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Dropping the Cass Trick and Extending Cass’ Theorem
to Asymmetric Information
Lionel Boisdeffre
To cite this version:
Lionel Boisdeffre. Dropping the Cass Trick and Extending Cass’ Theorem to Asymmetric Information. 2019. �halshs-02180836�
Documents de Travail du
Centre d’Economie de la Sorbonne
Dropping the Cass Trick and Extending Cass’ Theorem to Asymmetric Information
Lionel de BOISDEFFRE
Dropping the Cass Trick and Extending Cass’Theorem to Asymmetric Information
Lionel de Boisde¤re,1
(May 2019)
Abstract
In a celebrated 1984 paper, David Cass provided an existence theorem for …nan-cial equilibria in incomplete markets with exogenous yields. The theorem showed that, when agents had symmetric information and ordered preferences, equilibria existed on purely …nancial markets, supported by any collection of state prices. This theorem built on the so-called "Cass trick", along which one agent had an Arrow-Debreu budget set, with one single constraint, while the other agents were constrained a la Radner (1972), that is, in every state of nature. The current paper extends Cass’theorem to economies with asymmetric information and non-ordered preferences. It re…nes De Boisde¤re (2007), which characterized the existence of equilibria with asymmetric information by the no-arbitrage condition on purely …-nancial markets. The paper de…nes no arbitrage prices with asymmetric information. It shows that any collection of state prices, in the agents’commonly expected states, supports an equilibrium. This result is proved without using the Cass trick, in the sense that budget sets are de…ned symmetrically across all agents. Thus, the paper suggests, in the symmetric information case, an alternative proof to Cass’.
.
Key words: sequential equilibrium, perfect foresight, existence of equilibrium,
ra-tional expectations, incomplete markets, asymmetric information, arbitrage. JEL Classi…cation: D52
1 University of Paris 1 - Panthéon - Sorbonne, 106-112 Boulevard de l’Hôpital,
1 Introduction
When agents have incomplete or asymmetric information, they seek to infer in-formation from observing markets. A traditional response to that problem is given by the REE (rational expectations equilibrium) model by assuming, quoting Rad-ner (1979), that “agents have a ‘model’ or ‘expectations’ of how equilibrium prices are determined ”. Under this assumption, agents know the map between private information signals and equilibrium prices, along a so-called "forecast function".
In the simplest setting with two periods, no production and an uncertainty over future states to prevail, Cornet-De Boisde¤re (2002) suggests an alternative ap-proach to the REE, where asymmetric information is represented by private sig-nals, informing each agent that tomorrow’s true state will be in a subset of the state space. The latter paper generalizes the classical de…nitions of equilibrium, no-arbitrage prices and no-arbitrage condition to asymmetric information. In this model, De Boisde¤re (2007) shows that equilibria exist on purely …nancial markets if they preclude arbitrage. That no-arbitrage condition, which typically holds under asymmetric information, may always be reached by agents observing asset prices or available …nancial transfers. Along Cornet-De Boisde¤re (2009), or De Boisdef-fre (2016), that learning process requires no price model. Such results di¤er from Radner’s (1979) inferences and the related generic existence of fully revealing REE. In our setting, which drops rational expectations’inferences, the current paper provides new insights on the existence issue. It examines whether the so-called "Cass trick " is required, and Cass’theorem holds, under asymmetric information. The Cass trick (1984, 2006) is a device introduced in Radner’s (1972) budget sets and equilibria, which consists in replacing the budget constraints of one agent by a
single Arrow-Debreu constraint at the …rst period. The other agents’budget sets are left unchanged. This device enables to de…ne asset prices relative to individual state prices, as the (positively) weighted sum of payo¤s across states. It permits to show that any collection of state prices supports an equilibrium.
The current paper extends Cass’ results to asymmetric information and non-ordered preferences. Under asymmetric information, it also re…nes De Boisde¤re (2007), which characterized the existence of one equilibrium by the no-arbitrage condition. Our proof applies, as its main argument, the Gale-Mas-Collel (1975, 1979) …xed-point-like theorem to reaction correspondences, which are formal repre-sentations of the market and agents’behaviours. The proof extends Cass’theorem but drops the Cass trick, in the sense that budget sets are de…ned symmetrically across all agents. Under symmetric information, it thus suggests an alternative proof to Cass’. The paper is organized as follows: Section 2 presents the model. Section 3 states and proves the existence theorem. An Appendix proves Lemmas.
