![Strategic substitutes for A ≡ [0, a max ] ⊂ R. There exists a unique equilibrium and multiple fixed points for 2](https://www.researchgate.net/profile/Roger-Guesnerie/publication/226641653/figure/fig1/AS:393635125317634@1470861335398/Strategic-substitutes-for-A-0-a-max-R-There-exists-a-unique-equilibrium-and_Q320.jpg)
Content uploaded by Roger Guesnerie
Author content
All content in this area was uploaded by Roger Guesnerie on Feb 27, 2015
Content may be subject to copyright.
Econ Theory (2011) 47:205–246
DOI 10.1007/s00199-010-0556-8
SYMPOSIUM
Expectational coordination in simple economic contexts
Concepts and analysis with emphasis on strategic substitutabilities
Roger Guesnerie ·Pedro Jara-Moroni
Received: 16 November 2009 / Accepted: 24 June 2010 / Published online: 24 July 2010
© Springer-Verlag 2010
Abstract We consider an economic model that features (1) a continuum of agents
and (2) an aggregate state of the world over which agents have an infinitesimal
influence. We first review the connections between the “eductive” viewpoint on expec-
tational stability and standard game-theoretical rationalizability concepts. The “educ-
tive” reasoning selects different plausible beliefs that are a priori, and possibly a
posteriori, “diverse”. Such beliefs are associated with the sets of “Cobweb tâtonne-
ment” outcomes, “Rationalizable States” and “Point-Rationalizable States” (the latter
two being shown to be convex). In the case where our model displays strategic com-
plementarities, unsurprisingly, all our “eductive” criteria support similar conclusions,
particularly when the equilibrium is unique. With strategic substitutabilities, the suc-
cess of expectational coordination, in the case where a unique equilibrium does exists,
relates with the absence of cycles of order 2 of the “Cobweb” mapping: in this case,
full expectational coordination would be achieved. However, when cycles of order 2
Jara-Moroni gratefully acknowledges financial support from Centro de Modelamiento Matem ático and
Departamento de Ingeniería Matem ática of Universidad de Chile, Instituto Sistemas Complejos de
Ingeniería (ICM: P-05-004-F), Collège de France, French Ministry of Foreign Affairs and DICYT project
031062JM. We thank Jess Benhabib, Juan Carluccio, Christophe Chamley, Gabriel Desgranges, Georges
Evans, Stéphane Gauthier, Sayantan Ghosal, Ali Khan, Bruno Strulovici and Olivier Tercieux for helpful
comments on previous versions of this article. We would like to thank as well the audiences at the IESE
Conference on Complementarities and Information and at the Stanford Institute for Theoretical
Economics Workshop Segment 8: When Are Diverse Beliefs Central?. We are also grateful to an
anonymous referee for his careful reading and discussion of the paper and for his suggestions.
R. Guesnerie
Paris School of Economics and Collège de France, Paris, France
e-mail: roger.guesnerie@college-de-france.fr
P. Jara-Moroni (B
)
Depto. de Economía, Universidad de Santiago de Chile,
Av. Lib. Bdo. O’Higgins, 3363 Santiago, Chile
e-mail: pedro.jara@usach.cl
123
206 R. Guesnerie, P. Jara-Moroni
do exist, our different criteria predict larger sets of outcomes, although all tied with
cycles. Under differentiability assumptions, the Poincaré–Hopf method leads to other
global stability results. At the local level, the different criteria under scrutiny can be
adapted. They lead to the same expectational stability conclusions, only when there
are local strategic complementarities or strategic substitutabilities. However, for the
local stability analysis, it is demonstrated that the stochastic character of expectations
can most often be forgotten.
Keywords Expectational coordination ·Rational expectations ·Iterative
expectational stability ·Eductive stability ·Strong rationality ·Strategic
complementarities ·Strategic substitutabilities
JEL Classification D84 ·C72 ·C62
1 Introduction
This paper is concerned, as most of the papers of this conference, with the nature and
quality of expectational coordination in economic situations. The rational expectations
hypothesis, which provides a somewhat definitional description of what is “maximal”
or “full” coordination, generally rules out “Diverse Beliefs” in the sense of the title
of this conference. The present paper, which examines coordination in a variety of
economic contexts, does neither a priori rule out the occurrence of “full” coordina-
tion, nor take it for granted. It subjects the assessment of the plausibility of a rational
expectations outcome to a test, which is “eductive”, in a sense that will be made clear
later. When the test fails, the analysis not only supports the assumption of “diverse
beliefs”, but also provides an assessment of “how diverse” they are likely to be.
Hence, the viewpoint taken here is different, and complementary to other view-
points taken in this conference. The “endogenous uncertainty” we may discover is a
priori different from the one stressed in Kurz and Motolese (2010), and also the pos-
sible “heterogeneity of expectations” suggested here does not originate in “real time
learning” of the kind examined in Branch and Evans (2010)orBranch and McGough
(2010).
A few more words may be useful to clarify the connection between the logic of
the present analysis and the alternative logic of the two lines of research just evoked,
the “diverse beliefs” approach initiated by Kurz (1994) and the “evolutive” real-time
learning of the type examined in Evans and Honkapohja (2001). In the former, the root
of diversity is the disagreement concerning the distribution of exogenous variables, but
the equilibrium map, given the beliefs and the exogenous state is (or may be) known.1
In contrast, our approach questions the knowledge of the equilibrium map from argu-
ments that (informally) refer to the sensitivity of the actual realizations of the economic
state to (non-equilibrium) beliefs or (formally) rests on Common Knowledge (CK)
1An anonymous referee has rightly stressed that the two approaches (eductive theory and rational beliefs)
mainly occupy themselves with relaxing each one of the two basic Muthian hypotheses—that the actions
of others are known in equilibrium—that the distribution of the exogenous noise is Common Knowledge.
123
Expectational coordination in simple economic contexts 207
or rationalizability considerations. Concerning the former line of research, (real-time
learning) our approach provides an alternative critique of the rational expectations
assumption, which, as the real-time approach, may (under convergence) or may not
involve support. Furthermore, our “eductive” learning process, which does not suffer
from the arbitrariness of the choice of real-time processes, leads often to conclusions
that are likely to provide an efficient shortcut to studies of real-time learning (see for
example Guesnerie (2002) and Gauthier and Guesnerie (2005) for a defense of this
claim from a number of systematic comparisons).
Let us come to the paper. It focuses attention on a class of economic models with
“non-atomic” agents, i.e., agents that are too small to have a significant influence on
the aggregate state of the economic system.2Such an assumption often fits the need of
economic analysis, (in general equilibrium, macroeconomics, …) although it excludes
from our analysis oligopolistic competition models in which agents, (firms) have
market power.3Also, our framework refers to a context of perfect information.
Although a reinterpretation of the basic model allows it to encompass certain prob-
lems with imperfect information, a more systematic extension to general contexts of
imperfect information is in progress.
In the first section, we present the skeleton of the model under scrutiny, which we
interpret successively and equivalently as a game with a continuum of agents and as
an economic model (Sect. 2). We then switch attention to equilibria and on what may
be called the expectational quality (or plausibility, or robustness) of equilibria. Our
viewpoint is “eductive” in the sense that it refers to the reasonings of agents attempting
to guess the actions (or guesses) of others. Some of the concepts we use have purely
economic underpinnings: it is the case of the Cobweb tâtonnement outcomes or the
associated concept of Iterative Expectational Stability (IE-stability) that comes from
the macroeconomic literature of the eighties. Others have a general game theoreti-
cal inspiration: strategic point rationalizability or strategic plain rationalizability. We
show below how they can be adapted in a standard way,4in order to take advantage
of the specificities of our economic context. We then switch attention to what we
call State Point Rationalizability and State Rationalizability and on the corresponding
stability concepts of Strong Rationality. We also put emphasis on local counterparts
to the concepts that leads to stress local rather than global expectational stability of
equilibria (global or local strong rationality in the sense of Guesnerie (1992)).
Section 3defines and clarifies the connections between the different concepts under
review within our framework. The inclusion stressed in Proposition 2is unsurprising
but useful: it reflects the increasing demand of the “eductive” analysis when the sophis-
tication of the agents’ reasoning on others’ expectations increases. Another result has
to be stressed: the fact that state rationalizable sets are convex.
2Many existing studies on expectational coordination adopt such a framework with non-atomic agents.
For example, among others Guesnerie (1992,2002), Evans and Guesnerie (1993,2003,2005), Chamley
(1999,2004), Desgranges and Heinemann (2005). The same remark applies to part of the global games
literature starting from Morris and Shin (1998) and surveyed in Morris and Shin (2003).
3Not all, since the fashionable modelling of competition à la Dixit and Stiglitz conciliate market power
and smallness of agents. Note also that our main result has other implications for the theory of oligopolistic
competition that are examined in a forthcoming paper.
4See Guesnerie (2002).
123
208 R. Guesnerie, P. Jara-Moroni
Section 4comes to the main application of our general analysis, which concerns
economies with strategic substitutabilities. We first reformulate and prove in our setting
the standard results obtained when interactions are dominated by strategic comple-
mentarities (Proposition 3in Sect. 4.1). In such a context, a striking result is that
“uniqueness is the Grail”, in the sense that uniqueness of the equilibrium triggers
global stability of the equilibrium, for any of the criteria evoked in Sect. 3.How-
ever, this equivalence of expectational criteria is known to fail dramatically outside
the strategic complementarities world. Our main result of Sect. 4shows that in the
world with strategic substitutabilities under consideration, expectational stability is
still easy to analyse. Theorem 1asserts that uniqueness, not of fixed point of the best
response mapping, but uniqueness of its cycles of order two, is still the “Grail”: when
this occurs, the unique equilibrium does fit all stability criteria under consideration.
However, when cycles of order 2 do exist, our different criteria predict larger sets of
outcomes. Such outcomes may be viewed as delineating the extent of “diversity” of
beliefs, a diversity relating, however, to the characteristics of the cycles under exam-
ination. These somewhat surprising results have potentially many applications, in
particular, as we argue in Sect. 2.3, in a general equilibrium framework when strategic
substitutabilities often dominate strategic complementarities. We finally stress in this
section other global results, obtained under differentiability assumptions, from the use
of the Poincaré–Hopf method.
Section 5considers the adaptation at the local level of the different criteria under
scrutiny. They are shown to lead to the same expectational stability conclusions, only
when there are local strategic complementarities or strategic substitutabilities. How-
ever, for the local stability analysis, it is demonstrated that one can, most often, forget
about stochastic expectations and concentrate on (heterogeneous) point expectations.
We briefly conclude.
2 The canonical model
We can view our reference model either as a game with a continuum of players or
as a non-atomic economic model. This section has mainly an introductory purpose.
We present the different viewpoints successively, as well as the specific notation that
they call for. Existence of equilibria relies on earlier results, and a brief inspection of
relevant literature suggests that the model is relevant for shedding light on an a priori
wide range of economic questions.
2.1 The model as a game with a continuum of players
2.1.1 The setting
The players. Let us consider a game with a continuum of players.5In such games
the set of players is the measure space (I,I,λ
), where Iis the unit interval of R,
5The literature on non-atomic games goes back to Schmeidler (1973). For a comprehensive review of large
games see Khan and Sun (2002).
123
Expectational coordination in simple economic contexts 209
I≡[0,1], and λis the Lebesgue measure. Each player chooses a strategy s(i)∈S(i)
and we take S(i)⊆Rn. Strategy profiles in this setting are identified with integrable
selections of the set valued mapping i⇒S(i).6For simplicity, we will assume that
all the players have the same compact strategy set S(i)≡S⊂Rn
+. As a consequence,
the set of meaningful strategy profiles is the set of measurable functions from Ito S
7denoted from now on SI.
The aggregator In a game, players have payoff functions that depend on their own
strategy and the complete profile of strategies of the player π(i,·,·):S×SI→R.
The best reply correspondence Br(i,·):SI⇒Sis defined as
Br(i,s):= argmaxy∈Sπ(i,y,s).
The correspondence Br(i,·)describes the optimal response set for player i∈I
facing a strategy profile s.
In our particular framework the pay-off functions depend, for each player, on his
own strategy and an average of the strategies of all the other players. To obtain this
average we use the integral of the strategy profile, Is(i)di. This implies that all the
relevant information about the actions of the opponents is summarized by the values
of the integrals, which are points in the set8
A≡
I
S(i)di.
Since S(i)≡Swe have that9,10
A≡co {S}.
Payoffs π(i,·,·)in this setting are evaluated from an auxiliary utility function
u(i,·,·):S×co {S}→Rsuch that
6A selection is a function s:I→Rnsuch that s(i)∈S(i). We use the notation ⇒for set valued
mappings (also referred to as correspondences), and →for functions.
