# Equilibrium with Asymmetric Information and Restricted Participation: The No-Arbitrage Characterization

Lionel de Boisdeffre

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Lionel de Boisdeffre. Equilibrium with Asymmetric Information and Restricted Participation: The No-Arbitrage Characterization. 2025. halshs-05132840

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CES

Centre d'Économie de la Sorbonne
UMR 8174

Equilibrium with Asymmetric Information and Restricted
Participation: The No-Arbitrage Characterization

Lionel De BOISDEFFRE

2025.14
EQUILIBRIUM WITH ASYMMETRIC INFORMATION AND RESTRICTED
PARTICIPATION : THE NO-ARBITRAGE CHARACTERIZATION

Lionel de Boisdeffre,<fnref n="1" />

(june 2025)

Abstract

The paper presents a pure exchange economy, where consumers may have asymmetric information, non-ordered preferences and a restricted access to purely financial markets. Under standard conditions, it shows that the two concepts of no-arbitrage price and equilibrium asset price coincide. Namely, any collection of state prices in realizable states is proved to support a sequential equilibrium. This result extends Cass (2006), De Boisdeffre (2007) and similar results of symmetric information, such as Martins-da-Rocha and Triki's (2005) or Cornet and Gopalan's (2010). Moreover, the definition and existence of the sequential equilibrium do not rely on Radner's (1972, 1979) rational expectation assumptions, which may be dropped.

**Key words:** incomplete markets, restricted participation, arbitrage, existence of
equilibrium, rational expectations, perfect foresight, asymmetric information.

**JEL Classification:** D52

<footnote n="1">¹ University of Paris 1 - Panthéon - Sorbonne, 106-112 Boulevard de l'Hôpital, 75013 Paris, France. Email: lionel.de.boisdeffre@wanadoo.fr</footnote>
# 1 Introduction

The current paper extends Cornet-De Boisdeffre's (2002) step concept of purely financial equilibrium with asymmetric information to non-ordered preferences and restricted participation, and refines its existence characterization. Generalizing Cass (2006), it shows that equilibrium exists, not only for *one* collection of state prices, as in De Boisdeffre (2007), but for *every* collection of state prices in realizable states.

With nominal assets, the literature shows an essential real indeterminacy of equi-
librium (see, e.g., Balasko-Cass, 1989, and Geanakoplos-Mas-Colell, 1989), while
Cass (2006) proves that the sets of equilibrium asset prices and no-arbitrage prices
coincide. Moreover, equilibrium exists in standard conditions (see, e.g., Radner,
1972; Duffie, 1987; Werner, 1985, and Werner, 1989, for non-ordered preferences).
With numéraire assets, Geanakoplos and Polemarchakis (1986) show that equilibria
exist and are generically locally unique, but do not assess Cass' price equivalence.
To my best knowledge, the first extension of Cass' (2006) equivalence to numéraire
assets and asymmetric information is provided in De Boisdeffre (2021).

With real or mixed assets, the existence of equilibrium is not guaranteed as shown
by Hart (1975). Hart's counterexample is based on the collapse of the span of assets'
payoffs, that occurs exceptionally at market-clearing prices. One response to that
problem is to show, along Duffie and Shafer (1985), that it is indeed exceptional, in
the sense that equilibria exist generically in payoffs and endowments. This argument
is used by Geanakoplos and Shafer (1990) and Bich and Cornet (2004, 2009), with
symmetric information, or De Boisdeffre (2021), with asymmetric information.

When agents have asymmetric or incomplete information, they seek to learn in-
formation from observing prices and markets. The traditional inference mechanism,
referred to as the ‘Radner inferences’, is the foundation for the Radner (1979) ‘raditional expectations equilibrium’ (REE), generalized by the so-called ‘common knowledge of rationality and market clearing’ (CKRMC) models. These include, among others, McAllister’s (1990), Dutta-Morris’ (1997) and Ben Porath-Heifetz’ (2011).

Rational expectations, along Radner (1979) or CKRMC models, assume that agents are aware of a correspondence (or a map) between private information signals and the equilibrium prices observed on markets. Following Radner’s (1979) seminal paper, an abundant literature studied the (generic) existence and the information revealed by equilibrium prices (see, e.g., Allen, 1981; Mischel, Polemarchakis and Siconolfi, 1990; Polemarchakis and Siconolfi, 1993; Pietra and Siconolfi, 1997 and 1998; or Citanna and Villanacci, 2000a and 2000b, to quote only a few). I drop this assumption to show, along Cornet and De Boisdeffre (2009), that agents observing markets with no price model may refine their information until financial markets preclude arbitrage. The relevance of the latter inferences is argued in Section 6.

To show the equivalence between no-arbitrage and equilibrium prices on purely financial markets with symmetric information, Cass (2006) uses a ‘trick’, by which one agent has a single Arrow-Debreu budget constraint at the first period, while the other agents are constrained as la Radner (1972), that is, in every state of nature. That ‘trick’ implies that the first agent has a full access to financial markets. Not surprinsingly, Balasko, Cass and Siconolfi (1990) show that the equivalence between no-arbitrage and equilibrium prices fails under restricted participation, in general.

However, Cornet and Gopalan (2010) provide a condition on the accessibility of assets’ payoffs by agents, which guarantees the latter equivalence under restricted participation. The current paper proposes an alternative - in my view milder - accessibility condition, under which the latter equivalence also holds. Namely, it shows
that any collection of state prices in realizable states supports sequential equilibria.
The latter papers are compared in Section 5, in the symmetric information case.

None of Radner’s (1972 and 1979) rational expectation assumptions are assumed
in the current model. The sequential equilibrium is proved to exist, even when
these standard assumptions are dropped. This outcome has an interest of its own.
For expositional purposes, I present a two-period model. Its extension to multiple
periods is standard and straightforward. But it is differed as requiring additional
nodes and notations into the model, which would hamper its simplicity and focus.

The paper is organized as follows: Section 2 presents the model. Section 3 studies its arbitrage properties. Section 4 proves its existence theorem. With reference to existing results of the literature, Section 5 presents the latter theorem when one agent has full information. Section 6 concludes. An Appendix proves Lemmas.

2 The model

Throughout, a pure-exchange economy is considered, with two periods, $t \in \{0, 1\}$,
and an uncertainty, at $t = 0$, upon which state of nature will randomly prevail
at $t = 1$. Agents exchange consumption goods, on spot markets of both periods,
and assets, on financial markets at $t = 0$, which transfer wealth across periods and
states. The economy is finite, in the sense that the sets, $I$, $S$, $L$ and $J$, respectively,
of consumers, states of nature, consumption goods and assets are all finite. The
observed state, at $t = 0$, is denoted by $s = 0$ and $\Sigma'$ denotes $\{0\} \cup \Sigma$, whenever $\Sigma \subset S$.

**2.1 Markets and information**

Agents consume or exchange (the same) consumption goods, $l \in L$, on spot
markets at both periods. At $t = 0$, each agent, $i \in I$, receives privately the correct
information that tomorrow's true state will be in a subset, $S_i$, of $S$. It is costlessly assumed that $S = \cup_{i \in I} S_i$. Hence, the pooled information set, $\underline{S} := \cap_{i \in I} S_i$, always contains the true state, and the relation $\underline{S} = S$ characterizes symmetric information.

Agents need not be aware of the economy's or other agents' characteristics. Their beliefs and characteristics are private. They need not have a price model a la Radner (1972, 1979) or CKRMC models. The current paper therefore drops both the perfect foresight assumption, of symmetric information, and the Radner's inferences, of asymmetric information. Whenever appropriate, agents update their beliefs from observing exchange opportunities on financial markets, along Cornet and De Boisdeffre's (2009) inferences. The latter, which lead to arbitrage-free markets, do not depend on agents' beliefs, preferences or anticipations of future prices (see below).

Non restrictively, $P := \{p \in \mathbb{R}_{++}^L : \|p\| \le 1\}$ is the set of admissible commodity prices on every spot market, which each agent is assumed to observe, at $t=0$, or to expect privately, in each state she anticipates. Only the first period price is observed, therefore common to all agents. The generic $i^{th}$ agent's price expectation is henceforth denoted by $p_i := (p_{is}) \in P^{S_i}$. This vector is fixed at $t=0$, and exogenous, as in the temporary equilibrium model. For every $(i,s) \in I \times S_i$, $p_{is}$ represents the (unique) anticipation of the spot price in state $s$ by agent $i$. Whenever the relation $p_{is} = p_{js}$ holds for every triple $(i,j,s) \in I \times I \times \underline{S}$, agents are said to share the same beliefs. This outcome may occur 'by chance', or with no exchange between agents.

Agents may operate financial transfers across states in $S'$ (actually in $\underline{S}'$) by exchanging, at $t=0$, finitely many nominal assets, $j \in J$, which pay off, at $t=1$, conditionally on the realization of the state. Assets' payoffs define a $S \times J$ matrix, $V$, whose generic row in a state $s \in S$, denoted by $V(s) := (V_s^j)_{j \in J} \in \mathbb{R}^J$, does not depend on prices. Thus, at asset price, $q \in \mathbb{R}^J$, agents may buy or sell portfolios
of assets, $z = (z_j) \in \mathbb{R}^J$, for $q \cdot z$ units of account at $t = 0$, against the promise of delivery of a flow, $V(s) \cdot z$, of conditional payoffs across states, $s \in S$. The exchange of assets may be restricted. For each $i \in I$, the portfolio set, $Y_i \subset \mathbb{R}^J$, or the $i^{th}$ agent's set of admissible portfolios, is henceforth given. Restrictions to trade are typically encountered on financial markets. Some may be institutional, such as short sale constraints, transaction costs, bid-ask spreads or ratio regulations (see Elsinger and Summer's 2001 overview). Others may be behavioural or informational.