2 The model
We consider a pure-exchange …nancial economy with two periods,t 2 f0; 1g, and
an uncertainty, at t = 0, upon which state of nature will randomly prevail at t =
1. The economy is …nite in the sense that the sets, I, S, L and J, respectively,
of consumers, states of nature, consumption goods and assets are all …nite. The
observed state at t = 0 is denoted bys = 0 and we let 0 := f0g [ , whenever S.
2.1 Markets and information
Agents consume or exchange the consumption goods, l 2 L, on both periods’
that tomorrow’s true state will be in a subset, Si, of S. We assume costlessly that
S = [i2ISi. Thus, the pooled information set, S:= \i2ISi, contains the true state,
and the relation S= S characterizes symmetric information.
We let P := fp := (ps) 2 RL S
0
: kpk 6 1g be the set of admissible commodity prices, which each agent is assumed to observe, or anticipate perfectly, a la Radner (1972). Moreover, each agent with an incomplete information forms her private
forecasts in the unrealizable states she expects. Such forecasts, (s; pi
s), are pairs of
a state, s 2 SinS, and a price, pis 2 RL++, that the generic ith agent believes to be
the conditional spot price in state s. Thus, di¤erent agents are allowed to agree or
disagree on di¤erent forecasts, which are never self-ful…lling. Non restrictively, such idiosyncratic forecasts are exogeneously given, along De Boisde¤re (2007).
Agents may operate …nancial transfers across states in S0 (actually in S0) by
exchanging, at t = 0, …nitely many nominal assets, j 2 J, which pay o¤, at t = 1,
conditionally on the realization of the state. We assume that #J 6 #S, so that
…nancial markets be typically incomplete. Assets’payo¤s de…ne a S J matrix, V,
whose generic row in state s 2 S, denoted by V (s) 2 RJ, does not depend on prices.
Thus, at asset price, q 2 RJ, agents may buy or sell unrestrictively portfolios of
assets, z = (zj) 2 RJ, forq z units of account att = 0, against the promise of delivery
of a ‡ow, V (s) z, of conditional payo¤s across states,s 2 S.
2.2 The consumer’s behaviour and concept of equilibrium
Each agent, i 2 I, receives an endowment, ei := (eis), granting the commodity
bundles, ei0 2 RL+ at t = 0, and eis2 RL+, in each expected state, s 2 Si, if it prevails.
Given the market prices, p := (ps) 2 P and q 2 RJ, and her forecasts, the generic ith
agent’s consumption set isXi:= RL S
0 i
Bi(p; q) := f (x; z) 2 Xi RJ : p0(x0 ei0)6 q z and ps(xs eis)6 V (s) z; 8s 2 S
and pi
s(xs eis)6 V (s) z; 8s 2 SinS g.
Each consumer, i 2 I, is endowed with a complete preordering, -i, over her
consumption set, representing her preferences. Her strict preferences, i, are
rep-resented, for each x 2 Xi, by the set, Pi(x) := f y 2 Xi : x i y g, of
consump-tions which are strictly preferred to x. In the above economy, denoted by E =
f(I; S; L; J); V; (Si)i2I; (pis)(i;s)2I SinS; (ei)i2I; ( i)i2Ig, agents optimise their consumptions
in the budget sets. This yields the following concept of equilibrium:
De…nition 1 A collection of prices,p = (ps) 2 P, q 2 RJ, & decisions, (xi; zi) 2 Bi(p; q),
for each i 2 I, is an equilibrium of the economy, E, if the following conditions hold:
(a) 8i 2 I; (xi; zi) 2 Bi(p; q) and Pi(xi) RJ\ Bi(p; q) = ?;
(b) Pi2I (xis eis) = 0; 8s 2 S0;
(c) Pi2I zi= 0.
The economy, E, is called standard if it meets the following conditions:
Assumption A1 (monotonicity): 8(i; x; y) 2 I (Xi)2; (x6 y; x 6= y) ) (x iy);
Assumption A2 (strong survival): 8i 2 I; ei2 RL S
0 i
++ ;
Assumption A3: 8i 2 I, i is lower semicontinuous convex-open-valued and
such that x ix + (y x), whenever (x; y; ) 2 Xi Pi(x) ]0; 1];
Assumption A4: 9z 2 RJ
; 8s 2 S, V (s) z > 0.
