DECISION DANS L'INCERTAIN ET THEORIE DES JEUX

Stefano MORETTI1, David RIOS2 and Alexis TSOUKIAS1

1LAMSADE (CNRS), Paris Dauphine2Statistics and Operations Research Department, Rey Juan

Carlos University (URJC), Madrid

Tous les mercredis de 13h45 à 17h00 du 20/01 au 24/02

Dominant strategiesNash eq. (NE)Subgame perfect NENE & refinements…

CoreShapley valueNucleolusτ-valuePMAS….

Nash sol.Kalai-Smorodinsky….

CORENTU-valueCompromise value…

No binding agreementsNo side paymentsQ: Optimal behaviour in conflict situations

binding agreementsside payments are possible (sometimes)Q: Reasonable (cost, reward)-sharing

Cooperative games: a simple exampleAlone, player 1 (singer) and 2 (pianist) can

earn 100€ 200€ respect.Together (duo) 700€

How to divide the (extra) earnings?

I(v)700

400

600

200

100 300 500 700

x2

x1x1 +x2=700

“reasonable” payoff?

Imputation set: I(v)={xIR2|x1100, x2200, x1+x2 =700}

Simple exampleAlone, player 1 (singer) and 2 (pianist) can

earn 100€ 200€ respect.Together (duo) 700€

How to divide the (extra) earnings?

I(v)700

400

600

200

100 300 500 700

x2

x1x1 +x2=700

“reasonable” payoff

Imputation set: I(v)={xIR2|x1100, x2200, x1+x2 =700}

In this case I(v) coincides with

the core

(300, 400) =Shapley value = -value = nucleolus

COOPERATIVE GAME THEORYGames in coalitional formTU-game: (N,v) or vN={1, 2, …, n} set of playersSN coalition2N set of coalitions

DEF. v: 2NIR with v()=0 is a Transferable Utility (TU)-game with player set N.NB: (N,v)vNB2: if n=|N|, it is also called n-person TU-game, game in colaitional form, coalitional game, cooperative game with side payments...v(S) is the value (worth) of coalition S

Example(Glove game) N=LR, LR= iL (iR) possesses 1 left (right) hand gloveValue of a pair: 1€v(S)=min{| LS|, |RS|} for each coalition S2N\{} .

Example(Three cooperating communities)

source

1

2

3

100

30

30

8040

90

N={1,2,3}

v(S)=iSc(i) – c(S)

S= {1} {2} {3} {1,2} {1.3} {2,3} {1,2,3}

c(S) 0 100 90 80 130 110 110 140

v(S) 0 0 0 0 60 70 60 130

Example(flow games)

source

4,1

30

10,3

80

l1

5,2

N={1,2,3}

S= {1} {2} {3} {1,2} {1.3} {2,3} {1,2,3}

v(S) 0 0 0 0 0 4 5 9

sink

l2

l3

capacity owner

1€: 1 unit source sink

DEF. (N,v) is a superadditive game iff

v(ST)v(S)+v(T) for all S,T with ST=

Q.1: which coalitions form?Q.2: If the grand coalition N forms, how to divide v(N)?

(how to allocate costs?)

Many answers! (solution concepts)One-point concepts: - Shapley value (Shapley 1953)

- nucleolus (Schmeidler 1969)- τ-value (Tijs, 1981)…

Subset concepts: - Core (Gillies, 1954)- stable sets (von Neumann, Morgenstern, ’44)- kernel (Davis, Maschler)- bargaining set (Aumann, Maschler)…..

Show that v is superadditive and c is subadditive.

Example(Three cooperating communities)

source

1

2

3

100

30

30

8040

90

N={1,2,3}

v(S)=iSc(i) – c(S)

S= {1} {2} {3} {1,2} {1.3} {2,3} {1,2,3}

c(S) 0 100 90 80 130 110 110 140

v(S) 0 0 0 0 60 70 60 130

Claim 1: (N,v) is superadditiveWe show that v(ST)v(S)+v(T) for all S,T2N\{} with ST=60=v(1,2)v(1)+v(2)=0+070=v(1,3)v(1)+v(3)=0+060=v(2,3)v(2)+v(3)=0+060=v(1,2)v(1)+v(2)=0+0130=v(1,2,3) v(1)+v(2,3)=0+60130=v(1,2,3) v(2)+v(1,3)=0+70130=v(1,2,3) v(3)+v(1,2)=0+60

Claim 2: (N,c) is subadditiveWe show that c(ST)c(S)+c(T) for all S,T2N\{} with ST=130=c(1,2) c(1)+c(2)=100+90110=c(2,3) c(2)+v(3)=100+80110=c(1,2) c(1)+v(2)=90+80140=c(1,2,3) c(1)+c(2,3)=100+110140=c(1,2,3) c(2)+c(1,3)=90+110140=c(1,2,3) c(3)+c(1,2)=80+130

Example(Glove game) (N,v) such that N=LR, LR= v(S)=10 min{| LS|, |RS|} for all S2N\{}

Claim: the glove game is superadditive.

