Welfare Economy under Rough Sets Information
Takashi Matsuhisa
?
Department of Natural Sciences, Ibaraki National College of Technology
Nakane 866, Hitachinaka-shi, Ibaraki 312-8508, Japan.
E-mail:[email protected]
Abstract. This paper relates welfare economy and rough sets as information structure. We reconsider the fundamental welfare theorem for a
pure exchange economy under uncertainty from the logical point of view
in data mining theory; the traders are assumed to have a multi-modal
logic of belief and to make their decision under uncertainty represented
by rough sets. We propose a generalized notion of rational expectations
equilibrium, called expectation equilibrium in belief for the economy, and
we show an extension of the fundamental theorem for welfare: An allocation in the economy is ex-ante Pareto optimal if and only if it is an
expectations equilibrium allocation in belief for some initial endowment
with respect to some price system.
Keywords. Economy on belief, Rough sets, Information, Welfare theorem, Multi-modal logic of belief, Expectations equilibrium in belief,
Ex-ante Pareto optimum.
Journal of Economic Literature Classification: D51, D84, D52, C72.
1
Introduction
This paper relates economy and rough sets as information structure. The fundamental welfare theorem has shown that each competitive equilibrium for an
economy can be characterized as it is allocated ex-ante Pareto-optimally. In this
theorem the economy is under complete information. This paper explores the
extent to which the theorem is generalized in an economy under uncertainty.
We highlight the rough set theoretical aspect of the information structure for
traders which represent their uncertainty.
In recent years, several investigators have already generalized the theorem as
the no trade theorem. The no trade theorem shows that new information will
not give the traders any incentive to trade when their initial endowments are
allocated ex-ante Pareto-optimally. Milgrom and Stokey [8] shows this theorem
for an economy under uncertainty. They analyze the economy with a partition
information structure and characterize rational expectations equilibria from exante Pareto optimality.
?
This paper is presented at the seminar in Wroclaw University of Technology, 11
September 2006. It is a preliminary version, and the final form will be published
elsewhere.
Several variations of the no trade theorem have been developed. Neeman [10]
applies it in the case of p-beliefs, Luo and Ma [7] in the non-expected utility
case, Morris and Skiadas [9] in the case of rationalizable trades, and so on.
Geanakoplos [6] neatly analyzes non-partition information structure with
the introduction of a new concept, positive balancedness. With this concept, he
examines several classes of non-partition information and the relations among
them, and characterizes Nash equilibrium and rational expectations equilibrium
in those classes.
In the researches on economy either with complete information or with incomplete information, the role of traders’ knowledge (belief) remains obscured:1
The economy has not been investigated from the epistemic point of view. Here
this article aims to fill that gap.
This paper discusses the welfare theorem as the extension of the no trade
theorem with emphasis on the modal logical point of view, and captures different
features from their analysis. We need neither the reflexivity, the transitivity nor
the positive balancedness in the information structure. We focus the rough set
theoretical aspect of the non-partitional information structure. Specifically, we
seek the role of belief of traders in a pure exchange economy under uncertainty
from a rough set theoretical point of view.
Our model applies it to expected utility and rational expectations equilibrium, and uses the standard setting as Milgrom and Stokey [8]. Therefore we can
obtain the direct extension of the original welfare theorem for economy under
complete information.
We propose the notion of pure exchange economy based on the multi-modal
logic of belief B, by which the traders use making their decision, and we introduce the extended notion of equilibrium for the economy, called an expectation
equilibrium in belief. We establish the fundamental welfare theorem for the economy:
Main theorem. In a pure exchange economy under uncertainty, assume that the
traders have the multi-modal logic B and they are risk averse. Then an allocation
in the economy is ex-ante Pareto optimal if and only if it is an expectations
equilibrium allocation in belief for some initial endowment with respect to some
price system.
This article is organized as follows: In Section 2 we give a simple example
to illustrate our model, which is motivated by the global warming problem in
the earth. The emphasis is on the view point of rough sets. Section 3 presents
the multi-agent modal logic B and remarks the finite model property. Further,
in Section 4 we introduce an economy in logic B and the notion of expectations equilibrium, which is a generalization of rational expectations equilibrium.
Section 5 establishes the fundamental theorem for welfare economy. We give an
outline of the proof of the fundamental theorem. Finally we conclude by giving
remarks.
1
See the literatures cited in the survey of Forges et al [5] treated the topics and related
works from the standard view points of economic theory.
2
2
Illustrative Example of Rough Sets Information.
Let us consider the following situation: Three traders, Company L, Company F
and Company N are willing to buy and sell the L’s stocks, the F’s stocks and the
N’s stocks each other. Thus, there are three commodities L’s stocks, F’s stocks
and N’s stocks.
L has made his own stock price high by Window-dressing in order to make
his market capitalization high. But F and N have not realized L’s dirty tricks,
and it may come out that L commits the injustice.
We shall illustrate the situation as follows:
Example 1. Let Ω be the state space consisting of the two states {ω1 , ω2 }: The
state ω1 represents that L does not commit the injustice, the state ω2 represents
that L commits the injustice. So L can know which is the true state of either ω1
or ω2 occurs when each of the two states occurs. However traders F and N don’t
believe the state that L commits the injustice at all. Therefore, the traders L,
F, N have their information functions, PL (ω) = {ω} for ω = ω1 , ω2 , PF (ω1 ) =
{ω1 , ω2 }, PF (ω2 ) = {ω2 }, PF (ω1 ) = {ω1 , ω2 }, PF (ω2 ) = {ω2 }, PN = PF .
