THE NUCLEOLUS AND THE τ-VALUE OF INTERVAL GAMES Текст научной статьи по специальности «Математика»

Elena B. Yanovskaya

St.Petersburg Institute for Economics and Mathematics, Russian Academy of Sciences,

Tchaikovsky st. 1, St.Petersburg, 191187, Russia E-mail: eyanov@emi.nw.ru

Abstract Interval cooperative games model situations with cooperation in which the agents do not know for certain their coalitional payoffs, they know only bounds for the payoffs. Cooperative interval games have been introduced and studied in (Alparslan Gok, Branzei and Tijs, 2008, Branzei, Tijs and Alparslan Gok, 2008). Each interval game is defined by two cooperative games - the lower and the upper games - whose characteristic function values are bounds of the coalitional payoffs. Solutions for interval games are defined also in interval form. A TU game value p generates the interval value for the corresponding class of interval games if the value of the upper game dominates the value of the lower game. In (Alparslan Gok, Miquel and Tijs,

2009) it was shown how some monotonicity properties of some TU game values provide existence of the corresponding interval values for the class of convex interval games. However, the nucleolus and the r-value on this class do not possess such properties. Thus, in this paper the nucleolus for the interval values is defined as the result of the lexicographic minimization of the joint excess vector for upper and lower games. Its existence has been proved. The existence of the r-value is proved on the subclass of convex interval game generated by totally positive upper and lower games.

Keywords: interval cooperative game, convex game, totally positive game, nucleolus, r-value.

1. Introduction

There are many real-life situations in which people or businesses are uncertain about their coalitional payoffs. Situations with uncertain payoffs in which the agents cannot await the realizations of their coalition payoffs cannot be modeled according to classical game theory. Several models that are useful to handle uncertain payoffs exist in the game theory literature. We refer here to chance-constrained games (Charnes and Granot, 2007), cooperative games with stochastic payoffs (Suis, Borm, de Waegenaere and Tijs, 1999) cooperative games with random payoffs (Timmer, Borm and Tijs, 2005). In all these models stochastics plays an important role.

Interval cooperative games are models of cooperation where only bounds for payoffs of coalitions are known with certainty. Such games are called cooperative interval games. Let I(R) be the set of all compact intervals of the real line R. Formally, a cooperative interval game in coalitional form (Alparslan Gok, Miquel and Tijs, 2009) is an ordered pair (N,w) where N = {1, 2,...,n} is the set of players, and w : 2N ^ I(R) is the characteristic function such that w(0) = [0, 0], where I(R) is the set of all nonempty, compact intervals in R. For each S G 2N, the worth set (or worth interval) w(S) of the coalition S in the interval game (N, w) is of the form [w(S'), w(S')]. We denote by IGjy the family of all interval games

with player set N. Note that if all the worth intervals are degenerate intervals,

i.e. w(S) = w(S) for each S G 2N, then the interval game (N, w) corresponds in a natural way to the classical cooperative game (N, v) where v(S) = w(S) for all S G 2N. Some classical TU-games associated with an interval game w G IGN will play a key role, namely the border games (N,w}, (N,w) and the length game (N, |w|), where |m;| (S') = w(S) — w(S) for each S G 2N. Note that w = w + |w|. An interval solution concept F on IGN is a map assigning to each interval game (N, w) G IGn a subset of I(R)N.

Cooperative interval games are very suitable to describe real-life situations in which people or firms that consider cooperation have to sign a contract when they cannot pin down the attainable coalition payoffs, knowing with certainty only their lower and upper bounds. The contract should specify how the players’ payoff shares will be obtained when the uncertainty of the worth of the grand coalition is resolved at an ex post stage.

Note that the agreement on a particular interval allocation (I\,I2,...,In) based on an interval solution concept merely says that the payoff xi that player i will receive in the interim or ex post stage is in the interval Ii. This is a very weak contract to settle cooperation within the grand coalition. It can be considered as a first step of cooperation, where the following step should transform an interval allocation into a classical payoff vector. Such procedures are described in (Branzei, Tijs and Alparslan Gok, 2008).

The study of interval game solutions begins with extensions of classical theory of cooperative game solutions to interval games. For example, we can apply some single-valued solution concept to both border games, and in the case when the solution of the upper game weakly dominates that of the lower game, the corresponding interval vector could be admitted as the interval solution, generated by a classical cooperative game solution. Just in this manner the interval Shapley value for convex interval games was defined in (Alparslan Gok, Branzei and Tijs, 2009). The same approach can be applied to the extension of set-valued solutions as well (Alparslan Gok, Branzei and Tijs, 2008,2009).

