Amalie Solberg

Kurt I Draget

Christophe Schatz

Bjørn E Christensen
email: bjorn.e.christensen@ntnu.no

Alginate blocks and block polysaccharides: A review

Keywords: Alginate, G-blocks, diblocks, self-assembly, nanoparticles

Alginates consist of distinct blocks with different physical properties. This short review focuses on research carried out in Trondheim related to the early discovery of the block structure, their isolation, their different chemical and physical properties, and how they recently are utilized in diblock polysaccharides to obtain new nanostructuring properties.

Alginate structure

Alginates are naturally occurring block polysaccharides found in brown seaweeds and produced by some bacteria. The presence in alginates of the three principal block types (Figure 1) is a result of an enzymatic processes involving mannuronan-C5-epimerases. [1] These enzymes work on the homopolymeric mannuronan, converting at the polymer level in-chain 4linked β-D-mannuronate (M) to its 5-epimer, 4-linked α-L-guluronate (G). The enzymes govern both the extent of epimerization and the distribution of blocks in alginates, which depends on the biological source, growth conditions and even the type of plant tissue. [START_REF] Draget | Alginates[END_REF] Pure mannuronan, normally serving as an intermediate, is not found in nature but can be obtained from epimerasenegative mutants of alginate-producing bacteria. The resulting alginates have a blocky structure whose structure depends on the action of several epimerases. G-blocks are depicted in A by boxes. The epimerization changes the chair conformations from 4 C1 to 1 C4. G-blocks thus adopt a 'zig-zag' like helical structure with cavities or sites which is the basis for binding of Ca ++ with high selectivity (c). Calcium binding is associated with chain dimerization leading to the famous egg-box structure for junction zones in calcium alginate gels (d).

Isolation and characterisation of pure blocks

The understanding of the modular or block-wise structure of alginates preceded the discovery of epimerases and was long thought to be a result of a complex, sequence-dependent polymerization process, and the distribution of monomers can indeed to some extent be explained by a second order Markov model. [START_REF] Larsen | [END_REF] Although later being rejected as the key mechanism determining the block structures, the presence of distinct blocks was elegantly proven by studying the fragmentation of alginates by relatively mild acid hydrolysis combined with fractional precipitation of alginic acid. An overview is given in Figure 2. later been improved to provide very pure G-blocks. [4] (b) Blocks with narrow size distributions are generally obtained by size-exclusion chromatography. [5,6] (c) The pH dependence of block solubility. Reproduced with permission. [7] Copyright 1968, Royal Chemical Society. These findings also provide the key to methods for the excision of different block types. First, different glycosidic linkages in alginates hydrolyse at different rates when pH is near pKa of the uronic acids. [8,9] This is ascribed to the participation of the protonated carboxylic acids to intramolecular catalysis. [9,10] The conditions favour the hydrolysis of the G-M linkage. Holtan et al. [8] demonstrated that this linkage hydrolyses about one order of magnitude faster that the M-G linkage. As expected, the ratio decreased at higher pH when the carboxylic acids became deprotonated. On the other hand, the G-G linkages hydrolyse slower that the other linkage types, being in some way 'protected' by being insoluble below pH 3.2. Secondly, the solubility at low pH, i.e. of the acidic forms, is very different for the three block types (Figure 2c). G-blocks precipitate below pH 3.2 where other block types remain water-soluble. Together these observations (summarised in Figure 2a) clearly demonstrated the block-wise structure of alginates as well as providing methodologies to obtain the different block types. For the latter 4 several improvements have later taken place. Pure M-blocks are easily obtainable from pure mannuronan by partial hydrolysis and, analogously to G-blocks (Figure 2b), chromatographic separation according to DP. [6] G-blocks with high purity can be obtained from several seaweed alginates using a modification [4] of the hydrolysis/solubility method in Figure 2a. Alternatively, in vitro epimerization in combination with specific lyases can provide almost any block type. [11] The purity, composition, and chain length distribution of the block fractions in Figure 2a may vary depending on both the type of alginate and the isolation process. More detailed information is obtained by separation of individual DP as shown in Figure 2b. 1 H-NMR spectra provide the content and distribution of M and G residues as well as DP. [11] HPAED-PAD analysis of alginate fractions has been shown to be particularly useful due to its very high resolution. [6,11] during acid hydrolysis (95°C, pH 3.62). Minor unannotated peaks are ascribed to oligomers containing M residues. Reproduced from [12] . Copyright 2022, Royal Society of Chemistry.

