An existence theorem for a bargaining set* - brown.edu€¦ · Brown University, Providence, RI 02912, USA Submitted April 1989, accepted September 1989 Abstract: This paper considers

Journal of Mathematical Economics 20 (1991) 19-34. North-Holland

An existence theorem for a bargaining set*

Rajiv Vohra

Brown University, Providence, RI 02912, USA

Submitted April 1989, accepted September 1989

Abstract: This paper considers non-transferable utility games and provides sufftcient conditions for the existence of the bargaining set introduced in Mas-Cole11 (1989). Our main assumption is weak balancedness. This assumption is weaker than balancedness and is satisfied in all transferable utility games and all three-player games. The proof of our main result is based on an application of a coincidence theorem of Fan (1969).

1. Introduction

Mas-Cole11 (1989) proposed a solution concept which is a modification of the bargaining set introduced by Aumann and Maschler (1964). This solution concept differs from the core in the sense that coalitions, while blocking an outcome, are required to take account of their actions on the actions of other coalitions. More precisely, if a coalition finds that it can, on its own, improve the welfare of its members relative to some proposed outcome, this by itself does not constitute a justified objection; it must also be the case that there does not exist another coalition which can improve the welfare of its members, relative to the initial outcome, and also ensure that any player who is common to both the coalitions can, in fact, be attracted into this new coalition. Since blocking is made more difftcult, the bargaining set contains the core. Nevertheless, as Mas-Cole11 (1989) shows, in the context of a market game with a continuum of players, the bargaining set is equivalent to the set of Walrasian allocations.’

The fact that coalitions, in proposing improvements, are required to take account of the reactions of other coalitions may in itself be considered an

*I am grateful to Wanda Gorgol, Andreu Mas-Cole& Bezalel Peleg, Ajit Ranade, Debraj Ray, two anonymous referees and participants of a conference at Ohio State University and a seminar at Yale University for many useful comments. This paper has grown out of an earlier one titled ‘On A Bargaining Set with Restricted Coalition Size’ (1987). Support from NSF, grants SES-8605630 and SES-8646400, is gratefully acknowledged.

‘The equivalence problem has been further analyzed in Grodal (1986), Shitovitz (1989) and Yamazaki (1988).

03044068/90/$03.50 0 199O-Elsevier Science Publishers B.V. (North-Holland)

20 R. Vohra, An existence theorem for a bargaining set

advantage of the bargaining set over the core2 [see Maschler (1967)]. Moreover, in cases such as games with a finite number of players, where the equivalence theorem does not hold,3 and the bargaining set may be much larger than the core, there exists the possibility that one may be able to find sufficient conditions for the non-emptiness of the bargaining set which are weaker than those for the non-emptiness of the core. Indeed, in transferable utility games, the bargaining set is non-empty if the game is weakly superadditive (see section 3 below), while the core is non-empty if and only if the game is balanced.

Our main result provides sufficient conditions for the non-emptiness of the bargaining set in non-transferable utility games. Recall that the core is contained in the bargaining set. It then follows from a result of Scarf (1967) that every balanced game has a non-empty bargaining set. Our main assumption is weaker than balancedness and we refer to it as weak balancedness.4 The proof of our main result makes use of a coincidence theorem of Fan (1969). It would be possible to use Kakutani’s fixed point theorem instead, but at the expense of a much more complicated proof. We also show that weak balancedness is satisfied in a class of games we call weakly TU games. This is a class of games which includes all transferable- utility games and all three-player games. We do not know whether our hypothesis of weak balancedness is necessary for the non-emptiness of the bargaining set. In particular, we do not have an example of a game which does not satisfy weakly balancedness (but satisfies the other conditions of our main result) and has an empty bargaining set. This is clearly an important issue that remains to be investigated.