## 2.2 The consumer's behaviour and concept of equilibrium

Each agent, $i \in I$, receives an endowment, $e_i := (e_{is})$, granting the commodity bundles, $e_{i0} \in \mathbb{R}_+^L$ at $t = 0$, and $e_{is} \in \mathbb{R}_+^L$, in each state, $s \in S_i$, if it prevails. Given the observed market prices at $t = 0$, namely, $p \in P$, for goods, and $q \in \mathbb{R}^J$, for assets, and her expectation of future prices, $p_i = (p_{is}) \in P^{S_i}$, the generic $i^{th}$ agent's consumption set is $X_i := \mathbb{R}_+^{L \times S_i'}$, and her budget set is defined as follows:

$$B_i(p, q) := \{(x := (x_s), z) \in X_i \times Y_i : p \cdot (x_0 - e_{i0}) \le -q \cdot z \text{ and } p_{is} \cdot (x_s - e_{is}) \le V(s) \cdot z, \forall s \in S_i\}.$$

Each consumer, $i \in I$, is endowed with a complete preordering, $\prec_i$, on consumptions, representing her preferences. Her strict preferences, $\prec_i$, are represented, for all $x \in X_i$, by the set, $P_i(x) := \{ y \in X_i : x \prec_i y \}$, of consumptions that she strictly prefers to $x$. To simplify presentation, the latter set only depends on the agent's own consumption (and not on other agents' consumptions).

The economy is denoted by $\mathcal{E}_{(p_i)} = \{(I, S, L, J), V, (S_i), (p_i), (e_i), (\prec_i)\}$. Henceforth, the structural characteristics, $\{(I, S, L, J), V, (S_i), (e_i), (\prec_i)\}$, of the economy are fixed and always referred to. Only agents' exogenous expectations, $(p_i) \in P^{S_i \times I}$, may vary and distinguish economies. When agents share the same beliefs, a sequential equilibrium is defined as a market-clearing temporary equilibrium, namely:
**Definition 1** Given the expectations, $(p_i) \in P^{S_i \times I}$, of agents who share the same beliefs (i.e., $p_{is} = p_{js}$ whenever $(i, j, s) \in I \times I \times \underline{S}$), a collection of prices, $(p, q) \in P \times \mathbb{R}^J$, and decisions, $(x_i := (x_{is}), z_i) \in B_i(p, q)$, defined for each $i \in I$, is a (sequential) equilibrium of the economy, $\mathcal{E}_{(p_i)}$, if the following Assertions hold:

$$ (a) \quad \forall i \in I, (x_i, z_i) \in B_i(p, q) \text{ and } P_i(x_i) \times \mathbb{R}^J \cap B_i(p, q) = \emptyset; $$

$$ (b) \quad \forall s \in \underline{S}', \sum_{i \in I} (x_{is} - e_{is}) = 0; $$

$$ (c) \quad \sum_{i \in I} z_i = 0. $$

**Remark 1** Since no state $s \notin \underline{S} := \cap_{i \in I} S_i$ may prevail, no spot market may open or clear in any state $s \in S \setminus \underline{S}$. This explains Condition (b) of Definition 1, and why agents' expectations are all exogenous. In particular, a spot price in a state $s \in S \setminus \underline{S}$ is subjective and never observed. Since agents' exogenous forecasts are ex ante given, Definition 1 does not need to assume that common expectations are self-fulfilling. This outcome is natural and contradicted by no specification in a pure exchange economy, such as $\mathcal{E}_{(p_i)}$. It lets agents' decisions coincide ex ante and ex post.

For every $i \in I$, the following vector spaces are defined and henceforth given:

* $Z_i^o := \{z \in \mathbb{R}^J : V(s) \cdot z = 0, \forall s \in S_i\}$, and its orthogonal, $Z_i = \sum_{s \in S_i} \mathbb{R}V(s)$;

* $\underline{Z}^o := \{z \in \mathbb{R}^J : V(s) \cdot z = 0, \forall s \in \underline{S}\}$, and its orthogonal, $\underline{Z} = \sum_{s \in \underline{S}} \mathbb{R}V(s)$;

* $Z^o := \sum_{s \in S_i} Z_i^o$, its orthogonal complement, $Z = \cap_{i \in I} Z_i$, and $Z^* := Z \cap \underline{Z}^o$.

To the generic $i^{th}$ agent, $Z_i^o$ is the subjective space of redundant (or worthless) portfolios. The latter is always included in the actual set of redundant (or worthless) portfolios, $\underline{Z}^o$. All other vector spaces above are defined relative to the former and will serve in the proof of Section 4. Similarly, $\pi: \mathbb{R}^J \to \underline{Z}$ will henceforth denote the orthogonal projection of $\mathbb{R}^J$ on $\underline{Z}$, and serve later.

The economy, $\mathcal{E}_{(p_i)}$, is called 'standard' under the following Assumptions:
• **A1 (strict monotonicity):** $\forall (i,x,y) \in I \times X_i \times X_i, (x \le y, x \ne y) \Rightarrow (x <_i y)$;

• **A2 (strong survival):** $\forall i \in I, e_i \in \mathbb{R}_{++}^{L \times S'_i}$;

• **A3 for every i ∈ I, <_i is lower semi-continuous, convex-open-valued and such that** $x <_i x + \lambda(y - x)$, for every $(x, y, \lambda) \in X_i \times P_i(x) \times ]0, 1]$;

• **A4 for every i ∈ I, Y_i is closed, convex and contains zero;**

• **A5** $A := \{(z_i) \in \mathbb{R}^{J \times I} : \sum_{i \in I} z_i \in Z^o, V(s_i) z_i \ge 0, \forall (i, s_i) \in I \times S_i\} \subset \times_{i \in I} Y_i$ and $Z^* \subset \cap_{i \in I} \text{cone}(Y_i)$;

• **A6 (riskless asset):** $\forall i \in I, \exists z_i^+ \in Y_i : V(s) \cdot z_i^+ > 0, \forall s \in S$.

**Remark 2** Assumptions A1-A2-A3 are standard. Assumption A1, retained for convenience, could be weakened to the non-satiation of agents in any state. Assumption A4, made, e.g., by Siconolfi (1989) and Cornet-Gopalan (2010), is a minimal condition of most portfolio choice models with financial constraints (see Elsinger and Summer, 2001). Assumption A5 is technical, and dropped when one agent has full information (see Section 5). Under symmetric information, it would typically lead agents (from the implied relations $Z_i^o = \underline{Z}_i^o \subset Y_i$, for each $i \in I$) to be aware of, and eliminate, redundant portfolios. It is then weakened to a standard condition of restricted participations models, namely: $\underline{Z}_i^o \subset \cup_{i \in I} Y_i$. The latter condition always holds if redundant assets are eliminated ($\underline{Z}_i^o = \{0\}$). Why is Assumption A5 met? Because financial intermediaries, possibly better informed, typically emit and propose riskless clearing-market claims to agents when they exist (possibly freely), and other limited claims against their idiosyncratic risks. When it is met, agents typically refine their information from observing exchange opportunities on financial markets, until no arbitrage remains (see Claim 2). Along Assumption A6, agents have access to a safe saving instrument, which is always the case on actual markets.
# 3 The model's arbitrage concepts and properties

This Section defines no-arbitrage prices and shows their existence.

**Definition 2** For each $i \in I$, $As(Y_i) := \{z \in \mathbb{R}^J : \{z\} + Y_i \subset Y_i\}$ is the asymptotic cone of $Y_i$. A price $q \in \mathbb{R}^J$ is a no-arbitrage price if the following equivalent Conditions hold:

(i) $\forall (i,z) \in I \times As(Y_i), (-q \cdot z \ge 0 \text{ and } V(s) \cdot z \ge 0, \forall s \in S_i) \Rightarrow (z \in Z_i^o \text{ and } q \cdot z = 0)$

(ii) $\forall i \in I, \exists \lambda_i := (\lambda_{is}) \in \mathbb{R}_{++}^{S_i}, q \cdot z_i = \sum_{s \in S_i} \lambda_{is} V(s) \cdot z_i, \forall z_i \in As(Y_i)$.

Let $\mathcal{N}A$ be the set of no-arbitrage prices and $\mathcal{N}A(\lambda) := \{ q \in \mathcal{N}A : \pi(q) = \sum_{s \in \mathcal{S}} \lambda_s V(s) \}$, for every $\lambda := (\lambda_s) \in \mathbb{R}_{++}^{\mathcal{S}}$, be given. Financial markets, are said to be arbitrage-free if $\mathcal{N}A$ is non-empty. Given $q \in \mathcal{N}A$, scalars $\lambda_{is} \in \mathbb{R}_{++}$, for $(i,s) \in I \times S_i$, which meet the above Condition (ii), are called supporting state prices of price $q$. Scalars $\lambda_s \in \mathbb{R}_{++}$, for $s \in \mathcal{S}$, are called supporting state prices of the set $\mathcal{N}A((\lambda_s))$ and of its elements.
A price $q \in \mathcal{N}A$ is said to be frictionless, if the following equivalent Conditions hold:

(iii) $\forall (i,z) \in I \times \mathbb{R}^J, (-q \cdot z \ge 0 \text{ and } V(s) \cdot z \ge 0, \forall s \in S_i) \Rightarrow (z \in Z_i^o \text{ and } q \cdot z = 0)$

(iv) $\forall i \in I, \exists \lambda_i := (\lambda_{is}) \in \mathbb{R}_{++}^{S_i}, q \cdot z_i = \sum_{s \in S_i} \lambda_{is} V(s) \cdot z_i, \forall z_i \in \mathbb{R}^J$.

Markets are frictionless arbitrage-free if there exists a frictionless no-arbitrage price.

