The least core, kernel and bargaining sets of large games

The least core, kernel and bargaining sets

of large games

Ezra Einy1;2_{, Dov Monderer}3_{, and Diego Moreno}1 1_{Departamento de EconomõÂa, Universidad Carlos III de Madrid,}

E-28903 Getafe, Madrid, SPAIN

2_{Department of Economics, Ben-Gurion University of the Negev, Beer Sheva, ISRAEL} 3_{Faculty of Industrial Engineering and Management, Technion-Israel Institute of Technology,}

Haifa, ISRAEL

Received: June 6, 1996; revised version: March 1, 1997

Summary. We study the least core, the kernel and bargaining sets of coali-tional games with a countable set of players. We show that the least core of a continuous superadditive game with a countable set of players is a non-empty (norm-compact) subset of the space of all countably additive mea-sures. Then we show that in such games the intersection of the prekernel and the least core is non-empty. Finally, we show that the Aumann-Maschler and the Mas-Colell bargaining sets contain the set of all countably additive payo€ measures in the prekernel.

JEL Classi®cation Number:C71. 1 Introduction

Solution concepts for coalitional games with an in®nite set of players have been studied in many works. Most of these works deal with the core and the Shapley value. Bargaining sets and related solution concepts for games with a ®nite set of players have been studied intensively (for a comprehensive survey see Maschler [11]). There are a few works concerning bargaining sets (and related concepts) of games with an in®nite set of players. Wesley [21] deals with the kernel of games with a countable set of players. Bird [3] studies the nucleolus-like solutions for games with a measurable space of players. Mas-Colell [14] introduces his bargaining set in the context of pure exchange w_{The ®rst author thanks the Spanish Ministry of Education for its support. The third author}

gratefully acknowledges ®nancial support from the Ministerio de Trabajo y Asuntos Sociales through funds administered by the CaÂtedra Gumersindo AzcaÂrate, and from DGICYT grant PB93-2030. This work was done while Einy and Monderer visited the Department of Economics of the Universidad Carlos III de Madrid. The support of the Department is gratefully acknowledged.

economies with a continuum of agents. Shitovitz [19] deals with the Mas-Colell bargaining set in mixed market games. Einy et al. [8] study the Mas-Colell bargaining set in convex games with a measurable space of players.

The present work deals with the least core, the kernel, and the Aumann-Maschler and Mas-Colell bargaining sets for superadditive games with a countable set of players. The least core was introduced by Maschler, Peleg and Shapley [13], where they study its relation to the kernel and the nucle-olus. It is well known that the core of a continuous game with a measurable space of players, if it is non-empty, consists of countably additive payo€ measures (e.g., Schmeidler [17]). Here we show that the least core of a continuous superadditive game with a countable set of players is a non-empty norm-compact subset of the set of all countably additive measures de®ned on the set of coalitions (see Theorem A).

The kernel of a cooperative game was introduced by Davis and Maschler [5]. Since then it has been the subject of many studies. Originally it was regarded as an auxiliary solution concept whose main task was to illuminate the properties of the Aumann-Maschler bargaining set. Nevertheless, the kernel possesses interesting mathematical properties, and re¯ects in many ways the structure of the game. The prekernel is a simpli®ed version of the kernel which is not restricted to individually rational payo€s. We show that in continuous superadditive games with a countable set of players the pre-kernel (and hence the pre-kernel) and the least core have a non-empty intersec-tion. The proof of this result uses ®nite approximations. Wesley [21] proved by using non-standard analysis that under some conditions the kernel of a superadditive game with a countable set of players is non-empty (for every coalition structure). We show that Wesley's conditions imply that the game is continuous. We also give an example of a continuous game which does not satisfy one of Wesley's conditions (see Lemma 4.3 and Example 4.4). Thus, Wesley's result is a special case of our result when the coalition structure includes only the grand coalition.

The ®rst de®nition of a bargaining set for cooperative games was given in Aumann and Maschler [1]. Recently, several new concepts of bargaining set have been introduced (see Maschler [11] for a survey). Davis and Maschler [4] and Peleg [15] proved that the Aumann-Maschler bargaining set is non-empty in a coalitional game with a ®nite set of players. We show that in continuous superadditive games with a countable set of players, the Aum-ann-Maschler bargaining set contains the set of all countably additive payo€ measures in the prekernel, and thus it always contains a countably additive payo€ measure (see Theorem C).

Mas-Colell [14] proposed a bargaining set which is a modi®cation of the Aumann-Maschler bargaining set. One of the advantage of the Mas-Colell bargaining set is that it can be de®ned for games with a continuum of players. Mas-Colell [14] showed that in atomless pure exchange economies his bar-gaining set coincides with the set of competitive equilibria, and he pointed out that in ®nite coalitional games, the prekernel is always contained in his

bargaining set (see also Vohra [20]). In the de®nition of the Mas-Colell bargaining set it is not assumed that payo€s are individually rational. We show that in continuous superadditive games with a countable set of players the Mas-Colell bargaining set contains the set of all countably additive payo€ measures in the prekernel, and thus it always contains an individually rational countably additive payo€ measure.

The paper is organized as follows. Section 2 contains the basic de®nitions and the preliminary results which are relevant to our work. In Section 3 we prove that the least core of a continuous superadditive game with a count-able set of players is a non-empty norm-compact subset of the space of all countably additive measures on the set of coalitions. In Section 4 we show that in continuous superadditive games with a countable set of players the least core and the prekernel have a non-empty intersection. In Section 5 we prove that in continuous superadditive games the bargaining set of Aumann and Maschler and that of Mas-Colell contain the set of all countably additive payo€ measures in the prekernel.

