Influence and interaction indexes in cooperative games: a unified least squares approach

Influence and interaction indexes in cooperative games : a unified least squares approach

Jean-Luc Marichal* Pierre Mathonet**

*University of Luxembourg

**University of Li`ege

Cooperative games

Set of players

N={1, . . . ,n}

Game on N

f: 2^N →R (usuallyf(∅) = 0)

For every coalitionS ⊆N,

f(S) = theworth ofS

Cooperative games

We can identify

S ⊆N with 1S ∈ {0,1}ⁿ Example: N ={1,2,3}

S ={2,3} 1_S = (0,1,1)

Consequence

A gamef: 2^N →R can also be regarded as a function f:{0,1}ⁿ→R

Cooperative games

A game onN can always be represented as amultilinear polynomialof degree6n

f(x) = X

T⊆N

a(T) Y

i∈T

xi x∈ {0,1}ⁿ

M¨obius transforma: 2^N →R f(S) = X

T⊆S

a(T) a(S) = X

T⊆S

(−1)^|S|−|T|f(T)

Cooperative games

Multilinear extension of a game(Owen 1972) Given a gamef onN

f(x) = X

T⊆N

a(T) Y

i∈T

x_i x∈ {0,1}ⁿ we can define itsmultilinear extension

f(x) = X

T⊆N

a(T) Y

i∈T

xi x∈[0,1]ⁿ

Banzhaf power index

Marginal contributionof player i ∈N when joining a coalition T

∆_if(T) =f(T ∪ {i})−f(T) T ⊆N\ {i}

Banzhaf power indexfor playeri (Banzhaf 1965) IB(f,i) = 1

2ⁿ⁻¹ X

T⊆N\{i}

∆if(T)

Banzhaf power index

Discrete derivative off (resp. f) with respect to theith variable

∆if(x) =f(x |xi = 1)−f(x |xi = 0)

∆_if(x) =f(x |x_i = 1)−f(x |x_i = 0)

Banzhaf power index for playeri I_B(f,i) = 1

2ⁿ X

x∈{0,1}ⁿ

∆_if(x)

Banzhaf power index

Alternative forms(Owen 1972, Grabisch et al. 2000)

I_B(f,i) =X

T3i

1 2

|T|−1

a(T)

I_B(f,i) = ∆_if ₁

2, . . . ,¹₂

I_B(f,i) = Z

[0,1]ⁿ

∆if(x)dx

Banzhaf interaction index

Marginal interactionamong playersi and j conditioned to the presence of a coalitionT ⊆N\ {i,j}

∆ijf(T) =

f(T ∪ {i,j})−f(T ∪ {i})

| {z } marginal contr. ofj in the presence ofi

−

f(T ∪ {j})−f(T)

| {z } marginal contr. ofj

in the absence ofi

Banzhaf interaction index(Owen 1972) IB(f,{i,j}) = 1

2ⁿ⁻² X

T⊆N\{i,j}

∆ijf(T)

Banzhaf interaction index

In terms of discrete derivatives :

∆ijf(x) = ∆j∆if(x)

∆ijf(x) = ∆j∆if(x)

Banzhaf interaction index

I_B(f,{i,j}) = 1 2ⁿ

X

x∈{0,1}ⁿ

∆_ijf(x)

Banzhaf interaction index

Alternative forms(Grabisch et al. 2000)

I_B(f,{i,j}) = X

T⊇{i,j}

1 2

|T|−2

a(T)

I_B(f,{i,j}) = ∆_ijf ₁

2, . . . ,¹₂

IB(f,{i,j}) = Z

[0,1]ⁿ

∆ijf(x)dx

Banzhaf interaction index

Measure of the averageinteraction among players in coalition S (Roubens 1996)

I_B(f,S) = 1 2^n−|S|

X

T⊆N\S

∆_Sf(T)

I_B(f,S) = 1 2ⁿ

X

x∈{0,1}ⁿ

∆_Sf(x)

Special case: whenS ={i} −→power index

Banzhaf interaction index

Alternative forms(Grabisch et al. 2000)

IB(f,S) = X

T⊇S

1 2

|T|−|S|

a(T)

I_B(f,S) = ∆_Sf ₁

2, . . . ,¹₂

IB(f,S) = Z

[0,1]ⁿ

∆Sf(x)dx

S-approximation of a game

Given a gamef onN and a coalitionS ⊆N, thebest S -approximation of f is the unique game onS

f_S(x) = X

T⊆S

c(T) Y

i∈T

x_i

that minimizes the square distance X

x∈{0,1}ⁿ

f(x)−g(x)2

among all gamesg onS

Theorem(Hammer-Holzman 1992, Grabisch et al. 2000, M.-M. 2011) c(S) = IB(f,S)

Weighted S -approximation of a game

Toward a weighted interaction index ? Idea: consider a weighted square distance

X

x∈{0,1}ⁿ

w(x)

f(x)−g(x) 2

wherew:{0,1}ⁿ→]0,+∞[ is a weight function.