The paper is organized as follows. Section 2 presents the model and the main definitions. In section 3 we consider some special cases and briefly indicate the relationship between the bargaining set of Mas-Cole11 (1989), the bargaining set M’;” of Aumann and Maschler (1964) and the kernel.5 There should be no loss in continuity in moving from section 2 directly to section 4, which contains the main results. Section 5 is devoted to the proofs of the results stated in section 4.

2. Mas-Colell’s bargaining set

We shall consider a game with II players. Let N = { 1,. . . , n} denote the set

‘Indeed, it may be argued that consistency requires that reactions to objections loo should be made to take account of further reactions of other coalitions. The consequences of imposing such a consistency requirement are analyzed in Dutta et al. (1989).

3The equivalence between the core and the bargaining set does hold in ordinally convex games, even with a finite number of players. See Dutta et al. (1989).

4We also assume that the game is weakly superadditive and strongly comprehensive. Precise defmitions are provided in the following section, where we also show that these two assumptions are necessary to ensure that the bargaining set is non-empty.

%ee Dragan (1988) and Vind (1986) for other related concepts of the bargaining set.

R. Vohra, An existence theoremfor a bargaining sef 21

of players and 2N the set of all non-empty subsets of N. For any coalition

SEA*, (S( denotes the numbers of players in S and Rs is the Euclidean space of dimension (S( such that its coordinates have as subscripts the players of S. For x E RN, xs will denote its restriction on RS. Similarly, for a set B E RN, B, denotes its restriction on Rs. Each coalition S has a feasible set of payoffs or utilities denoted by V(S) E Rs. Let V(S) = {x E RN\x, E a(S)).

A non-transferable utilitp game in characteristic function form is defined as a pair (N, V), where V: 2N~ RN is a correspondence satisfying:

(i) for all SE 2*, V(S) is non-empty, closed and comprehensive, (ii) for all iEN,V({i})={xERN\xi~O}, (iii) for all SEAR, V(S), n R”, is bounded.

Let es denote the vector in RN whose ith coordinate is 1 if ig S and 0 otherwise. We shall also use the notation e for eN. For any vector x E RN, xi will denote its ith coordinate. For x, ye RN, x>y means that xi2 yi for all i and at least one of these inequalities is strict. x >> y means that xi > yi for all i. For a set B c RN, Co(B) denotes the convex hull of B and 8(B) denotes its boundary. For sets S, T, S\T denotes set theoretic subtraction.

An element x E V(N) is said to be an imputation if there does not exist ye V(N) such that y>x.

An element XE V(N) is said to be individually rational if x20.

The set of individually rational imputations of (N, V) is defined as

.Y(N, V)=(XE V(N)n RTI$~EV(N) such that y>x).

Given XE V(N), a pair (S, y), where SET* and ye V(S) is said to be an objection to x if y,>x,.

The core of a game (N, V) is defined as

C(N, V) = {x E V(N)/ g(S, y) which is an objection to x}.

Let (S, y) be an objection to x. A pair (T, z), where T E 2* and z E V(T) is said to be a counterobjection to (S, y) if (zSn T, zTIs) > ( ysn T, x~,~).

An objection (S,y) to x is to be a justified objection if there does not exist any counterobjection to (S, y).

The bargaining set of a game (N, V) is defined as

B(N, V) = {x E~(N, V)I $(S, y) which is a justified objection to x).


Notice that if we do not impose a strict inequality in defining a counterobjection, the bargaining set so defined would contain B(N, V) and its existence would follow from that of B(N, V). The same remark also applies if in the definition of an objection all inequalities are required to be strict.

Clearly, C(N, I’) c B(N, I’). Mas-Cole11 (1989) provides an example in which this inclusion is strict. In fact, in a game with identical players B(N, V) is always non-empty (see Proposition 3.1 below) while C(N, I’) may be empty.

Strictly speaking, our definition of the bargaining set is not identical to that of Mas-Cole11 (1989); the bargaining set B(N, I’) defined above is contained in the one defined by Mas-Colell. The following two paragraphs are devoted to a precise comparison.