**Remark 3** The equivalences between Conditions (i) and (ii), and Conditions (iii) and (iv), of the above Definition 2 follow from a standard separation argument. A frictionless no-arbitrage price is akin to a no-arbitrage price on frictionless financial markets, that is, where agents' participation is unrestricted ($Y_i = \mathbb{R}^J$, for each $i \in I$).

Martins-da-Rocha and Triki (2005) provide a sufficient condition for no-arbitrage prices to be frictionless. They require markets to be collectively frictionless, in the sense that $\cup_{i \in I} V Y_i = \text{Im } V$. A standard economy yields another interesting result:

**Claim 1** In a standard economy, $\mathcal{E}_{(p_i)}$, financial markets are arbitrage-free if and only if they are frictionless arbitrage-free.
**Proof** Let a standard economy, $\mathcal{E}_{(p_i)}$, be given. If financial markets are frictionless arbitrage-free, they are obviously arbitrage-free. To show the converse, the economy, $\mathcal{E}_{(p_i)}$, is henceforth assumed to have arbitrage-free markets. From Assumptions A4-A5, the relation $A := \{(z_i) \in \mathbb{R}^{J \times I} : \sum_{i \in I} z_i \in Z^o, V(s_i)z_i \ge 0, \forall (i, s_i) \in I \times S_i\} \subset \times_{i \in I} Y_i$ holds and implies, since $A$ is a cone: $\times_{i \in I} Z_i^o \subset A \subset \times_{i \in I} As(Y_i)$.

Let $q \in \mathcal{NA}$ be given. Then, the following relation (I) holds:

$$((z_i) \in \times_{i \in I} As(Y_i), \sum_{i \in I} z_i = 0 \text{ and } V(s_i) \cdot z_i \ge 0, \forall (i, s_i) \in I \times S_i) \Rightarrow ((z_i) \in \times_{i \in I} Z_i^o)$$

Indeed, assume by contraposition that there exists $(z_i) \in \times_{i \in I} As(Y_i)$, such that $\sum_{i \in I} z_i = 0$ and $V(s_i) \cdot z_i \ge 0$, for every $(i, s_i) \in I \times S_i$, with at least one strict inequality. Then, from Condition (ii) of Definition 2 (applied to $q \in \mathcal{NA}$), the relation $q \cdot \sum_{i \in I} z_i > 0$ holds, with $\sum_{i \in I} z_i = 0$. This contradiction proves relation (I), which implies the following relation (II):

$$((z_i) \in \mathbb{R}^{J \times I}, \sum_{i \in I} z_i = 0 \text{ and } V(s_i) \cdot z_i \ge 0, \forall (i, s_i) \in I \times S_i) \Rightarrow ((z_i) \in \times_{i \in I} Z_i^o)$$

Indeed, assume, by contraposition, that there exists $(z_i) \in \times_{i \in I} \mathbb{R}^{J \times I} \setminus \times_{i \in I} Z_i^o$, such that $\sum_{i \in I} z_i = 0$ and $V(s_i) \cdot z_i \ge 0$, for every $(i, s_i) \in I \times S_i$. Then, the definition of $A \subset \times_{i \in I} As(Y_i)$ implies: $(z_i) \in \times_{i \in I} As(Y_i)$. It follows from relation (I) above, that $(z_i) \in \times_{i \in I} Z_i^o$, which contradicts the opposite relation, $(z_i) \in \times_{i \in I} \mathbb{R}^{J \times I} \setminus \times_{i \in I} Z_i^o$, assumed above. This contradiction proves that relation (II) holds, which characterizes frictionless arbitrage-free markets, from Cornet and De Boisdeffre (2002, p. 401). □

**Remark 4** Claim 1 does not imply that all no-arbitrage prices are frictionless, in a standard economy. Consider a standard economy with two agents, three states and two assets, such that: $I = \{1, 2\}$, $S = \{1, 2, 3\}$, $S_1 = \{1, 2\}$, $S_2 = S$, $V(1) = (1, 1)$, $V(2) = (0, 1)$; $V(3) = (-1, 1)$, $Y_1 = \{0\} \times \mathbb{R}$ and $Y_2 = \mathbb{R}^2$. Then, obviously, $p = (1, 4)$ and $p = (0, 3)$ are, respectively, frictionless and non-frictionless no-arbitrage prices.
Though its sets of no-arbitrage and frictionless no-arbitrage prices differ, in general, any standard economy can be assumed to have arbitrage-free financial markets. This property generalizes Cornet and De Boisdeffre's (2009) to the economy $\mathcal{E}_{(p_i)}$:

**Claim 2** In a standard economy, the information structure, $(S_i)$, admits a coarsest refinement, $(S_i^*)$, which makes financial markets arbitrage-free. Agents with no price model can infer the refinement $(S_i^*)$ by ruling out unlimited arbitrage opportunities.

**Proof** Let a standard economy, $\mathcal{E}_{(p_i)}$, be given, whose information structure is denoted by $(S_i^0) = (S_i)$. Assume, that markets fail to be arbitrage-free. Then, from the proof of the above Claim 1, the following relation (I) fails:

$$((z_i) \in \times_{i \in I} \text{As}(Y_i), \sum_{i \in I} z_i = 0 \text{ and } V(s_i) \cdot z_i \ge 0, \forall (i, s_i) \in I \times S_i) \Rightarrow ((z_i) \in \times_{i \in I} Z_i^0)$$

Let $B^0 = \{(z_i) \in \times_{i \in I} \text{As}(Y_i) : \sum_{i \in I} z_i = 0 \text{ and } V(s_i) \cdot z_i \ge 0, \forall (i, s_i) \in I \times S_i^0\}$ be a non-empty cone. For every $j \in I$, let $\bar{S}_j^1 = \{s \in S_j^0 : \exists (z_i) \in B^0, V(s) \cdot z_j > 0\}$ be the $j^{th}$ agent's (possibly empty) set of arbitrage states. Since relation (I) fails, $\cup_{j \in I} \bar{S}_j^1$ is non-empty. Along Assumption A5, financial intermediation permits to implement exchanges, which reveal to agents their arbitrage states, when they exist. Those agents should rightly infer that such arbitrage states cannot prevail. As a result, agents infer the refinement $(S_i^1) = (S_i^0 \setminus \bar{S}_i^1)$ of their initial information structure, $(S_i^0) = (S_i)$. This refinement process proceeds as long as financial markets fail to preclude arbitrage.

After the initial sets above, we define by induction, on $n \in \mathbb{N}$, the sets: $(S_i^n) = (S_i^{n-1} \setminus \bar{S}_i^n)$, $B^n = \{(z_i) \in \times_{i \in I} \text{As}(Y_i) : \sum_{i \in I} z_i = 0 \text{ and } V(s_i) \cdot z_i \ge 0, \forall (i, s_i) \in I \times S_i^n\}$ and $\bar{S}_j^{n+1} = \{s \in S_j^n : \exists (z_i) \in B^n, V(s) \cdot z_j > 0\}$. The decreasing sequence, $\{(S_i^n)\}_{n \in \mathbb{N}}$, of finite sets is stationary and converges to a minimal element, say $(S_i^*)$. Since the refinement process stops at $(S_i^*)$, the following relation (III) holds from above:
$$((z_i) \in \times_{i \in I} As(Y_i), \sum_{i \in I} z_i = 0, V(s_i) \cdot z_i \ge 0, \forall (i, s_i) \in I \times S_i^*) \Rightarrow (V(s_i) \cdot z_i = 0, \forall (i, s_i) \in I \times S_i^*)$$

By the same token as for the proof of Claim 1, relation (III) implies the following:

$$((z_i) \in \mathbb{R}^{J \times I}, \sum_{i \in I} z_i = 0, V(s_i) \cdot z_i \ge 0, \forall (i, s_i) \in I \times S_i^*) \Rightarrow (V(s_i) \cdot z_i = 0, \forall (i, s_i) \in I \times S_i^*)$$

which characterizes frictionless arbitrage-free markets, from Cornet-De Boisdeffre (2002, p. 401). Moreover, by construction, the refinement $(S_i^*)$ inferred by rational agents is the coarsest refinement of $(S_i)$ making financial markets arbitrage-free. Thus, arbitrage-free markets at the outset reveal no information and let $(S_i^*) = (S_i)$. □

Henceforth, markets are assumed, at no cost from Claim 2, to be arbitrage-free. This condition neither depends on the observed or the expected prices. Once markets are arbitrage-free in the economy, $\mathcal{E}_{(p_i)}$, for some $(p_i) \in P^{S_i \times I}$, they are (frictionless) arbitrage-free in all economies $\mathcal{E}_{(p'_i)}$, for $(p'_i) \in P^{S_i \times I}$. Then, any collection of state prices in realizable states supports no-arbitrage prices, along Claim 3:

**Claim 3** For every $\lambda \in \mathbb{R}_{++}^{\mathcal{S}}$, the set $\mathcal{NA}(\lambda)$ of Definition 2 is non-empty.

**Proof** Let $\lambda := (\lambda_s) \in \mathbb{R}_{++}^{\mathcal{S}}$ be given. Since markets are arbitrage-free, there exists $\bar{q} \in \mathcal{NA}$ and $(\lambda_{is}) \in \times_{i \in I} \mathbb{R}_{++}^{S_i}$, such that $\bar{q} = \sum_{s \in S_i} \lambda_{is} V(s)$, for each $i \in I$, which implies $\bar{q} \cdot z = 0$ whenever $z \in Z^o$. For each $i \in I$, let $q_i \in \underline{Z}$ be defined by $q_i = \sum_{s \in S_i \setminus \underline{S}} \lambda_{is} \pi(V(s))$, if $S_i \ne \underline{S}$, and $q_i = 0$ otherwise.

