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1- A Presentation : 1- A Presentation : the « eductive » the « eductive »

viewpoint.viewpoint.

Rationalizability and the stability Rationalizability and the stability of expectations. of expectations.

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A simple game. A simple game. The rules of the game : The rules of the game :

write a number : [0,100]write a number : [0,100] Winner : 10 Euros : closest to 2/3 of the mean (of Winner : 10 Euros : closest to 2/3 of the mean (of

others)others) What happens in this game ? See Nagel (1995)What happens in this game ? See Nagel (1995)

Lessons : Lessons : 0 is the unique Nash equilibrium.0 is the unique Nash equilibrium. It is a rather « reasonable » predictor of what It is a rather « reasonable » predictor of what

happens. happens. Change the game :Change the game :

Announce : [0, + infinity) [0,100]...Announce : [0, + infinity) [0,100]... 3/2 instead of 2/3.3/2 instead of 2/3.

« Eductive » or « evolutive » explanation« Eductive » or « evolutive » explanation

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Expectational Expectational StabilityStability

A first theoretical A first theoretical perspective.perspective.

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The approach.The approach.

General inspiration : General inspiration : Expectational coordination is more or less Expectational coordination is more or less

stable.stable. Find criteria with « eductive » flavour.Find criteria with « eductive » flavour. Classify : type 2/3 or ...1/2, or 3/2 Classify : type 2/3 or ...1/2, or 3/2

Situations, economies ? globalSituations, economies ? global Equilibria : localEquilibria : local

The above argument is captured by The above argument is captured by rationnalisability ideasrationnalisability ideas Iterated elimination of dominated strategiesIterated elimination of dominated strategies Old wine : Luce Raiffa, Farqhason, Moulin.Old wine : Luce Raiffa, Farqhason, Moulin. New bottles : Bernheim, Pearce (1984).New bottles : Bernheim, Pearce (1984).

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Rationalizable solutions in Rationalizable solutions in games.games.

Game in normal formGame in normal form S(i), s(i), u(i,s(i), s(-i))S(i), s(i), u(i,s(i), s(-i)) Iterative elimination of non best response strategies :Iterative elimination of non best response strategies : S(0,i) =S(i)S(0,i) =S(i) S(1,i) = {S(0,i) \ strategies in S(0,i) non BR to some S(1,i) = {S(0,i) \ strategies in S(0,i) non BR to some

srategy in srategy in jj[[S(0,j)]}S(0,j)]} ........ S(S(,i) = {S(,i) = {S(-1,i) \ s(i) in S(-1,i) \ s(i) in S( - 1,i) non BR to - 1,i) non BR to jj[[((S(S( -1,j)]} -1,j)]} R = R = [[i(i((S((S(,i)],i)]

A number of technical variationsA number of technical variations : : Correlated or independant rationalizability.Correlated or independant rationalizability. Curb set, correlated equilibria, p-minimal setsCurb set, correlated equilibria, p-minimal sets

Note R contains N !Note R contains N !

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Rationalizable solutions in Rationalizable solutions in games.games.

The « 2/3 of the mean » game.The « 2/3 of the mean » game. S(i)={0,100}, u(i,s(i), s(-i))=…..;S(i)={0,100}, u(i,s(i), s(-i))=…..; Iterative elimination of non best response strategies :Iterative elimination of non best response strategies : S(0,i) = {0,100}, S(0,i) = {0,100}, S(1,i) = {0, 66,6666…}S(1,i) = {0, 66,6666…} ........ S(S(,i) = {0, (2/3),i) = {0, (2/3) 100} 100}

0 is 0 is the unique Nash equilibrium.the unique Nash equilibrium. The unique « rationalizable » outcome.The unique « rationalizable » outcome.

Dominant solvable Nash outcomeDominant solvable Nash outcome Strongly rational equilibrium, « edcutively » stableStrongly rational equilibrium, « edcutively » stable

We have « strategic complementarities »We have « strategic complementarities »

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Rationalizability and Rationalizability and expectational coordination. expectational coordination.

Rationalizability consequence of Rationalizability consequence of CK of the game, CK of rationality, CK of the game, CK of rationality, (Tan, Werlang)(Tan, Werlang)

The viewpoint taken here.The viewpoint taken here. Economic models. Economic models. No « market power » : a continuum of agents.No « market power » : a continuum of agents. Nash equilibria are Rational Expectations EquilibriaNash equilibria are Rational Expectations Equilibria

The equilibrium is unique.The equilibrium is unique. Strongly Rational Strongly Rational if it can be guessed through the mental process. if it can be guessed through the mental process.

There are several equilibriaThere are several equilibria A local viewpoint on the mental process.A local viewpoint on the mental process. Locally strongly rational. Locally strongly rational.