2 Basic de®nitions and preliminary results

In this section we de®ne some basic notions we use throughout and prove some preliminary results.

2.1 Mathematical preliminaries

LetNbe the set of natural numbers. The set of subsets ofN is denoted by 2N_.

The set of all functions f :N ! f0;1g is denoted by f0;1gN. Note that f0;1gN_{is a compact metric space (as the product of countable metric spaces),}

and convergence of sequences in this metric is identical to pointwise con-vergence. EveryS22N_{can be naturally identi®ed with its indicator function}

1S2 f0;1gN. The correspondence S !1S induces a metric on 2N under

which it is compact. It is well known that a sequencefSng1n 1converges toS (written limn!1SnˆS) in this metric if and only if

Sˆlimn!1Snˆlimn!1Sn; where limn!1SnˆS1n 1 T₁ k nSk;and limn!1SnˆT1n 1 S₁ k nSk.

IffSng1n 1is a sequence of sets in 2Nsuch thatSnSn‡1;andS1n 1SnˆS

we writeSn%S. Similarly, iffSng1n 1 is a sequence of sets in 2N such that Sn‡1Sn;andT1n 1SnˆS;then we writeSn&S. Note that iffSng1n 1 is a sequence such that Sn%S or Sn&S; then limn!1SnˆS. A function

v:2N_{! <ismonotonicifv…S† v…T†wheneverST. A functionvon 2}N_is

continuous if it is continuous with respect to the natural topology on 2N

de®ned above. For monotonic functions we have the following character-ization of continuity.

Lemma 2.1. Letv:2N_{! <be a monotonic function. Thenvis continuous if}

and only if for everyS22N_{we havelimn!1v…Sn† ˆv…S†wheneverSn%Sor}

Proof. We prove the non-obvious part of the lemma. LetS22N_{. Assume}

limn!1v…Sn† ˆv…S†wheneverSn%S orSn&S. We show thatvis

contin-uous atS. LetfSng1n 1 be such that limn!1SnˆS;that isSˆlimn!1Snˆ

limn!1Sn. We show that limn!1v…Sn† ˆv…S†. For eachnletAnˆT1k nSk;

andBnˆS1k nSk. Asvis monotonic andAnSnBnfor eachn;thenv…An†

v…Sn† v…Bn†for each n. Now,An‡1An andSˆlimSnˆS1n 1An. Also

Bn‡1Bn; andSˆlimSnˆT1n 1Bn. Thus, An%S andBn&S. Hence by

our assumption

lim

n!1v…An† ˆn!1limv…Bn† ˆv…S†;

and therefore limn!1v…Sn† ˆv…S†: u

We now recall some standard de®nitions and results from measure theory which are used throughout the paper. A®nitely additivemeasure on 2N _{is a}

functionl:2N_{! < which satis®esl…A[B† ˆl…A† ‡l…B†wheneverAand}

Bare two disjoint subsets ofN. The measurelis calledcountably additiveif for every countable family fAig1i 1 of disjoint subsets of N we have l…S1i 1Ai† ˆP1i 1l…Ai†. It is well known thatlis countably additive i€ it is

continuous atN. A ®nitely additive measurelon 2N _{is calledpurely ®nitely}

additiveif there is no non-negative and non-zero countably additive measure k such thatk…A† l…A† for all A22N_{. We note that if l is non-negative,}

then it is purely ®nitely additive i€ it vanishes at every ®nite subset ofN. A theorem of Yosida and Hewitt (e.g., Theorem 1.23 of Yosida and Hewitt [22]) asserts that any ®nitely additive measure can be uniquely decomposed into a sum of a countably additive measure and a purely ®nitely additive measure.

The Banach space of all bounded ®nitely additive measures on 2N _with

the variation norm is denoted by ba. Its closed subspace consisting of all countable additive measures is denoted by ca. It is well known that ba is naturally identi®ed with the dual space ofl1ˆl1…N†;which is the Banach

space of all bounded sequences of real numbers. Also,ca can be naturally identi®ed withl1ˆl1…N†;i.e., everyx2l1 corresponds tolx2ca, where

lx…S† ˆ

X

i2S xi ;

for everyS22N_.

The following lemma plays an important role in our work.

Lemma 2.2. Let K be a weak_{-compact subset of ba that contains only}

countably additive measures(i.e.,K ca).ThenK is norm-compact.

Proof. SinceKca;we can viewKas a subset of`1. Asbais the norm-dual of`1;which in turns is the norm-dual of`1;the weaktopology induced by baonK coincides with the weak topology induced by`1onK. HenceK is a weakly compact subset of`1. Since`1is a separable Banach space, the weak topology on its weakly compact subsets is metrizable (see e.g., Theorem

compact in`1. By Schur's Theorem (see Diestel [6], page 85, and Corollary 14 page 296 of Dunford-Schwartz [7]), weak convergence and norm conver-gence of sequences are equivalent in`1; henceK is norm-compact in`1: u We conclude this subsection with some de®nitions concerning nets which will be used in the proof of Theorem B.

Adirected system is a pair …D;†;where D is a set and is a binary relation onDsuch that the following holds:

(a) Ifabandbc;thenac.