Suppose S,T2N\{} with ST=. Then

v(S)+v(T)= min{| LS|, |RS|} + min{| LT|, |RT|}=min{| LS|+|LT|,|LS|+|RT|,|RS|+|LT|,|RS|+|RT|} min{| LS|+|LT|, |RS|+|RT|}since ST=

=min{| L(S T)|, |R (S T)|}=v(S T).

The imputation set

DEF. Let (N,v) be a n-persons TU-game. A vector x=(x1, x2, …, xn)IRN is called an imputation iff

(1) x is individual rational i.e. xi v(i) for all iN

(2) x is efficientiN xi = v(N)

[interpretation xi: payoff to player i]

I(v)={xIRN | iN xi = v(N), xi v(i) for all iN}Set of imputations

Example(N,v) such that N={1,2,3}, v(1)=v(3)=0, v(2)=3, v(1,2,3)=5.

x3

x2

X1

(5,0,0)

(0,5,0)

(0,0,5)

x 1+x 2+x 3=5

(x1,x2,x3)

(2,3,0)

(0,3,2)

I(v)

I(v)={xIR3 | x1,x30, x23, x1+x2+x3=5}

Claim: (N,v) a n-person (n=|N|) TU-game. Then

I(v) v(N)iNv(i)Proof ()Suppose xI(v). Thenv(N) = iNxi iNv(i)

EFF IR()Suppose v(N)iNv(i). Then the vector(v(1), v(2), …, v(n-1), v(N)- i{1,2, …,n-1}v(i)) is an imputation.

v(n)

The core of a game

DEF. Let (N,v) be a TU-game. The core C(v) of (N,v) is the setC(v)={xI(v) | iS xi v(S) for all S2N\{}}

stability conditionsno coalition S has the incentive to split off if x is proposed

Note: x C(v) iff(1) iN xi = v(N) efficiency(2) iS xi v(S) for all S2N\{} stability

Bad news: C(v) can be empty

Good news: many interesting classes of games have a non-empty core.

Example(N,v) such that N={1,2,3}, v(1)=v(3)=0, v(2)=3,v(1,2)=3, v(1,3)=1v(2,3)=4v(1,2,3)=5.

Core elements satisfy the following conditions:

x1,x30, x23, x1+x2+x3=5

x1+x23, x1+x31, x2+x34

We have that

5-x33x32

5-x21x34

5-x14x11

C(v)={xIR3 | 1x10,2x30, 4x23, x1+x2+x3=5}

Example(N,v) such that N={1,2,3}, v(1)=v(3)=0, v(2)=3,v(1,2)=3, v(1,3)=1v(2,3)=4v(1,2,3)=5.

x3

x2

X1

(5,0,0)

(0,5,0)

(0,0,5)

x 1+x 2+x 3=5(2,3,0)

(0,3,2)

C(v)(0,4,1)

(1,3,1)

(1,4,0)

C(v)={xIR3 | 1x10,2x30, 4x23, x1+x2+x3=5}

Example (Game of pirates) Three pirates 1,2, and 3. On the other side of the river there is a treasure (10€). At least two pirates are needed to wade the river…(N,v), N={1,2,3}, v(1)=v(2)=v(3)=0, v(1,2)=v(1,3)=v(2,3)=v(1,2,3)=10

Suppose (x1, x2, x3)C(v). Then

efficiency x1+ x2+ x3=10 x1+ x2 10

stability x1+ x3 10 x2+ x3 10

20=2(x1+ x2+ x3) 30 Impossible. So C(v)=.

Note that (N,v) is superadditive.

Example(Glove game with L={1,2}, R={3}) v(1,3)=v(2,3)=v(1,2,3)=1, v(S)=0 otherwise

Suppose (x1, x2, x3)C(v). Then

x1+ x2+ x3=1 x2=0 x1+x3 1 x1+x3 =1 x20 x2+ x3 1 x1=0 and x3=1

So C(v)={(0,0,1)}.