Naturally, the problem of existence of such interval solution arises. In fact if for some interval game (N, w) the characteristic function values of the lower and upper games on the grand coalition coincide, i.e., w(N) = w(N), then for any single-valued classical solution ip the (vector) inequality ip(N,w) < <f(N,w) is impossible, and this approach cannot be applied to the extension of the solution p to the interval game (N, w).

It is clear that the possibility of the extension of a classical cooperative game solution to interval games depends both on the class of interval games into consideration and on monotonicity properties of the classical cooperative game solution itself. Thus, in the paper by (Alparslan Gok, Branzei and Tijs, 2009) the class of convex interval games was introduced. It turned out that interval games was introduced. It turned out that the most known cooperative game solutions such as the core, the Shapley value, the Weber set, and are extendable to the class of convex interval games. At last, in (Yanovskaya, Branzei and Tijs, 2010) it was shown that the Dutta-Ray solution also can be extended to the interval games.

However, both the prenucleolus and the т-value are not aggregate monotonic on the class of convex TU games (Hokari, 2000, Hokari and van Gellekom, 2002)1. Therefore, interval analogues of these solutions either should be defined by another manner, or perhaps they exist in some other class of interval games. Both approaches are used in the paper: the nucleolus of a convex interval game is defined by lexicographical minimization of the lexmin relation on the set of joint excess vectors of lower and upper games. On the other hand, the т-value is shown to satisfy extend-ability condition on a subclass of convex games - on the class of totally positive convex games.

The interval nucleolus is determined in Section 2, and the proof of existence of the interval т-value on the class of interval totally positive games is given in Section 3.

2. The Nucleolus for interval games

An interval game is an ordered triple (N, (w,w)), where Ж is a finite set of players, w,w:2n ^ R are the lower and upper characteristic functions satisfying inequalities w(S) < w(S) for each coalition S C N. Cooperative game with transferable utilities (TU) (N,w}, (N,w) are called, respectively,the lower and the upper games of the interval game (N, (w,w)).

Denote be Gn an arbitrary class of TU games with the players’ set N, and by IGn denote the class of interval games with the players’ set N such that for every interval game (N, (w, w)) Є IGn both the lower and the upper games (N, w), (N, w) belong to the class Gn .

Denote by X(N, w), X(N, w) the sets of feasible payoff vectors of the lower and the upper games, and by Y(N, w), Y(N, w) - the set of efficient payoff vectors:

X(N,w) = {x є Rw | < w(N)},

X(N,w) = {x є RN | £ieN^ <

Y(N, w) = {x e X(N, w) | J2ieN xi =

Y (N, w) = {хе X (N, w) j Еієдг xi = w{N)}.

Definition 1. A single-valued solution (value) ф for the class IGn of interval games is a mapping assigning to every interval game (N, (w,w)) Є IGn я pair of payoff vectors (w, w)) = (x, у) Є Rw x Rw, satisfying the conditions x Є X(N, w), у Є X(N, w) and x < y.

It is clear that if Nu(N,w) < Nu(N,w), then (Nu(N,w), Nu(N,w)) Є INuS(N, (w,w)), and this definition is well-defined.

However the Interval Nucleolus Set, in general, is not single-valued. For example,

(^Nu(N,w),Nu(N,w,YNu(-N'm'>y (Nu(N, w, X^^'^), Nu(M, «;)) Є

INuS(N, (w,w)).

Thus, a more precise definition of the interval nucleoli is necessary to obtain its uniqueness.

1 Note that on the class of convex TU games the prenucleolus coincides with the nucleolus. Thus from now upon we will use the term nucleolus.

