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A Weak Bargaining Set for
Contract Choice Problems
Somdeb Lahiri
NOTA DI LAVORO 19.2006
JANUARY 2006
CTN – Coalition Theory Network
Somdeb Lahiri, Institute for Financial Management and Research
This paper can be downloaded without charge at:
The Fondazione Eni Enrico Mattei Note di Lavoro Series Index:
http://www.feem.it/Feem/Pub/Publications/WPapers/default.htm
Social Science Research Network Electronic Paper Collection:
http://ssrn.com/abstract=879717
The opinions expressed in this paper do not necessarily reflect the position of
Fondazione Eni Enrico Mattei
Corso Magenta, 63, 20123 Milano (I), web site: www.feem.it, e-mail: [email protected]
A Weak Bargaining Set for Contract Choice Problems
Summary
In this paper, we consider the problem of choosing a set of multi-party contracts, where
each coalition of agents has a non-empty finite set of feasible contracts to choose from.
We call such problems, contract choice problems. The main result of this paper states
that every contract choice problem has a non-empty weak bargaining set. The need for
such a solution concept which is considerably weaker than the core arises, since it is
well known that even for very simple contract choice problems, the core may be empty.
We also show by means of an example that the bargaining set due to Mas-Colell (1989),
as well as a weaker version of it, may be empty for contract choice problems, thereby
implying that the weakening we suggest is in some ways “tight”
Keywords: Weak bargaining set, Contract choice, NTU game, Matching
JEL Classification: C78, D71
I would like to thank the CTN Series Editor, for valuable suggestions, which have
subsequently been incorporated in the paper.
Address for correspondence:
Somdeb Lahiri
Institute for Financial Management and Research
24, Kothari Road
Nungambakkam
Chennai 600034
India
E-mail: [email protected], [email protected]
1
A Weak Bargaining Set For Contract Choice Problems
by
Somdeb Lahiri
Institute for Financial Management and Research
24, Kothari Road, Nungambakkam,
Chennai- 600 034,
Tamil Nadu, India.
Email: [email protected]
Or
[email protected]
June 2005.
Revised: October 2005.
Abstract
In this paper, we consider the problem of choosing a set of multi-party contracts,
where each coalition of agents has a non-empty finite set of feasible contracts to
choose from. We call such problems, contract choice problems. The main result
of this paper states that every contract choice problem has a non-empty weak
bargaining set. The need for a such a solution concept which is considerably
weaker than the core arises, since it is well known that even for very simple
contract choice problems, the core may be empty. We also show by means of an
example that the bargaining set due to Mas-Colell (1989), as well as a weaker
version of it, may be empty for contract choice problems, thereby implying that
the weakening we suggest is in some ways “tight”.
1. Introduction:
In this paper, we consider the problem of choosing a set of multi-party contracts,
where each coalition of agents has a non-empty finite set of feasible contracts to
choose from. We call such problems, contract choice problems. The economic
motivation behind the problem, arises from several real world "commons
problems", where agents can pool their initial resources and produce a marketable
surplus, which needs to be shared among themselves. There are clearly, two
distinct problems that arise out of such real world possibilities: (i) Coalition
Formation: Which are the disjoint coalitions that will form in order to pool in their
resources? (ii) Distribution: How will a coalition distribute the surplus within
itself? While, the possibility of an aggregate amount of surplus being generated
by a coalition is fairly common, there are many situations where more than one
aggregate surplus results from a cooperative activity, and the distribution of the
surplus depends on the particular aggregate that a coalition chooses to share.
In our model, which has been developed in a related paper [Lahiri (2003)] each
non-empty subset of agents has a non-empty finite set of pay-off vectors to
choose from. An outcome comprises a partition of the set of agents, and an
assignment for each coalition in the partition a feasible pay-off vector. Our model
is therefore a special kind of cooperative game with non-transferable utility. In the
2
context of our contract choice model, the Shapley-Scarf (1974) housing market
corresponds to a situation, where each individual assigns a monetary worth to
each object, and a feasible pay-off vector for a coalition, is the set of utility
vectors available to the coalition, when it re-allocates objects within itself,
without any one in the coalition retaining his initial endowment, unless the
coalition is a singleton. Lahiri (2004) proves the existence and Weak Pareto
Optimality of 'stable' outcomes in a two-sided contract choice problem. The
model studied in Lahiri (2004) is originally due to Roth and Sotomayor (1996).