Since $q_i \in \underline{Z} = \sum_{s \in \underline{S}} \mathbb{R}V(s)$, for every $i \in I$, there is a vector $(\mu_{is}) \in \mathbb{R}^{\mathcal{S}}$, such that $q_i = \sum_{s \in \underline{S}} \mu_{is} V(s)$. For $N \in \mathbb{N}$ large enough, the relations $|\mu_{is} / N| < \lambda_s$ hold, for every pair $(i, s) \in I \times \underline{S}$. Then, let $\gamma_i := (\gamma_{is}) \in \mathbb{R}_{++}^{S_i}$ be defined, for each $i \in I$, by $\gamma_{is} = \lambda_{is} / N$, for every $s \in S_i \setminus \underline{S}$, and $\gamma_{is} = \lambda_s - \mu_{is} / N$, for every $s \in \underline{S}$. By construction, the state prices, $\gamma_i := (\gamma_{is}) \in \mathbb{R}_{++}^{S_i}$, defined for all $i \in I$, support some no-arbitrage price, $q \in \mathcal{NA}(\lambda)$. □
# 4 The existence Theorem

Arbitrary vectors, $\lambda = (\lambda_s) \in \mathbb{R}_{++}^{\mathcal{S}}$ and $\bar{p}_i \in P^{S_i \setminus \mathcal{S}}$, for $i \in \{i \in I : S_i \neq \mathcal{S}\}$, are henceforth set as given. Under Assumptions A1 to A6, the following Theorem 1 guarantees the existence of a sequential equilibrium, consistent with the state prices, $(\lambda_s)$, and the subjective expectations $(\bar{p}_i)$:

**Theorem 1** Let arbitray vectors of state prices, $\lambda := (\lambda_s) \in \mathbb{R}_{++}^{\mathcal{S}}$, and subjective expectations, $\bar{p}_i := (\bar{p}_{is}) \in P^{S_i \setminus \mathcal{S}}$, of unfully informed agents, $i \in I^* := \{i \in I : S_i \neq \mathcal{S}\}$, be given. Under Assumptions A1 to A6, there exists a collection of market prices, $(p, p^1 = (p_s^1), q) \in P \times P^{\mathcal{S}} \times \mathcal{NA}(\lambda)$, and decisions $(x_i, z_i) \in B_i(p, q)$, defined for each $i \in I$, such that $(p, q, [(x_i, z_i)])$ is a sequential equilibrium of the economy $\mathcal{E}_{(p_i)}$, defined by $p_{is} = \bar{p}_{is}$, for every $(i, s) \in I^* \times S_i \setminus \mathcal{S}$, and $p_{is} = p_s^1$, for every $(i, s) \in I \times \mathcal{S}$.

The proof applies Gale-Mas-Colell's (1975, 1979) fixed-point-like theorem to lower semi-continuous correspondences representing agents' behaviours in an auxiliary compact economy, and derives the equilibrium from a pseudo fixed-point.

## 4.1 An auxiliary compact economy with modified budget sets

Let $Q := \{q \in Z^* : \|q\| \le 1\}$ and $P^* := \{p \in \mathbb{R}^L : \|p\| \le 1\}^{\mathcal{S}'}$ be given convex compact sets. From Assumptions A4-A5, for every $i \in I$, the portfolio set $Y_i$ contains $Z_i^o$ and its orthogonal projection on $Z_i$ (hence, equal to $Y_i \cap Z_i$). Therefore, the following convex sets are well defined for every $(p, p^1 = (p_s^1)_{s \in \mathcal{S}}) \in P^*$ and every $q \in Q$:

$$ 
\begin{aligned}
B_i^1(p, p^1, q) &:= \{ (x, z) \in X_i \times Y_i \cap Z_i : p \cdot (x_0 - e_{i0}) + q \cdot z + \sum_{s \in \mathcal{S}} \lambda_s p_s^1 \cdot (x_s - e_{is}) \le 1, \\
& \phantom{:=} p_s^1 \cdot (x_s - e_{is}) \le V(s) \cdot z, \forall s \in \mathcal{S}, \text{ and } \bar{p}_{is} \cdot (x_s - e_{is}) \le V(s) \cdot z, \forall s \in S_i \setminus \mathcal{S} \}; \\
\mathcal{A}(p, p^1, q) &:= \{ [(x_i, z_i)] \in \times_{i \in I} B_i^1(p, p^1, q) : \sum_{i \in I} (x_{is} - e_{is}) = 0, \forall s \in \mathcal{S}' \text{ and } \sum_{i \in I} z_i \in Z_i^o \}.
\end{aligned}
$$

**Lemma 1** $\exists r > 0 : \forall ((p, p^1), q) \in P^* \times Q, \forall [(x_i, z_i)] \in \mathcal{A}(p, p^1, q), \sum_{i \in I} (\|x_i\| + \|z_i\|) < r$
**Proof** : See the Appendix. □

Along Lemma 1, let $X_i^* := \{(x_s) \in X_i : \|x\| \le r\}$ and $Z_i^* := \{z \in Y_i \cap Z_i : \|z\| \le r\}$, and define, for every $(i, (p, p^1 := (p_s^1)), q) \in I \times P^* \times Q$, the following convex bounded sets:

$$ B'_i(p, p^1, q) := \{ (x, z) \in X_i^* \times Z_i^* : p \cdot (x_0 - e_{i0}) + q \cdot z + \sum_{s \in \mathcal{S}} \lambda_s p_s^1 \cdot (x_s - e_{is}) \le \gamma_{(p, p^1, q)} \\ p_s^1 \cdot (x_s - e_{is}) \le V(s) \cdot z, \forall s \in \mathcal{S}, \text{ and } \bar{p}_{is} \cdot (x_s - e_{is}) \le V(s) \cdot z, \forall s \in S_i \setminus \mathcal{S} \}; $$

$$ B_i''(p, p^1, q) := \{ (x, z) \in X_i^* \times Z_i^* : p \cdot (x_0 - e_{i0}) + q \cdot z + \sum_{s \in \mathcal{S}} \lambda_s p_s^1 \cdot (x_s - e_{is}) < \gamma_{(p, q)} \text{ and } p_s^1 \cdot (x_s - e_{is}) < V(s) \cdot z, \forall s \in \mathcal{S}, \text{ and } \bar{p}_{is} \cdot (x_s - e_{is}) < V(s) \cdot z, \forall s \in S_i \setminus \mathcal{S} \}; $$

where $\gamma_{(p,p^1,q)} = 1 - \min(1, \|p\| + \|p^1\| + \|q\|)$.

**Claim 4** For every $i \in I$, $B'_i$ is upper semicontinuous.

**Proof** Let $i \in I$ be given. The correspondences $B'_i$ is, as standard, upper semi-continuous, for having a closed graph in a compact set. □

**Claim 5** The following Assertions hold, for each $i \in I$:

(i) $\forall (p, p^1) \in P^*, \forall q \in Q, B_i''(p, p^1, q) \ne \emptyset$;

(ii) the correspondence $B_i''$ is lower semicontinuous (l.s.c.).

**Proof**

**Assertion (i):** let $(i, (p, p^1), q) \in I \times P^* \times Q$ be given. Assume, that $(p, p^1) \ne 0$ or that $\gamma_{(p,p^1,q)} = 1$. Then, from Assumptions A2-A4-A6, there exists $(\lambda, x) \in \mathbb{R}_{++} \times X_i^*$, such that $(x, \lambda z_i^+) \in B_i''(p, p^1, q)$. Assume, alternatively, that $(p, p^1) = 0$ and $q \in Z^* \setminus \{0\}$. The portfolios of $Z^* \subset \mathbb{Z}^\circ$ yield zero payoffs on $\mathcal{S}$. It follows from Assumptions A2-A4-A5-A6 that there exists $(\lambda, \mu) \in \mathbb{R}_{++}^2$, small enough, such that: $(0, \lambda z_i^+ - \mu q) \in B_i''(p, p^1, q)$. □

**Assertion (ii):** let $(i, (p, p^1), q) \in I \times P^* \times Q$, an open subset, $V$, of $X_i^* \times Z_i^*$, and $(x, z) \in X_i^* \times Z_i^*$ be given, such that $(x, z) \in V \cap B_i''(p, p^1, q)$. From the definition of $B_i''$,
there exists a neigbourhood, U, of $(p, p^1, q)$, such that $(x, z) \in V \cap B_i''(p', p'^1, q')$ holds whenever $(p', p'^1, q') \in U$, which means that $B_i''$ is lower semi-continuous at $(p, p^1, q)$. $\square$

## 4.2 The fixed-point-like argument

A convex compact set, $\Theta := P^* \times Q \times (\times_{i \in I} X_i^* \times Z_i^*)$, and l.s.c. correspondences are introduced. Namely, for each $i \in I$ and every $\theta := ([p, p^1 := (p_s^1)], q, [(x_i, z_i)]) \in \Theta$, we let:

$$ \Psi_i(\theta) := \begin{cases} B'_i(p, p^1, q) & \text{if } (x_i, z_i) \notin B'_i(p, p^1, q) \\ B''_i(p, p^1, q) \cap P_i(x_i) \times Z_i^* & \text{if } (x_i, z_i) \in B'_i(p, p^1, q) \end{cases} $$

$$ \Psi_0(\theta) := \{ ((p', p'^1), q') \in P^* \times Q : $$

$$ (q' - q) \cdot \sum_{i \in I} z_i + (p' - p) \cdot \sum_{i \in I} (x_{i0} - e_{i0}) + \sum_{s \in \mathcal{S}} (p_s'^1 - p_s^1) \cdot \sum_{i \in I} (x_{is} - e_{is}) > 0 \}. $$

**Lemma 2** For each $i \in I \cup \{0\}$, the correspondence $\Psi_i$ is lower semicontinuous.

**Proof** See the Appendix. $\square$

<figure><img src="image_2.png" /></figure>

The above correspondences display a remarkable element, $\theta^* \in \Theta$, which meets the properties of Claim 6, and is henceforth set as given and always referred to:

**Claim 6** There exists an element $\theta^* := ([p^*, p^{*1} = (p_s^{*1})], q^*, [(x_i^*, z_i^*)]) \in \Theta$, such that:

$$ (i) \forall (p, p^1 := (p_s^1)) \in P^*, \forall q \in Q, $$

$$ (q^* - q) \cdot \sum_{i \in I} z_i^* + (p^* - p) \cdot \sum_{i \in I} (x_{i0}^* - e_{i0}) + \sum_{s \in \mathcal{S}} (p_s^{1*} - p_s^1) \cdot \sum_{i \in I} (x_{is}^* - e_{is}) \geq 0; $$

$$ (ii) \forall i \in I, (x_i^*, z_i^*) \in B_i'(p^*, p^{*1}, q^*) \text{ and } B_i''(p^*, p^{*1}, q^*) \cap P_i(x_i^*) \times Z_i^* = \emptyset. $$

**Proof** Quoting Gale-Mas-Colell (1975, 1979): "Given $X = \times_{i=1}^m X_i$, where $X_i$ is a non-empty compact convex subset of $\mathbb{R}^n$, let $\varphi_i: X \to X_i$ be $m$ convex (possibly empty) valued correspondences, which are lower semicontinuous. Then, there exists $x := (x_i) \in X$ such that, for each $i$, either $x_i \in \varphi_i(x)$ or $\varphi_i(x) = \emptyset$". The correspondences $\Psi_i$, for $i \in I \cup \{0\}$, meet all conditions of the above theorem and yield Claim 6. $\square$

<figure><img src="image_3.png" /></figure>
**Claim 7** Given $\theta^* := ([p^*, p^{*1} = (p_s^{*1})], q^*, [(x_i^*, z_i^*)] ) \in \Theta$ along Claim 6, let $q := (q^* + \bar{q}) := (q^* + \sum_{s \in \mathcal{S}} \lambda_s V(s))$ be given. Then, the following Assertions hold:
(i) $\sum_{i \in I} z_i^* \in \mathcal{Z} \oplus \mathcal{Z}^o$ and $\sum_{i \in I} (x_{is}^* - e_{is}) = 0$, for every $s \in \mathcal{S}';$
(ii) $(x_i^*, z_i^*) \in B'_i(p^*, p^{*1}, q^*)$ and $B'_i(p^*, p^{*1}, q^*) \cap P_i(x_i^*) \times Z_i^* = \emptyset$, for every $i \in I$;
(iii) $\sum_{i \in I} z_i^* \in \mathcal{Z}^o$ and we let $(z_i^o) \in \times_{i \in I} \mathcal{Z}_i^o$ and $(z_i) = (z_i^* - z_i^o)$ be such that $\sum_{i \in I} z_i = 0$;
(iv) $(p^*, p^{*1}) \in P \times P^{\mathcal{S}}$ and $(p^*, q, [(x_i^*, z_i)])$ is a sequential equilibrium of the economy $\mathcal{E}_{(p_i)}$, defined by $p_{is} = \bar{p}_{is}$, for $(i, s) \in I^* \times S_i \setminus \mathcal{S}$, and $p_{is} = p_s^{*1}$, for $(i, s) \in I \times \mathcal{S}$;
(v) $q \in \mathcal{NA}(\lambda)$.

**Proof**

**Assertion (i):** the relations $q^* \cdot \sum_{i \in I} z_i^* \ge 0$, $p^* \cdot \sum_{i \in I} (x_{i0}^* - e_{i0}) \ge 0$ and $p_s^{*1} \cdot \sum_{i \in I} (x_{is}^* - e_{is}) \ge 0$, for every $s \in \mathcal{S}$, hold, from Claim 5-(i), with strict inequality whenever $\sum_{i \in I} z_i^* \notin Z^{*\perp}$, $\sum_{i \in I} (x_{i0}^* - e_{i0}) \ne 0$ or $\sum_{i \in I} (x_{is}^* - e_{is}) \ne 0$.

Assume, by contraposition, that $\sum_{i \in I} (x_{is}^* - e_{is}) \ne 0$ holds, for some $s \in \mathcal{S}'$, or that $\sum_{i \in I} z_i^* \notin Z^{*\perp} = \mathcal{Z} + \mathcal{Z}^o$. Then, from Claim 6-(i) and above, the relations $\gamma_{(p^*, p^{*1}, q^*)} = 0$ and $0 < p^* \cdot \sum_{i \in I} (x_{i0}^* - e_{i0}) + q^* \cdot \sum_{i \in I} z_i^* + \sum_{s \in \mathcal{S}} \lambda_s p_s^{1*} \cdot \sum_{i \in I} (x_{is}^* - e_{is})$ hold. From Claim 6-(ii), the budget constraints, $p^* \cdot (x_{i0}^* - e_{i0}) + q^* \cdot z_i^* + \sum_{s \in \mathcal{S}} \lambda_s p_s^{1*} \cdot (x_{is}^* - e_{is}) \le \gamma_{(p^*, p^{*1}, q^*)} = 0$, also hold, for each $i \in I$. Summing them up (for $i \in I$) yields a contradiction from above, namely: $0 < p^* \cdot \sum_{i \in I} (x_{i0}^* - e_{i0}) + q^* \cdot \sum_{i \in I} z_i^* + \sum_{s \in \mathcal{S}} \lambda_s \sum_{i \in I} p_s^{1*} \cdot (x_{is}^* - e_{is}) \le 0$. $\square$

**Assertion (ii):** from Claim 6, it suffices to show: $B'_i(p^*, p^{*1}, q^*) \cap P_i(x_i^*) \times Z_i^* = \emptyset$. By contraposition, assume that there exists a strategy $(x_i, z_i) \in B'_i(p^*, p^{*1}, q^*) \cap P_i(x_i^*) \times Z_i^*$. Let $(x'_i, z'_i) \in B''_i(p^*, p^{*1}, q^*)$ be given, along Claim 5. From Assumptions A3-A4 and convexity arguments, both relations $(x_i^n, z_i^n) := [\frac{1}{n}(x'_i, z'_i) + (1 - \frac{1}{n})(x_i, z_i)] \in B''_i(p^*, p^{*1}, q^*)$ and $(x_i^n, z_i^n) \in P_i(x_i^*) \times Z_i^*$ would hold, for $n \in \mathbb{N}$ big enough, contradicting Claim 6. $\square$

**Assertion (iii):** from Assumption A1 and Assertion (ii), the budget constraints of
$(x_i^*, z_i^*) \in B_i'(p^*, p^{*1}, q^*)$ (for $i \in I$) are all binding. Summing them up (at $t=1$) yields, from Assertion (i): $0 = p_s^* \cdot \sum_{i \in I} (x_{is}^* - e_{is}) = V(s) \cdot \sum_{i \in I} z_i^*$, for every $s \in \mathbb{S}$. The latter relations are written $\sum_{i \in I} z_i^* \in \mathbb{Z}^o$, which yields, from Assertion (i): $\sum_{i \in I} z_i^* \in \mathbb{Z}^o$. $\square$

Assertion (iv): summing up the binding (from above) budget constraints at $t=0$ yields, from Assertions (i)-(ii), $0 = p_0^* \cdot \sum_{i \in I} (x_{i0}^* - e_{i0}) + q^* \cdot \sum_{i \in I} z_i^* + \sum_{s \in \mathbb{S}} \lambda_s \sum_{i \in I} p_s^{1*} \cdot (x_{is}^* - e_{is}) = \#I \gamma_{(p^*, q^*)}$. The relations $(x_i^*, z_i) \in B_i(p^*, q)$, for each $i \in I$, in the economy $\mathcal{E}_{(p_i)}$ (defined in Assertion (iv)), follow from the definitions, Assumptions A4-A5 and the relation $q^* + \bar{q} \in Z^* \oplus \mathbb{Z}$, where $q := (q^* + \bar{q}) := q^* + \sum_{s \in \mathbb{S}} \lambda_s V(s)$ and $(z_i) \in \times_{i \in I} Y_i$ is defined along Assertion (iii). From Assertions (i)-(iii) and above, the collection, $\mathcal{C} := (p^*, q, [(x_i^*, z_i)])$, meets Conditions (b)-(c) of Definition 1 in the economy $\mathcal{E}_{(p_i)}$.

The collection $\mathcal{C}$ also meets Condition (a) of Definition 1. Indeed, the relation $[(x_i^*, z_i^*)] \in \mathcal{A}(p^*, p^{*1}, q^*)$ holds from Assertions (i)-(ii)-(iii) above and implies, from Lemma 1, that $(x_i^*, z_i^*)$ is interior to $X_i^* \times Z_i^*$, for each $i \in I$. The relation $(p^*, p^{*1}) \in P \times P^{\mathbb{S}}$ follows from Assertion (ii) and Assumption A1. Assume, by contraposition, that there exist $i \in I$ and $(x_i', z_i') \in B_i(p^*, q) \cap P_i(x_i^*) \times \mathbb{R}^J$ in the above economy $\mathcal{E}_{(p_i)}$. From Assumptions A2-A4-A5, the strategy $(x_i', z_i')$ may be chosen such that all budget constraints bind and $z_i' \in Z_i$. Then, for $n \in \mathbb{N}$ big enough, the relations $(x_i^n, z_i^n) := [\frac{1}{n}(x_i', z_i') + (1-\frac{1}{n})(x_i^*, z_i^*)] \in B_i'(p^*, p^{*1}, q^*) \cap P_i(x_i^*) \times Z_i^*$ would hold, from the definitions, Assumptions A3-A4 and above. The latter relations contradict Assertion (ii). $\square$

Assertion (v): Assertion (v) states the relations $\pi(q) = \bar{q}$ and $q \in \mathcal{NA}$. The former follows from the definitions, which imply: $q := q^* + \bar{q} \in Z^* \oplus \mathbb{Z}$. The latter is the standard no-arbitrage property of equilibrium prices: if there existed $(i, z) \in I \times As(Y_i)$, such that $-q \cdot z \ge 0$ and $V(s) \cdot z \ge 0$ for every $s \in S_i$, with one strict inequality, the strategy $(x_i^*, z_i)$ would not (from Assumption A1) be optimal in $B_i(p^*, q)$, in contradiction with Assertion (iv). Hence, $q \in \mathcal{NA}$. Theorem 1 is proved completely. $\square$
# 5 The model's outcomes when one agent is fully informed

Throughout this Section, one agent, denoted $i_0 \in I$, is fully informed (i.e., $S_{i_0} = \underline{s}$).

