Outline
Preliminaries
Reduced games
Axioms
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References
Consistent values and two player games
Miklós Pintér1
Faculty of Business and Economics, University of Pécs,
MTA-BCE ”Lendület” Strategic Interactions Research Group
Games and Optimization 2016, November 21-22, Saint-Etienne
1 The
author acknowledges the support by National Research, Development and
Innovation Office (NKFIH, K 101224, K 115538 and K 119930).
Miklós Pintér
Consistent values and two player games
University of Pécs
Outline
Preliminaries
Reduced games
Axioms
Results
Corollaries
References
Outline
1
Preliminaries
2
Reduced games
3
Axioms
4
Results
5
Corollaries
Miklós Pintér
Consistent values and two player games
University of Pécs
Outline
Preliminaries
Reduced games
Axioms
Results
Corollaries
References
TU games
N is the nonempty and finite players set,
T ⊆ N is a coalition,
v : P(N) → R with v (∅) = 0 is a TU game,
G N is the class of TU games with players set N,
vS ∈ G S , vS = v |S is the subgame of v on S.
Miklós Pintér
Consistent values and two player games
University of Pécs
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Some values
Definition
A value ψ on A ⊆ ΓN is function such that for all S ∈ N it holds that
ψ : A ∩ G S → RS .
Definition
The Shapley value (Shapley, 1953) of a game v ∈ G N is defined as
follows, i ∈ N:
Sh(v )i =
X
S⊆N\{i}
Miklós Pintér
Consistent values and two player games
vi0 (S)
|S|!(|N \ S| − 1)!
.
|N|!
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Definition
The nucleolus (Schmeidler, 1969) of a game v ∈ G N is defined as follows.
For each x ∈ RN let ev (x) ∈ RP(N) be defined
P as
e = {v (S) − x(S)}S∈P(N) , where x(S) = i∈S xi , and S ≤ T if
eS ≥ eT . Then the nucleolus of game v is the following:
{x ∈ I (v ) : ev (x) ≤lex ev (y ), y ∈ I (v )} ,
where ≤lex is the lexicographic ordering.
Miklós Pintér
Consistent values and two player games
University of Pécs
Outline
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Davies and Maschler
Definition
Take a game v ∈ A ∩ G N , a value ψ defined on A, and a coalition S ∈ N .
The DM-reduced game (Davis and Maschler, 1965) with respect to S
and ψ is the game

0


 v (N) − P ψ(v )i
DM
vS,ψ
(T ) =
i∈N\S
P


 max v (T ∪ Q) −
ψ(v )i
Q⊆N\S
Miklós Pintér
Consistent values and two player games
if T = ∅ ,
if T = S ,
if 0 < |T | < |S| .
i∈Q
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Imputation saving reduced game
Definition
Take a game v ∈ A ∩ G N , a value ψ defined on A, and a coalition S ∈ N .
The imputation saving reduced game (Snijders, 1995) with respect to S
and ψ is the game
IS
vS,ψ
(T ) =