(1,0,0)

(0,0,1)

(0,1,0)

I(v)

Example(flow games)

source

4,1

30

10,3

80

l1

5,2

N={1,2,3}

S= {1} {2} {3} {1,2} {1.3} {2,3} {1,2,3}

v(S) 0 0 0 0 0 4 5 9

sink

l2

l3

capacity owner

1€: 1 unit source sinkMin cut

Min cut {l1, l2}. Corresponding core element (4,5,0)

Non-emptiness of the coreNotation: Let S2N\{}.

1 if iS

eS is a vector with (eS)i=

0 if iS

DEF. A collection B2N\{} is a balanced collection if

there exist (S)>0 for SB such that:

eN= SB (S) eS

Example: N={1,2,3}, B={{1,2},{1,3},{2,3}}, (S)= for

SB

eN=(1,1,1)=1/2 (1,1,0)+1/2 (1,0,1) +1/2 (0,1,1)

Balanced games

DEF. (N,v) is a balanced game if for all balanced collections

B2N\{}

SB (S) v(S)v(N)

Example: N={1,2,3}, v(1,2,3)=10,

v(1,2)=v(1,3)=v(2,3)=8

(N,v) is not balanced

1/2v (1,2)+1/2 v(1,3) +1/2 v(2,3)>10=v(N)

Variants of duality theorem

Duality theorem.

min{xTc|xTAb}

| |

max{bTy|Ay=c, y0}

(if both programs feasible)

A

yT

bT

x c

Bondareva (1963) | Shapley (1967)Characterization of games with non-empty core

C(v)v(N)=min{xTeN | xTeSv(S) S2N\{}} duality

v(N)=max{vT | S2N\{}(S)eS=eN, 0}

0, S2

N\{}(S)eS=eN vT=S2

N\{}(S)v(S)v(N)

(N,v) is a balanced game

1 10 1

. eS .

. .

. .0 1

({1})……. (S)…….… (N)

V(1)………v(S)……..…v(N)

X1

X2

.

.

.x2

11...1

Theorem

(N,v) is a balanced game C(v)

Proof First note that xTeS=iS xi

Convex games (1)

DEF. An n-persons TU-game (N,v) is convex iffv(S)+v(T)v(ST)+v(ST) for each S,T2N.

This condition is also known as submodularity. It can be rewritten as

v(T)-v(ST)v(ST)-v(S) for each S,T2N

For each S,T2N, let C=(ST)\S. Then we have: v(C(ST))-v(ST)v(CS)-v(S)

Interpretation: the marginal contribution of a coalition C to a disjoint coalition S does not decrease if S becomes larger

Convex games (2)It is easy to show that submodularity is equivalent to

v(S{i})-v(S)v(T{i})-v(T)

for all iN and all S,T2N such that ST N\{i}interpretation: player's marginal contribution to a large

coalition is not smaller than her/his marginal contribution to a

smaller coalition (which is stronger than superadditivity)Clearly all convex games are superadditive (ST=…)A superadditive game can be not convex (try to find one)

An important property of convex games is that they are

(totally) balanced, and it is “easy” to determine the core

(coincides with the Weber set, i.e. the convex hull of all

marginal vectors…)

Example(N,v) such that N={1,2,3}, v(1)=v(3)=0, v(2)=3,v(1,2)=3, v(1,3)=1v(2,3)=4v(1,2,3)=5.Check it is convex

x3

x2

X1

(5,0,0)

(0,5,0)

(0,0,5)

x 1+x 2+x 3=5(2,3,0)

(0,3,2)

C(v)

Marginal vectors

123(0,3,2)

132(0,4,1)

213(0,3,2)

231(1,3,1)

321(1,4,0)

312(1,4,0)

(0,4,1)

(1,3,1)

(1,4,0)

C(v)={xIR3 | 1x10,2x30, 4x23, x1+x2+x3=5}

How to share v(N)…

The Core of a game can be used to exclude those allocations which are not stable.

But the core of a game can be a bit “extreme” (see for instance the glove game)

Sometimes the core is empty (pirates)And if it is not empty, there can be many

allocations in the core (which is the best?)

An axiomatic approach (Shapley (1953)Similar to the approach of Nash in bargaining:

which properties an allocation method should satisfy in order to divide v(N) in a reasonable way?

