Coalitional Games with Interval-Type Payoffs: A Survey

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

Cooperative Game Theory. Operations ResearchGames. Applications to Interval Games

Lecture 6: Coalitional Games with Interval-Type Payoffs: ASurvey

Sırma Zeynep Alparslan GokSuleyman Demirel University

Faculty of Arts and Sciences

Department of Mathematics

Isparta, Turkey

email:[email protected]

August 13-16, 2011


Outline

Introduction

Cooperative interval games

Classes of cooperative interval games

Economic and OR situations with interval data

Handling interval solutions

Final remarks and outlook

References


Introduction

Introduction

I This lecture is based on the book

Cooperative Interval Games: Theory and Applications

by Alparslan Gok published by

Lambert Academic Publishing (LAP).

I For more information please see:http://www.morebooks.de/store/gb/book/cooperative-interval-games/isbn/978-3-8383-3430-1

The book is the PhD thesis of Alparslan Gok entitled


from Middle East Technical University, Ankara-Turkey.


Introduction

Motivation

Game theory:

I Mathematical theory dealing with models of conflict andcooperation.

I Many interactions with economics and with other areas suchas Operations Research (OR) and social sciences.

I Tries to come up with fair divisions.

I A young field of study:The start is considered to be the book Theory of Games andEconomic Behaviour by von Neumann and Morgernstern(1944).

I Two parts: non-cooperative and cooperative.


Introduction

Motivation continued...

I Cooperative game theory deals with coalitions whichcoordinate their actions and pool their winnings.

I The main problem: How to divide the rewards or costs amongthe members of the formed coalition?

I Generally, the situations are considered from a deterministicpoint of view.

I Basic models in which probability and stochastic theory play arole are: chance-constrained games and cooperative gameswith stochastic/random payoffs.


Introduction

Motivation continued...

Idea of interval approach:

I In most economic and OR situations rewards/costs are notprecise.

I Possible to estimate the intervals to which rewards/costsbelong.

Why cooperative interval games are important?

I Useful for modeling real-life situations.

Aim: generalize the classical theory to intervals and apply it toeconomic situations and OR situations.

I In this study, rewards/costs taken into account are notrandom variables, but just closed and bounded intervals ofreal numbers with no probability distribution attached.


Introduction

Interval calculus

I (R): the set of all closed and bounded intervals in RI , J ∈ I (R), I =

[I , I], J =

[J, J], |I | = I − I , α ∈ R+

I addition: I + J =[I + J, I + J

]I multiplication: αI =

[αI , αI

]I subtraction: defined only if |I | ≥ |J|

I − J =[I − J, I − J

]I weakly better than: I < J if and only if I ≥ J and I ≥ J

I I 4 J if and only if I ≤ J and I ≤ J

I better than: I � J if and only if I < J and I 6= J

I I ≺ J if and only if I 4 J and I 6= J



Classical cooperative games versus cooperative intervalgames

I < N, v >, N := {1, 2, ..., n}: set of players

I v : 2N → R: characteristic function, v(∅) = 0

I v(S): worth (or value) of coalition S

GN : the class of all coalitional games with player set N

I < N,w >, N: set of players

I w : 2N → I (R): characteristic function, w(∅) = [0, 0]

I w(S) = [w(S),w(S)]: worth (value) of S

IGN : the class of all interval games with player set NExample (LLR-game): Let < N,w > be an interval game withw({1, 3}) = w({2, 3}) = w(N) = J < [0, 0] and w(S) = [0, 0]otherwise.



Arithmetic of interval games

w1,w2 ∈ IGN , λ ∈ R+, for each S ∈ 2N

I w1 4 w2 if w1(S) 4 w2(S)

I < N,w1 + w2 > is defined by (w1 + w2)(S) = w1(S) + w2(S).

I < N, λw > is defined by (λw)(S) = λ · w(S).

I Let w1,w2 ∈ IGN such that |w1(S)| ≥ |w2(S)| for eachS ∈ 2N . Then < N,w1 − w2 > is defined by(w1 − w2)(S) = w1(S)− w2(S).

Classical cooperative games associated with < N,w >

I Border games: < N,w > and < N,w >

I Length game: < N, |w | >, where |w | (S) = w(S)− w(S) foreach S ∈ 2N .



Preliminaries on classical cooperative games< N, v > is called a balanced game if for each balanced mapλ : 2N \ {∅} → R+ we have∑

S∈2N\{∅}

λ(S)v(S) ≤ v(N).

The core (Gillies (1959)) C (v) of v ∈ GN is defined by

C (v) =

{x ∈ RN |

∑i∈N

xi = v(N);∑i∈S

xi ≥ v(S),∀S ∈ 2N

}.

Theorem (Bondareva (1963), Shapley (1967)): Let < N, v > be ann-person game. Then, the following two assertions are equivalent:

(i) C (v) 6= ∅.(ii) < N, v > is a balanced game.



Selection-based solution concepts

Let < N,w > be an interval game.