2.3 A portfolio decomposition
This sub-Section introduces orthogonal sub-vector spaces of the portfolio set,RJ,
in relation to agents’information signals. It de…nes no-arbitrage prices, and relates them to supporting individual state prices.
For each i 2 I, we let Zi :=Ps2Si RV (s) R
J be the span of the payo¤ matrix’
rows in the range of the information set, Si. We let Zio:= fz 2 RJ: V (s) z = 0; 8s 2 Sig,
its orthogonal complement, and Zo:= f z 2 RJ : V (s) z = 0; 8s 2 S g, be the spaces of
useless portfolios in the eyes of, respectively, the ith agent and a (possibly missing)
fully informed agent.
We also de…ne Zo := P
i2IZio and Zo? := \i2IZi, its orthogonal complement.
Whereas the relation Zo Zo holds for any information structure, (S
i), from the
de…nition, the converse inclusion may fail, as shown on the heuristic example
be-low. We therefore let Z := Zo\ Zo? RJ be reduced to f0g, if and only if Zo= Zo.
For all portfolio, z 2 RJ, we henceforth denotez = z1+ z + z2 the so-called (with
slight abuse) "orthogonal decomposition" ofz onZo Z Zo?. We use this notation
throughout and may assume, from Assumption A4, that the lastJ2> 0assets belong
toZo?, and all other assets (if any) belong toZo. Thus,V (s) = V (s)2 holds for every
s 2 S. We now de…ne no-arbitrage prices and their supporting state prices.
De…nition 2 A no-arbitrage price is an asset price, q 2 RJ, which meets one of the
following equivalent conditions, and we let N A be their set:
(a) @(i; z) 2 I RJ, q z> 0 and V (s) z> 0, 8s 2 S
i, with one strict inequality;
(b) 8i 2 I, 9 i:= ( is) 2 RS++i , q =
P
s2Si isV (s).
Scalars, ( is) 2 i2IRS++i , which meet the above condition (b), are said to support the
no-arbitrage price, q 2 N A, and called (individual) state prices. We denoteN A( ) :=
fq 2 N A : q2=P
s2S sV (s)g, for every := ( s) 2 RS++, andN AC = [ 2RS
++N A( ).
Proof The equivalence between the Assertions (i) and (ii) of De…nition 2 is
standard and proved in Cornet-De Boisde¤re (2002, Lemma 1, p. 398).
in-formation structure,(Si), is arbitrage-free, i.e., admits a n-a price. Indeed, the latter
paper shows that agents, starting from any information structure, (Si), may always
infer, with no price model, a re…ned information structure, which is arbitrage-free.
The following heuristic example and Claim 1 show that Z may not be reduced
to f0g and that the sets N A and N AC do not coincide, in general, but that any
collection of symmetric state prices, := ( s) 2 RS++, supports a no-arbitrage price,
q 2 N A( ), and, from Theorem 1 below, supports an equilibrium.
Example Consider an economy, E, with two agents, i 2 f1; 2g, three states, s 2
f1; 2; 3g, an information stucture, S1 := f1; 2g and S2 := f2; 3g, and one asset, whose
price is q = 1. If the payo¤ matrix is V =
0 B B B B B B @ 1 0 1 1 C C C C C C A
, Assumption A4 fails and the
relations Z = R and q 2 N A \cN AC hold. If V =
0 B B B B B B @ 1 1 1 1 C C C C C C A , the relations Z Zo= f0g, q 2 N A \cN AC and N AC = R++ hold.
Claim 1 The following Assertions hold:
(i) 8 2 RS++; N A( ) 6= ?;
(ii) N A * N AC, in general.
