An Ordinal Banzhaf Index for Social Ranking

Hossein Khani, Stefano Moretti and Meltem Öztürk

CNRS UMR7243 - LAMSADE, Université Paris-Dauphine

IJCAI 2019- Macao, China

Social ranking problem

Power index :

Society : Company, Parliament,...

Power index indicates the in-fluence of individuals in the so-ciety, it can be cardinal or ordinal.

� ∼

Power Relation (Binary relation)

� ∼

Power Relation (Binary relation)

Social ranking solution\ Axiomatization

Total preorder

Objective

Input :I A set of individuals : N = {1, . . . ,n}I A power relation � on 2N :

S � T : The “team” S performs at least as good as T .

We suppose �∈ B(2N), set of all binary relations.Output :

I A solution R�(I� the symmetric part, P� the strict part), associates toevery power relation (�) a ranking (total preorder) over theset of individuals.

Ordinal Banzhaf solution : Motivation

I The Banzhaf value β(v) of TU-game (N, v) is the n-vectorβ(v) = (β1(v), β2(v), . . . , βn(v)), such that for each i ∈ N :

βi(v) =1

2n−1

∑S∈2N ,i /∈S

(v(S ∪ {i})− v(S)

I For a cooperative game ({ }, v) in order to rank any

two individuals and we have :, ,

Motivation : Robustness

1 − ε1 + ε

v2 v15/23

Motivation : Robustness

1 − ε1 + ε

ranked higher than

Solutions : Ordinal Banzhaf Solution

Ordinal Banzhaf rule :

Informative part : Marginalistic comparisons

Ordinal Marginal Contribution

Ordinal Marginal contribution of Red w.r.t

Ordinal Marginal Contribution

Ordinal Marginal contribution of Girl w.r.t

Banzhaf Score

I To compute the Banzhaf Score of individual Red wecompute the difference between the times its ordinalmarginal contribution is positive and number of times it isnegative.

Banzhaf Score

I To compute the Banzhaf Score of individual Red wecompute the difference between the times its ordinalmarginal contribution is positive and number of times it isnegative.

3− 1 = 29/23

Ordinal Banzhad solution

Banzhaf Scores :

PBanzhaf Scores :

Definition (Ordinal Banzhaf relation)

Let �∈ B(2N) and A ⊆ N. The ordinal Banzhaf relation is thebinary relation R̂�A ⊆ A× A such that for all i , j ∈ A :

iR̂�A j ⇔ s�i ≥ s�j .

I From the definition it immediately follows that the rankingresult is always transitive.

Axiomatization

Ordinal Banzhaf solution :

I Coalitional Anonymity

I Neutrality

I Monotonicity

Coalitional Anonymity

R� Rw

Definition (Coalitional Anonymity(CA))

Let A ⊆ N. A solution RA : C ⊆ B(2N)→ T (A) satisfies thecoalitional anonymity axiom on C if and only if for all powerrelations �,w∈ C, for all players i , j ∈ A and bijectionsπi : Ui → Ui and πj : Uj → Uj such thatS ∪ {i} � S ⇔ πi(S) ∪ {i} w πi(S) for all S ∈ Ui andS ∪ {j} � S ⇔ πj(S) ∪ {j} w πj(S) for all S ∈ Uj , then it holdsthat iR�A j ⇔ iRwA j .

In this definition :I Ui = {S ∈ 2N s.t i /∈ S}.

Neutrality

R� Rw

Neutrality(Ordinal Banzhaf)

Definition (Neutrality (N))

Let A ⊆ N. A solution RA : C ⊆ B(2N)→ T (A) satisfies theneutrality axiom on C if and only if for all power relations�,w∈ C and each bijection σ : N → N such that σ(A) = A andS � T ⇔ σ(S) w σ(T ) for all S,T ∈ 2N , then it holds thatiR�A j ⇔ σ(i)RwA σ(j) for every i , j ∈ A.

In this definition :I σ : N → N s.t

S = {i , j , k , ..., t} ⊆ N ⇒ σ(S) = {σ(i), σ(j), σ(k), ..., σ(t)}.

Monotonicity

R� Pw

Monotonicity

Definition (Monotonicity (M))

Let A ⊆ N. A solution RA : C ⊆ B(2N)→ T (A) satisfies themonotonicity axiom on C if and only if for all power relations�,w∈ C and i , j ∈ A such that :I there exists a coalition S ∈ Ui such that S � S ∪ i and

S ∪ i A S, andI T ∪ i � T ⇔ T ∪ i A T and T ∪ j � T ⇔ T ∪ j A T for all

the other coalitions T ∈ 2N ,T 6= S,then it holds that iR�A j ⇒ iPwA j .

In this definition :I Ui = {S ∈ 2N s.t i /∈ S}.

Characterization : Ordinal Banzhaf Solution

I For the set of all linear orders we have proved that :

Theorem

Let A ⊆ N. A solution RA : A(2N)→ T (A) is the ordinalBanzhaf solution if and only if it satisfies the three axioms CA,N and M on A(2N).

Ceteris Paribus Majority rule(IJCAI18)

Family of weighted majority rules

I Ceteris Paribus majority rule and Ordinal Banzhaf solutionbelong to the same family of weighted majority rules.

Ceteris Paribus Majority :

P21/23

Ordinal Banzhaf Solution Banzhaf Distance

P21/23

Family of weighted majority relations

Definition (Weighted majority relation)

Let �∈ B(2N), A ⊆ N and let w = [wSij ]i,j∈A,S∈2N :i,j /∈S be a

weight scheme such that wSij ≥ 0 for all i , j ∈ A and S ∈ Uij . The

weighted majority relation associated to w is the binary relationR�,wA ⊆ A× A such that for all i , j ∈ A ⊆ N :

iR�,wA j ⇔∑

S∈Uij

wSij d̄S

ij (�) ≥ 0.

In this definition :I dij(�) is the number of times that i performs better than j in

power relation �.I Uij = {S ∈ 2N s.t i , j /∈ S}.

Future work

I More general axioms composing two parts of Normativeand Informative.

I Find out other members of the Family of weighted majority.

I Looking for practical applications to model it as thementioned problem.

I To compare the solutions, applying the inverse of theproblem to retrieve the power relation.

An Ordinal Banzhaf Index for Social Ranking

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