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Welfare Economy under Welfare Economy under Rough Sets InformationRough Sets Information

Takashi MatsuhisaTakashi MatsuhisaIbaraki National College of TechnologyIbaraki National College of Technology

Ibaraki 312-8508, JapanIbaraki 312-8508, Japan

E-mail: mathisa@ge.ibaraki-ct.ac.jpE-mail: mathisa@ge.ibaraki-ct.ac.jp

OR ２００６ Karlsruhe

September 8, 2006

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BackgroundBackground

Economy under uncertaintyEconomy under uncertainty consists of consists of EconomyEconomy::

Trader set, Consumption set, Utility functionsTrader set, Consumption set, Utility functions UncertaintyUncertainty

1.1. ByBy Exact set information: Exact set information:

Partition structure on a state-space, or equiv. Partition structure on a state-space, or equiv.

Knowledge structure.Knowledge structure.

2.2. By By Rough sets information:Rough sets information:

Non-partition structure on a state-space, or Non-partition structure on a state-space, or equiv. Belief structure.equiv. Belief structure.

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Aim and ScopeAim and Scope Economy under Economy under Exact Sets InformationExact Sets Information1.1. Core equivalence theoremCore equivalence theorem:: There is no incentive among There is no incentive among

all traders to improve their equilibrium allocations.all traders to improve their equilibrium allocations.

2.2. Fundamental Theorem for Welfare Economy: Fundamental Theorem for Welfare Economy: Each Pareto Each Pareto optimal allocation is an equilibrium allocation.optimal allocation is an equilibrium allocation.

3.3. Others; e.g., No trade theoremOthers; e.g., No trade theorem: : There is no trade among There is no trade among traders if the initial endowments are an equilibrium.traders if the initial endowments are an equilibrium.

Economy under Economy under Rough Sets InformationRough Sets Information Can we extend these results into the economyCan we extend these results into the economy There are a few extensions of “No trade.”There are a few extensions of “No trade.” We extend the welfare theorem.We extend the welfare theorem.

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PurposePurpose

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.

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Matsuhisa and Ishikawa Matsuhisa and Ishikawa (2005) (2005)

GeanakoplosGeanakoplos

Author(s)Author(s)

Chronicle of ExtensionsChronicle of Extensions

AumannAumann

Einy et alEiny et al

MatsuhisaMatsuhisa

ResultResult

Core equivCore equiv

No TradeNo Trade

Core equivCore equiv

Core equivCore equiv

Welfare Welfare

EconomyEconomy

○

○

○

○

×PPt t (Exact set)

PPt t := non:= non PartitionPartition

（（ Ref, Trn: Rough setRef, Trn: Rough set ））

PPt t := := PartitionPartition （（ Exact Exact

setset ））

PPt t := non:= non Partition（（ None: Rough None: Rough setset ））

PPt t := non:= non PartitionPartition

（（ Ref: Rough setRef: Rough set ））

Uncertainty　　　　　 Information sets

(1989)(1989)

(1962)(1962)

(2000)(2000)

(2005)(2005)

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OutlineOutline

Belief structure and Rough sets Belief structure and Rough sets informationinformation

Economy on belief Economy on belief Expectations equilibrium in BeliefExpectations equilibrium in Belief Fundamental Theorem for WelfareFundamental Theorem for Welfare RemarkRemark

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Economy under UncertaintyEconomy under Uncertainty

〈〈 TT,, , , ee, , ((UUtt))t∈t∈TT, , (π(πtt))tt∈∈TT,, ((tt))tt∈∈TT,, 〉〉 ll : the number of : the number of commoditiescommodities RR++

ll : the : the consumption setconsumption set of trader of trader tt TT: : aa finite set of traders finite set of traders tt∈∈TT ee : : TT××→→RR++

ll : : anan initial endowmentinitial endowment UUtt :: RR++

ll××→→RR ： ： tt’s ’s utility functionutility function ππtt : subjective : subjective priorprior on on forfor tt∈∈TT tt : : partitionpartition on on whichwhichrepresents represents tradertrader t’ t’s s

uncertaintyuncertainty

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Economy on BeliefEconomy on Belief

