DP2021-15

T h e B a r g a i n i n g S e t a nd
Coalition Formation

Ken-Ichi SHIMOMURA

June 21, 2021

The Bargaining Set and Coalition Formation
Ken-Ichi ShimomuraJune, 2021
Forthcoming in the International Journal of Economic Theory
Abstract. We address the problem of predicting how rational agents will form coalitions in a
nontransferable utility game, and within each coalition how they will allocate the gains obtained
through cooperation. To answer these questions, we propose solution concepts according to which
the coalition structure and the payoff allocations are simultaneously determined. We prove thenonemptiness and partial efficiency of the steady bargaining set, a refinement of the Zhoubargaining set, for at least one coalition structure under the restrictive non-crossing condition. In
addition, we show the nonemptiness and possible inefficiency of the Mas-Colell bargaining set if
this condition is not assumed.
Keywords: Nontransferable utility game, coalition structure, bargaining set, restrictive noncrossing condition
JEL classification No: C71, D71

1. Introduction
We address the problem of predicting how rational agents, namely players, will form coalitions,and within coalitions how they will allocate the gains obtained through cooperation. We define a
coalition structure as a partition of the whole set of players, and formulate the underlying situation
as a coalitional game with coalition structure, in which the allocation of payoffs will depend on
which coalitions actually form. We consider a coalition structure as a collection of coalitions whose
members are always free to form a new coalition for higher payoffs. Then, the eventual formation
 RIEB, Kobe University, 2-1 Rokkodai, Nada, Kobe, Hyogo 657-8501, Japan.

(email: ken-ichi@rieb.kobe-u.ac.jp)
This is a revision of Brown University Economics Working Paper 95-11. I am grateful to Youngsub
Chun for giving me the opportunity of a publication, and Ryo-ichi Nagahisa for calling my attention
to the topic of the paper. Many helpful comments and discussions with Rajiv Vohra and WilliamThomson are greatly appreciated. Thanks are also due to the late Michael Maschler, Tatsuro Ichiishi,
and Roberto Serrano for encouragement, suggestions and conversations. I am indebted to Barry
Feldman for pointing out an error, and Matthew Coffey for reading carefully an earlier draft. This
research is partially supported by Kyoto University Foundation, Kikawada Foundation, and theJapanese Ministry of Education, Culture, Sports, Science and Technology (Grants-in-aid for Scientific
Research (A) 18H03641 and (C) 19K01558).

1

of coalitions will be influenced by what players expect to get in different coalitions. We thus search
for a solution in the family of bargaining sets to select the coalition structure and the payoff
allocation simultaneously, and study the existence and the efficiency of solution outcomes.
The idea of the bargaining set starts from a study by Aumann and Maschler (1964) of a stable(𝑖)solution for the class of coalitional games with coalition structures. Their bargaining set 𝑀1 isdefined in terms of an “objection from a player to another player given a proposal” and a“counterobjection from the objected player to the objecting player.” The notion of the bargaining
set depends on the particular definitions of objections and counterobjections being considered.
Several alternative definitions have been since. In this paper, we define an “objection from a
coalition to a proposal” and a “counterobjection from an objected coalition to the objecting
coalition.” Here is the essence: Let 𝑥 be a recommended payoff profile. Then, a coalition 𝑆 has an
objection if it can produce an alternative profile of payoffs for its members that make them strictly
better off than at 𝑥. A coalition T has a counterobjection to 𝑆 at 𝑥 if T can achieve a payoff profilethat makes its members at least as well off as in 𝑥, and makes players belonging to both S and T atleast as well off as they are in the objection. An objection to 𝑥 is “justified” if it has nocounterobjection.
Davis and Maschler (1963), and Peleg (1963) prove that, for transferable utility (TU) games,(𝑖)Aumann and Maschler’s bargaining set 𝑀1 is nonempty-valued for all coalition structures. Thus,(𝑖)

𝑀1

is not a solution to select stable coalition structures. Aumann and Drèze (1974) investigate(𝑖)

not only 𝑀1

but also the core of a coalitional game with a coalition structure, which is the set of

payoff profiles feasible for the coalition structure with no objections from any coalition. Aumann
and Drèze show that the core of a balanced TU game with a coalition structure is nonempty if andonly if the coalition structure maximizes the sum of the “worths” of the partitions of the grandcoalition [For details, see Shimomura (1997, P.295)]. This tells that the core selects stable pairs ofcoalition structure and payoff allocation for balanced TU games, but it does not work for the othertypes of TU games.
(𝑖)

For nontransferable utility (NTU) game, Mas-Colell (1989) proposes a modification of 𝑀1 ,now called the Mas-Colell bargaining set. Mas-Colell’s original is defined to be the set of efficient
payoff profiles produced by the grand coalition with no justified objection. The Mas-Colellbargaining set of a coalitional game with a coalition structure, discussed in this paper, is the set of
payoff profiles feasible for the coalition structure with no justified objection. In a TU game, it
contains efficient payoff profiles for at least one coalition structure [Zhou (1994), Shimomura(1997)]. In fact, Mas-Colell (1989, P.138) points out that for many games, his bargaining set “canbe quite large” regardless of the requirement of efficiency. Thus, the Mas-Colell bargaining set per
se is not a solution to select stable payoff allocations.
To our best knowledge, there has been no research on the Mas-Colell bargaining set of NTU
games with coalition structures although Vohra (1991) gives sufficient conditions for thenonemptiness, which implies efficiency, of Mas-Colell’s original bargaining set of NTU games. In

2this paper, we investigate refinements of the Mas-Colell bargaining set to identify stable coalition
structures and internal payoff allocations for a reasonably large class of NTU games including all
TU games. Zhou (1994) is the first author to challenge this project. We pursue a better notion ofstability for a bargaining set based on the fact that the outcomes become more sophisticated as therequirements imposed on counterobjection to qualify are strengthened.
To do so, we propose criterion to adopt counterobjections in the following ways. We first
impose the intersection criterion, intensively investigated by Zhou (1994). In the definition of acounterobjection by Mas-Colell (1989) as well as Aumann and Maschler's (1964), acounterobjecting coalition may not include any member of the objecting coalition. That is, thecoalitions making the objection and the counterobjection may have an empty intersection. Zhou
(1994) points out the importance of this requirement on counterobjections of a nonempty
intersection, what we call the "proper intersection condition." The resulting solution concept is theZhou bargaining set. We will investigate how strong of an intersection requirement one can impose
on counterobjections so as not to make the bargaining set empty for all coalition structures.
The second is the improvement criterion. Mas-Colell (1989) and Vohra (1991) define an
objection and a counterobjection as follows: an alternative payoff vector to the recommendation 𝑥
in a coalition is proposed which keeps its members as well off as at 𝑥, and at least one of themstrictly better off compared (and the objection to be counterobjected). Notice the symmetry in theirdefinitions of objections and counterobjections: both are defined in terms of Pareto-improvement.
We will examine the possibility of requring strict improvements, that is making all the members of
the objecting and counterobjecting coalitions strictly better off than at 𝑥. Note that the concepts ofobjection-counterobjection defined by Pareto-improvement and strict improvement are notlogically related.
The third criterion is the dominance criterion. Suppose that a payoff vector is proposed fora society, and coalition 𝑆 has an objection. Counterobjections are defined objection by objection;
which coalition is against 𝑆 depends on the objection proposed by 𝑆 . A coalition 𝑇 will bedisinclined to counterobject if they know that 𝑆 has an alternate option to which 𝑇 has nocounterobjection. The story is different if 𝑇 can make a counterobjection to every possible
objection that 𝑆 can make against 𝑥 . That is, 𝑇 can break any deviation by 𝑆 from 𝑥 . We will
require counterobjections to satisfy this criterion and define the steady bargaining set.
The steady bargaining set is contained in the Zhou bargaining set, which passes theintersection and dominance tests. We propose the subset of the steady bargaining set obtained by
imposing the improvement criterion for both objections and counterobjections. We call this thestrong steady bargaining set.
We discuss one more bargaining set. We refine the Mas-Colell bargaining set by requiring
the objection to be in the core of the “reduced game” on the objecting coalition as well as bothobjections and counterobjections to be defined by strict improvements. We call it the precise
bargaining set. The steady bargaining set is almost, but not quite, a subset of the precise bargaining3

set. The strong steady bargaining set is contained in both the precise bargaining set and the steadybargaining set.
We establish three theorems. First, for a NTU coalitional game satisfying the "restrictive noncrossing condition," the steady bargaining set contains at least one efficient payoff profile for at least
one coalition structure. Second, for a NTU coalitional game satisfying a “comprehensiveness”
assumption as well as the “restrictive non-crossing condition,” the strong steady bargaining set
contains at least one efficient payoff profile. The restrictive non-crossing condition, which all TU
games satisfy, enables us to avoid the "balanced family" argument, which is used by Scarf (1967),
Vohra (1991), and Zhou (1994). We also show by way of an example that the Zhou bargaining set
may be empty and the Mas-Colell bargaining set may be inefficient without the restrictive noncrossing condition. As corollaries, we deduce the possible emptiness of the steady bargaining set.
This tells us that a stable compromise does not generally lead to efficiency; it is a problem shared bycoalitional games and strategic games. Third, on the class of all NTU coalitional games, the precise
bargaining set is nonempty for at least one coalition structure although it may not contain any
efficient payoff vectors. To prove this, we need essentially no assumptions. The only condition usedin our proof is that for every coalition, the set of all individually rational payoff profiles is compact.
Zhou (1994) shows that on the class of TU games, the Zhou bargaining set contains at least
one efficient payoff profile for at least one coalition structure. Our first theorem extends his result to
NTU games similarly to the way Scarf (1967) had extended the core nonemptiness theorem of
Bondareva (1963) and Shapley (1967). Shimomura (1997) proves that on the class of TU games, thesteady bargaining set and the Zhou bargaining set are nonempty for every coalition structure that can
produce the maximal sum of payoffs to all the players. Combining this result with our first theorem,
we demonstrate that in a NTU coalitional game, it is not easily predictable which coalitions are likely
to form, but the idea of the bargaining set helps explain how coalition structures and payoff
allocations are determined.
The paper is organized as follows. In Section 2, we present a model of an NTU coalitional
game and define the Mas-Colell bargaining set. In Section 3, we propose, as refinements of the MasColell bargaining set, the Zhou bargaining set, the steady bargaining set, and the strong steadybargaining set, and investigate their nonemptiness and efficiency. In Section 4, we propose two
subsolutions of the Mas-Colell bargaining set, which are the precise bargaining set and the naturalbargaining set, and investigate their nonemptiness. Section 5 is for concluding remarks. The
appendix provides the proofs of the results stated in Sections 3 and 4.

2. The Model and the Mas-Colell Bargaining Set
In this section, we define fundamental concepts underlying the bargaining set in the model of a
coalitional game. The set of players is a nonempty finite set denoted by 𝑁, and a coalition is anonempty subset of 𝑁. For every coalition 𝑆, a payoff profile for 𝑆 is a point of ℝ𝑆 (throughout the
paper, we denote by 𝑋 𝐴 the set of functions from the set 𝐴 to the set 𝑋. If 𝐴 is finite, then every point𝑔 ∈ 𝑋 𝐴 is identified with the bundle (𝑔(𝑎))𝑎∈𝐴 ). We simply call a payoff profile for 𝑁 a payoff

4profile. Given a payoff profile 𝑥, we denote by 𝑥𝑖 ≡ 𝑥(𝑖) the payoff to player 𝑖 at profile 𝑥, and by
𝑥𝑆 the projection of 𝑥 on ℝ𝑆 . For each 𝐵 ⊂ ℝ𝑆 , we denote by int𝐵 and bd𝐵 respectively theinterior and the boundary of 𝐵 relative to ℝ𝑆 . Let 𝑆 ⊂ 𝑁, and 𝐹 ⊂ ℝ𝑆 . A representation for 𝐹 is afunction 𝑓: ℝ𝑆 → ℝ such that 𝐹 = {𝑥 ∈ ℝ𝑆 |𝑓(𝑥) ≤ 0} and int𝐹 = {𝑥 ∈ ℝ𝑆 |𝑓(𝑥) < 0}. We define
the following concepts:
Coalitional Game: A coalitional game, simply a game, is a correspondence 𝑉 that associates withevery coalition 𝑆 a set 𝑉(𝑆) ≡ {𝑥 ∈ ℝ𝑁 |𝑥𝑆 ∈ 𝑉𝑆 }, where 𝑉𝑆 ⊂ ℝ𝑆 , satisfying
continuity: there is a representation for 𝑉𝑆 that is continuous.
Moreover, there is a point 𝑥 ∈ ℝ𝑁 such that for every coalition 𝑆, the following hold:𝑆
𝒙-compactness: 𝑉𝑆 ∩ ({𝑥𝑆 } + ℝ+
) is compact;

