1

The background of The background of expectational stability expectational stability

studiesstudies..« Eductive stability ».« Eductive stability ».

Global versus local,Global versus local,

« High tech » versus « Low tech »…« High tech » versus « Low tech »…

2

Back to a simple game.Back to 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.

3

The logic of The logic of rationalizability…againrationalizability…again

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 »

4

The «eductive » viewpoint.The «eductive » 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 A « Low-tech » interpretation and alternative intuition.alternative 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 ?

6

An abstract framework.An abstract framework.

Games with a continuum of Games with a continuum of agents and aggregate summary agents and aggregate summary

statistics…statistics…

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The model from a game-The model from a game-

theoretical viewpointtheoretical viewpoint. . A continuum of players. A continuum of players.

A measure space : (A measure space : (II, , II, , λλ), with ), with II=[0,1], =[0,1], λλ Lebesgue measure Lebesgue measure Strategy sets : Strategy sets : S(i)=S , compact subset of RS(i)=S , compact subset of Rnn.. Strategy profile : Strategy profile : s: I―› S, s(i).s: I―› S, s(i).

An aggregation operator. An aggregation operator. A(s)= A(s)= ∫s(i) di∫s(i) di AA is the (convex) set of states, is the (convex) set of states, A A = ∫S(i)di = co{S}= ∫S(i)di = co{S}..

For each agent For each agent ii Utility Function: Utility Function: u(i, · , · ): u(i, · , · ): S S x x A A ―› R, ―› R, continuous continuous (C).(C).HM: mapping i-u(.,i) measurable.HM: mapping i-u(.,i) measurable.

The optimal strategy correspondence The optimal strategy correspondence B(i,·)B(i,·)::AASS is: is:B(i, a)B(i, a) := argmax := argmaxyySS { {u(i, y, a)u(i, y, a)} .} .

Nash equilibriumNash equilibrium.. Pure strategy Nash equilibrium Pure strategy Nash equilibrium s* is a strategy profile /s* is a strategy profile /

s*(i)s*(i) B(i, B(i, ∫s*(i)di∫s*(i)di)))) iiI, I, λλ-a.e.-a.e. Under assumptions C and HM, it exists, Rath (1992)Under assumptions C and HM, it exists, Rath (1992)

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The model from an economic The model from an economic viewpointviewpoint

Aggregate actions and best responseAggregate actions and best response A A = ∫S(i)di = co{S}= ∫S(i)di = co{S}.. B(i,a) = B(i,a) = argmaxargmaxyySS { {u(i, y, a)u(i, y, a)}. }. Def : Def : (a)= (a)= ∫ ∫ B(i,a) diB(i,a) di B(i, B(i, ) = ) = argmaxargmaxyySS {E {E[u(i, y, a)]} .[u(i, y, a)]} .

Equilibrium .Equilibrium . a* = a* = ∫∫ B(i,a*)di = B(i,a*)di = ∫∫ B(i, B(i,a*a*)di )di (a*)=a* (a*)=a* There exists an equilibrium.There exists an equilibrium. Equivalence for existence between the Nash viewpoint and Equivalence for existence between the Nash viewpoint and

the equilibrium viewpoint.the equilibrium viewpoint. Coordination.Coordination.

Focus on aggregate actions not on strategies.Focus on aggregate actions not on strategies.

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One example : strategic One example : strategic complementarities. complementarities.

The model :The model : The aggregate state a, The aggregate state a, proportion of people who join.proportion of people who join. {{u(i, y, a)u(i, y, a)}=}=a-c(i), a-c(i),

c(i)c(i) individual cost of joining. individual cost of joining. y= (0 or 1), y= (0 or 1), join, do not joinjoin, do not join

Distribution of costs : cumulative F(c).Distribution of costs : cumulative F(c). F(a) F(a) = = ∫∫ B(i,a)di = B(i,a)di = (a)(a)

Equilibrium a*=F(a*)Equilibrium a*=F(a*) Three orThree or One ?One ? How flat is the distributionHow flat is the distribution

c,a,

a

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Another example : the Another example : the linear Muth modellinear Muth model..

The Muth modelThe Muth model Sellers : firms) or farmers.Sellers : firms) or farmers.

Decide to-day about Decide to-day about production (wheat).production (wheat).

Cost C(f,q)).Cost C(f,q)). Buyers will buy to-morrow. Buyers will buy to-morrow. Demand curve : A-Bp.Demand curve : A-Bp. a=A-Bp, a=A-Bp, {{u(i, y, a)u(i, y, a)}= (}= (A/B-a/B)y- A/B-a/B)y-

yy22/2c(f),/2c(f), C= ∫ c(f)df. C= ∫ c(f)df.