Zhou (1994) introduced a concept of the bargaining set, which is a slight variation
of the original one due to Aumann and Maschler (1964). Yet another notion of a
bargaining set is due to Mas-Colell (1989). The Zhou(1994) bargaining set of a
marriage problem always contains its non-empty core. Klijn and Masso(2003)
introduced the concept of the weakly stable set for a marriage problem and
showed that the set of efficient and weakly stable matchings coincided with its
bargaining set as defined by Zhou (1994).
We introduce the concepts of the weak bargaining set for contract choice
problems. The basic idea behind the weak bargaining set is a set of feasible
allocations, which do not admit a credible objection (i.e. every strong objection
has a strong counter-objection). Our definition of a credible objection is
somewhat different from that of Zhou (1994) or Mas-Colell (1989), in that we
require a strong counter-objection to make none of its proponents worse off than
what they were at the time when the objection was raised. We further require that
no sub-coalition of an objecting coalition can block the objecting pay-off.
The main result of this paper states that every contract choice problem has a nonempty weak bargaining set. We show with the help of a three-agent example, that
a natural analog of the bargaining set due to Mas-Colell (1989), and hence the
bargaining set due to Mas-Colell (1989), may well be empty for room-mates
problems. This, in particular suggest that the weakening of the bargaining set we
suggest here, is indeed “tight”.
2. Contract Choice Problems:
Let X be a non-empty finite subset of ℵ(: the set of natural numbers), denoting
the set of participating agents. Let ℜ denote the set of all real numbers and ℜ+ the
set of non-negative real numbers. Let [X] denote the set of all non-empty subsets
of X. Members of [X] are called coalitions. Further let ℜS denote the set of all
functions from S to ℜ and [ℜS] denote the collection of all non-empty finite
subsets of ℜS. Given S∈[X], let #S denote the number of elements of S.
Given S∈[X], let C(S) = {µ/ µ is a bijection on X with µ(S) = S} and C0(S) =
{µ∈C(S)/ T is a non-empty proper subset of S implies µ(T) ≠T}.
Thus, if #S ≥ 2, then the function µ: X→ X, such that µ(a) = a for all a∈S,
belongs to C(S)\ C0(S).
A Contract Choice Problem (CCP) is a function G: [X]→( U[ℜS ] )∪{φ} such
S∈[ X ]
that for all S∈[X]: (i) G(S) ⊂ ℜ ; (ii) G({a}) = {0} for all a∈X.
G(S) is the set of all feasible allocations of pay-offs for agents in S.
S
3
Given a CCP G, a coalition structure for G is a partition of X.
A pay-off function is a function v : X → ℜ+. If v is a pay-off function and S∈[X],
then v|S denotes the restriction of v to the set S.
An outcome for a CCP G is a pair (f, v), where f is a coalition structure for G and
v is a pay-off function such that (i) for all a∈X: v(a) ≥ 0; (ii) for all S∈f: v|S∈
G(S).
The pair (f, v), where f = {{a}/ a∈X} and v(a) = 0 for all a∈X, is an outcome for
every CCP. Hence the set of outcomes is always non-empty.
Given an outcome (f, v) for a CCP G, a coalition S∈[X] is said to block (f, v) if
there exists x ∈ G(S): x(a) > v(a) for all a∈S.
An outcome (f, v) for a CCP G is said to belong to the core of
G, if it does not admit any blocking coalition. Let Core(G) denote the set of
outcomes in the core of G.
An outcome (f, v) for a CCP G is said to be Weakly Pareto Optimal if it does not
admit X as a blocking coalition.
Given a CCP G, an outcome (f, v) is said to be weakly blocked by a coalition
T∈[X], if there exists x∈G(T): x(a) ≥ v(a) for all a∈T, with strict inequality for at
least one a∈T. If an outcome (f,v) is weakly blocked by a coalition T∈[X], via
x∈G(T), then a∈T is said to be an active member of the weakly blocking coalition
T, if x(a) > v(a).
