Welfare Economy under Rough Sets Information

1
Welfare Economy under Rough Sets Information
OR 2006 Karlsruhe

• Takashi Matsuhisa
• Ibaraki National College of Technology
• Ibaraki 312-8508, Japan
• E-mail mathisa_at_ge.ibaraki-ct.ac.jp

September 8, 2006
2
Background

• 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.

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Aim and Scope

• 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.

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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.

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Chronicle of Extensions
Economy
Result
Author(s)
Information sets
Uncertainty
Aumann

Pt (?)??????(Exact set)
Core equiv
(1962)
Geanakoplos
?
Pt non Partition (Ref, Trn Rough set)
No Trade
(1989)
?
Einy et al
Core equiv
Pt Partition(Exact set)
(2000)
Pt non Partition (Ref Rough set)
?
Matsuhisa and Ishikawa (2005)
Core equiv
Pt non Partition(None Rough set)
?
Matsuhisa
Welfare
(2005)
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Outline

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

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Economy under Uncertainty

• ltT, S, m, W, e, (Ut)t?T, (pt)t?T, (Pt)t?T,gt
• l the number of commodities
• Rl the consumption set of trader t
• T a finite set of traders t?T
• e TW ? Rl an initial endowment
• Ut RlW ?R ts utility function
• pt subjective prior on W for t?T
• Pt partition on W which represents trader
  ts uncertainty

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Economy on Belief

• ltT, W, e, (Ut)t?T, (pt)t?T, (Bt)t?T, (Pt)t?Tgt
• l the number of commodities
• Rl the consumption set
• T a finite set of traders t
• e TW ? Rl an endowment
• Ut RlW ?R ts utility function initial
• pt subjective prior on W for t?T
• lt W, (Bt)t?T, (Pt) t?T gt the belief structure

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Belief structure
lt W, (Bt)t?T, (Pt)t?T gt

• W a non-empty finite set of states
• 2 W?E an event
• T a set of traders
• E ?w E occurs at w

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Belief structure
lt W, (Bt)t?T, (Pt)t?T gt

• ts belief operator Bt 2 W ? 2 W
• Bt E ?w t believes E at w
• ts possibility operator
• Pt 2 W ? 2 W , E ? Pt(E) W \ Bt (W \ E)
• Pt E ?w E is possible for t at w
• Pt(w) Pt(w) ts information set at w

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Livedoor v.s. Fuji TV Japan
L
F
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L-F Example
T L, F
Belief structure
E f w1 w2 W
BL E f w1 w2 W
BF E f w1 f W
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L-F Example
T L, F
The possibility operators
E f w1 w2 W
PL E f w1 w2 W
PF E f W w2 W
The Information Sets Pt(w) Pt(w)
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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 ltW, (Bt )gt 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.

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Economy on Belief
lt T, S, m, W, e, (Ut)t?T, (pt)t?T, (Bt)t?T,
(Pt)t?Tgt
(A-1) S e (t, w) ? 0
t?T

• Dom (Pt) w Pt(?) ? f
• the domain of Pt
• (A-2) For ?t, Dom (Pt) Dom (Ps) ? f

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Allocations

• An assignment x TW ? Rl
• An allocation a TW ? Rl
• S a (t, w) ? S e (t, w)

t ?T
t ?T
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Price and Budget

• Price system p W ? Rl ?0

?(p) the partition of W induced by p
?(p)(w) x p(?) p(w)
the information given by p at w
Budget set of t at w Bt(w, p) x p(w)x ?
p(w)e(t, w)
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Expectations in belief

• ts interim expectation
• EtUt(x (t, )) ?(p)nPt (w)
• ?Ut(x (t, x),x))pt(x ?(p) (w)nPt(w))
• ts ex-ant expectation
• EtUt(x (t, )(w) ?Ut(x (t, x),x))pt(x)

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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) ? EtUt(x(t, ))?(p)nPt(w) ?
EtUt(y(t, ))?(p)nPt(w) (EE3) S x(t, w) S
e(t, w)
if
t?T
t?T

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Existence Theorem for EE
Trader t is risk averse if (A-3) Ut(x , )
strictly increasing, quasi -concave on Rl,
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 Whats 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.
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Pareto Optimality
An allocation a ex-ante Pareto optimal
if there is no allocation x such that
(1) ?t?T EtUt(x (t, )) ?
EtUt(a (t, ))
(2) ?s?T EsUs(x (s, )) gtEsUs(a (s,
))
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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.
?
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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 players
  information can potentially yield important
  results in a world with imperfectly Bayesian
  agents

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• Thank you!
• Danken !