This illustration is very interesting from the view point of rough sets. Each
trader t’s information structure Pt : Ω → 2Ω assigning to each state ω in a state
space Ω the information set Pt (ω) that t possesses in ω entails the two operators
on 2Ω : t’s belief operator Bt : 2Ω → 2Ω and t’s possibility operator Pt : 2Ω → 2Ω .
These are defined by Bt (E) = {ω ∈ Ω | Pt (ω) ⊆ E} and Pt (E) = Ω \ Bt (Ω \ E)
respectively. According to the theory of rough sets, we call an event X exact if
Bt (X) = Pt (X), and call X rough if it is not exact.
We can observe that PL represents that trader L has the complete information with which each component PL (ω) is an exact set, because BL (PL (ω)) =
PL (PL (ω)). Each PF or PN represents that trader F and N have the incomplete
information, with which each component Pt (ω) is a rough set for t = F, N. This
suggests that the uncertainty of traders is modeled by rough sets.
The non-partition structure Pt with the rough sets components is equivalent
to the belief operator Bt satisfying ‘Truth’ axiom T: Bt (E) ⊆ E (what is known
is true). The partition structure PL with the exact sets components is equivalent to BL satisfying the two axioms 4 and 5 in addition to T: The ‘positive
introspection’ 4: BL (E) ⊆ BL (BL (E)) (we know what we do) and the ‘negative
introspection’5: Ω \ BL (E) ⊆ BL (Ω \ BL (E)) (we know what we do not know).
2
One of these requirements, symmetry (or the equivalent axiom 5), is indeed so
strong that describes the hyper-rationality of traders, and thus it is particularly
objectionable.
The recent idea of bounded rationality suggests dropping such assumptions
since real people are not complete reasoners. This raises the question to what
extent results on the information partition structure (or the equivalent postulates of knowledge). The answer is to weaken the conditions in the information
partition and knowledge, and we shall investigate the essential roles of trader’s
2
C.f.: Bacharach [2], Fagin, Halpern et al [4].
3
information represented by rough sets in the results. As has already been pointed
out in the literatures (e.g.: Bacharach [2]), this approach can potentially yield
important results in a world with imperfectly Bayesian agents. The idea has
been performed in different settings.3 However, all those researches have been
lacked the logic that represents the traders’ knowledge (or belief). In this article
we present the economy upon the logic of belief, and we extend the fundamental
welfare theorem into the economy.
3
Multi-Modal Logic of Belief
3.1
Syntax
Let T be a finite set of traders {1, 2, · · · , t, · · · , n}. The language is founded on as
follows: The sentences of the language form the least set containing each atomic
sentence Pm (m = 0, 1, 2, . . . ) and closed under the following operations: nullary
operators for falsity ⊥ and for truth >; unary and binary syntactic operations
for negation ¬, conditionality → and conjunction ∧, disjunction ∨, respectively;
two unary operations for modalities t , ♦t for t ∈ T . Other such operations
are defined in terms of those in usual ways. The intended interpretation of t ϕ
is the sentence that ‘trader t believes a sentence ϕ,’ and the sentence ♦t ϕ is
interpreted as the sentence that ‘trader a sentence ϕ is possible for t.’
3.2
Logic of belief and Belief structure
A multi-modal logic L is a set of sentences containing all truth-functional tautologies and closed under substitution and modus ponens. A multi-modal logic
L0 is an extension of L if L ⊆ L0 . A sentence ϕ in a modal logic L is a theorem
of L, written by `L ϕ. Other proof-theoretical notions such as L-deducibility,
L-consistency, L-maximality are defined in usual ways. (See, Chellas [3].)
A normal system of multi-modal logic L is a multi-modal logic containing
(Df♦ ) and is closed under the 2n rules of inference (RKt ) and (RK♦ ).
(Df♦ ) ♦t ϕ ←→ ¬t ¬ϕ;
1 ∧ϕ2 ∧···∧ϕk )−→ψ
(RKt ) (t ϕ1(ϕ
∧t ϕ2 ∧···∧t ϕk )−→t ψ
(RK♦ )
(ϕ1 ∧ϕ2 ∧···∧ϕk )−→ψ
(♦t ϕ1 ∧♦t ϕ2 ∧···∧♦t ϕk )−→♦t ψ
(k = 0)
(k = 0)
Definition. The logic of belief B is the minimal normal system of multi-modal
logic.
3
E.g.: Among other things J. Geanakoplos [6] showed the no speculation theorem
in the extended rational expectations equilibrium under the assumption that the
information structure is reflexive, transitive.
4
3.3
Belief structure, model and truth
Let Ω be a non-empty set called a state space and 2Ω the field of all subsets
of Ω. Each member of 2Ω is called an event and each element of Ω called a
state. A belief structure is a tuple hΩ, (Bt )t∈T , (Pt )t∈T i in which Ω is a state
space and Bt : 2Ω → 2Ω is trader t’s belief operator. The interpretation of the
event Bt E is that ‘t believes E.’ Pt is t’s possibility operator on 2Ω defined by
Pt E = Ω \ Bt (Ω \ E ) for every E in 2Ω . The interpretation of Pt E is that ‘E is
possible for t.’
