The Axiomatic Approach and

the Internet

Moshe Tennenholtz

Work carried out at:

Microsoft Research and Technion

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.

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.

Ranking, Trust, and Recommendation Systems

What is the right model / a good system?

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.

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 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.

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.

Social Choice - Example

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

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.

Ranking Systems

• Systems in which agents’ ranks (votes) for each other are aggregated into a social ranking.

• Examples:

Page Ranking Reputation System

Ranking Systems

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

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.

Ranking systems

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.

Transitive Effects

The rank of your voters should affect your own.

Transitivity

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

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:

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.

Strong TransitivityDoesn't always apply

?

?

Strong Transitivity too Strong?

My Homepage Seminar Homepage=

Many Pages

Assume: =

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]

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.

?

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.

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.

Impossibility Proof – Part 1

c

a

a

b d

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.

Impossibility Proof – Part 2

c

a

a

b d

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

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

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.

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.

Strong Quasi-Transitivity

• Here purple is preferable to green:

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.

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.

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.

Intermediate Conclusions:Ranking systems

• Basic impossibility results reveal limitations• Positive results by weakening requirements• New systems derived from the axiomatic

study• Quantifying Incentive Compatibility

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.

Examples of PRSs/Trust Systems

s

b

a c

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

Distances, a=b, c=d, e=f

α-Ranks, b, a, d, c, f, e

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

Classification of PRSs

Consistency IncentiveCompatibility

StrongQuasi-Transitivity

α-Rank

StrongTransitivity

PersonalizedPageRank

distance

?

? ??

? ?

PRS Distance P. PageRank α-RankSelf Confidence YES for d>1/2 YESConsistency YES NO NOTransitivity strong quasi none strongIncentive Comp. strong none none

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.

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

?

? ??

? ?

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.

Consistency IncentiveCompatibility

StrongQuasi-Transitivity

α-Rank

StrongTransitivity

PersonalizedPageRank

Weak Maximum

Transitivity

Full Classification

rec.indegree

pathcount

* Artificial Ranking Systems

*

* **

strong ??

?

?

Intermediate ConclusionsTrust/personalized ranking systems:• Full characterization of the systems that are

transitive, consistent, and incentive compatible.

• New trust systems offered. • Impossibility results are reversed.

Trust-Based Recommendation Systems

(MSR Collaboration) :

JenniferChayes

ChristianBorgs

Reid Anderson

Uri Feige

Abie Flaxman

Adam Kalai

Vahab Mirrokni

Moshe Tennenholtz

Ranking, Trust, and Recommendation Systems

What is the right model / a good system?

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

A Model

• “Trust graph”– Node set N, one node per agent– Edge multiset E N2

• 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. Rs(N,E,V+,V–) to each nonvoter

+

– ++0 +

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

6. Trust Propogation

–0

+

+1

– +

s

+

– ?

u

v

THM: Axioms 1-6 are satisfied uniquely by random walk system.

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 ps = Pr[walk stops at + voter]

– Let qs = Pr[walk stops at – voter]

• Output rec. for s = + if ps > qs

0 if ps = qs

– if ps < qs

+

– ++0 +

+

–

+0

–

Axiomatization #2No 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 impliesthe min-cut system

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 +

Intermediate ConclusionsTrust-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.

Affiliate/Viral MarketingWork by

Microsoft Research, Herzlyia &

ILDC Innovations Labs

Slides based on:

Mechanisms for multi-level marketing

Y. Emek, R. Karidi, MT, A. Zohar

The settingSell 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:

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?”

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

uv

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

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.

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

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

Summing contributions

• [SC] sequence {ck}k≥1 s.t. each buyer contributes ck to its kth ancestor in T, k = 1, 2, …reward = sum (over all descendents) of these contributions

• LEMMA: SC DLD + ADDu21 32)r( ccu

Geometric progression

• 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

(Darpa Baloon Challenge, Pyramid Schemes, etc.)

• We’ll show the “flavor” of the direction

SC + CD Geometric progression

• Require rational rewards (and contributions)– ck = xk / yk (integers)

2111

111

11211

r

1r

Tcyxcc

xxcc

cyxccT

kkkk

kkk

kkkk

k

kk yx 1

T2

1k

11 kk yx

T1

SC + CD Geometric progression

112

111

111111 1

kkk

kkkkkk

kkkkkkkk

ccc

cyxcyx

cyxcccyxcc

T1 T2

CD: r(T’1) = r(T’2)

1k

kk yx 1

T’2

k

11 kk yx

T’1

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

Split: from one node to…u

T1

T2 T3T4

T5

T6

T7

T8

Several replica

In our work: two notions of Weaker & Stronger Split

𝑢1

T1

T2 T3

T4

T5

T6

T7

𝑢2

𝑢4

𝑢3 𝑢6

𝑢5

T8

Split and reward (Geometric Progression)

Reward =

Children

Parent

Reward Gain in a Split

Reward

Children to parent

Children to grandparent

Parent to Grandparent Cost to buy the

good again

Children

Parent

(Fake) Grandparent

u

Dealing With Splits:Base mechanism

h = height(maximal perfect binary subtree rooted at u)

r(u)=h

h

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

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.

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).

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.

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.

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.

Selection systems (k=2)

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.

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.

vu=vu FG'

FG

1=vu FG

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.

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.

Probabilistic Mechanism

• A probabilistic mechanism outputs a distribution on the k-tuples selected.

• The agents care about their expected utilities.

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.

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.

}1,.....,2,1{},,....,2,1{ nknN

RPk 3/1

)/1(13/1

kO

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.