0


P


v (N) −
ψ(v )i



if T = ∅ ,
if T = S ,
i∈N\S
P
max
v
ψ(v )i
 Q⊆N\S (T ∪ Q) −

i∈Q


 min{ψ(v ) , max v ({i} ∪ Q) − P ψ(v ) }

i
i

Q⊆N\S
if 1 < |T | < |S| ,
if T = {i} .
i∈Q
Miklós Pintér
Consistent values and two player games
University of Pécs
Outline
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Hart and Mas-Colell
Definition
Take a game v ∈ A ∩ G N , a value ψ defined on A, and a coalition S ∈ N .
If for each T ⊆ S, T 6= ∅, the subgame of v on coalition T ∪ (N \ S),
v T ∪(N\S) is in A, then the HM-reduced game (Hart and Mas-Colell,
1989) with respect to S and ψ is the game
(
HM
vS,ψ
(T ) =
0
if T = ∅ ,
P
v (T ∪ (N \ S)) −
ψi (v T ∪(N\S) ) otherwise.
i∈N\S
Miklós Pintér
Consistent values and two player games
University of Pécs
Outline
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Moulin
Definition
Take a game v ∈ A ∩ G N , a value ψ defined on A, and a coalition S ∈ N .
The M-reduced game (Moulin, 1985) with respect to S and ψ is the
game
(
0
if T = ∅ ,
P
M
vT ,ψ (T ) =
v (T ∪ (N \ S)) −
ψi (v ) otherwise.
i∈N\S
Miklós Pintér
Consistent values and two player games
University of Pécs
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Reasonability
Definition
A notion of reduced game is reasonable, if for every game v ∈ A ∩ G N ,
value ψ defined on A, a coalition
P S ⊆ N, |S| = 2 we have a reduced
game that vS,ψ (S) = v (N) − i∈N\S ψ(v )i .
Remark
First of all, regarding an RGP property if the above reasonability property
does not hold, then an EFF solution cannot meet the RGP.
Second, all the above listed reduced game properties meet the above
reasonability property.
Miklós Pintér
Consistent values and two player games
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Axioms
Definition
A value ψ defined on A ⊆ ΓN meets
Aggregate Monotonicity (AM) if for all S ∈ N , v , w ∈ A ∩ G S such
that v (T ) = w (T ), T ⊂ S and v (S) ≥ w (S) we have that
ψ(v ) ≥ ψ(w ),
Core Selection (CS) if for all v ∈ A such that core (v ) 6= ∅ we have
that ψ(v ) ∈ core (v ),
Covariance (COV) if for all v ∈ A ∩ G T , T ∈ N , α ≥ 0 and β ∈ RT
such that αv ⊕ β ∈ A we have that ψ(αv ⊕ β) = αψ(v ) + β,
Covariance on singleton games (COVSING) if for all v ∈ A ∩ G T ,
T ⊆ N, |T | = 1, α ≥ 0 and β ∈ R such that αv ⊕ β ∈ A we have
that ψ(αv ⊕ β) = αψ(v ) + β.
Miklós Pintér
Consistent values and two player games
University of Pécs
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Axioms II
Definition
A value ψ defined on A ⊆ ΓN meets
Additive for additive games (ADAG) if for all v ∈ A ∩ G T , T ∈ N
and β ∈ RT such that v ⊕ β ∈ A we have that ψ(v ⊕ β) = ψ(v ) + β,
Efficiency (EFF) if for all v ∈ A we have that ψ(v )(N) = v (N),
Equal Treatment Property (ETP) if for all S ∈ N , v ∈ A ∩ G S and
players i, j ∈ S such that v (T ∪ {i}) = v (T ∪ {j}), T ⊆ S \ {i, j},
we have that ψi (v ) = ψj (v ),
Null-player Property (NP), if for all v ∈ A, the null players in v get 0
by ψ(v ),
Zero-game Property (ZGP), if 0 ∈ A ∩ G T , T ∈ N , then ψi (0) = 0,
i ∈ T.
Miklós Pintér
Consistent values and two player games
University of Pécs
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Axioms III
Definition
A value ψ defined on A ⊆ ΓN meets
Zero Singleton Game Property (ZSGP), if 0 ∈ A ∩ G T , T ∈ N ,
|T | = 1, then ψi (0) = 0, i ∈ T ,
R-Reduced Game Property (R-RGP) if for all S ∈ N , v ∈ A ∩ G S
and T ⊆ S such that vTR,ψ(v ) ∈ A we have that ψ(v )T = ψ(vTR,ψ(v ) ).
R-Weak Reduced Game Property (R-WRGP) if for all S ∈ N ,
v ∈ A ∩ G S and T ⊆ S, 1 ≤ |T | ≤ 2 such that vTR,ψ(v ) ∈ A we have