A model on a belief structure is a tuple M = hΩ, (Bt )t∈T , (Pt )t∈T , V i in
which hΩ, (Bt )t∈T , (Pt )t∈T i is a belief structure and a mapping V assigns either
true or false to every ω ∈ Ω and to every atomic sentence Pm . The model M
is called finite if Ω is a finite set.
Definition 1. By |=M
ω ϕ, we mean that a sentence ϕ is true at a state ω in a
model M. Truth at a state ω in M is defined by the inductive way as follows:
1.
2.
3.
4.
5.
6.
7.
8.
|=M
ω
|=M
ω
|=M
ω
|=M
ω
|=M
ω
|=M
ω
|=M
ω
|=M
ω
Pm if and only if V (ω, Pm ) = true, for m = 0, 1, 2, . . . ;
>, and not |=M
ω ⊥;
¬ϕ if and only if not |=M
ω ϕ;
M
ϕ −→ ψ if and only if |=M
ω ϕ implies |=ω ψ;
M
M
ϕ ∧ ψ if and only if |=ω ϕ and |=ω ψ;
M
ϕ ∨ ψ if and only if |=M
ω ϕ or |=ω ψ;
M
t ϕ if and only if ω ∈ Bt (||ϕ|| ), for t ∈ T ;
♦t ϕ if and only if ω ∈ Pt (||ϕ||M ), for t ∈ T :
Where ||ϕ||M denotes the set of all the states in M at which ϕ is true; this is
called the truth set of ϕ.
We say that a sentence ϕ is true in the model M and write |=M ϕ if |=M
ω ϕ for
every state ω in M. A sentence is said to be valid in a belief structure if it is
true in every model on the belief structure. Let Γ be a set of sentences. We say
that M is a model for Γ if every member of Γ is true in M. A belief structure
is said to be for Γ if every member of Γ is valid in it. Let C be a class of models
on a belief structure. A multi-modal logic L is sound with respect to C if every
member of C is a model for L. It is complete with respect to C if every sentence
valid in all members of C is a theorem of L. A multi-modal logic L is said to
have the finite model property if it is sound and complete with respect to the
class of all finite models in C. We denote by CF the class of all finite models in
C. We can establish that
Theorem 1. A normal system of multi-modal logic L has the finite model property; i.e., `L ϕ if and only if |=M ϕ for all M ∈ CF . In particular, B has the
finite model property.
Proof. Will be given in the line described in Chellas [3]. In Appendix we shall
give the proof for readers’ convenience.
By this fact, we shall consider hΩ, (Bt )t∈T , (Pt )t∈T , V i as a finite model for
a normal system L from now on.
5
4
4.1
Economy on Logic of Belief
Information structure
We shall give the generalized notion of information partition in the line of
Bacharach [2].
Definition 2. The associated information structure (Pt )t∈T with a model on
a belief structure hΩ, (Bt )t∈T , (Pt )t∈T , V i for a normal system of multi-modal
logic L is the class of t’s possibility operator Pt : 2Ω → 2Ω defined by Pt (E) =
Ω \ Bt (Ω \ E). t’s associated information function Pt : Ω → 2Ω is defined by
Pt (ω) = Pt ({ω}) = Ω \ Bt (Ω \ {ω})). We denote by Dom(Pt ) the set {ω ∈
Ω | Pt (ω) 6= ∅}, called the domain of Pt .
The information function Pt : Ω → 2Ω is called reflexive if ω ∈ Pt (ω) for
every ω ∈ Dom(Pt ), and it is said to be transitive if ξ ∈ Pt (ω) implies Pt (ξ) ⊆
Pt (ω) for any ξ, ω ∈ Dom(Pt ). Furthermore Pt is called symmetric if ξ ∈ Pt (ω)
implies Pt (ξ) 3 ω for any ω and ξ ∈ Dom(Pt ). It is noted that the operators Bt
is uniquely determined by the information structure Pt .
Definition 3. An event X is called exact if Bt (X) = Pt (X), and X is called
rough if it is not exact.
We let turn into the economy under uncertainty presented in Example 1, and
let the notations be the same in it. We shall illustrate the situation as follows:
Example 2. Let Ω be the state space consisting of the two states {ω1 , ω2 }: The
state ω1 represents that L does not commit the injustice, the state ω2 represents
that L commits the injustice. So L can know which is the true state of either
ω1 or ω2 occurs when each of the two states occurs. However traders F and N
don’t believe the state that L commits the injustice at all. Therefore, the belief
operators (Bt )t=L ,F ,N can be modeled as follows:
BL (E) = E for every E ∈ 2Ω ,
BF ({ω1 }) = {ω1 }, BF ({ω2 }) = ∅, BF (∅) = ∅, BF (Ω) = Ω,
BN E = BF E.
Then traders L, F, N have their information structure as follows:
PL (E) = E for every E ∈ 2Ω ,
PF ({ω1 }) = Ω, PF ({ω2 }) = {ω2 }, PF (∅) = ∅, PF (Ω) = Ω,
PN E = PF E.