I v is called a selection of w if v(S) ∈ w(S) for each S ∈ 2N .

I Sel(w): the set of selections of w

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

C (w) := ∪{C (v)|v ∈ Sel(w)} .



Selection-based solution conceptsAn interval game < N,w > is strongly balanced if for eachbalanced map λ it holds that∑

S∈2N\{∅}

λ(S)w(S) ≤ w(N).

Proposition: Let < N,w > be an interval game. Then, thefollowing three statements are equivalent:

(i) For each v ∈ Sel(w) the game < N, v > is balanced.(ii) For each v ∈ Sel(w), C (v) 6= ∅.(iii) The interval game < N,w > is strongly balanced.

Proof: (i)⇔ (ii) follows from Bondareva-Shapley theorem.(i)⇔ (iii) follows using w(N) ≤ v(N) ≤ w(N) and∑

S∈2N\{∅}

λ(S)w(S) ≤∑

S∈2N\{∅}

λ(S)v(S) ≤∑

S∈2N\{∅}

λ(S)w(S)

for each balanced map λ.



Interval solution conceptsI (R)N : set of all n-dimensional vectors with elements in I (R).The interval imputation set:

I(w) =

{(I1, . . . , In) ∈ I (R)N |

∑i∈N

Ii = w(N), Ii < w(i), ∀i ∈ N

}.

The interval core:

C(w) =

{(I1, . . . , In) ∈ I(w)|

∑i∈S

Ii < w(S), ∀S ∈ 2N \ {∅}

}.

Example (LLR-game) continuation:

C(w) =

{(I1, I2, I3)|

∑i∈N

Ii = J,∑i∈S

Ii < w(S)

},

C(w) = {([0, 0], [0, 0], J)} .



Classical cooperative games (Part I in Branzei, Dimitrovand Tijs (2008))

< N, v > is convex if and only if the supermodularity condition

v(S ∪ T ) + v(S ∩ T ) ≥ v(S) + v(T )

for each S ,T ∈ 2N holds.< N, v > is concave if and only if the submodularity condition

v(S ∪ T ) + v(S ∩ T ) ≤ v(S) + v(T )

for each S ,T ∈ 2N holds.



Convex and concave interval games

I < N,w > is supermodular if

w(S) + w(T ) 4 w(S ∪ T ) + w(S ∩ T ) for all S ,T ∈ 2N .

I < N,w > is convex if w ∈ IGN is supermodular and|w | ∈ GN is supermodular (or convex).

I < N,w > is submodular if

w(S) + w(T ) < w(S ∪ T ) + w(S ∩ T ) for all S ,T ∈ 2N .

I < N,w > is concave if w ∈ IGN is submodular and |w | ∈ GN

is submodular (or concave).



Illustrative examples

Example 1: Let < N,w > be the two-person interval game withw(∅) = [0, 0], w({1}) = w({2}) = [0, 1] and w(N) = [3, 4].Here, < N,w > is supermodular and the border games are convex,but |w | ({1}) + |w | ({2}) = 2 > 1 = |w | (N) + |w | (∅).Hence, < N,w > is not convex.Example 2: Let < N,w > be the three-person interval game withw({i}) = [1, 1] for each i ∈ N,w(N) = w({1, 3}) = w({1, 2}) = w({2, 3}) = [2, 2] andw(∅) = [0, 0].Here, < N,w > is not convex, but < N, |w | > is supermodular,since |w | (S) = 0, for each S ∈ 2N .



Example (unanimity interval games):

Let J ∈ I (R) such that J � [0, 0] and let T ∈ 2N \ {∅}. Theunanimity interval game based on T is defined for each S ∈ 2N by

uT ,J(S) =

{J, T ⊂ S[0, 0] , otherwise.

< N, |uT ,J | > is supermodular, < N, uT ,J > is supermodular:

T ⊂ A,T ⊂ BT ⊂ A,T 6⊂ BT 6⊂ A,T ⊂ BT 6⊂ A,T 6⊂ B

uT ,J(A ∪ B) uT ,J(A ∩ B) uT ,J(A) uT ,J(B)J J J JJ [0, 0] J [0, 0]J [0, 0] [0, 0] J

J or [0, 0] [0, 0] [0, 0] [0, 0].



Size monotonic interval games

I < N,w > is size monotonic if < N, |w | > is monotonic, i.e.,|w | (S) ≤ |w | (T ) for all S ,T ∈ 2N with S ⊂ T .

I SMIGN : the class of size monotonic interval games withplayer set N.

I For size monotonic games, w(T )− w(S) is defined for allS ,T ∈ 2N with S ⊂ T .

I CIGN : the class of convex interval games with player set N.

I CIGN ⊂ SMIGN because < N, |w | > is supermodular impliesthat < N, |w | > is monotonic.



I-balanced interval games

< N,w > is I-balanced if for each balanced map λ∑S∈2N\{∅}

λSw(S) 4 w(N).

IBIGN : class of interval balanced games with player set N.

