The Efficiency of Competitive Mechanisms Under Private Information · 2005-05-27 · Informal...
Transcript of The Efficiency of Competitive Mechanisms Under Private Information · 2005-05-27 · Informal...
Informal statement of the main theoremFormal modelMain theorem
Intuition and Interpretation
The Efficiency of Competitive MechanismsUnder Private Information
Nabil I. Al-Najjar1 Rann Smorodinsky2
1Kellogg School of ManagementNorthwestern University
2Technion
DIMACS Workshop
Northwestern University, April 2005
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
Informal statement of the main theoremFormal modelMain theorem
Intuition and Interpretation
Outline
1 Informal statement of the main theorem
2 Formal modelComplete information modelIncomplete information model
3 Main theoremFormal statementRelationship to the literatureThe price
4 Intuition and InterpretationIntuition for the proofConcluding remarks
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
Informal statement of the main theoremFormal modelMain theorem
Intuition and Interpretation
The problem
We look at large (but finite) exchange economies with1 private values2 private information3 strategic behavior
Trade is organized according to a competitive mechanismWe view any such mechanism as a black boxWe look for results that hold for all competitive mechanisms
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
Informal statement of the main theoremFormal modelMain theorem
Intuition and Interpretation
Type of questions one can ask
1 Do players report their types truthfully when the economyis large?
2 Do competitive mechanisms lead to (approximately)efficient allocations?
1 It is clear that (1) cannot be guaranteed in general2 We show that (2) obtains
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
Informal statement of the main theoremFormal modelMain theorem
Intuition and Interpretation
Informal statement of the main result
Main (and only) Theorem of this paper
Given any desired degree of approximation
There is N such that in any economy with N or moretraders
Any prior distribution on traders’ types
Any Bayesian-Nash equilibrium
Of any competitive mechanism for this economy
.. the resulting allocation is approximately efficient.
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
Informal statement of the main theoremFormal modelMain theorem
Intuition and Interpretation
Important features of the theorem
No replica structure is invoked
No need to make assumptions about the existence of awell defined limit economy
Distribution on types may be correlated (within limits)
Competitive mechanisms may be very sensitive toindividuals’ reports
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
Informal statement of the main theoremFormal modelMain theorem
Intuition and Interpretation
Methodology
Focus on a particular institution: competitive markets
..rather than search for the optimal institution.
Consider all competitive mechnaisms..
..and equilibria of these mechanisms
Why all competitive mechanisms?
We do not understand the process of price formation
Walrasian eq. is a consistency requirement a reasonableoutcome must satisfy; it is not a prediction of any particularoutcome
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
Informal statement of the main theoremFormal modelMain theorem
Intuition and Interpretation
Complete information modelIncomplete information model
Formal ModelComplete information economies
EN = (θN ,wN)
is an N-agent classical exchange economy with completeinformation where:
wN = {wn, n ∈ N} is the vector of endowments
θN = {θn, n ∈ N} is the vector of types
Each θ represents a classical utility function.
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
Informal statement of the main theoremFormal modelMain theorem
Intuition and Interpretation
Complete information modelIncomplete information model
Endowments
Endowments are common knowledge
There exist constants ξ+ > ξ− > 0 such that, uniformlyacross economies,
wn ∈ W ≡ [ξ−(1, . . . ,1), ξ+(1, . . . ,1)].
=⇒ ..each player holds a negligible fraction of totalendowments in the limit.
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
Informal statement of the main theoremFormal modelMain theorem
Intuition and Interpretation
Complete information modelIncomplete information model
Preferences
Finite set of types Θ, with θ ∈ Θ a classical utility functions
Sorting condition: Suppose that hyperplane H supports Uθ
at x , and hyperplane H ′ supports Uθ′ also at x . ThenH 6= H ′.
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
Informal statement of the main theoremFormal modelMain theorem
Intuition and Interpretation
Complete information modelIncomplete information model
Competitive mechanisms
A competitive mechanisms is any selection
σ : EN → ∆× (R l+)N
of the competitive correspondence.
Motivation....
