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On the accessibility of core-extensions
Games and Economic Behavior 74 (2012) 687–698
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Games and Economic Behavior www.elsevier.com/locate/geb
On the accessibility of core-extensions ✩ Yi-You Yang Department of Applied Mathematics, Aletheia University, New Taipei City 251, Taiwan
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 1 February 2011 Available online 30 August 2011
Sengupta and Sengupta (1996) study the accessibility of the core of a TU game and show that the core, if non-empty, can be reached from any non-core allocation via a finite sequence of successive blocks. This paper complements the result by showing that when the core is empty, a number of non-empty core-extensions, including the least core and the weak least core (Maschler et al., 1979), the positive core (Orshan and Sudhölter, 2001) and the extended core (Bejan and Gómez, 2009), are accessible in a strong sense, namely each allocation in each of the foregoing core-extensions can be reached from any allocation through a finite sequence of successive blocks. © 2011 Elsevier Inc. All rights reserved.
JEL classification: C71 C73 Keywords: Accessibility Core Core-extensions
1. Introduction The core is the most widely applied cooperative game theory solution concept in economics. A core allocation of a coalitional game with transferable utility (TU game) guarantees that each coalition of players obtains at least what it could gain on its own. This means that once players receive a core allocation, no coalition has an incentive to deviate from it. On the other hand, in case a non-core allocation x is proposed as a solution for the game, a coalition S whose members can gain by forming S may threaten to split off from the grand coalition and thereby suggest an alternative allocation y according to its myopic preferences. Again, if y stands outside the core, another blocking coalition could replace y with a third allocation, and so forth. A fundamental question concerning this dynamic process is: Which allocations may be reached via such a process of successive transitions? Sengupta and Sengupta (1996) address the issue for the class of games with non-empty cores. They adopt the notion of weak dominance1 to formulate the possible transitions among allocations and show that the core, if non-empty, is accessible in the sense that it can be reached from any non-core allocation through a sequence of successive blocks.2 Kóczy (2006) focuses on the time required to reach the core and shows that the accessibility of the core can be met in a bounded number of blocks. In addition, Yang (2010) and Béal et al. (2010), respectively, give alternative upper bounds for the number of blocks needed to access the core. The former is described as the number of active coalitions,3 and the latter is equal to the number of pairs of players, i.e., n(n − 1)/2, where n is the cardinality of the player set.
✩ The author is very grateful to two anonymous referees for their perceptive comments of the paper. Support by National Science Council of Republic of China under grant NSC 100-2410-H-156-008 is gratefully acknowledged. E-mail address: [email protected]. 1 A weak dominance is a dominance relation in which not every member of the blocking coalition is strictly better off. 2 Such an accessibility property of the core may not hold with strict dominance. 3 A coalition different from the grand coalition is said to be active if it can guarantee each of its members a payoff which is strictly individually rational.
0899-8256/$ – see front matter doi:10.1016/j.geb.2011.08.007
© 2011
Elsevier Inc. All rights reserved.
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Y.-Y. Yang / Games and Economic Behavior 74 (2012) 687–698
Nevertheless, numerous games derived from natural economic situations possess empty cores.4 This drawback leads to the development of relaxations of the core such as the least core and the weak least core (Maschler et al., 1979), the positive core (Orshan and Sudhölter, 2001), the extended core and its various refinements (Bejan and Gómez, 2009), each of which is always non-empty yet coincides with the core whenever the core is non-empty. A natural question is whether these core-extensions could inherit the accessibility of the core. In the present paper, we first attempt to extend Sengupta and Sengupta’s analysis to incorporate games with empty cores with the aid of an alternative notion of dominance relation, z-dominance, in which a payoff vector z = ( z i ) is adopted as a reference point which guarantees that each player i outside the blocking coalition obtains at least z i . Then we study conditions for an allocation to be an accessible allocation, i.e., an allocation that can be reached from any allocation via a sequence of successive transitions, and prove that a game possesses an accessible allocation if and only if the core is a singleton or an empty set. Finally, we apply the analysis to address the accessibility of the foregoing core-extensions by showing that when the core is empty, each allocation in each of these core-extensions is an accessible allocation. One feature of our approach is that players are assumed to be myopic in the sense that the blocking coalition ignores any possible further deviations, which may render some of its members lower payoffs than the former allocation does. Based on a similar myopia assumption, Roth and Vande Vate (1990) address the accessibility of the core for marriage markets and show that a decentralized process of successive blocks can terminate in the core. Their result is extended by Diamantoudi et al. (2004) and Klaus and Klijn (2007) to roommate markets and couples markets, respectively. The rest of the paper is organized as follows. Section 2 gives fundamental definitions and notations. Section 3 introduces a process of successive transitions and gives the main results. Section 4 concludes and a proof is presented in Appendix A. 2. Preliminaries Let N = {1, 2, . . . , n} be a finite set of players. A coalition is a non-empty subset of N. Let ζ denote the collection of coalitions except N, i.e., ζ := 2 N \{ N , ∅}. A coalitional game with transferable utility, or simply a game, is described by a v which assigns to each coalition S a real value v ( S ). A game v is said to be N-monotonic if for all coalitional function coalitions S, v ( S ) + i ∈ N \ S v ({i }) v ( N ).
i i A payoff vector x = (x1 , . . . , xn ) ∈ R N is feasible if i ∈ N x v ( N ). An allocation is a payoff vector x such that i∈N x = i v ( N ). Let A ( v ) denote the collection of allocations of a game v. An imputation is an allocation x such that x v ({i }) for all i ∈ N. We denote by I ( v ) the set of imputations of v. N i i For any coalition S, let x( S ) be a shorthand for For iany pair ofS vectors x, y ∈ R , we write x y if x y for allS i ∈ N. x and let x denote the projection of x on S. We also write x y S if xi y i for all i ∈ S. i∈ S Given an allocation x ∈ A ( v ), the excess of a coalition S is defined to be
e (x, S ) = v ( S ) − x( S ), which can be interpreted as a measure of the dissatisfaction of coalition S with respect to the allocation x. The core of a game v is defined to be
C ( v ) = x ∈ A ( v ) : e (x, S ) 0 for all S ∈ ζ , which collects allocations that cannot be improved upon by any coalition. Given a coalition S, we denote by e S the vector in R N whose i-th coordinate is 1 if i ∈ S and 0 otherwise. A collection of coalitions β ⊆ ζ is said to be balanced if there exists a system of positive weights {λ S } S ∈β such that S ∈β λ S e S = e N . A game v is called a balanced game if for any balanced collection β with weights {λ S } S ∈β ,
λ S v ( S ) v ( N ).
S ∈β
Bondareva (1963) and Shapley (1967) independently prove that a game is balanced if and only if it has a non-empty core. Let ε be a non-negative number. The strong ε -core of a game v is the set of all those allocations x ∈ A ( v ) such that e (x, S ) ε for all S ∈ ζ . The weak ε -core of a game v collects allocations x ∈ A ( v ) satisfying e (x, S ) | S |ε for all S ∈ ζ , where | S | denotes the cardinal number of the coalition S. The strong ε -core (or weak ε -core), introduced by Shapley and Shubik (1966), can be interpreted as the set of allocations that cannot be improved upon by any coalition S if forming S entails a cost of ε (or | S |ε ). Clearly, the strong ε -core and the weak ε -core are both non-empty when ε is large enough. The least core of a game v (Maschler et al., 1979), denoted by LC ( v ), is defined to be the intersection of all non-empty strong ε -cores. Similarly, the weak least core of v, denoted by LC w ( v ), is the intersection of all non-empty weak ε -cores. It is clear that LC ( v ) coincides with the strong ε0 -core with
ε0 = max 0, min max e(x, S ) . x∈ A ( v ) S ∈ζ
4
See, for example, Shapley and Shubik (1966), Wooders (1978), and Kaneko and Wooders (1982).
Y.-Y. Yang / Games and Economic Behavior 74 (2012) 687–698
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For any allocation x ∈ A ( v ), the excess vector θ( v , x) is defined as the vector that rearranges the coordinates of (e (x, S )) S ∈ζ in non-increasing order. The prenucleolus of a game v (Schmeidler, 1969) is the unique allocation PN ( v ) ∈ A ( v ) that lexicographically minimizes the excess vector θ( v , PN ( v )), i.e., θ( v , PN ( v )) lex θ( v , y ) for all y ∈ A ( v ), where lex denotes the lexicographic order. Similarly, the nucleolus of a game v is the unique imputation N ( v ) ∈ I ( v ) satisfying θ( v , N ( v )) lex θ( v , y ) for all y ∈ I ( v ). Note that PN ( v ) ∈ LC ( v ) for all game v, and PN ( v ) = N ( v ) if C ( v ) = ∅. For any allocation x ∈ A ( v ), let θ+ ( v , x) be the vector that rearranges the coordinates of (max{0, e (x, S )}) S ∈ζ in nonincreasing order. The positive core of v (Orshan and Sudhölter, 2001) is defined to be
C+ ( v ) = x ∈ A ( v ): ∀ y ∈ A ( v ), θ+ ( v , x) lex θ+ ( v , y ) , which is an intermediate concept between the prenucleolus and the least core, i.e., PN ( v ) ∈ C+ ( v ) ⊆ LC ( v ). Moreover, it is not difficult to see that all these core-like solution concepts, including the least core, the weak least core and the positive core, are non-empty extensions of the core. Namely, each of them is never empty and returns the core in case the core is non-empty. 3. A process of successive transitions Consider an allocation y that is proposed as a solution for the game v. If some coalition S can improve upon y, in order to obtain a higher total payoff, it may threaten to leave the grand coalition and thereby propose an alternative allocation x such that
v ( S ) x( S ) > y ( S )
and
xS y S .
(1)
In that case, we refer to S as a blocking coalition. Sengupta and Sengupta (1996) and Kóczy (2006) formulate such a transition from y to x as a notion of dominance relation. Definition 1. Let x and y be allocations of a game v. We say that x dominates y via a coalition S, denoted by x S y, if condition (1) holds. Nevertheless, it seems unrealistic that the blocking coalition S enjoys such a strong power that it can arbitrarily distribute the payoffs of the remaining players without any restriction. To make the transition more practical, Yang (2010) suggests to employ a pre-decided payoff vector z as a benchmark and argues that an “acceptable” counterproposal x should satisfy the remaining players in the sense that xi zi for all i ∈ N \ S. Combining with feasibility and the fact that the blocking coalition S can raise its aggregate payoff up to v ( S ) on its own, a proper benchmark vector z should be chosen such that v ( S ) + z( N \ S ) v ( N ) for all coalitions S. Following this logic, we formulate a restrictive notion of dominance as follows. Definition 2. A feasible payoff vector z is called a benchmark vector5 of a game v if
v ( S ) + z( N \ S ) v ( N ) for all S ∈ ζ.
(2)
Let x and y be allocations of a game v. We say that x dominates y via a coalition S with respect to a benchmark vector z (x z-dominates y via S for short) if x S y and x N \ S z N \ S . Remark 3. The notion of benchmark vector is closely related to the core and N-monotonic games. The following facts are easily checked. (a) An allocation z of a game v is a benchmark vector if and only if z is a member of the core.6 (b) The stand alone value vector v ∗ := ( v ({1}), . . . , v ({n})) ∈ R N of a game v is a benchmark vector if and only if v is N-monotonic. We say that an allocation x indirectly dominates another allocation y if there exist a sequence of allocations { y j }lj =0 and
a sequence of blocking coalitions { S j }lj =1 such that
x = y l S l y l −1 S l −1 · · · S 1 y 0 = y .
(3)
In addition, we say that x indirectly z-dominates y if z is a benchmark vector and for each j = 1, . . . , l, in the sequence (3), y j z-dominates y j −1 via S j . Such a sequence of blocks describes a possible trajectory of successive transitions. We are 5 Laussel and Le Breton (2001) analyze the Pareto efficient frontier of the set of benchmark vectors of a TU game to study the structure of equilibrium payoffs in a common agency game, which is introduced by Bernheim and Whinston (1986). 6 Sengupta and Sengupta (1996) and Béal et al. (2010) respectively give algorithms that pick a core allocation z as a reference point and generate sequences of successively z-dominating allocations that terminate in the core, although they do not formally introduce the notion of z-dominance.