CIGN ⊂ IBIGN

CIGN ⊂ (SMIGN ∩ IBIGN)

Theorem: Let w ∈ IGN . Then the following two assertions areequivalent:

(i) C(w) 6= ∅.(ii) The game w is I-balanced.



Solution concepts for cooperative interval gamesΠ(N): set of permutations, σ : N → N, of NPσ(i) =

{r ∈ N|σ−1(r) < σ−1(i)

}: set of predecessors of i in σ

The interval marginal vector mσ(w) of w ∈ SMIGN w.r.t. σ:

mσi (w) = w(Pσ(i) ∪ {i})− w(Pσ(i))

for each i ∈ N.

Interval Weber set W : SMIGN � I (R)N :

W(w) = conv {mσ(w)|σ ∈ Π(N)} .

Example: N = {1, 2}, w({1}) = [1, 3],w({2}) = [0, 0] and

w({1, 2}) = [2, 3 12 ]. This game is not size monotonic.

m(12)(w)is not defined.w(N)− w({1}) = [1, 1

2 ]: undefined since |w(N)| < |w({1})|.



The interval Shapley valueThe interval Shapley value Φ : SMIGN → I (R)N :

Φ(w) =1

n!

∑σ∈Π(N)

mσ(w), for each w ∈ SMIGN .

Example: N = {1, 2}, w({1}) = [0, 1],w({2}) = [0, 2],w(N) = [4, 8].

Φ(w) =1

2(m(12)(w) + m(21)(w));

Φ(w) =1

2((w({1}),w(N)− w({1})) + (w(N)− w({2}),w({2}))) ;

Φ(w) =1

2(([0, 1], [4, 7]) + ([4, 6], [0, 2])) = ([2, 3

1

2], [2, 4

1

2]).



Properties of solution concepts

I W(w) ⊂ C(w), ∀w ∈ CIGN and W(w) 6= C(w) is possible.Example: N = {1, 2}, w({1}) = w({2}) = [0, 1] andw(N) = [2, 4] (convex).W(w) = conv

{m(1,2)(w),m(2,1)(w)

}m(1,2)(w) = ([0, 1], [2, 4]− [0, 1]) = ([0, 1], [2, 3])m(2,1)(w) = ([2, 3], [0, 1]])m(1,2)(w) and m(2,1)(w) belong to C(w).([ 1

2 , 134 ], [1 1

2 , 214 ]) ∈ C(w)

no α ∈ [0, 1] exists satisfyingαm(1,2)(w) + (1− α)m(2,1)(w) = ([ 1

2 , 134 ], [1 1

2 , 214 ]).

I Φ(w) ∈ W(w) for each w ∈ SMIGN .

I Φ(w) ∈ C(w) for each w ∈ CIGN .



The square operator

I Let a = (a1, . . . , an) and b = (b1, . . . , bn) with a ≤ b.

I Then, we denote by a�b the vector

a�b := ([a1, b1] , . . . , [an, bn]) ∈ I (R)N

generated by the pair (a, b) ∈ RN × RN .

I Let A,B ⊂ RN . Then, we denote by A�B the subset ofI (R)N defined by

A�B := {a�b|a ∈ A, b ∈ B, a ≤ b} .

I For a multi-solution F : GN � RN we defineF� : IGN � I (R)N by F� = F(w)�F(w) for each w ∈ IGN .



Square solutions and related resultsI C�(w) = C (w)�C (w) for each w ∈ IGN .

Example: N = {1, 2}, w({1}) = [0, 1],w({2}) = [0, 2],w(N) = [4, 8].

(2, 2) ∈ C (w), (31

2, 4

1

2) ∈ C (w).

(2, 2)�(31

2, 4

1

2) = ([2, 3

1

2], [2, 4

1

2]) ∈ C (w)�C (w).

I C(w) = C�(w) for each w ∈ IBIGN .I W�(w) = W (w)�W (w) for each w ∈ IGN .

I C(w) ⊂ W�(w) for each w ∈ IGN .I C�(w) =W�(w) for each w ∈ CIGN .I W(w) ⊂ W�(w) for each w ∈ CIGN .



Classical cooperative gamesTheorem (Shapley (1971) and Shapley-Weber-Ichiishi (1981,1988)):Let v ∈ GN . The following five assertions are equivalent:

(i) < N, v > is convex.

(ii) For all S1,S2,U ∈ 2N with S1 ⊂ S2 ⊂ N \ U

v(S1 ∪ U)− v(S1) ≤ v(S2 ∪ U)− v(S2).

(iii) For all S1,S2 ∈ 2N and i ∈ N such that S1 ⊂ S2 ⊂ N \ {i}

v(S1 ∪ {i})− v(S1) ≤ v(S2 ∪ {i})− v(S2).

(iv) Each marginal vector mσ(v) of the game v with respect tothe permutation σ belongs to the core C (v).

(v) W (v) = C (v).



Basic characterizations for convex interval games

Theorem:Let w ∈ IGN be such that |w | ∈ GN is supermodular. Then, thefollowing three assertions are equivalent:

(i) w ∈ IGN is convex.

