# Rationalizability and the stability of expectations

advertisement ```The background of expectational
stability studies.
&laquo; Eductive stability &raquo;.
Global versus local,
&laquo; High tech &raquo; versus &laquo; Low tech &raquo;…
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Back to a simple game.
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The rules of the game :
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Lessons :
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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 &laquo; reasonable &raquo; predictor of what happens.
Change the game :
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Announce : [0, + infinity) [0,100]...
3/2 instead of 2/3.
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The logic of rationalizability…again
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The &laquo; 2/3 of the mean &raquo; game.
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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
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the unique Nash equilibrium.
The unique &laquo; rationalizable &raquo; outcome.
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Dominant solvable Nash outcome
Strongly rational equilibrium, &laquo; edcutively &raquo; stable
We have &laquo; strategic complementarities &raquo;
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The &laquo;eductive &raquo; viewpoint.
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A &laquo; high-tech &raquo; formal (global) definition.
 Definition (with a continuum of small agents)
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Let E* (in some vector space ) be an (Rat.Exp.) equilibrium.
Assertion A : It is CK that E   (rationality and the model are CK)
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Assertion B : It is CK that E=E*
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If A  B, the equ. is (globally) Strongly Rational.
A &laquo; high-tech &raquo; formal (local) definition.
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Let V(E*} be some non trivial neighbourhood
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Assertion It is CK that A : E is in V(E*)
Assertion B : It is CK that E=E*
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Same definition as before if V is the whole set of states.
E* is locally, (vis-&agrave;-vis V), Strongly Rational.
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The &laquo;eductive &raquo; stability criterion.
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Remarks on the generality.
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Remarks on the requirements.
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Potentially general.
Requires &laquo; rational &raquo; agents with some Common Knowledge on the
(working of) the system.
A &laquo; hyper-rationalistic &raquo; view of coordination.
A &laquo; Low-tech &raquo; interpretation and alternative intuition.
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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….?
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Local Expectational viewpoint.
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A Connection with &laquo; evolutive learning &raquo; (asymptotic stability of…)
Too demanding ?
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An abstract framework.
Games with a continuum of agents and
aggregate summary statistics…
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The model from a game-theoretical viewpoint.
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A continuum of players.
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An aggregation operator.
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A(s)= ∫s(i) di
A is the (convex) set of states, A = ∫S(i)di = co{S}.
For each agent i Utility Function:
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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, &middot; , &middot; ): S x A ―› R, continuous (C).HM: mapping i-u(.,i) measurable.
The optimal strategy correspondence B(i,&middot;):AS is:
B(i, a) := argmaxyS {u(i, y, a)} .
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Nash equilibrium.
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Pure strategy Nash equilibrium s* is a strategy profile /
s*(i)  B(i, ∫s*(i)di)) iI, λ-a.e.
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Under assumptions C and HM, it exists, Rath (1992)
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The model from an economic viewpoint
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Aggregate actions and best response
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Equilibrium .
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A = ∫S(i)di = co{S}.
B(i,a) = argmaxyS {u(i, y, a)}.
Def : (a)= ∫ B(i,a) di
B(i, ) = argmaxyS {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.
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Focus on aggregate actions not on strategies.
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One example : strategic
complementarities.
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The model :
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The aggregate state a,
proportion of people who join.
{u(i, y, a)}=a-c(i),
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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*)
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c,a,
Three or
One ?
How flat is the distribution
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Another example : the linear
Muth model.
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The Muth model
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Sellers : firms) or farmers.
 Decide to-day about production
(wheat).
 Cost C(f,q)).
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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.
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(a) = ∫ B(i,a)di= (CA)/B – (C/B)a.
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Strategic substitutabilities.
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C/B
More general case : D(p), C(p)
p=D-1(a),
(a) = C&deg;D-1(a).
a
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A reminder on Rationalizability.
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Game in normal form
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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.
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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
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Rationalizability 1.
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The (standard) game-theoretical
viewpoint.
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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 &laquo; economic viewpoint &raquo;
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Same process but conjectures on the
aggregate state.
Point expectations : Cobweb mapping.
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Point expectations: H, set of
strategy profiles.
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Recursive elimination of non best
responses.
…
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Def (a)= ∫ B(i,a) di
Cobweb t&acirc;tonnement outcome
=t0 t(A)
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Point expectations-ration.