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interested in which allocations the disequilibrium process may lead to. To address the issue, the notions of accessible sets and accessible allocations arise naturally. Definition 4. Let v be a game augmented with a benchmark vector z. A set of allocations ⊆ A ( v ) is called an acces/ , there exists an allocation x ∈ that indirectly dominates sible (respectively z-accessible) set if for any allocation y ∈ (respectively indirectly z-dominates) y. In particular, an allocation x ∈ A ( v ) is called an accessible (respectively z-accessible) allocation if the singleton {x} is an accessible (respectively z-accessible) set. 3.1. The core and z-accessible sets Sengupta and Sengupta (1996) focus on the class of balanced games and show that the core of a balanced game is an accessible set. The following theorem attempts to extend their analysis to incorporate unbalanced games. Theorem 5. Let v be a game with a benchmark vector z and let
z ( v ) := x ∈ A ( v ): x z . Then the union of C ( v ) and z ( v ) is a z-accessible set. Remark 6. The following observations are helpful to gain intuition behind Theorem 5. (a) (b) (c) (d)
The core is a subset of any z-accessible set. A sequence of successively z-dominating allocations terminates as soon as it enters the core. In case the core is empty, a sequence of successively z-dominating allocations never terminates. Once a sequence of successively z-dominating allocations enters z ( v ), from which it can never escape.
The proof of Theorem 5 requires the following lemma. N \T
Lemma 7. Let z be a benchmark vector of a game v. Let x0 be a non-core allocation such that x0 Then there exists a sequence of z-dominating allocations
z N \ T for some coalition T ∈ ζ .
xl S l xl−1 S l−1 · · · S 1 x0 N \T
such that xl ∈ C ( v ) or xl
z N \ T for some proper subset T of T .
/ C ( v ), there exists a coalition S 1 with v ( S 1 ) > x0 ( S 1 ). We consider two cases. Proof. Since x0 ∈ Case I. S 1 ∩ T = T . We define the allocation x1 by
x1i
=
x0i + zi +
v ( S 1 )−x0 ( S 1 ) , |S1| v ( N )− v ( S 1 )− z( N \ S 1 ) , |N \ S 1 |
if i ∈ S 1 , if i ∈ N \ S 1 . N \T
Let T = S 1 ∩ T , then x1 z-dominates x0 via S 1 , x1 z N \ T and hence the proof is done. Case II. S 1 ∩ T = T , i.e., T ⊆ S 1 . We define the allocation x1 by
⎧ i v ( S 1 )−x0 ( S 1 ) , if i ∈ T , ⎪ |T | ⎨ x0 + i i x1 = x , if i ∈ S 1 \ T , ⎪ ⎩ 0i i i z + t 1 (x0 − z ), if i ∈ N \ S 1 , v ( N )− z( N \ S )− v ( S )
N \T
where 0 t 1 = v ( N )−z( N \ S 1)−x ( S1 ) < 1. It is easily checked that x1 z-dominates x0 via S 1 , x1 ( S 1 ) = v ( S 1 ), and x1 z N \ T . 1 0 1 If x1 ∈ C ( v ), the proof is done. Otherwise, the proof will proceed by constructing the required z-dominating sequence inductively. Suppose that the sequence of z-dominating allocations xk S k xk−1 S k−1 · · · S 1 x0 has been constructed to satisfy (i) S j = S l for j = l; (ii) xk ( S j ) v ( S j ) and T ⊆ S j for j = 1, . . . , k; and N \T
(iii) xk
z N \ T and xk ∈ / C ( v ).
Then there exists a coalition S k+1 with v ( S k+1 ) > xk ( S k+1 ). Note that condition (ii) implies S k+1 = S j for j = 1, . . . , k. In case S k+1 ∩ T = T , we define the allocation xk+1 by
Y.-Y. Yang / Games and Economic Behavior 74 (2012) 687–698
xki +1 =
⎧ ⎨ xi + k ⎩z + i
v ( S k+1 )−xk ( S k+1 ) , | S k +1 |
691
if i ∈ S k+1 ,
v ( N )− v ( S k+1 )− z( N \ S k+1 ) , | N \ S k +1 |
if i ∈ N \ S k+1 . N \T
Let T = S k+1 ∩ T , then xk+1 z-dominates xk via S k+1 and xk+1 z N \ T and hence we obtain the desired result. In what follows, we assume S k+1 ∩ T = T , i.e., T ⊆ S k+1 , and define the allocation xk+1 by
⎧ v ( S k+1 )−xk ( S k+1 ) i ⎪ , if i ∈ T , ⎪ |T | ⎨ xk + i xk+1 = xi , if i ∈ S k+1 \ T , k ⎪ ⎪ ⎩ zi + t (xi − zi ), if i ∈ N \ S k+1 , k +1 k v ( N )− z( N \ S
(4)
N \T
where 0 tk+1 = v ( N )−z( N \ S k+1)−x ( Sk+1 ) < 1. Clearly, xk+1 z-dominates xk via S k+1 , xk+1 ( S k+1 ) = v ( S k+1 ) and xk+1 z N \ T . k+1 k k+1 We are going to show that xk+1 ( S j ) v ( S j ) for j = 1, . . . , k. )− v ( S
)
N \ S k+1
Let 1 j k. Since T ⊆ S k+1 , by condition (iii), we obtain that xk Thus, we have
N \ S k+1
z N \ S k+1 . This implies xk
N\S
xk+1k+1 by (4).
xk ( N \ S k+1 ) − xk+1 ( N \ S k+1 ) xk ( S j \ S k+1 ) − xk+1 ( S j \ S k+1 ).
(5)
Moreover, (4) also implies
xk+1 ( N \ S k+1 ) = v ( N ) − v ( S k+1 ).
(6)
The combination of (5) and (6) yields
xk+1 ( S j \ S k+1 ) xk ( S j \ S k+1 ) − v ( S k+1 ) + xk ( S k+1 ) and hence
xk+1 ( S j ) = xk+1 ( S j \ S k+1 ) + xk+1 ( S j ∩ S k+1 )\ T + xk+1 ( T )
xk ( S j \ S k+1 ) − v ( S k+1 ) + xk ( S k+1 ) + xk ( S j ∩ S k+1 )\ T + xk ( T ) + v ( S k+1 ) − xk ( S k+1 )
= xk ( S j \ S k+1 ) + xk ( S j ∩ S k+1 )\ T + xk ( T )
= xk ( S j ) v ( S j ). Note that to derive the above inequalities we use the facts that xk+1 (( S j ∩ S k+1 )\ T ) = xk (( S j ∩ S k+1 )\ T ), xk+1 ( T ) = xk ( T ) + v ( S k+1 ) − xk ( S k+1 ) and condition (ii). Since the number of those coalitions containing T is finite and the blocking coalitions at each step of the sequence are N \T
distinct, there exists some r such that the allocation xr constructed at the r-th step satisfies xr ∈ C ( v ) or xr where T = S r ∩ T is a proper subset of T . 2
z N \T ,
We are now ready to prove Theorem 5. Proof of Theorem 5. For any subset T of N, we denote
z ( v , T ) = x ∈ A ( v ): xN \T z N \T . Clearly, z ( v , ∅) = z ( v ). Let x0 be a non-core allocation. Then there exists a coalition T 1 such that v ( T 1 ) > x0 ( T 1 ). Since z is a benchmark vector, we obtain v ( T 1 ) + z( N \ T 1 ) v ( N ). Then there exists an allocation x1 ∈ z ( v , T 1 ) that z-dominates x0 via T 1 . In case x1 ∈ C ( v ), the proof is done. Otherwise, by Lemma 7, there exist a proper subset T 2 of T 1 and an allocation x2 ∈ C ( v ) ∪ z ( v , T 2 ) that indirectly z-dominates x1 . If x2 ∈ C ( v ) or T 2 = ∅, the proof is done. Otherwise, we may apply Lemma 7 to find a proper subset T 3 of T 2 and an allocation x3 ∈ C ( v ) ∪ z ( v , T 3 ) that indirectly z-dominates x2 . Since T 1 is a finite set and the sequence of sets
T1 T2 T3 · · · is strictly decreasing, we obtain the desired result by using Lemma 7 repeatedly.
2
The next result, originally proved by Sengupta and Sengupta (1996), is an immediate consequence of Theorem 5. Corollary 8. Let z be a core allocation of a game v. Then z is a benchmark vector and the core is z-accessible.
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Y.-Y. Yang / Games and Economic Behavior 74 (2012) 687–698
Proof. Since z ∈ C ( v ), we obtain that v ( S ) + z( N \ S ) = v ( S ) + v ( N ) − z( S ) v ( N ) for all coalitions S and hence z is a benchmark vector. Moreover, since z ( v ) = { z} ⊆ C ( v ), by Theorem 5, we obtain that C ( v ) = C ( v )∪z ( v ) is z-accessible. 2 The following result shows that for any benchmark vector z, the set z ( v ) plays a central role among z-accessible sets when the game v under consideration is unbalanced. Corollary 9. Let v be an unbalanced game with a benchmark vector z. (a) z ( v ) is a z-accessible set. (b) Let ⊆ A ( v ) be a z-accessible set, then ∩ z ( v ) is also z-accessible. Proof. (a) The core of an unbalanced game is empty and hence z ( v ) = C ( v ) ∪ z ( v ) is z-accessible by Theorem 5. (b) Let y be an allocation lying outside ∩ z ( v ). We are going to show that y is indirectly z-dominated by some allocation in ∩ z ( v ). Since z ( v ) is a z-accessible set, without loss of generality, we may assume that y ∈ z ( v )\. Note that, by definition, no member of z ( v ) can be z-dominated by an allocation outside z ( v ). Combining with the fact that is z-accessible, there exists x ∈ ∩ z ( v ) that indirectly z-dominates y. 2 3.2. z-Accessible allocations The purpose of this section is to study the existence of z-accessible (or accessible) allocations. In sight of Corollary 8, it is clear that when the core consists of a single allocation, the unique core allocation is the only accessible allocation. Theorem 10. Let x be an allocation of a balanced game v, then x is an accessible allocation if and only if C ( v ) = {x}. Proof. (⇒) By definition, the core is a subset of any accessible set. Assume that x ∈ A ( v ) is an accessible allocation. Then the singleton {x} is an accessible set and hence C ( v ) ⊆ {x}. Since v is balanced, C ( v ) = ∅. This implies C ( v ) = {x}. (⇐) Assume that the core of v consists of a unique core allocation x. By Corollary 8, x indirectly dominates any non-core allocation. 2 In what follows, we shall focus on unbalanced games. The next lemma establishes a method for finding z-accessible allocations. Lemma 11. Let v be an unbalanced game with a benchmark vector z. (a) z( N ) < v ( N ). (b) For any allocation x ∈ A ( v ), x is z-accessible if and only if x ∈ z ( v ) and x indirectly z-dominates all allocations y ∈ z ( v )\{x}. (c) Let β be a balanced collection with weights {λ S } S ∈β and let x ∈ z ( v ) be an allocation such that
x( S ) = z( S )
for all S ∈ β\βx ,
(7)
where βx := { S ∈ β : v ( S ) > x( S )}, then x is a z-accessible allocation. Proof. See Appendix A.
2
We are now ready to prove that each unbalanced game augmented with a benchmark vector z possesses z-accessible (or accessible) allocations. Theorem 12. Let v be an unbalanced game with a benchmark vector z. Let (t ∗ , ε∗ ) be an optimal solution for the linear programming problem
( P 1 ) min ε s.t. t ( S ) + ε v ( S ) − z( S ) for all S ∈ ζ, t ( N ) = v ( N ) − z( N ), N t ∈ R+ , ε 0. Then z + t ∗ ∈ z ( v ) is a z-accessible allocation. Proof. Since v is unbalanced, C ( v ) = ∅ and hence problem
ε∗ > 0. Let y ∗ = ( y ∗ ( S )) S ∈ζ ∪{N } be an optimal solution for ( P 1 )’s dual
Y.-Y. Yang / Games and Economic Behavior 74 (2012) 687–698
( D 1 ) max
S ∈ζ ∪{N }
s.t.
i ∈ S ∈ζ
693
y ( S ) v ( S ) − z( S )
y ( S ) + y ( N ) 0 for all i ∈ N ,
S ∈ζ
y ( S ) 1,
y ( N ) ∈ R , y ( S ) 0 for all S ∈ ζ. Let β = { S ∈ ζ : y ∗ ( S ) > 0} and let x∗ = z + t ∗ . It is easily checked that β = ∅ and for all S ∈ β , v ( S ) > x∗ ( S ). Indeed, in case β = ∅, i.e., y ∗ ( S ) = 0 for all S ∈ ζ , then y ∗ ( N ) 0 and hence
0 < ε∗ =
y ∗ ( S ) v ( S ) − z( S )
S ∈ζ ∪{ N }
= y ∗ ( N ) v ( N ) − z( N ) 0 by the duality theorem and Lemma 11(a). This is impossible. Moreover, by the complementary slackness condition, we see that for all S ∈ ζ ,
x∗ ( S ) + ε∗ − v ( S ) · y ∗ ( S ) = 0.