Note that in Definition 1 for each (x*,y*) G INuS(N, (w,w)) the determining of Nu(N,w, Xv ), Nu(N,w,Yx ) was fulfiled independently, in fact, both nucleoli were calculated under given y*, x* respectively. Since nucleoli express the idea of minimization of relative dissatisfaction of players and coalitions, i.e. excess vectors, in TU games, the interval nucleolus should minimize dissatisfactions at once both in lower and upper games. Thus, we can try to minimize lexicographically the vector of excesses of both games. For each pair of payoff vectors (x,y),x < y,x G X(N,w),y G X(N,w) denote by £{x,y) G M2rl+1~4 the vector of excesses w(S) — x(S), w(T) — y(T), S,T C N, S, T ^ N, 0, arranged in a weakly decreasing manner. Then we come to the following

Definition 2. The interval nucleolus (INu) of an interval game (N, (w,w)) is a pair (x*,y*) of payoff vectors x* G X(N,w),y* G X(N,w) such that x* < y*) and

-£(x*,y*) ^lexmin ~£(x, y) for all x G X(N,w),y G X(N,w),x <y. (1)

Theorem 1. There exists the unique interval nucleolus on the set of convex interval games.

Proof. The proof of the existence of the interval nucleolus is similar to that of Schmeidler (1969) of the existence of the nucleolus for TU games.

Let now (xi, yi) be the solution of the problem (1) without the condition x < y,

i.e.

-£{xi,yi) >~iexmin-£{x,y) for all x G X{N,w),y G X(N,w). (2)

Then x\ = Jf(N, w),yi = J\f(N, w), and there exists a solution of the problem (1). The uniqueness of the solution follows from convexity of the domain {(x, y) | x G X(N,w),y e X(N,w),x <y}. □

Corollary. The interval nucleolus belongs to the interval core.

Proof. For all (x, y) G C(N, (w,w)) we have x < y and

max(w(S') — x(S)) < 0, max(w(S') — y(S)) < 0.

Therefore, for each pair of vectors (z, u) £ C(N,w,w), z < u

£(x,y) ^lexmin E(z,u),

and the maximum of the lexmin relation cannot be out of the interval core.

For the interval nucleolus an analogue of Kohlberg’s characterization (Kohlberg, 1971) can be proved:

For each vectors x G X (N, w),y G X (w) and aGl denote

B0(x, y) = {i G N I xi = yi},

Ba(x) ={SciV| w(S) - x(S) > a},

Ba(y) = {S C N \w(S) - y(S) > a}.

Theorem 2. Given a convex interval game (N, (w,w)) and a pair of payoff vectors (x* ,y*) such that x* < y* of the lower and upper games respectively such that

the collections B_a(x*), Ba(y*) are empty or balanced for all a, then (x*,y*) =

IPN(N,(w,w)). ___

If(x*,y*) = INu(N,w,w), then for any a the collections B_a(x*) U Bo(x*, y*), Ba(y*)L>Bo(x*, y*)} are empty or weakly balanced with positive weights for coalitions from B_a(x*),B0(y*) respectively.

Proof. Let some pair of payoff vectors (x* ,y*) of the lower and upper games respectively and satisfying the inequality x* < y* the collections B_a(x*), Ba(y*) are empty or balanced for every a. Then there are no solutions of linear systems

x(S)>x*(S),SeRa(x*),

y{T)>y*{T),T&Bp{y*) w

for any a, [3.

If (x*,y*) ^ INu(N, (w,w)), then there would exist a pair of payoff vectors of the lower and upper games (x,y),x < y such that

£(x,y) ^lexmin £(x ,y ),

that would contradict unsolvedness of systems (3).

Now let (x*,y*) = INu(N, (w,w)). Then, in particular,

-£(x*,y*) >~ieXmin -£(x,y*) for all x G X(N, w), x < y*, (4)

-£(x*,y*) >~ieXmin -£(x*,y) for all y G X(N,w),x* < y. (5)

Kohlberg’s theorem (Kohlberg, 1971) and relation (4) imply that for every vector x G X(N,w) either the collections B_a(x*) are empty or balanced for all a, or for some a the collection 13a (x*) is not balanced but for any payoff vector of the lower game x, satisfying

-£(x,y*) yiexmin ~£(x, y*) for all x G X(N,w),x < y*,

the inequality x < y* does not hold. The last case means that the system of inequalities

x(S) > x*(S), S G Ba(x*), xj < x*,j G Bq(x*,y*), x(N) = w(N), x ^ x*.

is unsolvable.

This system is equivalent to the following one:

x(S) > x*(S), S G B_a{x*) U Bo(x*,y*), x(N) =w(N),x^x*. (6)

Unsolvedness of system (6) is equivalent to weakly balancedness of the collection

13a (x*) U Bo (x*, y*) with positive weights for coalitions from the collection 13a (x*).

