Cooperation Under Interval
Uncertainty∗
S.Z.Alparslan-Go k†S.Miquel‡S.Tijs
Abstract
In this paper,the classical theory of two-person cooperative games is extended to two-person cooperative games with interval uncertainty.
The core,balancedness,superadditivity and related topics are stud-
ied.Solutions called-values are introduced and characterizations
are given.Some economical situations with an interval character are
considered.
Keywords:cooperative game theory,interval 托福听力评分
uncertainty,core, value,balancedness.
1Introduction
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 situati寂寞如歌
ons here are considered from a deterministic point of view.For further information about classical cooperative game theory the reader is referred to the books by Branzei et al.(2005)and Tijs(2003).However,in most ∗The authors thank Rodica Branzei for her uful comments on earlier versions.
†Institute of Applied Mathematics,Middle East Technical University,06531Ankara, Turkey,e-mail:alzeynep@
‡Department of Mathematics,University of Lleida,Spain,e-mail: smiquel@matematica.udl.es
CentER and Department of Econometrics and OR,Tilburg University,P.O.Box90153, 5000LE Tilburg,The Netherlands,e-mail:S.H.Tijs@uvt.nl
1
economical situations potential rewards or costs are n卓越的英语
ot known precily,but often it is possible to estimate intervals to which they belong.In Yager and Kreinovich(2000)an algorithm for fair division un
der interval uncertainty is prented using the work on interval analysis by Moore(1979).Cooperative games arising from bankruptcy situations with interval uncertainty,called (cooperative)interval games,were introduced and analyzed by Branzei et al.(2003)and Branzei et al.(2004).In a classical bankruptcy situation a certain amount of money(estate)has to be divided among some people (claimants)who have individual claims on the estate,and the total claim is weakly larger than the estate(cf.Aumann and Maschler(1985),Curiel et al.(1987),O’Neill(1982)).When the estate and/or the claims may belong to intervals of real numbers we have bankruptcy situations under interval uncertainty.In Carpente et al.(2005)a method is propod to associate a coalitional interval game to each strategic game.Throughout the above lit-erature we canfind motivation交谊舞慢三步教学视频
s,from different points of view,for the study of interval games.Here,a cooperative interval game is defined as an ordered pair<N,w>where N is the t of players,and w is the characteristic function which assigns to each coalition S a clod interval w(S)in R.We introduce the notion of the core t of a cooperative interval game and various notions of balancedness.Then we focus on two-person(cooperative)interval games and extend to the games well-known results for classical two-person cooperative games.Moreover,we define and analyze specific solution con-cepts on the class of two-person interval games,such as the mini-core t and the-values.The mini-core t is determined by considering the upper bound of the worths of the one-player coalitions in the two-pers
on ca.If a mini-core allocation is propod,then no one-player coalition has any in-centive to split offfrom the grand coalition for each lection of the interval games.
The paper is organized as follows.In Section2we recall basic definitions and results on balancedness for classical cooperative games.In Section3we introduce some definitions for n-person cooperative games under interval un-certainty and focus on balancedness.Section4deals with two-person interval games and their solutions:balancedness,the mini-core t and its relation with the core t,the-values and their axiomatic characterizations.We conclude in Section5with some remarks on further rearch.
2
2Preliminaries on classical games in
coalitional form
We give in the following some definitions and a theorem concerning classical games in coalitional form.For an extensive description of classical games in coalitional form e Tijs (2003)and Branzei et al.(2005).
A cooperative n -person game in coalitional form is an ordered pair <N,v >,where N ={1,2,...,n }(the t of players)and v :2N →R is a map,as-signing to each coalition S ∈2N a real number,such that v (∅)=0.This function v is called the characteristic function of the game,v (S )is called the worth (or value)of coalition S .Often we identify a game <N,v >with its characteristic function v .
The t G N of coalitional games with player t N forms with the usual opera-tors of addition and scalar multiplication of functions a (2|N |−1)-dim装配工艺
ensional linear space;a basis of this space is supplied by the unanimity games u T (or <N,u T >),T ∈2N \{∅},which are defined by
u T (S )= 1if T ⊂S 0otherwi.
One can easily check that for each v ∈G N we have
v = T ∈2N \∅c T u T with c T = S :S ⊂T (−1)|T |−|S |v (S ).
A payoffvector x ∈R n is called an imputation for the game <N,v >if (i)x is individually rational ,i.e.,x i ≥v ({i })for all i ∈N ,
(ii)x is efficient (Pareto optimal),i.e., n
i =1x i =v (N ).
The t of imputations of <N,v >is denoted by I (v ).Note that I (v )=∅if and only if v (N )< i ∈N v ({i }).
The core of a game (cf.四季抗病毒口服液
Gillies (1953))is a central t-valued solution concept in game theory.
The core of a game <N,v >is the t
C (v )= x ∈I (v )|
i ∈S
x i ≥v (S )for all S ∈2N \{∅} .If x ∈C (v ),then no coalition S =N has any incentive to split offif x is the propod reward allocation in N ,becau the total amount i ∈S x i allocated
3
to S is not smaller than the amount v (S )which the players can obtain by forming the subcoalition.