An outcome (f,v) is said to be Pareto Optimal if it does not admit X as a weakly
blocking coalition.
A special case of a CCP is the room-mates problem of Gale and Shapley (1962),
where G(S) = φ, whenever #S > 2. The marriage problem of Gale and Shapley
(1962) is in turn a special case of their room-mates problem. If G(S) = φ,
whenever #S > 3, then we have a possible generalization of the man, woman and
child problem of Alkan (1988).
The following example due to Gale and Shapley (1962) shows that the core of a
room-mate problem may be empty.
Example 1 (Gale Shapley (1962)) : Let X = {1,2,3,4}. For a∈X, let ua: X→ℜ be
defined as follows:
u1: u1(2) = 3, u1(3) = 2, u1(4) = 1, u1(1) = 0;
u2: u2(3) = 3, u2(1) = 2, u2(4) = 1, u2(2) = 0;
u3: u3(1) = 3, u3(2) = 2, u3(4) = 1, u3(3) = 0;
u4: u4(1) = 3, u4(2) = 2, u4(3) = 1, u4(4) = 0.
Let, G be a CCP such that for all S∈[X]: (i)G(S) = {x∈ℜS/ for some µ∈C0(S),
x(a) = ua(µ(a)) for all a∈S}, if #S ∈{1,2}; (ii) G(S) = φ, otherwise.
Suppose (f,v) is an outcome such that v(4) ≠ 0. If v(4) = 1, then {3,4}∈f and v(3)
= 1. Thus,{2,3} blocks (f,v), since 2 can get 3 units and 3 can get 2 units in
G({2,3}); if v(4) = 2, then {2,4}∈f and v(2) = 1. Thus, {1,2} blocks (f,v) since 1
can get 3 units and 2 can get 2 units in G({1,2}); if v(4) = 3, then {1,4}∈f and
4
v(1) = 1. Thus, {1,3} blocks (f,v) since 3 can get 3 units and 1 can get 2 units in
G({1,3}). Thus, v(4) ≠ 0 implies (f,v) does not belong to Core(G). Hence suppose
v(4) = 0. If v(3) = 0, then both {2,3} and {3,4} block (f,v); if v(2) = 0, then both
{1,2}and {2,4} block (f,v); if v(1) = 0, then both {1,3} and {1,4} block (f,v).
Since v(4) = 0 requires v(a) = a for at least one a∈{1,2,3}, Core(G) = φ.
Note that the outcome (f*,v*) such that f* = {{1,2,3},{4}} and v*(1) = v*(2) =
v*(3) = 3, v*(4) = 0, belongs to the Core(G*), where G* is such that for all S∈[X]:
(i)G*(S) = {x∈ℜS/ for some µ∈C0(S), x(a) = ua(µ(a)) for all a∈S}, if #S
∈{1,2,3}; (ii) G*(S) =φ, otherwise.
Given a CCP G and a Pareto Optimal outcome (f,v), the pair ((f',v'), T) where
(f',v') is an outcome for G and T∈f' is said to be a strong objection against (f,v) if
v'(a) > v(a) for all a ∈T and no subset of T is a blocking coalition for (f',v').
Given a CCP G, a Pareto Optimal outcome (f,v) and a strong objection ((f',v'),T)
against (f,v), an ordered pair ((f'',v''),U) where (f'',v'') is an outcome for G and
U∈ f'' is said to be a strong counter-objection against ((f',v'),T) if: (a)U\T, T\U
and U∩T are all non-empty; (b)v''(a) > v'(a) for all a∈ U.
The strong objection ((f',v'),T) against the outcome (f,v) is said to be justified, if
((f',v'),T) has no strong counter-objection.
We define the weak bargaining set of a CCP G, to be the set WB(G) = {(f,v)/ (f,v)
is Pareto Optimal and such that no strong objection against (f,v) is justified}.
Example 1 (due to Gale and Shapley (1962)) is one that has an empty core, but a
non-empty weak bargaining set. We saw in Example 1, that Core(G) = φ.
