1. 1 The background of expectational stability studies. « Eductive stability ». Global versus local, « High tech » versus « Low tech »…

2. 2 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 others) Winner : 10 Euros : closest to 2/3 of the mean (of 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 happens. It is a rather « reasonable » predictor of what 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. 3 The logic of rationalizability…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( ,i) = {0, (2/3)  100} S( ,i) = {0, (2/3)  100} 0 is 0 is the unique Nash equilibrium. the unique Nash equilibrium. The unique « rationalizable » outcome. The unique « rationalizable » outcome. Dominant solvable Nash outcome Dominant solvable Nash outcome Strongly rational equilibrium, « edcutively » stable Strongly rational equilibrium, « edcutively » stable We have « strategic complementarities » We have « strategic complementarities »

4. 4 The «eductive » viewpoint. A « high-tech » formal (global) definition. A « high-tech » formal (global) definition. Definition (with a continuum of small agents) Definition (with a continuum of small agents) Let E* (in some vector space  ) be an (Rat.Exp.) equilibrium. Let E* (in some vector space  ) be an (Rat.Exp.) equilibrium. Assertion A : It is CK that E   (rationality and the model are CK) Assertion A : It is CK that E   (rationality and the model are CK) Assertion B : It is CK that E=E* Assertion B : It is CK that E=E* If A  B, the equ. is (globally) Strongly Rational. If A  B, the equ. is (globally) 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 It is CK that A : E is in V(E*) Assertion It is CK that A : E is in V(E*) Assertion B : It is CK that E=E* Assertion B : 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 locally, (vis-à-vis V), Strongly Rational. E* is locally, (vis-à-vis V), Strongly Rational.

5. 5 The «eductive » stability 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 Knowledge on the (working of) the system. Requires « rational » agents with some Common 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 intuition. A « Low-tech » interpretation and alternative intuition. Can we find a non-trivial nbd of equilibrium s.t if everybody believes tha the state will be in it, it will surely be….? Can we find a non-trivial nbd of equilibrium s.t if everybody believes tha the state will be in it, it will surely be….? Local Expectational viewpoint. Local Expectational viewpoint. A Connection with « evolutive learning » (asymptotic stability of…) A Connection with « evolutive learning » (asymptotic stability of…) Too demanding ? Too demanding ?

6. 6 An abstract framework. Games with a continuum of agents and aggregate summary statistics…

7. 7 The model from a game-theoretical viewpoint. A continuum of players. A continuum of players. A measure space : (I, I, λ), with I=[0,1], λ Lebesgue measure A measure space : (I, I, λ), with I=[0,1], λ Lebesgue measure Strategy sets : S(i)=S, compact subset of R n. Strategy sets : S(i)=S, compact subset of R n. Strategy profile : s: I―› S, s(i). Strategy profile : s: I―› S, s(i). An aggregation operator. An aggregation operator. A(s)= ∫s(i) di A(s)= ∫s(i) di A is the (convex) set of states, A = ∫S(i)di = co{S}. A is the (convex) set of states, A = ∫S(i)di = co{S}. For each agent i Utility Function: For each agent i Utility Function: u(i, ·, · ): S x A ―› R, continuous (C). HM: mapping i-u(.,i) measurable. u(i, ·, · ): S x A ―› R, continuous (C). HM: mapping i-u(.,i) measurable. The optimal strategy correspondence B(i,·): A  S is: The optimal strategy correspondence B(i,·): A  S is: B(i, a) := argmax y  S {u(i, y, a)}. Nash equilibrium. Nash equilibrium. Pure strategy Nash equilibrium s* is a strategy profile / Pure strategy Nash equilibrium s* is a strategy profile / s*(i)  B(i, ∫s*(i)di))  i  I, λ-a.e. Under assumptions C and HM, it exists, Rath (1992) Under assumptions C and HM, it exists, Rath (1992)