Mas-Cole11 (1989) in considering exchange economies with a continuum of agents, defines the bargaining set without restricting attention to individually rational imputations, i.e., he defines the bargaining set with V(N) instead of 9(N, V) in the definition of B(N, V) above. His equivalence result holds for this larger set. From the point of an existence result, however, it is important to impose some further restrictions. Otherwise, the bargaining set is always non-empty and is typically too large. The argument is as follows. If the core is non-empty then, of course, so is B(N, V). If C(N, V) =@, then consider any x<<O. This cannot have a justified objection against it. Any objection of the grand coalition has a counterobjection since C(N, I’) =8. If any smaller coalition objects there is a counterobjection from a player who does not belong to this coalition.

In considering games in characteristic function form, Mas-Cole11 (1986) restricts the bargaining set to be a subset of imputations6 without insisting, however, on individual rationality. If we restrict B(N, V) to be chosen from the imputations, without insisting on individual rationality then it is easy to see that any imputation which gives negative utility to all but one player is in the bargaining set if objecting coalitions are allowed to be no larger than n-2. If we are interested in obtaining individually rational imputations in the bargaining set, this requirement has to be made explicitly in the definition. In Example 2.1, below, there exists an imputation which is not individually rational and to which there does not exist a justified objection (see Remark 1 at the end of this section). Example 2 of Dutta et al. (1989) shows that this can happen even in a superadditive, transferable utility game.

From the standpoint of obtaining an existence result it is clearly preferable to consider only individually rational imputations. We shall, therefore, be concerned with the existence of B(N, V) as defined above.

6He defines imputations to correspond to weak Pareto optima rather than Pareto optima, i.e., XE V(N) is said to be an imputation if there does not exist YE Y(N) such that y>>.w. Similarly, he also defines counterobjections with strict inequalities. However, we shall be making an assumption [(SC) below] under which these distinctions disappear.

R. Vohra, An existence theorem for a bargaining set 23

We will make use of the following two assumptions.

(WS) Weak Superadditivity. For any SE 2N and i$ S, if x E V(S), then ye V(Su ii}), where y,=O and yj=xj for j#i.

(SC) Strong Comprehensiveness. For any SEAR, if x, ye V(S) and y,>x,z 0, there exists ZE V(S) such that zs>>xs.

Neither one of these assumptions can be dispensed with in proving the non-emptiness of B(N, V).

Assumption (WS) is weaker than superadditivity. It is necessary for our purposes in the sense that it is not satisfied, 4(N, V) may be empty. Moreover, assuming V(N) A !R”, #@I, which guarantees that .4(N, V) #0, is not sufficient for the bargaining set to be non-empty even if (SC) is satisfied. This is illustrated in the following example.

Example 2.1. Consider the game (N, V) where N = { 1,2,3},

V((2,3})={x~R~(x,+x,~2),

and for any other coalition SE 2N,

Clearly Y(N, V) = (0). But this does not belong to B(N, V) since there exists a justified objection (S, y) where S= {2,3) and y, = y, = 1.

Assumption (SC) ensures that given a feasible and individually rational utility allocation for a coalition, it is feasible for this coalition to increase the utility of any player in the coalition (possibly at the expense of some other player). This assumption is satisfied in all transferable utility games and in games derived from markets in which agents have strictly monotonic utility functions. Given (SC) the inequality > in the definition of imputations can be replaced with >>. In terms of an existence proof this is important since, along with (WS), it ensures that if 4(N, V) # (01, then it is homeomorphic to the unit simplex in RN.

If (SC) is not satisfied, B((N, V) may be empty even in a superadditive game, as the following example shows.’

Example 2.2. Consider the three-player game where N = { 1,2,3} and

7While this example does not appear in the published literature, it does not seem to be fairly well known and is similar to a four-player example due to Peleg.