Proof Assertion (i) Let := ( s) 2 RS++ be given. Non restrictively from
Cornet-De Boisde¤re (2009), we had assumed that the payo¤ and information structure,
[V; (Si)], was arbitrage-free (i.e., N A 6= ?). Hence, we letq 2 N A and ( is) 2 i2IRS++i
be given, such that q =Ps2Si isV (s), for eachi 2 I. Since q is a no-arbitrage price,
Assertion (i) From Assumption A4, J2 := dim Zo? > 0. We refer to the notations
of sub-Section 2.3 and let V (S) := (V (s))s2S be the S J extracted sub-matrix ofV,
de…ned by its rows,V (s), in statess 2 S. We show thatrank V (S) = J2or, equivalently,
that the relation V (S) z 6= 0 holds, wheneverz 2 Zo?nf0g. Indeed, the joint relations
V (S) z = 0and z 2 Zo? imply, from the de…nitions: z 2 Zo\ Zo? = f0g. For eachi 2 I,
we de…ne qi2 RJ byqi=Ps2SinS isV (s)
2, if S
i6= S, and qi= 0 otherwise.
Since qi 2 Zo? and rank V (S) = J2, there exists a vector ( is) 2 RS, which we set
as given, such that qi =Ps2S is V (s), for every i 2 I. For N 2 N large enough, the
relationsj is
N j < s hold, for every pair(i; s) 2 I S. Then, we let i:= ( is) 2 RS++i be
de…ned, for eachi 2 I, by is= Nis, for everys 2 SinS, and is= s Nis, for everys 2 S.
By construction, the individual state prices, i:= ( is) 2 R
Si
++, de…ned for each i 2 I,
support a no-arbitrage price,q 2 N A( ).
Assertion(ii) was shown on the heuristic example above.
3 The existence theorem and proof
Along Theorem 1, any symmetric state price collection supports an equilibrium:
Theorem 1Let 2 RS++ be given. A standard economy, E, admits an equilibrium,
(p; q; [(xi; zi]) 2 P N A( ) ( i2IBi(p; q)).
We henceforth set as given := ( s) 2 RS++ and letq =
P
s2S sV (s).
The proof’s main argument is the Gale-Mas-Colell (1975, 1979) …xed-point-like theorem. We apply the theorem to lower semi-continuous reaction correspondences, de…ned over a convex compact set, which formally represent agents’behaviours.
Thus, sub-Section 3.1 introduces an auxiliary compact economy, derived from the economy E. Sub-Section 3.2 de…nes the reaction correspondences in that economy and applies the GMC theorem. A so-called (with slight abuse) "…xed point" obtains. Sub-Section 3.3 derives from this …xed point an equilibrium of the initial economy. In particular, in the symmetric information case, the following proof, which de…nes budget sets symmetrically across agents, suggests an alternative to Cass’.
3.1 An auxiliary compact economy with modi…ed budget sets
Using the notations of sub-Section 2.3, we let Q := fq 2 Z : kqk 6 1g
and de…ne the following sets, for every(i; p; q) 2 I P Q:
B1
i(p; q) := f (x; z) 2 Xi Zi: p0 (x0 ei0) + q z +Ps2S sps(xs eis)6 1;
ps(xs eis)6 V (s) z; 8s 2 S; and psi (xs eis)6 V (s) z; 8s 2 SinS g;
A(p; q) := f [(xi; zi)] 2 i2I Bi(p; q + q) : Pi2I(xis eis) = 0; 8s 2 S0; (Pi2I zi) 2 Zog.
The latter set meets the following boundary condition:
Lemma 19r > 0 : 8(p; q) 2 P Q; 8 [(xi; zi)] 2 A(p; q); Pi2I(kxik + kzik) < r
Proof : See the Appendix.
Along Lemma 1, for every(i; p := (ps); q) 2 I P Q, we letXi := fx 2 Xi: kxk 6 rg
and Zi := fz 2 Zi: kxk 6 rg, and de…ne the following convex compact sets:
B0
i(p; q) := f (x; z) 2 Xi Zi : p0 (x0 ei0) + q z +Ps2S sps(xs eis)6 (p;q);
ps(xs eis)6 V (s) z; 8s 2 S; and psi (xs eis)6 V (s) z; 8s 2 SinS g,
where (p;q):= 1 min(1; kpk + kqk), so thatBi0(p; q) B1i(p; q).
The auxiliary economy is alike that of Section 2, up to the change in budget
correspondences, fromBitoBi0, for eachi 2 I, which meet the following Claim 2, and
Claim 2 For every i 2 I, B0
i is upper semicontinuous.