Properties and roles of the alginate blocks

The presence of G-blocks in alginates is intimately linked to their ability to form hydrogels with calcium ions. In brown algae the Ca-alginate (in equilibrium with sea water) provides both mechanical strength and elasticity. Extraction of water-soluble Na-alginate consequently involves calcium removal, predominantly by exchanging Ca ++ with acid, leading to insoluble alginic acid, followed by neutralisation using NaOH, NaHCO3 or Na2CO3.

Alternatively, calcium chelators can be used to remove calcium ions.

The interaction of alginate blocks with Ca ++ has been widely studied. This also includes studies on Sr ++ and Ba ++ because alginates tend to bind them strongly and selectively concentrate them (by exchange with Ca ++ ). Although practical work with alginate mostly starts with the Na + form, studies of the selectivity for divalent cations use Mg ++ as reference cation because it has the same charge as Ca ++ (allowing changes in ionic strength at constant Ca ++ /Mg ++ ratios) and because the strongly hydrated Mg ++ ions (like Na + and other monovalent cations) do not induce gelation or precipitation. Interactions with a divalent cation (Me ++ ) are often expressed [1] through the selectivity coefficient 𝐾𝐾 𝑀𝑀𝑀𝑀 𝑀𝑀𝑀𝑀 with Mg ++ as a reference ion:

𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 1. 𝐾𝐾 𝑀𝑀𝑀𝑀 𝑀𝑀𝑀𝑀 = 𝑋𝑋 𝑀𝑀𝑀𝑀 𝑋𝑋 𝑀𝑀𝑀𝑀 [𝑀𝑀𝑀𝑀 ++ ] [𝑀𝑀𝑀𝑀 ++ ]
X refers to the mole fractions of bound cation, whereas [Mg ++ ] and [Me ++ ] are the molar concentrations of cations in solution (at equilibrium). Studies [1] related to cation selectivity resulted in many important findings: a) G-blocks bind Ca ++ only moderately strong if prevented from subsequent aggregation (e.g. using an agarose gel). [13] b) G-blocks precipitate with Ca ++ but the crystal structure has been challenging to study by Xray crystallography. [14] c) In alginates containing G-blocks the Ca/Mg selectivity increases with increasing XCa reaching a maximum around XCa = 0.4-0.5 before decreasing towards XCa = 1. [1] d) The Ca-binding in alginates containing G-blocks shows strong hysteresis: After preparing fully saturated Ca-alginate (XCa = 1) the Ca/Mg-selectivity increases by lowering XCa and can react values close to 100. [1] e) Alginates lacking longer G-blocks, i.e. mannuronan or polyalternating (..MGMG.. type)

alginate have low Ca-selectivities. [1] These, as well as other considerations, have led to the famous 'egg-box model' for initial dimerization of alginate G-blocks with Ca ++ (schematically illustrated Figure 1D). Although first proposed [15] in 1973 it seems still to be the preferred model. [14,[START_REF] Donati | Material properties of alginates[END_REF] However, the dimeric state seems to be unstable and is followed by further growth, leading to large multi-chain junction zones in gels, [START_REF] Stokke | [END_REF] and precipitation for isolated G-blocks. [12] The addition of G-blocks (DPn in the range 20) during gelation of high molecular weight alginates has demonstrated complex interactions influencing both kinetics and equilibrium modulus, as well as degree of syneresis in the resulting gels. [18,[START_REF] Draget | Modification of gelling kinetics and elastic properties by nano structuring of alginate gels exploiting the properties of poly-guluronate[END_REF] In other words, by exploiting the ion binding properties of low molecular weight G-blocks it has become possible to decouple modification of alginate gel properties from viscosity changes. The presence of G-blocks always slows down the gelling kinetics of high molecular weight alginates and reduces syneresis. The apparent equilibrium modulus in the presence of G-blocks depends on the level of gelling ions present: At low Ca saturation the modulus will be reduced due to competitive ion binding, whereas at high Ca saturation the modulus will become higher than for the high molecular weight alginate alone. The latter is believed to result from G-block being able to interact with and connect topologically restricted G sequences within the high molecular weight alginate, thereby on average shortening the elastic segments. This is a reasonable approach as other studies have shown cross sectional dimensions of alginate gel junctions way above that for dimerization at high Ca saturation. [START_REF] Stokke | [END_REF] A rather surprising effect of G-blocks involves the ability to modify other ordered and complex biopolymer systems such as mucus, [START_REF] Nordgard | [END_REF]21] both in the pathological as well as in the non-pathological state. In the first instance, G-blocks have been shown to modulate and normalize abnormally viscous mucus as found in e.g. subjects suffering from cystic fibrosis. Gblocks are possibly able to disrupt complex mucus interactions through electrostatic interactions, although the exact mechanism remain unclear, resulting in a lower cross-linking density and increased mucus mobility. In the second instance it has been shown that guluronate oligomers are able to modify non-pathological mucus in terms of an overall altering of mucus network architecture (e.g. increased pore size) leading to a reduction in barrier function as reflected in an improved nanoparticle mobility and uptake in such systems. Hence, G-blocks may have a potential in mucosal delivery of nanomedicines. Under any circumstance there is no evidence that G-blocks function as a true mucolytic by de-polymerizing mucins but rather to modify mucus architecture by blocking macromolecular interactions.