𝒙-interiority: 𝑥𝑆 ∈ int 𝑉𝑆 ; and𝒙-compehensiveness: for every (𝑥𝑆 , 𝑦𝑆 ) ∈ ℝ𝑆 × 𝑉𝑆 , 𝑥𝑆 ≤ 𝑥𝑆 ≪ 𝑦𝑆 1 ⇒ 𝑥𝑆 ∈ int𝑉𝑆
Note that 𝑉(𝑆) ⊂ ℝ𝑁 and 𝑉𝑆 ⊂ ℝ𝑆 . The point 𝑥 plays a role similar to the reference point incooperative bargaining games à la Nash. We interpret 𝑥 as a profile of reservation payoffs. Then, 𝑥interiority corresponds to the “survival condition” in a general equilibrium model: every consumer
can reduce his utility and still live by discarding part of his endowment.
We assume the continuity of representations to deal with games that are possibly bounded(such as games with 𝑉𝑆 compact) and simplify the proofs. Continuity implies that every 𝑉𝑆 is a𝑆
closed subset of ℝ𝑆 . If 𝑉𝑆 is comprehensive (i.e., 𝑉𝑆 − ℝ+
⊂ 𝑉𝑆 ) and non-leveled (i.e., for every(𝑥𝑆 , 𝑦𝑆 ) ∈ bd𝑉𝑆 × bd𝑉𝑆 , 𝑥𝑆 ≤ 𝑦𝑆 ⇒ 𝑥𝑆 = 𝑦𝑆 ), then 𝑉𝑆 is 𝑥 -compehensive. Then for every
coalition 𝑆 , the frontier of 𝑉𝑆 is a connected surface. Note that, for every 𝑦𝑆 ∈ 𝑉𝑆 , the segment
connecting 𝑦𝑆 with 𝑥𝑆 , [𝑥𝑆 , 𝑦𝑆 [ ≡ {𝑥𝑆 ∈ 𝑉𝑆 |(∃𝑡 ∈ [0,1[)(𝑥𝑆 = (1 − 𝑡)𝑥𝑆 + 𝑦𝑆 )} , is included in
int𝑉𝑆 , namely [𝑥𝑆 , 𝑦𝑆 [ ⊂ int𝑉𝑆 . By continuity, 𝑥-comprehensiveness implies:
Weak 𝒙-comprehensiveness: for every (𝑥𝑆 , 𝑦𝑆 ) ∈ ℝ𝑆 × 𝑉𝑆 , 𝑥𝑆 ≤ 𝑥𝑆 ≤ 𝑦𝑆 ⇒ 𝑥𝑆 ∈ 𝑉𝑆
The converse is not always true: weak 𝑥-comprehensiveness does not imply 𝑥-comprehensiveness.
In addition, 𝑥-comprehensiveness is weaker than the following:
Strict 𝒙-comprehensiveness: for every coalition 𝑆 and (𝑥𝑆 , 𝑦𝑆 ) ∈ ℝ𝑆 × 𝑉𝑆 ,𝑥𝑆 ≤ 𝑥𝑆 ≪ 𝑦𝑆 & 𝑥𝑆 ≠ 𝑦𝑆 ⇒ 𝑥𝑆 ∈ int𝑉𝑆
A coalition structure is a partition of 𝑁. We denote by Π the class of all coalition structures. Let
𝑉 be a game. The feasible set for coalition structure 𝒫 is 𝑉 ∗ (𝒫) ≡ {𝑥 ∈ ℝ𝑁 |∀𝑃 ∈ 𝒫, 𝑥𝑃 ∈ 𝑉𝑃 } =⋂𝑃∈𝒫 𝑉(𝑃), and the feasible set of 𝑉 is 𝑉 ∗ ≡ ⋃𝒫∈Π 𝑉 ∗ (𝒫) . For every coalition 𝑆, a payoff profile𝑥𝑆 for 𝑆 is individually rational if 𝑥𝑖 ≥ sup𝑉{𝑖} for all 𝑖 ∈ 𝑆. The individually rational set for thecoalition structure 𝒫 is 𝐼(𝑉; 𝒫) ≡ {𝑥 ∈ 𝑉 ∗ (𝒫)|∀𝑖 ∈ 𝑁, 𝑥𝑖 ≥ sup𝑉{𝑖} } , and the individually
rational set of 𝑉 is 𝐼(𝑉) ≡ ⋃𝒫∈Π 𝐼(𝑉; 𝒫). A payoff profile 𝑥 is efficient if 𝑥 ∈ 𝑉 ∗ , and there is no𝑦 ∈ 𝑉 ∗ such that 𝑦𝑖 > 𝑥𝑖 for all 𝑖 ∈ 𝑁. A coalition structure 𝒫 is efficient if there is at least one
individually rational and efficient payoff profile 𝑥 ∈ 𝑉 ∗ (𝒫).

1

Vector inequalities: 𝑥 ≤ 𝑦 if 𝑥𝑖 ≤ 𝑦𝑖 for all 𝑖, 𝑥 ≪ 𝑦 if 𝑥𝑖 < 𝑦𝑖 for all 𝑖.

5A transferable utility (TU) game is a game 𝑉 such that for every coalition 𝑆 , there is a real
number 𝑣(𝑆) such that 𝑉𝑆 = {𝑥 ∈ ℝ𝑆 | ∑𝑖∈𝑆 𝑥𝑖 ≤ 𝑣(𝑆)}. The number 𝑣(𝑆) is the worth of 𝑆.
Let 𝑆 and 𝑇 be two coalitions, 𝑥 a payoff profile, 𝑦 ∈ 𝑉𝑆 , and 𝑧 ∈ 𝑉𝑇 . Then the pair (𝑇, 𝑧)
weakly improves upon (𝑆, 𝑦) at 𝑥 if (𝑇, 𝑧) ≠ (𝑆, 𝑦) and 𝑧 ≥ (𝑥𝑇\𝑆 , 𝑦𝑇∩𝑆 ) ; it Pareto-improves
upon (𝑆, 𝑦) at 𝑥 if 𝑧 ≥ (𝑥𝑇\𝑆 , 𝑦𝑇∩𝑆 ) and 𝑧 ≠ (𝑥𝑇\𝑆 , 𝑦𝑇∩𝑆 ); and it strictly improves upon (𝑆, 𝑦) at
𝑥 if 𝑧 ≫ (𝑥𝑇\𝑆 , 𝑦𝑇∩𝑆 ). If (𝑇, 𝑧) is such that 𝑧 ≫ 𝑥𝑇 , then (𝑇, 𝑧) strictly improves upon 𝑥 . Notethat (𝑇, 𝑧) weakly improves upon (𝑆, 𝑦) at 𝑥 even if (𝑇, 𝑧) ≠ (𝑆, 𝑦) and 𝑧 = (𝑥𝑇\𝑆 , 𝑦𝑇∩𝑆 ).
With these terms, we define the following concepts.
Objection: Let 𝑉 be a game and 𝑥 a payoff profile. Then the pair (𝑆, 𝑦) is an objection to 𝑥 if 𝑆
is a coalition and 𝑦 ∈ 𝑉𝑆 strictly improves upon 𝑥.
The Core: The core of a game 𝑉 for coalition structure 𝒫 is:
𝐶0 (𝑉; 𝒫) = {𝑥 ∈ 𝐼(𝑉; 𝒫)|There is no objection to 𝑥} .
The core of 𝑉 is: 𝐶0 (𝑉) = ⋃𝒫∈Π 𝐶0 (𝑉; 𝒫).
This is the definition of the core of a general coalitional game in terms of strict improvement [Scarf(1967), and Peleg (1985)]. Scarf (1967) proves a nonemptiness theorem for the core of a “balanced”
game with coalition structure {𝑁}. 2
Counterobjection: Let 𝑉 be a game, 𝑥 a payoff profile, and (𝑆, 𝑦) an objection to 𝑥. Then the pair(𝑇, 𝑧) is a counterobjection to (𝑆, 𝑦) at 𝑥 if 𝑇 is a coalition and 𝑧 ∈ 𝑉𝑇 weakly improves upon(𝑆, 𝑦) at 𝑥.
Mas-Colell bargaining set: The Mas-Colell bargaining set of a game 𝑉 for coalition structure 𝒫 is:
𝑀𝐵(𝑉; 𝒫)= {𝑥 ∈ 𝐼(𝑉; 𝒫)|For each objection to 𝑥, there is a counterobjection at 𝑥} .
The Mas-Colell bargaining set of 𝑉 is: 𝑀𝐵(𝑉) = ⋃𝒫∈Π 𝑀𝐵(𝑉; 𝒫).
Our definition of 𝑀𝐵 differs from Mas-Colell’s (1989, Section 6) original definition. Mas-Colell
does not require individually rationality, and defines both objections and counterobjections by
Pareto-improvement within coalitions 3 . We impose individual rationality constraint on 𝑀𝐵 anddefine counterobjections by weak improvement of counterobjecting coalitions. This is similar to the(𝑖)definition of the classical bargaining set 𝑀1 , proposed by Aumann and Maschler (1964) [see also(𝑖)Aumann (1989, P.77)]: Aumann and Maschler impose individual rationality constraint on 𝑀1 .
2

A class ℱ of coalitions is a balanced family if there is a function δ that assigns to eachcoalition 𝑆 a real number δ(𝑆) such that δ(𝑆) > 0 for 𝑆 ∈ ℱ , δ(𝑆) = 0 for 𝑆 ∉ ℱ , and∑𝑆∈ℱ(𝑖) δ(𝑆) = 1, where ℱ(𝑖) = {𝑆 ∈ ℱ|𝑆 ∋ 𝑖} for every player 𝑖 . A game 𝑉 with coalition
structure 𝒫 is balanced if ⋂𝑆∈ℱ 𝑉(𝑆) ⊂ ⋂𝑃∈𝒫 𝑉(𝑃) for every balanced family ℱ.
3

Mas-Colell (1989, Section 1) defines counterobjections by strict improvement in atomless
economies. He says that it “cannot simply be replaced by” weak improvement for theequivalence between his bargaining set and the set of Walrasian allocations in this environment.

6

They define an “objection of player 𝑖 against player 𝑗 at 𝑥” as strict improvement on 𝑖 via coalition
𝑆 such that 𝑖 ∈ 𝑆 and 𝑗 ∉ 𝑆 without making any members of 𝑆 worse off than at 𝑥. They also define
a “counterobjection from player 𝑗 to player 𝑖” as weak improvement on 𝑗 via coalition 𝑇 such that𝑗 ∈ 𝑇 and 𝑖 ∉ 𝑇 without hurting any members of 𝑇\𝑆 or 𝑇 ∩ 𝑆 than at 𝑥 or at the objection via 𝑆.

3. The Zhou Bargaining Set and the Steady Bargaining Set
In this section, we develop the refinement of the Mas-Colell bargaining set proposed by Zhou (1994)
in the NTU case. According to the above definition, a counterobjecting coalition may include no
player in the objecting coalition. This means that the counterobjection is not really against theobjection but the original proposal. To overcome this weakness, Zhou then proposes to require theintersection of the objection coalition and the counterobecting coalition to be a nonempty proper
subset of the two coalitions respectively to overcome this weakness. Based on this, Zhou defines arefined bargaining set now called the "Zhou bargaining set."
Further, we introduce a dominance relation between coalitions to identify powerfulcounterobjections. Given an initial recommendation, which coalitions have counterobjections to an
objection depends on what payoffs can be alternatively assigned to the members of the objecting
coalition. We select as "dominant" those coalitions having a counterobjection to every objection
from the objecting coalition. We impose on counterobjections both Zhou's intersection condition
and the dominance requirement. We then introduce the corresponding version, the "steadybargaining set," as a refinement of the Zhou bargaining set. We establish its nonemptiness under the"restrictive non-crossing condition. " In addition, we remove the asymmetry between the
improvement requirement underlying the objections and counterobjections. We selectcounterobjections which strictly improve upon the counterobjecting coalitions, and focus on theassociated subset of the steady bargaining set. This is the "strong steady bargaining set." We prove
its nonemptiness under the assumption of “strict 𝑥-comprehensiveness.”
Since the introduction of the bargaining set by Aumann and Maschler (1964), nointersection requirement has appeared in the definitions of alternative versions of the bargaining set,
including 𝑀𝐵 [see Zhou (1994, Sections 2 and 3) and Iñarra et al.. (2020, pp.254-253) for details].
Zhou (1994) proposes the following modification of the Mas-Colell bargaining set.
Proper Intersection/Proper counterobjection: Let 𝑉 be a game, and 𝑆 , 𝑇 ⊂ 𝑁 . Then, 𝑇 has aproper intersection with 𝑆 if 𝑇 ∩ 𝑆 ≠ Ø, 𝑆\𝑇 ≠Ø, and 𝑇\𝑆 ≠ Ø. Let 𝑥 be a payoff profile, (𝑆, 𝑦) an
objection to 𝑥, and (𝑇, 𝑧) a counterobjection to (𝑆, 𝑦) at 𝑥. Then, (𝑇, 𝑧) is proper if 𝑇 has a properintersection with 𝑆.
Zhou bargaining set: The Zhou bargaining set of a game 𝑉 for coalition structure 𝒫 is:
𝑍𝐵(𝑉; 𝒫)={𝑥 ∈ 𝐼(𝑉; 𝒫)|For each objection (𝑆, 𝑦) to 𝑥, there is a proper counterobjection to (𝑆, 𝑦) at 𝑥}

The Zhou bargaining set of 𝑉 is: 𝑍𝐵(𝑉) = ⋃𝒫∈Π 𝑍𝐵(𝑉; 𝒫).