7Equivalently, the set of measurable selections of the constant set valued mapping i⇒S.
8Following Aumann (1965) we define for a correspondence F:I⇒Rnits’ integral, IF(i)di, as:
I
F(i)di := ⎧
⎨
⎩
x∈Rn:x=
I
f(i)di and fis an integrable selection of F⎫
⎬
⎭
9Where co {X}stands for the convex hull of a set X.
10 If strategy sets differed across players, hypothesis over the correspondence i⇒S(i)that assure that
the set Ais well defined can be found in Aumann (1965) or in Chapter 14 of Rockafellar and Wets (1998).
In this case, using the Liapounov theorem, we get that Ais a convex set (Aumann 1965).
123
210 R. Guesnerie, P. Jara-Moroni
π(i,y,s)≡u⎛
⎝i,y,
I
s(i)di⎞
⎠(1)
2.1.2 Further preliminaries
Technical assumptions We assume
C:For all agent i ∈I, u(i,·,·)is continuous.
HM : The mapping that associates to each agent a utility function is measurable.11
Cis standard and does not deserve special comments. HM is technical but, in a
sense, natural in this setting. Adopting both assumptions on utility functions put us in
the framework of Rath (1992).
Nash equilibrium In the general notation, a Nash equilibrium is a strategy profile
s∗∈SIsuch that, ∀i∈Iλ-a.e., s∗(i)∈Br(i,s∗). In this setting, we write the
definition as follows:
Definition 1 A (pure strategy) Nash Equilibrium of a game is a strategy profile s∗∈SI
such that λ-almost-everywhere in I:
ui,s∗(i),s∗(i)di≥ui,y,s∗(i)di,∀y∈S
Under the previously mentioned hypothesis, Rath (1992) shows that for every such
game there exists a Nash Equilibrium.
2.2 Economies with a continuum of non-atomic agents
2.2.1 An economic reinterpretation of the game
The aggregate states of the system We now interpret the setting as a stylized eco-
nomic model in which there is a large number of small agents i∈I. In this economic
system, there is an aggregate variable or signal that represents the state of the system.
Now A⊆Rnis viewed as the set of all possible states of the economic system.
Agents take individual actions, which determine the state of the system through an
aggregation operator, A, the “ mediator” of the economic interaction of the agents.
The key feature of the system is that no agent, or small group of agents, can affect
unilaterally the state of the system.
The so-called economic system is then immediately imbedded onto the just defined
game with a continuum of players. We identify (individual) actions with (individual)
strategies so that the aggregation operator associates to each action or strategy profile
11 The set of functions for assumption HM is the set of real valued continuous functions defined on
S×co {S}endowed with the sup norm topology.
123
Expectational coordination in simple economic contexts 211
sa state of the model a=A(s)in the set of states A, the aggregation operator Abeing
the integral12 of the profile s:
A(s)≡
I
s(i)di.
with the state set Aequal to co {S}.
The variable a∈A, that is now viewed as the state of the system, determines, along
with each agents’ own action, his payoff. Each agent i∈Ithen, acts to maximize its
payoff function u(i,·,·):S×A→Ras already introduced in (1).
Useful mappings In our setting, and considering the auxiliary function u(i,·,·),
we can define the optimal strategy correspondence B(i,·):A⇒Sas the corre-
spondence which associates to each point a∈Athe set
B(i,a):= argmaxy∈S{u(i,y,a)}.(2)
Note that, since in this setting a=Is(i)di, then Br(i,s)=B(i,a).
In a situation where agents act in ignorance of the actions taken by the others or, for
what matters, of the value of the state of the system, they have to rely on forecasts. That
is, their actions must be a best response to some subjective probability distribution
over the space of aggregate data A. Mathematically, actions have to be elements of
the set of points that maximize expected utility, where the expectation is taken with
respect to this subjective probability. We can consider then the best reply to forecasts
correspondence B(i,·):P(A)⇒Sdefined by
B(i,μ
):= argmaxy∈SEμ[u(i,y,a)](3)
where μ∈P(A), and P(A)is the space of probability measures over A. Since the
utility functions are continuous, problems (2) and (3) are well defined and have always
a solution; so consequently the mappings B(i,·)and B(i,·)take non-empty com-
pact values for all a∈A. Clearly B(i,a)≡B(i,δ
a), where δais the Dirac measure
concentrated in a.
12 The aggregation operator can as well be the integral of the strategy profile with respect to any measure
¯
λthat is absolutely continuous with respect to the Lebesgue measure, or the composition of this result with
a continuous function. That is,
A(s)≡G⎛
⎝
I
s(i)f(i)di⎞
⎠
where G:IS(i)d¯
λ(i)→Ais a continuous function and fis the density of the measure ¯
λwith respect
to the Lebesgue measure.
However, not all the results in this work remain true if we choose such a setting.
123
212 R. Guesnerie, P. Jara-Moroni
2.2.2 Economic equilibrium
An equilibrium of this system is a state a∗generated by actions of the agents that are
optimal reactions to this state. We denote (a)=IB(i,a)di.
Definition 2 An equilibrium is a point a∗∈Asuch that
a∗∈a∗≡
I
Bi,a∗di ≡
I
B(i,δ
a∗)di (4)
Assumptions Cand HM assure that the integrals in Definition 2are well defined.13
The equilibrium conditions in (4) are standard description of self-fulfilling fore-
casts. That is, in an equilibrium a∗, agents must have a self-fulfilling point forecast
(Dirac measure) over a∗, i.e. in the economic terminology, if we take the model strictly
speaking, a perfect foresight equilibrium.
It is straightforward to see that an equilibrium as defined in (4) has as a counter-
part in the game-theoretical approach a Nash Equilibrium of the underlying game as
defined in (2). More precisely, for every (pure strategy) Nash Equilibrium s∗of the
system’s underlying game, there exists a unique equilibrium a∗given by a∗:= A(s∗),
and if a∗is an equilibrium of the system, then ∃s∗∈SIthat is a Nash Equilibrium of
the underlying game. As we already know that the game has a Nash equilibrium, we
also have an equilibrium existence result.
We will refer equivalently then, to equilibria as points a∗∈A, representing “eco-
nomic equilibria”, and s∗∈SI, as Nash Equilibria of the underlying game.
Such equilibria are associated with “perfect” expectational coordination and here,
by definition, rule out diverse beliefs. However, the plausibility or “quality” of the
expectational coordination that they assume, and hence their predictive power, for
what concerns for example the absence of diverse beliefs, have been challenged. The
critical assessment of expectational coordination has been made through different
glasses, for example, those of experimental economics, “evolutive” (real time) learn-
ing, and “eductive” assessment. The “eductive” viewpoint stresses arguments that do
not explicitly refer to real-time learning, although they may sometimes be interpreted
as “virtual time”learning. It is the one we will take here, in Sect. 3.
2.3 Examples
Many theoretical models of economic theory enter the above general framework,
whenever they have non-atomic agents.14 Let us give a few examples, going from
partial equilibrium, general equilibrium, finance, and macro-economics. In many of
these examples, the results we prove in the last section of the paper are directly useful.
13 See Lemma 3.2 in Jara-Moroni (2008).
14 We might replace “many” by “most” in the present sentence, if instead of the simplified version of the
aggregator under scrutiny here, we were dealing with the most sophisticated form alluded to above. Also,
our results have significant implications for models of the oligopolistic competition type with atomic agents
(work in progress).
123
Expectational coordination in simple economic contexts 213
The simplest example is a variant of Muth’s (1961) model presented in Guesnerie
(1992). In this partial equilibrium model, there is a group of “farmers” (or “firms”)
indexed by the unit interval I=[0,1]which we endow with the Lebesgue measure.
Farmers decide a positive production quantity q(i)and so strategy profiles are func-
tions from the set of agents to a compact subset (individual productions are bounded)
of the positive line R+(i.e. n=1), q:I→S(i)≡S⊂R+. The aggregate
variable in this case is aggregate production Qand the aggregation operator is the
integral of the production profile q,Q=Iq(i)di. Agents evaluate their payoff from
aggregate production through the price function u(i,q,Q)=P(Q)q−Ci(q), where
P:R+→R+is an inverse demand function that, given a quantity of good, gives the
price at which this quantity is sold.15 The model displays strategic substitutabilities
in the sense that a higher (expected) aggregate production triggers a lower individual
production decision.
The “n+2 goods” general equilibrium model under study in Guesnerie (2001a)
has similar features. The wage on the labour market is fixed, and individual firms must
take today production decisions. They face (strategic) uncertainty on the level of total
production, which triggers total income available tomorrow, and then determines the
market clearing prices for the goods. Production takes place in Lsectors indexed by l,
with Nlfirms in each sector. Firms hire workers at a fixed wage wand sell at market
clearing price pl.Firmiin sector lsupplies ql
iof good land 0 of the other goods, so
that aggregate supply is the vector q(i)di. We have then a L-dimensional version of
the present prototype model. It is shown in the just quoted article that if consumers’
demand satisfies the gross substitutability assumption, then strategic substitutabili-
ties, due to price effects, unambiguously dominate strategic complementarities, due to
income effects of the multiplier type. Thus, the model displays aggregate strategic sub-
stitutabilities, as in Sect. 4. One can also show that the Walrasian flexible wage version
of this L-goods model enters a somewhat similar framework, although less trivially.16
Note that the two just sketched categories of models strictly fall within our frame-
work, i.e., can be interpreted as having two periods, with the agents simultaneously
deciding at the first period. The conditions for “eductive” stability are obviously mod-
ified when the first period is divided in sub-periods allowing sequential decisions with
observations of previous decisions. But as substantiated in Guesnerie (2002) (Section
3-2-3), the flavour of the argument is not changed, and the precise conditions are
changed in a somewhat straightforward way.
In Finance, many models of transmission of information through prices have
non-atomic agents with different information, as in the pioneering work of
Grossman and Stiglitz (1980). The agents’ strategy, viewed in our framework, is a
15 On this example we can make the observation that the state of the game could be chosen to be the price
instead of aggregate production. In this one-dimensional setting, it is the case that most of the properties
herein presented are passed on from the aggregate production to the price variable.
16 The “action” or strategy of the agent in a Walrasian context is no longer his production decision, but his
reservation wage on the labour market (see Guesnerie 2001b, section 4). Strictly speaking, the aggregate
state of the model is no longer an additive function of individual decisions, but the present analysis is rather
easily adapted. The model has strategic substitutability whenever the best response to a higher reservation
wage of others is a lower reservation wage. For an expectational analysis of a general equilibrium model
along the same lines, see Ghosal (2006).
123
214 R. Guesnerie, P. Jara-Moroni
demand function, and the aggregate state of the system obtains by aggregation of
these individual demand functions, and determines a random actual price. The study
of “expectational stability”of the equilibria is done in Desgranges (2000); Desgranges
and Heinemann (2005) The models under scrutiny would fit our framework if this
framework were extended to an infinite-dimensional state variable (a demand func-
tion). However, special versions of the model do fit the present framework: it is the
case of the linear version of the Grossman and Stiglitz (1980) model under consid-
eration in Desgranges (2000) (indeed, the “state” of the system is there an aggregate
demand function, depending only on one parameter, and our strategic substitutability
assumption is satisfied17).
In macroeconomics, the analysis of expectational stability in standard OLG like
infinite horizon models, leads to put emphasis on the dynamics of growth rates, (in
the simplest models as those studied by Evans and Guesnerie (2003)): for example,
in a one-dimensional model one-step forward looking with memory one, the state of
the system may be viewed as an infinite sequence of growth rates, connected through
some equilibrium map. Local “eductive” stability raises the question of the possibil-
ity of CK of this map, starting from conjectures that are close to it. Although the
interpretation of the mental process invoked is more delicate, since in a strict interpre-
tation of the model, “just born” agents have to make conjectures on the conjectures
of “non-born” agents,18 it echoes the usual interpretation of the standard E-Stabil-
ity criterion that has been developed in the eighties in this literature. In others, more
complicated models, the argument is conceptually similar but bears on more com-
plex objects called “extended growth rates” (see Evans and Guesnerie 2005;Gauthier
2002), the dynamics of which has a reduced form that falls within the above frame-
work. However, the strategic complementarities or substitutabilities assumptions of
our last section may or may not be satisfied. Infinite horizon models with infinite hori-
zon agents, as described in macroeconomic models arising from the Real Business
Cycle tradition, also enter the infinite-dimensional extension of the aforementioned
framework (Guesnerie 2008), and this would be true of many existing macroeconomic
models with infinite horizon agents, for which the substitutability assumptions often
hold (work in progress).