(b) Ifa;b2D;then there existsc2Dsuch thatacandbc.

A net is a mapping of a directed system into a topological space X. We usually writexafor the value of the net ata2Dandfxaja2Dg(or simply fxag) for the net itself. We say thatfxagconverges to a point x2X if for every open set O containing x there exists a02D such that xa2O for all aa0. Letfxaja2Dg be a net. For everya2D letba2Dbe such that baa; then the setfxbaja2Dgis called asubnetof fxaja2Dg.

For a further discussion of nets and related notions the reader is referred to Kelley [10], p. 65 72.

2.2 Game theoretic preliminaries

We refer to the members ofNasplayers, and to each subset ofNas acoalition. Agameis a bounded functionv:2N_{! <satisfyingv…;† ˆ0;and for which}

there isk2casuch that v…S† k…S† for everyS22N_{. (Note that the later}

requirement is satis®ed for every non-negative set functionvwithk0:†A gamev issuperadditive if v…S[T† v…S† ‡v…T† for every two disjoint co-alitionsS;T 22N_{. Throughout this paper we assume that all games under}

consideration are superadditive, but to avoid confusion we emphasize it in the statements of the results. Note that everyk2ca is continuous, and that if v…S† k…S†for allS;thenvÿkis a monotonic function. Hence by Lemma 2.1 a gamev is continuous on 2N _{if and only if lim}

n!1v…Sn† ˆv…S† whenever

Sn%SorSn&S. Note that our de®nition of continuity coincides with that of

Schmeidler [17] (see also Aumann and Shapley [2]). For every gamevand every coalitionS22N_{;we de®ne}

rv…S† ˆinf

Xn i 1

v…Si†;

where in®mum is taken over all ®nite partitionsfS1;S2;. . .;Sngof S. Note

thatrv…i† ˆv…fig†for everyi2N. Note also thatrv2ba. Moreover,rv is

the largest measure in ba such that rv…S† v…S† for all S22N; that is, if

k2baandk…S† v…S† for allS22N_;thenkr

v (pointwise). Also, ifvis

continuous thenrv2ca.

A payo€ measure (or a preimputation) is a measure k2ba such that k…N† ˆv…N†. A payo€ measure k is individually rational if krv. An

im-putationis an individually rational payo€ measure. The set of all imputations is denoted byI…v†.

3 The least core

Recall that the core of a gamev, denoted byC…v†;is the set o€ all imputa-tions l2I…v† such that l…S† v…S† for each S22N_{. For everyk2ba let}

f…k† ˆsupfv…S† ÿk…S† jSNg. The least core of the game v; denoted LC…v†;is the set of all imputationsl2I…v†for whichf attains its minimal value onI…v†. Note that sincef is the supremum of ane weak_-continuous

functions, it is a convex and lower semicontinuous function onba. AsI…v†is a weak_{-compact subset of ba; f attains its minimal value v on I…v†.}

ThereforeLC…v†is a non-empty convex weak_{-compact subset ofI…v†. We}

note thatv0;andvˆ0 if and only ifC…v† 6ˆ ;. In this caseC…v† ˆLC…v†.

Theorem A.Let vbe a superadditive continuous game. ThenLC…v† is a non-empty convex norm-compact subset ofca.

Proof. By Lemma 2.2 it suces to prove thatLC…v† ca. Without loss of generality (w.l.o.g.) assume that rv is identically zero. (Alternatively, we

could replace the gamevwith the gamewˆvÿrv, which is also continuous

and satis®esl2LC…w† if and only ifl‡rv2LC…v†:†Also w.l.o.g. assume

that v…N† ˆ1. Finally, we may assume that v>0;as otherwise the result

follows from Schmeidler [17].

Let l2LC…v†. By Theorem 1.23 of Yosida and Hewitt [22], l can be uniquely decomposed into a sum of a non-negative countably additive measure l_c and a non-negative purely ®nitely additive measure l_p (i.e., a measurelpsatisfyinglp…S† ˆ0;for each ®niteS2R†. We must show thatlp

is identically zero. Aslpis non-negative, it suces to show thatlp…N† ˆ0.

Assume that lp…N†>0. We show that this implies that maxfv…S† ÿlc…S†

jS22N_g>

v. This leads to a contradiction since maxfv…S† ÿlc…S†

jS22N_{g(which exists becausevÿl}

cis continuous) is bounded above byv.

Indeed, letS22N_{;and for every n2N de®neSnˆ f1;. . .;ng. Then}

v…S† ÿlc…S† ˆ_n!1lim…v…S\Sn† ÿlc…S\Sn††

ˆ lim

n!1…v…S\Sn† ÿl…S\Sn††

v:

In the remaining of the proof of Theorem A assume thatlp…N†>0. The

proof proceeds in several steps. STEP1: We show thatl_c…N†>0.

Assume by way of contradiction that lc…N† ˆ0. Then lˆlp. As

l2LC…v†;v…S† ÿl…S† v; for every S22N. As v is continuous, we have

limn!1v…Sn† ˆv…N† ˆ1. Sincelˆlpandlp…Sn† ˆ0;we havev…Sn† ÿl…Sn†

ˆv…Sn†; for each n:Therefore vv…Sn† for each n;and thus vˆ1. Let

n2I…v†be given for eachS22N _{byn…S† ˆ}P

i2S2ÿi. Asvÿnis continuous,

there isS₂₂N _{such thatv…S† ÿn…S† ˆmaxfv…S† ÿn…S† jS22}N_{g. From}

v…S† 1;andn…S†>0 for eachS6ˆ ;in 2N_{. Summing up these inequalities}

we have

1ˆvv…S† ÿn…S† 1ÿn…S†<1 ;

which is a contradiction. Hencelc…N†>0.