This implies that v ( S ) = x∗ ( S ) + ε∗ > x∗ ( S ) for all S ∈ β . y (S) Since y ∗ satisfies the constraints of ( D 1 ), β = ∅ implies y ∗ ( N ) < 0. Let λ S = − y∗ ( N ) > 0 for all S ∈ β and let ∗
α = { i }:
y∗ ( S ) + y∗ (N ) < 0 .
i ∈ S ∈ζ
We consider two cases. Case I. α = ∅. Then β is a balanced collection with positive weights {λ S } S ∈β . By Lemma 11(c), x∗ is a z-accessible allocation. Case II. α = ∅. By the complementary slackness condition, for all i ∈ N,
i ∈ S ∈ζ
y ∗ ( S ) + y ∗ ( N ) · t ∗i = 0.
This implies x∗ ({i }) = z({i }) for all {i } ∈ α . For each {i } ∈ α , let
δ{ i } =
1− 1−
i ∈ S ∈β
λS ,
if {i } ∈ / β,
i ∈ S ∈β λ S + λ{i } , if {i } ∈ β.
Obviously, β ∪ α is a balanced collection with positive weights {λ S } S ∈β\α ∪ {δ{i } }{i }∈α . Using Lemma 11(c) again, we obtain the desired result. 2 Since each allocation of an unbalanced game is dominated by infinitely many allocations, by Lemma 11(b) and Theorem 12, we see that for any unbalanced game v with a benchmark vector z, the collection of z-accessible allocations is an infinite subset of z ( v ). A natural question is to explore the relation between the set of all z-accessible allocations and z ( v ). To shed some light on the issue, the next result identifies a sufficient condition for the coincidence of these two sets. Theorem 13. Let v be an unbalanced game and let z∗ be a benchmark vector that maximizes the amount z∗ ( N ). Then the set of all z∗ -accessible allocations coincides with z∗ ( v ). Proof. By Lemma 11(a), z∗ ( N ) < v ( N ) and hence z∗ is an optimal solution for the linear programming problem
( P 2 ) max z( N ) s.t. z( S ) v ( N ) − v ( N \ S ) for all S ∈ ζ, z ∈ RN . Let y ∗ = ( y ∗ ( S )) S ∈ζ be an optimal solution for ( P 2 )’s dual problem
( D 2 ) min s.t.
S ∈ζ
v (N ) − v (N \ S ) · y( S )
i ∈ S ∈ζ
y ( S ) = 1 for all i ∈ N ,
y ( S ) 0 for all S ∈ ζ.
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Y.-Y. Yang / Games and Economic Behavior 74 (2012) 687–698
Let β = { S ∈ ζ : y ∗ ( S ) > 0} and let θ = S ∈β y ∗ ( S ). Clearly, β is a balanced collection with positive weight { y ∗ ( S )} S ∈β and θ > 1. y∗ ( S ) Let β c = { N \ S: S ∈ β} and let λ N \ S = θ− for all N \ S ∈ β c . Since 1
λN \ S e N \ S =
S ∈β
y ∗ ( S ) S ∈β
=
θ −1
1
θ −1
eN\S =
1
θ −1
S ∈β
y ∗ ( S )e N −
y ∗ ( S )e S
S ∈β
(θ e N − e N ) = e N ,
β c is a balanced collection with weights {λ N \ S } N \ S ∈β c .
By the complementary slackness condition, we obtain that for all S ∈ ζ ,
z ∗ ( S ) − v ( N ) + v ( N \ S ) · y ∗ ( S ) = 0.
This implies that for all S ∈ β , z∗ ( S ) = v ( N ) − v ( N \ S ). N with For any x ∈ z∗ ( v ), we denote β(x) = { S ∈ β : x( S ) = z∗ ( S )}. Let x∗ ∈ z∗ ( v ) and let t ∗ = x∗ − z∗ . Then t ∗ ∈ R+ t ∗ ( N ) > 0. We will apply Lemma 11(c) to show that x∗ is z∗ -accessible. We consider two cases. Case I. β(x∗ ) = ∅. Then for all S ∈ β , x∗ ( S ) > z∗ ( S ) = v ( N ) − v ( N \ S ) and hence v ( N \ S ) > x∗ ( N \ S ). This implies that x∗ is a z-accessible allocation by Lemma 11(c). Case II. β(x∗ ) = ∅. We are going to show that x∗ is z-accessible by constructing a z-dominating sequence. Let x0 = x∗ and let S 1 ∈ β(x∗ ), then t ∗ ( S 1 ) = 0 and x∗ ( S 1 ) = z∗ ( S 1 ) = v ( N ) − v ( N \ S 1 ), or, equivalently, t ∗ ( N \ S 1 ) = t ∗ ( N ) and v ( N \ S 1 ) = x∗ ( N \ S 1 ). We define the allocation x1 ∈ z∗ ( v ) by
x1i
=
z∗i + 12 t ∗i , z∗i + 2|1S | t ∗ ( N ), 1
if i ∈ N \ S 1 , if i ∈ S 1 .
Clearly, x0 z∗ -dominates x1 via N \ S 1 and β(x1 ) ⊆ β(x0 )\{ S 1 } β(x0 ). Inductively, we assume that the z∗ -dominating sequence
x∗ = x0 N \ S 1 x1 N \ S 2 · · · N \ S l xl has been constructed such that for j = 1, . . . , l, β(x j ) ⊆ β(x j −1 )\{ S j } β(x j −1 ). If β(xl ) = ∅, by Case I, xl is z∗ -accessible and the proof is done. If β(xl ) = ∅, choose S l+1 ∈ β(xl ) and define the allocation xl+1 ∈ z∗ ( v ) by
xli+1
=
z∗i + 12 tli , z∗ + i
1 t ( N ), 2 | S l +1 | l
if i ∈ N \ S l+1 , if i ∈ S l+1 ,
N where tl = xl − z∗ ∈ R+ . It is easily checked that xl z∗ -dominates xl+1 via N \ S l+1 and β(xl+1 ) ⊆ β(xl )\{ S l+1 } β(xl ). Since the number of coalitions in β(x∗ ) is finite, we obtain that x∗ indirectly z∗ -dominates some allocation xr ∈ z∗ ( v ) with β(xr ) = ∅. Thus, we obtain the desired result. 2
We close this section with an integrated analysis on accessible sets. Corollary 14. Let v be a game with a benchmark vector z and let ⊆ A ( v ). (a) If v is balanced, then is an accessible set if and only if C ( v ) ⊆ . (b) If v is unbalanced, then is an accessible (respectively z-accessible) set if and only if contains at least an accessible (respectively z-accessible) allocation. (c) If v is unbalanced and z is a benchmark vector that maximizes z( N ), then is a z-accessible set if and only if ∩ z ( v ) = ∅. Proof. The combination of Corollary 8 and the fact that the core is a subset of any accessible set yields (a). (b) By Theorem 12, the collection of accessible (or z-accessible) allocations of an unbalanced game is non-empty. Clearly, is an accessible (respectively z-accessible) set if contains an accessible (respectively z-accessible) allocation. Conversely, let x be an accessible (respectively z-accessible) allocation and assume that is an accessible (respectively z-accessible) set. In case x ∈ , the proof of (b) is done. If x ∈ / , then there exists y ∈ that indirectly dominates (respectively zdominates) x. Clearly, y is accessible (respectively z-accessible) as well. Finally, putting (b) and Theorem 13 together yields (c). 2
Y.-Y. Yang / Games and Economic Behavior 74 (2012) 687–698
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3.3. Core-extensions In this section, we study the accessibility of a number of core-extensions, including the least core, the weak least core, the positive core, and the extended core. We first observe that for any unbalanced game v with a benchmark vector z, the collection of those z-accessible allocations contained in the least core (respectively the weak least core) exactly coincides with the intersection of the least core (respectively the weak least core) and z ( v ). Theorem 15. Let v be an unbalanced game with a benchmark vector z. For any allocation x ∈ LC ( v ) ∪ LC w ( v ), x is a z-accessible allocation if and only if x ∈ z ( v ). Proof. The “only if” part of the proof follows immediately from Lemma 11(b). To obtain the “if” part, we consider two cases. Case I. Let x∗ ∈ LC ( v ) ∩ z ( v ). Consider the linear programming problem
( P 3 ) min ε x( S ) + ε v ( S ) for all S ∈ ζ,
s.t.
x( N ) = v ( N ), x ∈ R N , ε 0, with optimum value ε∗ . Since x∗ ∈ LC ( v ), by definition, (x∗ , ε∗ ) is an optimal solution for ( P 3 ). Moreover, since v is unbalanced, C ( v ) = ∅ and hence ε∗ > 0. Let y ∗ = ( y ∗ ( S )) S ∈ζ ∪{ N } be an optimal solution for ( P 3 )’s dual problem
( D 3 ) max s.t.
y( S )v ( S )
S ∈ζ ∪{N } i ∈ S ∈ζ
y ( S ) + y ( N ) = 0 for all i ∈ N ,
S ∈ζ
y ( S ) 1,
y ( N ) ∈ R, y ( S ) 0 for all S ∈ ζ, and let β = { S ∈ ζ : y ∗ ( S ) > 0}. In case β = ∅, i.e., y ∗ ( S ) = 0 for all S ∈ ζ , then y ∗ ( N ) = 0 and hence
0 < ε∗ =
y∗ ( S )v ( S ) = 0
S ∈ζ ∪{ N }
by the duality theorem of linear programming. This is impossible. Thus we obtain β = ∅ and y ∗ ( N ) < 0. Moreover, we note y (S) that β is a balanced collection with positive weights {λ S } S ∈β , where λ S = − y∗ ( N ) for all S ∈ β . ∗ In sight of the complementary slackness condition, we obtain that for all S ∈ ζ,
x∗ ( S ) + ε∗ − v ( S ) · y ∗ ( S ) = 0.
This implies that v ( S ) = x∗ ( S ) + ε∗ > x∗ ( S ) for all S ∈ β . By Lemma 11(c), x∗ is a z-accessible allocation. Case II. Let x∗ ∈ LC w ( v ) ∩ z ( v ). Consider the linear programming problem
( P 4 ) min ε s.t.
(x + εe N )( S ) v ( S ) for all S ∈ ζ, x( N ) = v ( N ), x ∈ RN ,
ε 0,
with optimum value ε∗ . Since C ( v ) = ∅, ε∗ > 0. Note that w ∗ := x∗ + ε∗ e N is an optimal solution for the problem
( P 4 ) min w ( N ) s.t.
w ( S ) v ( S ) for all S ∈ ζ, w ∈ RN .
Let y ∗ = ( y ∗ ( S )) S ∈ζ be an optimal solution for ( P 4 )’s dual problem
( D 4 ) max s.t.
y( S )v ( S )
S ∈ζ i ∈ S ∈ζ
y ( S ) = 1 for all i ∈ N ,
y( S ) 0
for all S ∈ ζ.
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Y.-Y. Yang / Games and Economic Behavior 74 (2012) 687–698
Note that β := { S ∈ ζ : y ∗ ( S ) > 0} = ∅ is a balanced collection with positive weights { y ∗ ( S )} S ∈β . Moreover, by the complementary slackness condition, we have that for all S ∈ ζ ,
w ∗ ( S ) − v ( S ) · y ∗ ( S ) = 0.