Figure 1: Coalition 𝑇 having a properintersection with coalition 𝑆

𝑇\𝑆 ≠ Ø𝑆\𝑇 ≠Ø𝑆

7

𝑇∩𝑆 ≠Ø

𝑇

Figure 1 illustrates coalitions 𝑆 and 𝑇 having a proper intersection. This captures the idea that someplayers outside of 𝑆 tempt some members of 𝑆 to form a new coalition 𝑇. For the objection madeby 𝑆 to be valid, the players in 𝑇 ∩ 𝑆 should not have an incentive to move to 𝑇.
We now propose a refinement of the Zhou bargaining set by invoking a dominance relation
between coalitions: Consider a situation that, given a game 𝑉 and a coalition structure 𝒫, a payoff
profile 𝑥 is proposed. Suppose that some players are not satisfied. They form a coalition 𝑆, not in𝒫, and formulate an objection to 𝑥. According to the Zhou bargaining set, the objection is justified
unless at least one counterobjection is made by a coalition having a proper intersection with 𝑆. Then
another coalition 𝑇 cannot neutralize 𝑆 if 𝑆 propose payoffs for its members to which 𝑇 can findno counterobjection at 𝑥. Our idea is that 𝑆 will never form if 𝑇 can make a counterobjection to
every possible objection to 𝑥 that 𝑆 could make. We then say that " 𝑇 dominates 𝑆 at 𝑥.” Based on
this relation between coalitions, we define a new solution to assign to each game a payoff profile towhich an objecting coalition can form if and only if it is not "dominated" by any other coalition.
Dominating coalition: Let 𝑉 be a game, 𝑥 a payoff profile, and 𝑆, 𝑇 coalitions. Then, 𝑇 dominates
𝑆 at 𝑥 if for each objection (𝑆, 𝑦) to 𝑥, there is a proper counterobjection (𝑇, 𝑧) to (𝑆, 𝑦) at 𝑥.
Steady bargaining set: The steady bargaining set of a game 𝑉 for coalition structure 𝒫 is:
𝑆𝐵(𝑉, 𝒫)= {𝑥 ∈ 𝐼(𝑉; 𝒫)|For each objection to 𝑥, there is a dominant coalition at 𝑥} .
The steady bargaining set of a game 𝑉 is: 𝑆𝐵(𝑉) = ⋃𝒫∈Π 𝑆𝐵(𝑉; 𝒫)
For every game 𝑉, the inclusion relation among the core, the Mas-Colell bargaining set, the Zhoubargaining set, and the steady bargaining set is following: 𝐶0 (𝑉; 𝒫) ⊂ 𝑆𝐵(𝑉; 𝒫) ⊂ 𝑍𝐵(𝑉; 𝒫) ⊂𝑀𝐵(𝑉; 𝒫) for every coalition structure 𝒫 , and 𝐶0 (𝑉) ⊂ 𝑆𝑆𝐵(𝑉) ⊂ 𝑆𝐵(𝑉) ⊂ 𝑍𝐵(𝑉) ⊂ 𝑀𝐵(𝑉) .
The steady bargaining set is generally a proper subset of the Zhou bargaining set. For example, let
𝑉 be the three-person TU game such that the worth of 𝑆 is 0 if |𝑆| = 1, 𝑎 if |𝑆| = 2, and 1 if |𝑆| = 3.
Then 𝑆𝐵(𝑉; {𝑁}) is a proper subset of 𝑍𝐵(𝑉; {𝑁}) if 2/3 < 𝑎 ≤1[see Shimomura (1997, Section 4)
for details]. The Zhou bargaining set is also generally a proper subset of the Mas-Colell bargaining
set. Zhou (1994, Example 3.2) presents a TU game of which the Zhou bargaining set does notcontain the nucleolus while the Mas-Colell bargaining set does [see also Shimomura (1997,
Example 4)]. We exhibit a game in which the Zhou bargaining set is empty and the Mas-Colellbargaining set is nonempty later (Example 2). To compare properties of these two solutions, we
impose a new assumption on a game together with the basic four in the definition.
Restrictive non-crossing condition: For every 𝑥 ∈ 𝑃(𝑉 ∗ ) , there is no triple (𝑆, 𝑅, 𝑄) of disjoint
𝑆 ]𝑆 ]coalitions, and two profiles 𝑦𝑆 , 𝑦𝑆 ′ ∈ [{𝑥𝑆 } + ℝ++∩ [{𝑟𝑆 } + ℝ++
such that(𝑦𝑆 , 𝑥𝑅 ) ∈ int𝑉𝑆∪𝑅 & (𝑦𝑆 , 𝑥𝑄 ) ∉ 𝑉𝑆∪𝑄 & (𝑦𝑆 ′, 𝑥𝑄 ) ∈ int𝑉𝑆∪𝑄 & (𝑦𝑆 ′, 𝑥𝑅 ) ∉ 𝑉𝑆∪𝑅∗
where 𝑃(𝑉 ∗ ) ≡ {𝑥 ∈ 𝑉 ∗ ∩ [{𝑥} + ℝ𝑁
+ ]|There is no 𝑦 ∈ 𝑉 such that 𝑦 ≫ 𝑥 } and 𝑟𝑆 ≡ (sup𝑉{𝑖} )𝑖∈𝑆 .

8This condition says the following: given an efficient payoff vector 𝑥, suppose that the coalition 𝑆
can achieve new payoffs in collaboration with 𝑅 but not 𝑄 at which all the members of 𝑆 are strictly
better off than at 𝑥 and at the individually rational levels. Then 𝑆 can realize with 𝑅 any suchpayoffs that can be produced with 𝑄. See Figure 2.4The top panel represents a game that satisfies
the restrictive non-crossing condition, and the bottom panel describes a game that does not satisfy.
Vohra (1991) points out the importance of almost, but not exactly, the same condition for a game tohave an efficient payoff profile in the Mas-Colell bargaining set. He also gives pictures as in Figure
2 to illustrate a case in which the condition is satisfied and a case in which it is not satisfied; the𝑥frontiers of the slices, 𝑉𝑆 𝑅 (𝑆 ∪ 𝑅)𝑥

≡ {𝑦𝑆 ∈ ℝ𝑆 |(𝑦𝑆 , 𝑥𝑅 ) ∈ 𝑉𝑆∪𝑅 } and 𝑉𝑆 𝑄 (𝑆 ∪ 𝑄)≡𝑆{𝑦𝑆 ∈ ℝ |(𝑦𝑆 , 𝑥𝑄 ) ∈ 𝑉𝑆∪𝑄 } cross.

payoff to player 2F(𝑟1 , 𝑟2 )(𝑥1 , 𝑥2 )G
payoff to player 1

payoff to player 2F(𝑟1 , 𝑟2 ) (𝑥

1 , 𝑥2 )G
payoff to player 1

Figure 2: Game that satisfies the restrictive non-crossing condition and one that does not, where𝑥𝑥

𝑆 = {1,2}, 𝑥 ∈ 𝑃(𝑉 ∗ ), 𝐹 = 𝑉𝑆 𝑅 (𝑆 ∪ 𝑅), and 𝐺 = 𝑉𝑆 𝑄 (𝑆 ∪ 𝑄)

4

Demuynck and Potoms (2020) propose essentially the same condition for utility possibility

sets named the “no intersection property.”

9

Notice the restriction “𝑥 ∈ 𝑃(𝑉 ∗ ).” The restrictive non-crossing condition does not say anything
about the frontiers of slices through inefficient payoff profiles. If the feasible set 𝑉 ∗ is large enoughand payoffs 𝑥 ∈ 𝑃(𝑉 ∗ ) are too large, we may have either (𝑥𝑆 , 𝑥𝑅 ) ∉ int𝑉𝑆∪𝑅 or (𝑟𝑆 , 𝑥𝑅 ) ∉int𝑉𝑆∪𝑅 as long as 𝑆 ∪ 𝑅 ≠ 𝑁. Then the restrictive non-crossing condition is trivially satisfied eventhough the frontiers of the slices through some inefficient profiles may be crossing.
All TU games and all three-person comprehensive games satisfy the restrictive non-crossing
condition. A comprehensive game 𝑉 such that 𝑉(𝑆) ⊂ ⋂𝑖∈𝑆 𝑉({𝑖}) for every 𝑆 ⊂ 𝑁 with |𝑆| ≥ 3
and 𝑆 ≠ 𝑁 also satisfies the condition. These are games in which the two-person coalitions are
essential. "Roommate problems" are representable as such games, and may have an empty core [seeShubik (1984, p.221)]. We now examine the nonemptiness of the bargaining sets of a TU coalitional
game with coalition structures. Zhou (1994) proves the following theorem.
Theorem (Zhou, 1994): The Zhou Bargaining Set of a TU game contains at least one efficient
payoff profile for at least one coalition structure.
Zhou presents this theorem as an answer to the question of how players form coalitions. Our study
is inspired by his work. We present the following result to which Zhou’s theorem is a corollary:
Theorem 1: Given a game satisfying the non-crossing condition, the Steady Bargaining Set contains
at least one efficient payoff profile for at least one coalition structure.
We prove Theorem 1 in the appendix. By Theorem 3.2 in Shimomura (1997), we know thefollowing: Let 𝒫 be a coalition structure such that 𝑚𝑎𝑥{∑𝑖∈𝑁 𝑥𝑖 |𝑥 ∈ 𝑉 ∗ (𝒫)} ≥𝑚𝑎𝑥{∑𝑖∈𝑁 𝑥𝑖 |𝑥 ∈ 𝑉 ∗ (𝒬)} for all coalition structures 𝒬 . If 𝑉 is a TU game, then the steadybargaining set contains at least one efficient payoff profile for coalition structure 𝒫. Call the valueof max{∑𝑖∈𝑁 𝑥𝑖 |𝑥 ∈ 𝑉 ∗ (𝒬)} the worth of coalition structure 𝒬. If a coalition structure produces
maximal worth in a game, it is efficient and it admits a nonempty steady bargaining set. However,this holds only in the TU case. The following example shows an NTU game admitting a worthmaximizing coalition structure such that even the associated Zhou bargaining set is empty.
Example 1. A Couple and a Friend on Sunday. There are two women and a man: Andy, Beth,and Olive. We denote them by 𝑎, 𝑏, and 𝑜, respectively. The agents 𝑎 and 𝑏 are lovers, and 𝑜 istheir friend. The problem is how to spend the next Sunday. The (maximal) payoff each of them canachieve alone is zero. When 𝑎 and 𝑏 are together, they can achieve (3, 3). When 𝑎 and 𝑜 , or 𝑏
and 𝑜, are together, they can achieve (1, 1). When they all are together, they can achieve (2, 2, 2).
Let 𝑁 = {𝑎, 𝑏, 𝑜}. Then the game 𝑉 describing the above situations is as follows:
𝑉{𝑖} = {𝑥𝑖 ∈ ℝ{𝑖} |𝑥𝑖 ≤ 0} for every 𝑖 ∈ 𝑁,
𝑉{𝑎,𝑏} = {𝑥{𝑎,𝑏} ∈ ℝ{𝑎,𝑏} |(𝑥𝑎 , 𝑥𝑏 ) ≤ (3,3) },10

𝑉{𝑖,𝑜} = {𝑥{𝑖,𝑜} ∈ ℝ{𝑖,𝑜} |(𝑥𝑖 , 𝑥𝑜 ) ≤ (1,1) } for every 𝑖 ∈ {𝑎, 𝑏}, and
𝑉𝑁 = {𝑥 ∈ ℝ𝑁 |(𝑥𝑎 , 𝑥𝑏 , 𝑥𝑜 ) ≤ (2,2,2)}.
The efficient coalition structures are [{𝑎, 𝑏}, {𝑜}], and [𝑁], which are also worth-maximizing. The
payoff profile (𝑥𝑎 , 𝑥𝑏 , 𝑥𝑜 ) = (3, 3, 0), which is achieved in the coalition structure [{𝑎, 𝑏}, {𝑜}], has
no objection. Then it belongs to 𝑍𝐵(𝑉; {𝑎, 𝑏}, {𝑜}) as well as to 𝐶0 (𝑉; {𝑎, 𝑏}, {𝑜}). On the other
hand, ({𝑎, 𝑏}, (3,3)) is an objection to any payoff vector of 𝑉𝑁 with no counterobjection. Then,
𝑍𝐵(𝑉; 𝑁) as well as 𝑀𝐵(𝑉; 𝑁) is empty (if payoffs are transferable in 𝑁 , i.e., 𝑉𝑁 ={𝑥 ∈ ℝ𝑁 |𝑥𝑎 + 𝑥𝑏 + 𝑥𝑜 ≤ 2 + 2 + 2}, (3, 3, 0) also belongs to 𝐶0 (𝑉; 𝑁) and to 𝑍𝐵(𝑉; 𝑁)). ∎To establish a nonemptiness theorem for 𝑀𝐵(𝑉; 𝑁) , Vohra (1991) proposes the assumption of"weak balancedness,” which is implied by the restrictive no-crossing condition if the game 𝑉
satisfies strict 𝑥 -comprehensiveness. This assumption is weak, but as observed by Vohra, it may
not be easy to verify by itself since the concept of a "justified objection" appears in the definition.
Chang and Lee (1993) construct an example to show that weak balancedness is not necessary. Its
counterpart here is stated as follows: "for every 𝑥 ∈ 𝑃(𝑉 ∗ ) ∩ [{𝑥} + ℝ𝑁+ ], there is no pair (𝑆, 𝑆′) ofdistinct maximal coalitions having proper counterobjections such that 𝑆 ∩ 𝑆 ′ ≠ Ø." If we do not
impose such a restriction on a game, not only can the Zhou bargaining set be empty, but also theMas-Colell bargaining set may be inefficient. This means that some restrictions related to weakbalancedness has to be included for the theorem. See the following example.
Example 2: Four Researchers and a Research Project. There are four researchers: Alex, Billie,
Cameron, and Darian. We denote them by 𝑎, 𝑏, 𝑐 and 𝑑, respectively. They are involved in a research
project. It is not possible for any one of them or any two of them to carry it out. If three of themcooperate, they have two plans from which to choose. Every agent has a preference over options ofwhom to work with and how to divide the job. Let 𝑁 = {𝑎, 𝑏, 𝑐, 𝑑}.
coalition\agents𝑎 = Alex𝑏 = Billie𝑐 = Cameron𝑑 = Darian
6
4
1{𝑎, 𝑏, 𝑐}:plan 1
3
5
2{𝑎, 𝑏, 𝑐}:plan 2
6
4
1{𝑏, 𝑐, 𝑑}:plan 1
3
5
2{𝑏, 𝑐, 𝑑}:plan 2
1
6
4{𝑎, 𝑐, 𝑑}:plan 1
2
3
5{𝑎, 𝑐, 𝑑}:plan 2
4
1
6{𝑎, 𝑏, 𝑑}:plan 1
5
2
3{𝑎, 𝑏, 𝑑}:plan 2
1.5
1.5
1.5
1.5
𝑁 = {𝑎, 𝑏, 𝑐, 𝑑}others0.5
0.5
0.5
Table 1: Maximal payoffs players can get through cooperation

