Cooperation under Interval Uncertainty

6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011

Cooperative Game Theory. Operations Research
Games. Applications to Interval Games
Lecture 3: Cooperation under Interval Uncertainty

Sırma Zeynep Alparslan G¨k
o
S¨leyman Demirel University
u
Faculty of Arts and Sciences
Department of Mathematics
Isparta, Turkey
email:zeynepalparslan@yahoo.com

August 13-16, 2011


Outline

Introduction

Preliminaries on classical games in coalitional form

Cooperative games under interval uncertainty

On two-person cooperative games under interval uncertainty

Some economic examples

References

Introduction

Introduction

This lecture is based on the paper

Cooperation under interval uncertainty
by Alparslan G¨k, Miquel and Tijs
o

which was published in

Mathematical Methods of Operations Research.

Introduction

Introduction

Classical cooperative game theory deals with coalitions who
coordinate their actions and pool their winnings.
One of the problems is how to divide the rewards or costs among
the members of the formed coalition.
Generally, the situations are considered from a deterministic point
of view.
However, in most economical situations rewards or costs are not
known precisely, but it is possible to estimate intervals to which
they belong.

Introduction

define a cooperative interval game
introduce the notion of the core set of a cooperative interval
game and various notions of balancedness
focus on two-person cooperative interval games and extend to
these games well-known results for classical two-person
cooperative games
define and analyze specific solution concepts on the class of
two-person interval games, e.g., mini-core set, the ψ α -values



A cooperative n-person game in coalitional form is an ordered
pair < N, v >, where N = {1, 2, ..., n} (the set of players) and
v : 2N → R is a map, assigning to each coalition S ∈ 2N a
real number, such that v (∅) = 0.
v is the characteristic function of the game.
v (S) is the worth (or value) of coalition S.


A payoff vector x ∈ Rn is called an imputation for the game
< N, v > (the set is denoted by I (v )) if
x is individually rational: xi ≥ v ({i}) for all i ∈ N
n
x is efficient (Pareto optimal): i=1 xi = v (N)
The core (Gillies (1959)) of a game < N, v > is the set

C (v ) = x ∈ I (v )| xi ≥ v (S) for all S ∈ 2N {∅} .
i∈S

The idea of the core is by giving every coalition S at least their
worth v (S) so that no coalition has an incentive to split off.

For a two-person game I (v ) = C (v ).


An n-person game < N, v > is called a balanced game if for each
balanced map λ : 2N {∅} → R+ we have

λ(S)v (S) ≤ v (N).
S

Theorem (Bondareva (1963) and Shapley (1967)):

Let < N, v > be an n-person game. Then the following two
assertions are equivalent:
(i) C (v ) = ∅,
(ii) < N, v > is a balanced game.


π(N): the set of all permutations σ : N → N

The marginal vector mσ (v ) ∈ Rn has as i-th coordinate (i ∈ N)

miσ (v ) = v (P σ (i) ∪ {i}) − v (P σ (i))

P σ (i) = r ∈ N|σ −1 (r ) < σ −1 (i)
The Shapley value (Shapley (1953)) Φ(v ) of a game is the average
of the marginal vectors of the game:
1
Φ(v ) = mσ (v )
n!
σ∈π(N)


For a two-person game < N, v >:

m(12) (v ) = (v ({1}), v ({1, 2}) − v ({1}))
m(21) (v ) = (v ({1, 2}) − v ({1}), v ({2}))

v ({1, 2}) − v ({1}) − v ({2})
Φi (v ) = v ({i}) + (i = {1, 2})
2
Shapley value is the standard solution, in the middle of the core.
Marginal vectors are the extreme points of the core whose average
gives the Shapley value.


A game < N, v > is superadditive if v (S ∪ T ) ≥ v (S) + v (T )
for all S, T ∈ 2N with S ∩ T = ∅.
A two-person cooperative game < N, v > is superadditive if
and only if v ({1}) + v ({2}) ≤ v ({1, 2}) holds.
A two-person cooperative game < N, v > is superadditive if
and only if the game is balanced.
For further details on Cooperative game theory see Tijs (2003) and
Branzei Dimitrov and Tijs (2008).


