Welfare Economy under Rough Sets Information

1. 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

2. Economy under uncertainty consists of Economy: Trader set, Consumption set, Utility functions Uncertainty By Exact set information: Partition structure on a state-space, or equiv. Knowledge structure. By Rough sets information: Non-partition structure on a state-space, or equiv. Belief structure. Background

3. Economy under Exact Sets Information 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. 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. Aim and Scope

4. Purpose • “Rough sets” information structure induced from a belief structure • Economy with belief structure and expectation equilibria in belief • Characterization of the extended equilibria by Ex-post Pareto optimal allocations in traders.

5. Chronicle of Extensions Economy Result Author(s) Information sets Uncertainty Aumann × Pt ()(Exact set) (1962) Core equiv Geanakoplos ○ Pt := nonPartition （Ref, Trn: Rough set） No Trade (1989) ○ Einy et al Core equiv Pt := Partition（Exact set） (2000) Pt := nonPartition （Ref: Rough set） ○ Matsuhisa and Ishikawa (2005) Core equiv Pt := non Partition（None: Rough set） ○ Matsuhisa Welfare (2005)

6. Outline • Belief structure and Rough sets information • Economy on belief • Expectations equilibrium in Belief • Fundamental Theorem for Welfare • Remark

7. 〈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: afinite set of traders t∈T e : T×W → R+l : aninitial endowment Ut:R+l×W →R： t’sutility function πt: subjective prior on Wfort∈T Pt: partition on W whichrepresents trader t’s uncertainty Economy under Uncertainty

8. 〈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 : anendowment Ut:R+l×W →R： t’s utility function initial πt: subjective prior on W fort∈T 〈 W, (Bt)t∈T, (Pt) t∈T〉: the belief structure Economy on Belief

9. Belief structure • W : a non-empty finite set ofstates • 2W∋E : anevent • T : a set oftraders • E ∋w : “E occurs atw” 〈W, (Bt)t∈T, (Pt)t∈T〉

10. Belief structure • t’sbeliefoperator Bt:2W → 2W • BtE ∋w : “t believes E atw” • t’s possibility operator Pt : 2W → 2W , E → Pt(E):= W ∖ Bt (W ∖ E) • PtE ∋w : “E is possible for t atw” Pt(w):＝ Pt({w}) : t’s information setat w 〈W, (Bt)t∈T, (Pt)t∈T〉

11. Livedoor v.s. Fuji TV Japan L F

12. W = {w1, w2} w1 = L does not commit the injustice w2= L commits the injustice L-F Example T = {L, F} Belief structure:

13. w1 w2 PL PF w1 w2 L-F Example T = {L, F} The possibility operators The Information Sets： Pt(w)＝ Pt({w})

14. 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)|w∈W} is a partition of W, and every Pt(w) is exact. • Our interest is the case that Pt(w) does not make a partition, and so Pt(w) is rough in general.

15. Economy on Belief Dom (Pt) := { w | Pt(ω） ≠φ } = the domain of Pt (A-2) For ∀t, Dom (Pt) = Dom (Ps)≠ φ 〈 T, S, m, W, e, (Ut)t∈T, (πt)t∈T,(Bt)t∈T,(Pt)t∈T〉 (A-1) S e (t, w) ≩０ t∈T

16. An assignmentx: T×W →R+l An allocation a : T×W → R+l S a(t, w) ≦Se(t, w) Allocations t∈T t ∈T

17. Price systemp : W → R+l≠0 Price and Budget ⊿(p) = the partition of W induced by p; ⊿(p)(w) = { x | p(ξ) = p(w) } = “the information given by pat w” Budget set of t at w Bt(w, p) = { x | p(w)x ≦ p(w)e(t, w) }

18. t’s interim expectation Et[Ut(x(t, * )) | ⊿(p)∩Pt ](w) := ∑Ut(x(t, x),x))πt({x} | ⊿(p) (w)∩Pt(w)) t’s ex-ant expectation Et[Ut(x(t, * )](w):= ∑Ut(x(t, x),x))πt({x}) x x Dom( Dom( ) ) ∈ ∈ P P t t Expectations in belief

19. Expectation equilibrium in belief (p, x) : = an expectations equilibrium in belief (EE1) x(t, w) ∈Bt(w, p) (EE2) y(t, w) ∈Bt(w, p) ⇒ Et[Ut(x(t, *))|⊿(p)∩Pt](w) ≧ Et[Ut(y(t, *))|⊿(p)∩Pt](w) (EE3) S x(t, w) = S e(t, w) if t∈T t∈T

20. 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’’ Theorem1: Economy on belief with (A-1), (A-2), (A-3) and (A4). There exists an expectation equilibrium.

21. 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.

22. Pareto Optimality An allocation a = ‘‘ex-ante Pareto optimal’’ if there is no allocationxsuch that (1) ∀t∈T Et[Ut(x (t, * ))] ≧ Et[Ut(a (t, * ))] (2) ヨs∈T Es[Us(x (s, * ))] >Es[Us(a (s, * ))]

23. 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. ⇔

24. 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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