris

David

rank=5

rank=5

rank = 5

rank = 5 R(Chris) is more important than R(David)

Jane

rank=2


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).

Positive Reputation Systems: Transitivity

Transitivity [T]: Given a positive reputation systems setting G=(V,E) and a social ranking

S(G), then for every vi , vj V, if R(vi) is more important than R(vj) then vi > vj .

Positive Reputation Systems: Transitivity

M’Soft

Amazon

AliceBob

Chris

David

High rank=100

High rank=100

??

Low rank = 5

Low rank = 5 David should be ranked higher than Chris since his support is stronger

Positive Reputation Systems: Beyond Transitivity

M’Soft

AliceBob

Chris

David

High rank=100

??

Low rank = 5

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.

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 every vi , vj V, we have that if R(vi ) is not more important than R(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

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

An Impossibility Result

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

A Possibility Result

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

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.


Jon

Jane

Alice

Bob

Jeff Helen

Kim

Mark

2(I)

4(I)

5(I)3 (I)

6 (I)

5(I)

3(I)

3(I)


Jon

Jane

Alice

Bob

Jeff Helen

Kim

Mark

2(I+1)

4(I+1)

5(I+1)3.5(I+1)

6 (I+1)

5(I+1)

3(I+1)

3 (I+1)

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.


Bob

Jane

Chris

Alice David

2 01

1 1


Bob

Jane

Chris

Alice David

2 01.5

1 1


Bob

Jane

Chris

Alice David

2 01.5

1.4 1


Bob

Jane

Chris

Alice David

2 01.5

1.4 1.3


Bob

Jane

Chris

Alice David

03

2 1

4

Negative Reputation Systems

Alice

ChrisBob

Chris provides negative feedback about Bob, and Bob provides negative feedback about Alice.

Ranking agents based on such information is the basis of reputation based commerce!

Negative Reputation Systems

Alice

ChrisBob

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 f: R(vj) R(vi) such that 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).

Chris complains about Bob, and Bob complains about Alice.

Negative Reputation Systems: Transitivity

B-Transitivity [BT]: Given a negative reputation systems setting G=(V,E) and a social ranking

S(G), then for every vi , vj V, if R(vi ) is more reliable than R(vj ) then vi < vj .

Negative Reputation System: Transitivity

Alice

ChrisBob

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.

Chris > Alice > Bob

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 every vi , vj V, we have that if R(vi ) is not more reliable than R(vj )

but vi < vj then it must be the case that there exist v1 R(vi ) and v2 R(vj ) such that v1 > v2.

Negative Reputation Systems: Weak Monotonicity

Judith

AliceBob

Chris

David

Low rank=5

??

High rank = 100High rank = 100

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.

JonLow rank=5

An Impossibility Result – Negative Reputation Systems

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

A Possibility Result – Negative Reputation Systems

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

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.


Tc --- for every vi , vj V, if R(vi ) is socially stronger than R(vj ) then vi > vj .

Mc --- for every vi , vj V, we have that if R(vi ) is not socially stronger than R(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.

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”).

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 than

R(vj ) but vi > vj then it must be the case that there exist

v1 R(vi ) and v2 R(vj ) such that v1 > v2.

Relaxing the axioms

Theorem: There is no social reputation rule that satisfies WG,T,VWM.

Similar results can be obtained for negative reputation systems.

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.

Conclusion (reputation systems)

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.

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

Information aggregation and elicitation by brands based on voting technology

The above are based on viewing the Internet as a computational social system, where computers/people/organizations provide input about one another or about product/service features. As a result, the theory of social choice and game-theory may provide essential tools for understanding and improving upon these technologies.

Information Aggregation and Elicitation: Motivation

Voting about product or service features is a most popular tool in Internet sites of brand name companies, and widely exploit the power of the Internet.

When visiting a brand’s web-site a participant may be asked to learn about a new product/service feature and vote for or against it.

Visitors are typically interested in learning the public’s opinion but might not be willing to spend the time learning the required information, challenging very popular market research tools.

Example 1: should we offer the position?

A candidate to the economics department has already written 11 research papers, and the department would like to decide on whether to make her a job offer based on the quality of the papers.

There are 11 agents (committee members) who are each given one paper to read in order to make a recommendation.

Initially, each paper may be "good" or "bad" with equal probabilities, and the department has chosen to make an offer to a candidate if he has a majority of "good" papers.

Committee member values the correct recommendation of the committee, at 1000 USD to him, but values the time he needs to spend on reading the paper at 400 USD.


A simple mechanism asks all the agents, simultaneously, for their recommendations.

The strategy tuple where all agents choose to read the papers and report truthfully is not an equilibrium.