We can observe that the information functions Pt : Ω → 2Ω are: PL (ω) =
{ω} for ω = ω1 , ω2 , PF (ω1 ) = {ω1 , ω2 }, PF (ω2 ) = {ω2 }, PF (ω1 ) = {ω1 , ω2 },
PF (ω2 ) = {ω2 }.
6
These coincide with the information function appeared in Example 1.
Remark 1. M. Bacharach [2] introduces the strong epistemic model equivalent to
the Kripke semantics of the multi-modal logic S5.4 The strong epistemic model
is a tuple hΩ, (Kt )t∈T i in which t’s knowledge operator Kt : 2Ω → 2Ω satisfies
the postulates K, T, 4 and 5 with N: Kt Ω = Ω. t’s associated information
function Pt induced by Kt makes a partition of Ω, called t’s information partition Pt , which is reflexive, transitive and symmetric. This is just the Kripke
semantics corresponding to the logic S5; the postulates for Pt : reflexivity, transitivity and symmetry are respectively equivalent to the postulates T, 4 and 5
for Kt . The strong epistemic model can be interpreted as the belief structure
hΩ, (Bt )t∈T , (Pt )t∈T i such that Bt is the knowledge operator. It should be noted
that each Pt (ω) is an exact set in this model. Different approaches of knowledge
and possibility are given in Fagin et al [4].
4.2
Economy on belief
A pure exchange economy under uncertainty is a structure
E = hT, Ω, e, (Ut )t∈T , (πt )t∈T i
consisting of the following structure and interpretations: There are l commodities
in each state of the state space Ω; the consumption set of each trader t is Rl+ ;
an initial endowment is a mapping e : T × Ω → Rl+ with which e(t, ·) : Ω → Rl+
is called t’s initial endowment; Ut : Rl+ × Ω → R is t’s von-Neumann and
Morgenstern utility function; πt is a subjective prior on Ω for t ∈ T .
Let L be a normal system of multi-modal logic. A pure exchange economy
for L is a structure E L = hE, (Bt )t∈T , (Pt )t∈T , V i, in which E is a pure exchange
economy under uncertainty, and hΩ, (Bt )t∈T , (Pt )t∈T , V i is a finite model on a
belief structure for L. By the domain of the economy E L we mean Dom(E L ) =
∩t∈T Dom(Pt ). We always assume that Dom(E L ) 6= ∅, and that Dom(Pt ) ⊆
Supp(πt ) for all t.5
Definition 4. An economy on belief is the pure exchange economy for the logic
B, denoted by E B .
Remark 2. An economy under asymmetric information can be interpreted as
the economy E S5 for the multi-modal logic S5, in which the belief structure
hΩ, (Bt )t∈T , (Pt )t∈T , V i is given by the strong epistemic model, and that Dom(E B ) =
Ω.
We denote by Ft the field of Dom(Pt ) generated by {Pt (ω)| ω ∈ Ω} and
denote by Πt (ω) the atom containing ω ∈ Dom(Pt ). We denote by F the join
of all Ft (t ∈ T ) on Dom(E B ); i.e. F = ∨t∈T Ft , and denote by {Π(ω)| ω ∈
Dom(E B )} the set of all atoms Π(ω) containing ω of the field F = ∨t∈T Ft . We
shall often refer to the following conditions for E B : For every t ∈ T ,
4
5
The logic S5 is denoted by KT5 ( = KT45 ) in Chellas [3].
By the support of πwe mean Supp(πt ) := {ω ∈ Ω | πt (ω) 6= 0 }.
7
P
A-1
t∈T e(t, ω) 0 for each ω ∈ Ω.
A-2 e(t, ·) is F-measurable on Dom(Pt );
A-3 For each x ∈ Rl+ , the function Ut (x, ·) is at least F-measurable on Dom(E B ),
and the function: T ×Rl+ → R, (t, x) 7→ Ut (x, ω) is 2Ω ×B-measurable where
B is the σ-field of all Borel subsets of Rl+ .
A-4 For each ω ∈ Ω, the function Ut (·, ω) is strictly increasing on Rl+ , continuous, strictly quasi-concave and non-satiated on Rl+ .6
4.3
Expectations equilibrium in belief.
An assignment for an economy E B on belief is a mapping x : T × Ω → Rl+ such
that for each t ∈ T , the function x(t, ·) is at least F-measurable on Dom(E B ).
We denote by Ass(E B ) the set of all assignments for the economy E B .
By an allocation for E B we mean an P
assignment a such
P that a(t, ·) is Fmeasurable on Dom(E B ) for all t ∈ T and t∈T a(t, ω) 5 t∈T e(t, ω) for every
ω ∈ Ω. We denote by Alc(E B ) the set of all allocations for E B .
By t’s ex-ante expectation we mean
X
Et [Ut (x(t, ·)] :=
Ut (x(t, ω), ω)πt (ω)
ω∈Dom(Pt )
B
for each x ∈ Ass(E ). The interim expectation Et [Ut (x(t, ·)|Pt ] is defined by
X
Ut (x(t, ξ), ξ)πt ({ξ})|Pt (ω))
Et [Ut (x(t, ·)|Pt ](ω) :=
ξ∈Dom(Pt )
on Dom(Pt ).