V(N)=(x~iW3(x~(l,0,0)} u (xE:~3~x~(o,l,o)}

u {XER3(X~(0,0, I,).

Notice that (WS) is satisfied. There are three points in 9(N, V) correspond- ing to the vertices of the unit simplex in R3 and for each one of these there exists one and only one objection. Thus B(N, V)=@ It is easy to see that this example can be modified to ensure that (SC) and (WS) are satisfied but the core is still empty. However, in any such modification B(N, V) #$!I (see Proposition 4.3 below).

Remark 1. In both Examples 2.1 and 2.2, the fact that the bargaining set is empty depends crucially on our insistence that B(N, I’) E~(N, V). As we have already pointed out, existence is guaranteed if we drop this restriction altogether. In Example 2.1, existence would obtain if we dropped the requirement of individual rationality; the imputation (- 1,0.5,0.5) has no justified objection. In Example 2.2, existence would obtain if we weakened the definition of an imputation and required only weak Pareto optimality; 0 is weakly Pareto optimal and there does not exist any justified objection to it.

Remark 2. Notice that our definition of a counterobjection does not require the counterobjecting coalition to include some player from the objecting coalition. Imposing this additional requirement on a counterobjection seems quite natural and would lend further credence to the rationale that we offered for considering counterobjections. In general, modifying the definition of counterobjections in this manner would make for a bargaining set smaller than B(N, V). However, it is easy to see that in a superadditive game, restricting counterobjecting coalitions to have a non-empty intersection with the objecting coalition will yield B(N, V) as defined above. For superadditive games therefore, our existence results cover the existence of a bargaining set with this added restriction on counterobjections.

3. Some special cases

3.1. Games with identical players

Consider a game in which all players are identical in the sense that any


player in a coalition can be replaced by another player from outside the coalition. More precisely, we can define such a game as follows.

A game with identical players is a game (N, V) in which for any SEAR any XE V(S), and icS, j$S, there exists ZE V((S\{ij) u {j}) such that zk=xk for all keS\{i} and Zj.=Xi.

Proposition 3.1. If (N, V) is a game with identical players satisfying (WS) and (SC), then B(N, V) #8.

Proof. From (WS) and (SC) it follows that there exists XE~(N, V) such that xi= .xj for all i, j E N. We can now show that any objection to x has a counterobjection and, therefore, x E B(N, V). Suppose (S, y) is an objection to x. There must exist i E S such that yi > xi. Since (SI < n we can define T=(S\{i}) u {j}, where jE N\S which by assumption has ZE V(T) such that zk= y, for all k E T\(i) and zj= yi>xi=Xj. Thus, (T,z) is a counterobjection

to (S, Y). cl

3.2. Transferable utility games

A transferable utility game (TU game) is defined as a game (N, V) in which for each SEAR there exists a real number v(S) such that V(S) = {x E RN]

Cics xi s u(s)).

Notice that a TU game satisfies (SC). Consider a TU game (N, V). Given x E V(N) we define for each coalition S, its ‘excess’ at x as

e(S, x) = u(S) - 1 xi. ieS

We can also define the excess of player i against player j at x as

Sij(X)=max {e(S,X)liES,j$S}.

The pre-kernel of a TU game (N, V) is defined as

K”(N, V)= XE V(N) C x~=u(N) and Sij(X)=Sji(X) for all i,jeN I I isN

The kernel of a TU game (N, V) is

K(N, V)={XE$(N, V)I(~~j(X)-ssji(x))xj=O for all i,jEiV).

While the pre-kernel is always non-empty [see Remark 2.10 in Maschler, Peleg and Shapley (1972)], it is possible that K”(N, V) n Y(N, V)=@, as in


fact is the case in Example 2.1 of the previous section.* However, in weakly superadditive games, K”(N, V)=K(N, V) [see Maschler, Peleg and Shapley (1972)]. In particular, in such games, K”(N, V) c 9(N, V). Maschler and Peleg (1966) proved that K(N,V)#@ if9 u(S)>=0 for all SEAR. Thus, in weakly superadditive games, K”(N, V) n 4(N, V) #8.

