DECISION DANS L'INCERTAIN ET THEORIE DES JEUX
Stefano MORETTI1, David RIOS2 and Alexis TSOUKIAS1
1LAMSADE
(CNRS), Paris Dauphine
2Statistics and Operations Research Department, Rey Juan
Carlos University (URJC), Madrid
Tous les mercredis de 13h45 à 17h00 du 20/01 au 24/02
GAME THEORY
NONCOOPERATIVE
THEORY
Games in
extendive form
(tree games)
Games in strategic
form (normal form)
Dominant strategies
Nash eq. (NE)
Subgame perfect NE
NE & refinements
…
No binding agreements
No side payments
Q: Optimal behaviour in conflict
situations
COOPERATIVE
THEORY
Games in c.f.f. (TUgames or
coalitional games)
Core
Shapley
value
Nucleolus
τ-value
PMAS
….
Bargaining games
Nash sol.
KalaiSmorodinsky
….
NTU-games
CORE
NTU-value
Compromise
value
…
binding agreements
side payments are possible (sometimes)
Q: Reasonable (cost, reward)-sharing
Cooperative games: a simple example
Alone, player 1 (singer) and 2 (pianist) can
earn
100€
200€
Together (duo)
700€
respect.
How to divide the (extra) earnings?
x2
700
600
I(v)
“reasonable” payoff?
400
200
100
300
500
700
x1 +x2=700
x1
Imputation set: I(v)={xIR2|x1100, x2200, x1+x2 =700}
Simple example
Alone, player 1 (singer) and 2 (pianist) can
earn
100€
200€
Together (duo)
700€
respect.
How to divide the (extra) earnings?
x2
700
600
In this case I(v)
coincides with
the core
I(v)
“reasonable” payoff
400
(300, 400) =Shapley value
= -value = nucleolus
200
100
300
500
700
x1 +x2=700
x1
Imputation set: I(v)={xIR2|x1100, x2200, x1+x2 =700}
COOPERATIVE GAME THEORY
Games in coalitional form
TU-game: (N,v) or v
N={1, 2, …, n}
set of players
SN
coalition
2N
set of coalitions
DEF. v: 2NIR with v()=0 is a Transferable Utility (TU)-game with
player set N.
NB: (N,v)v
NB2: 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=LR,
LR= 
iL (iR) possesses 1 left (right) hand glove
Value of a pair: 1€
v(S)=min{| LS|, |RS|} for each coalition S2N\{} .
Example
(Three cooperating communities)
1
100
N={1,2,3}
30
source
3
80
40
90
30
2
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
v(S)=iSc(i) – c(S)
Example
(flow games)
4,1
capacity
owner
l1
10,3
source
N={1,2,3}
l3
sink
80
5,2
l2
1€: 1 unit source  sink
30
S=

{1}
{2}
{3}
{1,2}
{1.3}
{2,3}
{1,2,3}
v(S)
0
0
0
0
0
4
5
9
DEF. (N,v) is a superadditive game iff
v(ST)v(S)+v(T) for all S,T with ST=
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)
…..
Example
(Three cooperating communities)
1
100
N={1,2,3}
30
source
3
80
40
90
30
2
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
v(S)=iSc(i) – c(S)
Show that v is superadditive and c is subadditive.
Claim 1: (N,v) is superadditive
We show that v(ST)v(S)+v(T) for all S,T2N\{} with ST=
60=v(1,2)v(1)+v(2)=0+0
70=v(1,3)v(1)+v(3)=0+0
60=v(2,3)v(2)+v(3)=0+0
60=v(1,2)v(1)+v(2)=0+0
130=v(1,2,3) v(1)+v(2,3)=0+60
130=v(1,2,3) v(2)+v(1,3)=0+70
130=v(1,2,3) v(3)+v(1,2)=0+60
Claim 2: (N,c) is subadditive
We show that c(ST)c(S)+c(T) for all S,T2N\{} with ST=
130=c(1,2)  c(1)+c(2)=100+90
110=c(2,3)  c(2)+v(3)=100+80
110=c(1,2)  c(1)+v(2)=90+80
140=c(1,2,3)  c(1)+c(2,3)=100+110
140=c(1,2,3)  c(2)+c(1,3)=90+110
140=c(1,2,3)  c(3)+c(1,2)=80+130
Example
(Glove game) (N,v) such that N=LR, LR= 
v(S)=10 min{| LS|, |RS|} for all S2N\{}
Claim: the glove game is superadditive.
Suppose S,T2N\{} with ST=. Then
v(S)+v(T)= min{| LS|, |RS|} + min{| LT|, |RT|}
=min{| LS|+|LT|,|LS|+|RT|,|RS|+|LT|,|RS|+|RT|}
min{| LS|+|LT|, |RS|+|RT|}
since ST=
=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 iN
(2) x is efficient
iN xi = v(N)
[interpretation xi: payoff to player i]
I(v)={xIRN | iN xi = v(N), xi  v(i) for all iN}
Set of imputations
x3
Example
(N,v) such that
N={1,2,3},
v(1)=v(3)=0,
v(2)=3,
v(1,2,3)=5.
(0,0,5)
(0,3,2)
(x1,x2,x3)
I(v)
(0,5,0)
x2
(5,0,0)
X1
I(v)={xIR3 | x1,x30, x23, x1+x2+x3=5}
Claim: (N,v) a n-person (n=|N|) TU-game. Then
I(v)
 v(N)iNv(i)
Proof
()
Suppose xI(v). Then
v(N) =
iNxi

