Power indices on cooperative games: semivalues. · Cooperative game Cooperative game N set of...

Power indices on cooperative games:

semivalues.

Giulia Bernardi

Politecnico di Milano

3 dicembre 2014

Giulia Bernardi Indices on cooperative games 3 dicembre 2014 1 / 25

Cooperative game

Cooperative game

N set of players.

v : 2N → R utility function, such that v( /0) = 0.

Vector space given by all games on the �nite set N: GN ' R2n−1

Simple games

v is monotonic:

v(S)≤ v(T ) if S ⊆ T

v : 2N →{0,1}v(N) = 1

Unanimity games

For any coalition S

uS(T ) =

{1 if S ⊆ T

0 otherwise


Examples

Buyers and sellers (1)

A person wants to sell his car and evaluates it a value a. There are two

potential buyers, that evaluates the car b and c , respectively. (supposeb ≤ c and to avoid trivial game a< b).

v( /0) = 0 v(1) = a

v(2) = v(3) = v({2,3}) = 0

v({1,2}) = b v({1,3}) = v(N) = c

Buyers and sellers (2)

There are two people selling their own cars and only one potential buyer.

v( /0) = v(1) = v(2) = v(3) = 0

v({1,3}) = v({2,3}) = v(N) = 1 v({1,2}) = 0


Examples

Bankruptcy Problem

N = {1, . . . ,n} is the set of players (i.e. of creditors), c = (c1, . . . ,cn) such

that ci is the money claimed by player i and E is the available capital; the

bankruptcy condition is E < ∑i∈N ci .

v(S) = max{0,E − ∑i∈NrS

ci} for every S ⊆ N.

Talmud Bankruptcy Problem. A man, who was married to three wives,

died and the kethubah (the money the groom should give to his bride) of

the �rst wife was one-hundred zuz, the one of the second wife was

two-hundred zuz while the third one deserved three-hundred zuz; but the

estate was three-hundred zuz.

v( /0) = 0 v(N) = 300

v(1) = 0 v(2) = 0 v(3) = 0

v({1,2}) = 0 v({1,3}) = 100 v({2,3}) = 200

.


Examples

Weighted Majority Game

Given a positive number q and non negative integers w1, . . . ,wn the game

v = [q;w1,w2, . . . ,wn] is de�ned by

v(T ) =

{1 if w(T ) = ∑i∈T wi ≥ q

0 otherwise.

UN Security Council 5 permanent members and 10 non permanent

members. A motion is accepted if it gets at least 9 votes, including all the

votes of the permanent members.

v = [39;7,7,7,7,7,1,1,1,1,1,1,1,1,1,1].


Solution

Solution

ϕ : GN → Rn s.t. ϕi (v) is the amount given to player i in game v .

Properties:

linearity: ϕ(v +w) = ϕ(v) + ϕ(w) e ϕ(λv) = λϕ(v);

positivity: if v is monotonic then ϕi (v)≥ 0∀i ;e�ciency: ∑i∈N ϕi (v) = v(N);

symmetry: for any permutation ϑ on N

ϕi (ϑv) = ϕϑ(i)(v)

where (ϑv)(S) = v(ϑ(S));

dummy player: v(S ∪{i}) = v(S) + v({i}) for all S , thenϕi (v) = v({i}).


Solution concepts

Imputation

ϕ that are e�cient and individually rational, i.e.

I (v) = ∑i∈N

ϕi (v) = v(N) and ϕi (v)≥ v({i})

for all i ∈ N.

Core

ϕ that are e�cient and coalitionally rational, i.e.

C (v) = {ϕ : ∑i∈N

ϕi (v) = v(N) and ∑i∈S

ϕi (v)≥ v(S) for all S ⊆ N}


Power indices

Shapley

σi (v) = ∑S⊆Nr{i}

s!(n− s−1)!

n![v(S ∪{i})−v(S)]

It satis�es linearity, symmetry, dummy player and e�ciency.

Banzhaf

βi (v) = ∑S⊆Nr{i}

1

2n−1[v(S ∪{i})−v(S)]

It satis�es linearity, symmetry and the dummy player property.


Examples

Talmud Bankruptcy Problem

v( /0) = 0 v(i) = 0 v(N) = 300

v({1,2}) = 0 v({1,3}) = 100 v({2,3}) = 200.