11

0.5

Table 1 gives payoffs associated with the preferences. Take 𝟎 = (0,0,0,0) as 𝑥. The game 𝑉 defined
below represents the feasible sets of payoff profiles:
𝑉{𝑎,𝑏,𝑐} ={𝑥{𝑎,𝑏,𝑐} ∈ ℝ{𝑎,𝑏,𝑐} |(𝑥𝑎 , 𝑥𝑏 , 𝑥𝑐 ) ≤ (6,4,1)} ∪ {𝑥{𝑎,𝑏,𝑐} ∈ ℝ{𝑎,𝑏,𝑐}|(𝑥𝑎 , 𝑥𝑏 , 𝑥𝑐 ) ≤ (3,5,2)}
𝑉{𝑏,𝑐,𝑑} ={𝑥{𝑏,𝑐,𝑑} ∈ ℝ{𝑏,𝑐,𝑑} |(𝑥𝑏 , 𝑥𝑐 , 𝑥𝑑 ) ≤ (6,4,1)} ∪ {𝑥{𝑏,𝑐,𝑑} ∈ ℝ{𝑏,𝑐,𝑑} |(𝑥𝑏 , 𝑥𝑐 , 𝑥𝑑 ) ≤ (3,5,2)}
𝑉{𝑎,𝑐,𝑑} ={𝑥{𝑎,𝑐,𝑑} ∈ ℝ{𝑎,𝑐,𝑑}|(𝑥𝑎 , 𝑥𝑐 , 𝑥𝑑 ) ≤ (1,6,4)} ∪ {𝑥{𝑎,𝑐,𝑑} ∈ ℝ{𝑎,𝑐,𝑑}|(𝑥𝑏 , 𝑥𝑐 , 𝑥𝑑 ) ≤ (2,3,5)}
𝑉{𝑎,𝑏,𝑑} = {𝑥{𝑎,𝑏,𝑑} ∈ ℝ{𝑏,𝑐,𝑑} |(𝑥𝑏 , 𝑥𝑐 , 𝑥𝑑 ) ≤ (6,4,1)} ∪ {𝑥{𝑏,𝑐,𝑑} ∈ ℝ{𝑎,𝑏,𝑐} |(𝑥𝑏 , 𝑥𝑐 , 𝑥𝑑 ) ≤ (3,5,2)}
𝑉𝑁 = {𝑥 ∈ ℝ𝑁 |(𝑥𝑎 , 𝑥𝑏 , 𝑥𝑐 , 𝑥𝑑 ) ≤ (1.5,1.5,1.5,1.5)}, and𝑉𝑆 = {𝑥 ∈ ℝ𝑆 |𝑥𝑖 ≤ 0.5 for every 𝑖 ∈ 𝑆} for other coalitions 𝑆.
The game 𝑉 does not satisfy the restrictive non-crossing condition: let 𝑥 = (1.5,1.5,1.5,1.5) ∈𝑉(𝑁). We can verify 𝑥 ∈ 𝑃(𝑉 ∗ ). Note that𝑥𝑐({𝑎, 𝑏, 𝑐})={𝑦{𝑎,𝑏} ∈ ℝ{𝑎,𝑏} |(𝑦𝑎 , 𝑦𝑏 ) ≤ (3,5)}, and𝑉{𝑎,𝑏}

𝑥𝑑({𝑎, 𝑏, 𝑑})={𝑦{𝑎,𝑏} ∈ ℝ{𝑎,𝑏} |(𝑦𝑎 , 𝑦𝑏 ) ≤ (5,2)}
𝑉{𝑎,𝑏}

payoff to player b
5 F

2(1.5,1.5)

0G

payoff to player a53

Figure 3: Slices of 𝑉({𝑎, 𝑏, 𝑐}) with 𝑥𝑐 fixed and 𝑉({𝑎, 𝑏, 𝑑}) with 𝑥𝑑 fixed𝑥𝑐
𝑥𝑑({𝑎, 𝑏, 𝑐}), 𝐺 = 𝑉{𝑎,𝑏}({𝑎, 𝑏, 𝑑})
𝑆 = {𝑎, 𝑏}, 𝑥 ∈ 𝑃(𝑉 ∗ ), 𝐹 = 𝑉{𝑎,𝑏}These "slices" are drawn as in Figure 3. We see that
𝑥

{𝑎,𝑏}

{𝑎,𝑏}

𝑥

𝑥

{𝑎,𝑏}

{𝑎,𝑏}

𝑥

𝑐
𝑑
({𝑎, 𝑏, 𝑐}) ∩ [{𝑥{𝑎,𝑏} } + ℝ++ ] ∩ [{𝑟{𝑎,𝑏} } + ℝ++ ] \𝑉{𝑎,𝑏}({𝑎, 𝑏, 𝑑}) ≠ ∅, and𝑉{𝑎,𝑏}𝑑𝑐({𝑎, 𝑏, 𝑑}) ∩ [{𝑥{𝑎,𝑏} } + ℝ++ ] ∩ [{𝑟{𝑎,𝑏} } + ℝ++ ] \𝑉{𝑎,𝑏}({𝑎, 𝑏, 𝑐}) ≠ ∅𝑉{𝑎,𝑏}The efficient coalition structures are [{𝑎, 𝑏, 𝑐}, {𝑑}], [{𝑏, 𝑐, 𝑑}, {𝑎}], [{𝑎, 𝑐, 𝑑}, {𝑏}], [{𝑎, 𝑏, 𝑑}, {𝑐}],and [𝑁]. We can show that
𝑀𝐵(𝑉; {𝑎, 𝑏, 𝑐}, {𝑑}) = {𝑥 ∈ ℝ𝑁
+ |(𝑥𝑎 , 𝑥𝑏 , 𝑥𝑐 , 𝑥𝑑 ) ≤ (1,1,1,0.5)}.
(We give a proof in the appendix.) The point (1.5, 1.5, 1.5, 1.5) ∈ 𝑉 ∗ (𝑁) strictly dominates in allcomponents any payoff vector in 𝑀𝐵(𝑉; {𝑎, 𝑏, 𝑐}, {𝑑}), so that no profile is efficient. By
symmetry, there is no efficient profile either in 𝑀𝐵(𝑉; {𝑏, 𝑐, 𝑑}, {𝑎}), 𝑀𝐵(𝑉; {𝑎, 𝑐, 𝑑}, {𝑏}), or
𝑀𝐵(𝑉; {𝑎, 𝑏, 𝑑}, {𝑐}). We can also show that
𝑀𝐵(𝑉; {𝑎, 𝑏, 𝑐}, {𝑑}) = {𝑥 ∈ ℝ𝑁
+ |(𝑥𝑎 , 𝑥𝑏 , 𝑥𝑐 , 𝑥𝑑 ) ≤ (1,1,1,1)}.

12

Hence, there is no efficient profile in 𝑀𝐵(𝑉; 𝑁). Furthermore, given any payoff profile in all theabove Mas-Colell bargaining sets, (𝑁 , (1.5, 1.5, 1.5, 1.5)) is an objection that has no propercounterobjection. For every coalition structure 𝒫, 𝑍𝐵(𝑉; 𝒫) ⊂ 𝑀𝐵(𝑉; 𝒫), and for every efficientcoalition structure, 𝑍𝐵(𝑉; 𝒫) = ∅. Let 𝑥′ be an individually rational payoff vector in a coalition
structure that is not efficient. Such a payoff profile exists uniquely: 𝑥′ = (0.5, 0.5, 0.5, 0.5). Then,(𝑁, (1.5, 1.5, 1.5, 1.5)) is an objection to 𝑥′ that has no proper counterobjection. Hence, there isno coalition structure such that the Zhou bargaining set of 𝑉 is nonempty. ∎The above game is 0-comprehensive. We can perturb it to construct a strictly 0-comprehensive gamegiving rise to the same results. One would think that the improvement notion in a counterobjection
is rather weak. We now consider counterobjections that improve the payoff of every member of thecounterobjecting coalitions, and define the class of counterobjections that pass this strict
improvement test as well as the two we have already examined. We propose the associated versionof the bargaining set, which is contained in all of the other versions discussed in the paper.
Strict counterobjection: Let 𝑉 be a game, 𝑥 a payoff profile, (𝑆, 𝑦) an objection to 𝑥 , and 𝑇 a
coalition. Then a counterobjection (𝑇, 𝑧) to (𝑆, 𝑦) at 𝑥 is strict if (𝑇, 𝑧) strictly improves upon(𝑆, 𝑦) at 𝑥.
Strict dominating coalition: Let 𝑉 be a game, 𝑥 a payoff profile, and 𝑆, 𝑇 coalitions.. Then 𝑇 isstrictly dominates 𝑆 at 𝑥 if for each objection ( 𝑆 , 𝑦 ) to 𝑥 , there is a proper and strict
counterobjection (𝑇, 𝑧) to (𝑆, 𝑦) at 𝑥.
The Strong Steady Bargaining Set: The strong steady bargaining set of a game V for coalition
structure 𝒫 is:
𝑆𝑆𝐵(𝑉, 𝒫)= {𝑥 ∈ 𝐼(𝑉; 𝒫)|For each objection to 𝑥, there is a strictly dominant coalition at 𝑥} .
The strong steady bargaining set of a game 𝑉 is: 𝑆𝑆𝐵(𝑉) = ⋃𝒫∈Π 𝑆𝑆𝐵(𝑉; 𝒫)
Given a payoff vector 𝑥, the inclusion relation among the sets of objections 𝑂(𝑥), objections withno counterobjections 𝑀(𝑥), those with no proper counterobjections 𝑍(𝑥), those with no propercounterobjections 𝑆(𝑥), and those with no proper and strict counterobjections 𝑆𝑆(𝑥) satisfy thefollowing inclusion relations: 𝑂(𝑥) ⸧ 𝑆𝑆(𝑥) ⸧ 𝑆(𝑥)⸧ 𝑍(𝑥) ⸧ 𝑀(𝑥). Then for every game 𝑉, wededuce the following inclusion relations among the core, the Mas-Colell bargaining set, the Zhoubargaining set, the steady bargaining set, and the strong steady bargaining set: 𝐶0 (𝑉; 𝒫) ⊂𝑆𝑆𝐵(𝑉; 𝒫) ⊂ 𝑆𝐵(𝑉; 𝒫) ⊂ 𝑍𝐵(𝑉; 𝒫) ⊂ 𝑀𝐵(𝑉; 𝒫) for every coalition structure 𝒫 , and 𝐶0 (𝑉) ⊂𝑆𝑆𝐵(𝑉) ⊂ 𝑆𝐵(𝑉) ⊂ 𝑍𝐵(𝑉) ⊂ 𝑀𝐵(𝑉) . We establish a positive result for 𝑆𝑆𝐵(𝑉) by assuming
strict 𝑥-comprehensiveness additionally.
Theorem 2: Given a game satisfying the restrictive non-crossing condition and strict 𝑥 comprehensiveness, the strong steady bargaining set contains at least one efficient payoff profile forat least one coalition structure.

13

In the appendix, we give a proof of Theorem 2 that immediately follows from the proof of Theorem
1. Zhou (1994, p.520) argues that the definition of counterobjections by weak inequalities may be
the "price" we have to pay in order to obtain the nonemptiness of a bargaining set for every TU
game. Theorem 2 shows that this price is unnecessary as long as we assume the two properties all
TU games satisfy of restrictive non-crossing condition and strict 𝑥-comprehensiveness4. The Natural Bargaining Set and the Precise Bargaining Set
One might feel that the proper intersection condition is too restrictive. In this section, we consider
counterobjections defined by strict inequalities and weaker intersection conditions than the properintersection condition. We define two associated versions of the bargaining set, the "naturalbargaining set" and the "precise bargaining set." Their nonemptiness can be demonstrated underessentially no assumptions. We start with the weakest intersection condition to define the following
type of counterobjections and the associated bargaining set:
Natural counterobjection: Let 𝑉 be a game, 𝑥 a payoff profile, (𝑆, 𝑦) an objection to 𝑥 . Then acounterobjection (𝑇, 𝑧) to (𝑆, 𝑦) at 𝑥 is natural if 𝑇 ∩ 𝑆 ≠ Ø.
The Natural Bargaining Set: The natural bargaining set of a game V for coalition structure 𝒫 is:
𝑁𝐵(𝑉; 𝒫)={𝑥 ∈ 𝐼(𝑉; 𝒫)|For each objection to 𝑥, there is a natural and strict counterobjection at 𝑥}
The natural bargaining set of 𝑉 is: 𝑁𝐵(𝑉) = ⋃𝒫∈Π 𝑁𝐵(𝑉; 𝒫).
𝑆\𝑇 ≠Ø𝑆

𝑇\𝑆 ≠ Ø𝑇∩𝑆 ≠Ø

𝑇

𝑇\𝑆 ≠ Ø𝑆\𝑇 ≠Ø𝑆𝑇∩𝑆 ≠Ø

𝑇∩𝑆 ≠Ø

𝑇

𝑇

𝑆

Figure 4: Coalition 𝑆 making an objection and coalition 𝑇 making a natural counterobjection
Figure 4 illustrates the possible configurations of pairs of a coalition 𝑆 making an objection and acoalition 𝑇 making a natural counterobjection. We next propose a stronger version of the bargaining
set by imposing a restriction on the class of natural counterobjections:
Precise counterobjection: Let 𝑉 be a game, 𝑥 a payoff profile, (𝑆, 𝑦) an objection to 𝑥 . Then acounterobjection (𝑇, 𝑧) to (𝑆, 𝑦) at 𝑥 is precise if 𝑇 ∩ 𝑆 ≠ Ø and 𝑆\𝑇 ≠ Ø.
The Precise Bargaining Set: The precise bargaining set of a game V for coalition structure 𝒫 is:
𝑃𝐵(𝑉; 𝒫)= {𝑥 ∈ 𝐼(𝑉; 𝒫)|For each objection to 𝑥, there is a precise and strict counterobjection at 𝑥} .
The precise bargaining set of 𝑉 is: 𝑃𝐵(𝑉) = ⋃𝒫∈Π 𝑃𝐵(𝑉; 𝒫).