Terminology : « eductively stable ?Terminology : « eductively stable ?

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Strategic Strategic complementaritiecomplementaritie

s : 1.s : 1.First insights.First insights.

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One-dimensional One-dimensional aggregate : strategic aggregate : strategic complementarities.complementarities.

The specialized model.The specialized model. A continuum of (small) agents;A continuum of (small) agents; Each is concerned by his own action and an aggregate Each is concerned by his own action and an aggregate

of actions of others : a(i), A= of actions of others : a(i), A= a(i)dia(i)di Case 1 : the aggregate is one dimensional.Case 1 : the aggregate is one dimensional.

Strategic complementarities Strategic complementarities Muth etcMuth etc

The aggregate is not one dimensional.The aggregate is not one dimensional. Strategic complementarities : Strategic complementarities :

the best response a(i) increases with the (expected) A. the best response a(i) increases with the (expected) A.

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The « 2/3 of the mean » The « 2/3 of the mean » game.game.

A diagram :A diagram :

100 A

BR(A)

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Strategic Complementarities : Strategic Complementarities : a simple model.a simple model.

The model : The model : Population agents a :Population agents a : Individual Costs c, Individual Costs c,

F(c.), F(c.), Profitability = Profitability = da da Strategic Strategic

complementarities.complementarities. Equilibria Equilibria

Three orThree or One ?One ? How flat is the How flat is the

distribution.distribution.

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One dimensional aggregate One dimensional aggregate strategic complementarities. strategic complementarities.

Draw Draw aggregate = f((point) aggregate = f((point)

expectations of the expectations of the aggregate)aggregate)

Example : planned Example : planned production as a function production as a function of expected production of expected production (keynesian situation) (keynesian situation)

This function is This function is increasing. SCincreasing. SC

Possibly with jumps.Possibly with jumps. Equilibrium Equilibrium

Fixed point.Fixed point. unique. unique. or not or not What does What does

rationalizability bring ?rationalizability bring ?

S(1)

S(2) min max

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One-dimensional aggregate One-dimensional aggregate strategic strategic complementaritiescomplementarities

The unique The unique equilibrium is equilibrium is Strongly RationalStrongly Rational.. The argument.The argument. The Graal !… The Graal !…

In case of multiplicity In case of multiplicity The rationalizable set .The rationalizable set . Some equilibria are Some equilibria are

locally strongly locally strongly rational. rational.

S(1)

S(2) min max

R

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Local Strong Rationality.Local Strong Rationality.

The local viewpointThe local viewpoint Hypothetical Common Hypothetical Common

Knowledge of the nbhd. Knowledge of the nbhd. If slope >1, the If slope >1, the

hypothetical CK is hypothetical CK is dismissed.dismissed.

Question :Question : Is there a nbhd, Is there a nbhd,

distinct from the distinct from the equilibrium s.t. the fact equilibrium s.t. the fact that evby believes in that evby believes in that the outcome is in that the outcome is in the nbd implies surely the nbd implies surely that it will be ?that it will be ?

Locally Locally eductively stable eductively stable equilibriumequilibrium

It is CK That QV

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1-C1-C Expectational Expectational coordination in a simple coordination in a simple

partial equilibrium partial equilibrium contextcontext

The Muth model : The Muth model : strategic strategic

substitutabilities.substitutabilities.

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A simple partial A simple partial equilibrium modelequilibrium model

Very simple framework (Muth, 1961) :Very simple framework (Muth, 1961) : Sellers : firms) or farmersSellers : firms) or farmers

décide to-day about production (wheat).décide to-day about production (wheat). Cost C(f,q) ( =qCost C(f,q) ( =q22/2c(f) )/2c(f) ) ManyMany

Buyers will buy to-morrowBuyers will buy to-morrow.. Many : Many : demand curve : A-Bpdemand curve : A-Bp

« Walras, Marshall » partial equilibrium« Walras, Marshall » partial equilibrium : : Notional supplyNotional supply

Quadratic case : MaxQuadratic case : Maxqq [pq - q [pq - q22/2c(f)]/2c(f)] S(p,f) = c(f)p S(p,f) = c(f)p S(p) = CpS(p) = Cp

Equilibrium :Equilibrium : S(p) = A-BpS(p) = A-Bp héraut walrassien ? Is it Strongly Rational.héraut walrassien ? Is it Strongly Rational.

