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The background of expectational stability studies. « Eductive stability ». Global versus local, « High tech » versus « Low tech »… 1 Back to a simple game. The rules of the game : Lessons : write a number : [0,100] Winner : 10 Euros : closest to 2/3 of the mean (of others) What happens in this game ? See Nagel (1995) 0 is the unique Nash equilibrium. It is a rather « reasonable » predictor of what happens. Change the game : Announce : [0, + infinity) [0,100]... 3/2 instead of 2/3. 2 The logic of rationalizability…again The « 2/3 of the mean » game. S(i)={0,100}, u(i,s(i), s(-i))=…..; Iterative elimination of non best response strategies : S(0,i) = {0,100}, S(1,i) = {0, 66,6666…} .... S(,i) = {0, (2/3) 100} 0 is the unique Nash equilibrium. The unique « rationalizable » outcome. Dominant solvable Nash outcome Strongly rational equilibrium, « edcutively » stable We have « strategic complementarities » 3 The «eductive » viewpoint. A « high-tech » formal (global) definition. Definition (with a continuum of small agents) 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 B : It is CK that E=E* If A B, the equ. is (globally) Strongly Rational. A « high-tech » formal (local) definition. Let V(E*} be some non trivial neighbourhood Assertion It is CK that A : E is in V(E*) Assertion B : It is CK that E=E* Same definition as before if V is the whole set of states. E* is locally, (vis-à-vis V), Strongly Rational. 4 The «eductive » stability criterion. Remarks on the generality. Remarks on the requirements. Potentially general. Requires « rational » agents with some Common Knowledge on the (working of) the system. A « hyper-rationalistic » view of coordination. 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….? Local Expectational viewpoint. A Connection with « evolutive learning » (asymptotic stability of…) Too demanding ? 5 An abstract framework. Games with a continuum of agents and aggregate summary statistics… 6 The model from a game-theoretical viewpoint. A continuum of players. An aggregation operator. A(s)= ∫s(i) di A is the (convex) set of states, A = ∫S(i)di = co{S}. For each agent i Utility Function: A measure space : (I, I, λ), with I=[0,1], λ Lebesgue measure Strategy sets : S(i)=S , compact subset of Rn. Strategy profile : s: I―› S, s(i). u(i, · , · ): S x A ―› R, continuous (C).HM: mapping i-u(.,i) measurable. The optimal strategy correspondence B(i,·):AS is: B(i, a) := argmaxyS {u(i, y, a)} . Nash equilibrium. Pure strategy Nash equilibrium s* is a strategy profile / s*(i) B(i, ∫s*(i)di)) iI, λ-a.e. Under assumptions C and HM, it exists, Rath (1992) 7 The model from an economic viewpoint Aggregate actions and best response Equilibrium . A = ∫S(i)di = co{S}. B(i,a) = argmaxyS {u(i, y, a)}. Def : (a)= ∫ B(i,a) di B(i, ) = argmaxyS {E[u(i, y, a)]} . a* = ∫ B(i,a*)di = ∫ B(i,a*)di (a*)=a* There exists an equilibrium. Equivalence for existence between the Nash viewpoint and the equilibrium viewpoint. Coordination. Focus on aggregate actions not on strategies. 8 One example : strategic complementarities. The model : The aggregate state a, proportion of people who join. {u(i, y, a)}=a-c(i), a c(i) individual cost of joining. y= (0 or 1), join, do not join Distribution of costs : cumulative F(c). F(a) = ∫ B(i,a)di = (a) Equilibrium a*=F(a*) c,a, Three or One ? How flat is the distribution 9 Another example : the linear Muth model. The Muth model Sellers : firms) or farmers. Decide to-day about production (wheat). Cost C(f,q)). Buyers will buy to-morrow. Demand curve : A-Bp. a=A-Bp, {u(i, y, a)}= (A/B-a/B)y- y2/2c(f), C= ∫ c(f)df. (a) = ∫ B(i,a)di= (CA)/B – (C/B)a. Strategic substitutabilities. C/B More general case : D(p), C(p) p=D-1(a), (a) = C°D-1(a). a 10 A reminder on Rationalizability. Game in normal form S(i), s(i), u(i,s(i), s(-i)) Iterative elimination of non best response strategies : 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(,i) = {S(-1,i) \ s(i) in S( - 1,i) non BR to j[(S( -1,j)]} R = [i((S(,i)] Remarks. Consider Pr [(S(i)] = {S(i) \ s in S(i) non BR to j[(S,(j)]} R =Pr(R), and R is the largest set such that R =Pr(R), Other set N R 11 Rationalizability 1. The (standard) game-theoretical viewpoint. Pr(H)={ s is strategy profile such that s is a measurable selection of iBr(i,H)} Measurability of strategy profiles. The set of point rationalizable strategy profiles is the largest set such that: Pr(H)=H Equivalence with the economic viewpoint. The « economic viewpoint » Same process but conjectures on the aggregate state. Point expectations : Cobweb mapping. Point expectations: H, set of strategy profiles. Recursive elimination of non best responses. … Def (a)= ∫ B(i,a) di Cobweb tâtonnement outcome =t0 t(A) Point expectations-ration. Pr (X)= ∫ B(i,X) di The set of point-rationalizable states , is the largest set X A such that: Pr(X)=X Equivalence with the game viewpoint 12 Rationalizability 2. Point expectations: H, set of strategy profiles. Random expectations Same process but take random beliefs. Difficulty : measurability vis-à-vis probability distributions ? = non measurability vis-à-vis point expectations ? Point expectations:ration. Pr(H)={ s is strategy profile such that s is a measurable selection of iBr(i,H)} The set of point rationalizable strategy profiles is the largest set such that: Pr(H)=H Pr (X)= ∫ B(i,X) di The set of point-rationalizable states , is the largest set X A such that: Pr(X)=X Probabilistic expectations. Equivalence with the game viewpoint R(X) = ∫ B(i,P(X)) di The set of rationalizable states , is the largest set X A such that: R(X)=X Provides a substitute (equivalent) with the game viewpoint. Equilibria and rationalizable states. The state space : the concepts. Properties : E, , , E Co(E) The set of point rationalizable states is non-empty, convex, compact. The set of rationalizable states is non-empty and convex. Definitions and terminology. E= , Iteratively expectationally stable. (homogenous expectations) E = , Strongly point Rational. Heterogenous deterministic expectations E= Strongly Rational. Heterogenous probabilistic expectations. 