Given a subset C of GN (class of all TU-games with N as the set of players) a (point map) solution on C is a map Φ:C →IRN.

For a solution Φ we shall be interested in various properties…

Symmetry

PROPERTY 1(SYM) For all games vGN,

If v(S{i}) = v(S{j}) for all S2N s.t. i,jN\S,

then Φi(v) = Φj (v).

EXAMPLE

We have a TU-game ({1,2,3},v) s.t. v(1) = v(2) = v(3) = 0,

v(1, 2) = v(1, 3) = 4, v(2, 3) = 6, v(1, 2, 3) = 20.

Players 2 and 3 are symmetric. In fact:

v({2})= v({3})=0 and v({1}{2})=v({1}{3})=4

If Φ satisfies SYM, then Φ2(v) = Φ3(v)

Efficiency PROPERTY 2 (EFF) For all games vGN,

iNΦi(v) = v(N), i.e., Φ(v) is a pre-imputation.

Null Player Property DEF. Given a game vGN, a player iN s.t.

v(Si) = v(S) for all S2N will be said to be a null player.

PROPERTY 3 (NPP) For all games v GN, Φi(v) = 0 if i is a

null player.

EXAMPLE We have a TU-game ({1,2,3},v) such that v(1) =0,

v(2) = v(3) = 2, v(1, 2) = v(1, 3) = 2, v(2, 3) = 6, v(1, 2, 3) =

6. Player 1 is null. Then Φ1(v) = 0

EXAMPLE We have a TU-game ({1,2,3},v) such that

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

= 6, v(1, 2, 3) = 6. On this particular example, if Φ

satisfies NPP, SYM and EFF we have that

Φ1(v) = 0 by NPP

Φ2(v)= Φ3(v) by SYM

Φ1(v)+Φ2(v)+Φ3(v)=6 by EFF

So Φ=(0,3,3)

But our goal is to characterize Φ on GN. One more

property is needed.

Additivity PROPERTY 2 (ADD) Given v,w GN,

Φ(v)+Φ(w)=Φ(v +w).

.EXAMPLE Two TU-games v and w on N={1,2,3}

v(1) =3

v(2) =4

v(3) = 1

v(1, 2) =8

v(1, 3) = 4

v(2, 3) = 6

v(1, 2, 3) = 10

w(1) =1

w(2) =0

w(3) = 1

w(1, 2) =2

w(1, 3) = 2

w(2, 3) = 3

w(1, 2, 3) = 4

+

v+w(1) =4

v+w(2) =4

v+w(3) = 2

v+w(1, 2) =10

v+w(1, 3) = 6

v+w(2, 3) = 9

v+w(1, 2, 3) = 14

=

Φ ΦΦ

Theorem 1 (Shapley 1953) There is a unique map defined on GN that satisfies EFF,

SYM, NPP, ADD. Moreover, for any iN we have that

)(!

1)( vm

nv ii

Here is the set of all permutations σ:N →N of N, while mσi(v) is the

marginal contribution of player i according to the permutation σ, which is defined as: v({σ(1), σ(2), . . . , σ (j)})− v({σ(1), σ(2), . . . , σ (j −1)}),where j is the unique element of N s.t. i = σ(j).

Unanimity games (1)DEF Let T2N\{}. The unanimity game on T is

defined as the TU-game (N,uT) such that1 is TS

uT(S)=0 otherwise

Note that the class GN of all n-person TU-games is a vector space (obvious what we mean for v+w and v for v,wGN and IR).

the dimension of the vector space GN is 2n-1{uT|T2N\{}} is an interesting basis for the vector

space GN.

Unanimity games (2)Every coalitional game (N, v) can be written as a

linear combination of unanimity games in a unique

way, i.e., v =S2N λS(v)uS .

The coefficients λS(v), for each S2N, are called

unanimity coefficients of the game (N, v) and are

given by the formula: λS(v) = T2S (−1)s−t v(T ).