The model of game with a continuum of players is chosen not only because it cap-
tures the essential features of the aforementioned economic models. It is, in fact, of
wide use in the literature of non-atomic games, as Carmona (2009) states in a recent
article where he shows that several formulations of games with a continuum of players
turn out to be equivalent to Rath’s (Schmeidler 1973;Rath 1992,1995;Balder 2002).
3 “Eductive” criteria for assessing expectational stability
We have defined above equilibria, and we have argued that their plausibility should
be assessed from a viewpoint that we have called “eductive”. Our presentation of the
17 Fitting, to the best of our understanding, the one-dimensional version of the model of Sect. 4.2.
18 Here, the CK initial assumption describes collective beliefs, assumed to be shared by all generations:
the fact that it is CK that all these generations will be rational triggers the mental process at any time.
123
Expectational coordination in simple economic contexts 215
“eductive” criteria that we choose to present here refers to three broad categories:
purely game-theoretical criteria, “economic” criteria, and mixed criteria.
3.1 The game-theoretical viewpoint: rationalizability
3.1.1 Rationalizability
Rationalizability is associated with the work of Bernheim (1984) and Pearce (1984).
The set of rationalizable strategy profiles was defined and characterized by them in the
context of games with a finite number of players, continuous utility functions, and com-
pact strategy spaces. It has been argued that rationalizable strategy profiles are profiles
that cannot be discarded as outcomes of the game based on the premises of rationality
of players and common certainty of this rationality (see Tan and da Costa Werlang
1988;Brandenburger and Dekel 1987;Battigalli and Siniscalchi 2003).
First, agents only use strategies that are best responses to their forecasts and so
strategies in Sthat are never best response will never be used; second, agents know
that other agents are rational and so know that the others will not use the strategies
that are not best responses and so each agent may find that some of his remaining
strategies may no longer be best responses, since each agent knows that all agents
know, etc. This process continues ad-infinitum. The set of rationalizable solutions is
such that it is a “fixed point” of the elimination process, and it is the maximal set that
has such a property (Basu and Weibull 1991). Relying on the theory of convergence
of set valued mappings, these ideas can be formally stated in the context of a game
with a continuum of players, where players need to forecast strategy profiles in order
to take decisions. The formal construction is presented in Sects. A.1.1 and A.1.2 in
the appendix (see Sect. A.1 with references to Jara-Moroni (2008)).
However, in the present framework agents do not care about strategy profiles, but
on the value of their integrals. We present in Sect. 3.3 the concepts of Rationalizable
States and Point-Rationalizable States,19 where forecasts and the process of elimina-
tion are now taken over the set of states A. We argue in Sect. A.1.3 of the Appendix that
the state viewpoint that we take here in Sect. 3.3 and the game-theoretical viewpoint
presented in Sects. A.1.1 and A.1.2 are coherent (as stated in Proposition 6).
3.2 Expectational coordination from an “economic” viewpoint: iterative
expectations and Cobweb mapping
The Cobweb mapping brings back to ideas that make sense in our simplified “eco-
nomic” setting but that have not necessarily fruitful counterparts in an abstract
gametheoretical framework.
19 Following Bernheim (1984) we refer as point-rationalizability to the case of forecasts as points in the
set of strategies or states and plain rationalizability to the case of forecasts as probability distributions over
the corresponding set.
123
216 R. Guesnerie, P. Jara-Moroni
3.2.1 Cobweb mapping
Given the optimal strategy correspondence, B(i,·), defined in (2), we can define the
cobweb mapping or the best response mapping 20 :A⇒A:
(a):=
I
B(i,a)di (5)
This correspondence describes the actual possible states of the model when all agents
have the same point expectations on the state of the system a∈A.
In our framework, with the above assumptions, note that the cobweb mapping is
upper semi continuous as a set valued mapping, with non-empty, compact, and convex
values (a).21
In a sense, the Cobweb mapping provides a tool for assessing expectational stability
while ruling out heterogenous expectations. Indeed, the concept of IE-stability that has
been influential in the eighties in macroeconomics (see Lucas 1978;DeCanio 1979;
Evans 1985,1986) in the context of infinite horizon models, refers to a cobweb-like
mapping which is used for describing a mental collective process with homogenous
expectations.
Next, we can define the limit points of the Cobweb mapping.
3.2.2 Aggregate Cobweb Tâtonnement outcomes
Definition 3 The set of Aggregate Cobweb Tâtonnement Outcomes,CA, is defined
by
CA:=
t≥0
t(A)
where tis the tth iterate22 of the correspondence .
We immediately note that the equilibria of the economic system, denoted E, iden-
tify with the fixed points of the cobweb mapping and hence belong to the set CA.
The same is true for the cycles of any order of the mapping :a∗is a cycle of order
kof ⇔a∗∈k(a∗). Note that (a∗),...,
k−1(a∗), are also cycles of order k.
We denote Ckthe set of cycles of order kand =E∪∪+∞
k=1Ck, the set of periodic
equilibria, including standard equilibria. Our previous remarks involves ⊂CA.
20 The name cobweb mapping comes from the familiar cobweb tâtonnement although in this general context
the process of iterations of this mapping may not necessarily have a cobweb-like graphic representation.
21 See Lemma 3.2 in Jara-Moroni (2008).
22 This is:
0(A):= At+1(A):= t(A).
123
Expectational coordination in simple economic contexts 217
3.3 A mixed viewpoint: state rationalizability
This mixed viewpoint refers to the conceptual references of game-theoretical
inspiration, but, taking advantage of the added structure, adapt them to the “economic”
context under scrutiny.
Indeed, below, is a formal presentation of Point-Rationalizable States and Ratio-
nalizable States, which exploits the simplicity of our context. For the proofs of the
results herein stated and a more detailed treatment the reader is referred to Jara-Moroni
(2008).
3.3.1 Point rationalizable states
Analogously to what has been done in Sect. A.1.1, given the optimal strategy corre-
spondence defined in Eq. (2) we can define the process of elimination of unreasonable
states, considering forecasts as points in the set of states, as follows:
˜
Pr(X):=
I
B(i,X)di.
If initially agents’ CK about the actual state of the model is a subset X⊆Aand
if expectations are restricted to point-expectations, agents deduce that the possible
actions of each agent i∈IareinthesetB
(i,X):= a∈XB(i,a). Since all agents
know this, each agent can only discard the strategy profiles s∈SIthat are not selec-
tions of the mapping that assigns the above sets to each agent. Finally, they would
conclude that the actual state outcome will be restricted to the set obtained as the
integral of this set valued mapping. Thus, states that do not belong to ˜
Pr(X)defined
as above are excluded in this process.
Definition 4 The set of Point-Rationalizable States is the maximal subset X⊆Athat
satisfies the condition
X≡˜
Pr(X)
and we denote it PA.
We define similarly the set of Rationalizable States.
3.3.2 Rationalizable states
The difference between rationalizability and point-rationalizability is that in ration-
alizability forecasts are no longer constrained to be points in the set of outcomes.
Our approach to rationalizability leads to consider the correspondence B(i,·):
P(A)⇒Sdefined in (3). The process of elimination of non-maximizers of expected-
utility is described with the mapping ˜
R:B(A)→P(A):
123
218 R. Guesnerie, P. Jara-Moroni
˜
R(X):=
I
B(i,P(X)) di (6)
If it is CK that the actual state is restricted to a Borel subset X⊆A, then agents will
use strategies only in the set B(i,P(X)) := ∪μ∈P
(X)B(i,μ
)where P(X)stands as
before for the set of probability measures whose support is contained in X. Forecasts
of agents cannot give positive weight to points that do not belong to X.Strategypro-
files then will be selections of the correspondence i⇒B(i,P(X)). The state of the
system will be the integral of one of these selections.
Definition 5 The set of Rationalizable States is the maximal subset X⊆Athat
satisfies
X≡˜
R(X)
and we denote it RA.
We will make use of a result in Jara-Moroni (2008) restated in Proposition 1below,
which provides, in the continuum of agents framework, a key technical property of
the set of Rationalizable States.
Proposition 1 (Theorem 4.5 in Jara-Moroni (2008)) The set of Rationalizable States
can be computed as
RA≡
∞
t=0
˜
Rt(A)
An analogous result may be obtained for the set of Point-Rationalizable States
(Jara-Moroni 2008). The sets PAand RA, indeed obtain as the outcome of the itera-
tive elimination of unreachable states.
At this stage, let us make two remarks.
First, let us stress a technical point of importance, i.e., that for a given Borel
set X⊆Awe have ˜
Pr(X)⊆˜
R(X), a fact that is used below, for example in
Proposition 2.
Second, let us explain why, bypassing the game-theoretical difficulties occurring
in games with a continuum of players (and explicitly envisaged in the appendix), the
states set approach provides a fairly convincing view of rationalizability. To make
the connection with the strategic approach developed above, let us note that agents
in our economic model are unable to differentiate between strategy profiles that give
the same aggregate state, in mathematical terms, strategy profiles that have the same
integral. Looking at the integral as a function from the set SIto the set A,thesetof
values of the integrals, we see that the measurable space structure considered in Amay
induce a measurable space structure on SI.Theσ-field SI, introduced in Sect. A.1.2
is then the σ-field of the pre-images of the Borel subsets of Athrough the integral.
SI≡A−1(X):X∈B(A)
123
Expectational coordination in simple economic contexts 219
Any probability measure, ν, defined on SIinduces a probability measure in P(A)as
usual:
μ(·)≡νA−1(·).
Thus, when applying the process defined in (6), we are in fact considering at least all
probability measures that can be defined over SI. Reciprocally, for a given probability
measure in P(A)the function ν:SI→R+defined by23:
ν(H):= μ(A(H))
is in fact a probability measure on SI. The question arises as to whether a richer σ-field
would change the definition of rationalizable strategies in our context. This answer is
given by the structure of the model itself. Payoff functions are defined on the values of
the integrals of the strategy profiles. Consequently, when taking expectation, agents
precisely care only about the probabilities of events of the form s∈SI:s=a,
for each a∈Awhich brings us back to the induced σ-field SI.
With this in mind we can see that the process of elimination of profiles defined in
(11) is interlaced with the process of elimination of states.
3.4 Connecting the concepts
We have defined several concepts that all make sense in our prototype “economic”
model. In Proposition 6, in the appendix, we summarize the connections between the
infinite-players game and the “economic” viewpoints. The next proposition stresses
the connections between the concepts of “economic” inspiration.
Proposition 2 We have
(1)
E⊆⊆CA⊆PA⊆RA
(2) The sets PAand RAare convex and compact. CAis compact.
In (1), the first inclusions have already been noted: equilibria and cycles of any
order do belong to CA. We can obtain the two last inclusions of Proposition 2noting
that if a set satisfies X⊆˜
Pr(X)then it is contained in PAand equivalently if it
satisfies X⊆˜
R(X)then it is contained in RA. Then, the second inclusion is derived
from the fact that each point in CA, as a singleton, satisfies {a∗}⊆˜
Pr({a∗}), and the
third inclusion is true because the set PAsatisfies PA⊆˜
R(PA). Although in stan-
dard economic models some of these inclusions become equalities, in general they are
strict. This is clear for the first three inclusions; for the fourth, an example is given in
a forthcoming version of Jara-Moroni (2008).
23 By construction, if H∈SIthen A(H)∈B(A).
123
220 R. Guesnerie, P. Jara-Moroni
The above inclusions are unsurprising, in the sense that they reflect the decreasing
strength of the expectational coordination hypothesis, when going from equilibria
to Aggregate Cobweb outcomes, then to Point-Rationalizable States, and finally to
Rationalizable States. As argued above and recalled just below, the different concepts
reflect an enlargement of the complexity and diversity of expectations under scrutiny.
The statement also stresses that cycles of the best response mapping play a role into
the analysis of expectational coordination: this role will be shown to be crucial in the
next section.
3.5 From concepts to expectational stability criteria
The above concepts serve as a basis for assessing the expectational plausibility of the
economic equilibrium of our model, from a global and a local viewpoint.
3.5.1 The global criteria
Definition 6 An equilibrium a∗is said to be Globally Iteratively Expectationally Sta-
ble if, ∀a0∈A, any sequence at∈at−1satisfies limt→∞ at=a∗. Equivalently,
CA=E={a∗}.
The concept captures the idea that virtual coordination processes, referring to
homogenous deterministic expectations, converge globally. The terminology of IE-
stability is adopted from the literature on expectational stability in dynamical systems
(Evans and Guesnerie 1993,2003,2005).
Definition 7 The equilibrium state a∗is (globally) Strongly Point Rational if
PA≡a∗(=E).