STEP2: We show that the payo€ measurekˆ 1

lc…N†lcis a member ofLC…v†.

Letaˆ 1

lc…N†. Aslp…N†>0;a>1. For eachS22

N _{we have}

v…S† ÿk…S† ˆv…S† ÿal_c…S† ˆv…S† ÿl_c…S† ÿ …aÿ1†l_c…S† v…S† ÿl_c…S†:

Recall thatSnˆ f1;. . .;ng. Then for allS22N we have

v…S† ÿk…S† ˆ_n!1lim…v…S\Sn† ÿk…S\Sn†† lim n!1…v…S\Sn† ÿlc…S\Sn†† ˆ_n!1lim…v…S\Sn† ÿl…S\Sn†† v: Thereforek2LC…v†.

STEP3: We show that maxfv…S† ÿlc…S† jS22Ng> v;and we will get the

desired contradiction.

For everyS22N _{we havev…S† ÿl}

c…S† v…S† ÿl…S†. As l2LC…v†;we

have supfv…S† ÿl…S† jS22N_{g ˆv. Hence maxfv…S†ÿl}

c…S† jS22Ng v.

We show that this inequality is strict. Assume to the contrary that maxfv…S† ÿlc…S† jS22Ng ˆv. Letkˆ_lc1_N†lc;by Step 2,k2LC…v†. Now

if S22N _{is such that v…S† ÿk…S† ˆv; then k…S† ˆ0. For otherwise we}

would have v…S† ÿl_c…S†>v…S† ÿk…S† ˆv. Let S0ˆ fi2N jk…fig† ˆ0g. Because k2ca;k…S0† ˆ0. Let S22N be such that v…S† ÿ k…S† ˆ v (S6ˆ ;

becausev>0); hencek…S† ˆ 0;and thereforeSS0. Thus,S06ˆ ;. Let j2NnS0. Then k…fjg†>0. Let Cjˆ fS22Njj2Sg. The set

f1SjS2Cjg is a closed subset of f0;1gN in the topology of pointwise

convergence, and therefore it is compact. Let Q2Cj be such that v…Q†

ÿk…Q† ˆmaxfv…S† ÿk…S† jS2Cjg. Let dˆv…Q† ÿk…Q†. If dˆv; then

k…Q† ˆ0;which is impossible sincej2Q;and therefore k…Q† k…fjg†>0. Hence d< v. Let 0< <min…k…fjg†; vÿd†; and for each i2S0 let

iˆP2i

l2S₀2l. Also for eachi2N de®ne

^ k…fig† ˆ ki…fjg† ÿ ii2ˆSj0 k…fig† i2=S0[ fjg 8 < :

Note that ^k2ca‡. Moreover, since ^k…S0† ˆ, we have ^k…N† ˆ1; and therefore ^k2I…v†. We show that maxfv…S† ÿ^k…S† jS22N_g<

will contradict the de®nition of v. Let S22N, S6ˆ ;. We distinguish two

cases.

(a)SS0. In this case we havev…S† ÿ^k…S†<v…S† ÿk…S† v.

(b) SˆS1[S2; where S1S0; and S2NnS0 satis®es S26ˆ ;. Since S26ˆ ;;we have k…S†>0; hence v…S† ÿk…S†< v. Ifj2=S; then v…S† ÿ^k…S†

v…S† ÿk…S†< v. Ifj2S;thenv…S† ÿk…S† d. Thus

v…S† ÿ^k…S† ˆv…S† ÿk…S† ‡k…S† ÿ^k…S† d‡k…S† ÿ^k…S†

ˆk…S1† ÿ^k…S1† ‡d‡k…fjg† ÿ^k…fjg†

d‡ < v: (

The following example taken from Kannai [9] (Example 2.1) shows that when a game is not continuous its least core may contain only purely ®nitely additive payo€ measures.

Example 3.4. De®ne a gamevby

v…S† ˆ 1 NnSis finite

0 otherwise.

It is easy to see that the core ofvis not empty, and therefore it coincides with the least core. Since a payo€ measurel2bais in the core ofvif and only if l…S† ˆ0 for each ®niteS22N_{;the least core ofvcontains only purely ®nitely}

additive measures.

Remark. The least core for games with a ®nite set of players was de®ned by Maschler, Peleg and Shapley [13] as the set of all payo€ measureslfor which

min

k2I_{v†S6 ;}max_{;S6 N}…v…S† ÿk…S††

is attained atkˆl;whereI_{v†is the set of preimputations. It can be easily}

veri®ed that for games with an empty core our de®nition applied to a game with a ®nite set of players coincides with the above de®nition. However, if C…v† 6ˆ ;thenLC…v† ˆC…v†according to our de®nition, whereas the original de®nition may yield a strict subset of the core. In this work we restrict ourselves to solution concepts which are individually rational. Therefore we use our de®nition which also turn out to be more convenient from the technical point of view.