The Shapley value is

σ1 = 50 σ2 = 100 σ3 = 150

UN Security Council

v = [39;7,7,7,7,7,1,1,1,1,1,1,1,1,1,1].

If i is a permanent member and j is a non permanent member

σi (v) =421

2145σj(v) =

4

2145

βi (v) =53

1024βj(v) =

21

4096Giulia Bernardi Indices on cooperative games 3 dicembre 2014 9 / 25

Examples

Senato della Repubblica

v = [161;108,91,50,20,16,16,10,10]

Party % Seats Shapley Banzhaf

Partito Democratico 33.65 33.8 32.14

Popolo delle Libertà 28.35 26.19 25.00

5 Stelle 15.58 21.42 23.21

Scelta Civica 6.23 5.24 5.36

Lega Nord 4.98 3.33 3.57

Misto 4.98 3.33 3.57

Grandi Autonomie 3.11 3.33 3.57

Per le Autonomie 3.11 3.33 3.57


Probabilistic value

Proabilistic value

ϕ : GN → Rn that satisfies the following properties: linearity, positivity anddummy player.

Theorem

ϕ is a probabilistic value i� for all i ∈ N there are {piS}S⊆Nr{i} such that

piS ≥ 0, ∑S⊆Nr{i} piS = 1

ϕi (v) = ∑S⊆Nr{i}

piS [v(S ∪{i})−v(S)].

Probabilistic + e�ciency =⇒ quasivalue.

Probabilistic + symmetry =⇒ semivalue.


Semivalue

Semivalue

ϕ : GN → Rn linear, positive, symmetry and dummy player.

Theorem

ϕ is a semivalue i� there are {ps}s=0,...n−1 such that ps ≥ 0,

∑n−1s=0

(n−1s

)ps = 1


ps [v(S ∪{i})−v(S)].


Other semivalues

q-binomial

φi (v) = ∑S⊆Nr{i}

qs(1−q)n−s−1[v(S ∪{i})−v(S)]

Regular semivalues

ps > 0 for all s = 0, . . . ,n−1.

Dictatorial

p0 = 1 ps = 0

for all s = 1, . . . ,n−1.

Marginal

ps = 0 pn−1 = 1

for all s = 0, . . . ,n−2.


Other semivalues

Modi�ed Values

If ϕ is a semivalue with coe�cients ps , the semivalue ϕm2m1

is de�ned by the

coe�cients

p′s =

ps

∑m2j=m1

pj(n−1j )if s ∈ [m1,m2]

0 otherwise.

Modi�ed Shapley:

p′s =1

(m2−m1 +1)(n−1

s

)Modi�ed Banzhaf:

p′s =1

∑m2j=m1

(n−1j

)Giulia Bernardi Indices on cooperative games 3 dicembre 2014 14 / 25

Senato della Repubblica

PD PdL 5 Stelle Sc. Civica Altri

β = β 70

32.14 25 23.21 5.357 3.571

β 10,β 1

150 50 0 0 0

β 20,β 2

137.5 25 18.75 6.25 3.125

β 33,β 4

3,β 4

430.0 25.0 25.0 5.0 3.75

β 42,β 5

331.05 24.74 24.21 5.263 3.684

β 65,β 7

537.5 25 18.75 6.25 3.125

β 66,β 7

650 50 0 0 0

Table: Modi�ed Banzhaf index


Semivalues and Biology

Microarray game

Let M = (mij) be a matrix such that mij ∈ {0,1}. For every j de�neSj = {i : mij = 1} and v j

v j(T ) =

{1 if Sj ⊆ T

0 otherwise.

The microarray game associated to M is de�ned as

v =1

m

m

∑j=1

v j .

The columns represent the patients while the rows represent the genes,

that are the players of this game.

mij = 1 =⇒ the gene i is abnormally expressed in patient j ,mij = 0 =⇒ the gene i is normally expressed in patient j


Microarray Game

Example 1 1 0 0

0 1 1 1

0 0 1 0

v({1}) = v({2}) =1

4v({3}) = 0

v({1,2}) =3

4v({1,3}) =

1

4

v({2,3}) =2

4v(N) = 1


Semivalues and Biology

Shapley and Banzhaf values give di�erent rankings of genes, because they

have a di�erent behaviour on unanimity games.