All structural characteristics of a standard economy, $\{(I, S, L, J), V, (S_i), (e_i), (\prec_i)\}$, are unchanged from Section 2, except Assumption A5, which is dropped without changing the result of Theorem 1. With one informed agent, the relation $\underline{Z}^o = Z^o$ holds, and implies: $\mathcal{N}A \subset \underline{Z} = Z$. Hence, $\pi(q) = q$, for every $q \in \mathcal{N}A$. Thus, the no-arbitrage prices in the current economy identify to the standard no-arbitrage prices on frictionless markets of symmetric information. Theorem 1 states that the latter coincide with equilibrium prices, and may be supported by any collection of state prices of the informed agent. Assumption A5 is now replaced by a standard assumption of restricted participation models with symmetric information, namely:

* **A5'** $Z^o \subset Y_{i_0}$, or agents have symmetric information and, either $Z^o \subset \cup_{i \in I} Y_i$, or $F := \cup_{i \in I} Y_i$ is a vector space.

**Remark 5** The condition $Z^o \subset \cup_{i \in I} Y_i$ of Assumption A5' is equivalent, from A4, to $Z^o \subset \cup_{i \in I} As(Y_i) := \cup_{i \in I} \{z \in \mathbb{R}^J : \{z\} + Y_i \subset Y_i\}$ (Cornet and Gopalan, 2010, p. 247).

Under Assumptions A1-A2-A3-A4-A5'-A6, Section 5.1 proves that sequential equilibria exist consistently with any state prices in realizable states, and any given expectations in the other states. Section 5.2 relates this result to the literature.

## 5.1 Equilibrium with one fully informed agent

There exists an abundant literature on restricted participation. To quote only a few contributions, the reader may refer, e.g., to Balasko-Cass-Siconolfi (1990), Cass-Siconolfi-Villanacci (2001), or Polemarchakis-Siconolfi (1997). These papers
establish, in particular, that enough constraints on the trade of assets may lead
to a real indeterminacy of equilibrium similar to that of incomplete markets. The
present focus is different. It is on the conditions and steps used to extend Cass'
(2006) price equivalence theorem, and, in particular, to restricted participation.
As shown by Balasko, Cass and Siconolfi (1990), that price equivalence fails with
restricted participation, in general. Few contributions have investigated this issue.
Martins-da-Rocha and Triki (2005) and Cornet and Gopalan (2010) are exceptions.

In the latter papers, a weak form of equilibrium is proved to exist first, which
lets portfolios clear in value only. Namely, with two periods, Condition (c) of the
above Definition 1 ($\sum_{i \in I} z_i = 0$) would be replaced by: $q \cdot \sum_{i \in I} z_i = 0$. The latter
equilibrium is called a 'weak equilibrium' by Martins-da-Rocha and Triki (2005)
and an 'accounts clearing equilibrium' by Cornet and Gopalan (2010). Their proofs
derive the existence of the weaker equilibrium from a technical condition on agents'
accessibility to assets' payoffs. Then, from Assumption A5', the weaker equilibrium
yields a sequential equilibrium, consistent with any collection of given state prices.

Hereafter, Assumption A5' replaces A5 to suggest an alternative, direct, less
demanding proof (in my view) of Cass' price equivalence on purely financial markets,
when agents, with one informed, may have non-ordered preferences and restricted
participation. No other assumptions than A1-A2-A3-A4-A5'-A6 are required to
prove the existence of a sequential equilibrium and the desired price equivalence.

In Section 4 of the current model, owing to Assumptions A4-A5, the generic
$i^{th}$ agent's portfolios were restricted to the set $Y_i \cap Z_i$. This restriction was possible
because $Y_i$ contained its orthogonal projection on $Z_i$. Without Assumption A5, this
possibility vanishes, which may pose a boundary problem (see the Appendix). This
problem disappears when redundant assets are eliminated (i.e., $Z^o = \underline{Z}^o = \{0\}$).<fnref n="2" /> Yet, Won and Hahn (2003) show that redundant assets may contribute to risk sharing.

Moreover, restricted participation models typically keep the set $\underline{Z}^o$ non-trivial, so, I leave this possibility open. To circumvent the above problem, I introduce an auxiliary economy, which only changes portfolio sets. The latter are replaced, for each $i \in I$, by: $Y'_i := Y_i + Z_i^o$. When one agent is fully informed, both relations $Z^* = \{0\}$ and $A := \{(z_i) \in \mathbb{R}^{J \times I} : \sum_{i \in I} z_i = 0, V(s_i)z_i \ge 0, \forall (i, s_i) \in I \times S_i\} \subset \times_{i \in I} Z_i^o \subset \times_{i \in I} Y_i'$ hold, from the definitions and the above assumption that markets are arbitrage-free.<fnref n="3" />

The initial economy is henceforth assumed to meet Assumptions A1-A2-A3-A4-A5'-A6. From above, the auxiliary economy is then standard along Section 2. The latter is denoted by $\mathcal{E}'_{(p_i)}$, for every price expectations, $(p_i) \in P^{S_i \times I}$. It resumes all definitions and notations of Section 2, except budget sets' notations, which are now: $B_i'(p, q)$, for all $(i, p, q) \in I \times P \times Q$. The following Claim proves the desired price equivalence (in the initial economy) from Theorem 1 applied to the auxiliary economy:

**Claim 8** Let $\lambda \in \mathbb{R}_{++}^S$ and subjective expectations, $\bar{p}_i := (\bar{p}_{is}) \in P^{S_i \setminus \underline{S}}$, for every $i \in I^* := \{i \in I : S_i \neq \underline{S}\}$, be given. Along Theorem 1, let a sequential equilibrium, $(p, q, [(x_i, z_i)]) \in P \times \mathcal{NA}(\lambda) \times (\times_{i \in I} B_i'(p, q))$ of an auxiliary economy, $\mathcal{E}'_{(p_i)}$, be given, such that $p_{is} = \bar{p}_{is}$, for every $(i, s) \in I^* \times S_i \setminus \underline{S}$. Then, there exist portfolios, $(z_i^*) \in \times_{i \in I} Y_i$, such that $(p, q, [(x_i, z_i^*)])$ is a sequential equilibrium of the initial economy, $\mathcal{E}_{(p_i)}$.

**Proof** Let $\lambda \in \mathbb{R}_{++}^S$ and $\bar{p}_i := (\bar{p}_{is}) \in P^{S_i \setminus \underline{S}}$, for each $i \in I^* := \{i \in I : S_i \neq \underline{S}\}$, be

<footnote n="2">² From the definition, the set $A := \{(z_i) \in \mathbb{R}^{J \times I} : \sum_{i \in I} z_i = 0, V(s_i)z_i \ge 0, \forall (i, s_i) \in I \times S_i\}$ is always included in $\underline{Z}^oI$. In the current economy, the relations $Z^* = \{0\}$ and $\underline{Z}^o = \underline{Z}^o$ hold. Hence, if $\underline{Z}^o = \underline{Z}^o = \{0\}$ holds, Assumption A5 is implied by A4.</footnote>

<footnote n="3">³ The latter assumption is unnecessary, with symmetric information, since, in that case, the relation $A \subset \underline{Z}^oI = \underline{Z}^oI$ holds from the definition.</footnote>
given. Along Theorem 1, let $(p, q, [(x_i, z_i)]) \in P \times NA(\lambda) \times (\times_{i \in I} B'_i(p, q))$ be a sequential
equilibrium of an economy, $\mathcal{E}'_{(p_i)}$, such that $p_{is} = \bar{p}_{is}$, for every $(i, s) \in I^* \times S_i \setminus \mathbf{S}$. From
the definition of budget sets, there exists $(z_i^o) \in \times_{i \in I} Z_i^o$, such that $z^o := \sum_{i \in I} z_i^o \in Z^o$
and $(x_i, z_i + z_i^o) \in B_i(p, q)$, for all $i \in I$, since $q \in \mathcal{NA} \subset Z = Z^o \perp$ holds from above.

Assume, from Assumption A5' and Remark 5, that the relation $Z^o \subset Y_{i_0}$ holds or that $Z^o \subset \cup_{i \in I} As(Y_i)$ and agents have symmetric (full) information. Then, there exists a fully informed agent, $i' \in I$, such that $(z_{i'}^o + z_{i'}^o - z^o) \in Y_{i'}$. Let $(z_i^*) \in \times_{i \in I} Y_i$ be defined by $z_{i'}^* := (z_{i'}^o + z_{i'}^o - z^o)$ and $z_i^* := z_i + z_i^o$ for every $i \in I \setminus \{i'\}$. Then, $\mathcal{C} := (p, q, [(x_i, z_i^*)])$ is a sequential equilibrium of the economy $\mathcal{E}_{(p_i)}$. Indeed, $\mathcal{C}$ meets conditions (b)-(c) of Definition 1, by construction. Moreover, the relation $q \in Z$ and the optimality of $(x_i, z_i)$ in $B_i'(p, q) \supset B_i(p, q)$, imply that of $(x_i, z_i^*) \in B_i(p, q)$ in $B_i(p, q)$, for each $i \in I$.