〈〈 TT,, , , ee, , ((UUtt))t∈t∈TT, , (π(πtt))tt∈∈TT,, ((BBtt))tt∈∈TT,, ((PPtt))tt∈∈TT 〉〉 ll : the number of commodities: the number of commodities RR++

ll : the consumption set : the consumption set TT : a finite set of traders : a finite set of traders tt ee : : TT××→→RR++

ll : : anan endowment endowment UUtt :: RR++

ll××→→RR ： ： tt’s utility function initial’s utility function initial ππtt : subjective prior on : subjective prior on forfor tt∈∈TT 〈〈 , (, (BBtt))tt∈∈TT, (, (PPtt) ) tt∈∈TT 〉〉 :: the the belief structurebelief structure

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Belief structureBelief structure

: a non-empty finite set of : a non-empty finite set of statesstates 22 ∋∋EE : an : an eventevent TT : a set of : a set of traderstraders E E ∋∋: “: “E E occurs atoccurs at ””

〈〈 , , ((BBtt))tt∈∈TT, , ((PPtt))tt∈∈TT 〉〉

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Belief structureBelief structure

tt’s ’s beliefbelief operator operator BBtt :: 22 → → 22

BBtt E E ∋∋ : “ : “tt believes believes E E atat ””

tt’s ’s possibility operatorpossibility operator

PPtt : : 22 → → 22 EE → → PPtt((EE):= ):= ∖ ∖ BBt t ((∖ ∖ EE)) PPtt E E ∋∋ : “ : “E E isis possible possible forfor t t atat ””

PPtt(():): ＝ ＝ PPtt({({)) : : tt’s ’s information setinformation set at at 　　

〈〈 , , ((BBtt))tt∈∈TT, , ((PPtt))tt∈∈TT 〉〉

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Livedoor v.s. Fuji TV JapanLivedoor v.s. Fuji TV Japan

L F

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L-F ExampleL-F Example

11 = L does not commit the = L does not commit the

injustice injustice

= L commits the injustice= L commits the injustice

= {= { , , }}

T T = {= { LL, , FF }}

EE φφ {{11}} {{22}}

BBL L EE φφ {{11}} {{22}}

BBFF EE φφ {{11}} φφ

Belief structure:Belief structure:

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L-F ExampleL-F Example

PPFF

PPLL

The possibility operatorsThe possibility operators

T T = {= { LL, , FF }}

EE φφ {{11}} {{22}}

PPL L EE φφ {{11}} {{22}}

PPFF EE φφ {{22}}

The Information SetsThe Information Sets ： ： PPtt(()) ＝ ＝ PPtt({({})})

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Rough Set TheoryRough Set Theory

An event E is exact if Pt(E) = Bt (E) An event E is rough if Pt(E) ≠ Bt(E)

If 〈 , (Bt ) 〉 is the Kripke semantics for Modal logic S5 then {Pt()|∈} is a partition of and every Pt() is exact.

Our interest is the case that Pt() does not make a partition, and so Pt() is rough in general.

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Dom (Pt) := { | Pt(ω ） ≠ φ }

= the domain of Pt

(A-2) For ∀t, Dom (Pt) = Dom (Ps) ≠ φ

t∈T

Economy on Belief

(A-1) e (t, ) 　≩　０

〈 T, , , , e, (Ut)t∈T, (πt)t∈T, (Bt)t∈T, (Pt)t∈T 〉

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AllocationsAllocations

An assignmentAn assignment xx : : TT××→→RR++

ll

An allocation An allocation aa : : TT××→→RR++l l

aa ((tt, , ) ) ≦≦ ee ((tt, , )) t ∈T t ∈T

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Price and BudgetPrice and Budget

Price systemPrice system pp : : →→RR++ll ≠≠00

Budget setBudget set of of tt at at

BBtt((,, p p) = { ) = { xx | | pp(())xx ≦ ≦ pp(())ee((tt, , ) }) }

⊿⊿((pp))(() = { ) = { | | pp((ξξ) = ) = pp(() } ) }

= “the information given by = “the information given by pp at at ””