Taken together, G-blocks present a unique system based on abundant biomass that have important biological and technological properties. When part of alginates they explain, among other, the Ca ++ binding and gelation, making alginates attractive in biotechnology and biomedicine. However, other possibilities now emerge in the form of engineered block polysaccharides. This is described in the following section.

Terminal vs. lateral substitution: Block architectures

Lateral structural modifications (Figure 4a), presently comprising almost all polysaccharide derivatives, generally leads to pronounced changes in physical properties.

Lateral decoration of alginates by e.g. peptides or sulphate would, depending on the degree or pattern of substitution, gradually mask the unsubstituted regions necessary for Ca-induced egg-8 box formation or reduce the availability for alginate degrading enzymes. In contrast, terminal conjugation to a second chain would provide a diblock polysaccharide with laterally unsubstituted chains (Figure 4a). Conjugating a Ca-sensitive G-block to a second Ca-insensitive, but water-soluble, block should, according to the principles of synthetic block polymers, provide a system that could in the presence of Ca ++ self-assemble into a nanoparticle with a Caalginate core and an outer corona consisting of the second block (Figure 4b). diblock to form a core-corona nanoparticle. 4a is drawn with permission from [22] . Copyright 2022, Elsevier.

Engineered block polysaccharides

Initial work has focused on G-blocks (Gm, m = DP) terminally coupled to dextran (Dexn, n = DP) using PDHA (O,O′-1,3-propanediylbishydroxylamine) as the linker [12,22] . Dextrans are neutral (uncharged) polysaccharides with a flexible glucan backbone due to the a-1,6-linkages.

They may also contain some branches. Dextrans of suitable DP were obtained by partial hydrolysis to cover the DP-range 10 -200. The mixture was coupled at the reducing end to PDHA by reductive amination before the PDHA-activated dextrans were separated according to DP by SEC analogous to that shown in Figure 2b. Selected PDHA-activated dextran fractions were subsequently conjugated to G-block fractions of the desired DP, again using reductive amination at the reducing end. Gm-b-Dexn diblocks (fully conjugated PDHA-dextrans) were purified from unreacted G-blocks by semi-preparative SEC. Structures and chain lengths were confirmed by NMR and SEC-MALLS, respectively. As an alternative to PDHA-dextran Gblocks were terminally connected to oxyamine-PEG to obtain the Gm-b-PEGn diblock using a related protocol. The chemistry and kinetics of reactions of oligo-and polysaccharides with dioxyamines (PDHA) and dihydrazides was recently reviewed. [22] Self-assembly of the diblocks G11-b-Dex100 and G40-b-Dex100 with Ca ++ was studied [12] by dynamic light scattering using a dialysis setup with a nominal membrane cut-off of 100 -500 Da. Pure G-blocks led to precipitation, whereas the diblocks did not. For G40-b-Dex100 a transition from free, soluble diblocks to soluble nanoparticles was observed after ca. 3-4 days as an increase in the total scattering intensity. This was accompanied by the disappearance of the fast relaxation mode characteristic of free polyelectrolyte chains. After 7 days the scattering intensity levelled off and remained constant thereafter. Dynamic light scattering revealed two populations (intensity distributions) with diameters 25 and 150 nm, but only the former was detectable in the estimated number distributions. It thus seems this combination of chain lengths provide core-corona nanoparticles. The length of the G-block (m = 40) is well above the DP needed for strong binding of calcium ions [5,23] whereas the dextran block (n = 100) did not reach a volume large enough to prevent self-assembly. The lower limit with respect to the chain length of the dextran block remains to be identified. Lowering the G-block length to 11, which is closer to the critical length for strong calcium binding and chain dimerization, and which has a chain size appreciably smaller than the dextran did not provide well defined nanoparticles, but rather loose aggregates.