Assume, alternatively, that $F := \cup_{i \in I} Y_i$ is a vector space and that agents have symmetric information. The set $F$ contains $\cup_{i \in I} \{z_i + z_i^o\}$, hence also, $z^o = \sum_{i \in I} (z_i^o + z_i)$, and it may be assumed to exists a same fully informed agent, $i' \in I$, such that $-nz^o \in Y_{i'}$ for all $n \in \mathbb{N}$. Then, from Assumption A4, $z_{i'}^n := (1 - \frac{1}{n})(z_{i'} + z_{i'}^o) - \frac{1}{n}(nz^o) \in Y_{i'}$ hold for $n \in \mathbb{N}$, and $z_i^* := (z_{i'} + z_{i'}^o - z^o) \in Y_{i'}$ holds. The rest follows, identical to above. $\square$

**Remark 6** It follows from the above proof, resuming its notations, that the equi-
librium, $\mathcal{C}' := (p, q, [(x_i, z_i)])$, of the economy $\mathcal{E}'_{(p_i)}$, along Claim 8, yields an *accounts clearing* equilibrium of the economy $\mathcal{E}_{(p_i)}$, namely, $\mathcal{C}^o := (p, q, [(x_i, z_i + z_i^o)])$. The equilib-
rium, $\mathcal{C} := (p, q, [(x_i, z_i^*)])$, of the economy, $\mathcal{E}_{(p_i)}$, is derived from the latter, $\mathcal{C}^o$, owing to
Assumption A5'. The condition to apply Proposition 1 (p. 234) in Cornet-Gopalan
(2010), which derives an equilibrium from an *accounts clearing* equilibrium, is ex-
actly the same: Assumption A5'. The latter paper builds the existence of its *ac-
counts clearing* equilibrium on an additional condition on the accesssibility of asset
payoffs, which the current model replaces by the standard mild Assumption A6.
## 5.2 The above results with reference to previous ones of the literature

The literature is abundant on asymmetric information and restricted participation. When it introduces asymmetric information, agents would have the perfect beliefs, and make the Radner inferences, described in De Boisdeffre (2022). The two conditions are linked and highly theoretical. The current paper prefers to tackle asymmetric information with the Cornet-De Boisdeffre's (2009) inferences, which require no price model nor structural knowledge on the economy. On restricted participation, the literature typically assesses conditions under which equilibrium exists, the limits to Cass' (2006) theorem, and conditions leading, either to the generic regularity of equilibrium, or to its indeterminacy (see Section 1 and above).

The current paper has a different focus. It aims to extend Cass' (2006) theorem to a fairly general model of restricted participation, where agents may have non-ordered preferences, asymmetric information and no price expectations a la Radner or CKRMC models. Since the same issue was investigated, under symmetric information, by Martins-da-Rocha-Triki (2005) and Cornet-Gopalan (2010) it is relevant to compare the latter papers and the current one when agents are fully informed.

Cornet-Gopalan (2010) weakens slightly Martins-da-Rocha-Triki's (2005) assumptions for the existence of equilibrium prices, but uses a similar approach for the proof, which is not so far from the current paper's. The main differences stem from the fact that the former drop Assumption A6, at a significant cost. Technically, the authors have to insure that auxiliary budget sets, similar to those in Section 4, have non-empty interiors, which is far more complex than proving Claim 5, above. Replacing Assumption A6, Cornet-Gopalan (2010) assumes that the closed cone spanned by agents' accessible payoffs is a vector space, and Martins-da-Rocha-Triki (2005), that all available payoffs in the aggregate are accessible in reduced amount
to an agent. Those technical assumptions are sufficient conditions for the existence of the weaker equilibrium, but neither necessary nor transparent conditions.

With symmetric information, the current paper replaces the latter accessibility conditions by Assumption A6, which is met on all actual markets. Doing so yields a direct existence proof along Theorem 1, in the sense that no weak form of equilibrium is required as a first step. An auxiliary economy can be used instead, where an accounts clearing equilibrium is only implicit (Remark 6). The notion of sequential equilibrium does not require that frictions be technically restricted as above, but that an agent might purchase any accessible portfolio, which she desires at market prices. In either models, equilibrium prices adapt to insure that condition.<fnref n="4" /> Prices would prevent accessible transfers to take place, which cannot clear under frictions.

Another issue, which deserves attention, is the notion of no-arbitrage price. Definition 2 presents two standard concepts: the frictionless no-arbitrage price, and the no-arbitrage price, to which Martins-da-Rocha and Triki (2005) refer. Cornet and Gopalan (2010) introduce an intermediate concept of ‘aggregate no-arbitrage price’, which refers to the vector space, $\langle \cup_{i \in I} Y_i \rangle \subset \mathbb{R}^J$, spanned by $\cup_{i \in I} Y_i$. A price $\bar{q} \in \mathbb{R}^J$ is said to be ‘aggregate arbitrage-free’ if there exists a vector of state prices, $\lambda := (\lambda_s) \in \mathbb{R}_{++}^S$, such that $\bar{q} \cdot z = \sum_{s \in S} \lambda_s V(s) \cdot z$, for every $z \in \langle \cup_{i \in I} Y_i \rangle$. Cornet-Gopalan’s theorem states the equivalence between equilibrium prices and aggregate no-arbitrage prices, under the payoff accessibility condition above.

Theorem 1 states, more precisely, the equivalence between the sets of equilibrium prices (on markets with or without frictions) and of frictionless no-arbitrage prices (i.e., $\{\sum_{s \in S} \lambda_s V(s) : (\lambda_s) \in \mathbb{R}_{++}^S\}$), which also define no-arbitrage prices and

<footnote n="4">⁴ Consider the economy of Remark 4, where $s_1 = s_2 = s$ instead. This economy meets Cornet and Gopalan’s accessibility condition, above. But no trade in the first asset may take place at equilibrium. Prices need to insure that condition.</footnote>
aggregate no-arbitrage prices through a vector of state prices, $(\lambda_s) \in \mathbb{R}_{++}^S$. Up to mild changes, Theorem 1 thus embeds the latest developments on Cass' price equivalence given by Martins-da-Rocha and Triki (2005), that no-arbitrage prices characterize equilibrium prices *provided* financial markets be locally collectively frictionless (cone $\cup_{i \in I} VY_i = \text{Im } V$), or by Cornet and Gopalan (2010), who state that aggregate no-arbitrage prices characterize equilibrium prices *whenever* the closed cone spanned by agents' accessible payoffs is a linear space. Theorem 1 mainly replaces the above accessibility conditions by Assumption A6.

# 6 Concluding remarks

Section 5 shows that Theorem 1 and Claim 8 complement the existing results of the literature of symmetric information on Cass' price equivalence. Theorem 1 also refines and extends De Boisdeffre's (2007), which establishes that an equilibrium with asymmetric information on frictionless purely financial markets exists and is supported by *one* (unknown) collection of state prices. Theorem 1 now states that *any* collection of state prices in realizable states supports a sequential equilibrium, whether agents had a restricted or full access to financial markets. Finally, Theorem 1 proves the existence of a sequential equilibrium without assuming any form of rational expectations a la Radner (1972 and 1979). Agents' expectations are private exogenous in the current paper, as in the standard temporary equilibrium model.

The latter result is of self-interest. First, it contradicts the view that the sequential and temporary equilibria are two dichotomous concepts (Grandmont, 1982). The sequential equilibrium of Definition 1 is a temporary equilibrium, which lets agents' demands clear markets ex post.
Second, it suggests a response to the standard criticism of Radner’s (1972) perfect foresight assumption, stated, e.g., by Kurz and Wu (1996): “agents need to know the maps from states at future dates to prices in the future and it is entirely unrealistic to assume that agents can find out what this sequence of maps is.” Radner (1982) himself acknowledges that perfect price foresight “seems to require of the traders a capacity for imagination and computation far beyond what is realistic.” Theorem 1 makes perfect foresight unnecessary to insure the existence of sequential equilibria. This result had been intuitive so far, but, to my best knowledge, never proved.

Under asymmetric information, Theorem 1 avoids agents to detain the structural knowledge of the ex ante characteristics of the economy and of the other agents’ beliefs. This assumed knowledge is typically used to justify the Radner (1979) inferences in rational expectations’ and CKRMC models (see De Boisdeffre, 2022). Moreover, given the indeterminacy of equilibrium with nominal assets, the inferences in Radner (1979) are questionable from the theoretical viewpoint also. The latter are based on the assumption that there exists a map between the state of nature to be revealed and one selected equilibrium price to be observed - a map which all agents agree upon and reveals information.

With many other possible outcomes, how can one particular revealing price emerge as a market equilibrium outcome, Polemarchakis and Siconolfi (1993) ask, when alternative equilibrium prices are non-revealing? Why not agree on one of the latter prices, which would reveal no information? CKRMC models typically let agents anticipate finitely many possible equilibrium prices in any state of nature (instead of one price with certainty) and apply the same inferences as Radner’s to the correspondence relating states to multiple selected prices. But this new selection would not change the conundrum, when there are infinitely many equilibria.
In the current paper, agents endowed with no price model, but with private ex-
ogenous expectations, infer information from observing financial markets along the
Cornet-De Boisdeffre (2009) inferences. The latter demand very little in terms of ra-
tionality. Agents only have to realize that a lasting unlimited arbitrage opportunity
cannot exist, and, therefore, to rule out the states, which would offer a free lunch.
The latter inferences and their properties extend to arbitrary financial structures
and state spaces (possibly infinite). They seem to describe the actual refinement of
information, which takes place on most financial markets (De Boisdeffre, 2016).