Mas-Cole11 (1989) pointed out that the pre-kernel is always contained in the bargaining set (without the requirement of individual rationality) and the latter is, therefore, always non-empty in TU games. Since in weakly superadditive TU games, K”(N, V) = K(N, V), it follows that in such games B(N, V), as defined in the previous section, is non-empty. This result is also a special case of our main result in the following section. However, for the sake of completeness we state and prove:

Proposition 3.2. (K’(N, V) n 9(N, V)) G B(N, V) and, therefore, every TU game satisfying (WS) has a non-empty bargaining set.

Proof. Suppose XE(K’(N, V) n Y(N, V))\B(N, V). Then there must exist a justified objection (S, y) to x. Since x~4(N, V), S #N. Hence there exists J’EN\S and iES such that yi>xi. By hypothesis, there must exist a coalition TsuchthatjETandi$Tand

e(T,x)=sij(x)2e(S,x).

Since yi > Xi,

et&X)= 2 bk-Xk)> c (ykeXkh

kp.5 ksTn.5

This means that

‘tT)- 1 xk> c (Yk-Xk)r ksT ksTnS

or,

V(T)> c xk+ 1 Yk.

kcT\S kcTnS

This implies that T has a counterobjection to S, which contradicts the hypothesis that (S, y) is a justified objection to x. 0

We end this section by making a comparison between the Aumann- Maschler (1964) bargaining set M’;” and B(N, V). The former is defined over Y(N, V) and in this respect our modification of Mas-Colell’s bargaining set is in keeping with the definition of M 1. w While My’ is defined with respect to

‘In Example 2.1, KO(N, V)=(- l,O.S,O.S) and X(N, V)=K(N, V)={O). Wnder the normalization u( {ij) = 0 for all i E N.


coalition structures, the comparison with B(N, V) will be simpler if we restrict attention to the coalition structure {Nj. There still remain two differences between these notions of bargaining sets:

(i) counterobjections in My’ are defined with a weak inequality, (ii) there are restrictions on membership in a counterobjecting coalition.

As a result of (i), counterobjections are easier and this tends to make the set larger than B(N, V). However, (ii) has an opposite effect and, consequently, the relationship between the two notions of bargaining sets is one of non- comparability. To define My’, with the coalition structure {N), we need some more definitions.

Given x E~(N, I’), a pair (S, Y), where SEAR and YE V(S) is said to be an objection of i against j at x if i E S, j +! S, yi > Xi and ys 2 xs.

Let (S, Y) be an objection of i against j at x. A pair (7’, z), where TE 2N and ZE V(T) is said to be an My) counterobjection to (S, y) if jE T, i$ T and

(ZSnTIZT\S)~(YSnT,XT\S).

A pair (S, y) is said to be an My' justified objection to x if it is an objection of i against j at x and there does not exist an M’;” counterobjection to it.

The bargaining set My’, with the coalition structure {N}, is defined as

My’= {.xE.B(N, V)lJ(S, Y) which is an My' justified objection to x}.

Davis and Maschler (1963) and Peleg (1963) show that My’ is non-empty in TU games” satisfying u(S) 20 for all SEAL (under the normalization u({i>) =0 for all iE N). In particular assumption (WS) is not necessary. The fact that M’;” counterobjections are defined with a weak inequality is important for this existence result. In Example 2.1, (0) =.F(N, V) = K(N, V) = MY’. Any objection of players 2 or 3 against player 1 has an MY’ counterobjection ({l}, 0) - which would not be the case if counterobjections were required to satisfy a strict inequality.