4 The kernel

The set of all payo€ measures in a gamevis denoted byI_{v†; that is}

Fori;j2N;i6ˆj;de®ne the set

Tijˆ fS22N ji2S;j2=Sg:

Also fori;j2N;i6ˆj;andl2bade®ne

sij…l† ˆsupfv…S† ÿl…S† jS2Tijg:

Theprekernelofvis the set PK…v† ˆ fl2I_{v† js}

ij…l† ˆsji…l†;8i;j2N;i6ˆjg:

Thekernel ofvis the set

K…v† ˆ fl2I…v† j …sij…l† ÿsji…l††…l…fjg† ÿrv…fjg†† 0;8i;j2N;i6ˆjg:

The notion of the kernel of a coalitional game with a ®nite set of players was introduced by Davis and Maschler [5]. It is well-known that ifvis a super-additive game with a ®nite set of players, thenK…v† ˆPK…v† (see Theorem 2.7 in Maschler, Peleg and Shapley [12]). It is also well-known that for such games K…v† \LC…v† 6ˆ ;. In fact, the nucleolus ofv is contained in this in-tersection (see Corollary 6.7 in Maschler, Peleg and Shapley [13], and The-orem 3 in Schmeidler [16]). TheThe-orem B below establishes that the prekernel (and hence the kernel) and the least core have a non-empty intersection for continuous games.

Theorem B. Let v be a superadditive continuous game. Then PK…v†\ LC…v† 6ˆ ;. In particular, PK…v† contains an individually rational countably additive payo€ measure.

We need the following lemmas.

Lemma 4.1. Let v be a superadditive continuous game. Then each function sij;i;j2N; ;i6ˆj;is norm-continuous onca.

Proof. Let l2ca and let fl_ng1

n 1cabe such that limn!1klnÿlk ˆ0.

We show that for each i;j2N;i6ˆj;limn!1sij…ln† ˆsij…l†. Note that the

setf1SjS2Tijgis a closed (and therefore a compact) subset off0;1gN in

the topology of pointwise convergence. Since v is continuous and each

l_n2ca;vÿl_nattains its maximum on Tij. For eachn2N letSn2Tij be

such thatsij…ln† ˆv…Sn† ÿln…Sn†;and similarly letS2Tijsatisfyingsij…l† ˆ

v…S† ÿl…S†. Sincev…S† ÿl…S† v…Sn† ÿl…Sn†for everyn2N;we have

v…Sn† ÿln…Sn† v…Sn† ÿln…Sn† ‡ …v…S† ÿl…S†† ÿ …v…Sn† ÿl…Sn††

ˆv…S† ÿl…S† ‡l…Sn† ÿln…Sn†

v…S† ÿl…S† ‡ klÿlnk:

Also we havev…Sn† ÿln…Sn† v…S† ÿln…S†for eachn2N.

As klÿlnk !0;limn!1…v…Sn† ÿln…Sn†† ˆv…S† ÿl…S†. Hence,

lim

Lemma 4.2. Letwbe a superadditive game with a ®nite support(i.e.,for some n1;w…S\ f1;. . .;ng† ˆw…S†;for all S22N_{). ThenPK…w† \LC…w† 6ˆ ;.}

Proof. W.l.o.g. assume thatw0. Let Rˆ fS22N_{jS f1;. . .;ngg; and}

letube the restriction ofwtoR. Sincewis superadditive ,uis a superadditive game with a ®nite set of players. Therefore by Theorem 2.7 in Maschler, Peleg and Shapley [12], we have PK…u† ˆK…u†; and by Corollary 6.7 in Maschler, Peleg and Shapley [12], we havePK…u† \LC…u† 6ˆ ;.

Letl2PK…u† \LC…u†. De®ne

l…fig† ˆ l₀…fig†_; ; 1_otherwise.in

Obviouslyl2ca;and therefore there isS₂₂N _{such that}

w…S_{† ÿl…S† ˆmaxfw…S† ÿl…S† jS22}N_g:

Now,

ww…S† ÿl…S†

ˆu…S_{\ f1;. . .;ng† ÿl…S\ f1;. . .;ng†}

u:

Let k2LC…w†; since k minimizes the function f…n† ˆsupfw…S† ÿn…S† jS22N_{gonI…w†;we havek…S† ˆ0 for eachSNnf1;. . .;ng. Thus}

wˆmaxfw…S† ÿk…S† jS22Ng ˆmaxfu…Q† ÿk…Q† jQ2Rg u: Hencew…S_{† ÿl…S† ˆ} w;and thereforel2LC…w†. Now,

maxfw…S† ÿl…S† jS2Tijg ˆmaxfu…Q† ÿl…Q† jQ2R\Tijg; …4:1†

and

maxfw…S† ÿl…S† jS2Tjig ˆmaxfu…Q† ÿl…Q† jQ2R\Tjig: …4:2†

As l2PK…u† \LC…u†; we get from …4:1† and …4:2† that l2PK…w†\

LC…w†: u

With these results at hand we can now complete the proof of Theorem B. Proof of Theorem B. Assume, w.l.o.g., that rv is identically zero and

v…N† ˆ1. For everyn2NletSnˆ f1;. . .;ng;and letvnbe the game given by

vn…S† ˆv…S\Sn†. By Lemma 4.2, for each n2N there is ln2PK…vn†

\LC…vn†. As the gamesvn are continuous, then by Theorem A, ln2cafor

each n. LetM be the weak_{-closure inbaof the set Mˆ fl}

njn2Ng. We

show thatM is a norm-compact subset ofca. This yields that the sequence flng1n 1 has a subsequence which converges in the norm to a member of PK…v† \LC…v†. AsM is a weak_{-compact subset ofba;then by Lemma 2.2 it}

suces to show thatM ca. Moreover, sinceM ca;it suces to show that each l2MnM is a member of ca. Let l2MnM . Then there exists a net fln…a†ja2Dg in M which converges to l. Let k2LC…v†. By Theorem A,

k2ca. We distinguish two cases (note thatk0 andk…N† ˆ1). (1) At least one the following holds:

…1a† The netfk…Sn…a†† ja2Dgdoes not converge to 1.