(a)(a) = = ∫∫ B(i,a)di= (CA)/B – B(i,a)di= (CA)/B – (C/B)a.(C/B)a.

Strategic Strategic substitutabilities. substitutabilities. More general case : D(p), More general case : D(p),

C(p)C(p) p=Dp=D-1-1(a), (a), (a)(a) = C°D = C°D-1-1(a).(a).

C/B

a

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A rA reminder on eminder on Rationalizability.Rationalizability.

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 srategy in S(1,i) = {S(0,i) \ strategies in S(0,i) non BR to some 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)]

Remarks. Remarks. Consider Pr [Consider Pr [(S(i)] = (S(i)] = {S(i) \ s in S(i) non BR to {S(i) \ s in S(i) non BR to jj[[((S,(j)]}S,(j)]} R =Pr(R), R =Pr(R), and R is the largest set such that R =Pr(R),and R is the largest set such that R =Pr(R), Other set N Other set N RR

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Rationalizability 1.Rationalizability 1. The (standard) game-The (standard) game-

theoretical viewpoint. theoretical viewpoint. Recursive elimination of non Recursive elimination of non

best responses. best responses. ……

Point expectations: H, set Point expectations: H, set of strategy profiles.of strategy profiles. Pr(H)=Pr(H)={ { ss is strategy profile is strategy profile

such that such that ss is a measurable is a measurable selection of iselection of iBr(i,H)}Br(i,H)}

Measurability of strategy Measurability of strategy profiles.profiles.

The set of point rationalizable The set of point rationalizable strategy profiles is the largest strategy profiles is the largest set such that: Pr(H)=Hset such that: Pr(H)=H

Equivalence with the Equivalence with the economic viewpoint.economic viewpoint.

The « economic The « economic viewpoint »viewpoint » Same process but conjectures Same process but conjectures

on the aggregate state. on the aggregate state. Point expectations : Point expectations :

Cobweb mapping.Cobweb mapping. Def Def (a)= (a)= ∫ ∫ B(i,a) diB(i,a) di Cobweb tâtonnement Cobweb tâtonnement

outcomeoutcome

==tt00 tt((AA)) Point expectations-ration.Point expectations-ration.

Pr (X)= Pr (X)= ∫∫ B(i,X) di B(i,X) di The set of point-rationalizable The set of point-rationalizable

states states , is the largest set X, is the largest set X AA such that:such that:

Pr(X)=XPr(X)=X Equivalence with the Equivalence with the

game viewpointgame viewpoint

Rationalizability 2.Rationalizability 2. Point expectations: H, set Point expectations: H, set

of strategy profiles.of strategy profiles. Pr(H)=Pr(H)={ { ss is strategy profile is strategy profile

such that such that ss is a measurable is a measurable selection of iselection of iBr(i,H)}Br(i,H)}

The set of point rationalizable The set of point rationalizable strategy profiles is the largest strategy profiles is the largest set such that:set such that:

Pr(H)=HPr(H)=H Random expectationsRandom expectations

Same process but take Same process but take random beliefs. random beliefs.

Difficulty : measurability vis-Difficulty : measurability vis-à-vis probability à-vis probability distributions ?distributions ?

= non measurability vis-à-vis = non measurability vis-à-vis point expectations ?point expectations ?

Point expectations:rationPoint expectations:ration. . Pr (X)= Pr (X)= ∫∫ B(i,X) di B(i,X) di The set of point-rationalizable The set of point-rationalizable

states states , is the largest set X, is the largest set X AA such that:such that:

Pr(X)=XPr(X)=X Equivalence with the game Equivalence with the game

viewpointviewpoint Probabilistic expectations.Probabilistic expectations.

R(X) = ∫ B(i,R(X) = ∫ B(i,PP(X)) di(X)) di The set of rationalizable The set of rationalizable

states states , is the largest set X, is the largest set X AA such that: R(X)=Xsuch that: R(X)=X

Provides a substitute Provides a substitute (equivalent) with the game (equivalent) with the game viewpoint.viewpoint.

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Equilibria and Equilibria and rationalizable states.rationalizable states.

The state space : the concepts.The state space : the concepts. E, E, , , , , E E Co(E) Co(E)

Properties : Properties : The set of point rationalizable states is non-empty, convex, The set of point rationalizable states is non-empty, convex,

compact. compact. The set of rationalizable states is non-empty and convex.The set of rationalizable states is non-empty and convex.

Definitions and terminology.Definitions and terminology. E= E= , Iteratively expectationally stable. (homogenous , Iteratively expectationally stable. (homogenous

expectations)expectations) E = E = , Strongly point Rational. Heterogenous deterministic , Strongly point Rational. Heterogenous deterministic

expectationsexpectations E= E= Strongly Rational. Heterogenous probabilistic Strongly Rational. Heterogenous probabilistic

expectationsexpectations. .