Conclusions

Early research on the nature of alginates provided methods for excising and purifying the principal block types of alginates. More recent research has demonstrated that the Careactive G-blocks can modify both gelation kinetics, equilibrium elasticity and syneresis in alginate gels. They can further modify transport properties of mucus. G-blocks can further be terminally conjugated to other blocks to obtain a new generation of Ca-sensitive diblock polysaccharides which instead of precipitation or forming hydrogels instead form well-defined nanoparticles by spontaneous self-assembly without the need for mechanical treatments or complex mixing methods. To this end the diblock Gm-b-Dexn has been briefly investigated. It seems reasonable to assume that the mode of self-assembly of G40-b-Dex100 with Ca ++ corresponds to a core consisting of G-blocks crosslinked with calcium ions according to the egg-box-model whereas the dextran block for an outer corona endowed with stabilizing properties. Future research will focus on a wider range of block sizes, different rates and modes of calcium delivery, and investigation of other cations known to bind strongly to alginate Gblocks such as strontium and barium ions. Due to the nanometre size of the particles combined with the cation-binding properties these systems may be attractive in biomedical areas related to alginate-binding cations.

12 Assoc. Professor Christophe Schatz, University of Bordeaux, LCPO, conducts research aiming to better understand the interplay between the morphology, thermodynamics, and kinetics of non-covalent macromolecular assemblies with a special interest for out-ofequilibrium systems. He's also interested in various applications of macromolecular assemblies in drug and gene delivery systems with particular emphasis on polysaccharide-based systems.

Professor Bjørn E. Christensen, NTNU, has worked with a wide range of polysaccharides using mainly chemical and biophysical approaches related to polysaccharide stability and degradation, fractionation, purification, structure/modification, order/disorder/conformation and solution properties, and biopolymer self-assembly.

Figure 1 .

 1 Figure 1. Algal and bacterial alginates are obtained via the intermediate homopolymeric

Figure 2 .

 2 Figure 2. (a) Partial acid hydrolysis of alginate to DPn ca 20 provides a heterogeneous

Figure 3

 3 shows an example demonstrating the changes in chain length distribution of the G-block fraction during weak acid hydrolysis.

Figure 3 .

 3 Figure 3. HPAEC-PAD chromatograms demonstrating the distribution of guluronate oligomers

Figure 4 .

 4 Figure 4. a) Terminal versus lateral substation. b) Ca ++ -induced self-assembly of a Gm-b-Dexn

Author Biographies

Amalie Solberg is a PhD student at NTNU who will defend her PhD in February 2022 on the topic alginate-based block polysaccharides. She now works as a research scientist at Rise PFI, Trondheim, Norway.

Professor Kurt Ingar Draget, NTNU, has for 35 years had an interest in fundamental aspects of order/disorder transitions of biopolymeric systems, including how inclusion of oligomers can be used to manipulate properties of high Mw polymer organization. In this context, Draget has also had a wide domestic and international industrial collaboration focusing on the applications of biopolymers and -oligomers. He holds 100+ scientific articles and book chapters as well as 35+ original PCT patents and patent applications


Philippe Bich

Bernard Cornet

Steve Smale

Existence of pseudo-equilibria in a financial economy

Keywords: Mathematics Subject Classification (2000). Primary 47H10, 91B50 Pseudo-equilibrium, incomplete markets, nonordered preferences, fixed-point-like theorems, Grassmann manifold. 1. Introduction

).

2 Philippe Bich and Bernard Cornet analysis and economics" (see [START_REF] Smale | Global analysis and economics I, Pareto optimum and a generalization of Morse Theory[END_REF], [START_REF] Smale | Global analysis and economics, IIA, extension of a theorem of Debreu[END_REF], [START_REF] Smale | Global analysis and economics, III, Pareto optima and price equilibrium[END_REF], [START_REF] Smale | Global analysis and economics, IV, finiteness and stability of equilibria with general consumption sets and production[END_REF], [START_REF] Smale | Global analysis and economics, V, Pareto theory with constraints[END_REF]). This approach had a tremendous influence in the literature including the present paper and is presented in the books by Balasko [START_REF] Balasko | Foundations of the Theory of General Equilibrium[END_REF], Dierker [START_REF] Dierker | Topological Methods in Walrasian Economics[END_REF] and Mas-Colell [START_REF] Colell | The Theory of General Economic Equilibrium : a Differentiable Approach[END_REF]).