Consider the perspective of agent 1: assuming all agents replied (truthfully, or not), then agent 1 can alter the outcomeonly if the other 10 replies split evenly between 0 and 1 which has a probability of approximately 0.25.

Therefore, by guessing, and assuming all other agents compute, he will gain 0.25 X 500 + 0.75 X 1000 = 875 USD.

However, by computing an agent gains at most 1000-400=600 USD, and so player 1 has no incentive to compute (the same for all 11 agents).


This elicitation mechanism will also fail if only agent 1 has the above cost and all other agents have zero costs (the same analysis will hold for agent 1).

If however agents 2,3,….,11 are asked first for their recommendations, and agent 1 is approached only if there is a tie among the ten recommendations, then all agents will have enough incentive to invest the effort!

This illustrates the power of sequential mechanisms.

Model

N = {1, 2,…, n} -- a finite set of agents.

Each agent j has a unique secret sj {0,1} that he may compute.

Let 0.5 q <1 be the prior probability of sj=1 and assume these events are independent (the results apply for all 0<q<1).

Agents may compute their own secrets. However, computation is costly and agent j pays cj 0 for computing sj.

Assume wlog that c1 c2 ……. cn

Model (Cont.)

Agents are interested in computing some joint binary parameter (e.g., the majority vote or whether they have a consensus) that depends on the vector of private inputs, defined by an anonymous function

G: {0,1}n {0,1}.

Agent j has a utility of vj from learning the real value of G.

We will assume that vj > cj

We use wlog the convention that vj=1 (the more general case

is equivalent to the case where the value of agent j is 1 but the cost is cj/vj).

Sequential Mechanisms

Hi= {0,1}i -- the set of histories of length i H0 = null/empty history H = H0 H1 …… Hn

A sequential mechanism is a pair (g,f) where g: H N determines the agent to be approached, and f:H {0,1,*} is a function that expresses a decision about whether to halt and output either 0 or 1, or continue the elicitation process. We assume that if g(h)=j then g(h') j for every h' where h is a prefix of h‘.

Strategies and Equilibrium

= {don't compute and submit 0, don't compute and submit 1, compute and submit 0, compute and submit 1, compute and if 0 submit 0 and else submit 1, compute and if 1 submit 0 and else submit 1}

A pure strategy for agent j, xj:H

An equilibrium for the mechanism A = (f,g) is a vector of n strategies, one for each agent, such that each agent's strategy is the best response against the other agents' strategies.

A mechanism A is appropriate for G at 0.5 q<1 if there exists an equilibrium where G can surely be computed for all vector of agents' secrets.

Such an equilibrium is referred to as a computing equilibrium, and A is call q-appropriate.

The High Cost First (HCF) algorithmStep One - For each possible prefix, i.e. string of 0 and 1's, of

length in between 0 and n-1, compute the probability of being pivotal, conditional on agents being truthful. By knowing this probability we will have an upper bound on the cost of an agent that is expected to compute in equilibrium.

Step Two - Let us denote by Zj the event that j is pivotal, and consider the following recursive structure: For any given set of agents and costs choose the agent to move first as follows – choose agent j of maximal cost that still satisfies that 1-cj Prob(Zj) X q+P(Zj

c) X 1 Notice that this computation for an anonymous function is polynomial.

The High Cost First (HCF) algorithm (Cont.)

Use the procedure to allocate the first agent.

Depending on the reply of the first agent you end up with one of two trees. Apply the same procedure again to each tree, where at each time you are not allowed to allocate agents that have been already

allocated, and so on and so forth.

Theorem: Let G be an anonymous function. Assume it is common knowledge that there exists an appropriate mechanism for G, then the mechanism induced by the HCF algorithm has a computing equilibrium.

Notice that the HCF allows to make only polynomial on-line computation to determine at each point the next agent to be approached, if an appropriate mechanism exists.

A graph-theoretic representation Let G be an arbitrary anonymous function. Consider the graph G'=(V,E):

Let G=(V,E) be the graph induced by reducing G' to include only nodes where the value of the function G is still unknown.

\ where the set of nodes is$V=\{(i,k): i,k \in Z_{+}, 0 \leq k \leq i \leq n\}$, and the set of edges is $E=\{((i,k),(i+1,k)): 0 \leq i \leqn-1, k\leq i \} \cup \{((i,k),(i+1,k+1)): 0 \leq i \leq n-1, k\leq i \}$. Intuitively, this graph describes thepossible states of information, when approaching agents while computing the value of an anonymous

function, and the possible state transitions. Every state describes how many agents have been approached and how

many 1's havebeen heard so far. In particular, the node $v=(j,l)$ in $G'$ is interpreted as the event that $j$ agents

have beenapproached and $l$ 1's (and $j-l$ 0's) have been reported.