14

𝑇\𝑆 ≠ Ø
𝑇\𝑆 ≠ Ø𝑇∩𝑆 ≠Ø

𝑇∩𝑆 ≠Ø𝑇

𝑆

𝑆

𝑇

Figure 5: Coalition 𝑆 making an objection and coalition 𝑇 making a precise counterobjection
Figure 5 represents the possible configurations of coalitions making an objection and a precise
counterobjection. The reason we impose the above two intersection conditions is thatcounterobjections to (𝑆, 𝑦) at 𝑥 satisfying them coincide with objections to 𝑦 in the reduced gameon 𝑆 at 𝑥. This is a generalization of the definition of a reduced game in the TU case proposed by
Aumann and Drèze (1974) in a similar way that Peleg (1985) generalized the definition of a reduced
game introduced by Davis and Maschler (1965). The formulation is as follows: let 𝑉 be a game, 𝑆 a
coalition, and 𝑥 a payoff profile. The reduced game of 𝑉 on 𝑆 at 𝑥 is the game 𝑉 𝑥𝑁\𝑆 defined by𝑥𝑥

𝑉𝑆 𝑁\𝑆 = 𝑉𝑆 , and 𝑉𝑃 𝑁\𝑆 = ⋃𝑄⊂𝑁\𝑆{𝑦𝑃 |(𝑦𝑃 , 𝑥𝑄 ) ∈ 𝑉𝑃∪𝑄 } if 𝑃⊂𝑆 and 𝑃 ≠ 𝑆.5
We can see that (𝑆, 𝑦) is an objection to 𝑥 with no precise counterobjections if there is no objection𝑥

dominating 𝑦 in 𝑉𝑆 𝑁\𝑆 , that is, 𝑦 is in the core of the reduced game. The converse is true if thepossibility of improvement over (𝑦𝑃 , 𝑥𝑄 ) implies strict improvement (consider the case where 𝑥 ≫𝑥 and 𝑉 satisfies strict 𝑥 -comprehensiveness). In general, the precise bargaining set is strictly
included in the natural bargaining set. See the example below.
Example 3: Two Chinese Cooks and Two Japanese Cooks. There are four cooks. Two are Chinese,and the others are Japanese: Guang, Huan, Ichiro, and Jiro. We denote them by 𝑔, ℎ, 𝑖, and 𝑗. The
problem is how to run a restaurant. Neither a single individual nor a pair of cooks from different
countries can make a complete meal. Their capabilities are measured in terms of money. Let 𝑁 ={𝑔, ℎ, 𝑖, 𝑗}. The TU game 𝑉 describing their options is as follows: 𝑣({𝑔}) = 𝑣({ℎ}) = 𝑣({𝑖}) =𝑣({𝑗}) = 𝑣({𝑔, 𝑖}) = 𝑣({𝑔, 𝑗}) = 𝑣({ℎ, 𝑖}) = 𝑣({ℎ, 𝑗}) = 0, 𝑣({𝑔, ℎ}) = 2.3, 𝑣({𝑔, ℎ, 𝑖}) =𝑣({𝑔, ℎ, 𝑗}) = 3.5, 𝑣({𝑖, 𝑗})= 1.3, 𝑣({𝑔, 𝑖, 𝑗}) = 𝑣({ℎ, 𝑖, 𝑗}) = 2.5, and 𝑣(𝑁) = 4, where 𝑣(𝑆) is theworth of coalition 𝑆. Consider the payoff profile 𝑥 = (𝑥𝑔 , 𝑥ℎ , 𝑥𝑖 , 𝑥𝑗 ) = (1, 1, 1, 1). The coalitions
having objections are {𝑔, ℎ}, {𝑔, ℎ, 𝑖}, and {𝑔, ℎ, 𝑗}. Note that {𝑔, ℎ}⊂{𝑔, ℎ, 𝑖} and {𝑔, ℎ}⊂{𝑔, ℎ, 𝑗}.
We can see that 𝑥 is in 𝑁𝐵(𝑉; {𝑁}), but not in 𝑃𝐵(𝑉; {𝑁}). ∎Given a payoff profile 𝑥, the inclusion relation among the sets of objections 𝑂(𝑥), those with nocounterobjections 𝑀(𝑥), those with no proper and strict counterobjections 𝑆𝑆(𝑥), those with nonatural and strict counterobjections 𝑁(𝑥), and those with no precise objections 𝑃(𝑥) is as follows:5This is the reduced game in the NTU case proposed by Peleg (1985). For details about reduced

games in both the TU case and the NTU case, see Thomson (2021, forthcoming).

15

𝑂(𝑥) ⸧ 𝑆𝑆(𝑥) ⸧ 𝑃(𝑥)⸧ 𝑁(𝑥) ⸧ 𝑀(𝑥). Then for every game 𝑉, we deduce the following inclusion
relations among the core, the Mas-Colell Bargaining Set, the strong steady bargaining set, the naturalbargaining set, and the precise bargaining Set: 𝐶0 (𝑉; 𝒫) ⊂ 𝑆𝑆𝐵(𝑉; 𝒫) ⊂ 𝑃𝐵(𝑉; 𝒫) ⊂ 𝑁𝐵(𝑉; 𝒫) ⊂𝑀𝐵(𝑉; 𝒫) for every coalition structure 𝒫, and 𝐶0 (𝑉) ⊂ 𝑆𝑆𝐵(𝑉) ⊂ 𝑃𝐵(𝑉) ⊂ 𝑁𝐵(𝑉) ⊂ 𝑀𝐵(𝑉).
Neither the natural bargaining set nor the precise bargaining set contains the steadybargaining set. The counterobjections to determine the steady bargaining set satisfy a stronger
intersection condition than that of the precise bargaining set, but they are not defined by strict
inequalities. The intersection requirments make the natural, and the precise, bargaining sets larger,
and the strict inequalities make them smaller than the steady bargaining set.
The following theorem tells us that the above two refinements of the Mas-Colell bargaining
set are nonvacuous in every coalitional game.
Theorem 3: The Precise bargaining set of a game contains at least one payoff profile for at least
one coalition structure.
Corollary: The Mas-Colell Bargaining Set and the Natural Bargaining Set of a game contain atleast one payoff profile for at least one coalition structure.
𝑆)
As long as 𝑉𝑆 ∩ ({𝑟𝑆 } + ℝ+
is compact for every coalition 𝑆, we do not need all four conditions ofa coalitional game to establish the result. Notice that Theorem 3 does not say anything about
efficiency, and recall Example 2; the Mas-Colell bargaining set may not contain any of the efficientprofiles if the game does not satisfy the non-crossing condition. Thus, there may be no efficientprofile in 𝑃𝐵(𝑉) or 𝑁𝐵(𝑉).

5.

Concluding Remarks

In cooperative game theory, some solutions describe stable compromises in noncooperative
situations among coalitions [see Mas-Colell (1987), and Vohra (1995) for surveys]. Although there
are several studies of the bargaining set of NTU games [see Iñarra et al. (2020), p.260], little
research has been done on the bargaining set of NTU games with coalition structures. Our model,
as well as Zhou’s (1994), describes coalition formation as an outcome of the bargaining set. By
applying the bargaining set, we have explained how coalition formations and payoff allocations are
simultaneously determined in coalitional NTU games. The theory of the core successfully explains
coalition formation in limited classes of NTU games, such as one-to-one matchings. Even though
the core is empty for many problems, other classical solutions work for coalitional games. Hence,
there is no doubt that cooperative solutions will continue to be useful in considering multi-person
coalition formation together with payoff allocation in various contexts.

16Appendix
Proof of Theorem 1. We use a fixed-point argument. The key to the proof is the construction of
the fixed-point correspondence. In particular, the homeomorphism ℎ, defined in Step 1, plays acrucial role. The existence of ℎ heavily depends on where 𝑥 is, namely the conditions of 𝑥 interiority and 𝑥-comprehensiveness.
Step 1. We define functions 𝑓 𝑆 for 𝑆 ⊂ 𝑁 to use for construction of the fixed-point correspondence.
Let 𝒩 be the class of coalitions. For every coalition 𝑆 with 1 < |𝑆| < |𝑁|, define 𝒯(𝑆) ≡{ 𝑇 ∈ 𝒩|
𝑇 ∩ 𝑆 ≠ ∅ &𝑆\𝑇 ≠ ∅ &𝑇\𝑆 ≠ ∅ .. There is 𝑥 ∈ ℝ𝑁 such that 𝑉 satisfies 𝑥 -compactness, 𝑥 interiority and 𝑥-comprehensiveness. By 𝑥-interiority, 𝑥 ∈ ⋂𝑖∈𝑁 int 𝑉({𝑖}). Then 𝑥 ∈int𝑉*. By 𝑥𝑆
compactness, 𝑉𝑆 ∩ [{𝑥𝑆 } + ℝ+
] is compact for every 𝑆 ∈ 𝒩 . Hence, 𝑉+∗ ≡ 𝑉 ∗ ∩ [{𝑥} + ℝ𝑁
+ ] is𝑁
compact. Let 𝑍 ≡ {𝑧 ∈ ℝ+ |𝑧 ∙ 𝑒 ≤ 1} and ∆≡ {𝑧 ∈ ℝ𝑁
+ |𝑧 ∙ 𝑒 = 1}, where 𝑒 = (1, …,1).
Define the function ℎ: 𝑍 → 𝑉+∗ by𝑥
if 𝑧 = 0ℎ(𝑧) ≡ {𝑥 + 𝑐(𝑧/(𝑧 ∙ 𝑒))𝑧 if 𝑧 ≠ 0
for all 𝑧 ∈ 𝑍 , where 𝑐(𝑦) ≡ {𝑡 ∈ ℝ+|𝑥 + 𝑡𝑦 ∈ 𝑃(𝑉 ∗ )} for every 𝑦 ∈ ∆ . By 𝑥 -interiority and 𝑥 comprehensiveness, 𝑐(𝑧/(𝑧 ∙ 𝑒)) uniquely exists for each 𝑧 ∈ 𝑍\{0} and ℎ is a homeomorphismfrom 𝑍 onto 𝑉+∗ (See Figures 6). For every (𝑧, 𝑖) ∈ 𝑍 × 𝑁, note that 𝑧𝑖 = 0 if and only if ℎ𝑖 (𝑧) = 𝑥𝑖 .
In addition, ℎ(𝑧) ∈ 𝑃(𝑉 ∗ ) if 𝑧 ∈ ∆.
Let 𝑆 ∈ 𝒩. By continuity, there is a continuous representation 𝑣 𝑆 : ℝ𝑁 → ℝ for 𝑉𝑆 . Let 𝑖 ∈ 𝑁. By
𝑥 -compactness, max𝑉{𝑖} exists. Define 𝑟 ∈ ℝ𝑁 by 𝑟𝑖 ≡ max𝑉{𝑖} for all 𝑖 ∈ 𝑁 . For every 𝑥𝑆 ∈ ℝ𝑆 ,define [𝑥𝑆 ] ≡ {𝑧𝑆 ∈ ℝ𝑆 |𝑧𝑖 = max{𝑥𝑖 , 𝑟𝑖 } for each 𝑖 ∈ 𝑆}.
For all 𝑆 ∈ 𝒩, define 𝑓 𝑆 : 𝑍 → ℝ+ as follows: If |𝑆| = 1,or |𝑁|, then𝑓 𝑆 (𝑧) ≡max {−𝑣 𝑆 (ℎ𝑆 (𝑧)), 0}
for every 𝑧 ∈ 𝑍. If 1 < |𝑆| < |𝑁|, let 𝐷 𝑆 ≡ {𝑧 ∈ 𝑍|𝑣 𝑆 ([ℎ𝑆 (𝑧)]) < 0}, and define𝑆∩𝑇 𝑆
𝑂𝑇𝑆 (𝑧) ≡ {𝑦𝑆∩𝑇 ∈ {[ℎ𝑆∩𝑇 (𝑧)]} + ℝ++|𝑣 (𝑦𝑆∩𝑇 , ℎ𝑆\𝑇 (𝑧)) < 0} , where 𝑇 ∈ 𝒯(𝑆),𝑚𝑇 (𝑧) ≡ max {𝑣 𝑇 (𝑦𝑆∩𝑇 , ℎ𝑆\𝑇 (𝑧)) |𝑦𝑆∩𝑇 ∈ cl𝑂𝑇𝑆 (𝑧)}, where 𝑇 ∈ 𝒯(𝑆), and𝑒 𝑆 (𝑧) ≡ min{𝑚𝑇 (𝑧)|𝑇 ∈ 𝒯(𝑆)}where cl𝑂𝑇𝑆 (𝑧) is the closure of 𝑂𝑇𝑆 (𝑧), for every 𝑧 ∈ 𝐷 𝑆 . Finally, definemax {−𝑣 𝑆 ([ℎ𝑆 (𝑧)]), 0}max{𝑒 𝑆 (𝑧), 0} if 𝑧 ∈ 𝐷 𝑆
𝑓 𝑆 (𝑧) ≡ {max {−𝑣 𝑆 (ℎ𝑆 (𝑧)), 0}
if 𝑧 ∈ 𝑍\𝐷 𝑆
Step 2. For every 𝑆 ∈ 𝒩, 𝑓 𝑆 is continuous on 𝑍.
Proof of Step 2. Let 𝑆 ∈ 𝒩 be with |𝑆| = 1, or |𝑁|. Since 𝑣 𝑆 is continuous, so is 𝑓 𝑆 .
Next, fix 𝑆 ⊂ 𝑁 such that 1 < |𝑆| < |𝑁|. Let 𝑇 ∈ 𝒯(𝑆) and 𝑧 ∈ 𝐷 𝑆 . Since 𝑣 𝑆 ([ℎ𝑆 (𝑧)]) < 0 and[ℎ𝑆\𝑇 (𝑧)] ≥ ℎ𝑆\𝑇 (𝑧), 𝑥 -compehensiveness implies 𝑣 𝑆 ([ℎ𝑆∩𝑇 (𝑧)], ℎ𝑆\𝑇 (𝑧)) < 0. Since 𝑣 𝑆 is𝑆∩𝑇continuous, there exist 𝑦𝑆∩𝑇 ∈ {[ℎ𝑆∩𝑇 (𝑧)]} + ℝ++
such that 𝑣 𝑆 (𝑦𝑆∩𝑇 , ℎ𝑆\𝑇 (𝑧)) < 0. Thus,

𝑂𝑇𝑆 (𝑧) ≠ Ø. Then, cl𝑂𝑇𝑆 is the correspondence from 𝐷 𝑆 to ℝ𝑆∩𝑇 that assigns to each 𝑧 ∈ 𝐷 𝑆 theclosure of 𝑂𝑇𝑆 (𝑧).