A cooperative n-person interval game in coalitional form is an
ordered pair < N, w >:
N := {1, 2, . . . , n} is the set of players
w : 2N → I (R) is the characteristic function which assigns to
each coalition S ∈ 2N a closed interval w (S) ∈ I (R)
I (R) is the set of all closed intervals in R such that
w (∅) = [0, 0]
The worth interval w (S) is denoted by [w (S), w (S)].
Here, w (S) is the lower bound and w (S) is the upper bound.
If all the worth intervals are degenerate intervals, i.e.,
w (S) = w (S), then the interval game < N, w > corresponds to
the classical cooperative game < N, v >, where v (S) = w (S).


Let < N, w > be an interval game; then v is called a selection of
w if v (S) ∈ w (S) for each S ∈ 2N .
The set of selections of w is denoted by Sel(w ).
The imputation set of an interval game < N, w > is deﬁned by

I (w ) := ∪ {I (v )|v ∈ Sel(w )}.

The core set of an interval game < N, w > is deﬁned by

C (w ) := ∪ {C (v )|v ∈ Sel(w )}.

An interval game < N, w > is strongly balanced if for each
balanced map λ it holds that

λ(S)w (S) ≤ w (N).
S


Proposition: Let < N, w > be an interval game. Then, the
following three statements are equivalent:
(i) For each v ∈ Sel(w ) the game < N, v > is balanced.
(ii) For each v ∈ Sel(w ), C (v ) = ∅.
(iii) The interval game < N, w > is strongly balanced.
Proof: (i) ⇔ (ii) follows from Bondareva and Shapley theorem.
(i) ⇔ (iii) follows using the inequalities:

w (N) ≤ v (N) ≤ w (N)

λ(S)w (S) ≤ λ(S)v (S) ≤ λ(S)w (S)


Balancedness and related topics
Let < N, w > be a two-person interval game:
the pre-imputation set

I ∗ (w ) := x ∈ R2 |x1 + x2 ∈ w (1, 2)

the imputation set

I (w ) := x ∈ R2 |x1 ≥ w (1), x2 ≥ w (2), x1 + x2 ∈ w (1, 2)

the mini-core set

MC (w ) := x ∈ R2 |x1 ≥ w (1), x2 ≥ w (2), x1 + x2 ∈ w (1, 2)

the core set

C (w ) := x ∈ R2 |x1 ≥ w (1), x2 ≥ w (2), x1 + x2 ∈ w (1, 2)


Example 1
The mini-core set and the core set of a strongly balanced game

x2

12
d
10 d core set
d ©
d
d d
d d mini-core set
d ©d
5 d d
d d
d d
2 d d
d d
d d
1 3 10 12 x1


w (1) + w (2) = 3 + 5 ≤ w (1, 2) = 10 (a strongly balanced game)

x2

12
d
10 d core set
d ©
d
d d
d d mini-core set
©
d d
5 d d
d d
d d
2 d d
d d
d d
1 3 10 12 x1


Balancedness and related topics

s1 ∈ w (1) = [w (1), w (1)], s2 ∈ w (2) = [w (2), w (2)]
t ∈ w (1, 2) = [w (1, 2), w (1, 2)]

w s1 ,s2 ,t : the selection of w corresponding to s1 , s2 and t

C (w ) = ∪ C (w s1 ,s2 ,t )|(s1 , s2 , t) ∈ w (1) × w (2) × w (1, 2)

MC (w ) = ∪ C (w s1 ,s2 ,t )|s1 ∈ [w (1), w (1)], s2 ∈ [w (2), w (2)]
MC (w ) ⊂ ∪ C (w s1 ,s2 ,t )|s1 ∈ w (1), s2 ∈ w (2), t ∈ w (1, 2)
The mini-core set is interesting because for each s1 , s2 and t all
points in MC (w ) with x1 + x2 = t are also in C (w s1 ,s2 ,t ).


Superadditivity

Let A and B be two intervals. We say that A is left to B, denoted
by A B, if for each a ∈ A and for each b ∈ B, a ≤ b.

A two-person interval game < N, w > is called superadditive, if

w (1) + w (2) w (1, 2)


If < N, w > is a superadditive game, then for each s1 , s2 and t we
have s1 + s2 ≤ t. So, each selection w s1 ,s2 ,t of w is balanced.