General Equilibrium Theory with Incomplete markets aims to study the interactions of the financial part of the economy and its commodity part in a world where time and uncertainty play a fundamental role (see, for example, Radner [START_REF] Radner | Existence of equilibrium plans, prices, and price expectations[END_REF], Magill-Quinzii [START_REF] Magill | Theory of Incomplete Markets[END_REF], or Geanakoplos [START_REF] Geanakoplos | An introduction to general equilibrium with incomplete asset markets[END_REF] for a survey on GEI model). The mathematical tools used to prove the existence still rely on fixed-point-like theorems but of a different kind since some variables may belong to a Grassmann manifold G k (V ), that is, the set of vector spaces of fixed dimension k in a given Euclidean space V . The main difficulty to deal with this extended framework comes from the fact that, in general, the set G k (V ) does not satisfy properties of convexity or acyclicity that would allow to use standard fixed-point theorems of Brouwer, Kakutani or Eilenberg-Montgomery [START_REF] Eilenberg | Fixed point theorems for multivalued transformations[END_REF] to deduce the existence of equilibria as in the standard case.

The aim of this paper is to prove the existence of a pseudo-equilibrium in a financial economy with incomplete markets in which the agents may have nonordered preferences and thus extend the existence result by Duffie-Shafer [START_REF] Duffie | Equilibrium in incomplete markets I: basic model of generic existence[END_REF]. We will use a fixed-point-like theorem of [START_REF] Bich | Fixed-point-like theorems on subspaces[END_REF] that generalizes the results by Hirsch, Magill, Mas-Colell [START_REF] Hirsch | A geometric approach to a class of equilibrium existence theorems[END_REF] and Husseiny, Lasry, Magill [START_REF] Husseini | Existence of equilibrium with incomplete markets[END_REF] to encompass the framework considered by Gale and Mas-Colell ( [START_REF] Gale | An equilibrium existence theorem for a general model without ordered preferences[END_REF], [START_REF] Gale | Corrections to an equilibrium existence theorem for a general model without ordered preferences[END_REF]). The article is organized as follows. In Section 2, we present the general equilibrium model with incomplete markets and recall the definitions of an equilibrium and of a pseudo-equilibrium. We also state the main existence result (Theorem 1) for the existence of a pseudoequilibrium. The proof of the main existence result is given in Section 3.

The model and the main result 2.1. The model

We first give the formal definition of the model. There is a finite set I of consumers (by abuse of notation, I will also denote the cardinal of the set I) and we consider the simplest case of intertemporal model with two dates, t = 0 (today) and t = 1 (tomorrow) (see [START_REF] Magill | Theory of Incomplete Markets[END_REF], [START_REF] Angeloni | Existence of financial equilibria in a multi-period stochastic economy[END_REF], [START_REF] Cornet | Restricted participation and arbitrage in a multiperiod model[END_REF] for a generalization to several periods). There is no uncertainty at date 0, and the uncertainty at date t = 1 is represented by finitely many states of nature S (s = 1, . . . , S); only one state happens at t = 1 and it is only known at date t = 1. For convenience, the unique state of nature at t = 0 will be denoted s = 0. There is a positive number of divisible goods available at each date and each state s = 0, . . . , S; hence, the number of commodities available either at t = 0 (with certainty) or at t = 1 (contingent on each possible state of nature) is L = (1 + S). An economy E can be summarized by a list E = (X i , P i , e i ) i∈I , V where for every i ∈ I: Document de Travail du Centre d'Economie de la Sorbonne -2009.60

• X i ⊂ R L
is the consumption set of the i-th consumer, that is the set of possible consumptions of this consumer. • For all x ∈ X := Π i∈I X i , P i (x) ⊂ X i is the set of consumptions which are strictly preferred to x i by the i-th consumer, given the consumptions (x j ) j =i of the other consumers. 1 Thus, P i is a correspondence from X to X i . • e i ∈ R L is the initial endowment vector of the i-th consumer.