A price system is a non-zero function p : Ω → Rl+ which is F-measurable on
Dom(E B ). We denote by ∆(p) the partition on Ω induced by p, and denote by
σ(p) the field of Ω generated by ∆(p). The budget set of a trader t at a state
ω for a price system p is defined by Bt (ω, p) := { x ∈ Rl+ | p(ω) · x 5 p(ω) ·
e(t, ω) }. Define the mapping ∆(p) ∩ Pt : Dom(Pt ) → 2Ω by (∆(p) ∩ Pt )(ω) :=
∆(p)(ω) ∩ Pt (ω). We denote by Dom(∆(p) ∩ Pt ) the set of all states ω in which
∆(p)(ω) ∩ Pt (ω) 6= ∅. Let σ(p) ∨ Ft be the smallest σ-field containing both the
fields σ(p) and Ft .
Definition 5. An expectations equilibrium in belief for an economy E B on
belief is a pair (p, x), in which p is a price system and x is an assignment for E B
satisfying the following conditions:
EB1 x is an allocation for E B ;
EB2 For all t ∈ T and for every ω ∈ Ω, x(t, ω) ∈ Bt (ω, p);
EB3 For all t ∈ T , if y(t, ·) : Ω → Rl+ is F-measurable on Dom(E B ) with
y(t, ω) ∈ Bt (ω, p) for all ω ∈ Ω, then
Et [Ut (x(t, ·))|∆(p) ∩ Pt ](ω) = Et [Ut (y(t, ·))|∆(p) ∩ Pt ](ω)
pointwise on Dom(∆(p) ∩ Pt );
6
That is, for any x ∈ Rl+ there exists an x0 ∈ Rl+ such that Ui (x0 , ω) Ui (x, ω).
8
EB4 For every ω ∈ Dom(E B ),
The allocation x in E
for E B .
B
P
t∈T
x(t, ω) =
P
t∈T
e(t, ω).
is called an expectations equilibrium allocation in belief
We denote by EB(E B ) the set of all the expectations equilibria of a pure exchange economy E B , and denote by A(E B ) the set of all the expectations equilibrium allocations in belief for the economy.
5
The Results
Let E B be an economy on belief and E B (ω) the economy with complete information hT, (e(t, ω))t∈T , (Ut (·, ω))t∈T i for each ω ∈ Ω. We denote by W(E B (ω))
the set of all competitive equilibria for E B (ω).
5.1
Fundamental theorem for welfare economy
An allocation x for E B is said to be ex-ante Pareto-optimal if there is no allocation a such that Et [Ut (a(t, ·)] = Et [Ut (x(t, ·)] for all t ∈ T with at least one
inequality strict. We can now state explicitly our main theorem.
Theorem 2. Let E B be an economy on belief satisfying the conditions A-1,A2, A-3 and A-4. An allocation is ex-ante Pareto optimal if and only if it is an
expectations equilibrium allocation in belief
P endowment w with
P for some initial
respect to some price system such that
t∈T e(t, ω) for each
t∈T w(t, ω) =
ω ∈ Dom(E B ).
Proof. will be given in the next section.
Example 3. We let turn into the situation in Example 2, and let the notations
be the same in it.
Now, suppose L, F, N have risk averse utilities:
1
1
2
UL (x, y, z; ω) = x 10 y 2 z 5 for every ω ∈ Ω,
3
2
3
1
1
2
1
2
1
UF (x, y, z; ω1 ) = x 10 y 5 z 10 , UF (x, y, z; ω2 ) = x 10 y 2 z 5 ,
3
3
2
UN (x, y, z; ω1 ) = x 10 y 10 z 5 , UN (x, y, z; ω2 ) = x 10 y 5 z 2 .
In the economy, equilibrium price p(ω) = (p1 , p2 , p3 ) is given by: For ω = ω1 ,
(p1 /p2 ) = 13053/19760, (p2 /p3 ) = 1040/899, and for ω = ω2 , (p1 /p2 ) = 153/732,
(p2 /p3 ) = 244/215. The expectations equilibrium allocation x(t, ω) is given by
x(L, ω) = (wL /10p1 , wL /2p2 , 2wL /5p3 ),
x(F, ω1 ) = (3wF /10p1 , 2wF /5p2 , 3wF /10p3 ),
x(F, ω2 ) = (wF /10p1 , wF /2p2 , 2wF /5p3 ),
x(N, ω1 ) = (3wN /10p1 , 3wN /10p2 , 2wN /5p3 ),
x(N, ω2 ) = (wN /10p1 , 2wN /5p2 , wN /2p3 ),
We note that the equilibrium allocation x is ex-ante Pareto optimal, and the
converse is also true.
9
5.2
Proof of Theorem 2
Theorem 2 immediately follows from propositions 1 and 2 as follows.
Proposition 1. Notations and assumptions being the same in Theorem 2, an
allocation x is ex-ante Pareto optimal if it is an expectations equilibrium allocation in belief with respect to some price system.
Proposition 2. Let notations and assumptions be the same in Theorem 2. If
an allocation x is ex-ante Pareto optimal in E B then there are a price system
and an initial P
endowment e0 such
P that x is an expectations equilibrium allocation
in belief with t∈T e0 (t, ω) = t∈T e(t, ω) for each ω ∈ Dom(E B ).