8. 8 The model from an economic viewpoint Aggregate actions and best response Aggregate actions and best response A = ∫S(i)di = co{S}. A = ∫S(i)di = co{S}. B(i,a) = argmax y  S {u(i, y, a)}. B(i,a) = argmax y  S {u(i, y, a)}. Def :  (a)= ∫ B(i,a) di Def :  (a)= ∫ B(i,a) di B(i,  ) = argmax y  S {E  [u(i, y, a)]}. B(i,  ) = argmax y  S {E  [u(i, y, a)]}. Equilibrium. Equilibrium. a* = ∫ B(i,a*)di = ∫ B(i,  a* )di a* = ∫ B(i,a*)di = ∫ B(i,  a* )di  (a*)=a*  (a*)=a* There exists an equilibrium. There exists an equilibrium. Equivalence for existence between the Nash viewpoint and the equilibrium viewpoint. Equivalence for existence between the Nash viewpoint and the equilibrium viewpoint. Coordination. Coordination. Focus on aggregate actions not on strategies. Focus on aggregate actions not on strategies.

9. 9 One example : strategic 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)}=a-c(i), {u(i, y, a)}=a-c(i), c(i) individual cost of joining. c(i) individual cost of joining. y= (0 or 1), join, do not join y= (0 or 1), join, do not join Distribution of costs : cumulative F(c). Distribution of costs : cumulative F(c). F(a) = ∫ B(i,a)di =  (a) F(a) = ∫ B(i,a)di =  (a) Equilibrium a*=F(a*) Equilibrium a*=F(a*) Three or Three or One ? One ? How flat is the distribution How flat is the distribution c,a, a

10. 10 Another example : the linear Muth model. The Muth model The Muth model Sellers : firms) or farmers. Sellers : firms) or farmers. Decide to-day about production (wheat). Decide to-day about 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)}= (A/B-a/B)y- y 2 /2c(f), {u(i, y, a)}= (A/B-a/B)y- y 2 /2c(f), C= ∫ c(f)df. C= ∫ c(f)df.  (a) = ∫ B(i,a)di= (CA)/B – (C/B)a.  (a) = ∫ B(i,a)di= (CA)/B – (C/B)a. Strategic substitutabilities. Strategic substitutabilities. More general case : D(p), C(p) More general case : D(p), C(p) p=D -1 (a), p=D -1 (a),  (a) = C°D -1 (a).  (a) = C°D -1 (a). C/B a 

11. 11 A r eminder on Rationalizability. Game in normal form Game 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  j [  S(0,j)]} S(1,i) = {S(0,i) \ strategies in S(0,i) non BR to some srategy in  j [  S(0,j)]}........ S( ,i) = {S(  -1,i) \ s(i) in S(  - 1,i) non BR to  j [  (S(  -1,j)]} S( ,i) = {S(  -1,i) \ s(i) in S(  - 1,i) non BR to  j [  (S(  -1,j)]} R = [  i(   (S( ,i)] R = [  i(   (S( ,i)] Remarks. Remarks. Consider Pr [  (S(i)] =  {S(i) \ s in S(i) non BR to  j [  (S,(j)]} Consider Pr [  (S(i)] =  {S(i) \ s in S(i) non BR to  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  R Other set N  R

12. 12 Rationalizability 1. The (standard) game-theoretical viewpoint. The (standard) game-theoretical viewpoint. Recursive elimination of non best responses. Recursive elimination of non best responses. … Point expectations: H, set of strategy profiles. Point expectations: H, set of strategy profiles. Pr(H)={ s is strategy profile such that s is a measurable selection of i  Br(i,H)} Pr(H)={ s is strategy profile such that s is a measurable selection of i  Br(i,H)} Measurability of strategy profiles. Measurability of strategy profiles. The set of point rationalizable strategy profiles is the largest set such that: Pr(H)=H The set of point rationalizable strategy profiles is the largest set such that: Pr(H)=H Equivalence with the economic viewpoint. Equivalence with the economic viewpoint. The « economic viewpoint » The « economic viewpoint » Same process but conjectures on the aggregate state. Same process but conjectures on the aggregate state. Point expectations : Cobweb mapping. Point expectations : Cobweb mapping. Def  (a)= ∫ B(i,a) di Def  (a)= ∫ B(i,a) di Cobweb tâtonnement outcome Cobweb tâtonnement outcome  =  t  0  t ( A ) Point expectations-ration. Point expectations-ration. Pr (X)= ∫ B(i,X) di Pr (X)= ∫ B(i,X) di The set of point-rationalizable states , is the largest set X  A such that: The set of point-rationalizable states , is the largest set X  A such that:Pr(X)=X Equivalence with the game viewpoint Equivalence with the game viewpoint