.EXAMPLE Two TU-games v and w on N={1,2,3}

v(1) =3

v(2) =4

v(3) = 1

v(1, 2) =8

v(1, 3) = 4

v(2, 3) = 6

v(1, 2, 3) = 10

1(v) =3

2(v) =4

3(v) = 1

{1,2}(v) =-3-4+8=1

{1,3}(v) = -3-1+4=0

{2,3}(v) = -4-1+6=1

{1,2,3}(v) = -3-4-1+8+4+6-10=0

v=3u{1}(v)+4u{2}(v)+u{3}(v)+u{1,2}(v)+u{2,3}(v)

Sketch of the Proof of Theorem1 Shapley value satisfies the four properties (easy).Properties EFF, SYM, NPP determine on the class

of all games αv, with v a unanimity game and αIR.Let S2N. The Shapley value of the unanimity game

(N,uS) is given by

/|S| if iS

i(αuS)=

0 otherwiseSince the class of unanimity games is a basis for the

vector space, ADD allows to extend in a unique way to GN.

Example(N,v) such that N={1,2,3}, v(1)=v(3)=0, v(2)=3,v(1,2)=3, v(1,3)=1,v(2,3)=4,v(1,2,3)=5.

Permutation 1 2 31,2,3 0 3 2

1,3,2 0 4 12,1,3 0 3 2

2,3,1 1 3 1

3,2,1 1 4 0

3,1,2 1 4 0

Sum 3 4 0

(v) 3/6 21/6 6/6

Probabilistic interpretation: (the “room parable”)Players gather one by one in a room to create the “grand coalition”, and each one who enters gets his marginal contribution.Assuming that all the different orders in which they enter are equiprobable, the Shapley value gives to each player her/his expected payoff.

Example(N,v) such that N={1,2,3}, v(1)=v(3)=0, v(2)=3,v(1,2)=3, v(1,3)=1v(2,3)=4v(1,2,3)=5.

x3

x2

X1

(5,0,0)

(0,5,0)

(0,0,5)

x 1+x 2+x 3=5(2,3,0)

(0,3,2)

C(v)

Marginal vectors

123(0,3,2)

132(0,4,1)

213(0,3,2)

231(1,3,1)

321(1,4,0)

312(1,4,0)

(0,4,1)

(1,3,1)

(1,4,0)

(v)=(0.5, 3.5,1)

Example(N,v) such that N={1,2,3}, v(1) =3v(2) =4 v(3) = 1v(1, 2) =8 v(1, 3) = 4v(2, 3) = 6 v(1, 2, 3) = 10

Permutation 1 2 31,2,3

1,3,22,1,3

2,3,1

3,2,1

3,1,2

Sum

(v)

Example(Glove game with L={1,2}, R={3}) v(1,3)=v(2,3)=v(1,2,3)=1, v(S)=0 otherwise

(1,0,0)

(0,0,1)

I(v)

Permutation 1 2 31,2,3 0 0 1

1,3,2 0 0 12,1,3 0 0 1

2,3,1 0 0 1

3,2,1 0 1 0

3,1,2 1 0 0

Sum 1 1 4

(v) 1/6 1/6 4/6

C(v)

(0,1,0)

(v)(1/6,1/6,2/3)

Let mσ’i(v)=v({σ’(1),σ’(2),…,σ’(j)})− v({σ’(1),σ’(2),…, σ’(j −1)}), where

j is the unique element of N s.t. i = σ’(j). Let S={σ’(1), σ’(2), . . . , σ’(j)}. Q: How many other orderings do we have in which

{σ(1), σ(2), . . . , σ (j)}=S and i = σ’(j)? A: they are precisely (|S|-1)!(|N|-|S|)! Where (|S|-1)! Is the number of orderings of S\{i} and (|N|-|

S|)! Is the number of orderings of N\S We can rewrite the formula of the Shapley value as the

following:

SiSi N iSvSvn

snsv

:2}){\()(

!

)!()!1()(

An alternative formulation

Alternative properties PROPERTY 5 (Anonymity, ANON) Let (N, v)GN σ : N →N be a

permutation. Then, Φσ(i)(σ v) = Φi(v) for all iN.

Here σv is the game defined by: σv(S) = v(σ(S)), for all S2N.Interpretation: The meaning of ANON is that whatever a player gets

via Φ should depend only on the structure of the game v, not on his “name”, i.e., the way in which he is labelled.

DEF (Dummy Player) Given a game (N, v), a player iN s.t.

v(S{i}) =v(S)+v(i) for all S2N will be said to be a dummy player.

• PROPERTY 6 (Dummy Player Property, DPP) If i is a dummy

player, then Φi(v) =v(i).

NB: often NPP and SYM are replaced by DPP and ANON,

respectively.