If w (1) + w (2) ≤ w (1, 2) is satisﬁed, then each selection w s1 ,s2 ,t of
w is superadditive.

A two-person interval game < N, w > is superadditive if and only
if < N, w > is strongly balanced.


ψ α -values and their axiomatization

α = (α1 , α2 ) ∈ [0, 1] × [0, 1]: the optimism vector
α
s1 1 (w ) := α1 w (1) + (1 − α1 )w (1)

α
s2 2 (w ) := α2 w (2) + (1 − α2 )w (2)


α α
If α = (1, 1), then s1 1 (w ) = w (1), s2 2 (w ) = w (2) which are
the optimistic (upper) points.
α α
If α = (0, 0), then s1 1 (w ) = w (1), s2 2 (w ) = w (2) which are
the pessimistic (lower) points.
If α = ( 1 , 1 ), then s1 1 (w ) = w (1)+w (1) , s2 2 (w ) = w (2)+w (2)
2 2
α
2
α
2
which are the middle points of the intervals w (1) and w (2),
respectively.


A map F : IG {1,2} → K(R2 ) assigning to each interval game w
a unique curve

F (w ) : [w (1, 2), w (1, 2)] → R2

for t ∈ [w (1, 2), w (1, 2)], i ∈ {1, 2}, in K(R2 ) is called a solution.

IG {1,2} : the family of all interval games with player set {1, 2}


F has the following properties:
(i) eﬃciency (EFF), if for all w ∈ IG {1,2} and
t ∈ [w (1, 2), w (1, 2)]; i∈N F (w )(t)i = t.

(ii) α-symmetry (α-SYM), if for all w ∈ IG {1,2} and
α α
t ∈ [w (1, 2), w (1, 2)] with s1 1 (w ) = s2 2 (w );
F (w )(t)1 = F (w )(t)2 .

(iii) covariance with respect to translations (COV), if for all
w ∈ IG {1,2} , t ∈ [w (1, 2), w (1, 2)] and a = (a1 , a2 ) ∈ R2
F (w + ˆ)(a1 + a2 + t) = F (w )(t) + a.
a


ψ α -values and their axiomatization

ˆ ∈ IG {1,2} is deﬁned by
a

ˆ({1}) = [a1 , a1 ], ˆ({2}) = [a2 , a2 ]
a a

ˆ({1, 2}) = [a1 + a2 , a1 + a2 ]
a
w + ˆ ∈ IG {1,2} is deﬁned by
a

(w + ˆ)(s) = w (s) + ˆ(s)
a a

for
s ∈ {{1} , {2} , {1, 2}}


For each w ∈ IG {1,2} and t ∈ [w (1, 2), w (1, 2)]
we deﬁne the map

ψ α : IG {1,2} → K(R2 )

such that
α α
ψ α (w )(t) := (s1 1 (w ) + β, s2 2 (w ) + β),

where
1 α α
β = (t − s1 1 (w ) − s2 2 (w )).
2


Proposition: The ψ α -value satisfies the properties EFF, α-SYM
and COV.
Proof:
(i) EFF property is satisfied since
α α
ψ α (w )(t)1 + ψ α (w )(t)2 = s1 1 (w ) + s2 2 (w ) + 2β = t.
α α
(ii) α-SYM property is satisfied since s1 1 (w ) = s2 2 (w ) implies

α α
ψ α (w )(t)1 = s1 1 (w ) + β = s2 2 (w ) + β = ψ α (w )(t)2 .


(iii) COV property is satisﬁed since
α ˆ α ˆ
ψ α (w + ˆ)(a1 + a2 + t) = (s1 1 (w + ˆ) + β, s2 2 (w + ˆ) + β)
a a a

Then,
α α
ψ α (w +ˆ)(a1 +a2 +t) = (s1 1 +β, s2 2 +β)+(a1 , a2 ) = ψ α (w )(t)+a.
a

Note that

ˆ 1 t α α
β = β = (ˆ − s1 1 (w + ˆ) − s2 2 (w + ˆ)),
a a
2
where ˆ = a1 + a2 + t.
t


Theorem: The ψ α -value is the unique solution satisfying EFF,
α-SYM and COV properties.

Proof: Suppose F satisﬁes the properties above. Show F = ψ α .