…1b† The netfv…Sn…a††gdoes not converge to 1.

…1c† There exists a coalition S such that the net fvn…a†…S†g does not

converge tov…S†.

We show that if…1a†is satis®ed thenlis countably additive (the proof thatl is countably additive when …1b† or …1c† are satis®ed is similar). Now …1a† implies that there exists a subnet of fk…Sn…a†† ja2Dg that converges to a

numbera2[0,1). Assume, w.l.o.g., thatfk…Sn…a†† ja2Dgitself converges to

a. Let >0 be such thata‡ <1;then there isa02Dsuch that for each aa0(heredenotes the order relation onD) we havek…Sn…a††<a‡ <1.

As limn!1k…Sn† ˆ1;there isn02N such thatk…Sn†>a‡for everyn>n0.

Thus, for eachaa0we haven…a† n0:LetS22N. Asfln…a†gconverges in

the weak_{-topology tol;the netfl}

n…a†…S†gconverges tol…S†. Hence there is

b02D such that for each bb0 we have jln…b†…S† ÿl…S†j< . From the

de®nition of a net it follows that there isc2D;such thatca0;andcb0. Therefore n…c† n0;and jln…c†…S† ÿl…S†j< . Thus we have shown that if

0< <1ÿa is given, then for each S22N _{there is 1nn}

0 such that jln…S† ÿl…S†j< . Since the set f1;. . .;n0g is ®nite and can be chosen arbitrarily small, there is 1nn0such thatln…S† ˆl…S†.

We now show that this implies thatl2ca. Assume thatfTlg1l 12Nis a non-decreasing sequence of coalitions such thatS1l 1TlˆN. Then for every

l2N there is 1nn0such thatln…Tl† ˆl…Tl†. Therefore there is a

sub-sequence flkg1k 1N; and 1nn0; such that l…Tlk† ˆln…Tlk† for each

k2N:Asln2I…vn† \ca‡;limk!1ln…Tlk† ˆ1. Therefore limk!1l…Tlk† ˆ1.

Nowfl…Tl†g1l 1is a non-decreasing and bounded sequence of real numbers, and therefore it converges. As fl…Tlk†g1k 1 is a subsequence of fl…Tl†g1l 1

which converges to 1, we have liml!1l…Tl† ˆ1. Thus,l2ca.

(2) Case (1) does not hold. We show that in this casel2LC…v†and thus by Theorem A it is countably additive. Ask2LC…v†;then for eacha2Dand all

SN we have

v…S\Sn…a†† ÿk…S\Sn…a†† v:

Sincek…Sn…a†† converges to 1, we assume, w.l.o.g., thatk…Sn…a††>0 for each

a2D. For anySN anda2Dlet

kn…a†…S† ˆ_kv…S_Sn…a††

n…a††k…S\Sn…a††;

an…a†ˆ1 v_kS_Sn…a†† n…a††:

Then kn…a†2I…vn…a††; and the net fan…a†g converges to zero. Also for every

SN we have

vn…a†…S† kn…a†…S† v…S\Sn…a†† k…S\Sn…a†† ‡an…a†:

Therefore

vn…a†v‡an…a†:

Asln…a†2LC…vn…a††;for everySN we have

vn…a†…S† ln…a†…S† v‡an…a†:

Hence for everySN we have

v…S† l…S† v:

Thus l2LC…v†. Asvis continuous by Theorem A, LC…v† ca. Therefore l2ca.

We have shown that in any casel2ca. Thus,M ca, and therefore it is norm compact in ca. As the sequence flnjn2Ng M; there is a subse

quence flnkg 1

k 1 which converges in the norm to a member l of ca. As limk!1l…Snk† ˆlimk!1v…Snk† ˆ1; and limk!1vnk… † ˆS v S… † for each

S22N_{;an argument identical to the one given in (2) above yieldsl2LC…v†.}

Leti;j2N;i6ˆj. Sincelnk 2PK…vnk†for eachk2N;we havesij…lnk† ˆsji …lnk†. By Lemma 4.1, limk!1sij lnk

ÿ

ˆsij… †l; and limk!1sji…lnk† ˆsji…l†.

Thereforesij…l† ˆsji…l†;and thusl2PK…v†. Hence,PK…v† \LC…v† 6ˆ ;: (

Wesley [21] showed by using non standard analysis that a non negative gamevhas a non empty kernel (for every coalition structure) if it satis®ed the following conditions:

For eachS22N_;lim

n!1…v…S† v…S\ f1;. . .ng†† ˆ0;and …4:3† X1 i 1 ri<1;whereriˆsupfv…S[ fig† v…S† jSNnfigg: …4:4†

We show that …4:3†and…4:4†imply thatvis continuous;and then we give an example of a continuous game which does not satisfy …4:4†. Thus, Wesley's Theorem is a special case of Theorem B when the coalition structure is fNg.