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Pr (X)= ∫ B(i,X) di
The set of point-rationalizable states
, is the largest set X A such that:
Pr(X)=X
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Equivalence with the game
viewpoint
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Rationalizability 2.
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Point expectations: H, set of
strategy profiles.
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Random expectations
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Same process but take random beliefs.
Difficulty : measurability vis-&agrave;-vis
probability distributions ?
= non measurability vis-&agrave;-vis point
expectations ?
Point expectations:ration.
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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
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Pr (X)= ∫ B(i,X) di
The set of point-rationalizable states
, is the largest set X A such that:
Pr(X)=X
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Probabilistic expectations.
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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.
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The state space : the concepts.
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Properties :
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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.
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E= , Iteratively expectationally stable. (homogenous expectations)
E = , Strongly point Rational. Heterogenous deterministic expectations
E=  Strongly Rational. Heterogenous probabilistic expectations.
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The local viewpoint.
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The local transposition.
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The connections.
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a* is locally iteratively stable…
a* is locally Strongly point Rational…
a* is locally strongly rational….
32 1.
1 weaker than 3
The equivalence between 2 and 3
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Reinforcing locally strongly rational in Strictly locally strongly point
rational (The contraction V-Prn(v) is strict).
Strictly locally strongly rational = locally point rational.
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Strategic Complementarities.
Attempt at generalisation.
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Economies with strategic
complementarities.
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Strategic complementarities in the state space.
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Properties.
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1B, S is the product of n compact intervals in R+.
2B, u(i, &middot; , 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
a*min and a*max, smallest and largest equilibria.
a*minE       a*max
All these sets but the first are convex.
 =  ??
Comments.
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Uniqueness equivalent to Strong Rationality, Strong point
rationalizability, IE stability. … the Graal.
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Locally, criteria equivalent.
Heterogeneity does not matter so much, neither probabilistic beliefs.
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Back to one-dimensional Strategic
complementarities.
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The model :
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The aggregate state a,
proportion of people who join.
{u(i, y, a)}=a-c(i),
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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*)
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Three or one ?
How flat is the distribution.
The Equilibrium is either a SREE
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Or [a*min ,a*max ]===.
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a
a*min
c,a,
a*max
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Strategic Complementarities with
A  R2 and multiple equilibria.
a2
a0max
A
a1max
a2max
a*max

a*min
a2min
a1min
a0min
a1
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Economies with Strategic
subsitutabilities.
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Economies with Strategic substitutabilities.
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Results.
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1B, S is the product of n compact intervals in R+.
2B, u(i, &middot; , 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.
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.
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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.
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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
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The Muth model with two crops
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The Model :
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A variant of Muth :
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Independant demands
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D(p(1)), D(p(2)
S(p(1),p(2))
Strategic substitutes…
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Two crops : wheat and corn…
If a(1), a(2) increases, the vector S(D-1(a(1),a(2)) decreases.
&laquo; Eductive stability &raquo;: the local viewpoint.
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S’12/ D’1D’2 &lt;1-k, k=(assumption) (S1’ /D’1)= (S2 ’ /D’2).
1-k is the index of &laquo; eductive stability &raquo; in case of indemendant markets.
The interaction between the markets is destabilizing…
One issue of the present crisis…
Provisional conclusions
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Simple worlds: global coordination
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With strategic complementarities, uniqueness is the &laquo; Graal &raquo;.
With strategic substituabilities,
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Uniqueness is no longer the Graal,
But absence of cycle of order two. Absence of self-defeating pair of
expectations…
Outside simple worlds.
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More complex, cycles of any order matter..
Local &laquo; eductive &raquo; stability and local properties of the best
response mapping..
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Appendix 1a : supermodular games
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Tarsky Theorem :
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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 :
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G : Rn  R, (strictly) supermodular
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2G/xi xj &gt;0,
i #j
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Let f(t) = maxx G(x,t),
G (strictly) supermodular on X t
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Then, the mapping f is 
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X compact and G USC in x, f compact.
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Appendix 1b : visualizations.
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An increasing function
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…has a fixed point.
Even with discontinuities.
See the left diagram..
With a supermodular function :
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U(a,t)
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a (planned production)
t (expected total production)
..keynesian situation
Best response are increasing in t()
Possibly with jumps.
Inspect the second left diagram..
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Appendix 1c : Supermodular games.
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Definition
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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,
&laquo; eductively &raquo; stable
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