This implies that for all S ∈ β , v ( S ) = w ∗ ( S ) = x∗ ( S ) + | S |ε∗ > x∗ ( S ). Using Lemma 11(c) again, we obtain the desired result. 2 Next, putting Theorems 12 and 15 together yields the following results. Theorem 16. Let v be an unbalanced game. (a) Each allocation in the least core (or the positive core) is accessible. In particular, the prenucleolus is an accessible allocation. (b) Each allocation in the weak least core is accessible. (c) If v is N-monotonic, then the nucleolus is an accessible allocation. Proof. Let x be an allocation in LC ( v ) ∪ LC w ( v ). Since LC ( v ) ∪ LC w ( v ) is a bounded set, there exists a benchmark vector z such that LC ( v ) ∪ LC w ( v ) ⊆ z ( v ). Combining with Theorem 15, we obtain that every allocation in LC ( v ) ∪ LC w ( v ) is an accessible allocation. Moreover, since PN ( v ) ∈ C+ ( v ) ⊆ LC ( v ), it follows that the prenucleolus as well as every allocation in the positive core is an accessible allocation. We still have to prove (c). Assume that v is an N-monotonic game. It is clear that the stand alone value vector v ∗ = ( v ({i }))i ∈ N is a benchmark vector. Consider the linear programming problem ( P 1 ) with z = v ∗ . By definition, there exists an optimal solution (t ∗ , ε∗ ) for the problem ( P 1 ) such that N ( v ) = v ∗ + t ∗ . This implies that N ( v ) is an accessible allocation by Theorem 12. 2 Theorem 16, together with Corollary 8, shows that the least core as well as the weak least core can be reached from any allocation via a sequence of successive blocks, and when the underlying game is unbalanced, such an accessibility property can be strengthened in the sense that each member of the (weak) least core can be reached from any allocation. Finally, we analyze the accessibility of the extended core, which is a non-empty core-extension introduced by Bejan and Gómez (2009), by establishing an analogue of Theorem 16. The notion of extended core is based on a method of financing a subsidy via individual taxes. Formally, the extended core of a game v is defined to be
EC ( v ) = x ∈ A ( v ): T ( v , x) T ( v , y ) for all y ∈ A ( v ) , where
N T ( v , x) := min t ( N ): t ∈ R+ and e (x, S ) t ( S ) for all S ∈ ζ
is the least subsidy needed to prevent coalitional deviations from x. In other words, the extended core is the set of those allocations that require a minimal subsidy. Clearly, the extended core is a non-empty extension of the core and the inclusion LC w ( v ) ⊆ EC ( v ) always holds. This means that the weak least core is actually a refinement of the extended core and the result of Theorem 16(b) could be subsumed as a special case of the following theorem.7 Theorem 17. Each allocation in the extended core of an unbalanced game is accessible. N Proof. Let v be an unbalanced game and let x∗ ∈ EC ( v ). Since C ( v ) = ∅, there exists t ∗ ∈ R+ with t ∗ ( N ) > 0 such that w ∗ := x∗ + t ∗ is an optimal solution for the problem ( P 4 ). Let y ∗ = ( y ∗ ( S )) S ∈ζ be an optimal solution for ( D 4 ). Then β := { S ∈ ζ : y ∗ ( S ) > 0} = ∅ is a balanced collection with positive weights { y ∗ ( S )} S ∈β . By the duality theorem, we obtain
y ∗ ( S ) v ( S ) = w ∗ ( N ) = x∗ ( N ) + t ∗ ( N ) > v ( N ).
(8)
S ∈β
Note that for each S ∈ ζ , by the complementary slackness condition,
w ∗ ( S ) − v ( S ) · y ∗ ( S ) = 0.
(9)
Thus, v ( S ) = w ∗ ( S ) for all S ∈ β . We consider two cases. Case I. t ∗ ( S ) > 0 for all S ∈ β . Then v ( S ) = w ∗ ( S ) = x∗ ( S ) + t ∗ ( S ) > x∗ ( S ) for all S ∈ β . Let z be a benchmark vector such that x ∈ z ( v ). By Lemma 11(c), x∗ is a z-accessible allocation. Case II. t ∗ ( S¯ ) = 0 for some S¯ ∈ β . Then v ( S¯ ) = w ∗ ( S¯ ) = x∗ ( S¯ ). Let x¯ = w ∗ − ε∗ e N , where ε∗ = t ∗ ( N )/n. Then x¯ ∈ LC w ( v ) ¯ By Theorem 16(b), x¯ is an accessible allocation and so is x∗ . 2 and x∗ dominates x¯ via S. 7
In addition, Bejan and Gómez (2009) define various refinements of the extended core using the concept of a tax problem.
Y.-Y. Yang / Games and Economic Behavior 74 (2012) 687–698
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4. Concluding remarks This paper contributes to the literature on the accessibility of cooperative solution concepts. We identify a sufficient and necessary condition for existence of an accessible allocation, namely, a game possesses an accessible allocation if and only if the core is a singleton or an empty set, and show that the collection of accessible allocations, if non-empty, contains a number of core-extensions. In this concluding section, we briefly discuss the implications of our main results and some further research directions. We first note that, whenever a solution concept contains the core and intersects with either the least core or the extended core, the combination of Corollary 8, Theorems 16 and 17 shows that the solution must be an accessible set. For example, Einy et al. (1999) analyze the relation between the least core and the Mas-Colell bargaining set (Mas-Colell, 1989) and show that the former is always contained in the latter. Thus, a direct application of our results guarantees the accessibility of the Mas-Colell bargaining set. Another issue is concerned with the collection of accessible allocations. Our analysis on its relation to various coreextensions sheds some light on the structure of accessible allocations. However, we cannot fully characterize the region of accessible allocations and leave the question for further works. Appendix A. Proof of Lemma 11 (a) Since a benchmark vector must be feasible, we obtain z( N ) v ( N ). Suppose, to the contrary, that z( N ) = v ( N ). For each coalition S ∈ ζ , by definition, z( S ) + v ( N \ S ) v ( N ), or equivalently, v ( N \ S ) z( N \ S ). Thus, z ∈ C ( v ). This contradicts to the fact that v is an unbalanced game. (b) (⇒) Assume that x is a z-accessible allocation. It suffices to show that x lies in z ( v ). Indeed, since the singleton {x} is a z-accessible set, by Corollary 9(b), we obtain {x} ∩ z ( v ) = ∅. This implies x ∈ z ( v ). (⇐) Since indirect dominance is transitive, by Corollary 9(a), we immediately obtain the desired result. (c) Let y 0 ∈ z ( v ) be an arbitrary allocation. By (b), it suffices to show that x indirectly z-dominates y 0 . We first note that βx is non-empty, otherwise x( S ) = z( S ) for all S ∈ β and hence
v ( N ) = x( N ) =
λ S x( S ) =
S ∈β
λ S z( S ) = z( N ).
S ∈β
By (a), this is impossible. Moreover, it follows that
λS v (S ) +
S ∈βx
λ S z( S ) >
S ∈β\βx
λ S x( S ) = x( N ) = y 0 ( N ) =
S ∈β
λS y0 ( S ) +
S ∈βx
λ S y 0 ( S ).
S ∈β\βx
Since y 0 z, there exists a coalition S 1 ∈ βx such that v ( S 1 ) > y 0 ( S 1 ). We define the allocation y 1 by
y 1i
=
where t 1 =
v ( S 1 )− y 0 ( S 1 ) , |S1| i i (1 − t 1 )x + t 1 z ,
y 0i +
v ( S 1 )−x( S 1 ) . x( N \ S 1 )− z( N \ S 1 )
if i ∈ S 1 , if i ∈ N \ S 1 ,
Since z is a benchmark vector and S 1 ∈ βx , it follows that
x( N \ S 1 ) − z( N \ S 1 ) = v ( N ) − x( S 1 ) − z( N \ S 1 ) v ( S 1 ) − x( S 1 ) > 0 N\S
and hence 1 t 1 > 0. Clearly, y 1 z-dominates y 0 via S 1 , y 1 ( S 1 ) = v ( S 1 ) > x( S 1 ), y 1 ∈ z ( v ) and x N \ S 1 y 1 1 . We now recursively construct the coalition S r +1 ∈ βx and the allocation y r +1 ∈ z ( v ) for r 1 assuming that the sequence of allocations { y i }ri=1 ⊆ z ( v ) and the sequence of coalitions { S 1 , S 2 , . . . , S r } ⊆ βx have already been constructed to satisfy (i) y j z-dominates y j −1 via S j for j = 1, . . . , r; r (ii) j =1 S j = ∅, y r ( S r ) = v ( S r ) > x( S r ); and r r i (iii) x y ri for all i ∈ j =1 ( N \ S j ) = N \( j =1 S j ). Since y r z, y r ( S r ) > x( S r ) and
S ∈βx
λ S yr ( S ) +
S ∈β\βx
λ S yr ( S ) = yr ( N ) = x( N ) =
S ∈βx
λ S x( S ) +
λ S z( S ),
S ∈β\βx
there exists a coalition S r +1 ∈ βx such that v ( S r +1 ) > x( S r +1 ) > y r ( S r +1 ). r +1 r r Note that j =1 S j . Indeed, if j =1 S j is a subset of S r +1 , by (iii), we obtain j =1 S j must be a proper subset of x( N \ S r +1 ) y r ( N \ S r +1 ) and hence
v ( N ) = x( S r +1 ) + x( N \ S r +1 ) > y r ( S r +1 ) + y r ( N \ S r +1 ) = v ( N ).
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Y.-Y. Yang / Games and Economic Behavior 74 (2012) 687–698
This is impossible. We consider two cases.
r
r +1
S
S r +1 y r r+1 by (iii). Let y r +1 = x. Since v ( S r +1 ) > x( S r +1 ) > Case I. j =1 S j and hence x j =1 S j = ∅. Then S r +1 ⊆ N \ y r ( S r +1 ) and x z, it follows that x z-dominates y r via S r +1 . r +1 Case II. j =1 S j = ∅. Let y r +1 be the allocation defined by
y ri +1
=
⎧ ⎪ yi + ⎪ ⎨ r
v ( S r +1 )− y r ( S r +1 )
|
r +1
j =1 S j |
,
if i ∈
r +1
j =1
S j,
r y ri , if i ∈ S r +1 \ j =1 S j , ⎪ ⎪ ⎩ (1 − tr +1 )xi + tr +1 zi , if i ∈ N \ S r +1 , v(S
)−x( S
)
where 1 t r +1 = x( N \ S r +1)−z( Nr\+S1 ) > 0. It is easily checked that y r +1 z-dominates y r via S r +1 , y r +1 ( S r +1 ) = v ( S r +1 ) > r +1 r +1 r +1 r +1 x( S r +1 ), y r +1 ∈ z ( v ) and xi y ri +1 for all i ∈ j =1 ( N \ S j ) = N \( j =1 S j ). Since | S 1 | n − 1 and
r ∗
r +1
r ∗ +1
j =1
S j is a proper subset of
r
j =1
S j for each r, there exists a positive integer r ∗ n − 1 such
∗ that j =1 S j = ∅ and j =1 S j = ∅. Consequently, we obtain a sequence of allocations { y 0 , y 1 , . . . , y r +1 = x} ⊆ z ( v ) and a sequence of coalitions { S 1 , S 2 , . . . , S r ∗ +1 } ⊆ βz such that y j z-dominates y j −1 via S j for j = 1, . . . , r ∗ + 1. This completes the proof.