Proof Leti 2 I be given. The correspondences B0
i is, as standard, upper
semicon-tinuous, for having a closed graph in a compact set. 3.2 The …xed-point-like argument
Budget sets were modi…ed in sub-section 3.1, so that their interiors be non-empty. This was required to prove the lower semi-continuity of the reaction correspondences
of Lemma 2, below. For every (p; q) 2 P Q, these interior budget sets are as follows:
B00
i(p; q) := f (x; z) 2 Xi Zi : p0 (x0 ei0) + q z +Ps2S sps(xs eis) < (p;q);
ps(xs eis) < V (s) z; 8s 2 S; and pis(xs eis) < V (s) z; 8s 2 SinS g, for every i 2 I.
Claim 3 The following Assertions hold, for each i 2 I:
(i) 8(p; q) 2 P Q, B00i(p; q) 6= ?;
(ii) the correspondence B00
i is lower semicontinuous.
Proof Let (p; q) 2 P Q and i 2 I be given. Assertion (i) From Assumptions
A2-A4 and the de…nition, we may always choose(x; z) 2 B00
i(p; q).
Assertion(ii) The convexity ofB00
i(p; q)yields, from Assertion(i),B0i(p; q) = B00i(p; q).
Then,B00
i is lower semicontinuous for having an open graph in a compact set.
We now introduce an agent representing markets (i = 0) and a reaction
corre-spondence, for each agent, on the convex compact set, := P Q ( i2IXi Zi), so as
to apply the GMC theorem. Thus, we let, for eachi 2 I and all := (p; q; [(xi; zi)]) 2 :
0( ) := f (p0; q0) 2 P Q : (q0 q) Pi2Izi+Ps2S0 [(p0s ps) Pi2I(xis eis)] > 0 g; i( ) := 8 > > < > > : Bi0(p; q) if (xi; zi) =2 Bi0(p; q) Bi00(p; q) \ Pi(xi) Zi if (xi; zi) 2 Bi0(p; q)
Lemma 2 For each i 2 I [ f0g, i is lower semicontinuous.
Proof See the Appendix.
Claim 4 There exists := (p ; q ; [(xi; zi)]) 2 , such that:
(i) 8(p; q) 2 P Q; (q q) Pi2Izi +Ps2S0 [(ps ps) Pi2I (xis eis)]> 0;
(ii) 8i 2 I; (xi; zi) 2 B0i(p ; q ) and Bi00(p ; q ) \ Pi(xi) Zi = ?.
Proof Quoting Gale-Mas-Colell (1975, 1979): “Given X = m
i=1Xi, where Xi is
a non-empty compact convex subset of Rn, let '
i : X ! Xi be m convex (possibly
empty) valued correspondences, which are lower semicontinuous. Then, there exists
x in X such that for each i either xi2 'i(x) or 'i(x) = ?”. The correspondences i,
for each i 2 I [ f0g, meet all conditions of the above theorem and yield Claim 4.
3.3 An equilibrium of the economy E
The above …xed point, , meets the following properties, proving Theorem 1:
Claim 5 Given := (p ; q ; [(xi; zi)] ) 2 , along Claim 4, the following holds:
(i) Pi2I zi = 0 and Pi2I (xis eis) = 0, 8s 2 S0;
(ii)for every i 2 I; (xi; zi) 2 Bi0(p ; q ) and Bi0(p ; q ) \ Pi(xi) Zi = ?;
(iii)there exist (zi) 2 RJ I and q 2 N A( ), such that (p ; q; [(xi; zi)] ) is an
equili-brium of the economy E and p 2 RL S++ 0.
Proof Assertion(i)We show, …rst, thatPi2I zi = 0(where, for eachi 2 I,zi is
the orthogonal projection ofzi onZ ). Assume, by contraposition, thatPi2I zi 6= 0.
Then, from Claim 4-(i), the relations q Pi2I zi = q Pi2I zi > 0 and (p ;q ) = 0
hold. Moreover, from Claim 4-(i), the relations06Pi2I ps (xis eis)hold, for every
s 2 S0. From Claim 4-(ii), the relationsp0 (xi0 ei0) + q zi +
P
s2S sps (xis eis)6 0
0 < p0 Pi2I (xi0 ei0) + q Pi2I zi +
P
s2S s Pi2I ps (xis eis)6 0.