17

payoff to player 2𝒉(𝒛)𝒉(𝒛′ )𝒛𝒛′𝒙

payoff to player 1

payoff to player 3
𝒉(𝒛)𝒛payoff to player 2𝒉(𝒛′ )𝒛𝒛′𝒙

payoff to player 1
Figures 6: Homeomorphisms for two − person and three − person games18

We show that the correspondence cl𝑂𝑇𝑆 is compact-valued, upper-hemicontinuous, and lower𝑆
hemicontinuous on 𝐷 𝑆 . Since 𝑉𝑆 ∩ [{𝑥𝑆 } + ℝ+
] is compact and ℎ𝑆 (𝑧) ≥ 𝑥𝑆 , cl𝑂𝑇𝑆 is compactvalued. Since 𝑍 is compact and ℎ is continuous, ℎ𝑆\𝑇 (𝑍) is compact, so that the range of cl𝑂𝑇𝑆 is
included in a compact subset of ℝ𝑆∩𝑇 .
To show the upper-hemicontinuity of cl𝑂𝑇𝑆 , we prove that the graph of cl𝑂𝑇𝑆 is closed in𝑞

𝐷 𝑆 × ℝ𝑆∩𝑇 . Let (𝑧, 𝑦𝑆∩𝑇 ) ∈ 𝐷 𝑆 × ℝ𝑆∩𝑇 and {(𝑧 𝑞 , 𝑦𝑆∩𝑇 )} be a sequence in 𝐷 𝑆 × ℝ𝑆∩𝑇 such that𝑞𝑞𝑦𝑆∩𝑇 ∈ cl𝑂𝑇𝑆 (𝑧 𝑞 ) and (𝑧 𝑞 , 𝑦𝑆∩𝑇 ) → (𝑧, 𝑦𝑆∩𝑇 ) . We show 𝑦𝑆∩𝑇 ∈ cl𝑂𝑇𝑆 (𝑧) . For each 𝜀 > 0 andevery 𝑥𝑆∩𝑇 ∈ ℝ𝑆∩𝑇 , define 𝐵(𝜀; 𝑥𝑆∩𝑇 ) ≡ {𝑦𝑆∩𝑇 ∈ ℝ𝑆∩𝑇 |‖𝑦𝑆∩𝑇 − 𝑥𝑆∩𝑇 ‖ < 𝜀} (the open ball with𝑞

𝑥𝑆∩𝑇 center and 𝜀 radius). Since 𝑦𝑆∩𝑇 → 𝑦𝑆∩𝑇 , there is a subsequence {𝑞(𝑘)} of {𝑞} such that𝑞∀𝑘 ∈ ℕ, ∀𝑞 ∈ ℕ[𝑞 ≥ 𝑞(𝑘) ⇒ 𝑦𝑆∩𝑇 ∈ 𝐵(1/𝑘; 𝑦𝑆∩𝑇 )]𝑞(𝑘)Since 𝑦𝑆∩𝑇 ∈ cl𝑂𝑇𝑆 (𝑧 𝑞(𝑘) ), it follows that
𝑆∩𝑇
𝐵(1/𝑘; 𝑦𝑆∩𝑇 ) ∩ ({[ℎ𝑆∩𝑇 (𝑧 𝑞(𝑘) )]} + ℝ++) ∩ {𝑦𝑆∩𝑇 ∈ ℝ𝑆∩𝑇 |𝑣 𝑆 (𝑦𝑆∩𝑇 , ℎ𝑆\𝑇 (𝑧 𝑞(𝑘) )) < 0} ≠ ∅.

Take
𝑘
𝑆∩𝑇
𝑦𝑆∩𝑇
∈ 𝐵(1/𝑘; 𝑦𝑆∩𝑇 ) ∩ ({[ℎ𝑆∩𝑇 (𝑧 𝑞(𝑘) )]} + ℝ++) ∩ {𝑦𝑆∩𝑇 ∈ ℝ𝑆∩𝑇 |𝑣 𝑆 (𝑦𝑆∩𝑇 , ℎ𝑆\𝑇 (𝑧 𝑞(𝑘) )) < 0}𝑘𝑘Then, 𝑦𝑆∩𝑇
∈ 𝐵(1/𝑘; 𝑦𝑆∩𝑇 ), and 𝑦𝑆∩𝑇
→ 𝑦𝑆∩𝑇 . Since 𝑧 𝑞 → 𝑧, we have 𝑧 𝑞(𝑘) → 𝑧. In addition,
𝑘
𝑆∩𝑇
𝑦𝑆∩𝑇
∈ ({[ℎ𝑆∩𝑇 (𝑧 𝑞(𝑘) )]} + ℝ++) ∩ {𝑦𝑆∩𝑇 ∈ ℝ𝑆∩𝑇 |𝑣 𝑆 (𝑦𝑆∩𝑇 , ℎ𝑆\𝑇 (𝑧 𝑞(𝑘) )) < 0} .

Since

𝑘
𝑦𝑆∩𝑇
≫ [ℎ𝑆∩𝑇 (𝑧 𝑞(𝑘) )], it follows by the continuity of ℎ𝑆∩𝑇 that 𝑦𝑆∩𝑇 ≥ [ℎ𝑆∩𝑇 (𝑧)] . Note that𝑣 𝑆 (𝑦𝑆∩𝑇 , ℎ𝑆\𝑇 (𝑧 𝑞(𝑘) )) < 0. By the continuity of ℎ𝑆\𝑇 and 𝑣 𝑆 , 𝑣 𝑆 (𝑦𝑆∩𝑇 , ℎ𝑆\𝑇 (𝑧)) ≤ 0. Thus,𝑦𝑆∩𝑇 ∈ cl𝑂𝑇𝑆 (𝑧). Hence cl𝑂𝑇𝑆 has a closed graph, so that cl𝑂𝑇𝑆 is upper-hemicontinuous.
We next show that cl𝑂𝑇𝑆 is lower-hemicontinuous on 𝐷 𝑆 . Let 𝑧 ∈ 𝐷 𝑆 , and 𝑈 be an open subset
of ℝ𝑆∩𝑇 such that 𝑂𝑇𝑆 (𝑧) ∩ 𝑈 ≠ Ø. Then there is 𝑦𝑆∩𝑇 ∈ 𝑈 such that 𝑦𝑆∩𝑇 ≫ [ℎ𝑆∩𝑇 (𝑧)] and𝑣 𝑆 (𝑦𝑆∩𝑇 , ℎ𝑆\𝑇 (𝑧)) < 0. Since ℎ is continuous, there is an open neighborhood 𝑊 of 𝑧 such that𝑦𝑆∩𝑇 ≫ [ℎ𝑆∩𝑇 (𝑧′)] and 𝑣 𝑆 (𝑦𝑆∩𝑇 , ℎ𝑆\𝑇 (𝑧′)) < 0 for all 𝑧′ ∈ 𝑊 . Hence, 𝑦𝑆∩𝑇 ∈ 𝑂𝑇𝑆 (𝑧′) ∩ 𝑈 ,namely 𝑂𝑇𝑆 (𝑧′) ∩ 𝑈 ≠ Ø for every 𝑧′ ∈ 𝑊. Thus, 𝑂𝑇𝑆 is lower-hemicontinuous, and so is cl𝑂𝑇𝑆 .
Recall 𝑚𝑇 (𝑧) = max {𝑣 𝑇 (𝑦𝑆∩𝑇 , ℎ𝑆\𝑇 (𝑧)) |𝑦𝑆∩𝑇 ∈ cl𝑂𝑇𝑆 (𝑧)}. Since 𝑣 𝑇 is continuous, by the
Berge maximum theorem, 𝑚𝑇 is continuous. Recall 𝑒 𝑆 (𝑧) ≡ min{𝑚𝑇 (𝑧)|𝑇 ∈ 𝒯(𝑆)}, then 𝑒 𝑆 is
also continuous. Since there is a compact subset of ℝ𝑆 including the range of cl𝑂𝑇𝑆 and 𝒯(𝑆) is
finite, 𝑒 𝑆 has an upper bound.
We finally show that 𝑓 𝑆 is continuous on 𝑍. Let 𝑧 ∈ 𝐷 𝑆 , then 𝑒 𝑆 and 𝑓 𝑆 is continuousat 𝑧. Next choose 𝑧 ∗ ∈ 𝑍\𝐷 𝑆 , then 𝑣 𝑆 ([ℎ𝑆 (𝑧 ∗ )]) ≥ 0 and 𝑓 𝑆 (𝑧) =max {−𝑣 𝑆 ([ℎ𝑆 (𝑧 ∗ )]), 0} = 0.
Let {𝑧 𝑞 } be a sequence in 𝑍 such that 𝑧 𝑞 → 𝑧 ∗ , then max{−𝑣 𝑆 ([ℎ𝑆 (𝑧 𝑞 )]), 0} → 0. Since 𝑒 𝑆 has
an upper bound, max {−𝑣 𝑆 ([ℎ𝑆 (𝑧 𝑞 )]), 0}max{𝑒 𝑆 (𝑧 𝑞 ), 0} → 0, so that 𝑓 𝑆 (𝑧 𝑞 ) → 0. Hence, 𝑓 𝑆 iscontinuous at 𝑧 ∗ . Thus, 𝑓 𝑆 is continuous on 𝑍. ∎Step 3. We construct the fixed-point correspondence φ by 𝑓 𝑆 , and apply the Kakutani theorem.
Define the function 𝐹: 𝑍 → ℝ𝑁
+ by

19

𝐹𝑖 (𝑧) ≡ max{𝑓 𝑆 (𝑧)|𝑆 ∈ 𝒩 with 𝑖 ∈ 𝑆}
for every 𝑧 ∈ 𝑍, then 𝐹 is continuous. Define the correspondence φ: ∆→ ∆ byφ(𝑧) ≡ {𝑧 ∗ ∈ ∆|(∀𝑧′ ∈ ∆)(𝑧 ∗ ∙ 𝐹(𝑧) ≥ 𝑧 ′ ∙ 𝐹(𝑧)) }for every 𝑧 ∈ ∆. Note that φ(𝑧) = ∆ if 𝐹(𝑧) = 0. By the Berge maximum theorem, φ is compactvalued, and upper-hemicontinuous. Further, φ is convex-valued. By the Kakutani fixed-point
theorem [Kakutani (1941)], there exists 𝑧 ∗ ∈ ∆ such that 𝑧 ∗ ∈ φ(𝑧 ∗ ) . That is, 𝑧 ∗ ∙ 𝐹(𝑧 ∗ ) ≥ 𝑧 ∙𝐹(𝑧 ∗ ) for all 𝑧 ∈ ∆.
Step 4. The payoff profile 𝑥 ∗ associated with a fixed-point is efficient.
Proof of Step 4.Let 𝑥 ∗ = ℎ(𝑧 ∗ ). Since 𝑧 ∗ ∈ ∆, 𝑥 ∗ ∈ 𝑃(𝑉 ∗ ). Thus, 𝑥 ∗ is an efficient payoff profile. ∎Step 5. If there are two distinct maximal coalitions 𝑄 and 𝑅 such that 𝑓 𝑄 (𝑧 ∗ ) > 0 and 𝑓 𝑅 (𝑧 ∗ ) > 0,then they are disjoint, namely 𝑄 ∩ 𝑅 = Ø.
Proof of Step 5. Let 𝑄, 𝑅 ∈ 𝒩 be two distinct maximal coalitions such that 𝑓 𝑄 (𝑧 ∗ ) > 0 and 𝑓 𝑅 (𝑧 ∗ )>0. We show that 𝑄 ∩ 𝑅 = Ø. We must have that |𝑄| < |𝑁|, and |𝑅| < |𝑁|. If |𝑄| = 1 or |𝑅| = 1, thenit follows by maximality that 𝑄 ∩ 𝑅 = Ø. If |𝑄| > 1 and |𝑅| > 1, then∗𝑣 𝑄 ([𝑥𝑄∗ ]) < 0 & (∀𝑇 ∈ 𝒯(𝑄)) (∃𝑦𝑄 ∈ cl𝑂𝑇𝑄 (𝑧 ∗ )) (𝑣 𝑇 (𝑦𝑇∩𝑄 , 𝑥𝑇\𝑄
) > 0)∗′𝑣 𝑅 ([𝑥𝑅∗ ]) < 0 & (∀𝑇 ∈ 𝒯(𝑅)) (∃𝑦𝑅′ ∈ cl𝑂𝑇𝑄 (𝑧 ∗ )) (𝑣 𝑇 (𝑦𝑇∩𝑅, 𝑥𝑇\𝑅) > 0).