For a two-person game <N,v >,I (v )=C (v ).A map :2N \{∅}→R +is called a balanced map if S ∈2N \{∅}(S )e S =e N .Here e S is the characteristic vector for coaliton S with
e S i = 1i
f i ∈S 0if i ∈N \S.
A collection
B of coalitions is called a balanced collection if there is a balanced map such that B = S ∈2N \{∅}|(S )>0 .
An n -person game <N,v >is called a balanced game if for each balanced map :2N \{∅}→R +we have S (S )v (S )≤v (N ).
The importance of this notion becomes clear in the following theorem proved by Bondareva (1963)and Shapley (1967).This theorem characterizes games with a non-empty core.
Theorem 2.1.Let <N,v >be an n -person game.Then the following two asrtions are equivalent:
(i)C (v )=∅,
(ii)<N,v >is a balanced game.
Let (N )be the t of all permutations :N →N .The t
P
(i )={r ∈N |−1(r )<−1(i )}consists of all predecessors of i with respect to the permutation .
Let v ∈G N and ∈(N ).The marginal vector m (v )∈R n with respect to
and v has as i -th coordinate m i (v )=v (P (i )∪{i })−v (P (i ))for each
i ∈N .
The Shapley value (cf.Shapley (1967))is one of the most interesting one-point solution concepts in classical cooperative game theory.The Shapley value associates to each n -person game one (payoff)vector in R n .
The Shapley value (v )of a game v ∈G N is the average of the marginal vectors of the (v )=1n ! ∈(N )
m (v ).4
Marginal vectors of a two-person game<N,v>are
m(12)(v)=(v({1}),v({1,2})−v({1})), and
m(21)(v)=(v({1,2})−v({1}),v({2})). For a two person game<N,v>we have
i(v)=v({i})+v({1,2})−v({1})−v({2})
2
,i={1,2}.
Note that for a two-person game<N,v>,the Shapley value is the standard solution which is in the middle of the core and the marginal vectors are the extreme points of the core who 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=∅.In a superadditive game it is advantageous for the players to cooperate.
A two-person cooperative game<N,v>is superadditive if and only if v({1})+v({2})≤v({1,2})holds.Note that a two-person cooperative game <N,v>is superadditive if and only if the game is balanced.
3Cooperative games under interval
uncertainty
In the following we will develop a theory of cooperation under interval un-certainty,inspired by the classical cooperative game theory(cf.Branzei et al.(2005)and Tijs(2003)).
A cooperative n-person interval game in coalitional form is an ordered pair <N,w>where N:={1,2,...,n}is the t of players,and w:2N→I(R) is the characteristic function which assigns to each coalition S∈2N a clod interval w(S)∈I(R)where I(R)is the t of all clod intervals in R such that w(∅)=[0,0].
For each S∈2N,the worth t(or worth interval)of the coalition S in the in-terval game,w(S),is a clod interval which will be denoted by[w(S),w(S)], where w(S)is the lower bound and w(S)is the upper bound of w(S). Note that if all the worth intervals are degenerate ,w(S)=w(S), then the interval game<N,w>corresponds to the classical cooperative game<N,v>where v(S)=w(S).
5
Let <N,w >be an interval game;then v :2N →R is called a lection of w if v (S )∈w (S )for each S ∈2N .We denote the t of lections of w by Sel (w ).
The imputation t of an interval game <N,w >is defined by
I (w )=∪{I (v )|v ∈Sel (w )}.
The core t of an interval game <N,w >is defined by
C (w )=∪{C (v )|v ∈Sel (w )}.
C (w )=∅if and only if there exists a v ∈Sel (w )with C (v )=∅.
The family of all interval games with player t N is denoted by IG N .
If all the worth intervals of an interval game w ∈IG N are degenerate inter-vals,then I (w )=I (w )=I (w )and C (w )=C (w )=C (w ).
Note that v (S )∈w (S )is a real number,but w (S )=[w (S ),w (S )]is a de-generate interval which is a t consisting of one point.
An interval game <N,w >is strongly balanced if for each balanced map it holds that (S )w (S )≤w (N ).The family of all strongly balanced interval games with player t N is denoted by BIG N .
Proposition 3.1.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 Theorem 2.1.
(i )⇔(iii )follows using the inequalities w (N )≤v (N )≤w 盐酸伊托必利片
(N ), (S )w (S )≤ (S )v (S )≤ (S )w (S )for each balanced map .
It follows from Proposition 3.1that for a strongly balanced game <N,w >,C (w )=∅since for all v ∈Sel (w ),C (v )=∅.
We call an interval game <N,w >strongly unbalanced,if there ex怀孕可以吃木瓜吗
ists a balanced map such that (S廉洁廉政
)w (S )>w (N ).Then,C (v )=∅for all v ∈Sel (w ),which implies that C (w )=∅.
If all the worth intervals of an interval game <N,w >are degenerate in-tervals then strongly balancedness corresponds to balancedness and strongly unbalancedness corresponds to unbalancedness in classical cooperative game <N,v >.
6