13. Rationalizability 2. Point expectations: H, set of strategy profiles. Point expectations: H, set of strategy profiles. Pr(H)={ s is strategy profile such that s is a measurable selection of i  Br(i,H)} Pr(H)={ s is strategy profile such that s is a measurable selection of i  Br(i,H)} The set of point rationalizable strategy profiles is the largest set such that: The set of point rationalizable strategy profiles is the largest set such that:Pr(H)=H Random expectations Random expectations Same process but take random beliefs. Same process but take random beliefs. Difficulty : measurability vis-à-vis probability distributions ? Difficulty : measurability vis-à-vis probability distributions ? = non measurability vis-à-vis point expectations ? = non measurability vis-à-vis point expectations ? Point expectations:ration. Point expectations:ration. Pr (X)= ∫ B(i,X) di Pr (X)= ∫ B(i,X) di The set of point-rationalizable states , is the largest set X  A such that: The set of point-rationalizable states , is the largest set X  A such that:Pr(X)=X Equivalence with the game viewpoint Equivalence with the game viewpoint Probabilistic expectations. Probabilistic expectations. R(X) = ∫ B(i, P (X)) di R(X) = ∫ B(i, P (X)) di The set of rationalizable states , is the largest set X  A such that: R(X)=X The set of rationalizable states , is the largest set X  A such that: R(X)=X Provides a substitute (equivalent) with the game viewpoint. Provides a substitute (equivalent) with the game viewpoint.

14. 14 Equilibria and rationalizable states. The state space : the concepts. The state space : the concepts. E, , ,  E, , ,  E  Co(E)       E  Co(E)       Properties : Properties : The set of point rationalizable states is non-empty, convex, compact. The set of point rationalizable states is non-empty, convex, 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= , Iteratively expectationally stable. (homogenous expectations) E= , Iteratively expectationally stable. (homogenous expectations) E = , Strongly point Rational. Heterogenous deterministic expectations E = , Strongly point Rational. Heterogenous deterministic expectations E=  Strongly Rational. Heterogenous probabilistic expectations. E=  Strongly Rational. Heterogenous probabilistic expectations.

15. 15 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. 3  2  1. 3  2  1. 1 weaker than 3 1 weaker than 3 The equivalence between 2 and 3 The equivalence between 2 and 3 Reinforcing locally strongly rational in Strictly locally strongly point rational (The contraction V-Pr n (v) is strict). Reinforcing locally strongly rational in Strictly locally strongly point rational (The contraction V-Pr n (v) is strict). Strictly locally strongly rational = locally point rational. Strictly locally strongly rational = locally point rational.

16. 16 Strategic Complementarities. Attempt at generalisation.

17. 17 Economies with strategic complementarities. Strategic complementarities in the state space. Strategic complementarities in the state space. 1B, S is the product of n compact intervals in R +. 1B, S is the product of n compact intervals in R +. 2B, u(i, ·, a) is supermodular for all a  A and all i  I. 2B, u(i, ·, a) is supermodular for all a  A and all i  I. 3B,  i  I, the function u(i, y, a) has increasing differences in y and a. 3B,  i  I, the function u(i, y, a) has increasing differences in y and a. B(i,a) est croissant en a, comme B(i,  ) …comme  (a)= ∫ B(i,a) di B(i,a) est croissant en a, comme B(i,  ) …comme  (a)= ∫ B(i,a) di Properties. Properties. a* min and a* max, smallest and largest equilibria. a* min and a* max, smallest and largest equilibria. a* min  E        a* max a* min  E        a* max All these sets but the first are convex. All these sets but the first are convex.  =  ??  =  ?? Comments. Comments. Uniqueness equivalent to Strong Rationality, Strong point rationalizability, IE stability. … the Graal. Uniqueness equivalent to Strong Rationality, Strong point rationalizability, IE stability. … the Graal. Locally, criteria equivalent. Locally, criteria equivalent. Heterogeneity does not matter so much, neither probabilistic beliefs. Heterogeneity does not matter so much, neither probabilistic beliefs.