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
Informal statement of the main theoremFormal modelMain theorem
Intuition and Interpretation
Complete information modelIncomplete information model
Efficiency
A feasible allocation xN is (ex post) η-efficient if there is nofeasible allocation xN such that
Uθn(xn) > Uθn(xn) + η, ∀n = 1 . . . ,N.
Eff (θN ,wN) and Effη(θN ,wN) will denote the sets of all efficientand η-efficient allocations respectively.
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
Informal statement of the main theoremFormal modelMain theorem
Intuition and Interpretation
Complete information modelIncomplete information model
Type space: Strategic vs. non-strategic types
Strategic types: May report any type in Θ = Θ
Non-strategic types: reports his true type Θ = Θ
For each player there is a 0-1 random variable that determineswhether he is strategic or not
Type space is: [Θ× Θ× {0,1}]N
Details on definition of the type space
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
Informal statement of the main theoremFormal modelMain theorem
Intuition and Interpretation
Complete information modelIncomplete information model
Type distributions
Prior distribution on types: ψN ∈ ΨN(εc , εχ)
εc , εχ ∈ (0,1]
ψNs , ψ
Nc , ψ
Nχ are the marginals on ΘN , ΘN and {0,1}N
Assumptions:1 ψN
c and ψNχ have independent marginals across n;
2 For all n, ψNc (θn = θ) ≥ εc ∀θ ∈ Θ;
3 For all n, ψNχ (χn = 1) ≥ εχ.
A trader’s true type is: θn = θn if χn = 0 and θn = θn if χn = 1.
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
Informal statement of the main theoremFormal modelMain theorem
Intuition and Interpretation
Complete information modelIncomplete information model
The Market Game
We define an N-player game Γ(σ, ψN ,wN) as follows:
Types are drawn according to ψN
Strategies are reporting functions
θn : Θ× Θ× {0,1} → ∆(Θ)
such that non-strategic types report truthfully
The competitive mechanism σ picks a competitiveequilibrium σ(θN ,wN).
Payoffs are Uθn(σxn(θN ,wN))
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
Informal statement of the main theoremFormal modelMain theorem
Intuition and Interpretation
Formal statementRelationship to the literatureThe price
Main Theorem
For any η > 0, there is ε > 0 such that for any 0 < ε < ε andany pair (εc , εχ), satisfying εc · εχ ≥ ε:
there exists N such that for any N > N
any private information economy (ψN ,wN), withψN ∈ ΨN(εc , εχ)
any competitive mechanism σ
and any Bayesian-Nash equilibrium of Γ(σ, ψN ,wN)
P{σxN (θN ,wN) ∈ Effη(θN ,wN)
}> 1− η
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
Informal statement of the main theoremFormal modelMain theorem
Intuition and Interpretation
Formal statementRelationship to the literatureThe price
What we do and don’t do
We show that any competitive equilibrium of the reportedeconomy must be approximately efficient relative to the trueeconomy
...but we do NOT show that all agents truthfully report theirtypes or that agents’ influence on prices vanishes with large N
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
Informal statement of the main theoremFormal modelMain theorem
Intuition and Interpretation
Formal statementRelationship to the literatureThe price
The literature: the mechanism design approach
Much of the literature is concerned with designing goodmechanisms, i.e., ones that possess at least one equilibrium inwhich all agents truthfully report their information
Gul-Postlewaite (Econometrica 92)
McLean-Postlewaite (several papers)
Mas-Colell & Vives (RES 93)
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
Informal statement of the main theoremFormal modelMain theorem
Intuition and Interpretation
Formal statementRelationship to the literatureThe price
The literature: focus on specific institutions
Other papers consider specific auction mechanisms, typicallyauctions, and establish their properties as theN →∞:
Gresik-Satterthewaite, Rustichini-Satterthewaite-Williams
Swinkels, Cripps-Swinkels
Reny-Perry
...
We also look at the properties of a specific institution, butcompetitive markets rather than auctions.
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
Informal statement of the main theoremFormal modelMain theorem
Intuition and Interpretation
Formal statementRelationship to the literatureThe price
The price: Types must be finite
This ensures that: “if a player has small influence, then he mustreport truthfully.”This issue appears in:
Gul-Postlewaite
Jackson-Manelli
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
Informal statement of the main theoremFormal modelMain theorem
Intuition and Interpretation
Formal statementRelationship to the literatureThe price
The price: Sorting condition
Needed if we want to conclude something about allBayes-Nash equilibria.
compare with Gul-postlewaite..
