OR ２００６ Karlsruhe
Welfare Economy under
Rough Sets Information
Takashi Matsuhisa
Ibaraki National College of Technology
Ibaraki 312-8508, Japan
E-mail: mathisa@ge.ibaraki-ct.ac.jp
September 8, 2006
1
Background
Economy under uncertainty consists of

Economy:
Trader set, Consumption set, Utility functions
 Uncertainty
1. By Exact set information:
Partition structure on a state-space, or equiv.
Knowledge structure.
2. By Rough sets information:
Non-partition structure on a state-space, or
equiv. Belief structure.
2
Aim and Scope

Economy under Exact Sets Information
1.
Core equivalence theorem: There is no incentive among
all traders to improve their equilibrium allocations.
Fundamental Theorem for Welfare Economy: Each Pareto
optimal allocation is an equilibrium allocation.
Others; e.g., No trade theorem: There is no trade among
traders if the initial endowments are an equilibrium.
2.
3.

Economy under Rough Sets Information
 Can we extend these results into the economy
 There are a few extensions of “No trade.”
 We extend the welfare theorem.
3
Purpose
1. “Rough sets” information structure
induced from a belief structure
2. Economy with belief structure and
expectation equilibria in belief
3. Characterization of the extended
equilibria by Ex-post Pareto optimal
allocations in traders.
4
Chronicle of Extensions
Author(s)
Aumann
Result
(1962) Core equiv
Geanakoplos
(1989)
No Trade
Einy et al (2000) Core equiv
Matsuhisa and
Core equiv
Ishikawa (2005)
Matsuhisa (2005)
Welfare
Economy
Uncertainty
Information sets
×
Pt ()(Exact set)
○
Pt := non Partition
○
Pt := Partition（Exact set）
○
Pt := non Partition
○
（Ref, Trn: Rough set）
（Ref: Rough set）
Pt := non Partition（None:
Rough set）
5
Outline
Belief structure and Rough sets
information
 Economy on belief
 Expectations equilibrium in Belief
 Fundamental Theorem for Welfare
 Remark

6
Economy under Uncertainty
〈T, S,m,W, e, (Ut)t∈T, (πt)t∈T, (Pt)t∈T,〉







l : the number of commodities
R+l : the consumption set of trader t
T: a finite set of traders t∈T
e : T×W R+l : an initial endowment
Ut : R+l×W→R ： t’s utility function
πt : subjective prior on W for t∈T
Pt : partition on Wwhich represents trader t’s
uncertainty
7
Economy on Belief
〈T, W, e, (Ut)t∈T, (πt)t∈T, (Bt)t∈T, (Pt)t∈T〉

l : the number of commodities
R+l : the consumption set

T : a finite set of traders t

e : T×WR+l : an endowment
Ut : R+l×W→R ： t’s utility function initial
πt : subjective prior on Wfor t∈T
〈 W, (Bt)t∈T, (Pt) t∈T 〉: the belief structure




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Belief structure
〈 W, (Bt)t∈T, (Pt)t∈T 〉




W : a non-empty finite set of states
2 W∋E : an event
T : a set of traders
E ∋: “E occurs at ”
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Belief structure
〈 W, (Bt)t∈T, (Pt)t∈T 〉
t’s belief operator Bt : 2 W → 2 W
 Bt E ∋ : “t believes E at ”
 t’s possibility operator
Pt : 2 W → 2 W,E → Pt(E):= W∖ Bt (W∖ E)
 Pt E ∋ : “E is possible for t at ”

Pt():＝ Pt({) : t’s information set at 
10
Livedoor v.s. Fuji TV Japan
L
F
11
L-F Example
T = { L, F }
1 = L does not commit the
injustice
W= {1 , 2 }
2 = L commits the injustice
Belief structure:
E
BL E
φ
φ
{1}
{1}
{2}
{2}
W
W
BF E
φ
{1}
φ
W
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L-F Example T = { L, F }
The possibility operators
E
PL E
φ
φ
{1}
{1}
{2}
{2}
W
W
PF E
φ
W
{2}
W
The Information Sets： Pt()＝ Pt({})
1
2
PL
1
2
PF
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Rough Set Theory
An event E is exact if Pt(E) = Bt (E)
 An event E is rough if Pt(E) ≠ Bt(E)