that ψ(v )T = ψ(vTR,ψ(v ) ).
Miklós Pintér
Consistent values and two player games
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Axioms IV
Definition
A value ψ defined on A ⊆ ΓN meets
R-Very Weak Reduced Game Property (R-VWRGP) if for all S ∈ N ,
v ∈ A ∩ G S and T ⊆ S, |T | = 2 such that vTR,ψ(v ) ∈ A we have that
ψ(v )T = ψ(vTA,ψ(v ) ).
R-Extremely Weak Reduced Game Property (R-EWRGP) if for all
S ∈ N , v ∈ A ∩ G S and T ⊆ S, |T | = 1 such that vTR,ψ(v ) ∈ A we
have that ψ(v )T = ψ(vTA,ψ(v ) ).
Miklós Pintér
Consistent values and two player games
University of Pécs
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Closed classes of games
Definition
Consider a class of games A ⊆ ΓN , a value ψ defined on A and a reduced
game notion R-reduced game. Then the class of games A is R-reduced
game closed for ψ if for all v ∈ A ∩ G S , S ⊆ N, and T ⊆ S, T 6= ∅ we
have that vTR,ψ(v ) ∈ A.
Definition
Consider a class of games A ⊆ ΓN . Then the class of games A is ADAG
closed if for all v ∈ A ∩ G T , T ∈ N , and β ∈ RT we have that v ⊕ β ∈ A.
Miklós Pintér
Consistent values and two player games
University of Pécs
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The main result
Theorem
Consider two values φ and ψ on A ⊆ ΓN and a reasonable reduced game
notion R-reduced game such that the class of games A is R-reduced
game closed for both φ and ψ, and ADAG closed. If
both values meet ADAG and R-VWRGB,
for all S ⊆ N, |S| = 2, v ∈ G S it holds that φ(v ) = ψ(v ) and both
meet EFF.
S
Then φ = ψ on S⊆N, |S|≥2 A ∩ G S .
Miklós Pintér
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Proof
Proof.
Let v ∈ A ∩ G T , T ⊆ N. If |T | = 2, then by assumption we have that
ψ(v ) = φ(v ). Suppose that |T | ≥ 3.
Then there exists β ∈ RT such that
ADAG
ψ(v ) = φ(v ) + β = φ(v ⊕ β) .
Let S ⊆ T be such that |S| = 2. Then by ADAG, R-VWRGB and by that
ψ = φ on G S we have that
R
R
R
R
R
φ(vS,φ(v
)+β ) = φ(vS,ψ(v ) ) = ψ(vS,ψ(v ) ) = φ(vS,φ(v ) )+βS = φ(vS,φ(v ) ⊕βS ) .
Miklós Pintér
Consistent values and two player games
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Proof II
Proof.
By EFF and by the reasonability of the reduced game for all we have that
X
X
R
v (N) −
φi (v ) + βi =
φ(vS,φ(v
)+β )i
i∈T \S
=
X
R
φ(vS,φ(v
)
⊕ βS )i = v (N) −
X
φi (v ) +
i∈T \S
i∈S
i∈S
X
βi ,
S ⊆ T , |S| = 2 ,
i∈S
that is,
X
i∈T \S
βi =
X
βi ,
S ⊆ T , |S| = 2 .
i∈S
It is ?easy? to see that the above system of linear equalities has a unique
solution: β = 0.
Miklós Pintér
Consistent values and two player games
University of Pécs
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A corollary
It is easy to see that if we change R-VWRGP for R-WRGP in Theorem
16, then we get that the two values equal on the whole (one player
games are included) considered class of games. Formally,
Corollary
Consider two values φ and ψ on A ⊆ ΓN and a reasonable reduced game
notion R-reduced game such that the class of games A is R-reduced
game closed for both φ and ψ, and ADAG closed. If
both values meet ADAG and R-WRGB,
for all S ⊆ N, |S| = 2, v ∈ G S it holds that φ(v ) = ψ(v ) and both
meet EFF.
Then φ = ψ on A.
Miklós Pintér
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Lemma
Consider a class of games A ⊆ ΓN , a value ψ defined on A and a reduced
game notion R-reduced game. Suppose that the class of games A is
R-reduced game closed for ψ and the R-reduced game notion is
reasonable. If the value ψ meets COVSING and R-EWRGP, then it meets
EFF.
Miklós Pintér
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Proof.