This contradiction proves that Pi2I zi = 0.
Similarly, from Claim 4-(i), ps Pi2I (xis eis) > 0 holds, for every s 2 S0, and
p0 Pi2I (xi0 ei0)+Ps2S sPi2I ps(xis eis) > 0holds whenever(Pi2I (xis eis))s2S0 6= 0.
Assume, by contraposition, thatPi2I(xis eis) 6= 0, for somes 2 S0. Then, from Claim
4, the relation (p ;q )= 0 and the following budget constraints hold for each i 2 I:
p0 (xi0 ei0) + q zi +
P
s2S sps(xis eis)6 0.
Summing them up yields, from above:
0 < p0 Pi2I (xi0 ei0) +Ps2S s Pi2I ps (xis eis)6 0.
This contradiction proves that Pi2I (xis eis) = 0, for every s 2 S0.
Assertion(ii). Leti 2 Ibe given. From Claim 4-(ii), we need only show: B0
i(p ; q )\
Pi(xi) Zi = ?. Assume, by contraposition, there exists(xi; zi) 2 B0i(p ; q ) \ Pi(xi) Zi.
From Claim 3, there exists (x0
i; zi0) 2 Bi00(p ; q ) Bi0(p ; q ). By construction, the
relations (xni; zni) := [1n(x0i; zi0) + (1 n1)(xi; zi)] 2 B00i(p ; q ) hold, for every n 2 N. From
Assumption A3, the relation (xNi ; ziN) 2 Pi(xi) Zi also holds, for N 2 Nbig enough,
which implies: (xN
i ; ziN) 2 Bi00(p ; q ) \ Pi(xi) Zi. The latter contradicts Claim 4-(ii).
Assertion(iii)The relation ps2 RL S++ 0 is standard from Assertions(i)-(ii)and
As-sumptions A1-A2. From Assertions (i)-(ii)and Assumption A1, agents’budget
con-straints hold with equality. Then, from Assertion(i), the relations P
i2I (xis eis) = 0 ,
for s 2 S, yield: 0 =Pi2I ps(xis eis) =Pi2IV (s) zi = V (s)
P
i2I zi, for every s 2 S.
Referring to the notations of sub-Section 2.3, the latter relations are written
(Pi2I zi) 2 Zo, whereas, from Assertion (i), Pi2I zi = 0. It follows that
P
(Pi2I z 1
i ) 2 Zo. Letq := q + q := (q +
P
s2S sV (s)) 2 N A( ) be a given. From above,
we set as given(zo
i) 2 i2IZio, such that
P
i2I(zi zoi) = 0, and denotezi= (zi zio), for
eachi 2 I. From the de…nitions, the following relations hold: q zi= q zi = q zi +q zi
and V (s) zi= V (s) zi, for each (i; s) 2 I Si.
For eachi 2 I, the satiated budget constraints of(xi; zi) 2 B0
i(p ; q )in statess 2 S
imply: Ps2S s ps (xis eis) = q zi. At the …rst period, the same constraints are
written, for each i 2 I and from above: p0(xi0 ei0) + q zi +
P
s2S s ps (xis eis) =
p0(xi0 ei0) + q zi= (p ;q ). Summing up the latter relations (for i 2 I) yields, from
Assertion(i): 0 =Pi2I p0(xi0 ei0)+q Pi2Izi+
P
s2S s Pi2I ps (xis eis) = #I: (p ;q ).
Then, all above relations, Assertion (ii) and the de…nitions of q 2 N A( ) and of
buget sets (namely, B0
i(p ; q ) and Bi(p ; q)) imply: (xi; zi) 2 Bi(p ; q), for each i 2 I,
and [(xi; zi)] 2 A(p ; q ). The latter implies, from Lemma 1: Pi2I(kxik + kzik) < r.
We now let i 2 I be given and show that (xi; zi) is optimal in Bi(p ; q). If not,
there exists (xi; zi0) 2 Bi(p ; q) \ Pi(xi) RJ. From the de…nitions of q, Zio,Zi, from
As-sumption A3 and above, we may assume that (xi; zi0) 2 Bi0(p ; q ) \ Pi(xi) Zi, which
contradicts Assertion (ii). Thus, we have shown that Conditions (a)-(b)-(c) of
De…n-ition 1 hold for the collection, C := (p ; q; [(xi; zi)]) 2 P N A( ) ( i2IBi(p ; q)).