Before proving the propositions we shall establish
Proposition 3. Notations and assumptions being the same as above, we obtain
that
A(E B ) = {x ∈ Alc(E B ) | There is a price system p such that
(p(ω), x(·, ω)) ∈ W(E B (ω)) for all ω ∈ Dom(E B )}.
Proof. Let x ∈ A(E B ) and (p, x) ∈ EB(E B ). We shall show that (p(ω), x(·, ω)) ∈
W(E B (ω)) for any ω ∈ Dom(E B ). Suppose to the contrary that there exist a trader s ∈ T and states ω 0 ∈ Dom(E B ), ω0 ∈ (∆(p) ∩ Ps )(ω 0 ) with
the property: There is an a(s, ω0 ) ∈ Bs (ω0 , p) such that Us (a(s, ω0 ), ω0 ) Us (x(s, ω0 ), ω0 ). Define the F-measurable function y : T ×Ω → Rl+ by y(t, ξ) :=
a(t, ω0 ) for ξ ∈ Π(ω0 ), and y(t, ξ) := x(t, ξ) otherwise. It follows immediately
that Es [Us (x(s, ·))|∆(p) ∩ Ps ](ω 0 ) Es [Us (y(s, ·))|∆(p) ∩ Ps ](ω 0 ), in contradiction.
The converse will be shown: Let x ∈ Ass(E B ) with (p(ω), x(·, ω)) ∈ W(E B (ω))
for any ω ∈ Dom(E B ). Set the price system p∗ : Ω → Rl+ by p∗ (ξ) := p(ω) for all ξ ∈
Π(ω) and ω ∈ Dom(E B ), and p∗ (ξ) := p(ω) for ω ∈
/ Dom(E B ). We shall
∗
B
show that (p , x) ∈ EB(E ): x(t, ·) is F-measurable and x(t, ω) ∈ Bt (ω, p∗ )
on Dom(E B ) for all t ∈ T . It can be plainly observed that EB1, EB2 and
EB4 are all valid. For EB3: Let y(t, ·) : Ω → Rl+ be an F-measurable function with y(t, ω) ∈ Bt (ω, p∗ ) for all ω ∈ Dom(E B ). Since (p∗ (ω), x(·, ω)) ∈
W(E B (ω)) it follows that Ut (x(t, ω), ω) = Ut (y(t, ω), ω) for all t ∈ T and for
each ω ∈ Dom(E B ). It can be easily observed that Et [Ut (x(t, ·))|∆(p∗ )∩Pt ](ω) =
Et [Ut (y(t, ·))|∆(p∗ ) ∩ Pt ](ω) for all ω ∈ Dom(∆(p∗ ) ∩ Pt ), and so (p∗ , x) ∈
EB(E B ), in completing the proof.
Proof of proposition 1: Let x ∈ A(E B ). It follows from proposition 3 that
there is a price system p such that (p(ω), x(·, ω)) ∈ W(E B (ω)) at each ω ∈
Dom(E B ). By the fundamental theorem of welfare in the economy E B (ω), we can
plainly observe that for all ω ∈ Dom(E B ), x(·, ω) is Pareto optimal in E B (ω),
and thus x is ex-ante Pareto optimal.
10
Proof of proposition 2: It can be shown that for each ω ∈ Ω there exists
p∗ (ω) ∈ Rl+ P
such that (p∗ (ω),P
x(·, ω)) ∈ W(E B (ω)) for some initial endowment
0
0
e (·, ω) with t∈T e (t, ω) = t∈T e(t, ω). Proof: First it can be observed that
∗
l
∗
for each ωP∈ Ω there exists
· v 5 0 for all v ∈
P p (ω) ∈ R+ lsuch that p (ω)
G(ω) = { t∈T x(t, ω) − t∈T y(t, ω) ∈ R | y ∈ Ass(E B ) and Ut (y(t, ω), ω) =
Ut (x(t, ω), ω) for all t ∈ T } for each ω ∈ Dom(E B ): In fact, on noting that that
G(ω) is convex and closed in Rl+ by the conditions A-1, A-2, A-3 and A-4,
the assertion immediately follows from the fact that v 5 0 for all v ∈ G(ω)
by the separation theorem7 : Suppose to the contrary. Let ω0 ∈ Ω and v0 ∈
0
G(ω0 ) with v0 0. Take P
y0 ∈ Ass(E B ) such
0) =
P that0 for all t, Ut (y (t, ω), ω
Ut (x(t, ω0 ), ω0 ) and v0 = t∈T x(t, ω0 ) − t∈T y (t, ω0 ). Let z ∈ Alc(E B ) be
v0
defined by z(t, ξ) := y0 (t, ω0 ) + |T
| if ξ ∈ Π(ω0 ), z(t, ξ) := x(t, ξ) if not. By A-4
it follows that for all t ∈ T , Et [Ut (z)] = Et [Ut (x)], in contradiction to which
x is ex-ante Pareto optimal. By a similar argument in the proof of the second
∗
fundamental theorem of welfare economics,8 we P
can verify that (p
P (ω), x(·, ω)) ∈
W(E B (ω)) for some initial endowment e0 with t∈T e0 (t, ω) = t∈T e(t, ω).