The proof goes by induction on the number of the players. For any game
v ∈ G T , |T | = 1,, by COVSING (by positive homogeneity) we have that
for all α ≥ 0 we have that
ψ(0) = ψ(α0) = αψ(0) ,
that is, ψ(0) = 0.
Moreover, by COVSING (ADAG)
ψ(v ) = ψ(0 + v (T )) = ψ(0) + v (T ) = v (T ) ,
that is, ψ meets EFF on any game with one player.
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Proof.
Now take a game v ∈ A ∩ G S , S ⊆ T . Then for each player i ∈ S by
R-EWRGP and by that the R-reduced game notion is reasonable and by
the previous point (ψ meets EFF on any game with one player) we have
that
X
ψ(v )i = ψ(v{i},ψ(v ) )i = v{i},ψ(v ) ({i}) = v (S) −
ψ(v )j ,
j∈S\{i}
that is, v (S) =
P
Miklós Pintér
Consistent values and two player games
i∈S
ψ(v )i .
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Remark
Notice that in Lemma 18 we need only ZSGP and ADAG on the
singleton games instead of COVSING.
Miklós Pintér
Consistent values and two player games
University of Pécs
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Lemma
Consider a class of games A ⊆ G N , |N| = 2 such that it is ADAG closed.
Then there exists only one solution on A that meets ADAG, EFF and
ETP.
Miklós Pintér
Consistent values and two player games
University of Pécs
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Proof.
Let v ∈ A and two ADAG, EFF and ETP values ψ and φ be defined on
A, and let N = {i, j}. If v ({i}) = v ({j}), then i ∼v j, therefore by that
both ψ and φ meet EFF and ETP we have that ψ(v ) = φ(v ). Otherwise,
w.l.o.g. we can assume that v ({i}) > v ({j}). Let β ∈ RN be such that
βi = 0 and βj = v ({i}) − v ({j}). Then i ∼v ⊕β j, hence
ψ(v ⊕ β) = φ(v ⊕ β). Therefore, by ADAG we have that
ψ(v ) = ψ(v ⊕ β) − β = φ(v ⊕ β) − β = φ(v ) .
Miklós Pintér
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The next proposition shows that our result is a generalization of the
characterization results known in the literature.
Proposition
A value defined on ΓN meets ADAG, ETP, ZSGP and
DM-RGP, if and only if it is the prenucleolus,
IS-RGP, if and only if it is the nucleolus,
HM-RGP, if and only if it is the Shapley value,
M-RGP, if and only if it is the Moulin value.
Miklós Pintér
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Proof
Proof.
If: It is well known that these values meet ADAG, ETP, ZSGP and
DM-RGP/IS-RGP/HM-RGP/M-RGP respectively.
Only if: Apply Theorem 16 and Lemmata 18 and 20.
Miklós Pintér
Consistent values and two player games
University of Pécs
Outline
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Thank you for the attention!
Miklós Pintér
Consistent values and two player games
University of Pécs
Outline
Preliminaries
Reduced games
Axioms
Results
Corollaries
References
References
Davis M, Maschler M (1965) The kernel of a cooperative game. Naval
Research Logistics Quarterly 12(3):223–259
Hart S, Mas-Colell A (1989) Potential, value, and consistency.
Econometrica 57:589–614
Moulin H (1985) The separability axiom and equal-sharing methods.
Journal of Economic Theory 36:120–148
Schmeidler D (1969) The Nucleolus of a Characteristic Function Game.
SIAM Journal on Applied Mathematics 17:1163–1170
Shapley LS (1953) A value for n-person games. In: Kuhn HW, Tucker AW
(eds) Contributions to the Theory of Games II, Annals of Mathematics
Studies, vol 28, Princeton University Press, Princeton, pp 307–317
Snijders C (1995) Axiomatization of the Nucleolus. Mathematics of
Operations Research 20(1):189–196
Miklós Pintér
Consistent values and two player games
University of Pécs