Appendix
LetA(p; q) := f [(xi; zi)] 2 i2I Bi(p; q +q) : Pi2I(xis eis) = 0; 8s 2 S0; (Pi2I zi) 2 Zog,
for every (p; q) 2 P Q. These sets are bounded as follows:
Lemma 19r > 0 : 8(p; q) 2 P Q; 8 [(xi; zi)] 2 A(p; q); Pi2I(kxik + kzik) < r
Proof Let =Pi2Ikeik,(p; q) 2 P Qand[(xi; zi)] 2 A(p; q)be given. The relations
of A(p; q). From the fact that there exists > 0, such that pi
s 2 [ ; +1[L, for every
forecast, (i; s) 2 I SinS, it su¢ ces to prove the following Assertion:
9r0 > 0; 8(p; q) 2 P Q; 8 [(xi; zi)] 2 A(p; q); P
i2I kzik < r0.
Assume, by contraposition, that, for every k 2 N, there exist (pk; qk) 2 P Q
and [(xki; zki)] 2 A(pk; qk), such that kzkk := Pi2I kzikk := k > k. For every k 2 N, let
z0k:= zk=
k:= (zik= k). The bounded sequence,fz0kg, may be assumed to converge in a
closed set, say toz := (zi) 2 i2IZi, such thatkzk = 1. The relations[(xki; zik)] 2 A(pk; qk)
hold, for every k 2 N, and imply, for every(i; s; k) 2 I Si N:
V (s) zk
i > , hence, V (s) zi0k> =k and, in the limit, V (s) zi> 0, for each s 2 Si;
P
i2I zi0k2 Zo, hence,
P
i2I zi 2 Zo.
LetPi2I zi=Pi2I zoi for some (zoi) 2 i2IZio, and(zi) := (zi) (zio) be given. The
following relations hold from above: V (s) zi > 0, for all(i; s) 2 I SiandPi2Izi = 0.
The latter relations imply, from Cornet-De Boisde¤re (2002, p. 401): (zi)2 i2IZio,
that is, z := (zi) = 0. This contradicts the relation kzk = 1and proves Lemma 1.
Lemma 2 For each i 2 I [ f0g, i is lower semicontinuous.
Proof The correspondences 0 is lower semicontinuous for having an open graph.
We now seti 2 I and := (p; q; [(xi; zi)]) 2 as given.
Assume that (xi; zi) =2 Bi0(p; q). Then, i( ) = Bi0(p; q).
Let V be an open set in Xi Zi, such that V \ B0
i(p; q) 6= ?. It follows from the
convexity ofB0
?. From Claim 3, there exists a neighborhood U of (p; q), such that V \ B0
i(p0; q0)
V \ B00
i(p0; q0) 6= ?, for every (p0; q0) 2 U.
SinceB0
i(p; q)is nonempty, closed, convex in the compact setXi Zi, there exist
open setsV1andV2inXi Zi, such that(xi; zi) 2 V1,Bi0(p; q) V2andV1\V2= ?. From
Claim 2, there exists a neighborhood U1 U of (p; q), such that Bi0(p0; q0) V2, for
every (p0; q0) 2 U1. Let W = U1 ( j
2IWj), where Wi:= V1, and Wj := Xj Zj, for each
j 2 Infig. Then,W is a neighborhood of , such that i( 0) = Bi0(p0; q0), andV \ i( 0) 6=
?, for every 02 W. Thus, iis lower semicontinuous at .
Assume that (xi; zi) 2 Bi0(p; q), i.e., i( ) = Bi00(p; q) \ Pi(x) Zi.
Lower semicontinuity results from the de…nition if i( ) = ?. Assume i( ) 6= ?.
We recall that Pi (from Assumption A3 ) is lower semicontinuous with open values
and that Bi00 has an open graph. As a corollary, the correspondence (p0; q0[(x0i; z0i)]) 2
7! B00i(p0; q0) \ Pi(xi0) Zi Bi0(p0; q0) is lower semicontinuous at . Then, from the
latter inclusions, i is lower semicontinuous at .
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