(ii) For all S1,S2,U ∈ 2N with S1 ⊂ S2 ⊂ N \ U

w(S1 ∪ U)− w(S1) 4 w(S2 ∪ U)− w(S2).

(iii) For all S1,S2 ∈ 2N and i ∈ N such that S1 ⊂ S2 ⊂ N \ {i}

w(S1 ∪ {i})− w(S1) 4 w(S2 ∪ {i})− w(S2).



Basic characterizations of convex interval games

Proposition:Let w ∈ IGN . Then the following assertions hold:

(i) A game < N,w > is supermodular if and only if its bordergames < N,w > and < N,w > are convex.

(ii) A game < N,w > is convex if and only if its length game< N, |w | > and its border games < N,w >, < N,w > areconvex.

(iii) A game < N,w > is convex if and only if its border game< N,w > and the game < N,w − w > are convex.




Theorem: Let w ∈ IBIGN . Then, the following assertions areequivalent:

(i) w is convex.

(ii) |w | is supermodular and C(w) =W�(w).

Proof: By (ii) of Proposition, w is convex if and only if |w | ,w andw are convex. Clearly, the convexity of |w | is equivalent with itssupermodularity.Further, w and w are convex if and only if W (w) = C (w) andW (w) = C (w).These equalities are equivalent with W�(w) = C�(w). Finally,since w is I-balanced by hypothesis, we have C(w) =W�(w).




Theorem: Let w ∈ IGN . Then, the following assertions areequivalent:

(i) w is convex.

(ii) |w | is supermodular and mσ(w) ∈ C(w) for all σ ∈ Π(N).

Proposition: Let w ∈ CIGN . Then, W(w) ⊂ C(w).Proof: By the above theorem we have mσ(w) ∈ C(w) for eachσ ∈ Π(N). Now, we use the convexity of C(w).



Population interval monotonic allocation schemes (pmias)(inspired by Sprumont (1990))

For a game w ∈ IGN and a coalition S ∈ 2N \ {∅}, the intervalsubgame with player set T is the game wT defined bywT (S) := w(S) for all S ∈ 2T .

TIBIGN : class of totally I-balanced interval games (intervalgames whose all subgames are I-balanced) with player set N.

We say that for a game w ∈ TIBIGN a schemeA = (AiS)i∈S ,S∈2N\{∅} with AiS ∈ I (R)N is a pmias of w if

(i)∑

i∈S AiS = w(S) for all S ∈ 2N \ {∅},(ii) AiS 4 AiT for all S ,T ∈ 2N \ {∅} with S ⊂ T and for each

i ∈ S .



Population interval monotonic allocation schemes

A pmias allocates a larger payoff to each player as the coalitionsgrow larger.

In order to take the possibility of partial cooperation a pmiasspecifies not only how to allocate w(N) but also how to allocatew(S) of every coalition S ∈ 2N \ {∅}.

I We say that for a game w ∈ CIGN an imputationI = (I1, . . . , In) ∈ I(w) is pmias extendable if thereexists a pmas A = (AiS)i∈S ,S∈2N\{∅} such thatAiN = Ii for each i ∈ N.

Theorem: Let w ∈ CIGN . Then each element I of W(w) isextendable to a pmias of w .



Population interval monotonic allocation schemesExample: Let w ∈ CIGN with w(∅) = [0, 0],w({1}) = w({2}) = w({3}) = [0, 0],w({1, 2}) = w({1, 3}) = w({2, 3}) = [2, 4] and w(N) = [9, 15]. Itis easy to check that the interval Shapley value for this gamegenerates the pmias depicted as

N{1, 2}{1, 3}{2, 3}{1}{2}{3}

1 2 3[3, 5] [3, 5] [3, 5][1, 2] [1, 2] ∗[1, 2] ∗ [1, 2]∗ [1, 2] [1, 2]

[0, 0] ∗ ∗∗ [0, 0] ∗∗ ∗ [0, 0]

.



Classical big boss games versus big boss interval games

Classical big boss games (Muto et al. (1988), Tijs (1990)):< N, v > is a big boss game with n as big boss if

(i) v ∈ GN is monotonic, i.e.,v(S) ≤ v(T ) if for each S ,T ∈ 2N with S ⊂ T ;

(ii) v(S) = 0 if n /∈ S ;

(iii) v(N)− v(S) ≥∑

i∈N\S(v(N)− v(N \ {i}))for all S ,T with n ∈ S ⊂ N.

Big boss interval games:< N,w > is a big boss interval game if < N,w > and< N,w − w > are classical (total) big boss games.BBIGN : the class of big boss interval games.Marginal contribution of each player i ∈ N to the grand coalition:Mi (w) := w(N)− w(N \ {i}).



Properties of big boss interval games

Theorem: Let w ∈ SMIGN . Then, the following conditions areequivalent:

(i) w ∈ BBIGN .

(ii) < N,w > satisfies

(a) Veto power property:w(S) = [0, 0] for each S ∈ 2N with n /∈ S .