 If 〈W, (Bt )〉 is the Kripke semantics for
Modal logic S5 then {Pt()|∈W} is a
partition of W,and every Pt() is exact.
 Our interest is the case that Pt() does
not make a partition, and so Pt() is rough
in general.
14
Economy on Belief
〈 T, S, m, W, e, (Ut)t∈T, (πt)t∈T, (Bt)t∈T, (Pt)t∈T〉
(A-1) Se (t, ) ≩ ０
t∈T
Dom (Pt) := { | Pt(ω） ≠ φ }
= the domain of Pt
(A-2) For ∀t, Dom (Pt) = Dom (Ps) ≠ φ
15
Allocations

An assignment x : T×W R+l

An allocation a : T×WR+l
Sa (t, ) ≦ S e (t, )
t ∈T
t ∈T
16
Price and Budget
Price system p : WR+l ≠0
⊿(p) = the partition of W induced by p;
⊿(p)() = { x | p(ξ) = p() }
= “the information given by p at ”
Budget set of t at 
Bt(, p) = { x | p()x ≦ p()e(t, ) }
17
Expectations in belief

t’s interim expectation
Et[Ut(x (t, * )) | ⊿(p)∩Pt ]()
:= ∑Ut(x (t, x),x))πt({x} | ⊿(p) ()∩Pt())
x∈Dom( Pt )

t’s ex-ant expectation
Et[Ut(x (t, * )]():= ∑Ut(x (t, x),x))πt({x})
x∈Dom( Pt )
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Expectation equilibrium in belief
(p, x) : = an expectations equilibrium in belief
(EE1) x(t, ) ∈ Bt(, p)
(EE2) y(t, ) ∈ Bt(, p)
if
⇒ Et[Ut(x(t, *))|⊿(p)∩Pt]()
≧ Et[Ut(y(t, *))|⊿(p)∩Pt]()
(EE3) Sx(t, ) = Se(t, )
t∈T
t∈T
19
Existence Theorem for EE
Trader t is risk averse if:
(A-3) Ut(x , ∙) = ‘‘strictly increasing, quasi concave on R+l, etc’’
Measurability of Utility:
(A-4) Ut(x , ∙) = ‘‘measurable for the finest
field generated by Pt for all t ∈ T ’’
Theorem 1: Economy on belief with (A-1),
(A-2), (A-3) and (A4). There exists an
expectation equilibrium.
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Question
Question : What’s characteristics of the
expectations equilibrium in belief?
Answer 1. Welfare theorem: The expectations
equilibrium is an ex-ante Pareto-optimal
Answer 2. Core equivalence: The expectations
equilibrium is a core allocation, and vice versa.
21
Pareto Optimality
An allocation a = ‘‘ex-ante Pareto optimal’’
if there is no allocation x such that
(1) ∀t∈T
Et[Ut(x (t, * ))] ≧ Et[Ut(a (t, * ))]
(2) ヨs∈T
Es[Us(x (s, * ))] >Es[Us(a (s, * ))]
22
Welfare Theorem
Economy with belief structure:
(A-1), (A-2), (A-3), and (A-4)
An allocation a = ‘‘ex-ante Pareto optimal’’
For ∃p = price,
⇔ (p, a) = ‘‘an expectations equilibrium’’
for some initial endowments.
23
Concluding remark

Propose an extended economy under rough sets
information.
 Emphasize with the epistemic aspect of belief of
the traders
 Remove out: Partition structure of traders’
information.
 Extend Fundamental Theorem for Welfare.
Bounded rationality point of view :The relaxation of the
partition structure for player’s information can
potentially yield important results in a world with
imperfectly Bayesian agents
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Thank you!
Danken !
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