References Béal, S., Rémila, E., Solal, P., 2010. On the number of blocks required to access the core. MPRA Paper 26578, University Library of Munich, Germany. Bejan, C., Gómez, J.C., 2009. Core extensions for non-balanced TU-games. Int. J. Game Theory 38, 3–16. Bernheim, B.D., Whinston, M.D., 1986. Menu auctions, resource allocation, and economic influence. Quart. J. Econ. 101, 1–31. Bondareva, O.N., 1963. Some applications of the methods of linear programming to the theory of cooperative games. Problemy Kibernet. 10, 119–139 (in Russian). Diamantoudi, E., Miyagawa, E., Xue, L., 2004. Random paths to stability in the roommate problem. Games Econ. Behav. 48, 18–28. Einy, E., Holzman, R., Monderer, D., 1999. On the least core and the Mas-Colell bargaining set. Games Econ. Behav. 28, 181–188. Kaneko, M., Wooders, M.H., 1982. Cores of partitioning games. Math. Soc. Sci. 3, 313–327. Klaus, B., Klijn, F., 2007. Paths to stability for matching markets with couples. Games Econ. Behav. 58, 154–171. Kóczy, L.Á., 2006. The core can be accessed with a bounded number of blocks. J. Math. Econ. 43, 56–64. Laussel, D., Le Breton, M., 2001. Conflict and cooperation; the structure of equilibrium payoffs in common agency games. J. Econ. Theory 100, 93–128. Maschler, M., Peleg, B., Shapley, L.S., 1979. Geometric properties of the kernel, nucleolus, and related solution concepts. Math. Oper. Res. 4, 303–338. Mas-Colell, A., 1989. An equivalence theorem for a bargaining set. J. Math. Econ. 18, 129–139. Orshan, G., Sudhölter, P., 2001. The positive core of a cooperative game. Discussion Paper 268, Center for Rationality and Interactive Decision Theory, Hebrew University of Jerusalem. Roth, A.E., Vande Vate, J.H., 1990. Random paths to stability in two-sided matching. Econometrica 58, 1475–1480. Schmeidler, D., 1969. The nucleolus of a characteristic function game. SIAM J. Appl. Math. 17, 1163–1170. Sengupta, A., Sengupta, K., 1996. A property of the core. Games Econ. Behav. 12, 266–273. Shapley, L.S., 1967. On balanced set and cores. Naval Res. Logist. Quart. 14, 453–460. Shapley, L.S., Shubik, M., 1966. Quasi-cores in a monetary economy with nonconvex preferences. Econometrica 34, 805–827. Wooders, M.H., 1978. Equilibria, the core, and jurisdiction structures in economies with a local public good. J. Econ. Theory 18, 328–348. Yang, Y.-Y., 2010. On the accessibility of the core. Games Econ. Behav. 69, 194–199.
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Games and Economic Behavior www.elsevier.com/locate/geb
On the accessibility of core-extensions ✩ Yi-You Yang Department of Applied Mathematics, Aletheia University, New Taipei City 251, Taiwan
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 1 February 2011 Available online 30 August 2011
Sengupta and Sengupta (1996) study the accessibility of the core of a TU game and show that the core, if non-empty, can be reached from any non-core allocation via a finite sequence of successive blocks. This paper complements the result by showing that when the core is empty, a number of non-empty core-extensions, including the least core and the weak least core (Maschler et al., 1979), the positive core (Orshan and Sudhölter, 2001) and the extended core (Bejan and Gómez, 2009), are accessible in a strong sense, namely each allocation in each of the foregoing core-extensions can be reached from any allocation through a finite sequence of successive blocks. © 2011 Elsevier Inc. All rights reserved.
JEL classification: C71 C73 Keywords: Accessibility Core Core-extensions
1. Introduction The core is the most widely applied cooperative game theory solution concept in economics. A core allocation of a coalitional game with transferable utility (TU game) guarantees that each coalition of players obtains at least what it could gain on its own. This means that once players receive a core allocation, no coalition has an incentive to deviate from it. On the other hand, in case a non-core allocation x is proposed as a solution for the game, a coalition S whose members can gain by forming S may threaten to split off from the grand coalition and thereby suggest an alternative allocation y according to its myopic preferences. Again, if y stands outside the core, another blocking coalition could replace y with a third allocation, and so forth. A fundamental question concerning this dynamic process is: Which allocations may be reached via such a process of successive transitions? Sengupta and Sengupta (1996) address the issue for the class of games with non-empty cores. They adopt the notion of weak dominance1 to formulate the possible transitions among allocations and show that the core, if non-empty, is accessible in the sense that it can be reached from any non-core allocation through a sequence of successive blocks.2 Kóczy (2006) focuses on the time required to reach the core and shows that the accessibility of the core can be met in a bounded number of blocks. In addition, Yang (2010) and Béal et al. (2010), respectively, give alternative upper bounds for the number of blocks needed to access the core. The former is described as the number of active coalitions,3 and the latter is equal to the number of pairs of players, i.e., n(n − 1)/2, where n is the cardinality of the player set.
✩ The author is very grateful to two anonymous referees for their perceptive comments of the paper. Support by National Science Council of Republic of China under grant NSC 100-2410-H-156-008 is gratefully acknowledged. E-mail address: [email protected]. 1 A weak dominance is a dominance relation in which not every member of the blocking coalition is strictly better off. 2 Such an accessibility property of the core may not hold with strict dominance. 3 A coalition different from the grand coalition is said to be active if it can guarantee each of its members a payoff which is strictly individually rational.
0899-8256/$ – see front matter doi:10.1016/j.geb.2011.08.007
© 2011
Elsevier Inc. All rights reserved.
688
Y.-Y. Yang / Games and Economic Behavior 74 (2012) 687–698
Nevertheless, numerous games derived from natural economic situations possess empty cores.4 This drawback leads to the development of relaxations of the core such as the least core and the weak least core (Maschler et al., 1979), the positive core (Orshan and Sudhölter, 2001), the extended core and its various refinements (Bejan and Gómez, 2009), each of which is always non-empty yet coincides with the core whenever the core is non-empty. A natural question is whether these core-extensions could inherit the accessibility of the core. In the present paper, we first attempt to extend Sengupta and Sengupta’s analysis to incorporate games with empty cores with the aid of an alternative notion of dominance relation, z-dominance, in which a payoff vector z = ( z i ) is adopted as a reference point which guarantees that each player i outside the blocking coalition obtains at least z i . Then we study conditions for an allocation to be an accessible allocation, i.e., an allocation that can be reached from any allocation via a sequence of successive transitions, and prove that a game possesses an accessible allocation if and only if the core is a singleton or an empty set. Finally, we apply the analysis to address the accessibility of the foregoing core-extensions by showing that when the core is empty, each allocation in each of these core-extensions is an accessible allocation. One feature of our approach is that players are assumed to be myopic in the sense that the blocking coalition ignores any possible further deviations, which may render some of its members lower payoffs than the former allocation does. Based on a similar myopia assumption, Roth and Vande Vate (1990) address the accessibility of the core for marriage markets and show that a decentralized process of successive blocks can terminate in the core. Their result is extended by Diamantoudi et al. (2004) and Klaus and Klijn (2007) to roommate markets and couples markets, respectively. The rest of the paper is organized as follows. Section 2 gives fundamental definitions and notations. Section 3 introduces a process of successive transitions and gives the main results. Section 4 concludes and a proof is presented in Appendix A. 2. Preliminaries Let N = {1, 2, . . . , n} be a finite set of players. A coalition is a non-empty subset of N. Let ζ denote the collection of coalitions except N, i.e., ζ := 2 N \{ N , ∅}. A coalitional game with transferable utility, or simply a game, is described by a v which assigns to each coalition S a real value v ( S ). A game v is said to be N-monotonic if for all coalitional function coalitions S, v ( S ) + i ∈ N \ S v ({i }) v ( N ).
i i A payoff vector x = (x1 , . . . , xn ) ∈ R N is feasible if i ∈ N x v ( N ). An allocation is a payoff vector x such that i∈N x = i v ( N ). Let A ( v ) denote the collection of allocations of a game v. An imputation is an allocation x such that x v ({i }) for all i ∈ N. We denote by I ( v ) the set of imputations of v. N i i For any coalition S, let x( S ) be a shorthand for For iany pair ofS vectors x, y ∈ R , we write x y if x y for allS i ∈ N. x and let x denote the projection of x on S. We also write x y S if xi y i for all i ∈ S. i∈ S Given an allocation x ∈ A ( v ), the excess of a coalition S is defined to be
e (x, S ) = v ( S ) − x( S ), which can be interpreted as a measure of the dissatisfaction of coalition S with respect to the allocation x. The core of a game v is defined to be
C ( v ) = x ∈ A ( v ) : e (x, S ) 0 for all S ∈ ζ , which collects allocations that cannot be improved upon by any coalition. Given a coalition S, we denote by e S the vector in R N whose i-th coordinate is 1 if i ∈ S and 0 otherwise. A collection of coalitions β ⊆ ζ is said to be balanced if there exists a system of positive weights {λ S } S ∈β such that S ∈β λ S e S = e N . A game v is called a balanced game if for any balanced collection β with weights {λ S } S ∈β ,
λ S v ( S ) v ( N ).
S ∈β
Bondareva (1963) and Shapley (1967) independently prove that a game is balanced if and only if it has a non-empty core. Let ε be a non-negative number. The strong ε -core of a game v is the set of all those allocations x ∈ A ( v ) such that e (x, S ) ε for all S ∈ ζ . The weak ε -core of a game v collects allocations x ∈ A ( v ) satisfying e (x, S ) | S |ε for all S ∈ ζ , where | S | denotes the cardinal number of the coalition S. The strong ε -core (or weak ε -core), introduced by Shapley and Shubik (1966), can be interpreted as the set of allocations that cannot be improved upon by any coalition S if forming S entails a cost of ε (or | S |ε ). Clearly, the strong ε -core and the weak ε -core are both non-empty when ε is large enough. The least core of a game v (Maschler et al., 1979), denoted by LC ( v ), is defined to be the intersection of all non-empty strong ε -cores. Similarly, the weak least core of v, denoted by LC w ( v ), is the intersection of all non-empty weak ε -cores. It is clear that LC ( v ) coincides with the strong ε0 -core with
ε0 = max 0, min max e(x, S ) . x∈ A ( v ) S ∈ζ
4
See, for example, Shapley and Shubik (1966), Wooders (1978), and Kaneko and Wooders (1982).
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For any allocation x ∈ A ( v ), the excess vector θ( v , x) is defined as the vector that rearranges the coordinates of (e (x, S )) S ∈ζ in non-increasing order. The prenucleolus of a game v (Schmeidler, 1969) is the unique allocation PN ( v ) ∈ A ( v ) that lexicographically minimizes the excess vector θ( v , PN ( v )), i.e., θ( v , PN ( v )) lex θ( v , y ) for all y ∈ A ( v ), where lex denotes the lexicographic order. Similarly, the nucleolus of a game v is the unique imputation N ( v ) ∈ I ( v ) satisfying θ( v , N ( v )) lex θ( v , y ) for all y ∈ I ( v ). Note that PN ( v ) ∈ LC ( v ) for all game v, and PN ( v ) = N ( v ) if C ( v ) = ∅. For any allocation x ∈ A ( v ), let θ+ ( v , x) be the vector that rearranges the coordinates of (max{0, e (x, S )}) S ∈ζ in nonincreasing order. The positive core of v (Orshan and Sudhölter, 2001) is defined to be
C+ ( v ) = x ∈ A ( v ): ∀ y ∈ A ( v ), θ+ ( v , x) lex θ+ ( v , y ) , which is an intermediate concept between the prenucleolus and the least core, i.e., PN ( v ) ∈ C+ ( v ) ⊆ LC ( v ). Moreover, it is not difficult to see that all these core-like solution concepts, including the least core, the weak least core and the positive core, are non-empty extensions of the core. Namely, each of them is never empty and returns the core in case the core is non-empty. 3. A process of successive transitions Consider an allocation y that is proposed as a solution for the game v. If some coalition S can improve upon y, in order to obtain a higher total payoff, it may threaten to leave the grand coalition and thereby propose an alternative allocation x such that
v ( S ) x( S ) > y ( S )
and
xS y S .
(1)
In that case, we refer to S as a blocking coalition. Sengupta and Sengupta (1996) and Kóczy (2006) formulate such a transition from y to x as a notion of dominance relation. Definition 1. Let x and y be allocations of a game v. We say that x dominates y via a coalition S, denoted by x S y, if condition (1) holds. Nevertheless, it seems unrealistic that the blocking coalition S enjoys such a strong power that it can arbitrarily distribute the payoffs of the remaining players without any restriction. To make the transition more practical, Yang (2010) suggests to employ a pre-decided payoff vector z as a benchmark and argues that an “acceptable” counterproposal x should satisfy the remaining players in the sense that xi zi for all i ∈ N \ S. Combining with feasibility and the fact that the blocking coalition S can raise its aggregate payoff up to v ( S ) on its own, a proper benchmark vector z should be chosen such that v ( S ) + z( N \ S ) v ( N ) for all coalitions S. Following this logic, we formulate a restrictive notion of dominance as follows. Definition 2. A feasible payoff vector z is called a benchmark vector5 of a game v if
v ( S ) + z( N \ S ) v ( N ) for all S ∈ ζ.