iNv(i)
EFF
IR
()
Suppose v(N)iNv(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
set
C(v)={xI(v) | iS xi  v(S) for all S2N\{}}
stability conditions
no coalition S has the incentive to split off
if x is proposed
Note: x  C(v) iff
(1) iN xi = v(N) efficiency
(2) iS xi  v(S) for all S2N\{} stability
Bad news: C(v) can be empty
Good news: many interesting classes of games have a nonempty 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)=1
v(2,3)=4
v(1,2,3)=5.
Core elements satisfy the
following conditions:
x1,x30, x23, x1+x2+x3=5
x1+x23, x1+x31, x2+x34
We have that
5-x33x32
5-x21x34
5-x14x11
C(v)={xIR3 | 1x10,2x30, 4x23, 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)=1
v(2,3)=4
v(1,2,3)=5.
x3
(0,0,5)
(0,3,2)
(1,3,1)
C(v)
(0,4,1)
(0,5,0)
x2
(5,0,0)
X1
C(v)={xIR3 | 1x10,2x30, 4x23, 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
x 1+ x 2
10
stability
x1+
x3 10
x2+ x3 10
20=2(x1+ x2+ x3) 30
Note that (N,v) is superadditive.
Impossible. So C(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
Suppose (x1, x2, x3)C(v). Then
x1+ x2+ x3=1
x2=0
x1+x3 1
x1+x3 =1
x20
x2+ x3 1
x1=0 and
So C(v)={(0,0,1)}.
(0,0,1)
I(v)
(1,0,0)
(0,1,0)
x3=1
Example
(flow games)
4,1
capacity
owner
l1
10,3
source
N={1,2,3}
l3
sink
80
l2
5,2
Min cut
1€: 1 unit source  sink
30
S=

{1}
{2}
{3}
{1,2}
{1.3}
{2,3}
{1,2,3}
v(S)
0
0
0
0
0
4
5
9
Min cut {l1, l2}. Corresponding core element (4,5,0)
Non-emptiness of the core
Notation: Let S2N\{}.
1 if iS
eS is a vector with (eS)i=
0 if iS
DEF. A collection B2N\{} is a balanced collection if
there exist (S)>0 for SB such that:
eN= SB (S) eS
Example: N={1,2,3}, B={{1,2},{1,3},{2,3}}, (S)= for
SB
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
B2N\{}
 SB (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|xTAb}
||
max{bTy|Ay=c, y0}
yT
x
A
bT
(if both programs feasible)
c
Bondareva (1963) | Shapley (1967)
Characterization of games with non-empty core
Theorem
(N,v) is a balanced game  C(v)
Proof First note that xTeS=iS xi
({1})……. (S)…….… (N)
1
1
C(v)
X
0
1
X
.

.
.
eS
.
.
.
T
N
T
S
N
.
v(N)=min{x e | x e v(S) S2 \{}}
.
.
x
0
1
 duality
V(1)………v(S)……..…v(N)
v(N)=max{vT | S2 \{}(S)eS=eN, 0}

0, S2 \{}(S)eS=eN vT=S2 \{}(S)v(S)v(N)