The current model drops rational expectations completely, that is, both the per-
fect foresight and Radner inference assumptions. But this does not solve one major
problem of the standard sequential equilibrium concept, which stems from the de-
generate (i.e., unique) price expectations. Because of insufficiently many observed
parameters, no rational agent could infer, in general, which particular spot prices
might obtain tomorrow, amongst all equilibrium possibilities. In case of indetermi-
nacy, uncountably many market-clearing prices exist. Yet, the standard definition
of sequential equilibrium in the literature lets agents commonly expect one price on
every spot market. The common selection of a unique equilibrium price, amongst
uncountably many, is a mysterious process. It could only be explained, theoreti-
cally, if the unique spot prices were the necessary outcomes of the economy's ex
ante characteristics and agents' beliefs, under their common knowledge of rational-
ity and unlimited forecast abilities. Such an assumption is, indeed, at the core of
CKRMC models, but would not lead to degenerate expectations in such models.

In the current model, no such rationality is assumed. Future market-clearing
prices always exist, from Theorem 1. But the probability that agents had expected
them uniquely ex ante (as required at equilibrium), say ‘by chance’, is as negligible
as above. This problem vanishes when agents anticipate exogenous sets of spot
prices (endowed with a probability distribution), which always contain (but may
not be limited to) the true equilibrium price. Facing uncountably many equilibrium
possibilities, rational agents should also have uncountable anticipation sets. When
the latter always contain the true spot price, agents are said to have correct foresight.

Correct foresight may replace perfect foresight in the definition of the sequential
equilibrium. With smooth preferences, the set of all market-clearing prices is shown
to be non-empty, fixed and bounded above zero (whatever agents' beliefs). Having
no price model, and no awareness of other agents' characteristics and beliefs, the
latter set is argued, in De Boisdeffre (2022), to represent agents' minimum uncer-
tainty about the future spot prices. To agents, the location of tomorrow's spot
price within the minimum uncertainty set is undetermined until it is observed. In
the latter paper, a correct foresight equilibrium is shown to exist whenever agents'
anticipation sets embed the minimum uncertainty set. By a same technique of proof,
the result of Theorem 1 could be used and extended to an economy, where agents'
degenerate anticipations are replaced by anticipation sets. The extended theorem
would state that any collection of state prices in realizable states supports a 'correct
foresight' sequential equilibrium, in an economy which fully discards rational ex-
pectations. The latter equilibrium would exist when agents' anticipation sets embed
the minimum uncertainty set. Theorem 1 is also a step in the proof of this outcome.

Appendix

**Lemma 1** $\exists r > 0 : \forall ((p, p^1), q) \in P^* \times Q, \forall [(x_i, z_i)] \in \mathcal{A}(p, p^1, q), \sum_{i \in I} (\|x_i\| + \|z_i\|) < r,$

where $\mathcal{A}(p, p^1, q) := \{[(x_i, z_i)] \in \times_{i \in I} B_i^1(p, p^1, q) : \sum_{i \in I} (x_{is} - e_{is}) = 0, \forall s \in S', \sum_{i \in I} z_i \in Z^o\}$.
**Proof** Referring to the notations of Theorem 1, let $\delta := (1 + \sum_{i \in I^*} \|\bar{p}_i\|) \sum_{i \in I} \|e_i\|$ be given. From the market-clearing conditions in $\mathcal{A}(p, p^1, q)$, the relations $x_{is} \in [0, \delta]^L$ hold, for every $(p, p^1, q) \in P^* \times Q$, every $[(x_i, z_i)] \in \mathcal{A}(p, p^1, q)$ and every $(i, s) \in I \times S'$. Moreover, the idiosyncratic expectations, $(\bar{p}_{is})$, for $i \in I^*$, have a positive lower bound. Lemma 1 will, hence, be implied by the following Assertion:

$$ \exists r' > 0, \forall (p, p^1, q) \in P^* \times Q, \forall [(x_i, z_i)] \in \mathcal{A}(p, p^1, q), \sum_{i \in I} \|z_i\| < r'. $$

Assume, by contraposition, that the above Assertion fails, that is, for every $k \in \mathbb{N}$ ($k > 0$), there exist prices, $(p^k, p^{1k}, q^k) \in P^* \times Q$, and strategies, $[(x_i^k, z_i^k)] \in \mathcal{A}(p^k, p^{1k}, q^k)$, such that $\alpha_k := \sum_{i \in I} \|z_i^k\| \ge k$. Then, the relations $z^k := (z_i^k) \in \times_{i \in I} Y_i \cap Z_i$ and $z'^k := z^k/\alpha_k := (z_i^k/\alpha_k) \in \times_{i \in I} Y_i \cap Z_i$ hold, for every $k \in \mathbb{N}$, from the definition of $\mathcal{A}(p^k, p^{1k}, q^k)$ and Assumption A4. The bounded sequence, $\{z'^k\}$, may be assumed to converge in a closed set, say to $z := (z_i) \in \times_{i \in I} Y_i \cap Z_i$, such that $\|z\| = 1$. The relations $[(x_i^k, z_i^k)] \in \mathcal{A}(p^k, p^{1k}, q^k)$ hold, for every $k \in \mathbb{N}$, and imply, for every $(i, s) \in I \times S_i$:

$V(s) \cdot z_i^k \ge -\delta$, that is, $V(s) \cdot z_i'^k \ge -\frac{\delta}{k}$, hence, in the limit, $V(s) \cdot z_i \ge 0$, for each $s \in S_i$, while $\sum_{i \in I} z_i^k \in Z^o$ implies $\sum_{i \in I} z_i'^k \in Z^o$, hence, in the limit, $\sum_{i \in I} z_i \in Z^o$.

Let $(z_i^o) \in \times_{i \in I} Z_i^o$ satisfy $\sum_{i \in I} z_i^k = \sum_{i \in I} z_i^o \in Z^o$ and define $(z_i^*) = (z_i - z_i^o)$. Then, the relations $V(s) \cdot z_i^* \ge 0$, for each $(i, s) \in I \times S_i$, and $\sum_{i \in I} z_i^* = 0$ hold and imply, from Cornet and De Boisdeffre (2002, p. 401) and the fact that markets are frictionless arbitrage-free along Claim 1: $(z_i^*) \in \times_{i \in I} Z_i^o$. The relation $(z_i) \in \times_{i \in I} (Z_i^o \cap Z_i) = \{0\}$ follows by construction, contradicting the above: $\|z\| = 1$. This completes the proof. $\square$

**Lemma 2** For each $i \in I \cup \{0\}$, $\Psi_i$ is lower semicontinuous.

**Proof** The correspondence $\Psi_0$ is lower semicontinuous for having an open graph.

We now set as given $i \in I$ and $\theta := (p, p^1, q, [(x_i, z_i)]) \in \Theta := P^* \times Q \times (\times_{i \in I} X_i^* \times Z_i^*)$.
• Assume that $(x_i, z_i) \notin B'_i(p, p^1, q)$. Then, $\Psi_i(\theta) = B'_i(p, p^1, q)$.

Let $V$ be an open set in $X_i^* \times Z_i^*$, such that $V \cap B_i'(p, p^1, q) \neq \emptyset$. It follows from the convexity of $B_i'(p, p^1, q)$ and the non-emptyness of the open set $B_i''(p, p^1, q) \subset B_i'(p, p^1, q)$ that $V \cap B_i''(p, p^1, q)$ is non-empty. From Claim 5, there exists a neighborhood $U$ of $(p, p^1, q)$, such that $V \cap B_i'(p', p'^1, q') \supset V \cap B_i''(p', p'^1, q') \neq \emptyset$, for every $(p', p'^1, q') \in U$.

Since $B_i'(p, p^1, q)$ is non-empty, closed, convex in the compact set $X_i^* \times Z_i^*$, there exist two open sets, $V_1$ and $V_2$, in $X_i^* \times Z_i^*$, such that $(x_i, z_i) \in V_1$, $B_i'(p, p^1, q) \subset V_2$ and $V_1 \cap V_2 = \emptyset$. From Claim 4, there exists a neighborhood, $U_1 \subset U$, of $(p, p^1, q)$, such that $B_i'(p', p'^1, q') \subset V_2$, for every $(p', p'^1, q') \in U_1$. Let $W = U_1 \times (\times_{j \in I} W_j)$, where $W_i := V_1$, and $W_j := X_j^* \times Z_j^*$, for each $j \in I \setminus \{i\}$, be a neighbourhood of $\theta$ in $\Theta$. Then, the joint relations, $\Psi_i(\theta') = B_i'(p', p'^1, q')$ and $V \cap \Psi_i(\theta') \neq \emptyset$, hold, for every $\theta' := (p', p'^1, q', [(x'_i, z'_i)]) \in W$, and imply that $\Psi_i$ is lower semicontinuous at $\theta$. $\square$

• Assume that $(x_i, z_i) \in B_i'(p, p^1, q)$. Then, $\Psi_i(\theta) = B_i''(p, p^1, q) \cap P_i(x_i) \times Z_i^*$.

The lower semicontinuity of $\Psi_i$ at $\theta$ results from the definition if $\Psi_i(\theta) = \emptyset$. Assume, now, that $\Psi_i(\theta) \neq \emptyset$. The correspondence $P_i$ is lower semicontinuous with open values, from Assumption A3, and $B_i''$ has an open graph in $P^* \times Q \times X_i^* \times Z_i^*$. As a corollary, the correspondence $(p', p'^1, q', [(x'_i, z'_i)]) \in \Theta \mapsto B_i''(p', p'^1, q') \cap P_i(x'_i) \times Z_i^* \subset B_i'(p', p'^1, q')$ is lower semicontinuous at $\theta$; and so is also $\Psi_i$, from the latter inclusions. $\square$

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