Weakly balancedness of the collections Ba U Bo (x*, y*) with positive weights for coalitions from Ba(y*) is proved analogously. q

Example 1 Let us consider the example of convex TU game from Hokari’s paper (Hokari, 2000), showing the absence of aggregate monotonicity of its nucleolus.

N = {1, 2, 3, 4}, v({i}) = 0Vi G N,

-y({1, 3}) = 0, v(S) = 2 for other S, |S| = 2, (7)

v({1, 2, 3} = 4, v(S) = 6 for other S, IS| = 3, (7)

v(N) = 10.

Then the nucleolus Nu(N,v) = (2, 2, 2, 4). Let (N,v') be the game whose characteristic function v' differs from v only on the grand coalition: v'(N) = 12, v'(S) = v(S) for other S C N.

Then the nucleolus Nu(N, v') = (3, 3, 3, 3).

Consider the interval game {N, (v,v')). For this game the interval nucleolus set

Remark. It is known that the per capita nucleolus of convex TU games satisfies aggregate monotonicity. However, this property does not provide existence of the interval per capita nucleolus as the pair of the nucleoli of the lower and upper games. The example from (Hokari, 2000) shows this fact:

Let us consider the four-person convex game (7). It is easy to show that the per capita nucleolus for this game Nupc(N, v) coincides with its nucleolus and equals Nu(N,v) = (2, 2, 2, 4).

Let {N,vs) be the following four-person symmetric game:

In symmetric games the players have equal gains both for the nucleolus and the per capita nucleolus, i.e. Nupc(N,vs) = (3, 3, 3, 3).

Evidently, the game {N,vs) is convex, and vs(S) > v(S) for all S C N. The difference game {N, vs — v) is determined by

Therefore, this game is also convex, and the triple {N, (v,vs)) is a convex interval game. However

is equal to

INuS(N, (v, v'))

10 — a 10 — a 10 — a 3 ’ 3 ’ 3 ,a

12 — a 12 — a 12 — a 3 ’ 3 ’ 3 ,a

aE [3,4]

and the interval nucleolus

INu(N, (v, v')) = ((2, 2, 2,4), (8/3, 8/3, 8/3, 4))

vs(S)

0, for S, S = 1,

2, for S, |S| = 2,

6, for S, |S| = 3,

12, for S = N.

for S = {1, 3}, {1, 2, 3},N. for other S C N.

Nupc(N, v) = (2, 2, 2, 4) < (3, 3, 3, 3) = Nupc(N, vs).

2. The t-value

For the class of convex TU games the T-value is defined as follows. If {N, v) G Gc then for each i G N

Ti(N, v) = A(v(N) — v(N \ {i})) + (1 — A)v({i}), (8)

where the number A G [0,1] is defined from the equation

£ (a(v(N) — v(N \ {j})) + (1 — A)v({j})) = v(N).

jEN

Hokari (2000) gave an example of convex seven-person game whose T-value is not aggregate monotonic. It is the game with the characteristic function defined as follows:

10 for S = N,

6 f»r |S| = <S,S 3 1,

V 7 '2, for S| = 5,S 3 1,

^0 otherwise.

Putting these values in (8) we obtain ti(N, v) = 100/34. If the value v(N) increases

up to 10.1, and other values v(S), S = N are not changed, then in the new game

(N,v)

n(N,v) = 102.01/34.7 < 100/34 = ti(N, v).

Note that the difference game (N, v — v) is also convex. Therefore, the interval rvalue for the class of convex interval games defined as pair of T-values of lower and upper games does not exist. However, it turns out that it exists for a subclass of interval convex games whose lower and upper games are totally positive. In this section we prove this result.

Definition (8) implies that the T-value for the class of convex TU games is covariant. Hence, for investigation of the other properties of T-value we may restrict ourselves by games in 0-reduced form, i.e. games whose characteristic functions satisfy equalities v({i}) = 0 for all singletons i G N. Then formula (8) takes the form

n(N,v) = V{V(n){N)n\\-\) 'V{NI (10)

EjEN(v(N) — v(N \{J})

and t(N, v) depends only on the values v(N \ {i}), v(N), i.e.

t(N, v) = t(N, v°),

where

V <iS) fo,- 5,|S|>„-1,

0 for other S N.

Let us consider the class of totally convex games Gtp :

Gtp = {(N, v) | AS > 0 VS C N}, where AS, S C N are the dividends of coalitions satisfying the equations

v(S) = £ At.

T CS

It is known that

Gtc C Gc.