Suppose that 𝑄 ∩ 𝑅 ≠ Ø. By maximality, 𝑄\𝑅 ≠ Ø and 𝑅\𝑄 ≠ Ø, so that 𝑅 ∈ 𝒯(𝑄) and 𝑄 ∈𝒯(𝑅). Further, recall the definition of 𝑂𝑇𝑆 (𝑧). By continuity, we may assume that𝑄∗∗′(∃𝑦𝑄 ∈ 𝑂𝑅 (𝑧 ∗ )) (𝑣 𝑅 (𝑦𝑄∩𝑅 , 𝑥𝑅\𝑄
, 𝑥𝑄\𝑅) > 0)&(∃𝑦𝑅′ ∈ cl𝑂𝑄𝑅 (𝑧 ∗ )) (𝑣 𝑄 (𝑦𝑄∩𝑅) > 0).
∗
Note that 𝑦𝑄 ∈ 𝑂𝑅𝑄 (𝑧 ∗ ) means 𝑦𝑄 ≫ [ℎ𝑄 (𝑧 ∗ )] = 𝑥𝑄∗ and 𝑣 𝑄 (𝑦𝑄∩𝑅 , 𝑥𝑄\𝑅
) < 0 . Similarly, 𝑦𝑅′ ∈∗cl𝑂𝑄𝑅 (𝑧 ∗ ) means 𝑦𝑅 ′ ≫ [ℎ𝑅 (𝑧 ∗ )] = 𝑥𝑅∗ and 𝑣 𝑅 (𝑦𝑄∩𝑅 , 𝑥𝑅\𝑄
) < 0. Then
∗
∗
∗
[𝑥𝑄∩𝑅
] ≪ 𝑦𝑄∩𝑅 & 𝑣 𝑄 (𝑦𝑄∩𝑅 , 𝑥𝑄\𝑅
) < 0 & 𝑣 𝑅 (𝑦𝑄∩𝑅 , 𝑥𝑅\𝑄
)>0
∗
∗
∗
′
′
′
& 𝑣 𝑅 (𝑦𝑄∩𝑅, 𝑥𝑅\𝑄, 𝑥𝑄\𝑅[𝑥𝑄∩𝑅] ≪ 𝑦𝑄∩𝑅) < 0 & 𝑣 𝑄 (𝑦𝑄∩𝑅)>0Hence,
∗
∗
∗
∗
′
′
′
𝑣 𝑄 (𝑦𝑄∩𝑅 , 𝑥𝑄\𝑅
, 𝑥𝑄\𝑅, 𝑥𝑅\𝑄, 𝑥𝑅\𝑄) < 0 < 𝑣 𝑄 (𝑦𝑄∩𝑅)&𝑣 𝑅 (𝑦𝑄∩𝑅) < 0 < 𝑣 𝑅 (𝑦𝑄∩𝑅
),
∗
∗
∗
∗
′
′
′
i.e., (𝑦𝑄∩𝑅 , 𝑥𝑄\𝑅
, 𝑥𝑄\𝑅, 𝑥𝑅\𝑄, 𝑥𝑅\𝑄) ∈ int𝑉𝑄 & (𝑦𝑄∩𝑅) ∉ 𝑉𝑄 &(𝑦𝑄∩𝑅) ∈ int𝑉𝑅 & (𝑦𝑄∩𝑅) ∉ 𝑉𝑅∗′∗
Note that 𝑦𝑄∩𝑅 ≫ [𝑥𝑄∩𝑅≫ [𝑥𝑄∩𝑅] ≥ 𝑟𝑄∩𝑅 and 𝑦𝑄∩𝑅] ≥ 𝑟𝑄∩𝑅 . Since 𝑥 ∗ ∈ 𝑃(𝑉 ∗ ), this contradicts

the restrictive non-crossing condition. Thus, 𝑄 ∩ 𝑅 = Ø. ∎Step 6. For all 𝑆 ∈ 𝒩, 𝑓 𝑆 (𝑧 ∗ ) = 0.
Proof of Step 6. The proof is by contradiction. Suppose there is 𝑆 ∈ 𝒩 with 𝑓 𝑆 (𝑧 ∗ )> 0.
We first prove that there is a partition 𝒫 of 𝑁 such that 𝑓 𝑄 (𝑧 ∗ ) > 0 for all 𝑄 ∈ 𝒫 : Let𝑓 𝑆 (𝑧 ∗ ) > 0, then 𝐹(𝑧 ∗ ) ≥ 0 and 𝐹(𝑧 ∗ ) ≠ 0 . Let 𝑖 ∈ 𝑁 . If 𝐹𝑖 (𝑧 ∗ ) = 0, then 0 = 𝑓 {𝑖} (𝑧 ∗ ) =max {−𝑣 {𝑖} (ℎ𝑖 (𝑧 ∗ )), 0} = max {−𝑣 {𝑖} (𝑥𝑖∗ ), 0}. Thus, 𝑣 {𝑖} (𝑥𝑖∗ ) ≥ 0. Since 𝐹(𝑧 ∗ ) ≥ 0, 𝐹(𝑧 ∗ ) ≠ 0

20and 𝑧 ∗ ∈ φ(𝑧 ∗ ) , we have 𝑧𝑖∗ = 0 . By the construction of ℎ and 𝑥 -interiority, we have 𝑥𝑖∗ =ℎ𝑖 (𝑧 ∗ ) = 𝑥𝑖 ∈ int𝑉{𝑖} . This contradicts the inequality 𝑣 {𝑖} (𝑥𝑖∗ ) ≥ 0 . Thus, for every 𝑖 ∈ 𝑁 ,𝐹𝑖 (𝑧 ∗ ) > 0. Hence, for every 𝑖 ∈ 𝑁, there is 𝑆(𝑖) ∈ 𝒩 such that 𝑖 ∈ 𝑆(𝑖) and 𝑓 𝑆(𝑖) (𝑧 ∗ ) > 0. Notethat ⋃𝑖∈𝑁 𝑆(𝑖) = 𝑁. Then, there exist a class 𝒫 of maximal coalitions such that 𝑓 𝑄 (𝑧 ∗ ) > 0 for
all 𝑄 ∈ 𝒫 and ⋃𝑄∈𝒫 𝑄 = 𝑁. By Step 5, 𝒫 is a partition of 𝑁.
Let 𝑆 ∈ 𝒫. Since 𝑣 𝑆 ([𝑥𝑆∗ ]) < 0 and 𝑥𝑆∗ ≤ [𝑥𝑆∗ ], we have 𝑣 𝑆 (𝑥𝑆∗ ) < 0, so that 𝑥𝑆∗ ∈ int𝑉𝑆 . Thus,𝑥𝑆∗ ∈ int𝑉𝑆 for all 𝑆 ∈ 𝒫. Hence, 𝑥 ∗ ∈ int𝑉 ∗. Recall 𝑥 ∗ ∈ 𝑃(𝑉 ∗ ). This is a contradiction. Hence,
there is no 𝑆 ∈ 𝒩 such that 𝑓 𝑆 (𝑧 ∗ )> 0. That is, all 𝑆 ∈ 𝒩, 𝑓 𝑆 (𝑧 ∗ ) = 0. ∎Step 7. 𝑥 ∗ ∈ 𝑆𝐵(𝑉).
Proof of Step 7. Let 𝑖 ∈ 𝑁. Since 0 = 𝑓 {𝑖} (𝑧 ∗ ) , 𝑣 {𝑖} (𝑥𝑖∗ ) ≥ 0, namely 𝑥𝑖∗ ∉ int𝑉{𝑖} . Thus, 𝑥𝑖∗ ≥ 𝑟𝑖 ,and 𝑥𝑆∗ = [𝑥𝑆∗ ] ≫ 𝑥𝑆 for all 𝑆 ∈ 𝒩. Since 0= 𝑓 𝑁 (𝑧 ∗ ) = max {−𝑣 𝑁 (ℎ(𝑧 ∗ )), 0} = max {−𝑣 𝑁 (𝑥 ∗ ), 0},
we also have 𝑥 ∗ ∉ int𝑉𝑁 . Let 𝑆 ∈ 𝒩 with 1 < |𝑆| < |𝑁| satisfy 0 > 𝑣 𝑆 ([𝑥𝑆∗ ]) = 𝑣 𝑆 (𝑥𝑆∗ ) . Then,max {𝑒 𝑆 (𝑧 ∗ ), 0} = 0, so that 𝑒 𝑆 (𝑧 ∗ ) ≤ 0. By 𝑥 -comprehensiveness, 𝑦𝑆∩𝑇 ∈ cl𝑂𝑇𝑆 (𝑧 ∗ ) if 𝑇 ∈∗∗𝒯(𝑆), 𝑦𝑆∩𝑇 ≫ 𝑥𝑆∩𝑇
and 𝑣 𝑆 (𝑦𝑆∩𝑇 , 𝑥𝑆\𝑇
) ≤ 0 . By the definition of 𝑒 𝑆 , there is 𝑇 ∈ 𝒯(𝑆) such that
∗
∗
∗
𝑆∩𝑇
} + ℝ++∀𝑦𝑆∩𝑇 ∈ {𝑥𝑆∩𝑇
s.t. 𝑣 𝑆 (𝑦𝑆∩𝑇 , 𝑥𝑆\𝑇
) ≤ 0, 𝑣 𝑇 (𝑦𝑆∩𝑇 , 𝑥𝑇\𝑆
) ≤ 0 (**)∗∗
Let (𝑆, 𝑦) be an objection to 𝑥 ∗ . Then 𝑦 ≫ 𝑥𝑆∗ . Since 𝑦𝑆∩𝑇 ≫ 𝑥𝑆∩𝑇
, 𝑣 𝑇 (𝑦𝑆∩𝑇 , 𝑥𝑇\𝑆
) ≤ 0. Thus,∗
there is a proper counterobjection (𝑇, (𝑦𝑆∩𝑇 , 𝑥𝑇\𝑆)) to (𝑆, 𝑦) at 𝑥 ∗ . Hence, 𝑥 ∗ ∈ 𝑆𝐵(𝑉). ∎This completes the proof of Theorem 1. ∎Proof of Theorem 2. Suppose 𝑉 satisfies strict 𝑥-comprehensiveness. Let 𝑥 ∗ , (𝑆, 𝑦), and 𝑇 be
respectively the same profile, objection, and dominant coalition to 𝑆 at 𝑥 ∗ as in the proof of∗∗∗
Theorem 1. Recall 𝑦 = (𝑦𝑆∩𝑇 , 𝑦𝑆\𝑇 ) ≥ (𝑦𝑆∩𝑇 , 𝑥𝑆\𝑇), 𝑦 ≠ (𝑦𝑆∩𝑇 , 𝑥𝑆\𝑇), and (𝑦𝑆∩𝑇 , 𝑥𝑆\𝑇
) ≥ 𝑥𝑆 .
∗∗′By strict 𝑥 -comprehensiveness, (𝑦𝑆∩𝑇 , 𝑥𝑆\𝑇
, 𝑥𝑆\𝑇) ∈ int𝑉𝑆 , and there is 𝑦′ = (𝑦𝑆∩𝑇) ∈ 𝑉𝑆 such∗′′∗that 𝑦𝑆∩𝑇≫ 𝑦𝑆∩𝑇 . Note that 𝑦𝑆∩𝑇
≫ 𝑦𝑆∩𝑇 ≫ 𝑥𝑆∩𝑇
and 𝑣 𝑆 (𝑦𝑆∩𝑇 , 𝑥𝑆\𝑇
) < 0. By (**) in the proof
∗
∗
∗
∗
′
′
′
of Theorem 1, 𝑣 𝑇 (𝑦𝑆∩𝑇, 𝑥𝑇\𝑆, 𝑥𝑇\𝑆, 𝑥𝑇\𝑆) ≤ 0 . Note that (𝑦𝑆∩𝑇) ≥ (𝑦𝑆∩𝑇 , 𝑥𝑇\𝑆
) , (𝑦𝑆∩𝑇)≠∗∗∗(𝑦𝑆∩𝑇 , 𝑥𝑇\𝑆), and (𝑦𝑆∩𝑇 , 𝑥𝑇\𝑆
) ≥ 𝑥𝑇 . By strict 𝑥-comprehensiveness, (𝑦𝑆∩𝑇 , 𝑥𝑇\𝑆
) ∈ int𝑉𝑇 , so∗that there is 𝑧 ∈ 𝑉𝑇 such that 𝑧 ≫ (𝑦𝑆∩𝑇 , 𝑥𝑇\𝑆
) . Thus, (𝑇, 𝑧) is a strict and propercounterobjection to 𝑦 at 𝑥 ∗ , and 𝑇 is strictly dominant to 𝑆 at 𝑥 ∗ . Hence, 𝑥 ∗ ∈ 𝑆𝑆𝐵(𝑉). ∎Proof of (*) in Example 2. We have 𝐼(𝑉; {𝑎, 𝑏, 𝑐}, {𝑑}) = {𝑥 ∈ ℝ𝑁
+ |(𝑥𝑎 , 𝑥𝑏 , 𝑥𝑐 , 𝑥𝑑 ) ≤ (6,4,1,0)..
∗Let 𝑥 ∈ 𝑀𝐵(𝑉; {𝑎, 𝑏, 𝑐}, {𝑑}). Note that 𝑀𝐵(𝑉; {𝑎, 𝑏, 𝑐}, {𝑑}) ⊂ 𝐼(𝑉; {𝑎, 𝑏, 𝑐}, {𝑑}) . The pair({𝑏, 𝑐, 𝑑}, (6, 4, 1)) is an objection to any payoff profile of 𝐼(𝑉; {𝑎, 𝑏, 𝑐}, {𝑑}). For 𝑥 ∗ to be in𝑀𝐵(𝑉; {𝑎, 𝑏, 𝑐}, {𝑑}), there need to be a counterobjection to ({𝑏, 𝑐, 𝑑}, (6, 4, 1)) at 𝑥 ∗ . We seethat {𝑎, 𝑐, 𝑑} is the only coalition that can make such a counterobjection if and only if 𝑥𝑎∗ ≤ 1.
We then suppose 𝑥𝑎∗ ≤1. The pair ({𝑎, 𝑐, 𝑑}, (2, 3, 5)) is an objection to any payoff profile of21∗𝐼(𝑉; {𝑎, 𝑏, 𝑐}, {𝑑}) ∩ {𝑥 ∈ ℝ𝑁
+ |(𝑥𝑎 , 𝑥𝑏 , 𝑥𝑐 , 𝑥𝑑 ) ≤ (1,1,1,0)}. For 𝑥 ∈ 𝑀𝐵(𝑉; {𝑎, 𝑏, 𝑐}, {𝑑}) ,there need to be a counterobjection to ({𝑎, 𝑐, 𝑑}, (2, 3, 5)) at 𝑥 ∗ . We see that {𝑎, 𝑏, 𝑑} is the only
coalition that can make such a counterobjection if and only if 𝑥𝑏∗ ≤1. Hence,
𝑀𝐵(𝑉; {𝑎, 𝑏, 𝑐}, {𝑑}) ⊂ 𝐼(𝑉; {𝑎, 𝑏, 𝑐}, {𝑑})..
∗Conversely, let 𝑥 ∗ ∈ {𝑥 ∈ ℝ𝑁
+ |(𝑥𝑎 , 𝑥𝑏 , 𝑥𝑐 , 𝑥𝑑 ) ≤ (1,1,1,0)}. Then given 𝑥 , any objection from{𝑎, 𝑏, 𝑐} (resp., {𝑏, 𝑐, 𝑑} , {𝑎, 𝑐, 𝑑} , and {𝑎, 𝑏, 𝑑}) has the counterobjection ({𝑏, 𝑐, 𝑑} , (6, 4, 1))(resp., ({𝑎, 𝑐, 𝑑}, (1, 6, 4)), ({𝑎, 𝑏, 𝑑}, (4, 1, 6)), and ({𝑎, 𝑏, 𝑐}, (6, 4, 1)). In addition, given 𝑥 ∗ ,any objection from 𝑁 has the counterobjection ( {𝑎, 𝑏, 𝑐} , (3, 5, 2)). Hence, we have{𝑥 ∈ ℝ𝑁+ |(𝑥𝑎 , 𝑥𝑏 , 𝑥𝑐 , 𝑥𝑑 ) ≤ (1,1,1,0)} ⊂ 𝑀𝐵(𝑉; {𝑎, 𝑏, 𝑐}, {𝑑}). It therefore follows that
𝑀𝐵(𝑉; {𝑎, 𝑏, 𝑐}, {𝑑}) = {𝑥 ∈ ℝ𝑁
+ |(𝑥𝑎 , 𝑥𝑏 , 𝑥𝑐 , 𝑥𝑑 ) ≤ (1,1,1,0)}. ∎Proof of Theorem 5. We define the following concept:
P-justified objection: Let 𝑉 be a game, 𝑥 a payoff profile, (𝑆, 𝑦) an objection to 𝑥. Then,(𝑆, 𝑦) is P-justified if there is no precise and strict counterobjection to (𝑆, 𝑦) at 𝑥.
Let 𝑉 be a game. For convenience, we identify 𝒫 ∪ {Ø} with 𝒫 for every coalition structure 𝒫, anddefine 𝑉(Ø) ≡ ℝ𝑁 and {{𝑖}|𝑖 ∈ Ø} ≡ Ø . Let 𝑥 0 ≡ 𝑟 ∈ ℝ𝑁 , where 𝑟𝑖 = max𝑉{𝑖} for all 𝑖 ∈ 𝑁 . If
there is no P-justified objection to 𝑥 0 , then 𝑥 0 ∈ 𝑃𝐵(𝑉; {{𝑖}|𝑖 ∈ 𝑁}). Otherwise, go to Stage 1.
Stage 1. Choose a P-justified objection (𝑆(0), 𝑦 0 ) to 𝑥 0 so that there is no 𝑦′ ∈ 𝑉𝑆(0) such that 𝑦′ ≫𝑦 0 . Consider the coalition structure {𝑆(0)} ∪ {{𝑖}|𝑖 ∈ 𝑁\𝑆(0)} , and define 𝑥 1 ≡ (𝑦 0 , 𝑟𝑁\𝑆(0) ) .
Then, 𝑥 0 ≤ 𝑥 1 , 𝑥 0 ≠ 𝑥 1 and 𝑥 1 ∈ 𝑉(𝑆(0)) ∩ ⋂𝑖∈𝑁\𝑆(0) 𝑉({𝑖}). If there is no P-justified objection
to 𝑥 1 , then 𝑥 1 ∈ 𝑃𝐵(𝑉; {𝑆(0)} ∪ {{𝑖}|𝑖 ∈ 𝑁\𝑆(0)}). Otherwise, go to Stage 2.
Stage 2. Choose a P-justified objection (𝑆(1), 𝑦1 ) to 𝑥 1 in the same way as above. Since 𝑥 0 ≤ 𝑥 1 ,(𝑆(1), 𝑦1 ) is also an objection to 𝑥 0 and strictly improves upon (𝑆(0), 𝑦 0 ) at 𝑥 0 . Since theobjection (𝑆(0), 𝑦 0 ) to 𝑥 0 is P-justified, we have 𝑆(1) ∩ 𝑆(0) = Ø or 𝑆(0)\𝑆(1) = Ø .
Consider the coalition structure {𝑆(1), 𝑆(0)\𝑆(1)} ∪ {{𝑖}|𝑖 ∈ 𝑁\(𝑆(1) ∪ 𝑆(0))}. The possible0
features of {𝑆(0)\𝑆(1)} are {𝑆(0)} and {∅}. Define 𝑥 2 ≡ (𝑦1 , 𝑦𝑆(0)\𝑆(1), 𝑟𝑁\(𝑆(1)∪𝑆(0)) ), where(𝑦 0 , 𝑟𝑁\(𝑆(1)∪𝑆(0)) )0(𝑦𝑆(0)\𝑆(1), 𝑟𝑁\(𝑆(1)∪𝑆(0)) ) ≡ {𝑟𝑁\𝑆(1)