18. 18 Back to one-dimensional Strategic 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)}=a-c(i), {u(i, y, a)}=a-c(i), c(i) individual cost of joining. c(i) individual cost of joining. y= (0 or 1), join, do not join y= (0 or 1), join, do not join Distribution of costs : cumulative F(c). Distribution of costs : cumulative F(c). F(a) = ∫ B(i,a)di =  (a) F(a) = ∫ B(i,a)di =  (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 SREE The Equilibrium is either a SREE Or [a* min,a* max ]=  =  = . Or [a* min,a* max ]=  =  = . c,a, a a*mina*max

19. 19 a* max a 1 min a 2 max A a 0 min a 0 max a* min a 2 min Strategic Complementarities with A  R 2 and multiple equilibria.  a 1 max a1a1a1a1 a2a2a2a2

20. 20 Economies with Strategic subsitutabilities. Economies with Strategic substitutabilities. Economies with Strategic substitutabilities. 1B, S is the product of n compact intervals in R +. 1B, S is the product of n compact intervals in R +. 2B, u(i, ·, a) is supermodular for all a  A and all i  I. 2B, u(i, ·, a) is supermodular for all a  A and all i  I. 3–B’,  i  I, the function u(i, y, a) has decreasing differences in y and a. 3–B’,  i  I, the function u(i, y, a) has decreasing differences in y and a. The cobweb mapping  is decreasing The cobweb mapping  is decreasing The second iterate of ,  2 is increasing. The second iterate of ,  2 is increasing. Results. Results. a* min and a* max, cycles of order 2 of  a* min and a* max, cycles of order 2 of        [a* min +R n, a* max - R n ]       [a* min +R n, a* max - R n ] 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 equilibrium, Strong Rationality, Strong point rationalizability, IE stability. The Graal : no cycle of order 2 and a unique equilibrium, 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.

21. 21 A a max a min A a max Muthian Strategic substitutes for A  R with unique equilibrium and multiple fixed points of  2  2 (a* max )= a* max  a* min  2 (a* min )= a* min a* max   (a*)= a* 

22. The Muth model with two crops 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(D -1 (a(1),a(2)) decreases. If a(1), a(2) increases, the vector S(D -1 (a(1),a(2)) decreases. « Eductive stability »: the local viewpoint. « Eductive stability »: the local viewpoint. S’ 12 /  D’ 1 D’ 2 <1-k, k=(assumption) (S 1 ’ /D’ 1 )= (S 2 ’ /D’ 2 ). S’ 12 /  D’ 1 D’ 2 <1-k, k=(assumption) (S 1 ’ /D’ 1 )= (S 2 ’ /D’ 2 ). 1-k is the index of « eductive stability » in case of indemendant markets. 1-k is the index of « eductive stability » in case of 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…

23. 23 Provisional conclusions Simple worlds: global coordination Simple worlds: global coordination With strategic complementarities, uniqueness is the « Graal ». With strategic complementarities, uniqueness 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-defeating pair of expectations… But absence of cycle of order two. Absence of self-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 properties of the best response mapping.. Local « eductive » stability and local properties of the best response mapping..

24. 24 Appendix 1a : supermodular games Appendix 1a : supermodular games Tarsky Theorem : Tarsky Theorem : F, function  from S to S, S complete lattice F, function  from S to S, S complete lattice The set of fixed points E is non empty and is a complete lattice. The set of fixed points E is non empty and is a complete lattice. Applications : S  R n, product of intervals in R, Applications : S  R n, product of intervals in R, sup E and inf E are fixed points sup E and inf E are fixed points Super modular functions : Super modular functions : G : R n  R, (strictly) supermodular G : R n  R, (strictly) supermodular  2 G/  x i  x j >0, i #j  2 G/  x i  x j >0, i #j Let f(t) = max x G(x,t), Let f(t) = max x G(x,t), G (strictly) supermodular on X  t G (strictly) supermodular on X  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.

25. 25 Appendix 1b : vi 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..

26. 26 Appendix 1c : 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 smallest element. The set of equilibria is non empty, has a greatest and a smallest element. Comments Comments Serially dominated strategies converge to the set Min [], Max [] Serially dominated strategies converge to the set Min [], Max [] Expectational coordination on this set. Expectational coordination on this set. If the equilibrium is unique, it is dominant solvable, globally SREE, « eductively » stable If the equilibrium is unique, it is dominant solvable, globally SREE, « eductively » stable