Sincew is defined up to a multiplicative constant r >0, we can assume that

X

x∈{0,1}ⁿ

w(x) = 1

−→probability distribution over {0,1}ⁿ

Weighted S -approximation of a game

A possible interpretation ofw LetC denote a random coalition inN Define

w(S) = Pr(C =S) i.e., the probability that coalitionS forms

Independence: Suppose that the players behave independently of each other to form coalitions

This means that the events “C 3i ”,i ∈N, are independent

Weighted S -approximation of a game

Example: N={1,2,3}

w({2,3}) = Pr(C ={2,3})

= Pr(C 631) Pr(C 32) Pr(C 33)

= (1−p₁)p₂p₃

In general: Introducing p= (p₁, . . . ,p_n) with p_i = Pr(C 3i), we have

w(S) = Y

i∈S

pi

Y

i∈N\S

(1−pi)

Weighted Banzhaf interaction index

Thebest S -approximation of f is the unique game onS f_S(x) = X

T⊆S

c(T) Y

i∈T

x_i

that minimizes the weighted square distance X

x∈{0,1}ⁿ

w(x)

f(x)−g(x)2

among all gamesg onS Definition

I_B,p(f,S) = c(S)

Weighted Banzhaf interaction index

Theorem We have

I_B,p(f,S) = X

T⊆N\S

p_T^S ∆_Sf(T) with

p_T^S = Y

i∈T

p_i Y

i∈N\(S∪T)

(1−p_i) = Pr(T ⊆C ⊆S∪T)

I_B,p(f,S) = X

x∈{0,1}ⁿ

w(x) ∆_Sf(x)

Weighted Banzhaf interaction index

Non-weighted least squares (uniform probability) w(S) = 1

2ⁿ ⇐⇒ pi = Pr(C 3i) = 1 2 In this special case:

I_B,p(f,S) =I_B(f,S)

Theorem We have

I_B(f,S) = Z

[0,1]ⁿ

I_B,p(f,S)dp

Weighted Banzhaf interaction index

Alternative forms

IB,p(f,S) = X

T⊇S

a(T) Y

i∈T\S

pi

IB,p(f,S) = ∆Sf (p)

IB,p(f,S) = Z

[0,1]ⁿ

∆Sf(x) dF1(x1)· · ·dFn(xn) with p_i =

Z 1 0

x dF_i(x)

Weighted Banzhaf interaction index

Example (majority game): N={1,2,3}

f(x₁,x₂,x₃) =x₁x₂+x₂x₃+x₃x₁−2x₁x₂x₃ We have

f(0,0,0) = 0 f(1,0,0) = 0 f(1,1,0) = 1 f(1,1,1) = 1

∆₁₂f(x₁,x₂,x₃) = ∆₂(x₂+x₃−2x₂x₃) = 1−2x₃ IB,p(f,{1,2}) = ∆₁₂f

(p1,p2,p3) = 1−2p3

Weighted Banzhaf interaction index

Theorem

The mapf 7→ {I_B,p(f,S) :S ⊆N} is a linear bijection The inverse bijection is given by

f(x) = X

T⊆N

I_B,p(f,T) Y

i∈T

(x_i −p_i)

We also have a conversion formula betweenI_B,p(f,·) andI_B,p⁰(f,·) for everyp⁰

I_B,p⁰(f,S) = X

T⊇S

I_B,p(f,T) Y

i∈T\S

(p_i⁰−p_i)

Weighted Banzhaf interaction index

Define

E(f) = X

x∈{0,1}ⁿ

w(x)f(x)

σ²(f) = X

x∈{0,1}ⁿ

w(x) f(x)−E(f)2

Theorem We have

I_B,p(f,S)

6 σ(f) Q

i∈S

pp_i(1−p_i) and the inequality is tight

Banzhaf influence index

Marginal contributionof{i,j} when joining a coalitionT σ_ijf(T) =f(T ∪ {i,j})−f(T) T ⊆N\ {i,j}