(2)
Let x and y be allocations of a game v. We say that x dominates y via a coalition S with respect to a benchmark vector z (x z-dominates y via S for short) if x S y and x N \ S z N \ S . Remark 3. The notion of benchmark vector is closely related to the core and N-monotonic games. The following facts are easily checked. (a) An allocation z of a game v is a benchmark vector if and only if z is a member of the core.6 (b) The stand alone value vector v ∗ := ( v ({1}), . . . , v ({n})) ∈ R N of a game v is a benchmark vector if and only if v is N-monotonic. We say that an allocation x indirectly dominates another allocation y if there exist a sequence of allocations { y j }lj =0 and
a sequence of blocking coalitions { S j }lj =1 such that
x = y l S l y l −1 S l −1 · · · S 1 y 0 = y .
(3)
In addition, we say that x indirectly z-dominates y if z is a benchmark vector and for each j = 1, . . . , l, in the sequence (3), y j z-dominates y j −1 via S j . Such a sequence of blocks describes a possible trajectory of successive transitions. We are 5 Laussel and Le Breton (2001) analyze the Pareto efficient frontier of the set of benchmark vectors of a TU game to study the structure of equilibrium payoffs in a common agency game, which is introduced by Bernheim and Whinston (1986). 6 Sengupta and Sengupta (1996) and Béal et al. (2010) respectively give algorithms that pick a core allocation z as a reference point and generate sequences of successively z-dominating allocations that terminate in the core, although they do not formally introduce the notion of z-dominance.
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interested in which allocations the disequilibrium process may lead to. To address the issue, the notions of accessible sets and accessible allocations arise naturally. Definition 4. Let v be a game augmented with a benchmark vector z. A set of allocations ⊆ A ( v ) is called an acces/ , there exists an allocation x ∈ that indirectly dominates sible (respectively z-accessible) set if for any allocation y ∈ (respectively indirectly z-dominates) y. In particular, an allocation x ∈ A ( v ) is called an accessible (respectively z-accessible) allocation if the singleton {x} is an accessible (respectively z-accessible) set. 3.1. The core and z-accessible sets Sengupta and Sengupta (1996) focus on the class of balanced games and show that the core of a balanced game is an accessible set. The following theorem attempts to extend their analysis to incorporate unbalanced games. Theorem 5. Let v be a game with a benchmark vector z and let
z ( v ) := x ∈ A ( v ): x z . Then the union of C ( v ) and z ( v ) is a z-accessible set. Remark 6. The following observations are helpful to gain intuition behind Theorem 5. (a) (b) (c) (d)
The core is a subset of any z-accessible set. A sequence of successively z-dominating allocations terminates as soon as it enters the core. In case the core is empty, a sequence of successively z-dominating allocations never terminates. Once a sequence of successively z-dominating allocations enters z ( v ), from which it can never escape.
The proof of Theorem 5 requires the following lemma. N \T
Lemma 7. Let z be a benchmark vector of a game v. Let x0 be a non-core allocation such that x0 Then there exists a sequence of z-dominating allocations
z N \ T for some coalition T ∈ ζ .
xl S l xl−1 S l−1 · · · S 1 x0 N \T
such that xl ∈ C ( v ) or xl
z N \ T for some proper subset T of T .
/ C ( v ), there exists a coalition S 1 with v ( S 1 ) > x0 ( S 1 ). We consider two cases. Proof. Since x0 ∈ Case I. S 1 ∩ T = T . We define the allocation x1 by
x1i
=
x0i + zi +
v ( S 1 )−x0 ( S 1 ) , |S1| v ( N )− v ( S 1 )− z( N \ S 1 ) , |N \ S 1 |
if i ∈ S 1 , if i ∈ N \ S 1 . N \T
Let T = S 1 ∩ T , then x1 z-dominates x0 via S 1 , x1 z N \ T and hence the proof is done. Case II. S 1 ∩ T = T , i.e., T ⊆ S 1 . We define the allocation x1 by
⎧ i v ( S 1 )−x0 ( S 1 ) , if i ∈ T , ⎪ |T | ⎨ x0 + i i x1 = x , if i ∈ S 1 \ T , ⎪ ⎩ 0i i i z + t 1 (x0 − z ), if i ∈ N \ S 1 , v ( N )− z( N \ S )− v ( S )
N \T
where 0 t 1 = v ( N )−z( N \ S 1)−x ( S1 ) < 1. It is easily checked that x1 z-dominates x0 via S 1 , x1 ( S 1 ) = v ( S 1 ), and x1 z N \ T . 1 0 1 If x1 ∈ C ( v ), the proof is done. Otherwise, the proof will proceed by constructing the required z-dominating sequence inductively. Suppose that the sequence of z-dominating allocations xk S k xk−1 S k−1 · · · S 1 x0 has been constructed to satisfy (i) S j = S l for j = l; (ii) xk ( S j ) v ( S j ) and T ⊆ S j for j = 1, . . . , k; and N \T
(iii) xk
z N \ T and xk ∈ / C ( v ).
Then there exists a coalition S k+1 with v ( S k+1 ) > xk ( S k+1 ). Note that condition (ii) implies S k+1 = S j for j = 1, . . . , k. In case S k+1 ∩ T = T , we define the allocation xk+1 by
Y.-Y. Yang / Games and Economic Behavior 74 (2012) 687–698
xki +1 =
⎧ ⎨ xi + k ⎩z + i
v ( S k+1 )−xk ( S k+1 ) , | S k +1 |
691
if i ∈ S k+1 ,
v ( N )− v ( S k+1 )− z( N \ S k+1 ) , | N \ S k +1 |
if i ∈ N \ S k+1 . N \T
Let T = S k+1 ∩ T , then xk+1 z-dominates xk via S k+1 and xk+1 z N \ T and hence we obtain the desired result. In what follows, we assume S k+1 ∩ T = T , i.e., T ⊆ S k+1 , and define the allocation xk+1 by
⎧ v ( S k+1 )−xk ( S k+1 ) i ⎪ , if i ∈ T , ⎪ |T | ⎨ xk + i xk+1 = xi , if i ∈ S k+1 \ T , k ⎪ ⎪ ⎩ zi + t (xi − zi ), if i ∈ N \ S k+1 , k +1 k v ( N )− z( N \ S
(4)
N \T
where 0 tk+1 = v ( N )−z( N \ S k+1)−x ( Sk+1 ) < 1. Clearly, xk+1 z-dominates xk via S k+1 , xk+1 ( S k+1 ) = v ( S k+1 ) and xk+1 z N \ T . k+1 k k+1 We are going to show that xk+1 ( S j ) v ( S j ) for j = 1, . . . , k. )− v ( S
)
N \ S k+1
Let 1 j k. Since T ⊆ S k+1 , by condition (iii), we obtain that xk Thus, we have
N \ S k+1
z N \ S k+1 . This implies xk
N\S
xk+1k+1 by (4).
xk ( N \ S k+1 ) − xk+1 ( N \ S k+1 ) xk ( S j \ S k+1 ) − xk+1 ( S j \ S k+1 ).
(5)
Moreover, (4) also implies
xk+1 ( N \ S k+1 ) = v ( N ) − v ( S k+1 ).
(6)
The combination of (5) and (6) yields
xk+1 ( S j \ S k+1 ) xk ( S j \ S k+1 ) − v ( S k+1 ) + xk ( S k+1 ) and hence
xk+1 ( S j ) = xk+1 ( S j \ S k+1 ) + xk+1 ( S j ∩ S k+1 )\ T + xk+1 ( T )
xk ( S j \ S k+1 ) − v ( S k+1 ) + xk ( S k+1 ) + xk ( S j ∩ S k+1 )\ T + xk ( T ) + v ( S k+1 ) − xk ( S k+1 )
= xk ( S j \ S k+1 ) + xk ( S j ∩ S k+1 )\ T + xk ( T )
= xk ( S j ) v ( S j ). Note that to derive the above inequalities we use the facts that xk+1 (( S j ∩ S k+1 )\ T ) = xk (( S j ∩ S k+1 )\ T ), xk+1 ( T ) = xk ( T ) + v ( S k+1 ) − xk ( S k+1 ) and condition (ii). Since the number of those coalitions containing T is finite and the blocking coalitions at each step of the sequence are N \T
distinct, there exists some r such that the allocation xr constructed at the r-th step satisfies xr ∈ C ( v ) or xr where T = S r ∩ T is a proper subset of T . 2
z N \T ,
We are now ready to prove Theorem 5. Proof of Theorem 5. For any subset T of N, we denote
z ( v , T ) = x ∈ A ( v ): xN \T z N \T . Clearly, z ( v , ∅) = z ( v ). Let x0 be a non-core allocation. Then there exists a coalition T 1 such that v ( T 1 ) > x0 ( T 1 ). Since z is a benchmark vector, we obtain v ( T 1 ) + z( N \ T 1 ) v ( N ). Then there exists an allocation x1 ∈ z ( v , T 1 ) that z-dominates x0 via T 1 . In case x1 ∈ C ( v ), the proof is done. Otherwise, by Lemma 7, there exist a proper subset T 2 of T 1 and an allocation x2 ∈ C ( v ) ∪ z ( v , T 2 ) that indirectly z-dominates x1 . If x2 ∈ C ( v ) or T 2 = ∅, the proof is done. Otherwise, we may apply Lemma 7 to find a proper subset T 3 of T 2 and an allocation x3 ∈ C ( v ) ∪ z ( v , T 3 ) that indirectly z-dominates x2 . Since T 1 is a finite set and the sequence of sets
T1 T2 T3 · · · is strictly decreasing, we obtain the desired result by using Lemma 7 repeatedly.
2
The next result, originally proved by Sengupta and Sengupta (1996), is an immediate consequence of Theorem 5. Corollary 8. Let z be a core allocation of a game v. Then z is a benchmark vector and the core is z-accessible.
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Proof. Since z ∈ C ( v ), we obtain that v ( S ) + z( N \ S ) = v ( S ) + v ( N ) − z( S ) v ( N ) for all coalitions S and hence z is a benchmark vector. Moreover, since z ( v ) = { z} ⊆ C ( v ), by Theorem 5, we obtain that C ( v ) = C ( v )∪z ( v ) is z-accessible. 2 The following result shows that for any benchmark vector z, the set z ( v ) plays a central role among z-accessible sets when the game v under consideration is unbalanced. Corollary 9. Let v be an unbalanced game with a benchmark vector z. (a) z ( v ) is a z-accessible set. (b) Let ⊆ A ( v ) be a z-accessible set, then ∩ z ( v ) is also z-accessible. Proof. (a) The core of an unbalanced game is empty and hence z ( v ) = C ( v ) ∪ z ( v ) is z-accessible by Theorem 5. (b) Let y be an allocation lying outside ∩ z ( v ). We are going to show that y is indirectly z-dominated by some allocation in ∩ z ( v ). Since z ( v ) is a z-accessible set, without loss of generality, we may assume that y ∈ z ( v )\. Note that, by definition, no member of z ( v ) can be z-dominated by an allocation outside z ( v ). Combining with the fact that is z-accessible, there exists x ∈ ∩ z ( v ) that indirectly z-dominates y. 2 3.2. z-Accessible allocations The purpose of this section is to study the existence of z-accessible (or accessible) allocations. In sight of Corollary 8, it is clear that when the core consists of a single allocation, the unique core allocation is the only accessible allocation. Theorem 10. Let x be an allocation of a balanced game v, then x is an accessible allocation if and only if C ( v ) = {x}. Proof. (⇒) By definition, the core is a subset of any accessible set. Assume that x ∈ A ( v ) is an accessible allocation. Then the singleton {x} is an accessible set and hence C ( v ) ⊆ {x}. Since v is balanced, C ( v ) = ∅. This implies C ( v ) = {x}. (⇐) Assume that the core of v consists of a unique core allocation x. By Corollary 8, x indirectly dominates any non-core allocation. 2 In what follows, we shall focus on unbalanced games. The next lemma establishes a method for finding z-accessible allocations. Lemma 11. Let v be an unbalanced game with a benchmark vector z. (a) z( N ) < v ( N ). (b) For any allocation x ∈ A ( v ), x is z-accessible if and only if x ∈ z ( v ) and x indirectly z-dominates all allocations y ∈ z ( v )\{x}. (c) Let β be a balanced collection with weights {λ S } S ∈β and let x ∈ z ( v ) be an allocation such that
x( S ) = z( S )
for all S ∈ β\βx ,
(7)
where βx := { S ∈ β : v ( S ) > x( S )}, then x is a z-accessible allocation. Proof. See Appendix A.