The idea captured by the concept is now that virtual coordination processes, refer-
ring to heterogenous but deterministic expectations, converge globally.
Then comes the most demanding concept referring to heterogenous and stochastic
expectations.
Definition 8 The equilibrium state a∗is (globally) Strongly Rational if
RA≡a∗(=E).
The criteria and terminology used here closely follow Guesnerie (1992) and Evans
and Guesnerie (1993), as well as Chamley (2004, chapter 11).24 With a less precise
terminology, when one of the above criteria is satisfied, we say that the equilibrium is
(globally) “eductively” stable.
24 The equilibrium might also be viewed and called dominant-solvable. The terminology is intended to sug-
gest the strength or robustness of what is generally called in an “economic” context a rational expectations
equilibrium.
123
Expectational coordination in simple economic contexts 221
The reader will remember here that studies on “evolutive learning” most often
assume that agents have identical point expectations (that they revise according to a
universally agreed upon rule). In a similar way, IE-Stability, where iterations of the
cobweb mapping describe agents’ reactions to the same point forecast over the set of
states, rules out expectational heterogeneity. And let us repeat that “eductive” coordi-
nation, as assessed from the last two definitions of Strong Point Rationality or Strong
Rationality, takes into account the fact that agents25 have heterogenous expectations
that may be point expectations or stochastic expectations.
It is straightforward that these concepts are increasingly demanding: strong Ratio-
nality implies Strong Point Rationalizability that implies IE-stability.
We shall turn later to the local version of these concepts.
4 Economies with strategic substitutabilities
Games with strategic complementarities have been the focus of intensive research
particularly since the end of the eighties (see Milgrom and Roberts 1990). We adapt
the standard argument and re-assess the standard findings within our “economic”
framework with a continuum of agents. This is Proposition 3in the next subsection.
We then wonder whether the striking findings on global stability in an “economic”
model with strategic complementarities can be extended. We then focus attention on
another polar world dominated by strategic substitutabilities and show that we still
have powerful results to analyse “eductive” global stability.
4.1 Preliminaries: economic models with strategic complementarities
Our economic system presents Strategic Complementarities if the individual best
response mappings of the underlying game are increasing for each i∈I.26
We could define such properties in the framework of the underlying game of
Sect. 2.1, while directly referring to the theory of supermodular games as studied
in Milgrom and Roberts (1990) and Vives (1990)27 (see as well Topkis 1998).28 How-
ever, since agents cannot affect the state of the system, all agents have forecasts over
25 Even if these agents were homogeneous (with the same utility function).
26 For definitions related to lattice structure of sets and monotonic properties of operators please refer to
Sect. A.7 in the appendix, and Topkis (1998).
27 The related literature is still active and a new recent contribution weakens the standard increasing dif-
ferences and single crossing conditions (Quah and Strulovici 2009).
28 Supermodularity (and of course submodularity as in the next section) could be studied in the context of
games with continuum of agents with a broad generality using the strategic approach (using for instance
the tools available from Riesz spaces). However, as we noted before, working with a continuum of agents
and an aggregate variable, allows us to focus our attention to the case where agents forecast the outcome
of this aggregate variable, instead of strategy profiles. This does not occur in the context of “small” games
since then the forecast of different agents would be in different sets, namely the set of aggregate values of
“the others” which could well be a different set for each agent. Another difficulty is passing from strategies
to states in terms of complementarity. An important result related to this issue is treated in Lemma 6in the
appendix.
123
222 R. Guesnerie, P. Jara-Moroni
the same set, namely the set of states A, and we shall directly proceed within the
“economic” framework.
For that, let us make the following assumptions over the strategy set Sand the utility
functions: u(i,·,·):
1.B Sis the product of ncompact intervals in R+.
2.B u(i,·,a)is supermodular for all a∈Aand all i∈I.
3.B ∀i∈I, the function u(i,y,a)has increasing differences in yand a. That is, ∀
y,y∈S, such that y≥yand ∀a,a∈Asuch that a≥a:
u(i,y,a)−ui,y,a≥ui,y,a−ui,y,a(7)
Assumption 2.B is straightforward. Assumption 1.B implies that the set of strat-
egies is a complete lattice in Rn. One implication of our setting is that since Sis a
convex complete lattice, then A≡co {S}≡Sis as well a complete lattice.
From now on we will refer to the above supermodular setting as G. The technical
results we use in the proof in the appendix are recalled as lemmas.
Lemma 1 Under assumptions 1.B through 3.B, we have
1. the mappings B(i,·)are increasing in a in the set A, and the sets B(i,a)are
complete sublattices of S,
2. the correspondence is increasing and (a)is subcomplete for each a ∈A,
From the previous Lemma, an existence result follows, but what is most important
is that the set of equilibria has a complete lattice structure. In particular we know that
there exist points a∗∈Aand ¯a∗∈A(that could be the same point) such that if
a∗∈Eis an equilibrium, then a∗≤a∗≤¯a∗.
We also mention another intermediate result.
Lemma 2 In G,fora
∈Aand μ∈P(A),ifa
≤a, ∀a∈sup(μ), then ∀i∈I
Bi,aB(i,μ
);
equivalently, if a≥a, ∀a∈sup(μ), then ∀i∈I
Bi,aB(i,μ
).
That is, if the forecast of an agent has support on points that are larger than a point
a∈A, then his optimal strategy set is larger than the optimal strategy associated with
a(for the induced set ordering29) and analogously for the second statement. Here is
the statement that summarizes our results.
Proposition 3 In the economic system with Strategic Complementarities we have
(i) The set of equilibria E⊆Ais a non-empty complete lattice.
29 See Sect. A.7.
123
Expectational coordination in simple economic contexts 223
(ii) There exist a greatest equilibrium and a smallest equilibrium, that is ∃a∗∈E
and ¯a∗∈Esuch that ∀a∗∈E,a
∗≤a∗≤¯a∗.
(iii) The sets of Rationalizable and Point-Rationalizable States are convex sets
“tightly contained” in the interval a∗,¯a∗. That is,30
CA⊆PA⊆RA⊆a∗,¯a∗.
and ¯a∗∈CAand a∗∈CAso that a∗,¯a∗is the smallest interval that contains
these sets.
The proof is relegated to the appendix. The intuitive interpretation of the proof is as
follows: Originally, agents know that the state of the system will be greater than inf A
and smaller than sup A. Since the actual state is in the image through ˜
Pr of A,the
monotonicity properties of the forecasts to state mappings allow agents to deduce that
the actual state will be in fact greater than the image through of the constant forecast
a0=inf Aand smaller than the image through of the constant forecast ¯a0=sup A.
That is, it suffices to consider the cases where all the agents have the same forecasts
inf Aand sup A. The eductive procedure then can be secluded on each iteration, only
with iterations of . Since is increasing, we get an increasing sequence that starts at
a0and a decreasing sequence that starts at ¯a0. These sequences converge and upper
semi-continuity of implies that their limits are fixed points of .31
Our results are unsurprising. In the context of an economic game with a continuum
of agents, they mimic, in an expected way, the standard results obtained in a game-the-
oretical framework with a finite number of agents and strategic complementarities.32
Additional convexity properties reflect the use of a continuum setting.
The most striking feature of the result, reinterpreted within our categories is that
all the global stability criteria defined above are equivalent as soon as the equilibrium
is unique. In this sense, uniqueness is the Grail, as stated formally below.
Corollary 1 In G, the four following statements are equivalent:
(i) an equilibrium a∗is globally Strongly Rational.
(ii) an equilibrium a∗is globally Strongly Point Rational.
(iii) an equilibrium a∗is globally IE-Stable.
(iv) there exists a unique equilibrium a∗.
This last statement may be interpreted as the fact that in the present setting, heter-
ogeneity of expectations does not play any role in expectational coordination, at least
30 On any ordered space(E,≥), we understand an interval [a,b]⊆Eas the set of points x∈Esuch that
x≥aand b≥x.
31 Note that there are three key features to keep in mind, that lead to the conclusion. First, the fact that
there exists a set Athat, being a complete lattice and having as a subset the whole image of the mapping A,
allows the eductive process to be initiated. Second, monotonic structure of the model implies that it suffices
to use to seclude, in each step, the set obtained from the eductive process into a compact interval. Third,
continuity properties of the utility functions and the structure of the model allow the process to converge.
32 Note, however, that, to the best of our understanding, our results do not follow from previous results
obtained in the finite agents setting. Shedding full light on the relationship requires a theory connecting the
finite context and the continuum one.
123
224 R. Guesnerie, P. Jara-Moroni
when the equilibrium is unique. This is a very special feature of expectational coordi-
nation as argued in Evans and Guesnerie (1993). Surprisingly enough, the logic (and
simplicity) of the analysis somewhat extends within the next class of models under
consideration.
It should also be noted that this strong result requires that the mapping has no
cycle, whatever the order of the cycle, a fact that follows trivially from monotonicity.
In the next subsection devoted to economies with strategic substitutabilities, it will be
shown that the mapping may have cycles of order 2, but no cycles of other orders.
As we shall see, such cycles will play a key role in the analysis.
4.2 Economies with strategic substitutabilities
We turn to the case of Strategic Substitutabilities. This is done by replacing assumption
3.B with assumption 3.B’ below.
1.B Sis the product of ncompact intervals in R+.
2.B u(i,·,a)is supermodular for all a∈A
3.B’ u(i,y,a)has decreasing differences in yand a. That is, ∀y,y∈S, such that
y≥yand ∀a,a∈Asuch that a≥a:
u(i,y,a)−ui,y,a≤ui,y,a−ui,y,a(8)
Assumptions 1.B through 3.B’ turn the underlying game of our model into a sub-
modular game with a continuum of agents which we will denote by G. The relevant
difference with the previous section is that now the monotonicity of the mapping A
along with assumption 3.B’ implies that the best response mappings are decreasing
on the strategy profiles (note that two examples of economic models fitting the above
assumptions are mentioned in Sect. 2).
The following Lemmas and Corollary are the counterparts of Lemmas 1and 2:
Lemma 3 Under assumptions 1.B, 2.B, and 3.B’, the mappings B(i,·)are decreas-
ing in a in the set A, and the sets B(i,a)are complete sublattices of S. The corre-
spondence is decreasing and (a)is subcomplete for each a ∈A.
We denote 2for the second iterate of the cobweb mapping, that is 2:A⇒A,
2(a):= ∪a∈
(a)a.
Corollary 2 In Gthe correspondence 2is increasing and 2(a)is subcomplete for
each a ∈A.
Proof Is a consequence of being decreasing.
The correspondence 2will be our main tool for the case of strategic substitutabilities.
This is because, in the general context, the fixed points of 2are point-rationalizable
just as the fixed points of are. Actually, one checks immediately that the fixed points
of any iteration of the mapping are as well point-rationalizable. The relevance of stra-
tegic substitutabilities is that under their presence it suffices to use the second iterate of
123
Expectational coordination in simple economic contexts 225
the cobweb mapping to seclude the set of Point-Rationalizable States. Using Lemma 1
we get that under assumptions 1.B, 2.B, and 3.B’, the set of fixed points of 2, shares
the properties that the set of equilibria Ehad under strategic complementarities.
Lemma 4 The set of fixed points of 2is a non-empty complete lattice.
Proof Apply Lemma 1to 2.
The relevance of Lemma 4is that, as in the case of strategic complementarities, under
strategic substitutabilities it is possible to seclude the set of Point-Rationalizable States
into a tight compact interval. This interval is now obtained from the complete lattice
structure of the set of fixed points of 2, which can be viewed, in a multi-period
context, as cycles of order 2 of the system.
We also need, as above
Lemma 5 In G,fora
∈Aand μ∈P(A),ifa
≤a, ∀a∈sup(μ), then ∀i∈I
Bi,aB(i,μ
);
equivalently, if a≥a, ∀a∈sup(μ), then ∀i∈I
Bi,aB(i,μ
).
which can be proved by adapting the proof of Lemma 2, in the appendix, to the
decreasing differences case.
We are now able to state the main result of the strategic substitutabilities case,
which, together with its corollaries, is also the main result of the paper.
Theorem 1 In economies with Strategic Substitutabilities we have the following:
(i) There exists at least one equilibrium a∗.
(ii) There exist a greatest and a smallest rationalizable state, that is ∃a∈RAand
¯a∈RAsuch that ∀a∈RA,a≤a≤¯a, where a and ¯a are cycles of order 2
of the Cobweb mapping.
(iii) The sets of Rationalizable and Point-Rationalizable States are convex sets
“tightly contained” in the interval a,¯a. That is,33
CA⊆PA⊆RA⊆a,¯a.
and ¯a∈CAand a ∈CAso that a,¯ais the smallest interval that contains
these sets.