Lemma 4.3. Assume thatvis a non negative superadditive game which satis®es …4:3†and…4:4†.Thenvis continuous.

Proof. LetS22N_{. We ®rst show that iffS}

ng1n 1is a non decreasing sequence of coalitions such that S1

fSng1n 1 is such a sequence then for each n2N there is m…n† such that mm n… †impliesS\ f1;. . .;ng Sm. Asvis monotonic we have

v…S\ f1;. . .;ng† v…Sm† v…S†;

for everymm…n† . Thus by…4:3†, limn!1v…Sn† ˆv…S†.

Assume now thatfSng1n 1is a non-increasing sequence of coalitions such that T1n 1SnˆS. We show that limn!1v…Sn† ˆv…S†. For AN we de®ne

r…A† ˆPi2Ari. Then r2ca‡. Let n2N; and let B22N be such that

B\Sˆ ;. We show thatv…S[B† ÿv…B† r…B†. De®neBnˆB\ f1;. . .;ng.

Then by (4.4)

v…S[Bn† ÿv…S† r…Bn†:

As Bn‡1Bn; and S1n 1BnˆB; by what we have just shown,

limn!1v…S[Bn† ˆv…S[B†. Since limn!1r…Bn† ˆr…B), we have

v…S[B† ÿv…S† r…B†:

AsBwas an arbitrary coalition, for eachn2N we have 0v…Sn† ÿv…S† ˆv…S[ …SnnS†† ÿv…S† r…SnnS†:

As limn!1r…SnnS† ˆ0;we have limn!1v…Sn† ˆv…S†: u

We give an example of a non-negative continuous game which does not satisfy (4.4).

Example 4.4. For every 0x1 let f…x† ˆ1ÿp1ÿx. De®ne a measure l2ca by l…S† ˆcPi2S

1

i2; where cˆ …P1_{i 1}i12† 1. For every SN let

v…S† ˆf…l…S††. Sincef is continuous on [0,1],vis continuous. Moreover, as f is convex on [0,1],vis convex; that is,

v…S1[S2† ‡v…S1\S2† v…S1† ‡v…S2†

for eachS1;S2N (see Shapley [18]), and in particular, vis superadditive.

For everyi2N we have

riˆsupfv…S[ fig† ÿv…S† jSNnfigg v…N† ÿv…Nnfig†:

As v…N† ÿv…Nnfig† ˆ c p i ; we have X1 i 1ri c p X1 i 1 1 i: ThusP1i 1riˆ 1;and (4.4) is not satis®ed.

As it was mentioned above, Maschler, Peleg, and Shapley [13] showed that for superadditive games with a ®nite set of players the kernel coincides with the prekernel. By Theorems B there are always countably additive payo€s measures in the kernel and the prekernel of a continuous

superad-ditive game on 2N_{. Using the compactness of 2}N _{the same proofs as those of}

Theorem 2.4 and Theorem 2.7 in Maschler, Peleg, and Shapley [13] yield. Proposition 4.5.Letvbe a superadditive continuous game. Then every count-ably additive payo€ measure inPK…v† is individually rational.

Proposition 4.6.Letvbe a superadditive continuous game. Then K…v† \caˆPK…v† \ca:

5 Bargaining sets

In this section we establish that the Aumann-Maschler and the Mas-Colell bargaining sets of a continuous game with a countable set of players are non-empty sets which contain the set of all countably additive payo€ measures in the prekernel.

Letv be a game, letl2I…v†, and leti;j2N;i6ˆj;be two players. An objection of i to j in l is a pair …A;k† such that k2ba;krv;AN;

i2A;j2=A;k…A† v…A†; andk…fkg†>l…fkg† for each k2A. A counterob-jection of j to …A;k† is a pair …B;n† such that n2ba;n rv;BN;j2B;

i2=B;n…B† v…B†;n…fkg† l…fkg† for k2BnA; and n…fkg† k…fkg† for k2A\B. Ajusti®ed objectionofi2N tojinlis an objection which does

not have a counterobjection. The Aumann-Maschler bargaining set (see

Aumann and Maschler [1]) ofvis the setB…v†of all imputationsl2I…v†such that no playerihas a justi®ed objection to another player inl.

Davis and Maschler [4], and Peleg [15] showed that ifvis a game with a ®nite set of players, thenB…v† is a non-empty set. For this class of games, Davis and Maschler [5] proved that K…v† is a subset of B…v†. Theorem C below establishes that ifvis a continuous game, then the set of all countably additive payo€ measures in the prekernel is containedB…v†.

Theorem C. Let v be a superadditive continuous game. Then PK…v† \ca B…v†.In particular,B…v†contains a countably additive payo€ measure. Proof. W.l.o.g. assume that rv is identically zero. Let l2PK…v† \ca. We

show that l2B…v†. Assume to the contrary that l2=B…v†. Then there are i;j2N;i6ˆj; such that j has a justi®ed objection …A;k† to iinl. By the Yosida and Hewitt Theorem [22],kcan be uniquely decomposed into a sum of a non-negative countably additive measurekc;and a non-negative purely