14 The local viewpoint. The local transposition. The connections. a* is locally iteratively stable… a* is locally Strongly point Rational… a* is locally strongly rational…. 32 1. 1 weaker than 3 The equivalence between 2 and 3 Reinforcing locally strongly rational in Strictly locally strongly point rational (The contraction V-Prn(v) is strict). Strictly locally strongly rational = locally point rational. 15 Strategic Complementarities. Attempt at generalisation. 16 Economies with strategic complementarities. Strategic complementarities in the state space. Properties. 1B, S is the product of n compact intervals in R+. 2B, u(i, · , a) is supermodular for all aA and all iI. 3B, iI, 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 a*min and a*max, smallest and largest equilibria. a*minE a*max All these sets but the first are convex. = ?? Comments. Uniqueness equivalent to Strong Rationality, Strong point rationalizability, IE stability. … the Graal. Locally, criteria equivalent. Heterogeneity does not matter so much, neither probabilistic beliefs. 17 Back to one-dimensional Strategic complementarities. The model : The aggregate state a, proportion of people who join. {u(i, y, a)}=a-c(i), c(i) individual cost of joining. y= (0 or 1), join, do not join Distribution of costs : cumulative F(c). F(a) = ∫ B(i,a)di = (a) Equilibrium a*= F(a*) Three or one ? How flat is the distribution. The Equilibrium is either a SREE Or [a*min ,a*max ]===. a a*min c,a, a*max 18 Strategic Complementarities with A R2 and multiple equilibria. a2 a0max A a1max a2max a*max a*min a2min a1min a0min a1 19 Economies with Strategic subsitutabilities. Economies with Strategic substitutabilities. Results. 1B, S is the product of n compact intervals in R+. 2B, u(i, · , a) is supermodular for all aA and all iI. 3–B’, iI, the function u(i, y, a) has decreasing differences in y and a. The cobweb mapping is decreasing The second iterate of , 2 is increasing. a*min and a*max , cycles of order 2 of [a*min+Rn, a*max- Rn] All these sets but the first are convex. = ?? Comments. The Graal : no cycle of order 2 and a unique equilibrium, Strong Rationality, Strong point rationalizability, IE stability. Locally, criteria equivalent. Heterogeneity does not matter so much, neither probabilistic beliefs. 20 Muthian Strategic substitutes for A R with unique equilibrium and multiple fixed points of 2 amax 2(a*max)= a*max A (a*)= a* 2(a*min)= a*min amin a*min a*max amax A 21 The Muth model with two crops The Model : A variant of Muth : Independant demands D(p(1)), D(p(2) S(p(1),p(2)) Strategic substitutes… Two crops : wheat and corn… If a(1), a(2) increases, the vector S(D-1(a(1),a(2)) decreases. « Eductive stability »: the local viewpoint. S’12/ D’1D’2 <1-k, k=(assumption) (S1’ /D’1)= (S2 ’ /D’2). 1-k is the index of « eductive stability » in case of indemendant markets. The interaction between the markets is destabilizing… One issue of the present crisis… Provisional conclusions Simple worlds: global coordination With strategic complementarities, uniqueness is the « Graal ». With strategic substituabilities, Uniqueness is no longer the Graal, But absence of cycle of order two. Absence of self-defeating pair of expectations… Outside simple worlds. More complex, cycles of any order matter.. Local « eductive » stability and local properties of the best response mapping.. 23 Appendix 1a : supermodular games Tarsky Theorem : F, function from S to S, S complete lattice The set of fixed points E is non empty and is a complete lattice. Applications : S Rn , product of intervals in R, sup E and inf E are fixed points Super modular functions : G : Rn R, (strictly) supermodular 2G/xi xj >0, i #j Let f(t) = maxx G(x,t), G (strictly) supermodular on X t Then, the mapping f is X compact and G USC in x, f compact. 24 Appendix 1b : visualizations. An increasing function …has a fixed point. Even with discontinuities. See the left diagram.. With a supermodular function : U(a,t) a (planned production) t (expected total production) ..keynesian situation Best response are increasing in t() Possibly with jumps. Inspect the second left diagram.. 25 Appendix 1c : Supermodular games. Definition Compact strategy space. U(i, s(i), s(-i)) (strict.) supermodular (see above) Equilibria in supermodular games : Best response Fn The set of equilibria is non empty, has a greatest and a smallest element. Comments Serially dominated strategies converge to the set Min [], Max [] Expectational coordination on this set. If the equilibrium is unique, it is dominant solvable, globally SREE, « eductively » stable 26