The proof is relegated to the appendix. Keeping in mind the proof of Proposition
3, we can follow the idea of the proof of Theorem 1. As usual, CK says that the state
of the system will be greater than inf Aand smaller than supA. In first-order basis
then, the actual state is known to be in the image through ˜
Pr of A. Since now the
33 See Footnote 30
123
226 R. Guesnerie, P. Jara-Moroni
cobweb mapping is decreasing, the structure of the model allows the agents to deduce
that the actual state will be in fact smaller than the image through of the constant
forecast a0=inf Aand greater than the image through of the constant forecast
¯a0=sup A. That is, again it suffices to consider the cases where all the agents have
the same forecasts inf Aand sup A, and this will give a1, associated with ¯a0, and ¯a1,
associated with a0. However, now we have a difference with the strategic comple-
mentarities case. In the previous section, the iterations started in the lower bound of
the state set were lower bounds of the iterations of the eductive process. As we see,
this is not the case anymore. Nevertheless, here is where the second iterate of gains
relevance. In a second-order basis, once we have a1and ¯a1obtained as above, we can
consider the images through of these points, and we get new points ¯a2,froma1, and
a2, from ¯a1, that are, respectively, upper and lower bounds of the second step of the
eductive process. That is, in two steps we obtain that the iterations started at the upper
(resp. lower) bound of the states set is an upper (resp. lower) bound of the second step
of the eductive process. Moreover, the sequences obtained by the second iterates are
increasing when started at a0and decreasing when started at ¯a0. The complete lattice
structure of Aagain implies the convergence of the monotone sequences while 2
inherits upper semi-continuity from . This implies that the limits of the sequences
are fixed points of 2.
The three key features that lead to the conclusion are analogous to the strategic
complementarity case. First, Ais a complete lattice that has as a subset its image
through the function Aand thus allows the eductive process to be initiated. Second,
monotonic structure of the model implies that it now suffices to use 2to seclude,
every second step, the set obtained from the eductive process into a compact interval.
Third, continuity properties of the utility functions and the monotonic structure of the
model allow the process to converge.
Note that, also as in the case of strategic complementarities, since the limits of the
interval a,¯aare point-rationalizable, this is the smallest interval that contains the
set of Point-Rationalizable States.
The full equivalence of Global Stability criteria obtains
Corollary 3 If in G,2has a unique fixed point a∗, then
E=CA=RA≡PA≡a∗.
This can be restated as
Corollary 4 The four following statements are equivalent:
(i) an equilibrium a∗is globally Strongly Rational.
(ii) an equilibrium a∗is globally Strongly Point Rational.
(iii) an equilibrium a∗is globally IE-Stable.
(iv) there exists a unique cycle of order two a∗.
Proof Note that if 2has a unique fixed point, it is necessarily a degenerate cycle,
i.e., an equilibrium (since there exists at least an equilibrium and equilibria are cycles
of any order). Observe that both limits of the interval presented in Theorem 1,aand
¯a, are fixed points of 2. Hence the result.
123
Expectational coordination in simple economic contexts 227
Fig. 1 Strategic substitutes for
A≡[0,amax]⊂R.There
exists a unique equilibrium and
multiple fixed points for 2
b
Γ
2b b
¯
b
Γ
2¯
b¯
b
a
Γ
a a
0amax
amax
Under strategic substitutabilities, the strong equivalence result of Corollary 1does not
obtain any longer. If the sequences ¯
btand btdefined in the proof of Theorem 1(in
the appendix) converge to the same point, i.e., b∗=¯
b∗=a∗, then a∗is the unique
equilibrium of the system, it is strongly rational, and IE-stable. However, uniqueness
of equilibrium does not imply this situation, there could well be a unique equilibrium
that is not necessarily strongly rational. Think of the case of A⊂R, where a continu-
ous decreasing function has a unique fixed point, that could well be part of a bigger
set of Point-Rationalizable States (see Fig. 1).
The Grail is no longer uniqueness of the equilibrium, as in the strategic comple-
mentarities case, but existence of a unique cycle of order two. Uniqueness of cycles
of order two or uniqueness of equilibrium are equally demanding in the case of stra-
tegic complementarities but the former is more demanding here. Note that the results
immediately apply to the models studied before and presented in Sect. 2and that it
enriches the known results.34
In particular, the diagram in Fig. 1, that displays a cycle of order two, in a one-
dimensional model with strategic substitutabilities, may be viewed as visualizing the
rationalizable set of the Muth model, (a fact that was not known to the best of our
understanding). Note also that according to the Figure, and along the lines of the
34 The fact that, in a partial equilibrium à la Muth, as studied by Guesnerie (1992), the absence of cycles
of order 2 is sufficient for the global “eductive” stability of the equilibrium was not known (to the best of
our knowledge). The same remark applies to the general equilibrium models mentioned in the introduction.
Theorem 1enriches immediately the results obtained in the fixed wage version of Guesnerie (2001a,b)and
less immediately those obtained in the flexible wage version of these models.
It is less immediate, but extremely enlightening to use the findings of the Theorem in order to discuss the
transmission of information through prices. Let us finally note that early insights into the role of cycles
in the mechanics of expectational coordination, along lines related to the present ones, can be found in
Benhabib and Bull (1988), who focus on what they call conjectural equilibrium bounds in an OLG frame-
work. (Note that in a dynamical setting, the close connections between cycles and sunspot equilibria have
been established in Azariadis and Guesnerie (1986)).
123
228 R. Guesnerie, P. Jara-Moroni
present analysis, all individual beliefs with support in b∗,¯
b∗, are examples of the
“diverse beliefs” that may emerge.
4.3 The differentiable case
Here, we add an assumption concerning the cobweb mapping :
H1 :A→Ais a C1-differentiable function.
Remark 1 Note that from the definition of , in both cases, the vector-field(a−(a))
points outwards on A: formally, this means that if p(a)is a supporting vector at a
boundary point of A(p(a)·A≤0), then p(a)·(a−(a)) ≥0. When, as in most
applications Ais the product of intervals, for example, [0,Mh], this means h(a)≥0,
whenever ah=0, and Mh−h(a)≥0, whenever ah=Mh.
The jacobian of the function ,∂, can be obtained from the first-order conditions
of problem (2) along with (5).
∂(a)=
I
∂B(i,a)di
when the optimal strategy (now) function has a jacobian ∂B(i,a). This jacobian is
equal to
∂B(i,a)≡−
[Duss(i,B(i,a),a)]−1Dusa(i,B(i,a),a)(9)
where Duss(i,B(i,a),a)is the matrix of second derivatives with respect to sof the
utility functions and Dusa(i,B(i,a),a)is the matrix of cross second derivatives, at
the point(B(i,a),a).
4.3.1 The strategic complementarities case
Under assumptions 1.B to 3.B, along with C2differentiability of the utility functions
u(i,·,·), we get from (9) that the matrices ∂B(i,a)are positive,35 and conse-
quently so is ∂(a).
The properties of positive matrices are well known. When there exists a positive
vector x, such that Ax <x,the matrix Ais said “productive”, its eigenvalue of highest
modulus is positive and smaller than one. When ais one-dimensional, the condition
says that the slope of is smaller than 1.
In this special case, as well as in our more general framework, the intuition behind
the condition is that actions do not react too wildly to expectations.
In this case, we obtain
35 It is a well-known fact that increasing differences implies positive cross derivatives on
Dusa(i,B(i,a),a), and it can be proved that for a supermodular function the matrix −[Duss(i,·,·)]−1
is positive at(B(i,a),a).
123
Expectational coordination in simple economic contexts 229
Theorem 2 (Uniqueness) If ∀a∈A,∂(a)is a “productive” matrix, then there
exists a unique Strongly Rational Equilibrium.
Proof Compute in any equilibrium a∗the sign of det[I−∂(a∗)].If∂(a∗)is pro-
ductive, its eigenvalue of highest modulus is real positive and smaller than 1. Hence,
the real eigenvalues of [I−∂(a∗)]are all positive.36 It follows that the sign of
det[I−∂(a∗)]is the sign of the characteristics polynomial det{[I−∂(a∗)]−λI}
for λ→−∞, i.e., is plus. The index of ϕ(a)=a−(a)is then +1. The Poincaré–
Hopf theorem for vector fields pointing inwards implies that the sum of indices must
be equal to +1; hence the conclusion of uniqueness. Strong Rationality follows from
Corollary 1.
The above statements generalize in a reasonable way the intuitive findings easily
obtainable from the one-dimensional model. It is a clearly unsurprising statement,
although it does not seem to have been stressed in previous literature.
4.3.2 The strategic substitutabilities case
Let us turn to the Strategic Substitutabilities case.
When passing from 3.B to 3.B’, we get that now the matrix ∂ has negative37
entries. And I−∂(a)is a positive matrix. Again, it has only positive eigenvalues,
whenever the positive eigenvalue of highest modulus of −∂ is smaller than 1.
Theorem 3 Let us assume that ∀a1,a
2∈A,∂(a1)∂(a2)is “productive”; then
(1) There exists a unique equilibrium.
(2) It is globally Strongly Rational.
Proof The assumption implies that ∀a∈A,−∂(a)is productive.
Hence I−∂ is a positive matrix, and whenever the positive eigenvalue of highest
modulus of −∂ is smaller than 1, it has only positive eigenvalues. Then its determi-
nant is positive.
Then the above Poincaré–Hopf argument applies to the first and second iterations
of .
It follows that there exists a unique equilibrium and no two-cycle.
Then, Theorem 1applies.
Again, this seems a potentially most useful result for assessing expectational sta-
bility in the context of strategic substitutabilities. The sufficient condition found here
has a flavour similar to the one found elsewhere, and in particular in the strategic
complementarity case just considered, i.e. that realizations should not be too sensi-
tive to expectations. Note, however, that the same intuitive requirement turns out, in
the strategic substitutabilities case, to a condition which is, in a sense, formally more
demanding condition (since the condition requires more that ∂2being productive).
36 It has at least one real eigenvalue, associated with the eigenvalue of highest modulus of a∗.
37 Since the matrix Dusa (i,B(i,a),a)has only non-positive entries under strategic substitutabilities (see
footnote 35).
123
230 R. Guesnerie, P. Jara-Moroni
5 The local viewpoint
5.1 The local stability criteria and their general connections
We now give the local version of the above stability concepts. Outside the cases where
global stability results may be expected (and they may be expected for well under-
stood reasons, as just precisely shown in the economies under consideration, either with
strategic complementarities or substitutabilities), the local analysis becomes one most
natural entry point of any analysis of expectational stability along the present “educ-
tive” lines : first, the absence of local stability for every equilibrium would strongly
signal expectational instability (making implausible, with the “eductive” glasses, the
rational expectations coordination); second, the fact that one equilibrium is locally
stable, provides argument for its occurrence, less strong than in the case of global
stability, but still sound (for a discussion presenting both “high tech” and “low tech”
appraisals of the relevance of the local criterion, see Guesnerie (2008)). Consequently,
and in view of the limited cases in which global stability may be expected, the present
section may have more bite on general expectational coordination problems than the
previous one.
Again, the definition of (local) IE-Stability, stated below, is similar to the one given
in Evans and Guesnerie (1993).
Definition 9 An equilibrium a∗is said to be Locally Iterative Expectationally Stable
if there is a neighbourhood Va∗such that ∀a0∈Vany sequence at∈at−1
satisfies limt→∞ at=a∗.
Definition 10 An equilibrium state a∗is Locally Strongly Point Rational if there exists
a neighbourhood Va∗such that the process governed by ˜
Pr started at Vgenerates
a nested family that leads to a∗. That is, ∀t≥1,
˜
Prt(V)⊂˜
Prt−1(V)
and
t≥0
˜
Prt(V)≡a∗.
Definition 11 An equilibrium state a∗is Locally Strongly Rational (Guesnerie 1992)
if there exists a neighbourhood Va∗such that the eductive process governed by ˜
R
started at Vgenerates a nested family that leads to a∗. That is, ∀t≥1,
˜
Rt+1(V)⊂˜
Rt(V)
and
t≥0
˜
Rt(V)≡a∗.
123
Expectational coordination in simple economic contexts 231
Note that beyond the sophistication of the definition (high tech!), one can find a
simpler and more immediately accessible (low tech!) reading. Loosely speaking, the
reader will easily check that the definition is (almost) equivalent to the fact that there is
a neighbourhood of the equilibrium, such that the collective belief that the economic
state is within this neighbourhood, cannot be falsified (whatever the precise form of
the belief).
There are straightforward connections between the local concepts stressed above.