(b) Monotonicity property:w(S) 4 w(T ) for each S ,T ∈ 2N with n ∈ S ⊂ T .

(c) Union property:

w(N)− w(S) <∑

i∈N\S

(w(N)− w(N \ {i}))

for all S with n ∈ S ⊂ N.



T -value (inspired by Tijs(1981))

I the big boss interval point: B(w) := ([0, 0], . . . , [0, 0],w(N));

I the union interval point:

U(w) := (M1(w), . . . ,Mn−1(w),w(N)−n−1∑i=1

Mi (w)).

I The T -value T : BBIGN → I (R)N is defined by

T (w) :=1

2(U(w) + B(w)).



Holding situations with interval data

Holding situations: one agent has a storage capacity and otheragents have goods to store to generate benefits.In classical cooperative game theory, holding situations aremodeled by using big boss games (Tijs, Meca and Lopez (2005)).For a holding situation with interval data one can construct aholding interval game which turns out to be a big boss intervalgame.

Example: Player 3 is the owner of a holding house which hascapacity for one container. Players 1 and 2 have each onecontainer which they want to store. If player 1 is allowed to storehis/her container, then the benefit belongs to [10, 30] and if player2 is allowed to store his/her container, then the benefit belongs to[50, 70].



Example continues ...

The situation described corresponds to an interval game as follows:

I The interval game < N,w > with N = {1, 2, 3} andw(S) = [0, 0] if 3 /∈ S , w(∅) = w({3}) = [0, 0],w({1, 3}) = [10, 30] and w(N) = w({2, 3}) = [50, 70] is a bigboss interval game with player 3 as big boss.

I B(w) = ([0, 0], [0, 0], [50, 70]) andU(w) = ([0, 0], [40, 40], [10, 30]) are the elements of theinterval core.

I T (w) = ([0, 0], [20, 20], [30, 50]) ∈ C(w).



Bi-monotonic interval allocation schemes (inspired byBranzei, Tijs and Timmer (2001))

I Pn: the set {S ⊂ N|n ∈ S} of all coalitions containing the bigboss.

Take a game w ∈ BBIGN with n as a big boss.We call a scheme B := (BiS)i∈S ,S∈Pn an (interval) allocationscheme for w if (BiS)i∈S is an interval core element of thesubgame < S ,w > for each coalition S ∈ Pn. Such an allocation

scheme B = (BiS)i∈S,S∈Pn is called a bi-monotonic (interval)allocation scheme (bi-mias) for w if for all S ,T ∈ Pn with S ⊂ Twe have BiS < BiT for all i ∈ S \ {n}, and BnS 4 BnT .Remark: In a bi-mias the big boss is weakly better off in largercoalitions, while the other players are weakly worse off.



Bi-monotonic interval allocation schemesI We say that for a game w ∈ BBIGN with n as a big boss, an

imputation I = (I1, . . . , In) ∈ I(w) is bi-mias extendable ifthere exists a bi-mas B = (BiS)i∈S,S∈Pn such that BiN = Ii foreach i ∈ N.

Theorem: Let w ∈ BBIGN with n as a big boss and let I ∈ C(w).Then, I is bi-mias extendable.

Example continues: The T -value generates a bi-mias representedby the following matrix:

N{1, 3}{2, 3}{3}

1 2 3

[0, 0] [20, 20] [30, 50][5, 15] ∗ [5, 15]∗ [25, 35] [25, 35]∗ ∗ [0, 0]

.



Airport situations with interval dataIn airport situations, the costs of the coalitions are considered(Driessen (1988)):

I One runway and m types of planes (P1, . . . ,Pm pieces of therunway: P1 for type 1, P1 and P2 for type 2, etc.).

I Tj < [0, 0]: the interval cost of piece Pj .

I Nj : the set of players who own a plane of type j .

I nj : the number of (owners of) planes of type j .

I < N, d > is given byN = ∪mj=1Nj : the set of all users of the runway;

d(∅) = [0, 0], d(S) =∑j

i=1 Ti

if S ∩ Nj 6= ∅, S ∩ Nk = ∅ for all j + 1 ≤ k ≤ m.

S needs the pieces P1, . . . ,Pj of the runway. The interval cost of

the used pieces of the runway is∑j

i=1 Ti .



Airport situations with interval data

Formally, d =∑m

k=1 Tku∗∪mr=kNr

, where

u∗K (S) :=

{1, K ∩ S 6= ∅0, otherwise.

Interval Baker-Thompson allocation for a player i of type j :

γi :=

j∑k=1

(m∑

r=k

nr )−1Tk .

Proposition: Interval Baker-Thompson allocation agrees with theinterval Shapley value Φ(d).



Airport situations with interval data

Proposition: Let < N, d > be an airport interval game. Then,< N, d > is concave.

Proof: It is well known that non-negative multiples of classical dualunanimity games are concave (or submodular). By formaldefinition of d the classical games d =

∑mk=1 T ku

∗k,m and

|d | =∑m

k=1 |Tk | u∗k,m are concave because T k ≥ 0 and |Tk | ≥ 0for each k , implying that < N, d > is concave.