However, consider v(4) = 1, v(3) = 1, v(2) = 2, v(1) = 3, f = {{1,2},{3,4}}. The
pair ((f', v'), {2,3}) is a strong objection against (f,v), where v'(2) = 3, v'(3) = 2,
v'(1) = v'(4) = 0 and f' = {{1},{4},{2,3}}. Let f'' = {{2}, {4}, {1,3}}, v''(1) = 2,
v''(3) = 3, v'' (2) = v''(4) = 0.Then the pair ((f'',v''), {1,3}) is a strong counterobjection against ((f',v'),{2,3}). Further, (f,v) admits no blocking coalition other
than {1,3}. Since no subset of {1,3} blocks (f',v'), (f,v) belongs to WB(G).
Note that it is possible to provide a definition of the weak bargaining set modified
along the lines suggested in Mas-Colell (1989).
Given a CCP G an outcome (f,v) and a strong objection ((f',v'),T) against (f,v), an
ordered pair ((f'',v''),U) is said to be a classical strong counter-objection against
((f',v'),T) if: (a) U∈f''; (b) U\T, U\S and U∩T are all non-empty; (c) v''(a) ≥ v(a)
for all a∈U|T; (d) v''(a) > v'(a) for all a∈U.
The strong objection ((f',v'),T) against the outcome (f,v) is said to be classically
justified, if ((f',v'),T) has no classical strong counter-objection.
We define the classical weak bargaining set of a CCP G, to be the set WB*(G) =
{(f,v)/ (f,v) is Pareto Optimal, and such that no strong objection against (f,v) is
classically justified}.
5
However, the following example reveals that even for room-mates problems,
WB*(G) may be empty.
Example 2: Let X = {1,2,3}. For a∈X, let ua: X→ℜ be defined as follows:
u1: u1(2) = 2, u1(3) = 1, u1(1) = 0;
u2: u2(3) = 2, u2(1) = 1, u2(2) = 0;
u3: u3(1) = 3, u3(2) = 2, u3(3) = 0.
Let, G be the CCP such that for all S∈[X]: (i)G(S) = {x∈ℜS/ for some µ∈C0(S),
x(a) = ua(µ(a)) for all a∈S}, if #S ∈{1,2}; (ii) G(S) =φ, otherwise.
Since (f,v) such that v(a) = 0 for all a∈X is not Pareto Optimal, it cannot belong
to WB*(G).
Let (f,v) be the outcome such that f = {{1,3},{2}}, v(1) = 1, v(2) = 0, v(3) = 2 and
(f',v') be the outcome such that f' = {{1,2},{3}}, v'(1) = 2, v'(2) = 1, v'(3) = 0.
Thus, ((f',v'), {1,2}) is a strong objection against (f,v). Any strong counterobjection or classical strong counter-objection cannot contain agent 1, since agent
1 gets 2 units of pay-off at (f',v'). The only possibility is (({{2,3}, {1}}, v''),
{2,3}) where v''(1) = 0, v''(3) = 1, v''(2) = 2,which is a strong counter-objection
though not a classical strong counter-objection, since agent 3 is worse off at
(f'',v'') than at (f,v). Thus, (f,v) ∉WB*(G).
Let (f,v) be the outcome such that f = {{1},{2,3}}, v(1) = 0, v(2) = 2, v(3) = 1 and
(f',v') be the outcome such that f' = {{1,3},{2}}, v'(1) = 1, v'(2) = 0, v'(3) = 2.
Thus, ((f',v'), {1,3}) is a strong objection against (f,v). Any strong counterobjection or classical strong counter-objection cannot contain agent 3, since agent
3 gets 2 units of pay-off at (f',v'). The only possibility is (({{1,2}, {3}}, v''),
{1,3}) where v''(1) = 2, v''(3) = 0, v''(2) = 1,which is a strong counter-objection
though not a classical strong counter-objection, since agent 2 is worse off at
(f'',v'') than at (f,v). Thus, (f,v) ∉WB*(G). Thus, WB*(G) = φ.
Hence Bar*(G) = φ.
3. The non-emptiness of the weak bargaining set:
Theorem 1: Let G be a CCP. Then, WB(G) ≠ φ.