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
Informal statement of the main theoremFormal modelMain theorem
Intuition and Interpretation
Formal statementRelationship to the literatureThe price
The price: non-strategic types
We need enough strategic uncertainty about the opponents’reports and about the outcome of the mechanism that isbounded away from zero uniformly in the number of traders andthe strategies they play
Using the terminology we introduced in Al-Najjar andSmorodinsky (2000), define the “influence” of a trader on anabstract outcome function as the maximum change in theexpected outcome that can be caused by changes in thistrader’s actions.
For related ideas, so Fudenberg-Levine-Pesendorer (1998) andSwinkels (2001)for related concepts and results.
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
Informal statement of the main theoremFormal modelMain theorem
Intuition and Interpretation
Intuition for the proofConcluding remarks
Why is this true?
Stopped here!!Results on influence in abstract environments in Al-Najjarand Smorodinsky. We use the version for correlatedenvironments.Role of noiseRole of the assumption of finite types
HOWEVER: the fraction of influential agents decreases as Ngrows=⇒ the distribution of characteristics in the true and reported
economies must be close=⇒ an equilibrium of the reported economy is an approximate
equilibrium of the true economy=⇒ approximate efficiency obtains
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
Informal statement of the main theoremFormal modelMain theorem
Intuition and Interpretation
Intuition for the proofConcluding remarks
Outline of the proof: 1-Influence arguments
Key lemma
Given any δ > 0 and ε > 0 there exist integers M and N suchthat for
any economy EN , with N > N
any competitive mechanism σ
any (Bayes-Nash) equilibrium of Γ(σ, ε,N)
any prior distribution ψ on types (redundant)
P{θN : #{n : θn 6= θn} < M · |Θ| · N3/4 | θN
}> 1− δ
Details on definition of influence
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
Informal statement of the main theoremFormal modelMain theorem
Intuition and Interpretation
Intuition for the proofConcluding remarks
Outline of the proof:2-Continuity
1 Use a standard metric on economies2 standard result in general equilibrium is that the excess
demand function is continuous in prices and economies3 The influence arguments show that EN and EN are close4 law of large numbers implies that for small ε, EN and EN
are also close5 The outcome of the game is always a competitive
equilibrium for EN
PROPOSITION 2: Given Proposition 1, the outcome of themechanism is an approximate equilibrium of the true economy,hence approximately efficient.
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
Informal statement of the main theoremFormal modelMain theorem
Intuition and Interpretation
Intuition for the proofConcluding remarks
Concluding thoughts: Why not require competitivemechanisms to be continuous?
Mas-Colell and Vives consider anonymous and continuousmechanisms in a continuum economy
This requirement makes little sense in finite economies
Even it makes sense, our ignorance of the process of priceformation makes it unreasonable in our view to require theselection to be continuos
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
Informal statement of the main theoremFormal modelMain theorem
Intuition and Interpretation
Intuition for the proofConcluding remarks
Concluding thoughts: Common values
HAS TO BE REVIEWED THOROUGHLYKey issue: How much information do prices reveal?
Interpretation of the fully revealing REE literature: Find σso that information about θN is fully revealed
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
Informal statement of the main theoremFormal modelMain theorem
Intuition and Interpretation
Intuition for the proofConcluding remarks
THE END !!
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
AppendixInfluenceType space
Appendix
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
AppendixInfluenceType space
Influence idea
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information
AppendixInfluenceType space
A generic type realization (θ, θ, i) means that:
There is a random draw of the player’s strategic type,which turned up to be θ;
..a separate draw of this player’s non-strategic type θ;
..and finally a draw of whether the player is strategic or not.
This is a more complicated type space than: Θ× {0,1}, but wecannot see see why this is substantively different from the typespace we use, which is Θ×Θ× {0,1}.
The latter is what we use in our proofs.
Al-Najjar & Smorodinsky The Efficiency of Competitive Mechanisms Under Private Information