Characterization on a subclass Shapley and Shubik (1954) proposed to use the Shapley value as a power

index,

ADD property does not impose any restriction on a solution map defined

on the class of simple games SN, which is the class of games such that v(S)

{0,1} (often is added the requirement that v(N) = 1).

Therefore, the classical conditions are not enough to characterize the

Shapley–Shubik value on SN.

A condition that resembles ADD and can substitute it to get a

characterization of the Shapley–Shubik (Dubey (1975) index on SN:

PROPERTY 7 (Transfer, TRNSF) For any v,w S(N), it holds:

Φ(v w)+Φ(v w) = Φ(v)+Φ(w).

Here v w is defined as (v w)(S) = (v(S) w(S)) = max{v(S),w(S)}, and v w is defined as (v w)(S) = (v(S) w(S)) = min{v(S),w(S)},

.EXAMPLE Two TU-games v and w on N={1,2,3}

+v(1) =0

v(2) =1

v(3) = 0

v(1, 2) =1

v(1, 3) = 1

v(2, 3) = 0

v(1, 2, 3) = 1

=w(1) =1

w(2) =0

w(3) = 0

w(1, 2) =1

w(1, 3) = 0

w(2, 3) = 1

w(1, 2, 3) = 1

Φ Φ vw(1) =0

vw(2) =0

vw(3) = 0

vw(1, 2) =1

vw(1, 3) = 0

vw(2, 3) = 0

vw(1, 2, 3) = 1

Φ vw(1) =1

vw(2) =1

vw(3) = 0

vw(1, 2) =1

vw(1, 3) = 1

vw(2, 3) = 1

vw(1, 2, 3) = 1

Φ

+

ReformulationsOther axiomatic approaches have been provided for the

Shapley value, of which we shall briefly describe those by

Young and by Hart and Mas-Colell.

PROPERTY 8 (Marginalism, MARG) A map Ψ : GN→IRN

satisfies MARG if, given v,wGN, for any player iN s.t.

v(S{i}) −v(S) = w(S{i}) −w(S) for each S2N,

the following is true:

Ψi(v) = Ψi(w).

Theorem 2 (Young 1988) There is a unique map Ψ defined on G(N) that satisfies EFF, SYM, and MARG. Such a Ψ coincides with the Shapley value.

.EXAMPLE Two TU-games v and w on N={1,2,3}

v(1) =3

v(2) =4

v(3) = 1

v(1, 2) =8

v(1, 3) = 4

v(2, 3) = 6

v(1, 2, 3) = 10

w(1) =2

w(2) =3

w(3) = 1

w(1, 2) =2

w(1, 3) = 3

w(2, 3) = 5

w(1, 2, 3) = 4

w({3})- w() = v({3})- v()=1

w({1}{3})- w({1}) = v({1}{3})- v({1})=1

w({2}{3})- w() = v({2}{3})- v()=1

w({1,2}{3})- w({1,2}) = v({1,2}{3})- v({1,2)=1

Ψ3(v) = Ψ3(w).

Potential A quite different approach was pursued by Hart and Mas-Colell

(1987). To each game (N, v) one can associate a real number P(N,v) (or,

simply, P(v)), its potential. The “partial derivative” of P is defined as

Di(P )(N, v) = P(N,v)−P(N\{i},v|N\{i})

Theorem 3 (Hart and Mas-Colell 1987) There is a unique map

P, defined on the set of all finite games, that satisfies:

1) P( , v∅ 0) = 0,

2) For each iN, DiP(N,v) = v(N).

Moreover, Di(P )(N, v) = i(v). [(v) is the Shapley value of v]

there are formulas for the calculation of the potential.

For example, P(N,v)=S2N S/|S| (Harsanyi dividends)

Example

v(1) =3

v(2) =4

v(3) = 1

v(1, 2) =8

v(1, 3) = 4

v(2, 3) = 6

v(1, 2, 3) = 10

1(v) =3

2(v) =4

3(v) = 1

{1,2}(v) =1

{1,3}(v)=0

{2,3}(v)=1

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

P({1,2,3},v)=3+4+1+1/2+1/2=9

P({1,2},v)=3+4+1/2=15/2

P({1,3},v)=3+1=4

P({2,3},v)=4+1+1/2=11/2

1(v)=P({1,2,3},v)-P({2,3},v|{2,3})=9-11/2=7/2

2(v)=P({1,2,3},v)-P({1,3},v|{2,3})=9-4=5

3(v)=P({1,2,3},v)-P({1,2},v|{2,3})=9-15/2=3/2