Proposition 4 We have
(i) a∗is (Locally) Strongly Rational ⇒ a∗is (Locally) IE-Stable.
(ii) a∗is Locally Strongly Rational ⇒ a∗is Locally Strongly Point Rational.
A sufficient condition for the converse of statement (ii) to be true is that there
exists a neighborhood V of a∗such that for almost all i ∈I , for any Borel subset
X⊆V
B(i,P(X)) ⊆co {B(i,X)}(10)
At a first glance the hypothesis in the second part of Proposition 4appears to be very
restrictive; however, it involves only local properties of the agents’ utility functions.
It intuitively states that given a restriction on CK (subsets of the set V), when agents
evaluate all the possible actions to take when facing probability forecasts with support
“close” to the equilibrium, these actions are somehow “not to far” and “surrounded”
by the set of actions that could be taken when facing point forecasts (B(i,μ
)⊆
co {B(i,X)}if sup(μ)⊆X). The assumption is true in a number of applications
where it can be derived from standard assumptions over utility functions.
5.2 More on the general connections between the local stability criteria
In general contexts and outside the case stressed in Evans and Guesnerie (2003), the
three local concepts are unlikely to be equivalent: Local IE-stability is in general a
weaker requirement than Local Strong Rationality (see Guesnerie 2002;Evans and
Guesnerie 2005).
Concerning the connection between Local Point Strong Rationality and Local
Strong Rationality, we have noted that condition (10) above, which relates the individ-
ual reactions of agents facing non-degenerate subjective forecasts, with their reactions
when facing point (Dirac) forecasts, is sufficient for the equivalence. An alternative
approach consists in focusing attention on the convergence of the process generated
by point forecasts. When this convergence is sufficiently fast, then we shall say that
the equilibrium is Strictly Locally Point Rational, and we shall argue it drags the
convergence to equilibrium of the stochastic eductive process, as well.
For a positive number α>0 and a set V⊆Athat contains a unique equilibrium
a∗we will denote by Vαthe set
Vα:= x∈A:x=αv−a∗+a∗,v∈V
123
232 R. Guesnerie, P. Jara-Moroni
Definition 12 We say that an equilibrium state a∗is Strictly Locally Point Rational
if it is Locally Strongly Point Rational and there is a number ¯
k<1 such that, ∀
0<α≤1,
sup
v∈˜
Pr(Vα)
v−a∗
<¯
ksup
v∈Vα
v−a∗
.
Strict Locally Point Rationality introduces the consideration of the speed of con-
vergence of the point forecast process. Under this property, we have that ˜
Pr(V)⊂V¯
k,
with ¯
k<1, and so ˜
Prt(V)⊂V¯
kt.
Theorem 4 If the utility functions are twice continuously differentiable, a∗∈intA,
B(i,μ
)is single valued for all μwith support in a neighborhood of a∗, and Duss(s,a)
is non-singular in an open set V a∗, then a∗is Locally Strongly Rational ⇐⇒ a∗
is Strictly Locally Point Rational.
The idea of the theorem is that if the process ˜
Pr associated with point forecasts is
sufficiently fast, then, although the eductive process ˜
Rmay be slower, it is anyhow
dragged to the equilibrium state.
We have called the result theorem, since the local analysis of stability is consider-
ably simplified when the rather weak assumption stated above holds. Being allowed
to forget about stochastic expectations and concentrating on (local) point-rationaliz-
ability, makes an operationally decisive difference for the easiness of the analysis.
5.3 Local stability in economies with strategic complementarities
and strategic substitutabilities
We now stress that the equivalence of the three local criteria, generally hopeless, is
easy to assess whenever we are in the cases of strategic complementarities and substit-
utabilities. In both cases, the assessment of local stability can be made without explicit
reference to the heterogeneity of expectations, i.e. by referring only to the map .
Proposition 5 Under the assumptions of Theorem 4, assume either strategic substit-
utabilities or strategic complementarities. Consider an equilibrium a∗. If there exists
a neighbourhood V of a∗, such that (V)⊂intV , then the equilibrium is locally
stable in the three different senses of stability just introduced.
The proof is left to the reader who will show that, under the condition stressed,
(V)⊂intV, the model restricted to the neighbourhood V , fits the assumption of
the theorem. It is easy to check then that there is no cycle in V: the conclusion follows
from the above theorem. The reader will also notice that the condition is almost nec-
essary for stability in the three cases, but only almost since in border cases, the strict
equivalence of the three criteria is not warranted.
Note that in fact the local version of our theorems relying on differentiability allows
to prove the next corollary to Theorem 4(the proof again left to the reader, immediate
with Strategic Complementarities, easy with Strategic Substitutabilities).
123
Expectational coordination in simple economic contexts 233
Corollary 5 Under Strategic Complementarities or Substitutabilities, if ∂(a∗)is
“productive” , for some equilibrium a∗, then a∗is locally Strongly Rational.
This corollary is nothing but the differentiable version of the previous proposition:
since ∂(a∗)productive implies the existence of Vsuch that (V)⊂intV.
6 Comments and conclusions
The rational expectations hypothesis has been subject of scrutiny in recent years
through the assessment of expectational coordination. Although the terminology is
still fluctuating, the ideas behind what we call here Strong Rationality or Eductive Sta-
bility have been at the heart of the study of diverse macroeconomic and microeconomic
models of standard markets with one or several goods, see Guesnerie (1992,2001a),
models of information transmission (Desgranges 2000;Desgranges and Heinemann
2005;Ben-Porath and Heifetz 2006).
This paper comes back on the conceptual basis of previously used eductive stability
concepts, by connecting a now standard line of research in game-theory-games with a
continuous of players - with what may be called the economic viewpoint. The frame-
work under consideration, the so-called economic model with non-atomic agents, is
potentially useful in broad economic contexts. The paper explores in a pedestrian way,
the connections between the different concepts that have been proposed for analysing
the expectational plausibility of equilibria: cycles, Cournot tâtonnement outcomes,
Point-Rationalizable States, Rationalizable States. (the two latter sets being proved to
be convex) and the corresponding stability criteria: IE-Stability, Strong Point Ratio-
nality, Strong Rationality.
In Sect. 4, we first derive, in a world with strategic complementarities, results that
reformulate, in our setting, the classical game-theoretical findings of Milgrom and
Roberts (1990) and Vives (1990). In the opposite polar case of strategic substitutabili-
ties, (much less documented although economically most relevant), we are still able to
exhibit results that are still strong although somewhat less strong. The flavour of those
results is, however, strikingly different. For example, when in the strategic comple-
mentarities case, uniqueness triggers stability along the lines of all expectational sta-
bility criteria under scrutiny, this is no longer the case with strategic substitutabilities:
uniqueness does not imply global “expectational stability”, whatever the exact sense
given to the assertion. We give, however, simple and appealing conditions (absence
of cycle of order 2) implying both uniqueness of equilibria and stability in the most
demanding sense of Strong Rationality. Hence, in the strategic substitutabilities set-
ting, the analysis, particularly when differentiability assumptions are introduced, leads
to analytically tractable sufficient conditions for global stability. Note here that outside
these cases of global stability, the present analysis suggests plausible “diverse beliefs”,
associated with the different versions of state rationalizability that we have introduced.
Local stability is in itself a subject of high interest, particularly, when we fall outside
the polar worlds under consideration in Sect. 4, so that global stability is unlikely.38
38 As stressed earlier, many economic models that fit our framework, such as the one associated with the
analysis of expectational stability in a class of general dynamical systems (Evans and Guesnerie 2005)have
123
234 R. Guesnerie, P. Jara-Moroni
Absence of local stability suggests there that diverse beliefs would be pervasive.
Theorem 4provides an operationally important simplification for the analysis, when
Proposition 5shows that, again in the case of complementarities and substitutabilities,
the local analysis is strikingly simpler.
Let us finally come back to the complementarities and differences of this paper
with the other papers of the conference. Here, as in the whole conference, some kind
of “endogenous uncertainty” appears, although the one we find here, is not, at least
directly, tied to the careful assessment of empirical regularities by agents that are cen-
tral to the analysis of Kurz (1994), Kurz and Motolese (2010). Similar remarks hold
for the connections with the “evolutive” learning literature. Although the “eductive”
virtual time tests we use have a screening power which is reminiscent of the screening
power of real-time learning,39 the “diverse beliefs”40 that emerge are a priori differ-
ent: the “heterogenous expectations” that our approach may suggest are different from
those arising along the path of a real-time learning process, (whether it converges or
does not) or at the limiting part of the path, (when it converges but not to a standard
rational expectations outcome, as is the case in this conference in the paper by Branch
and Evans (2010)).
One final words on lines of research, likely to provide improvements on our present
findings. We have mentioned in the introduction the implications of our paper for the
theory of oligopolistic competition. Also, incorporating incomplete information in our
setting, along the line of argument of the global games literature (Carlsson and van
Damme 1993;Morris and Shin 1998,2003), looks a potentially fruitful undertaking.
Indeed, when the informational random variable can take a finite number of values,
our basic setting can be reinterpreted to describe an incomplete information world. In
general, however, the reinterpretation requires that an infinite-dimensional version of
our basic model be considered. As we already argued, such an infinite-dimensional
variant would also be most useful in the context of models of theoretical finance, and
provide potentially useful links with other literature.
A Appendix
A.1 Rationalizable strategies in games with a continuum of players
A.1.1 Point rationalizability with a continuum of players
Following Jara-Moroni (2008), and coming to our setting, in a game-theoretical per-
spective, the recursive process of elimination of non-best responses, when agents have
footnote 38 continued
neither strategic complementarities nor substitutabilities. The complexity of the findings that has increased
when going from the first case to the second one, will still increase. Hence, the local Stability analysis that
is presented in the last section, may turn out to provide the most important operational insights.
39 See Evans and Honkapohja (2001) for the study of real-time learning in an intertemporal context and,
for example Gauthier and Guesnerie (2005), for the comparison of the results of real-time learning tests
and the “virtual time” “eductive” tests within a large class of infinite horizon models.
40 Also, the diverse beliefs of the Wu and G uo (2010) paper, refers to difference of opinions on exogenous
risks.
123
Expectational coordination in simple economic contexts 235
point expectations, is associated with the mapping Pr :PSI→PSI, where
PSIis the set of all subsets of SI, which to each subset H⊆SIassociates the set
Pr(H)defined by
Pr(H):= s∈SI:sis a measurable selection of i⇒Br(i,H).
The operator Pr represents the process under which we obtain strategy profiles that are
constructed as the reactions of agents to strategy profiles contained in the set H⊆SI.
If it is known that the outcome of the game is in a subset H⊆SI, with point expecta-
tions, the strategies of agent i∈Iare restricted to the set Br(i,H)≡s∈HBr(i,s)
and so actual strategy profiles must be measurable selections of the set valued map-
ping i⇒Br(i,H). It has to be kept in mind that strategies of different agents within
a strategy profile in Pr(H)describe the individual reactions to (generally) different
strategy profiles in H.
We then define
Definition 13 The set of Point-Rationalizable Strategy Profiles is the maximal subset
H⊆SIthat satisfies
H≡Pr(H).
and we denote it PS.
A.1.2 Rationalizability with a continuum of players
Rationalizable Strategies should be obtained from a similar exercise but considering
forecasts as probability measures over the set of strategies of the opponents. A diffi-
culty in a context with continuum of players relates to the continuity or measurability
properties that must be attributed to subjective beliefs, as a function of the agent’s
name. It is not a priori clear whether it is better to describe stochastic individual
beliefs as measures on the set of the considered (measurable) strategy profiles or as
measurable profiles of distributions of beliefs on strategies. In light of the structure of
our model, we “stick” to the former.
Consider then a measurable structure in the space of strategy profiles. That is, con-
sider the set SIand a σ-field SI. As above, we may consider a process of elimination
of strategies that are not the best response to any forecast that are now in the form of
probability measures on SI. Consider then the mapping R:SI→PSIwhich to
each set Hassociates the set R(H)given by
R(H):= s∈SI:sis a measurable selection of i⇒IB r (i,P(H)).(11)
where P(H)stands for the set of probability measures over SIwith support on Hand
IB r (i,ν)is the set of optimal solutions of the problem of maximization of expected
utility:
123
236 R. Guesnerie, P. Jara-Moroni
IB r (i,ν):= argmaxy∈S⎧
⎪
⎨
⎪
⎩
SI
u⎛
⎝i,y,
I
sdi⎞
⎠dν(s)⎫
⎪
⎬
⎪
⎭
.