Now, let p be the price system defined by: p(ξ) := p∗ (ω)for all ξ ∈ Π(ω) and ω ∈
Dom(E B ), p(ξ) := p∗ (ω) for ω ∈
/ Dom(E B ). Further we extend e0 to the initial
B
endowment w for E by w(t, ξ) := e0 (t, ω) for all ξP
∈ Π(ω) and ω ∈PDom(E B ). It
can be observed that w(t, ·) is F-measurable with t∈T w(t, ω) = t∈T e0 (t, ω).
To conclude the proof we shall show that (p, x) ∈ EB(E B ). Proof: For each
ω ∈ Dom(E B ), there exists ξ such that ξ ∈ (∆(p) ∩ Pt )(ω) = ∆(p)(ξ) = Π(ξ),
and so we can observe by A-3 that for each x ∈ Alc(E B ), Et [Ut (x(t, ·))|(∆(p) ∩
Pt )](ω) = Ut (x(t, ξ), ξ). We shall verify EB3 only: Suppose to the contrary
that there exists s ∈ T with the two properties: (i) there is an F-measurable
function y(s, ·) : Ω → Rl+ such that y(s, ω) ∈ Bs (ω, p) for all ω ∈ Ω; and
(ii) Es [Us (y(s, ·))|(∆(p) ∩ Ps )](ω0 ) Es [Us (x(s, ·)|(∆(p) ∩ Ps )](ω0 ) for some
ω0 ∈ Dom(∆(p) ∩ Ps ). In view of the above equation it follows from (ii) that
there exists ξ ∈ (∆(p) ∩ Pt )(ω0 ) with Us (y(s, ξ), ξ) Us (x(s, ξ), ξ), and thus
y(s, ξ) x(s, ξ) by A-4. Thus p(ξ) · y(s, ξ) p(ξ) · x(s, ξ), in contradiction.
This completes the proof of Theorem 2.
6
Concluding Remarks.
More recently, researchers in Economics, AI, and Computer Science become
entertained lively concerns about relationships between knowledge (belief) and
actions. At what point does an economic agent (a trader) sufficiently know to
stop gathering information and make decisions? At the heart of any analysis
of such situations as a conversation, a bargaining session or a protocol run by
processes is the interaction between agents. An agent in a group must take into
account not only events that have occurred in the world but also the knowledge
(belief) of the other agents in the group. The most interest to us is the emphasis
7
8
C.f.: Lemma 8, Chapter 4 in K. J. Arrow and F. H. Hahn [1]
C.f.: proposition 16.D.1 in A. Mas-Colell et al [11], pp. 552–554.
11
on considering the situation involving the distributed knowledge (belief) of multiagents rather than of just a single agent.
On considering the situation that each agent makes his/her decisions under
uncertainty, the uncertainty is modeled by a partition structure on a state-space
by which the traders receive information. The partition structure just corresponds to the Kripke semantics for the modal logic S5; and the logic is closely
related to the theory of rough sets, which is applied to many practical fields in
data base theory, especially in data mining.
Our real concern in this article is about the role of traders’ belief on their
decision making, especially when and how the traders take corporate actions
under their decisions. We focus on giving the logical foundation of the welfare
theorem in an economy with emphasis on logic of belief related to the theory
of rough sets. The traders are expectations utility maximizers under their information. We have shown that when each player receives the information of
prices from the market, the nature of the theorem depends not on the partition
structure of traders’ information consisting of exact sets components but on the
non-partition structure of their information consisting of rough sets components
associated with the logic of belief.
References
1. K. J. Arrow and F. H. Hahn: General competitive analysis. North-Holland Publishing Company, Amsterdam (1971) xii + 452pp.
2. M. O. Bacharach: Some extensions of a claim of Aumann in an axiomatic model
of knowledge, Journal of Economic Theory 37 (1985) 167–190.
3. B. F. Chellas, Modal Logic: An introduction. Cambridge University Press, Cambridge, London (1980) x + 295pp.
4. R. Fagin, J.Y. Halpern, Y Moses and M.Y. Vardi: Reasoning about Knowledge.
The MIT Press, Cambridge, Massachusetts, London, England (1995) xiii + 477pp.
5. F. Forges, E. Minelli, and R. Vohra: Incentive and core of an exchange economy:
a survey, Journal of Mathematical Economics 38 (2002) 1–41.
6. J. Geanakoplos: Game theory without partitions, and applications to speculation
and consensus, Cowles Foundation Discussion Paper No.914 (1989) Yale University. (Available in http://cowles.econ.yale.edu)
7. Luo, X., and C. Ma: “Agreeing to disagree” type results: a decision-theoretic approach, Journal of Mathematical Economics 39 (2003) 849–861.
8. Milgrom, P. and N. Stokey: Information, trade and common knowledge, Journal
of Economic Theory 26 (1982) 17–27.
9. Morris, S. and C. Skiadas: Rationalizable trade, Games and Economic Behavior
31 (2000) 311–323.
10. Neeman, Z.; Common beliefs and the existence of speculative trade, Games and
Economic Behavior 16 (1996) 77–96.