Proposition: Let (N, (Tk)k=1,...,m) be an airport situation withinterval data and < N, d > be the related airport interval game.Then, the interval Baker-Thompson rule applied to this airportsituation provides an allocation which belongs to C(d).



Example:< N, d > airport interval game interval costs: T1 = [4, 6],T2 = [1, 8],d(∅) = [0, 0], d(1) = [4, 6], d(2) = d(1, 2) = [4, 6] + [1, 8] = [5, 14],d = [4, 6]u∗{1,2} + [1, 8]u∗{2},

Φ(d) = ( 12 ([4, 6] + [0, 0]), 1

2 ([1, 8] + [5, 14])) = ([2, 3], [3, 11]),γ = ( 1

2 [4, 6], 12 [4, 6] + [1, 8]) = ([2, 3], [3, 11]) ∈ C(d).

Figure:



Sequencing situations with interval data

Sequencing situations with one queue of players, each with onejob, in front of a machine order. Each player must have his/her jobprocessed on this machine, and for each player there is a costaccording to the time he/she spent in the system (Curiel, Pederzoliand Tijs (1989)).

A one-machine sequencing interval situation is described as a4-tuple (N, σ0, α, p),σ0: a permutation defining the initial order of the jobsα = ([αi , αi ])i∈N ∈ I (R+)N , p = ([p

i, pi ])i∈N ∈ I (R+)N : vectors

of intervals with αi , αi representing the minimal and maximalunitary cost of the job of i , respectively, p

i, pi being the minimal

and maximal processing time of the job of i , respectively.




I To handle such sequencing situations, we propose to use eitherthe approach based on urgency indices or the approach basedon relaxation indices. This requires to be able to compute

either ui =[αipi

, αipi

](for each i ∈ N) or ri =

[piαi, piαi

](for each

i ∈ N), and such intervals should be pair-wise disjoint.

Interval calculus: Let I , J ∈ I (R+).We define · : I (R+)× I (R+)→ I (R+) by I · J := [I J, I J].Let Q :=

{(I , J) ∈ I (R+)× I (R+ \ {0}) | I J ≤ I J

}.

We define ÷ : Q → I (R+) by IJ := [ IJ ,

IJ

] for all (I , J) ∈ Q.



Sequencing situations with interval dataExample (a): Consider the two-agent situation withp1 = [1, 4], p2 = [6, 8], α1 = [5, 25], α2 = [10, 30]. We can computeu1 =

[5, 25

4

], u2 =

[53 ,

154

]and use them to reorder the jobs as the

intervals are disjoint.

Example (b): Consider the two-agent situation withp1 = [1, 3], p2 = [4, 6], α1 = [5, 6], α2 = [11, 12]. Here, we cancompute r1 =

[15 ,

12

], r2 =

[4

11 ,12

], but we cannot reorder the jobs

as the intervals are not disjoint.

Example (c): Consider the two-agent situation withp1 = [1, 3], p2 = [5, 8], α1 = [5, 6], α2 = [10, 30]. Now, r1 is definedbut r2 is undefined. On the other hand, u1 is undefined and u2 isdefined, so no comparison is possible; consequently, the reorderingcannot take place.



Sequencing situations with interval dataLet i , j ∈ N. We define the interval gain of the switch of jobs i andj by

Gij :=

{αjpi − αipj , if jobs i and j switch[0,0], otherwise.

The sequencing interval game:

w :=∑

i ,j∈N:i<j

Giju[i ,j].

Gij ∈ I (R) for all switching jobs i , j ∈ N andu[i ,j] is the unanimity game defined as:

u[i ,j](S) :=

{1, if {i , i + 1, ..., j − 1, j} ⊂ S0, otherwise.




I The interval equal gain splitting rule is defined byIEGSi (N, σ0, α, p) = 1

2

∑j∈N:i<j

Gij + 12

∑j∈N:i>j

Gij , for each

i ∈ N.

Proposition: Let < N,w > be a sequencing interval game. Then,i) IEGS(N, σ0, α, p) = 1

2 (m(1,2...,n)(w) + m(n,n−1,...,1)(w)).ii) IEGS(N, σ0, α, p) ∈ C(w).

Proposition: Let < N,w > be a sequencing interval game. Then,< N,w > is convex.Example: Consider the interval situation with N = {1, 2},σ0 = {1, 2}, p = (2, 3) and α = ([2, 4], [12, 21]).The urgency indices are u1 = [1, 2] and u2 = [4, 7], so that the twojobs may be switched.We have:G12 = [18, 30], IEGS(N, σ0, α, p) = ([9, 15], [9, 15]) ∈ C(w).



Bankruptcy situations with interval data

In a classical bankruptcy situation, a certain amount of moneydhas to be divided among some people, N = {1, . . . , n}, who haveindividual claims ci , i ∈ N on the estate, and the total claim isweakly larger than the estate. The corresponding bankruptcy gamevE ,d : vE ,d(S) = (E −

∑i∈N\S di )+ for each S ∈ 2N , where

x+ = max {0, x} (Aumann and Maschler (1985)).