Proof: Let G be a CCP and let (f,v) be a Pareto Optimal outcome for G. If (f,v)
does not admit a strong objection then clearly, (f,v) ∈WB(G). Suppose ((f1,v1),
S1) is a strong objection against (f,v) which further does not admit a strong
counter-objection. Then, no member of S1 is part of a strong objection against
(f1,v1). Since ((f1,v1), S1) is a strong objection against (f,v) which further does not
admit a strong counter-objection, there can be no strong objection ((f2,v2), S2)
against (f1,v1) such that S2∩S1≠φ. If (f1,v1) does not admit any strong objection,
then (f1,v1)∈WB(G). Suppose ((f2,v2), S2) is a strong objection against (f1,v1)
which further does not admit a strong counter-objection. Thus, S2∩S1= φ. Without
loss of generality suppose S1∈f2 and v2(a) = v1(a) for all a∈S1. This is possible,
since S2∩S1 = φ. Then, no member of S1∪S2 is part of a strong objection against
(f2,v2).
6
Having constructed a strong objections ((fp,vp), Sp) against (fp-1,vp-1) for p = 1,….,
k, where (f0,v0) = (f,v), such that no member of
k
US
p
is part of a blocking
p =1
coalition against (fk,vk) there are two possibilities: there does exist a strong
objection against (fk,vk) in which case (fk,vk)∈WB(G); there exists a strong
objection ((fk+1, vk+1), Sk+1) against (fk,vk). If every such strong objection admits a
strong counter-objection, then (fk,vk)∈WB(G). If not then there exists a strong
objection ((fk+1,vk+1), Sk+1), which further does not admit a strong counterk
objection. Clearly, Sk+1∩( U S p ) = φ. Without loss of generality suppose, Sp∈ fk+1
p =1
k
for p = 1,…,k and vk+1(a) = vk(i) for all a∈ U S p . Then no member of
p =1
k +1
US
p
is
p =1
part of a strong objection against (fk+1, vk+1).
Since X is a finite set, there is a smallest positive integer K, such that either every
K
objection ((f',v'), T) against (f ,v ) admits a strong counter-objection, or [ U S p =
K
K
p =1
K
K
X or no member of X is part of a blocking coalition against (f ,v ). In either case,
(fK,vK)∈WB(G). Q.E.D.
Acknowledgment: I would like to thank the CTN Series Editor, for valuable
suggestions, which have subsequently been incorporated in the paper.
References:
1. Alkan, A. (1988): " Non-Existence of Stable Threesome Matchings",
Mathematical Social Sciences, 16, 207-209.
2. Aumann, R. and M. Maschler (1964): "The Bargaining Set for Cooperative
Games", in M. Dresher, L. Shapley and A. Tucker (eds.) Advances in Game
Theory. Princeton, N.J.: Princeton University Press.
3. Gale, D. and L. Shapley (1962): “College Admissions and the Stability of
Marriage”, American Mathematical Monthly, 69, 9-15.
4. F. Klijn and J. Masso (2003): "Weak Stability and a Bargaining Set for the
Marriage Model", Games and Economic Behavior 91, 91-100.
5. Lahiri, S. (2004): "Stable Outcomes for Two-Sided Contract Choice Problems",
Computational and Mathematical Organization Theory 10, 323-334.
6. Lahiri, S. (2003): "Stable Outcomes for Contract Choice Problems", (mimeo)
WITS University.
7. Mas-Colell, A. (1989): "An Equivalence Theorem for a Bargaining Set",
Journal of Mathematical Economics 18, 129-139.
8. Roth, A.E. and M. Sotomayor (1996): “Stable Outcomes In Discrete And
Continuous Models Of Two-Sided Matching: A Unified Treatment”, R.de
Econometrica, 16, 1-24.
9. Shapley, L. and H. Scarf (1974): “ On Cores and Indivisibility”, Journal of
Mathematical Economics, 1, 23- 28.
7
10. Zhou, L. (1994): "A New Bargaining Set of an N-Person Game and Endogenous
Coalition Formation", Games and Economic Behavior 6, 512-526.
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