(N,v) is a balanced game
1
2
2
N
N
N
1
1
.
.
.
1
Convex games (1)
DEF. An n-persons TU-game (N,v) is convex iff
v(S)+v(T)v(ST)+v(ST) for each S,T2N.
This condition is also known as submodularity. It can be
rewritten as
v(T)-v(ST)v(ST)-v(S) for each S,T2N
For each S,T2N, let C=(ST)\S. Then we have:
v(C(ST))-v(ST)v(CS)-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 iN and all S,T2N such that ST  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 (ST=…)
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)=1
v(2,3)=4
v(1,2,3)=5.
Check it is convex
x3
Marginal vectors
123(0,3,2)
132(0,4,1)
(0,0,5)
213(0,3,2)
231(1,3,1)
321(1,4,0)
(0,3,2)
(1,3,1)
312(1,4,0)
C(v)
(0,4,1)
(0,5,0)
x2
(5,0,0)
X1
C(v)={xIR3 | 1x10,2x30, 4x23, 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 vGN,
If v(S{i}) = v(S{j}) for all S2N s.t. i,jN\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 vGN,
 iNΦi(v) = v(N), i.e., Φ(v) is a pre-imputation.
Null Player Property
DEF. Given a game vGN, a player iN s.t.
v(Si) = v(S) for all S2N 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
Φ
w(1) =1
v(2) =4
w(2) =0
v(3) = 1
w(3) = 1
v(1, 2) =8
v(1, 3) = 4
+
w(1, 2) =2
w(1, 3) = 2
Φ
v+w(1) =4
Φ
v+w(2) =4
v+w(3) = 2
=
v+w(1, 2) =10
v+w(1, 3) = 6
v(2, 3) = 6
w(2, 3) = 3
v+w(2, 3) = 9
v(1, 2, 3) = 10
w(1, 2, 3) = 4
v+w(1, 2, 3) = 14
Theorem 1 (Shapley 1953)
There is a unique map  defined on GN that satisfies EFF,
SYM, NPP, ADD. Moreover, for any iN we have that
1
i (v )    mi (v )
n!
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 T2N\{}. The unanimity game on T is
defined as the TU-game (N,uT) such that
1 is TS
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,wGN and IR).
 the dimension of the vector space GN is 2n-1
{uT|T2N\{}} 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 =S2N λS(v)uS .
The coefficients λS(v), for each S2N, are called
unanimity coefficients of the game (N, v) and are
given by the formula: λS(v) = T2S (−1)s−t v(T ).
.EXAMPLE Two TU-games v and w on N={1,2,3}
v(1) =3
1(v) =3
v(2) =4
3(v) = 1
v(3) = 1
v(1, 2) =8
2(v) =4
{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(1, 3) = 4
v(2, 3) = 6
v(1, 2, 3) = 10
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 αIR.
Let S2N. The Shapley value of the unanimity game
(N,uS) is given by
/|S| if iS
i(αuS)=
0
otherwise
Since the class of unanimity games is a basis for the
vector space, ADD allows to extend  in a unique
way to GN.
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)=1,
v(2,3)=4,
v(1,2,3)=5.
Permutation
1,2,3
1
0
2
3
3
2
1,3,2
2,1,3
0
0
4
3
1
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
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.
x3
Marginal vectors
123(0,3,2)
132(0,4,1)
(0,0,5)
213(0,3,2)
231(1,3,1)
321(1,4,0)
312(1,4,0)
(0,3,2)
(v)=(0.5, 3.5,1)
(1,3,1)
C(v)
(0,4,1)
(0,5,0)
x2
(5,0,0)
X1
Example
(N,v) such that
N={1,2,3},
Permutation
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
1,3,2
2,1,3
2,3,1
3,2,1
3,1,2
Sum
(v)
1
2
3
Example
(Glove game with L={1,2}, R={3})
v(1,3)=v(2,3)=v(1,2,3)=1,
v(S)=0 otherwise
Permutation
1,2,3
1
0
2
0
3
1
1,3,2
2,1,3
0
0
0
0
1
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
C(v)
(0,0,1)
(1/6,1/6,2/3)
(v)
I(v)
(1,0,0)
(0,1,0)
4/6
An alternative formulation
 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:
( s  1)! (n  s )!
v( S )  v( S \ {i})
i (v )  S2 N :iS
n!
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 iN.
 Here σv is the game defined by: σv(S) = v(σ(S)), for all S2N.
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 iN s.t.
v(S{i}) =v(S)+v(i) for all S2N 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
Φ
+
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
=
vw(1) =0
vw(2) =0
vw(3) = 0
vw(1, 2) =1
vw(1, 3) = 0
vw(2, 3) = 0
vw(1, 2, 3) = 1

Φ
+
vw(1) =1
vw(2) =1
vw(3) = 0
vw(1, 2) =1
vw(1, 3) = 1
vw(2, 3) = 1
vw(1, 2, 3) = 1
Φ
Reformulations
Other 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,wGN, for any player iN s.t.
v(S{i}) −v(S) = w(S{i}) −w(S) for each S2N,
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}
w({3})- w() = v({3})- v()=1
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({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(∅, v0) = 0,
2) For each iN, 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)=S2N 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
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
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