I A bankruptcy interval situation with a fixed set of claimantsN = {1, 2, . . . , n} is a pair (E , d) ∈ I (R)× I (R)N , whereE = [E ,E ] < [0, 0] is the estate to be divided and d is thevector of interval claims with the i-th coordinate di = [d i , d i ](i ∈ N), such that [0, 0] 4 d1 4 d2 4 . . . 4 dn andE <

∑ni=1 d i .

BRIN : the family of bankruptcy interval situations with set ofclaimants N.




We define a subclass of BRIN , denoted by SBRIN , consisting of allbankruptcy interval situations such that

|d(N \ S)| ≤ |E | for each S ∈ 2N with d(N \ S) ≤ E .

I We call a bankruptcy interval situation in SBRIN a strongbankruptcy interval situation. With each (E , d) ∈ SBRIN weassociate a cooperative interval game < N,wE ,d >, defined by

wE ,d(S) := [vE ,d(S), vE ,d(S)] for each S ⊂ N.

SBRIGN : the family of all bankruptcy interval games wE ,d

with (E , d) ∈ SBRIN .




Example: Let (E , d) be a two-person bankruptcy situation.We suppose that the claims of the players are closed intervalsd1 = [70, 70] and d2 = [80, 80], respectively,and the estate is E = [100, 140].Then, the corresponding game < N,wE ,d > is given by

wE ,d(∅) = [0, 0],wE ,d(1) = [20, 60],wE ,d(2) = [30, 70]

and wE ,d(1, 2) = [100, 140].

I This game is supermodular, but is not convex because|wE ,d | ∈ GN is not convex.



How to use interval games and their solutions ininteractive situations

Stage 1 (before cooperation starts):with N = {1, 2, . . . , n} set of participants with interval data ⇒interval game < N,w > and interval solutions ⇒ agreement forcooperation based on an interval solution ψ and signing a bindingcontract (specifying how the achieved outcome by the grandcoalition should be divided consistently with Ji = ψi (w) for eachi ∈ N.

Stage 2 (after the joint enterprise is carried out):The achieved reward R ∈ w(N) is known; apply the agreed uponprotocol specified in the binding contract to determine theindividual shares xi ∈ Ji .Natural candidates for rules used in protocols are bankruptcy rules.




Example:w(1) = [0, 2], w(2) = [0, 1] and w(1, 2) = [4, 8].Φ(w) = ([2, 4 1

2 ], [2, 3 12 ]). R = 6 ∈ [4, 8]; choose proportional rule

(PROP) defined by

PROPi (E , d) :=di∑j∈N dj

E

for each bankruptcy problem (E , d) and all i ∈ N.(Φ1(w),Φ2(w)) +

PROP(R − Φ1(w)− Φ2(w); Φ1(w)− Φ1(w),Φ2(w)− Φ2(w))= (2, 2) + PROP(6− 2− 2; (2 1

2 , 112 ))

= (3 14 , 2

34 ).



Conclusion and future work

The State-of-the-art of interval game literature:

I Branzei R., Dimitrov D. and Tijs S., “Shapley-like values forinterval bankruptcy games”, Economics Bulletin 3 (2003) 1-8.

I Alparslan Gok S.Z., Branzei R., Fragnelli V. and Tijs S.,“Sequencing interval situations and related games”, to appearin Central European Journal of Operations Research (CEJOR).

I Alparslan Gok S.Z., Branzei O., Branzei R. and Tijs S.,“Set-valued solution concepts using interval-type payoffs forinterval games”, to appear in Journal of MathematicalEconomics (JME).



I Alparslan Gok S.Z., Branzei R. and Tijs S., “Convex intervalgames”, Journal of Applied Mathematics and DecisionSciences, Vol. 2009, Article ID 342089, 14 pages (2009a)DOI: 10.1115/2009/342089.

I Alparslan Gok S.Z., Branzei R. and Tijs S., “Big boss intervalgames”, International Journal of Uncertainty, Fuzziness andKnowledge-Based Systems (IJUFKS), Vol. 19, no:1 (2011)pp.135-149.

I Branzei R. and Alparslan Gok S.Z., “Bankruptcy problemswith interval uncertainty”, Economics Bulletin, Vol. 3, no. 56(2008) pp. 1-10.



I Branzei R., Mallozzi L. and Tijs S., “Peer group situationsand games with interval uncertainty”, International Journal ofMathematics, Game Theory, and Algebra, Vol.19, Issues 5-6(2010).

I Branzei R., Tijs S. and Alparslan Gok S.Z., “Somecharacterizations of convex interval games”, AUCO CzechEconomic Review, Vol. 2, no.3 (2008) 219-226.