Rrepresents the process under which we obtain strategy profiles that are constructed
as the reactions of players to probabilistic forecasts contained in their supports to
the set H∈SI. The same reasoning as in the previous Subsection applies. We are
able then to formally define a set of (correlated) rationalizable strategy profiles as the
maximal set of strategy profiles that is contained in its image through the process of
elimination of strategies:41
Definition 14 The set of (Correlated) Rationalizable Strategy Profiles is the maximal
subset H∈SIthat satisfies
H≡R(H),
and we denote it RSI.
A.1.3 Connection with state rationalizability
Proposition 6 The “game-theoretical” and the “economic” appraisal of (point-)
rationalizability are coherent.
PS≡s∈SI:sis a measurable selection of i ⇒B(i,PA)(12)
PA≡⎧
⎨
⎩
a∈A:a=
I
s(i)di and sis a measurable function in PS⎫
⎬
⎭
.(13)
RSI≡s∈SI:sis a measurable selection of i ⇒B(i,P(RA))(14)
RA≡⎧
⎨
⎩
a∈A:a=
I
s(i)di and sis a measurable function in RSI⎫
⎬
⎭
.(15)
Equations (12) through (15) stress the equivalence for (point-)rationalizability between
the state approach and the strategic approach in games with continuum of players: the
sets of (Point-)Rationalizable States can be obtained from the set of (Point-)(corre-
lated)Rationalizable Strategies and vice versa. For instance, in (12) we see that the
strategy profiles in PSare profiles of best responses to PA. Conversely, in (13) we get
that the points in PAare obtained as integrals of the profiles in PS. For the proof and
further discussion, see Jara-Moroni (2008).
41 What we are considering in terms of strategy profiles resembles more to “correlated strategic rational-
izability” (Brandenburger and Dekel 1987).
123
Expectational coordination in simple economic contexts 237
A.2 Technical Lemmas
Lemma 6 If S ⊂Rnis a complete lattice for the product order in Rn, then for a
measurable correspondence F :I⇒S with non-empty, closed and subcomplete
values, the functions s:I→S and ¯
s:I→S, defined by
s(i):= inf
SF(i),
¯s(i):= sup
S
F(i),
are measurable selections of F.
Proof Since F(i)is subcomplete, s(i)and ¯s(i)belong to F(i).Wehavetoprovethat
sand ¯
sare measurable.
Since Fis measurable, it has a Castaing representation. That is, there exists a count-
able family of measurable functions sν:I→Rn,ν∈IN , such that sν(i)∈F(i)
and,
F(i)≡clsν(i):ν∈IN .(16)
For s, consider then for each ν∈IN the set valued mappings Fν:I⇒Rn, defined
by 42
Fν(i):= F(i)−∞,sν(i)
Since Fis measurable and closed valued, and we can write ]−∞,sν(i)]=sν(i)−Rn
+
which is as well measurable and closed valued, the correspondences Fνare measurable
and closed valued.43
Note that ∀ν∈IN ,s(i)∈Fν(i). Defining the closed valued correspondence
F:I⇒Rn:
F(i):=
ν∈IN
Fν(i)
we get then that s(i)∈F(i). The correspondence Fis as well measurable.43
We now prove that actually F(i)≡s(i), which completes the proof. Indeed,
suppose that y∈F(i). Then, by definition of F,y∈F(i)and y≤sν(i),∀ν∈IN .
From equality (16) we get that any point in F(i)can be obtained as the limit of a
subsequence of {sν(i):ν∈IN }, so in the limit the inequality is maintained, that
is, ∀s∈F(i),y≤s. That is, yis a lower bound for F(i). This implies, by the
definition of inf SF(i), that y≤inf SF(i),but y∈F(i),soinf
SF(i)≤y. Thus,
y≤s(i)=inf SF(i)≤y.
42 The interval ]−∞,x]is the set of points of Rnthat are smaller than x∈Rn; similarly, [x,+∞ [is
the set of points in Rnthat are greater than x.
43 See Proposition 14.11 in Rockafellar and Wets (1998).
123
238 R. Guesnerie, P. Jara-Moroni
Analogous arguments applied to the mapping ¯
F:I⇒Rn:
¯
F(i):= F(i)
ν∈IN sν(i),+∞
prove the statement for ¯
s.
Proof of Lemma 1In 1 the first part is a consequence of Theorem 2 in Milgrom and
Roberts (1990) and the second part we apply Theorem 2.8.1 in Topkis (1998) consid-
ering the constant correspondence Sa≡S∀a∈A
Part 2 is a consequence of Lemma 6above.
Proof of Lemma 2Observe first that supermodularity of u(i,·,a)is preserved44
when we take expectation on a.
Now consider y∈Bi,aand y∈B(i,μ
)we show that min y,y∈Bi,a
and max y,y∈B(i,μ
). Since y∈Bi,awe have that
0≤ui,y,a−ui,min y,y,a.
Increasing differences of u(i,y,a)in(y,a)implies that ∀a∈sup(μ),
ui,y,a−ui,min y,y,a≤ui,y,a−ui,min y,y,a
and so if on the right-hand side we take expectation with respect to μwe get
ui,y,a−ui,min y,y,a≤Eμ[ui,y,a]−Eμ[ui,min y,y,a].
Supermodularity of u(i,·,a)implies that
Eμ[ui,y,a]−Eμui,min y,y,a
≤Eμui,max y,y,a−Eμ[u(i,y,a)]
and the last term is less or equal to 0 since y∈B(i,μ
).
All these inequalities together imply that max y,y∈B(i,μ
)and min y,y∈
Bi,a
The second statement is proved analogously.
44 If u(i,·,a)is supermodular, then for s,s∈S, we have for each a∈A:
ui,min s,s,a+ui,max s,s,a−u(i,s,a)+ui,s,a≥0
Taking expectation we get the result.
123
Expectational coordination in simple economic contexts 239
A.3 Proof of Proposition 3
Proof Let us denote the fixed points of by E. As a consequence of the Theorem
2.5.1 in Topkis (1998), Eis a non-empty complete lattice and so has a greatest and
a smallest element. Convexity of the rationalizable sets is a general property and a
consequence of Proposition 1and Theorems 3.6 and 4.5 in Jara-Moroni (2008). We
will first prove the following statement:
In Gwe have
PA⊆[inf
E
{E},sup
E
{E}]
and inf E{E}and supE{E}are equilibria.
Following the structure of the proof of Theorem 5 in Milgrom and Roberts we prove
that ˜
Prt(A)is contained in some interval [at,¯at]and that the sequences atand ¯at
satisfy at→inf E{E}and ¯at→supE{E}.
Define a0and atas
a0:= inf A
at:= inf
Aat−1∀t≥1
–˜
Prt(A)⊆at,+∞
Clearly it is true for t=0.
Suppose that it is true for t≥0. That is, at≤a∀a∈˜
Prt(A). Since B(i,·)
is increasing, we get that Bi,atB(i,a)∀a∈˜
Prt(A). In particular ∀a∈
˜
Prt(A)and ∀y∈B(i,a),wehaveinf
SBi,at≤y. From Lemma 6, the corre-
spondence i⇒inf SBi,atis measurable. This implies that for any measurable
selection s∈SIof i⇒Bi,˜
Prt(A),
inf
SBi,atdi ≤s(i)di.(17)
Since Bi,atis subcomplete, inf SBi,at∈Bi,atand so we get that:
inf
Aat
≡inf
Ab∈A:∃smeasurable selection of i⇒Bi,atsuch that, b=A(s)
≤inf
SBi,atdi (18)
We conclude then that
at+1≡inf
Aat≤inf
SBi,atdi ≤a∀a∈˜
Prt+1(A).
123
240 R. Guesnerie, P. Jara-Moroni
The equality is the definition of at+1, the first inequality comes from (18) and the
second one is obtained from (17) and the definition of ˜
Pr.
– The sequence is increasing:
By definition of a0,a0≤a1. Suppose that at−1≤at, then from Lemma 2.4.2 in
Topkis (1998), at≡infAat−1≤infAat≡at+1.
– The sequence has a limit and limt→+∞ atis a fixed point of :
Since the sequence is increasing and Ais a complete lattice, it has a limit a∗.
Furthermore, since is subcomplete, upper semi-continuity of implies that
a∗∈a∗.
–a∗≡inf E{E}:
According to the previous demonstration, since the fixed points of are in the set
PA, all fixed points must be in a∗,+∞and so a∗is the smallest fixed point.
Defining ¯a0and ¯atas
¯a0:= sup A
¯at:= sup
A
¯at−1∀t≥1
In an analogous way we obtain that PA⊆−∞,¯a∗, with ¯a∗being the greatest fixed
point of .
This proves the statement. Proposition 3obtains then using Lemma 2. We can see
that ˜
Rt(A)⊆[at,¯at]and so we get the result.
A.4 Proof of Theorem 1
Proof Following the proof of Proposition 3, consider the sequence at∞
t=0therein
defined, but let us change the definition of atwhen tis odd to
at:= sup
A
at−1.
By the definition of a0, we know that ∀a∈A,a≥a0. Since the mappings B(i,·)
are decreasing we have Bi,a0B(i,a)∀a∈Aand in particular
sup
S
Bi,a0≥y∀y∈B(i,a)∀a∈A
Since Bi,a0is subcomplete supSBi,a0∈Bi,a0and from Lemma 6the func-
tion i→supSBi,a0is measurable, so supSBi,a0∈a0thus
sup
A
a0≥sup
S
Bi,a0di ≥s(i)di
123
Expectational coordination in simple economic contexts 241
for any measurable selection sof i⇒B(i,A). That is, a1≥a∀a∈˜
Pr1(A);or
equivalently,
˜
Pr1(A)⊆ −∞,a1 .
A similar argument leads to conclude that ˜
Pr2(A)⊆a2,+∞ .
Let us define then the sequence bt:= a2t,t≥0. This sequence satisfies
1. ˜
Pr2t⊂bt,+∞ . This can be obtained as above by induction over t.
2. btt≥0is increasing.
As before, we get that btt≥0has a limit b∗. Since is u.s.c. and Ais compact, we
obtain that the second iteration of ,2is as well u.s.c.. Moreover, from Proposition
2, we get that bt∈2bt−1. This implies that b∗is a fixed point of 2and so it is a
point-rationalizable state. Consequently, we get
1. PA⊂b∗,+∞
2. b∗∈2b∗and b∗is a point-rationalizable state.
Considering the analogous sequence to obtain the upper bound for PA
¯a0:= sup A
¯at:= inf
A¯at−1when tis odd
¯at:= sup
A
¯at−1when tis even
We generate a decreasing sequence ¯
btt≥0defined by ¯
bt:= ¯a2t,t≥0, whose limit
¯
b∗, is a point-rationalizable state and an upper bound for PA; that is,
1. PA⊆−∞,¯
b∗
2. ¯
b∗∈2¯
b∗. Which implies that ¯
b∗is a point-rationalizable state.
As a summary, we get
PA≡
t≥0
˜
Prt(A)≡
t≥0
˜
Pr2t(A)⊆[b∗,¯
b∗]
Then again using Lemma 5we can see that ˜
R2t(A)⊆[a2t,¯a2t]and so we get (ii)
and (iv).
Assertion (iii) is a consequence of the general setting of Rath (1992). The existence
Theorem therein gives the existence of equilibrium.
123
242 R. Guesnerie, P. Jara-Moroni
A.5 Proof of Proposition 4
Proof For (i), note that
(a)≡
I
B(i,a)di ≡!˜
Pr({a})
IB(i,δ
a)di ≡˜
R({a})
and use Proposition 2.
For (ii), from Proposition 2we see that we only need to prove that under condition
(10):
a∗is Locally Strongly Point Rational ⇒ a∗is Locally Strongly Rational.
For a subset X⊆Acall P(X):= "t≥0˜
Prt(X)and note that if P(X)≡{a∗}, then
∀X⊆X,PX≡{a∗}.
Take V, the neighbourhood of the proposition. For a Borel subset X⊆Vthe
hypothesis implies that the integral of i⇒B(i,P(X)) is contained in the integral of
i⇒co {B(i,X)}.FromAumann (1965) we know that
I
co {B(i,X)}di ≡
I
B(i,X)di
and so
˜
R(X)≡
I
B(i,P(X)) di ⊆
I
B(i,X)di ≡˜
Pr(X)(19)
If a∗is Locally Strongly Point Rational, then there exists a neighbourhood Vsuch
that PV={a∗}. So now take an open ball of radius ε>0 around a∗that is
contained in both Vand V. To ensure that the process under probability forecasts
is well defined, we can take the closed ball of radius ε/2, Ba∗,ε
2, that is strictly
contained in the previous ball and of course in the intersection of both neighborhoods.