α α
Let a = (s1 1 (w ), s2 2 (w )). Then, s α (w − ˆ) = (0, 0).
a
By α-SYM and EFF, for each ˜ = t − a1 − a2
t

F (w − ˆ)(˜) = ( 1 ˜, 2 ˜) = ψ α (w − ˆ)(˜).
a t 2t
1
t a t

By COV of F and ψ α ,
F (w )(t) = F (w − ˆ)(˜) + a = ψ α (w − ˆ)(˜) + a = ψ α (w )(t).
a t a t

From Proposition it follows ψ α satisﬁes EFF, α-SYM and COV.


Marginal curves and the Shapley-like solution

The marginal curves for a two-person game < N, w > are deﬁned
by
mσ,α (w ) : [w (1, 2), w (1, 2)] → R2 ,
where
α α
m(1,2),α (w )(t) = (s1 1 (w ), t − s1 1 (w )),
α α
m(2,1),α (w )(t) = (t − s2 2 (w ), s2 2 (w )).
The Shapley-like solution ψ α is equal to
1
ψ α (w ) = (m(1,2),α (w ) + m(2,1),α (w )).
2


Example 2

A bankruptcy situation with two claimants with demands d1 = 70
and d2 = 90 on (uncertain) estate E = [100, 120].

w (∅) = [0, 0], w (1) = [(E − d2 )+ , (E − d2 )+ ] = [10, 30]

w (2) = [(E − d1 )+ , (E − d1 )+ ] = [30, 50], w (1, 2) = [100, 120].

w (1) + w (2) = 30 + 50 ≤ w (1, 2) = 100

(a strongly balanced game)


1
ψ (0,0) (w )(t) = (10 + β, 30 + β) with β = (t − 40) and t ∈ [100, 120]
2

 
t 100 106 110 114 120
β  30 33 35 37 40 
ψ (0,0) (w )(t) (40, 60) (43, 63) (45, 65) (47, 67) (50, 70)


The mini-core set and the ψ (0,0) -values of the game < N, w >,
where L = {ψ α (w )(t)|t ∈ [100, 120]}.

x2

120
d
100 d
d d
d d
d Ld
d d mini-core
50 d ©d
d d
30 d d
d d
d d
d d
10 30 100 120 x1


Example 3

A reward game with N = {1, 2} which is not strongly balanced.

w (∅) = [0, 0], w (1) = [1, 3], w (2) = [2, 5], w (1, 2) = [6, 10].

ψ (0,0) (w )(t) = (1 + β, 2 + β) with 3 + 2β = t and t ∈ [6, 10].

 
t 6 7 8 9 10
β  11 2 2 21 2 3 31 2

(0,0) (w )(t) 1 1 1 1 1 1
ψ (2 2 , 3 2 ) (3, 4) (3 2 , 4 2 ) (4, 5) (4 2 , 5 2 )


x2

10
d
d
d
6 d
© mini-core
5 d d
d d
dL
d
2 d d
d d
d d
1 3 6 10 x1
The mini-core set and the ψ α -values of an interval game, where

L = {ψ α (w )(t)|t ∈ [6, 10]}.

References

References

[1] Alparslan G¨k S.Z., Miquel S. and Tijs S., “Cooperation under
o
interval uncertainty”, Mathematical Methods of Operations
Research, Vol. 69 (2009) 99-109.
[2] Bondareva O.N., “Certain applications of the methods of linear
programming to the theory of cooperative games”, Problemly
Kibernetiki 10 (1963) 119-139 (in Russian).
[3] Branzei R., Dimitrov D. and Tijs S., “Models in Cooperative
Game Theory”, Springer-Verlag Berlin (2008).
[4] Gillies D.B., “Some Theorems on n-person Games”, Ph.D.
Thesis, Princeton University Press (1953).

References

[5] Shapley L.S., “A value for n-person games”, Annals of
Mathematics Studies 28 (1953) 307-317.
[6] Shapley L.S., “On balanced sets and cores”, Naval Research
Logistics Quarterly 14 (1967) 453-460.
[7] Tijs S., “Introduction to Game Theory”, SIAM, Hindustan
Book Agency, India (2003).

Cooperation under Interval Uncertainty

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