if 𝑆(0)\𝑆(1) = 𝑆(0)if 𝑆(0)\𝑆(1) = ∅Then, 𝑥 2 ∈ 𝑉(𝑆(1)) ∩ 𝑉(𝑆(0)\𝑆(1)) ∩ ⋂𝑖∈𝑁\(𝑆(1)∪𝑆(0)) 𝑉({𝑖}). If there is no P-justified objection,𝑥 2 ∈ 𝑃𝐵 (𝑉; {𝑆(1), 𝑆(0)\𝑆(1)} ∪ {{𝑖}|𝑖 ∈ 𝑁\(𝑆(1) ∪ 𝑆(0))}). Otherwise, go to Stage 3.
Stage 3. Choose a P-justified objection (𝑆(2), 𝑦 2 ) to 𝑥 2 in the same way as above. Since 𝑥 0 ≤ 𝑥 2 ,(𝑆(2), 𝑦 2 ) is also an objection to 𝑥 0 and strictly improves upon (𝑆(0), 𝑦 0 ) at 𝑥 0 . Since theobjection (𝑆(0), 𝑦 0 ) to 𝑥 0 is P-justified, we have 𝑆(2) ∩ 𝑆(0) = Ø or 𝑆(0)\𝑆(2) = Ø. Similarly,22

(𝑆(2), 𝑦 2 ) is an objection to 𝑥 1 ≤ 𝑥 2 and the objection (𝑆(1), 𝑦1 ) to 𝑥 1 is P-justified, so that
𝑆(2) ∩ 𝑆(1) = Ø or 𝑆(1)\𝑆(2) = Ø. Consider the coalition structure {𝑆(2), 𝑆(1)\𝑆(2), 𝑆(0)\(𝑆(2) ∪ 𝑆(1))} ∪ {{𝑖}|𝑖 ∈ 𝑁\(𝑆(2) ∪ 𝑆(1) ∪ 𝑆(0))}. The possible features of {𝑆(1)\𝑆(2), 𝑆(0)\(𝑆(2) ∪ 𝑆(1))} are {𝑆(1), 𝑆(0)}, {𝑆(1)} , {𝑆(0)} and {∅}.
0
1
Define 𝑥 3 ≡ (𝑦 2 , 𝑦𝑆(1)\𝑆(2), 𝑦𝑆(0)\(𝑆(2)∪𝑆(1)), 𝑟𝑁\(𝑆(2)∪𝑆(1)∪𝑆(0)) ), where01(𝑦𝑆(1)\𝑆(2), 𝑦𝑆(0)\(𝑆(2)∪𝑆(1)), 𝑟𝑁\(𝑆(2)∪𝑆(1)∪𝑆(0)) )(𝑦1 , 𝑦 0 , 𝑟𝑁\(𝑆(2)∪𝑆(1)∪𝑆(0)) ) if {𝑆(1)\𝑆(2), 𝑆(0)\(𝑆(2) ∪ 𝑆(1))} = {𝑆(1), 𝑆(0)}≡(𝑦1 , 𝑟𝑁\(𝑆(2)∪𝑆(1)) )if {𝑆(1)\𝑆(2), 𝑆(0)\(𝑆(2) ∪ 𝑆(1))} = {𝑆(1)}(𝑦 0 , 𝑟𝑁\(𝑆(2)∪𝑆(0)) )if {𝑆(1)\𝑆(2), 𝑆(0)\(𝑆(2) ∪ 𝑆(1))} = {𝑆(0)}if {𝑆(1)\𝑆(2), 𝑆(0)\(𝑆(2) ∪ 𝑆(1))} = {∅}{ 𝑟𝑁\𝑆(2)Then, 𝑥 3 ∈ 𝑉(𝑆(2)) ∩ 𝑉(𝑆(1)\𝑆(2)) ∩ 𝑉 (𝑆(0)\(𝑆(2) ∪ 𝑆(1))) ∩ ⋂𝑖∈𝑁\(𝑆(2)∪𝑆(1)∪𝑆(0)) 𝑉({𝑖}).
If there is no P-justified objection, then𝑥 3 ∈ 𝑃𝐵 (𝑉; {𝑆(2), 𝑆(1)\𝑆(2), 𝑆(0)\(𝑆(2) ∪ 𝑆(1))} ∪ {{𝑖}|𝑖 ∈ 𝑁\(𝑆(2) ∪ 𝑆(1) ∪ 𝑆(0))}).
Otherwise, go to Stage 4.
Suppose that this process does not terminate at Stage 𝑘 − 1. Then we go to Stage 𝑘.
Stage 𝑘. For every 𝑡 = 1, ⋯ , 𝑘 − 1, (𝑆(𝑡), 𝑦 𝑡 ) is a P-justified objection to0
𝑡−2
𝑥 𝑡 ≡ (𝑦 𝑡−1 , 𝑦𝑆(𝑡)\𝑆(𝑡−1), ⋯ , 𝑦𝑆(0)\(𝑆(𝑡−1)∪⋯∪𝑆(1)), 𝑟𝑁\(𝑆(𝑡−1)∪⋯∪𝑆(1)∪𝑆(0)) ),which is chosen in the same way as above.
Note that 𝑆(𝑡′) ∩ 𝑆(𝑡) = ∅ or 𝑆(𝑡′)\𝑆(𝑡) = ∅ for each 𝑡 ′ , 𝑡 = 1, ⋯ , 𝑘 − 1 with 𝑡 ′ < 𝑡 . For each𝑡 = 0,1, ⋯ , 𝑘 − 1, let 𝑈(𝑡) ≡ 𝑆(𝑘 − 1) ∪ ⋯ ∪ 𝑆(𝑡). Consider the coalition structure {𝑆(𝑘 −1), 𝑆(𝑘 − 2)\𝑈(𝑘 − 1), ⋯ , 𝑆(0)\𝑈(1)} ∪ {{𝑖}|𝑖 ∈ 𝑁\𝑈(0)}, and define𝑘−20𝑥 𝑘 ≡ (𝑦 𝑘−1 , 𝑦𝑆(𝑘−2)\𝑈(𝑘−1), ⋯ , 𝑦𝑆(0)\𝑈(1), 𝑟𝑁\𝑈(0) ),Then𝑥 𝑘 ∈ 𝑉(𝑆(𝑘 − 1)) ∩ 𝑉(𝑆(𝑘 − 1)\𝑈(𝑘 − 1)) ∩ ⋯ 𝑉(𝑆(0)\𝑈(1)) ∩ ⋂𝑉({𝑖}).
𝑖∈𝑁\𝑈(0)If there is no P-justified objection to 𝑥 𝑘 , then𝑥 𝑘 ∈ 𝑃𝐵(𝑉; {𝑆(𝑘 − 1), 𝑆(𝑘 − 1)\𝑈(𝑘 − 1), ⋯ , 𝑆(0)\𝑈(1)} ∪ {{𝑖}|𝑖 ∈ 𝑁\𝑈(0)})
Otherwise, go to Stage 𝑘 + 1.
By construction, each coalition can make a P-justified objection at most once. Since thenumber of coalitions is finite, this process terminates in finite stages. Then, we can find a point of
𝑃𝐵(𝑉; 𝒫) for at least one coalition structure 𝒫. ∎23

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