For each such node we may ask whether the value of $G$can be computed, or not.. Notice that $\bar{G}$ is a DAG.

(0,0)

(1,0) (1,1)

(2,1)(2,0) (2,2)

The figure refers to the case n=2. The pair (i,j) refers to the fact i agents have revealed their bits, j of them declared the bit 1).

An Existence Criterion

Theorem: Let G=(V,E) be the directed graph associated with an anonymous function G, 0.5 q < 1,

and let c(v) for any v V be defined as the largest i, for which 1-ci Prob(Zi|v) X q+ Prob(Zi

c|v) where Zi is the event that agent i is pivotal for the function G.

Then, there exists a q-appropriate mechanism iff there does not exist a path in G originating from (0,0) and leading to some end-node v (v1=(0,0), v2,……, vl=v), for which there exists some 1 i n such that |{vj:c(vj) i}|>i.

Verifying Existence is Polynomial

Theorem: Let G be an anonymous function, and consider a set of n agents, with costs c1,….,cn, and a parameter 0< q < 1, then there exists a polynomial algorithm for checking the existence of an appropriate mechanism

The results complement one another:the above theorem tells that we can efficiently check whether there is an appropriate mechanism,and the HCF algorithm tells us how to construct in an efficient manner the desired behavior if an appropriate mechanism exists.

Sequential Information Elicitation: Additional Topics

The study of information elicitation with private communication channels.

A characterization of appropriate mechanisms for the more general setting.

The study of probabilistic mechanisms. A general theory of non-cooperative

computing, where the agents’ incentives may refer not only to their costs, but to externalities such as the desire that other agents will not know the final outcome.

Conclusion (information elicitation)

Information elicitation in multi-agent systems can be viewed as a form of a multi-party computation game.

Sequential elicitation is helpful. There exists an efficient algorithm for checking the

existence of an appropriate sequential elicitation mechanism, which relies on a necessary and sufficient existence criterion we provide.

There exists an algorithm that allows to (on-line) efficiently generate the actions of an appropriate sequential elicitation mechanism, if such a mechanism exists.

Privacy and probabilistic mechanisms are helpful.

Conclusion

New remarkable technologies and tools have been introduced by Internet companies, and the Internet has become a useful tool for both brands and personal users.

Computational social systems can and should be tackled from social choice and game-theoretic perspectives.

New problems need to be tackled, leading to influential and powerful theories, bridging the gap between CS/AI and game theory: reputation systems and non-cooperative computation are appropriate examples.

Private Communication Channels

Partial information is modelled by having mechanisms where the information provided is also a function of the history.

Theorem: In a setting with private channels (where the elicitation protocol specifies the information to be communicated), if there is an appropriate mechanism then there is also a yes/no appropriate mechanism, where the information provided is whether the function can be already computed or not.

Proposition: There exist anonymous functions, ci's and q, for which there does not exist an appropriate mechanism in the full information setting, but there exists an appropriate mechanism in the partial information setting.

Probabilistic Mechansims When we consider private channels there may be also a role to

probabilistic mechanisms.

Proposition: There exists an anonymous function, ci ‘s and q, for which there does not exist an appropriate deterministic mechanism in the partial information setting, but there exists an appropriate probabilistic mechanism.

Theorem: If all the ci ‘s are equal and there exists an appropriate deterministic mechanism then the mechanism where all agents are approached simultaneously is an appropriate mechanism.

The result showing that probabilistic mechanisms help remains valid also when we require all ci ‘s to be identical. This also shows that sequential mechanisms are helpful even when all the ci‘s are identical.

A More General Setting: Non-Cooperative Computation:

A set of agents (e.g. computers serving different Internet users/sites), each of which holds a piece of private information (e.g. whether they support offering a job to Jon or not).

A function of the private secrets which is of interest (e.g. what does the majority think?)

A reliable center who offers and run an elicitation procedure

Alice Chris David

AA

Bob

A reliable center runs a mechanism

S(A)=1 S(B)=0S(C)=1

S(D)=1

A More General Setting: Non-Cooperative Computation

In non-cooperative computation the agents are strategic, and the aim is to devise mechanisms that will allow the computation of the function in equilibrium. Particular cases:

1. Free riding (discussed in this talk) Accessing the private information may be

costly, and each agent may prefer not doing it, if the fact the other do so may be good enough.

2. Externalities Each agent knows his secret, and is interested in knowing the function’s value (when applied to the agents’ secretes), but has a secondary objective of the others not knowing this value.

A full characterization of the boolean functionthat can be non-cooperatively computable in thatsetting has been provided by Shoham and Tennenholtz

computational social systems: reputation and non-cooperative computation

Documents

internet reputation

computational social

social preference

social ranking sg

social ranking of agents

low rank

theory of social choice

v agents e v2