2
We are now ready to prove that each unbalanced game augmented with a benchmark vector z possesses z-accessible (or accessible) allocations. Theorem 12. Let v be an unbalanced game with a benchmark vector z. Let (t ∗ , ε∗ ) be an optimal solution for the linear programming problem
( P 1 ) min ε s.t. t ( S ) + ε v ( S ) − z( S ) for all S ∈ ζ, t ( N ) = v ( N ) − z( N ), N t ∈ R+ , ε 0. Then z + t ∗ ∈ z ( v ) is a z-accessible allocation. Proof. Since v is unbalanced, C ( v ) = ∅ and hence problem
ε∗ > 0. Let y ∗ = ( y ∗ ( S )) S ∈ζ ∪{N } be an optimal solution for ( P 1 )’s dual
Y.-Y. Yang / Games and Economic Behavior 74 (2012) 687–698
( D 1 ) max
S ∈ζ ∪{N }
s.t.
i ∈ S ∈ζ
693
y ( S ) v ( S ) − z( S )
y ( S ) + y ( N ) 0 for all i ∈ N ,
S ∈ζ
y ( S ) 1,
y ( N ) ∈ R , y ( S ) 0 for all S ∈ ζ. Let β = { S ∈ ζ : y ∗ ( S ) > 0} and let x∗ = z + t ∗ . It is easily checked that β = ∅ and for all S ∈ β , v ( S ) > x∗ ( S ). Indeed, in case β = ∅, i.e., y ∗ ( S ) = 0 for all S ∈ ζ , then y ∗ ( N ) 0 and hence
0 < ε∗ =
y ∗ ( S ) v ( S ) − z( S )
S ∈ζ ∪{ N }
= y ∗ ( N ) v ( N ) − z( N ) 0 by the duality theorem and Lemma 11(a). This is impossible. Moreover, by the complementary slackness condition, we see that for all S ∈ ζ ,
x∗ ( S ) + ε∗ − v ( S ) · y ∗ ( S ) = 0.
This implies that v ( S ) = x∗ ( S ) + ε∗ > x∗ ( S ) for all S ∈ β . y (S) Since y ∗ satisfies the constraints of ( D 1 ), β = ∅ implies y ∗ ( N ) < 0. Let λ S = − y∗ ( N ) > 0 for all S ∈ β and let ∗
α = { i }:
y∗ ( S ) + y∗ (N ) < 0 .
i ∈ S ∈ζ
We consider two cases. Case I. α = ∅. Then β is a balanced collection with positive weights {λ S } S ∈β . By Lemma 11(c), x∗ is a z-accessible allocation. Case II. α = ∅. By the complementary slackness condition, for all i ∈ N,
i ∈ S ∈ζ
y ∗ ( S ) + y ∗ ( N ) · t ∗i = 0.
This implies x∗ ({i }) = z({i }) for all {i } ∈ α . For each {i } ∈ α , let
δ{ i } =
1− 1−
i ∈ S ∈β
λS ,
if {i } ∈ / β,
i ∈ S ∈β λ S + λ{i } , if {i } ∈ β.
Obviously, β ∪ α is a balanced collection with positive weights {λ S } S ∈β\α ∪ {δ{i } }{i }∈α . Using Lemma 11(c) again, we obtain the desired result. 2 Since each allocation of an unbalanced game is dominated by infinitely many allocations, by Lemma 11(b) and Theorem 12, we see that for any unbalanced game v with a benchmark vector z, the collection of z-accessible allocations is an infinite subset of z ( v ). A natural question is to explore the relation between the set of all z-accessible allocations and z ( v ). To shed some light on the issue, the next result identifies a sufficient condition for the coincidence of these two sets. Theorem 13. Let v be an unbalanced game and let z∗ be a benchmark vector that maximizes the amount z∗ ( N ). Then the set of all z∗ -accessible allocations coincides with z∗ ( v ). Proof. By Lemma 11(a), z∗ ( N ) < v ( N ) and hence z∗ is an optimal solution for the linear programming problem
( P 2 ) max z( N ) s.t. z( S ) v ( N ) − v ( N \ S ) for all S ∈ ζ, z ∈ RN . Let y ∗ = ( y ∗ ( S )) S ∈ζ be an optimal solution for ( P 2 )’s dual problem
( D 2 ) min s.t.
S ∈ζ
v (N ) − v (N \ S ) · y( S )
i ∈ S ∈ζ
y ( S ) = 1 for all i ∈ N ,
y ( S ) 0 for all S ∈ ζ.
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Let β = { S ∈ ζ : y ∗ ( S ) > 0} and let θ = S ∈β y ∗ ( S ). Clearly, β is a balanced collection with positive weight { y ∗ ( S )} S ∈β and θ > 1. y∗ ( S ) Let β c = { N \ S: S ∈ β} and let λ N \ S = θ− for all N \ S ∈ β c . Since 1
λN \ S e N \ S =
S ∈β
y ∗ ( S ) S ∈β
=
θ −1
1
θ −1
eN\S =
1
θ −1
S ∈β
y ∗ ( S )e N −
y ∗ ( S )e S
S ∈β
(θ e N − e N ) = e N ,
β c is a balanced collection with weights {λ N \ S } N \ S ∈β c .
By the complementary slackness condition, we obtain that for all S ∈ ζ ,
z ∗ ( S ) − v ( N ) + v ( N \ S ) · y ∗ ( S ) = 0.
This implies that for all S ∈ β , z∗ ( S ) = v ( N ) − v ( N \ S ). N with For any x ∈ z∗ ( v ), we denote β(x) = { S ∈ β : x( S ) = z∗ ( S )}. Let x∗ ∈ z∗ ( v ) and let t ∗ = x∗ − z∗ . Then t ∗ ∈ R+ t ∗ ( N ) > 0. We will apply Lemma 11(c) to show that x∗ is z∗ -accessible. We consider two cases. Case I. β(x∗ ) = ∅. Then for all S ∈ β , x∗ ( S ) > z∗ ( S ) = v ( N ) − v ( N \ S ) and hence v ( N \ S ) > x∗ ( N \ S ). This implies that x∗ is a z-accessible allocation by Lemma 11(c). Case II. β(x∗ ) = ∅. We are going to show that x∗ is z-accessible by constructing a z-dominating sequence. Let x0 = x∗ and let S 1 ∈ β(x∗ ), then t ∗ ( S 1 ) = 0 and x∗ ( S 1 ) = z∗ ( S 1 ) = v ( N ) − v ( N \ S 1 ), or, equivalently, t ∗ ( N \ S 1 ) = t ∗ ( N ) and v ( N \ S 1 ) = x∗ ( N \ S 1 ). We define the allocation x1 ∈ z∗ ( v ) by
x1i
=
z∗i + 12 t ∗i , z∗i + 2|1S | t ∗ ( N ), 1
if i ∈ N \ S 1 , if i ∈ S 1 .
Clearly, x0 z∗ -dominates x1 via N \ S 1 and β(x1 ) ⊆ β(x0 )\{ S 1 } β(x0 ). Inductively, we assume that the z∗ -dominating sequence
x∗ = x0 N \ S 1 x1 N \ S 2 · · · N \ S l xl has been constructed such that for j = 1, . . . , l, β(x j ) ⊆ β(x j −1 )\{ S j } β(x j −1 ). If β(xl ) = ∅, by Case I, xl is z∗ -accessible and the proof is done. If β(xl ) = ∅, choose S l+1 ∈ β(xl ) and define the allocation xl+1 ∈ z∗ ( v ) by
xli+1
=
z∗i + 12 tli , z∗ + i
1 t ( N ), 2 | S l +1 | l
if i ∈ N \ S l+1 , if i ∈ S l+1 ,
N where tl = xl − z∗ ∈ R+ . It is easily checked that xl z∗ -dominates xl+1 via N \ S l+1 and β(xl+1 ) ⊆ β(xl )\{ S l+1 } β(xl ). Since the number of coalitions in β(x∗ ) is finite, we obtain that x∗ indirectly z∗ -dominates some allocation xr ∈ z∗ ( v ) with β(xr ) = ∅. Thus, we obtain the desired result. 2
We close this section with an integrated analysis on accessible sets. Corollary 14. Let v be a game with a benchmark vector z and let ⊆ A ( v ). (a) If v is balanced, then is an accessible set if and only if C ( v ) ⊆ . (b) If v is unbalanced, then is an accessible (respectively z-accessible) set if and only if contains at least an accessible (respectively z-accessible) allocation. (c) If v is unbalanced and z is a benchmark vector that maximizes z( N ), then is a z-accessible set if and only if ∩ z ( v ) = ∅. Proof. The combination of Corollary 8 and the fact that the core is a subset of any accessible set yields (a). (b) By Theorem 12, the collection of accessible (or z-accessible) allocations of an unbalanced game is non-empty. Clearly, is an accessible (respectively z-accessible) set if contains an accessible (respectively z-accessible) allocation. Conversely, let x be an accessible (respectively z-accessible) allocation and assume that is an accessible (respectively z-accessible) set. In case x ∈ , the proof of (b) is done. If x ∈ / , then there exists y ∈ that indirectly dominates (respectively zdominates) x. Clearly, y is accessible (respectively z-accessible) as well. Finally, putting (b) and Theorem 13 together yields (c). 2
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3.3. Core-extensions In this section, we study the accessibility of a number of core-extensions, including the least core, the weak least core, the positive core, and the extended core. We first observe that for any unbalanced game v with a benchmark vector z, the collection of those z-accessible allocations contained in the least core (respectively the weak least core) exactly coincides with the intersection of the least core (respectively the weak least core) and z ( v ). Theorem 15. Let v be an unbalanced game with a benchmark vector z. For any allocation x ∈ LC ( v ) ∪ LC w ( v ), x is a z-accessible allocation if and only if x ∈ z ( v ). Proof. The “only if” part of the proof follows immediately from Lemma 11(b). To obtain the “if” part, we consider two cases. Case I. Let x∗ ∈ LC ( v ) ∩ z ( v ). Consider the linear programming problem
( P 3 ) min ε x( S ) + ε v ( S ) for all S ∈ ζ,
s.t.
x( N ) = v ( N ), x ∈ R N , ε 0, with optimum value ε∗ . Since x∗ ∈ LC ( v ), by definition, (x∗ , ε∗ ) is an optimal solution for ( P 3 ). Moreover, since v is unbalanced, C ( v ) = ∅ and hence ε∗ > 0. Let y ∗ = ( y ∗ ( S )) S ∈ζ ∪{ N } be an optimal solution for ( P 3 )’s dual problem
( D 3 ) max s.t.
y( S )v ( S )
S ∈ζ ∪{N } i ∈ S ∈ζ
y ( S ) + y ( N ) = 0 for all i ∈ N ,
S ∈ζ
y ( S ) 1,
y ( N ) ∈ R, y ( S ) 0 for all S ∈ ζ, and let β = { S ∈ ζ : y ∗ ( S ) > 0}. In case β = ∅, i.e., y ∗ ( S ) = 0 for all S ∈ ζ , then y ∗ ( N ) = 0 and hence
0 < ε∗ =
y∗ ( S )v ( S ) = 0
S ∈ζ ∪{ N }
by the duality theorem of linear programming. This is impossible. Thus we obtain β = ∅ and y ∗ ( N ) < 0. Moreover, we note y (S) that β is a balanced collection with positive weights {λ S } S ∈β , where λ S = − y∗ ( N ) for all S ∈ β . ∗ In sight of the complementary slackness condition, we obtain that for all S ∈ ζ,
x∗ ( S ) + ε∗ − v ( S ) · y ∗ ( S ) = 0.
This implies that v ( S ) = x∗ ( S ) + ε∗ > x∗ ( S ) for all S ∈ β . By Lemma 11(c), x∗ is a z-accessible allocation. Case II. Let x∗ ∈ LC w ( v ) ∩ z ( v ). Consider the linear programming problem
( P 4 ) min ε s.t.
(x + εe N )( S ) v ( S ) for all S ∈ ζ, x( N ) = v ( N ), x ∈ RN ,
ε 0,
with optimum value ε∗ . Since C ( v ) = ∅, ε∗ > 0. Note that w ∗ := x∗ + ε∗ e N is an optimal solution for the problem
( P 4 ) min w ( N ) s.t.
w ( S ) v ( S ) for all S ∈ ζ, w ∈ RN .
