The background of expectational stability studies.

1
The background of expectational stability studies.

•  Eductive stability .
• Global versus local,
•  High tech  versus  Low tech 

2
Back to a simple game.

• The rules of the game
• 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)
• Lessons
• 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.

3
The logic of rationalizabilityagain

• 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 

4
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 EE
• 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 EE
• Same definition as before if V is the whole set
  of states.
• E is locally, (vis-à-vis V), Strongly Rational.

5
The eductive  stability criterion.

• Remarks on the generality.
• Potentially general.
• Remarks on the requirements.
• 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 ?

6
An abstract framework.

• Games with a continuum of agents and aggregate
  summary statistics

7
The model from a game-theoretical viewpoint.

• A continuum of players.
• A measure space (I, I, ?), with I0,1, ?
  Lebesgue measure
• Strategy sets S(i)S , compact subset of Rn.
• Strategy profile s I? S, s(i).
• An aggregation operator.
• A(s) ?s(i) di
• A is the (convex) set of states, A ?S(i)di
  coS.
• For each agent i Utility Function
• u(i, , ) S x A ? R, continuous (C).HM
  mapping i-u(.,i) measurable.
• The optimal strategy correspondence B(i,)A?S
  is
• B(i, a) argmaxy?S u(i, y, a) .
• Nash equilibrium.
• Pure strategy Nash equilibrium s is a strategy
  profile /
• s(i) ? B(i, ?s(i)di)) ?i?I, ?-a.e.
• Under assumptions C and HM, it exists, Rath
  (1992)

8
The model from an economic viewpoint

• Aggregate actions and best response
• A ?S(i)di coS.
• B(i,a) argmaxy?S u(i, y, a).
• Def ?(a) ? B(i,a) di
• B(i, ?) argmaxy?S E?u(i, y, a) .
• Equilibrium .
• 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.

9
One example 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 aF(a)
• Three or
• One ?
• How flat is the distribution

a
c,a,
10
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.
• aA-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.
• More general case D(p), C(p)
• pD-1(a),
• ?(a) CD-1(a).

?
11
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

12
Rationalizability 1.

• The (standard) game-theoretical viewpoint.
• Recursive elimination of non best responses.
• 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)
• 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.
• Def ?(a) ? B(i,a) di
• Cobweb tâtonnement outcome
• ??t?0 ?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

13
Rationalizability 2.

• 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)
• The set of point rationalizable strategy profiles
  is the largest set such that
• Pr(H)H
• Random expectations
• Same process but take random beliefs.
• Difficulty measurability vis-à-vis probability
  distributions ?
• non measurability vis-à-vis point expectations ?

• Point expectationsration.
• 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
• Probabilistic expectations.
• 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.

14
Equilibria and rationalizable states.

• The state space the concepts.
• E, ?, ?, ?
• E ? Co(E) ? ? ? ? ? ?
• Properties
• 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.

15
The local viewpoint.

• The local transposition.
• a is locally iteratively stable
• a is locally Strongly point Rational
• a is locally strongly rational.
• The connections.
• 3?2 ?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.

16
Strategic Complementarities.

• Attempt at generalisation.

17
Economies with strategic complementarities.

• Strategic complementarities in the state space.
• 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.
• 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
• Properties.
• amin and amax, smallest and largest equilibria.
• amin?E ? ? ? ? ? ? ?amax
• 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.

18
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 amin ,amax ???.

a
c,a,
amin
amax
19
Strategic Complementarities withA ? R2 and
multiple equilibria.
a2
a0max
A
a1max
a2max
amax
?
amin
a2min
a1min
a0min
a1
20
Economies with Strategic subsitutabilities.

• Economies with Strategic substitutabilities.
• 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.
• 3B, ?i?I, 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.
• Results.
• amin and amax , cycles of order 2 of ?
• ? ? ? ? ? ? aminRn, amax- 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.

21
Muthian Strategic substitutes for A ? R with
unique equilibrium and multiple fixed points of ?2
?2(amax) amax
?
?(a) a
?
?2(amin) amin
?
amax
amin
22
The Muth model with two crops

• The Model
• A variant of Muth
• Two crops wheat and corn
• Independant demands
• D(p(1)), D(p(2)
• S(p(1),p(2))
• Strategic substitutes
• If a(1), a(2) increases, the vector
  S(D-1(a(1),a(2)) decreases.
•  Eductive stability  the local viewpoint.
• S12/ ?D1D2 lt1-k, k(assumption) (S1 /D1)
  (S2 /D2).
• 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

23
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..

24
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 gt0, 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.

25
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..

26
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