⊿⊿((pp) = the partition of ) = the partition of induced by induced by pp; ;

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Expectations in beliefExpectations in belief

tt’s’s interim interim expectation expectation

EEtt[[UUtt((xx ((tt, , ** )) | )) | ⊿(⊿(pp)∩)∩PPt t ]](())

:= := ∑∑UUtt((xx ((tt, , )),,))π))πtt({({} | } | ⊿(⊿(pp) ) ((∩∩PPtt(())))

tt’s’s ex-ant ex-ant expectation expectation

EEtt[[UUtt((xx ((tt, , ** )] )](()):= := ∑∑UUtt((xx ((tt, , )),,))π))πtt({({})}) 　　

∈Dom(Pt)

∈Dom(Pt)

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((pp, , xx) : = an ) : = an expectations equilibrium in beliefexpectations equilibrium in belief

if

(EE1) (EE1) xx((tt, , ) ) ∈∈ BBtt((, , pp))

(EE2) (EE2) yy((tt, , ) ) ∈∈ BBtt((, , pp))

⇒ ⇒ EEtt[[UUtt((xx((tt, , **))| (⊿))| (⊿ pp))∩∩PPtt](]())

≧ ≧ EEtt[[UUtt((yy((tt, , **))| (⊿))| (⊿ pp))∩∩PPtt](]())

(EE3) (EE3) xx((tt, , ) = ) = ee((tt, , ))t∈T t∈T

Expectation equilibrium in beliefExpectation equilibrium in belief

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Existence Theorem for EE

Theorem 1: Economy on belief with (A-1), (A-2), (A-3) and (A4). There exists an expectation equilibrium.

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∈ ’’

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QuestionQuestion

Answer 1. Answer 1. Welfare theoremWelfare theorem: The expectations : The expectations equilibrium is an ex-ante Pareto-optimalequilibrium is an ex-ante Pareto-optimal

Answer 2. Answer 2. Core equivalenceCore equivalence: The expectations : The expectations equilibrium is a core allocation, and vice versa.equilibrium is a core allocation, and vice versa.

Question Question : What’s characteristics of the : What’s characteristics of the expectations equilibrium in belief?expectations equilibrium in belief?

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An allocation An allocation aa = ‘‘ = ‘‘ex-anteex-ante Pareto optimal’’ Pareto optimal’’

Pareto OptimalityPareto Optimality

if there ishere is no no allocationallocation xx such thatsuch that (1) (1) ∀∀tt∈∈TT EEtt[[UUtt((xx ( (tt, , ** ))] ≧ ))] ≧ EEtt[[UUtt((aa ( (tt, , ** ))] ))]

(2) (2) ヨヨ ss∈∈TT EEss[[UUss((xx ( (ss, , ** ))] > ))] >EEss[[UUss((aa ( (ss, , ** ))] ))]

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Welfare TheoremWelfare Theorem

⇔⇔

　 Economy Economy with belief structure:with belief structure:

(A-1), (A-2), (A-3), and (A-4)(A-1), (A-2), (A-3), and (A-4)

　　 An allocation An allocation aa = ‘‘ex-ante Pareto = ‘‘ex-ante Pareto optimal’’optimal’’

ForFor ∃∃pp = price, = price,

((pp, , aa) = ‘‘an expectations equilibrium’’ ) = ‘‘an expectations equilibrium’’ for some initial endowments. for some initial endowments.

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Concluding remarkConcluding remark

Propose an extended economy under rough sets Propose an extended economy under rough sets information.information.

Emphasize with the epistemic aspect of belief of Emphasize with the epistemic aspect of belief of the tradersthe traders

Remove out: Partition structure of traders’ Remove out: Partition structure of traders’ information.information.

Extend Fundamental Theorem for Welfare.Extend Fundamental Theorem for Welfare.

Bounded rationality point of viewBounded rationality point of view :The relaxation of the :The relaxation of the partition structure for player’s information can partition structure for player’s information can potentially yield important results in a world with potentially yield important results in a world with imperfectly Bayesian agentsimperfectly Bayesian agents

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Thank you!Thank you!Danken !Danken !