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An « eductive » processAn « eductive » process

A collective A collective « cognitive » process« cognitive » process CK : P < pCK : P < pmaxmax Evbd knows that Evbd knows that

S(f) < S(pS(f) < S(pmaxmax,f),f) S< S< S(pS(pmaxmax,f)df,f)df p>Dp>D(-1) (-1) 0 0 S = pS = p11 Then Then S> S> S(p S(p11,f)df,f)df

If evbd knows that evbd If evbd knows that evbd knows :knows :

evbd knows that evbd knows that p< Dp< D(-1) (-1) 0 0 S(pS(p11) = p) = p2....2....EtcEtc

conclusion conclusion Cvgce iif C<BCvgce iif C<B

p0

q

p

A-Bp

C

B

C p

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The « eductive » process :The « eductive » process :A variantA variant..

Another (almost) Another (almost) equivalent story :equivalent story : Expectations Q, Expectations Q, Q=A-Bp, Price =A/B-Q=A-Bp, Price =A/B-

Q/B, Q/B, Realisations :Q(R) = Realisations :Q(R) =

CA/B – (C/B)Q(e)CA/B – (C/B)Q(e) Hence the processHence the process

Strategic Strategic substitutabilities. substitutabilities.

Alternate optimism and Alternate optimism and pessimism.pessimism.

C/B

Q

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The « eductive » process :The « eductive » process :Two variantsTwo variants..

Price diagramPrice diagram

p1

q

p

A-Bp

C

B

C p

pmax

C/B

QStrong RationalityEductive Stability :

C<B

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Comparison with evolutive Comparison with evolutive

learning.learning.

C/B

alpha

p(e,t,t+1)=p(t)+(1- )p(e,t-1,t)

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An insight into the An insight into the algebraalgebra..

The equations :The equations : Learning equations : a= Learning equations : a= p(e,t,t+1)=ap(t)+(1-a)p(e,t-1,t)p(e,t,t+1)=ap(t)+(1-a)p(e,t-1,t) Equilibrium equations.Equilibrium equations. A-Bp(t)=Cp(e,t-1,t)A-Bp(t)=Cp(e,t-1,t) p(t)=(1/B)(A-C p(e,t-1,t))p(t)=(1/B)(A-C p(e,t-1,t)) p(e,t,t+1)= ..[-a(C/B)+(1-a)]p(e,t-1,t)p(e,t,t+1)= ..[-a(C/B)+(1-a)]p(e,t-1,t) A(C/B) +a -1<1A(C/B) +a -1<1

Conclusions.Conclusions. Small a is good. (inf. learning process)Small a is good. (inf. learning process) Small C/B is good.(inf system) Small C/B is good.(inf system) =captured by the «eductive » viewpoint=captured by the «eductive » viewpoint

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The «eductive » The «eductive » viewpoint. viewpoint.

A « high-tech » formal (global) definition. A « high-tech » formal (global) definition. Definition Definition (with a continuum of small agents)(with a continuum of small agents) Let E* (in some vector space Let E* (in some vector space ) be an (Rat.Exp.) equilibrium) be an (Rat.Exp.) equilibrium. . Assertion A : Assertion A : It is CK that It is CK that E E (rationality and the model are (rationality and the model are

CK)CK) Assertion BAssertion B : It is CK that E=E* : It is CK that E=E* If A If A B, the equ. is ( B, the equ. is (globally)globally) Strongly Rational.Strongly Rational.

A « high-tech » formal (local) definition. A « high-tech » formal (local) definition. Let V(E*} be some non trivial neighbourhood Let V(E*} be some non trivial neighbourhood Assertion Assertion It is CK that It is CK that AA : E is in V(E*) : E is in V(E*) Assertion BAssertion B : It is CK that E=E* : It is CK that E=E* Same definition as before if V is the whole set of states.Same definition as before if V is the whole set of states. E* is E* is locallylocally, (vis-à-vis V), Strongly Rational., (vis-à-vis V), Strongly Rational.

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The «eductive » stability The «eductive » stability criterion. criterion.

Remarks on the generality. Remarks on the generality. Potentially general. Potentially general.

Remarks on the requirements.Remarks on the requirements. Requires « rational » agents with some Common Requires « rational » agents with some Common

Knowledge on the (working of) the system.Knowledge on the (working of) the system. A « hyper-rationalistic » view of coordination. A « hyper-rationalistic » view of coordination.

A « Low-tech » interpretation and alternative A « Low-tech » interpretation and alternative intuition.intuition. Can we find a non-trivial nbd of equilibrium s.t if Can we find a non-trivial nbd of equilibrium s.t if

everybody believes tha the state will be in it, it will everybody believes tha the state will be in it, it will surely be….?surely be….?

Local Expectational viewpoint. Local Expectational viewpoint. A Connection with « evolutive learning » (asymptotic A Connection with « evolutive learning » (asymptotic

stability of…)stability of…) Too demanding ?Too demanding ?