Therefore, if for games {N,v), {N,v') G Gtp the inequalities A'S > AS > 0 hold for all S C N, then {N, (v, v')) is a convex interval game.

It is clear that if a game {N,v) is convex (totally positive ), then the game {N, v°) (11) is also convex (totally positive), and the dividends of the game {N, v°) differ from zero only on coalitions the coalitions S, |S| = n — 1,n (n = N|):

( A0S, for S = n — 1,

v0(S) = < AN + EjEN AN j for S = N, (12)

I 0, for other S C N.

Proposition 1 The t-value of every totally positive game in the form (11) is mono-tonically increasing for all i G N in v(N), v(N \ {j}),j G N.

Proof. Note that if a game {N, v) is totally positive then it remains to be totally positive when the value v(N) is increasing. Thus, to prove monotonicity of the T-value in v(N) it only suffices to show non-negativeness of the corresponding derivatives of Ti(N, v) for all i G N.

Formula (10) takes the following form:

n(N, v)= , ■ v(N), VteN (13)

EjEN(v(N) — v(N \{J})

Let us differentiate Ti(N,v) in v(N). Then we obtain

dri(N,v) v(N)—v(N\{i})

9v(N) - E3eNMN)-v(N\{j})) +

(E,eiv(v(N)-v(N\{j})r

(14)

To know the sign of the fraction , it suffices only to check the sign of its

numerator, since the denominator is positive.

(v(N) — v(N \ {i})) (EjEN(v(N) — v(N \ {j})) — nv(N)) + v(N)J2jEN(v(N) — v(N \{j})) = v(N)(^nv(N) — 2 EjEN v(N\{j})) +

v(N \ {i}) EjE_N EjEN v(N \ {j}) = v(N)(nAN + (n — 2) EjEN v(N \ {j}^ +

v(N \{i})Z jEN v(N \{j})).

(15)

Since the game {N, v) is totally positive, the dividends AN, v(N), v(N \ {j}) > 0 for all j G N, and expression (15) are non negative for n > 2, and from (15) it follows that Ti(N, v) is non decreasing in v(N) for all i G N.

Now let us calculate the derivative yiew °f the game (N,v) is

totally positive, the values v(N \ {j}),v(N) are not independent, since v(N) = AN + EjEN v(N \ {j}). Put this this equality in (10), then we obtain

s (an + \ {j}))(AN + Y,jeNv(N \ {j}))

T* ,V n\N + (n-l)Y,jeNv(N\{j}) ’

dT(N,v) _ (an + Ej=i v(N \{j}))(AN + EjEN v(N \{j}))

dv(N \{i}) (nAN + (n — ^EjEN v(N \{j})2

The numerator of the derivative equals

(An + ]Tv(N \ {j}))an > °.

j=i

It remains to find the derivatives dt|jv\{j}) ^or •?' ^ * •

Ti(N,v) (iAN + J2jEN

V(N \ {i})) (n\N + (n

1) SjeAT

dv(N\{j}) (n\N + (n-l)J2jeNv(N\{j})2 '

The numerator of this derivative equals

2nAN + 2AN(n — 1) EjEN v(N \ {j}) + nAN EjEN v(N \ {j}) +

(n — 1) (EjEN v(N \ {j})) + nAN Ej=i v(N \ {j}) +

(n — 1) EjEN v(N \ {j}) Ej=i v(N \ {j}) — (n — 1)AN — AN Ej=i v(N \ {j}) ,

— (n — 1) Ej=i v(N \ {j}) EjE_N v(N \ {j}) =

(n + 1)AN + 2(n — 1)AN EjE_N v(N \ {j}) + (n — 1)AN Ej=i v(N \ {j}) > 0. and in this case we have also obtained the nonnegativeness of the derivative

dTi(N,v) . / .

dv(N\{j})’ J I □

Note that the game (9) was not totally positive, since the equalities v(S) = 0 for all S, |S| < 5, v(S) = 2 = AS for S, |S| = 5,S 3 1, imply that for S 3 1, |S| = 6 the following equalities hold

v(S) = 6 = AS + 5AT, where \T| = 5,1 G T

showing negativeness As or Ay. q

Proposition 1 and formula (13) immediately imply

Theorem 3. There exists the interval t-value for the class of totally positive interval games.

Acknowlegments. Research for this paper was supported by the Russian Foundation for Basic Research, grant n0 09-06-00155a.

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