Similarly, because of the restrictions on membership in a counterobjection, it is possible that there exists XE B(N, V)\MY’. This can be illustrated through example 1 of Mas-Cole11 (1969). Consider the TU game which is the minimal superadditive game compatible with: N ={ 1,2,3,4}, u({i>)=O for all iEN, u(N)=4, u((1,2,3})=3.1, v({2,4})=~((3,4})=2.06. The individually

“Peleg (1963) considers non-transferable utility games and shows that My’ is non-empty in games of pairs, but, in general, it can be empty. Billera (1970) provides sufficient conditions for the non-emptiness of M’i’ in non-transferable utility games (see also footnote 12 below). It is also worth pointing out that the kernel was originally constructed in order to establish the existence of IV’;“. K(N, V) c My’ and from our earlier remarks it now follows that in TU games satisfying (WS) K(N V) 5 B(N, V) n M”‘. , 3 1


rational imputation (1, 1, 1, 1) E B(N, V). It does not belong to M’;” since there exists a justified objection of player 3 against player 1. This is the objection (S, y), where S= {3,4} and y, = 1.06 and y,= 1. Since player 1 cannot counterobject, except through a coalition which also contains player 3, this is an MY’ justified objection.

4. Main results

Throughout this section we consider games which satisfy (WS) and (SC). In order to state our assumption which, in addition to (WS) and (SC), is sufficient for the non-emptiness of B(N, V), we need to recall the definition of a balanced collection.

A collection of sets 98 G ZN is said to be balanced if for each SE@ there exists a non-negative real number As, called a balancing coefficient, such that

We will also find it useful to consider the last condition in its following form:

for all i E N, c AS=l. {SeBJiES)

We can now state our main assumption.

A game (N, V) is said to be weakly balanced if there does not exist an individually rational imputation x and a balanced collection %? such that every SE 2I has a justified objection against x.

It is worthwhile to compare weak balancedness with balancedness.

A game (N, V) is said to be balanced if nSEl V(S) E V(N) for any balanced collection B.

The difference between weak balancedness and balancedness becomes transparent if we restate the definition of balancedness in its following, equivalent form:”

A game (N, V) is balanced if and only if there does not exist an imputation x and a balanced collection 99 such that every SEW has an objection against x.

Notice that balancedness and weak balancedness differ in only two

“This equivalence is a simple consequence of (SC), and we leave it for the reader to verify it.


respects. Firstly, balancedness imposes a restriction on imputations while

weak balancedness imposes a restriction only on individually rational imputations. Secondly, balancedness requires that there exist no balanced collection of objections while weak balancedness requires that there exist no

balanced collection of justified objections. On both counts, therefore weak balancedness is a weaker assumption than balancedness.

Since the core is contained in the bargaining set, it follows from a result of Scarf (1967) that every balanced game has a non-empty bargaining set. Our main result is the following.

Theorem 4.1. Every weakly balanced game has a non-empty bargaining set.

This result is a straightforward consequence of the following proposition.

Proposition 4.1. lf B(N, V) =@, then there exists 5 ~le;(N, V) and a balanced collection 93 such that every S E G? has a justified objection against X.

While weak balancedness by itself may not be easy to verify, it does follow from the condition that for every imputation there exists a player who belongs to every coalition which has a justified objection.” It turns out that this condition is satisfied in all TU games and all three-player games. We now define a class of games which includes such games and is a subset of the class of weakly balanced games.

A game is said to be weakly TU if it satisfies the following condition:

(WTU) Suppose S, TE 2N\N and S n T#@. If there exist x E V(S) n R”, and y E a( V( T)) n R”, such that xs n r >hnT, then for any (h\snT3y$nT)E W)n

RN, there exists (x N\Snr,.$nr)E1/(S)nR~ such that Xi,r>yi,,.