Let y ∗ = ( y ∗ ( S )) S ∈ζ be an optimal solution for ( P 4 )’s dual problem
( D 4 ) max s.t.
y( S )v ( S )
S ∈ζ i ∈ S ∈ζ
y ( S ) = 1 for all i ∈ N ,
y( S ) 0
for all S ∈ ζ.
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Y.-Y. Yang / Games and Economic Behavior 74 (2012) 687–698
Note that β := { S ∈ ζ : y ∗ ( S ) > 0} = ∅ is a balanced collection with positive weights { y ∗ ( S )} S ∈β . Moreover, by the complementary slackness condition, we have that for all S ∈ ζ ,
w ∗ ( S ) − v ( S ) · y ∗ ( S ) = 0.
This implies that for all S ∈ β , v ( S ) = w ∗ ( S ) = x∗ ( S ) + | S |ε∗ > x∗ ( S ). Using Lemma 11(c) again, we obtain the desired result. 2 Next, putting Theorems 12 and 15 together yields the following results. Theorem 16. Let v be an unbalanced game. (a) Each allocation in the least core (or the positive core) is accessible. In particular, the prenucleolus is an accessible allocation. (b) Each allocation in the weak least core is accessible. (c) If v is N-monotonic, then the nucleolus is an accessible allocation. Proof. Let x be an allocation in LC ( v ) ∪ LC w ( v ). Since LC ( v ) ∪ LC w ( v ) is a bounded set, there exists a benchmark vector z such that LC ( v ) ∪ LC w ( v ) ⊆ z ( v ). Combining with Theorem 15, we obtain that every allocation in LC ( v ) ∪ LC w ( v ) is an accessible allocation. Moreover, since PN ( v ) ∈ C+ ( v ) ⊆ LC ( v ), it follows that the prenucleolus as well as every allocation in the positive core is an accessible allocation. We still have to prove (c). Assume that v is an N-monotonic game. It is clear that the stand alone value vector v ∗ = ( v ({i }))i ∈ N is a benchmark vector. Consider the linear programming problem ( P 1 ) with z = v ∗ . By definition, there exists an optimal solution (t ∗ , ε∗ ) for the problem ( P 1 ) such that N ( v ) = v ∗ + t ∗ . This implies that N ( v ) is an accessible allocation by Theorem 12. 2 Theorem 16, together with Corollary 8, shows that the least core as well as the weak least core can be reached from any allocation via a sequence of successive blocks, and when the underlying game is unbalanced, such an accessibility property can be strengthened in the sense that each member of the (weak) least core can be reached from any allocation. Finally, we analyze the accessibility of the extended core, which is a non-empty core-extension introduced by Bejan and Gómez (2009), by establishing an analogue of Theorem 16. The notion of extended core is based on a method of financing a subsidy via individual taxes. Formally, the extended core of a game v is defined to be
EC ( v ) = x ∈ A ( v ): T ( v , x) T ( v , y ) for all y ∈ A ( v ) , where
N T ( v , x) := min t ( N ): t ∈ R+ and e (x, S ) t ( S ) for all S ∈ ζ
is the least subsidy needed to prevent coalitional deviations from x. In other words, the extended core is the set of those allocations that require a minimal subsidy. Clearly, the extended core is a non-empty extension of the core and the inclusion LC w ( v ) ⊆ EC ( v ) always holds. This means that the weak least core is actually a refinement of the extended core and the result of Theorem 16(b) could be subsumed as a special case of the following theorem.7 Theorem 17. Each allocation in the extended core of an unbalanced game is accessible. N Proof. Let v be an unbalanced game and let x∗ ∈ EC ( v ). Since C ( v ) = ∅, there exists t ∗ ∈ R+ with t ∗ ( N ) > 0 such that w ∗ := x∗ + t ∗ is an optimal solution for the problem ( P 4 ). Let y ∗ = ( y ∗ ( S )) S ∈ζ be an optimal solution for ( D 4 ). Then β := { S ∈ ζ : y ∗ ( S ) > 0} = ∅ is a balanced collection with positive weights { y ∗ ( S )} S ∈β . By the duality theorem, we obtain
y ∗ ( S ) v ( S ) = w ∗ ( N ) = x∗ ( N ) + t ∗ ( N ) > v ( N ).
(8)
S ∈β
Note that for each S ∈ ζ , by the complementary slackness condition,
w ∗ ( S ) − v ( S ) · y ∗ ( S ) = 0.
(9)
Thus, v ( S ) = w ∗ ( S ) for all S ∈ β . We consider two cases. Case I. t ∗ ( S ) > 0 for all S ∈ β . Then v ( S ) = w ∗ ( S ) = x∗ ( S ) + t ∗ ( S ) > x∗ ( S ) for all S ∈ β . Let z be a benchmark vector such that x ∈ z ( v ). By Lemma 11(c), x∗ is a z-accessible allocation. Case II. t ∗ ( S¯ ) = 0 for some S¯ ∈ β . Then v ( S¯ ) = w ∗ ( S¯ ) = x∗ ( S¯ ). Let x¯ = w ∗ − ε∗ e N , where ε∗ = t ∗ ( N )/n. Then x¯ ∈ LC w ( v ) ¯ By Theorem 16(b), x¯ is an accessible allocation and so is x∗ . 2 and x∗ dominates x¯ via S. 7
In addition, Bejan and Gómez (2009) define various refinements of the extended core using the concept of a tax problem.
Y.-Y. Yang / Games and Economic Behavior 74 (2012) 687–698
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4. Concluding remarks This paper contributes to the literature on the accessibility of cooperative solution concepts. We identify a sufficient and necessary condition for existence of an accessible allocation, namely, a game possesses an accessible allocation if and only if the core is a singleton or an empty set, and show that the collection of accessible allocations, if non-empty, contains a number of core-extensions. In this concluding section, we briefly discuss the implications of our main results and some further research directions. We first note that, whenever a solution concept contains the core and intersects with either the least core or the extended core, the combination of Corollary 8, Theorems 16 and 17 shows that the solution must be an accessible set. For example, Einy et al. (1999) analyze the relation between the least core and the Mas-Colell bargaining set (Mas-Colell, 1989) and show that the former is always contained in the latter. Thus, a direct application of our results guarantees the accessibility of the Mas-Colell bargaining set. Another issue is concerned with the collection of accessible allocations. Our analysis on its relation to various coreextensions sheds some light on the structure of accessible allocations. However, we cannot fully characterize the region of accessible allocations and leave the question for further works. Appendix A. Proof of Lemma 11 (a) Since a benchmark vector must be feasible, we obtain z( N ) v ( N ). Suppose, to the contrary, that z( N ) = v ( N ). For each coalition S ∈ ζ , by definition, z( S ) + v ( N \ S ) v ( N ), or equivalently, v ( N \ S ) z( N \ S ). Thus, z ∈ C ( v ). This contradicts to the fact that v is an unbalanced game. (b) (⇒) Assume that x is a z-accessible allocation. It suffices to show that x lies in z ( v ). Indeed, since the singleton {x} is a z-accessible set, by Corollary 9(b), we obtain {x} ∩ z ( v ) = ∅. This implies x ∈ z ( v ). (⇐) Since indirect dominance is transitive, by Corollary 9(a), we immediately obtain the desired result. (c) Let y 0 ∈ z ( v ) be an arbitrary allocation. By (b), it suffices to show that x indirectly z-dominates y 0 . We first note that βx is non-empty, otherwise x( S ) = z( S ) for all S ∈ β and hence
v ( N ) = x( N ) =
λ S x( S ) =
S ∈β
λ S z( S ) = z( N ).
S ∈β
By (a), this is impossible. Moreover, it follows that
λS v (S ) +
S ∈βx
λ S z( S ) >
S ∈β\βx
λ S x( S ) = x( N ) = y 0 ( N ) =
S ∈β
λS y0 ( S ) +
S ∈βx
λ S y 0 ( S ).
S ∈β\βx
Since y 0 z, there exists a coalition S 1 ∈ βx such that v ( S 1 ) > y 0 ( S 1 ). We define the allocation y 1 by
y 1i
=
where t 1 =
v ( S 1 )− y 0 ( S 1 ) , |S1| i i (1 − t 1 )x + t 1 z ,
y 0i +
v ( S 1 )−x( S 1 ) . x( N \ S 1 )− z( N \ S 1 )
if i ∈ S 1 , if i ∈ N \ S 1 ,
Since z is a benchmark vector and S 1 ∈ βx , it follows that
x( N \ S 1 ) − z( N \ S 1 ) = v ( N ) − x( S 1 ) − z( N \ S 1 ) v ( S 1 ) − x( S 1 ) > 0 N\S
and hence 1 t 1 > 0. Clearly, y 1 z-dominates y 0 via S 1 , y 1 ( S 1 ) = v ( S 1 ) > x( S 1 ), y 1 ∈ z ( v ) and x N \ S 1 y 1 1 . We now recursively construct the coalition S r +1 ∈ βx and the allocation y r +1 ∈ z ( v ) for r 1 assuming that the sequence of allocations { y i }ri=1 ⊆ z ( v ) and the sequence of coalitions { S 1 , S 2 , . . . , S r } ⊆ βx have already been constructed to satisfy (i) y j z-dominates y j −1 via S j for j = 1, . . . , r; r (ii) j =1 S j = ∅, y r ( S r ) = v ( S r ) > x( S r ); and r r i (iii) x y ri for all i ∈ j =1 ( N \ S j ) = N \( j =1 S j ). Since y r z, y r ( S r ) > x( S r ) and
S ∈βx
λ S yr ( S ) +
S ∈β\βx
λ S yr ( S ) = yr ( N ) = x( N ) =
S ∈βx
λ S x( S ) +
λ S z( S ),
S ∈β\βx
there exists a coalition S r +1 ∈ βx such that v ( S r +1 ) > x( S r +1 ) > y r ( S r +1 ). r +1 r r Note that j =1 S j . Indeed, if j =1 S j is a subset of S r +1 , by (iii), we obtain j =1 S j must be a proper subset of x( N \ S r +1 ) y r ( N \ S r +1 ) and hence
v ( N ) = x( S r +1 ) + x( N \ S r +1 ) > y r ( S r +1 ) + y r ( N \ S r +1 ) = v ( N ).
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Y.-Y. Yang / Games and Economic Behavior 74 (2012) 687–698
This is impossible. We consider two cases.
r
r +1
S
S r +1 y r r+1 by (iii). Let y r +1 = x. Since v ( S r +1 ) > x( S r +1 ) > Case I. j =1 S j and hence x j =1 S j = ∅. Then S r +1 ⊆ N \ y r ( S r +1 ) and x z, it follows that x z-dominates y r via S r +1 . r +1 Case II. j =1 S j = ∅. Let y r +1 be the allocation defined by
y ri +1
=
⎧ ⎪ yi + ⎪ ⎨ r
v ( S r +1 )− y r ( S r +1 )
|
r +1
j =1 S j |
,
if i ∈
r +1
j =1
S j,
r y ri , if i ∈ S r +1 \ j =1 S j , ⎪ ⎪ ⎩ (1 − tr +1 )xi + tr +1 zi , if i ∈ N \ S r +1 , v(S
)−x( S
)
where 1 t r +1 = x( N \ S r +1)−z( Nr\+S1 ) > 0. It is easily checked that y r +1 z-dominates y r via S r +1 , y r +1 ( S r +1 ) = v ( S r +1 ) > r +1 r +1 r +1 r +1 x( S r +1 ), y r +1 ∈ z ( v ) and xi y ri +1 for all i ∈ j =1 ( N \ S j ) = N \( j =1 S j ). Since | S 1 | n − 1 and
r ∗
r +1
r ∗ +1
j =1
S j is a proper subset of
r
j =1
S j for each r, there exists a positive integer r ∗ n − 1 such
∗ that j =1 S j = ∅ and j =1 S j = ∅. Consequently, we obtain a sequence of allocations { y 0 , y 1 , . . . , y r +1 = x} ⊆ z ( v ) and a sequence of coalitions { S 1 , S 2 , . . . , S r ∗ +1 } ⊆ βz such that y j z-dominates y j −1 via S j for j = 1, . . . , r ∗ + 1. This completes the proof.
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