11. A. Mas-Colell, M. Whinston, and J. Green: Microeconomics Theory. Oxford University Press (1995) xvii + 981pp. .
12
Proof of Theorem 19
A
We establish the finite model property for the logic B.
A.1
Proof Sets
Let L be a system of multi-modal logic. We recall the Lindenbaum’s lemma:
Lemma 1. Let L be a system of multi-modal logic. Every L-consistency set of
sentences has an L-maximal extension.
Proof. This is because L includes the ordinary propositional logic.
We call the extension an L-maximally consistent set. As a consequence, we
can observe the fact that a sentence in L is deducible from a set of sentences Γ
if and only if it belongs to every L-maximally consistent set of Γ , and thus
Theorem 3. Let L be a system of multi-modal logic. A sentence is a theorem of
L if and only if it is a member of every L-maximally consistent set of sentences.
We denote by |ϕ|L the class of L-maximally consistent sets of sentences containing the sentence ϕ; this is called the proof set of ϕ. We note that
Corollary 1. Let L be a system of multi-modal logic.
(i) A sentence ϕ is a theorem of L if and only if |ϕ|L = ΩL ;
(ii) A sentence ϕ → ψ is a theorem of L if and only if |ϕ|L ⊆ |ψ|L .
A.2
Canonical Model
Let L be a system of multi-modal logic. The canonical model for L is the model
ML = hT, ΩL , (BtL )t∈T , (AL
t )t∈T , VL i for L consisting of:
1.
2.
ΩL is the set of all the L-maximally consistent sets of sentences;
BtL : 2ΩL → 2ΩL is given by
ω ∈ BtL (E) if and only if there exists ϕ ∈ L such that |ϕ|L = E and t ϕ ∈ ω
3.
ΩL
AL
→ 2ΩL is given by
t :2
ω ∈ AL
t (E) if and only if there exists ϕ ∈ L such that |ϕ|L = E and t ϕ ∈ ω
4.
VL is the mapping such that VL (ω, Pm ) = {ω ∈ Ω L | Pm ∈ ω} for
m = 0, 1, 2, . . . .
We can easily observe that
Proposition 4. The canonical model ML is a model for a system L of multimodal logic.
9
The proof is given by the same way in Chellas [3], and the appendix will be deleted
in the final version.
13
A.3
Filtration of Model
Let M be a model hT, ΩL , (Bt )t∈T , (Pt )t∈T , V i for a normal system L of multimodal logic. For each set of sentences Γ , we define the equivalence relation ≡ on
M
Ω as follows: ω ≡ ξ if and only if for every sentence ψ of Γ , |=M
ω ψ ⇐⇒|=ξ ψ .
We denote by [ω]Γ the equivalence class of ω and denote by [X]Γ the set of
equivalence classes [ω]Γ for all ω of X whenever X is a subset of Ω.
By the Γ -filtration MΓ (or filtration of MΓ through Γ ), we mean the model
Γ
M = hT, Ω Γ , (BtΓ )t∈T , (PtΓ )t∈T , V Γ i for L, which consists of: For each t ∈ T ,
Ω Γ = [Ω]Γ ;
BtΓ : 2ΩL → 2ΩL is given by BtΓ ([E]Γ ) = [Bt (E)]Γ ;
PtΓ : 2ΩL → 2ΩL is given by PtΓ ([E]Γ ) = [Pt (E)]Γ ;
V Γ ([ω]Γ , Pm ) = VL (ω, Pm ) .
1.
2.
3.
4.
Remark 3. The Γ -filtration MΓ is a well-defined model for the system L; i.e., it
is actually a model for L in which the both mappings BtΓ and PtΓ are independent
of the choices of events in each equivalence class, and V Γ is independent of the
choices of states in each equivalence class
By induction on the complexity of a sentence ϕ we can plainly verify that
Proposition 5. Let M be a model for a normal system L and Γ a set of sentences closed under subsentences. Then the following two properties are true:
Γ
(i) For every sentence ϕ in Γ , |=M ϕ if and only if |=M ϕ .
(ii) The model MΓ is finite if so is Γ .
The important result about a canonical model is the following:
Basic theorem. Let ML be the canonical model for a system L of traders’
knowledge. Then for every sentence ϕ, |=ML ϕ if and only if `L ϕ . In other
words, ||ϕ||ML = |ϕ|ML .
Proof. By induction on the complexity of ϕ. We treat only that ϕ is t ψ. As an
inductive hypothesis we assume that ||ψ||ML = |ψ|ML . Then for every ω ∈ ΩL ,
L
|=M
t ψ if and only if ω ∈ BtL (||ϕ||ML ), by the definition of validity ;
ω
if and only if ω ∈ BtL (|ϕ|ML ), by the inductive hypothesis as above;
if and only if `L t ψ, by the definition of canonical model.
A.4
Proof of Theorem 1
Soundness has been already observed in Proposition 4. The completeness will
be shown by the way of contradiction as follows: Suppose that some sentence
ϕ is not a theorem in L. In view of Basic theorem A.3, it follows that ϕ is not
valid for a canonical model ML . Let Γ be the set of all subsentences of ϕ. By
Proposition 5 we can observe that ϕ is not valid for the Γ -filtration MΓL ∈ CF ,
in contradiction.
14