Condition (WTU) may be explained as follows. Consider two coalitions which have a non-empty intersection and neither one is the grand coalition. Given the amounts allocated to players which are not common to the two coalitions we can consider the feasible utilities that can be allocated to the common players in the two coalitions. Condition (WTU) is the requirement

‘“An analog of this condition in the Aumann-Maschler setting was used by Billera (1970). However, an application of Billera’s result to the present context is not straightforward. In particular, he uses an additional assumption. This is assumption (b) of Billera (1970) which states if x is an imputation and .x,=0, then player i belongs to every coalition which has a justified objection against x. It is easy to check that the analog of this assumption, in the context of Mas-Colell’s bargaining set, may not hold even in superadditive TU games. For example, consider the TU game which is the minimal superadditive game compatible with N = ( 1,2,3,4) and u{1,2}= 1. Let x=(0,0.5,0.5,0). Now (S,y) is a justified objection, where S=j1,2), y,=O.5, y, = 0.5. Moreover x4 = 0 and 4 4 S.

R. Vohra, An existence theorem for a bargaining set

. Ul 0

,

S,T # N, I E l’(S), y E a(V(T)), 5 n T = {1,2}

A= {U E V(S) I UN\S~T = XN\SC-T), B = (U E V(T) IUN\S~T =YN\s~T}

Fig. 1

that the boundaries of these sets must not cross. Fig. 1 illustrates a case in which (WTU) is not satisfied.

Proposition 4.2. All weakly TU games are weakly balanced and, therefore, have a non-empty bargaining set.

The following result is easy to verify and we leave its proof to the reader.

Proposition 4.3. All TU games and all three-player games are weakly TU and, therefore, have a non-empty bargaining set.

5. Proofs

To prove Proposition 4.1 we will use Fan’s Coincidence theorem (1969, theorem 6). We could make do with Kakutani’s fixed point theorem, but then the argument would be more involved.

Theorem 5.1 (Fan). Let X be a compact, convex subset of RN and u:XwRN


and p: X t+lRN be non-empty, convex valued, upper hemicontinuous correspon- dences. Suppose

for all x E X, there exists a real number ,a > 0, a E a(x) and b E p(x), such that x + p(a -b) E X. (1)

Then there exists X E X such that a(X) n j?(X) #0.

Proof of Proposition 4.1. Under the hypothesis that B(N, lf) =@, it is easy to see that (WS) implies Y(N, V)#(O}. G’ iven (WS) and (SC) it can now be

shown that 9(N, V) is homeomorphic to A, the unit simplex in RN. In particular, it can be shown that the correspondence f: AwY(lV, V) defined as

f(s)={xd(N,V)(x=t f s or some positive real number t)

is a well defined, continuous function; see Vohra (1987). Let J: 9(N, V)H~~ be a correspondence defined as

J(x)={S~2~/3y~ V(S) such that (S, y) is a justified objection to x).

Let G: 9( N, V)H A be a correspondence defined as

If B(N, V) is empty, then G is non-empty. We shall now verify that G is upper hemicontinuous. Suppose x4 -+x, gq -P

g, and that x4 EY(N, V) and g4~ G(x4) for all q. We need to establish that ge G(x). Since G(4(N, V)) is a finite set, there exists 4 such that for all q>q, gq=g. This implies that for all q>q, (S, y”) is a justified objection to x4. We need to show that S also has a justified objection to x. Without loss of generality, we may assume that the vector y4,,, is fixed at some arbitrary level for all q. Since y’J > XJ z-0, for all q > 4, V(S) is closed and V(S), n R”, is bounded, we can assert that yq has a subsequence which tends to y. Without loss of generality we may assume that this subsequence is yq itself. Of course, YE V(S) and y,Lx,. Suppose (S, y) is not a justified objection to x. This means that either it is not an objection to x (and ys=xs) or that there exists some other coalition which has a counterobjection to (S,y). In either case, given the hypothesis that there does exist a justified objection to x, this implies that there exists (T, z) such that z E V(T) and zi >= yi for all i E S n T, Zip xi for all i E T\S and one of these inequalities is strict. By (SC), there is no loss of generality in assuming that all of these inequalities are strict. Since yq + y and x9+x, this means that for q large enough, and greater than g, Zi > ys for all ic S A T and zi > xp for all iE T\S, i.e., (T, z) is a counter-


objection to (S, y4). But this contradicts the fact that (S, y”) is a justified objection to x4 for all q> 4. Thus, (S, y) is a justified objection to x, and

g E G(x). Define p: AHA such that p(s) =e/n. Let a: AHA be defined as

a(s) =Co(G(f(s))). Both r~. and /? are non-empty, convex valued and upper hemicontinuous. We will now show that condition (1) of Theorem 5.1 is satisfied.