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The local viewpoint.The local viewpoint. The local transposition.The local transposition.

a* is locally iteratively stable…a* is locally iteratively stable… a* is locally Strongly point Rational…a* is locally Strongly point Rational… a* is locally strongly rational….a* is locally strongly rational….

The connections. The connections. 332 2 1. 1. 1 weaker than 31 weaker than 3

The equivalence between 2 and 3The equivalence between 2 and 3 Reinforcing locally strongly rational in Strictly locally Reinforcing locally strongly rational in Strictly locally

strongly point rational (The contraction V-Prstrongly point rational (The contraction V-Prnn(v) is (v) is strict). strict).

Strictly locally strongly rational = locally point rational. Strictly locally strongly rational = locally point rational.

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Strategic Strategic

ComplementaritiesComplementarities..Attempt at generalisation.Attempt at generalisation.

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Economies with strategic Economies with strategic complementarities.complementarities.

Strategic complementarities in the state space.Strategic complementarities in the state space. 1B, 1B, SS is the product of is the product of nn compact intervals in R compact intervals in R++.. 2B, 2B, u(i, · , a)u(i, · , a) is supermodular for all is supermodular for all aaAA and all and all iiII.. 3B, 3B, iiII, the function , the function u(i, y, a)u(i, y, a) has has increasingincreasing differences in differences in yy and and aa.. B(i,a) est croissant en a, comme B(i,B(i,a) est croissant en a, comme B(i,) …comme ) …comme (a)= (a)= ∫ ∫ B(i,a) diB(i,a) di

Properties. Properties. a*a*min min andand a*a*maxmax, smallest and largest equilibria., smallest and largest equilibria. a*a*minminE E a*a*maxmax All these sets but the first are convex.All these sets but the first are convex. = = ?? ??

Comments.Comments. Uniqueness equivalent to Strong Rationality, Strong Uniqueness equivalent to Strong Rationality, Strong

point rationalizability, IE stability. … the Graal.point rationalizability, IE stability. … the Graal. Locally, criteria equivalent.Locally, criteria equivalent. Heterogeneity does not matter so much, neither probabilistic Heterogeneity does not matter so much, neither probabilistic

beliefs.beliefs.

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Back to one-dimensional Back to one-dimensional Strategic Strategic

complementarities. complementarities. The model :The model :

The aggregate state a, The aggregate state a, proportion of people who join.proportion of people who join. {{u(i, y, a)u(i, y, a)}=}=a-c(i), a-c(i),

c(i)c(i) individual cost of joining. individual cost of joining. y= (0 or 1), y= (0 or 1), join, do not joinjoin, do not join

Distribution of costs : cumulative F(c).Distribution of costs : cumulative F(c). F(a) F(a) = = ∫∫ B(i,a)di = B(i,a)di = (a)(a)

Equilibrium a*= F(a*)Equilibrium a*= F(a*) Three or one ?Three or one ? How flat is the distribution. How flat is the distribution. The Equilibrium is either a SREEThe Equilibrium is either a SREE Or [Or [a*a*minmin , ,a*a*maxmax ]= ]=====..

c,a,

a

a*mina*min a*maxa*max

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a*a*maxmax

aa11minmin

aa22maxmax

AA

aa00minmin

aa00maxmax

a*a*minmin

aa22minmin

Strategic Strategic Complementarities withComplementarities with

AA R R22 and multiple and multiple equilibria.equilibria.

aa11maxmax

aa11

aa22

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Economies with Strategic Economies with Strategic subsitutabilities.subsitutabilities.

Economies with Strategic substitutabilities.Economies with Strategic substitutabilities. 1B, 1B, SS is the product of is the product of nn compact intervals in R compact intervals in R++.. 2B, 2B, u(i, · , a)u(i, · , a) is supermodular for all is supermodular for all aaAA and all and all iiII.. 3–B’, 3–B’, iiII, the function , the function u(i, y, a)u(i, y, a) has has decreasingdecreasing differences in differences in yy and and aa.. The cobweb mapping The cobweb mapping is decreasing is decreasing The second iterate of The second iterate of , , 22 is increasing. is increasing.

Results.Results. a*a*min min and a*and a*max max ,, cycles of order 2 of cycles of order 2 of [a*[a*minmin+R+Rnn, a*, a*maxmax- R- Rnn]] All these sets but the first are convex.All these sets but the first are convex. = = ?? ??

Comments.Comments. The Graal : no cycle of order 2 and a unique equilibriumThe Graal : no cycle of order 2 and a unique equilibrium, ,

Strong Rationality, Strong point rationalizability, IE stability. Strong Rationality, Strong point rationalizability, IE stability. Locally, criteria equivalent.Locally, criteria equivalent. Heterogeneity does not matter so much, neither probabilistic beliefs.Heterogeneity does not matter so much, neither probabilistic beliefs.