®nitely additive measurekp(note thatkrv;and thusk0†. Askpvanishes

on ®nite subsets of N; we have kc…fkg†>l…fkg† for each k2A; and

kc…A† k…A† v…A†. Since any counterobjection to …A;kc† is also a

counterobjection to…A;k†;…A;kc† is a justi®ed objection. Sincev is

continu-ous andl2ca;there isB22N _{such thati2B;j2=B;and}

Asl2PK v… †; v…B† ÿl…B† ˆsji…l† v…A† ÿl…A†: Sincekc…fjg†>l…fjg†; v…A† ÿl…A† X k2A …kc…fkg† ÿl…fkg†† > X k2A\B …kc…fkg† ÿl…fkg††: Therefore v…B† ÿX k2Al…fkg†> X k2A\B…kc…fkg† ÿl…fkg††; i.e., v…B†> X k2A\B kc…fkg† ‡ X k2BnA l…fkg†:

For eachSNletn… † ˆS kc…S\A† ‡l…S\ …BnA††. Thenn…B† v…B†by

the last inequality, and ifk2A\B;n…fkg† ˆkc…fkg†;and if l2BnA;then

n…flg† ˆl…flg†. Therefore each member ofB(and in particular playeri) has a counterobjection to the objection …A;kc† ofj, which contradicts the fact

that …A;kc† is a justi®ed objection. Now by Theorem A, PK…v† \LC…v†

PK…v† \ca. Therefore by Theorem B, B…v† contains a countably additive

payo€ measure. u

Mas Colell [14] proposed a notion of bargaining set di€erent from that of Aumann and Maschler bargaining set, and showed that in the context of a market with a continuum of players, this new bargaining set coincides with the set of walrasian allocations. The advantage of the Mas-Colell bargaining set is that it can be de®ned for games with an uncountable set of players.

The following de®nitions are taken from Einy et al. [8], who provide a straightforward generalization of the Mas-Colell [14] de®nition to games with an in®nite set of players. Letvbe a game, and letlbe a payo€ measure inv. Anobjectiontol(in the sense of Mas-Colell) is a pair…A;k†such that A22N_{, and k2ba satis®es k…A† v…A†, k…A†>l…A†, and k…B† l…B† for}

every coalitionBA. Acounterobjection(in the sense of Mas-Colell) to the objection…A;k†is a pair…C;n† such that

n2ba;andn…C† v…C†; …5:1†

For everyBA\C;n…B† k…B†;and for everyDCnA;

…5:2†

n…D† l…D†;and n…C†>k…A\C† ‡l…CnA†:

…5:3†

Ajusti®ed objectionis an objection which has no counterobjection. The Mas-Colell bargaining setofvis the setMB…v†of all payo€ measures which have no justi®ed objection.

Mas-Colell [14] in considering exchange economies with a continuum of agents, de®nes the bargaining set without restricting attention to individually rational allocations. Thus, his equivalence result holds for a large set. From the point of view of an existence result it will be interesting to show that the Mas-Colell bargaining set always contains an individually rational payo€ measure. Theorem D below establishes that the Mas-Colell bargaining set of a continuous game contains the set of all countably additive payo€ measures in the prekernel. As a consequence, for these games the Mas-Colell bar-gaining set contains an individually rational countably additive payo€ measure.

Theorem D. Let v be a superadditive continuous game. Then PK…v† \ca MB…v†. In particular, MB…v† contains an individually rational countably additive payo€ measure.

The following lemma will be useful in the proof of Theorem D.

Lemma 5.1.Letv be game,and letlbe a payo€ measure inv:If…A;k†is a justi®ed objection tol,thenv…B† k…B\A† ‡l…BnA†for eachB2R. Proof. Assume to the contrary that there isB2Rsuch thatv…B†>k…B\A† ‡l…BnA†. Then B6ˆ ;. Let ˆv…B† ÿ …k…B\A† ‡l…BnA††. Choose t2B;

and letdt be the probability measure concentrated onftg. De®nen2baby

n…S† ˆk…S\ …B\A†† ‡l…S\ …BnA†† ‡dt…S†:

Thus, ifS2Rsatis®esSA\Bwe haven…S† k…S†;and ifS2Rsatis®es SBnAwe haven…S† l…S†. Also

n…B† ˆv…B†>k…B\A† ‡l…BnA†:

Thus…B;n†is a counterobjection to…A;k†;which is a contradiction. u Proof of Theorem D. W.l.o.g. assume that rv is identically zero. Let

l2PK…v† \ca. We show that l2MB…v†. Assume by way of contradiction that l2=MB…v†. Then there is a justi®ed objection …A;k† to l. Since lis a payo€ measure,A6ˆN. Assume, w.l.o.g., thatk…A† ˆv…A†. Since …A;k†is a justi®ed objection tol, by Lemma 5.1 for eachBAwe havek…B† v…B†. For eachS22N_{letvA…S† ˆv…S\A†;andkA…S† ˆk…S\A†. ThenkAis in the}

core of the gamevA. SincevAis continuous,kA2ca. Ask…A†>l…A†, there is

j2A such that kA…fjg†>l…fjg†. Let i2NnA. As l2PK…v†; there is

CN;i2C;j2=C; such that v…C† ÿl…C† ˆsij…l† v…A† ÿl…A†. Since

kA…fjg†>l…fjg†, an argument identical to the one given in the proof of

Theorem C yields v…C†> X k2A\C kA…fkg† ‡ X k2CnA l…fkg† ˆk…A\C† ‡l…CnA†:

By Lemma 5.1 this contradicts the assumption that …A;k†is a justi®ed

ob-jection to l. Now by Theorem A, PK…v† \LC…v† MB…v†, and thus by

Theorem B, MB…v† contains an individually rational countably additive

payo€ measure. u

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