Consider SEA. Let x=f(s) and K={~EN~s~=O}={~EN~X~=O). Suppose (S,y) is a justified objection to x. By assumption (WS), it follows that ZE V(S u K), where z,=O and zs=xp It is also easy to see that (S u K, z) is a justified objection to x. This means that if SEJ(X), then (S u K)EJ(x). Thus, for any s E A we can find a E a(s) such that ak 2 l/n for all k E K. Notice that cjEN(aj- l/n)=O. It is now easy to see that there must exist a real number p> 0 such that s + ~(a -e/n) E A, i.e., condition (1) of Theorem 5.1 is satisfied. Thus, we can appeal to Theorem 5.1 to assert that there exists SE A

such that e/n E Co (G(f(F))). Let X=f(F). This implies that there exist non- negative numbers 6’ such that

C #e,=e/n.

SEJW)

Clearly, then 9?=.I(X) is a balanced collection of justified objections to x. 0

We shall prove Proposition 4.2 by showing that if there is a balanced collection of justified objections to an imputation in a weakly TU game (N, V), then N must have a justified objection to x. This provides the necessary contradiction.

Proof of Proposition 4.2. Suppose the result is false. Then for a weakly TU game (N, V) there exists XEY(N, V), a balanced collection 9I= (S’,. . .,S”} with balancing coefficients (A’, . . . , A”) and y’ E V(S’) for i= 1,. . . , m such that (Si, y’) is a justified objection to x for all i = 1,. . . , m.

Since x is an imputation,

NfgB. (2)

Since 99 is balanced, this means that it must contain at least two coalitions S’,Sj#N. Since (S’, y’) and (Sj, yj) are justitied objections, it follows that S’ n Sj#@. Moreover, since x 20, y’>=O and yjz0. We now claim that

(Yh- &lnsj)EV(Si) for all i,jE{l,...,m). (3)


Suppose not. Since ~‘20, this must mean that there exists

(Vi N\sinsjry~,ns,)Ea(v(Si))nrWN, such that y&insj>y$znsj. By (WTU) this implies that there exists ( y&sLn sj, y$ n s,) E V(Sj) and y$ n s, > y$ nS,. But this provides Sj with a counterobjection to (S’, y’) and contradicts the hypothesis

that (Si, yi) is a justified objection to x. Let S* = {k~S’/y~ >xk). Certainly, S* ~8. Suppose there exists keS* and

Sj~.%9 such that k#Sj. Then by (SC) and (3), we would find an objection

(S’,y”) such that ~i:~~~>>yy$,,,~,, which contradicts the fact that (Sj, y’) is a justified objection. We can, therefore, claim that

S*ESj for all jE{l,...,mj. (4)

Let keS*. From (4) and the definition of a balanced collection, it follows that

IjlkeSJk j=l

There must exist SE&~ for which the balancing coefficient iLs >O. Suppose there exists k’e N such that k’$S. This means that &j\k,Esj)~J<~~zr 2’. Given (5), this contradicts the hypothesis that .%? is balanced. Thus, we have shown that there exist Se99 such that S = N. But this contradicts (2) and completes the proof that a weakly TU game is weakly balanced. 0

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An existence theorem for a bargaining set* - brown.edu€¦ · Brown University, Providence, RI 02912, USA Submitted April 1989, accepted September 1989 Abstract: This paper considers