σi (uS) =1

sβi (uS) =

1

2s−1

a-value

σai (uS) =

{1

sa if i ∈ S

0 otherwise

where uS is the simple game such that uS(T ) = 1 i� S ⊆ T .


Semivalues on G

G space of all �nite games

U universe of players.

v : 2U → R and there is a �niteN such that ∀S ⊂ U, v(S) = v(S ∩N).

A G - Additive games: v(S ∪T ) = v(S) + v(T )

Semivalues on G

ϕ : G →A G that satis�es:

linearity;

symmetry: ϕ ◦ϑ = ϑ ◦ϕ for all permutation ϑ ;

monotonicity: v monotonic =⇒ ϕ(v) monotonic;

axis projection: if v ∈A G then ϕ(v) = v .


Semivalues on G

Theorem

ϕ is a semivalue on G i� there is a probability distributiion ξ over (0,1)such that


(∫1

0

ts(1− t)n−s−1dξ (t)

)[v(S ∪{i})−v(S)]

for each game v with support N.

Unanimity Game

For any coalition S :

uS(T ) =

{1 S ⊆ T

0 otherwise


Generating Semivalues

Theorem

The function ϕ de�ned on the basis of unanimity games

ϕαi (uS) =

{αs if i ∈ S

0 otherwise

and extended by linearity is a semivalue on G i� α1 = 1 and the sequence

αs is completely monotonic.

A sequence {µn}∞n=0

is completely monotonic if µn ≥ 0 and

(−1)k∆kµn = (−1)k

k

∑j=0

(−1)j(k

j

)µn+k−j ≥ 0

for all k,n = 0,1,2, . . . .


Proof

pns =n−s−1

∑k=0

(n− s−1

k

)(−1)kαs+k+1

So we get

n−1

∑s=0

(n−1

s

)pns = 1 ⇐⇒ α1 = 1

pns ≥ 0 ⇐⇒ (−1)k∆kαt ≥ 0

a-values

σai (uS) =

{1

sa if i ∈ S

0 otherwise

Shapley value if a = 1.

q-binomial values

φi (uS) =

{qs−1 if i ∈ S

0 otherwise

Banzhaf value if q = 1

2.


Not regular semivalues

Proposition

Let {µk}+∞

k=0be a completely monotonic sequence. Then

(−1)k∆kµn > 0

for every n,k unless the sequence is constant except at most for the �rst

term.

A semivalue ϕ on G is regular i� ϕ|N is a regular semivalue on GN for all

N. A semivalue that is not regular is irregular.

Irregular semivalues

The irregular semivalues are generated by α = (1,q, . . .q) for any q ∈ [0,1]and their weighting coe�cients are pn = (1−q,0, . . . ,0,q) for all n.If q = 1 we �nd the marginal value.

If q = 0 we �nd the dictatorial value.


Generating Semivalues

A function f (x) is completely monotonic if (−1)nf (n)(x)≥ 0, ∀n ≥ 0.

We can use completely monotonic functions to de�ne completely

monotonic sequences:

Theorem

Let f (x) be a completely monotonic function in [a,+∞) and let δ be any

�xed positive number, then

µn = {f (a+nδ )}+∞

n=0

is a completely monotonic sequence.

f (x) = eax f (x) = ln(b+

c

x) f (x) =

1

(d + ex)γ


Bibliography

Giulia Bernardi and Roberto Lucchetti.

Generating semivalues via unanimity games.

Journal of Optimization Theory and Applications, pages 1�12, 2014.

Francesc Carreras and Josep Freixas.

Semivalue versatility and applications.

Annals of Operations Research, 109(1-4):343�358, 2002.

Pradeep Dubey, Abraham Neyman, and Robert James Weber.

Value theory without e�ciency.

Mathematics of Operations Research, 6(1):122�128, 1981.

Roberto Lucchetti, Paola Radrizzani, and Emanuele Munarini.

A new family of regular semivalues and applications.

International Journal of Game Theory, 40(4):655�675, 2011.

Robert James Weber.

Probabilistic values for games.

The Shapley Value. Essays in Honor of Lloyd S. Shapley, 1988.


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