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



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




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




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




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.




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




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




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.





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.





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 ′.





Competitive mechanisms

A competitive mechanisms is any selection

σ : EN → ∆× (R l+)N

of the competitive correspondence.

Motivation....





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.





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





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.





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))




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





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





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)





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.





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





The price: Sorting condition

Needed if we want to conclude something about allBayes-Nash equilibria.

compare with Gul-postlewaite..





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.




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





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





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.





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





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





THE END !!


AppendixInfluenceType space

Appendix



Influence idea



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.


The Efficiency of Competitive Mechanisms Under Private Information · 2005-05-27 · Informal...

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