Banzhaf influence indexfor {i,j} I_B(f,{i,j}) = 1

2ⁿ⁻² X

T⊆N\{i,j}

σijf(T)

Banzhaf influence index

Marginal contributionofS when joining a coalition T σ_Sf(T) =f(T ∪S)−f(T) T ⊆N\S

Banzhaf influence indexfor S I_B(f,S) = 1

2^n−|S|

X

T⊆N\S

σSf(T)

Banzhaf influence index

Define

σ_Sf(x) =f(x |x_i = 1, i ∈S)−f(x |x_i = 0, i ∈S)

Banzhaf influence indexfor S I_B(f,S) = 1

2ⁿ X

x∈{0,1}ⁿ

σ_Sf(x)

Banzhaf influence index

Alternative forms

I_B(f,S) = X

T⊆N T∩S6=∅

1 2

|T\S|

a(T)

I_B(f,S) = σSf ₁

2, . . . ,¹₂

I_B(f,S) = Z

[0,1]ⁿ

σ_Sf(x)dx

Least squares approach

LetfS be the S -approximation of a game f onN (with a non-weighted distance)

fS(x) = X

T⊆S

c(T) Y

i∈T

xi

Theorem

I_B(f,S) = f_S(N)−f_S(∅)

Weighted version ofI_B ? −→ use a weighted distance

Weighted least squares

LetfS be the S -approximation of a game f onN

with a distance weighted by a weight functionw: 2^N →]0,+∞[, defined by

w(S) = Y

i∈S

p_i Y

i∈N\S

(1−p_i)

Definition

I_B,p(f,S) = f_S(N)−f_S(∅)

Weighted Banzhaf influence index

Theorem We have

I_B,p(f,S) = X

T⊆N\S

p^S_Tσ_Sf(T) with

p^S_T = · · · (as before)

I_B,p(f,S) = X

x∈{0,1}ⁿ

w(x)σSf(x)

Weighted Banzhaf influence index

Non-weighted least squares (uniform probability) w(S) = 1

2ⁿ ⇐⇒ pi = 1 2 In this special case:

I_B,p(f,S) =I_B(f,S)

Theorem We have

I_B(f,S) = Z

[0,1]ⁿ

I_B,p(f,S)dp

Weighted Banzhaf influence index

Alternative forms

I_B,p(f,S) = X

T⊆N T∩S6=∅

a(T) Y

i∈T\S

p_i

I_B,p(f,S) = σ_Sf (p)

I_B,p(f,S) = Z

[0,1]ⁿ

σSf(x) dF1(x1)· · ·dFn(xn) with pi =

Z 1 0

x dFi(x)

Weighted Banzhaf influence index

Can we reconstruct the game from the influence index ? We have

I_B(f,S) = X

T⊆S

|T|odd 1

2 |T|−1

IB(f,T)

Weighted Banzhaf influence index

When|S|is even, we have I_B(f,S) =−X

T S

E_|S|−|T|(0) 2^|S|−|T|I_B(f,T)

whereE_nis the nth Euler polynomial (withEn(0) = 0 for evenn>1)

The “even” influences can be obtained from the “odd” influences

=⇒ The mapf 7→ {I_B(f,S) :S ⊆N} is not a bijection (half of the information is lost)

Weighted Banzhaf influence index

In the weighted case: We haveσ_∅f ≡0 and hence I_B,p(f,∅) = 0

=⇒ The mapf 7→ {I_B,p(f,S) :S ⊆N}is not a bijection (still a piece of the information is lost)

Weighted Banzhaf influence index

However... we can reconstructI_B,p(f,S) whenever Y

i∈S

(1−pi)−Y

i∈S

(−pi)6= 0

I_B,p(f,S) = 1 Q

i∈S(1−p_i)−Q

i∈S(−pi) X

T⊆S

(−1)^|S|−|T|I_B,p(f,T)

Thus, for almost everyp, the knowledge of I_B,p(f,·) enables us to reconstruct almost allIB,p(f,·) and hencef

Open problem

An interesting question:

Define weighted interaction and influence indexes in the nonindependent case

Thank you for your attention !

Best approximation theorem

Consider a finite-dimensional subspaceW of an inner product spaceV

Theorem

Ifu ∈ V, then proj_Wu is the best approximation to u from W in the sense that

ku−proj_Wuk<ku−wk for everyw∈W such thatw6=proj_Wu Theorem

If{v₁, . . . ,vr}is an orthonormal basis for W, then for everyu∈V, we have

proj_Wu=

r

X

i=1

hu,viivi