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AA

aamaxmax

aaminmin

AA

aamaxmax

Muthian Strategic substitutes for Muthian Strategic substitutes for AA R R with unique equilibrium and multiple with unique equilibrium and multiple

fixed points of fixed points of 22

22(a*(a*maxmax)= a*)= a*maxmax

a*a*minmin

22(a*(a*minmin)= a*)= a*minmin

a*a*maxmax

(a*)= a*(a*)= a*

The Muth model with two The Muth model with two cropscrops

The Model :The Model : A variant of Muth : A variant of Muth :

Two crops : wheat and corn…Two crops : wheat and corn… Independant demands Independant demands

D(p(1)), D(p(2)D(p(1)), D(p(2) S(p(1),p(2))S(p(1),p(2))

Strategic substitutes…Strategic substitutes… If a(1), a(2) increases, the vector S(DIf a(1), a(2) increases, the vector S(D-1-1(a(1),a(2)) decreases.(a(1),a(2)) decreases.

« Eductive stability »: the local viewpoint.« Eductive stability »: the local viewpoint. S’S’1212/ / D’D’11D’D’22 <1-k, k=(assumption) (S <1-k, k=(assumption) (S11’ /D’’ /D’11)= (S)= (S22 ’ /D’ ’ /D’22).). 1-k is the index of « eductive stability » in case of 1-k is the index of « eductive stability » in case of

indemendant markets.indemendant markets. The interaction between the markets is destabilizing…The interaction between the markets is destabilizing… One issue of the present crisis…One issue of the present crisis…

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Provisional conclusionsProvisional conclusions Simple worlds: global coordination Simple worlds: global coordination

With strategic complementarities, uniqueness With strategic complementarities, uniqueness is the « Graal ». is the « Graal ».

With strategic substituabilities, With strategic substituabilities, Uniqueness is no longer the Graal, Uniqueness is no longer the Graal, But absence of cycle of order two. Absence of self-But absence of cycle of order two. Absence of self-

defeating pair of expectations…defeating pair of expectations… Outside simple worlds. Outside simple worlds.

More complex, cycles of any order matter..More complex, cycles of any order matter.. Local « eductive » stability and local Local « eductive » stability and local

properties of the best response mapping..properties of the best response mapping..

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Appendix 1a : supermodular Appendix 1a : supermodular gamesgames

Tarsky Theorem :Tarsky Theorem : F, function F, function from S to S, S complete lattice from S to S, S complete lattice The set of fixed points E is non empty and is a complete The set of fixed points E is non empty and is a complete

lattice.lattice. Applications : S Applications : S R Rn n , product of intervals in R, , product of intervals in R, sup E and inf E are fixed pointssup E and inf E are fixed points

Super modular functionsSuper modular functions : : G : RG : Rn n R, (strictly) supermodular R, (strictly) supermodular

22G/G/xxi i xxj j >0, i #j>0, i #j Let f(t) = maxLet f(t) = maxxx G(x,t), G(x,t), G (strictly) supermodular on XG (strictly) supermodular on X t t Then, the mapping f is Then, the mapping f is X compact and G USC in x, f compact.X compact and G USC in x, f compact.

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Appendix 1b : Appendix 1b : vivisualizations.sualizations.

An increasing function An increasing function ……has a fixed point.has a fixed point. Even with discontinuities.Even with discontinuities. See the left diagram.. See the left diagram..

With a supermodular function : With a supermodular function : U(a,t) U(a,t)

a (planned production) a (planned production) t (expected total production) t (expected total production) ..keynesian situation..keynesian situation

Best response are increasing in t() Best response are increasing in t() Possibly with jumps.Possibly with jumps. Inspect the second left diagram..Inspect the second left diagram..

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Appendix 1c : Appendix 1c : Supermodular games.Supermodular games.

Definition Definition Compact strategy space.Compact strategy space. U(i, s(i), s(-i)) U(i, s(i), s(-i))

(strict.) supermodular (see above)(strict.) supermodular (see above) Equilibria in supermodular games :Equilibria in supermodular games :

Best response Fn Best response Fn The set of equilibria is non empty, has a greatest and a The set of equilibria is non empty, has a greatest and a

smallest element.smallest element. Comments Comments

Serially dominated strategies converge to the set Min Serially dominated strategies converge to the set Min [], Max [][], Max []

Expectational coordination on this set.Expectational coordination on this set. If the equilibrium is unique, it is dominant solvable, If the equilibrium is unique, it is dominant solvable,

globally SREE, « eductively » stable globally SREE, « eductively » stable