I Branzei R., Tijs S. and Alparslan Gok S.Z., “How to handleinterval solutions for cooperative interval games”,International Journal of Uncertainty, Fuzziness andKnowledge-based Systems, Vol.18, Issue 2, (2010) 123-132.

I Branzei R., Branzei O., Alparslan Gok S.Z., Tijs S.,“Cooperative interval games: a survey”, Central EuropeanJournal of Operations Research (CEJOR), Vol.18, no.3(2010) 397-411.



I Moretti S., Alparslan Gok S.Z., Branzei R. and Tijs S.,“Connection situations under uncertainty and cost monotonicsolutions”, Computers and Operations Research, Vol.38, Issue11 (2011) pp.1638-1645.

I Branzei R., Alparslan Gk S.Z. and Branzei O., “On theConvexity of Interval Dominance Cores”, to appear in CentralEuropean Journal of Operations Research (CEJOR), DOI:10.1007/s10100-010-0141-z.

I Alparslan Gok S.Z., Branzei R. and Tijs S., “Airport intervalgames and their Shapley value”, Operations Research andDecisions, Issue 2 (2009).

I Alparslan Gok S.Z., Miquel S. and Tijs S., “Cooperationunder interval uncertainty”, Mathematical Methods ofOperations Research, Vol. 69, no.1 (2009) 99-109.



I Alparslan Gok S.Z., “Cooperative interval games”, PhDDissertation Thesis, Institute of Applied Mathematics, MiddleEast Technical University, Ankara-Turkey (2009).

I Alparslan Gok S.Z., Branzei R. and Tijs S., “The intervalShapley value: an axiomatization”, Central European Journalof Operations Research (CEJOR), Vol.18, Issue 2 (2010) pp.131-140.

Future work:

I Promising topic (interesting open problems exist to begeneralized in the theory of cooperative interval games).

I Other OR situations and combinatorial optimization problemswith interval data can be modeled by using cooperativeinterval games, e.g., flow, linear production, holdingsituations, financial and energy markets.


References

References[1]Alparslan Gok S.Z., Cooperative Interval Games: Theory andApplications, Lambert Academic Publishing (LAP), Germany(2010) ISBN:978-3-8383-3430-1.[2]Aumann R. and Maschler M., Game theoretic analysis of abankruptcy problem from the Talmud, Journal of EconomicTheory 36 (1985) 195-213.[3] Bondareva O.N., Certain applications of the methods of linearprogramming to the theory of cooperative games, ProblemlyKibernetiki 10 (1963) 119-139 (in Russian).[4] Branzei R., Dimitrov D. and Tijs S., Models in CooperativeGame Theory, Springer, Game Theory and Mathematical Methods(2008).[5] Branzei R., Tijs S. and Timmer J., Information collectingsituations and bi-monotonic allocation schemes, MathematicalMethods of Operations Research 54 (2001) 303-313.


References

References

[6]Curiel I., Pederzoli G. and Tijs S., Sequencing games, EuropeanJournal of Operational Research 40 (1989) 344-351.[7]Driessen T., Cooperative Games, Solutions and Applications,Kluwer Academic Publishers (1988).[8] Gillies D. B., Solutions to general non-zero-sum games. In:Tucker, A.W. and Luce, R.D. (Eds.), Contributions to the theoryof games IV, Annals of Mathematical Studies 40. PrincetonUniversity Press, Princeton (1959) pp. 47-85.[9]Ichiishi T., Super-modularity: applications to convex games andto the greedy algorithm for LP, Journal of Economic Theory 25(1981) 283-286.


References

References

[10] Muto S., Nakayama M., Potters J. and Tijs S., On big bossgames, The Economic Studies Quarterly Vol.39, No. 4 (1988)303-321.[11]Shapley L.S., On balanced sets and cores, Naval ResearchLogistics Quarterly 14 (1967) 453-460.[12] Shapley L.S., Cores of convex games, International Journal ofGame Theory 1 (1971) 11-26.[13] Sprumont Y., Population Monotonic Allocation Schemes forCooperative Games with Transferable Utility, Games and EconomicBehavior 2 (1990) 378-394.[14] Tijs S., Bounds for the core and the τ -value, In: MoeschlinO., Pallaschke D. (eds.), Game Theory and MathematicalEconomics, North Holland, Amsterdam (1981) pp. 123-132.


References

References

[15] Tijs S., Big boss games, clan games and information marketgames. In:Ichiishi T., Neyman A., Tauman Y. (eds.), Game Theoryand Applications. Academic Press, San Diego (1990) pp.410-412.[16]Tijs S., Meca A. and Lopez M.A., Benefit sharing in holdingsituations, European Journal of Operational Research 162 (1)(2005) 251-269.[17] von Neumann, J. and Morgernstern, O., Theory of Games andEconomic Behaviour, Princeton: Princeton University Press(1944).[18] Weber R., Probabilistic values for games, in Roth A.E. (Ed.),The Shapley Value: Essays in Honour of Lloyd S. Shapley,Cambridge University Press, Cambridge (1988) 101-119.

Coalitional Games with Interval-Type Payoffs: A Survey

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