In particular, we have that PBa∗,ε
2≡{a∗}and that ˜
RtBa∗,ε
2is well defined
and closed for all t≥1. The last assertion, along with (19), implies that for all t≥1
˜
RtBa∗,ε
2≡˜
PrtBa∗,ε
2. We conclude that,
t≥0
˜
RtBa∗,ε
2≡PBa∗,ε
2≡a∗
A.6 Proof of Theorem 4
Proof We give the proof for the case where all the agents have the same utility function
u:S×A→R.
Consider then a convex neighbourhood Vof a∗and the space of probability
measures P(V). Take a probability measure with finite support, μ, in this space, that
123
Expectational coordination in simple economic contexts 243
is μ=#L
l=1μlδal, with {al}L
l=1⊂V. For this measure, under the differentiability
hypothesis, we can prove that if the support of μ,{a1,...,aL}, is contained in a ball
45 B(a∗,ε
1), then
B(μ)−BEμ[a]
<ε
2
1.
Since Eμ[a]∈Vwe get that BEμ[a]∈B(V). Using a density argument we may
conclude that B(μ)is “close” to B(V)⊆co {B(V)}≡˜
Pr(V)for any measure in
P(V). We can take then ε1>0 small, related to the neighbourhood V, such that,
˜
R(V)⊂˜
Pr(V)+B0,ε
2
1(20)
From the hypothesis we get that we can choose a number ¯
k<k<1 such that the
following inclusions hold:
˜
Pr(V)⊂V¯
k⊂Vk⊂V(21)
˜
R(V)⊂˜
Pr(V)+B0,ε
2
1⊂Vk(22)
Moreover, taking the second iterate of ˜
Rstarting at V,using(22) and (20)onVk,
˜
R2(V)⊂˜
Pr(Vk)+B0,ε
2
2
where ε2depends on k. However, it can be chosen in such a way that the following
inclusions hold. Using (21) we get
˜
Pr(Vk)+B0,ε
2
2⊂V¯
kk+B0,ε
2
2
⊂Vk2.
We have then,
˜
R2(V)⊂Vk2
Using the same argument, choosing εtrelated to the powers of k,kt−1, we get that
for all t,
˜
Rt(V)⊂˜
PrVkt−1+B0,ε
2
t⊂V¯
kkt−1+B0,ε
2
t⊂Vkt
We conclude then that the eductive process converges to the equilibrium a∗.
45 Since Ais compact Vis bounded.
123
244 R. Guesnerie, P. Jara-Moroni
A.7 Lattices theory tools
All following definitions and related properties can be found with more detail in Topkis
(1998):
In an ordered set (E,≥), given a non-empty subset X⊆Ean upper bound of X
is an element x∈Esuch that x≥x,∀x∈X, analogously a lower bound of Xis
an element x∈Esuch that x≤x,∀x∈X.If ¯xis an upper bound of Xand ¯x≤x
for any upper bound of X, then ¯xis the supremum of Xand we note it supEX.The
infimum of X, is defined analogously and we note it inf EX.
The set Eis a lattice if for each two element subset {x,y}⊆E, the elements
supE{x,y}∈Eand inf E{x,y}∈E. The lattice Eis a complete lattice if any non-
empty subset X⊆Ehas a greatest and smallest bound on E, that is, supEX∈Eand
inf EX∈E. A subset Xof Eis a sublattice of Eif for all x,y∈X,sup
E{x,y}∈X
and inf E{x,y}∈X. The sublattice Xis subcomplete if for any non-empty subset X
of X,sup
EX∈Xand inf EX∈X.
We say that a function f:E→Ris supermodular if ∀x,y∈E,
f(x)+f(y)≤finf
E{x,y}+fsup
E
{x,y}
All functions defined on Rare supermodular.
With ≥we can induce a set ordering in the set of subsets of E,P(E), as follows:
for X,Y⊆Ewe say that Xis greater than Y, denoted XY,if∀(x,y)∈X×Y,
supE{x,y}∈Xand inf E{x,y}∈Y. With this definition we are able to define the
concept of increasing (decreasing) set valued mapping. We will say that a mapping
F:E⇒Yis increasing (decreasing) if x≥xthen F(x)Fx(F(x)Fx).
Note that if Fis single valued we obtain the usual definition of increasing (decreasing)
function.
References
Aumann, R.J.: Integrals of set-valued functions. J Math Anal Appl 12, 1–12 (1965)
Azariadis, C., Guesnerie, R.: Sunspots and cycles. Rev Econ Stud 53(5), 725–737 (1986)
Balder, E.J.: A unifying pair of cournot-nash equilibrium existence results. J Econ Theory 102, 437–
470 (2002)
Basu, K., Weibull, J.W.: Strategy subsets closed under rational behavior. Econ Lett 36(2), 141–146 (1991)
Battigalli, P., Siniscalchi, M.: Rationalization and incomplete information. Adv Theor Econ 3(1):Article 3.
http://www.bepress.com/bejte/advances/vol3/iss1/ art3 (2003)
Ben-Porath, E., Heifetz, A.: Rationalizable expectations. Discussion Paper Series dp461, Center for Ratio-
nality and Interactive Decision. Hebrew University, Jerusalem (2006)
Benhabib, J., Bull, C.: Conjectural equilibrium bounds. In: Galeotti, M, Gerolazzo, L., Gori, F. (eds.) Non
Linear Dynamics in Economics and Social Sciences, pp. 2–28.Bologna: Pitagora Editrice (1988)
Bernheim, B.D.: Rationalizable strategic behavior. Econometrica 52(4), 1007–1028 (1984)
Branch, W.,Evans, G.: Monetary policy and heterogeneous expectations. Econ. Theory (2010, this volume).
doi:10.1007/s00199-010-0539- 9
Branch, W., McGough, B.: Business cycle amplification with heterogeneous expectations. Econ Theory
(2010, this volume). doi:10.1007/s00199-010- 0541-2
Brandenburger, A., Dekel, E.: Rationalizability and correlated equilibria. Econometrica 55(6), 1391–1402
(1987)
123
Expectational coordination in simple economic contexts 245
Carlsson, H., van Damme, E.: Global games and equilibrium selection. Econometrica 61(5), 989–1018
(1993)
Carmona, G.: A remark on the measurability of large games. Econ Theory 39(3), 491–494 (2009)
Chamley, C.: Coordinating regime switches. Q J Econ 114(3), 869–905 (1999)
Chamley, C.P.: Rational Herds: Economic Models of Social Learning. Cambridge: Cambridge University
Press (2004)
DeCanio, S.J.: Rational expectations and learning from experience. Q J Econ 93(1), 47–57 (1979)
Desgranges, G.: Ck-equilibria and informational efficiency in a competitive economy. Econometric Society
World Congress 2000 Contributed Papers 1296, Econometric Society (2000)
Desgranges, G., Heinemann, M.: Strongly rational expectations equilibria with endogenous acquisition
of information. In: Working Paper Series in Economics, vol. 9. University of Lüneburg, Institute of
Economics (2005)
Dixit, A.K., Stiglitz, J.E.: Monopolistic competition and optimum product diversity. Am Econ
Rev 67(3), 297–308 (1977)
Evans, G.: Expectational stability and the multiple equilibria problem in linear rational expectations mod-
els. Q J Econ 100(4), 1217–1233 (1985)
Evans, G.: The stability of rational expectations in macro-economic models. In: Frydman, R., Phelps,
E.S. (eds.) Individual Forecasting and Aggregate Outcomes ‘Rational Expectations’ Examined,Cam-
bridge, England: Cambridge University Press (1986)
Evans, G.W., Guesnerie, R.: Rationalizability, strong rationality and expectational stability. Games Econ
Behav 5, 632–646 (1993)
Evans, G.W., Guesnerie, R.: Coordination on saddle-path solutions: the eductive viewpoint-linear univariate
models. Macroecon Dyn 7(1), 42–62 (2003)
Evans, G.W., Guesnerie, R.: Coordination on saddle-path solutions: the eductive viewpoint-linear multi-
variate models. J Econ Theory 124(2), 202–229 (2005)
Evans, G., Honkapohja, S.: Learning and Expectations in Macroeconomics. Princeton: Princeton University
Press (2001)
Gauthier, S.: Determinacy and stability under learning of rational expectations equilibria. J Econ
Theory 102(2), 354–374 (2002)
Gauthier, S., Guesnerie, R.: Comparing expectational stability criteria in dynamical models: a preparatory
overview. In: Assesing Rational Expectations: “Eductive” Stability in Economics, vol. 2. MIT Press
(2005)
Ghosal, S.: Intertemporal coordination in two-period markets. J Math Econ 43(1), 11–35 (2006)
Grossman, S.J., Stiglitz, J.E.: On the impossibility of informationally efficient markets. Am Econ
Rev 70(3), 393–408 (1980)
Guesnerie, R.: An exploration on the eductive justifications of the rational-expectations hypothesis. Am
Econ Rev 82(5), 1254–1278 (1992)
Guesnerie, R.: On the robustness of the analysis of expectational coordination: from 3 to n+2 goods. In:
Debreu, G., Neuefeind, W.T.W. (eds.) Economic Essays, a Festschrift for Werner Hildenbrand, pp. 141–
158. New York: Springer (2001a)
Guesnerie, R.: Short-run expectational coordination: fixed versus flexible wages. Q J Econ 116(3), 1115–
1147 (2001b)
Guesnerie, R.: Anchoring economic predictions in common knowledge. Econometrica 70(2), 439 (2002)
Guesnerie, R.: Macroeconomic and monetary policies from the “eductive” viewpoint. In: Schmidt-Hebbel,
K., Walsh, C. (eds.) Monetary Policy Under Uncertainty and Learning,Santiago: Central Bank of
Chile (2008)
Jara-Moroni, P.: Rationalizability in games with a continuum of players. PSE Working Papers 2007-25.
PSE (Ecole normale supérieure) (2008)
Khan, M.A., Sun, Y.: Non-cooperative games with many players. In: Aumann, R., Hart, S. (eds.) Handbook
of Game Theory, vol. 3, pp. 1761–1808.Amsterdam : Elsevier Science (2002)
Kurz, M.: On the structure and diversity of rational beliefs. Econ Theory 4:877–900 (1994). An edited
version appears as Chapter 2 of Kurz, M. (ed.) (1997)
Kurz, M., Motolese, M.: Diverse beliefs and time variability of risk premia. Econ Theory (2010, this
volume). doi:10.1007/s00199-010- 0550-1
Lucas, J.: Asset prices in an exchange economy. Econometrica 46(6), 1429–1445 (1978)
Milgrom, P., Roberts, J.: Rationalizability, learning, and equilibrium in games with strategic complemen-
tarities. Econometrica 58(6), 1255–1277 (1990)
123
246 R. Guesnerie, P. Jara-Moroni
Morris, S., Shin, H.S.: Unique equilibrium in a model of self-fulfilling currency attacks. Am. Econ.
Rev. 88(3), 587–597 (1998)
Morris, S., Shin, H.S. (2003) Global games: theory and applications. In: Dewatripont, M., Hansen, L.,
Turnovsky, S. (eds.) Advances in Economics and Econometrics: Theory and Applications, Eighth
World Congress, vol. I, pp. 57–114. Cambridge University Press
Muth, J.F.: Rational expectations and the theory of price movements. Econometrica 29(3), 315–335 (1961)
Pearce, D.G.: Rationalizable strategic behavior and the problem of perfection. Econometrica 52(4), 1029–
1050 (1984)
Quah, J.K.H., Strulovici, B.: Comparative statics, informativeness, and the interval dominance order.
Econometrica 77(6), 1949–1992 (2009)
Rath, K.P.: A direct proof of the existence of pure strategy equilibria in games with a continuum of play-
ers. Econ Theory 2, 427–433 (1992)
Rath, K.P.: Representation of finite action large games. Int J Game Theory 24, 23–35 (1995)
Rockafellar, R., Wets, R.J.B.: Variational Analysis. New York: Springer (1998)
Schmeidler, D.: Equilibrium points of nonatomic games. J Stat Phys 7(4), 295–300 (1973)
Tan, T.C.C., da Costa Werlang, S.R.: The Bayesian foundations of solution concepts of games. J Econ
Theory 45(2), 370–391 (1988)
Topkis, D.M.: Supermodularity and Complementarity. Frontiers of Economic Research. Princeton:
Princeton University Press (1998)
Vives, X.: Nash equilibrium with strategic complementarities. J Math Econ 19, 305–321 (1990)
Wu, H.M., Guo, W.C.: Financial leverage and market volatility with diverse beliefs. Econ Theory (2010,
this volume). doi:10.1007/s00199-010- 0548-8
123