• At time t = 0, there exist financial markets for a positive number J of assets, J ≤ S. An asset j (j = 1, . . . , J) can be bought at time t = 0 and delivers at time t = 1 a financial payoff V j s (p) (in unit of account) if state s prevails, given the commodity price vector p ∈ R L . We denote by V (p) = V j s (p) the S × J payoff matrix across states s = 1, . . . , S. Given the asset price vector q = (q 1 , . . . , q J ) ∈ R J specifying the price q j of the j-th asset for j = 1, . . . , J, we denote by W (p, q) the (1 + S) × J total payoff matrix across states s = 0, . . . , S, that is

W (p, q) = -q V (p) .
The vector z = (z 1 , . . . , z J ) ∈ R J denotes the portfolio of a consumer and specifies the quantity |z j | of each asset j in the portfolio, with the usual convention that if z j > 0 then |z j | represents the quantity of asset j bought at date 0 and if z j < 0 then |z j | represents the quantity of asset j sold at date 0. Then W (p, q)z is a (1 + S)-column vector (or by extension a vector in R 1+S ) which describes the financial stream of the portfolio z across the 1 + S states of nature.

Equilibria and pseudo-equilibria

We now formally define the notions of equilibrium and of pseudo-equilibrium. 2 Definition 2.1. An equilibrium of the economy E is a list (x, z, p, q) in R LI × R JI × R L × R J such that, if we let x = (x i ) i∈I and z = (z i ) i∈I , one has: (i) for every i ∈ I, (x i , zi ) ∈ B i (p, q) and (P i (x) × R J ) ∩ B i (p, q) = ∅, where B i (p, q) = {(x i , z i ) ∈ X i × R J : p (x i -e i ) ≤ W (p, q)z i };

(ii) i∈I xi = i∈I e i ;

1 This general way to describe the tastes of the consumers encompasses the case where the consumer i has a preference relation i which is a complete preorder on X i . In this case, for all x ∈ Π i∈I X i , P i (x) = {x i ∈ X i : x i ≺ i x i } where the strict preference relation ≺ i is defined by

x i ≺ i x i if [ x i i x i and not x i i x i ].
2 In this paper, we will often denote a vector x ∈ R L by x = (x(0), x(1), . . . , x(S)) where x(s) ∈ R for s = 0, 1, . . . , S. If p = (p(0), . . . , p(S)) ∈ R L [resp. λ = (λ(0), . . . , λ(S)) ∈ R 1+S ] and x = (x(0), . . . , x(S)) ∈ R L , we denote p x [resp. λ p] the vector in R 1+S [R L ] defined by p x = (p(0) • x(0), p(1) • x(1), . . . , p(S) • x(S))[resp. λ p = (λ(0)p(0), λ(1)p(1), . . . , λ(S)p(S))],

where x • y := P i=1 x i y i denotes the dot product of two vectors x = (x i ), y = (y i ) in R . Besides, for every integer n and every λ ∈ R n , λ ⊥ := {(x 1 , ..., xn) ∈ R n :

P n i=1 λ i x i = 0}. Last, if x = (x(0), . . . , x(S)) and y = (y(0), . . . , y(S)) are two vectors of R 1+S , or by extension two (1 + S)-column vectors, then x ≤ y (resp. x << y) means that for every s = 0, 1, . . . , S, one has x(s) ≤ y(s) (resp. x(s) < y(s)). Document de Travail du Centre d'Economie de la Sorbonne -2009.60 (iii) i∈I zi = 0. Condition (i) states that (x i , zi ) is optimal in consumer i's budget set B i (p, q). The so-called Market Clearing Conditions (ii) and (iii) guarantee the feasibility of the equilibrium allocation, and of the portfolio allocation.

We recall that even under strong assumptions on the economy, an equilibrium may fail to exist, because of possible discontinuities of the column span of W (p, q), denoted ImW (p, q) when prices vary (see, for example, Hart [START_REF] Hart | On the optimality of equilibrium when the market structure is incomplete[END_REF]). A standard answer (see Duffie, Shaffer [START_REF] Duffie | Equilibrium in incomplete markets I: basic model of generic existence[END_REF]) is to relax the previous equilibrium notion by the notion of pseudo-equilibrium and we refer to Aouani, Cornet [START_REF] Aouani | Existence of financial equilibria with restricted participation[END_REF] for another approach that assumes that the dimension of ImW (p, q) remains constant when prices vary. In the following, if V is a Euclidean space and k is an integer such that 0 ≤ k ≤dim(V ), then we denote by G k (V ) the set consisting of all the linear subspaces of V of dimension k.

Definition 2.2. A pseudo-equilibrium of the economy E is a list (x, p, q, Ē) in R LI × R L × R J × G J (R 1+S ) such that, if we let x = (x i ) i∈I , one has: (i) for every i ∈ I, xi ∈ B i (p, Ē) and P i (x) ∩ B i (p, Ē) = ∅, where B i (p, Ē) := x i ∈ X i : ∃t i ∈ Ē, p (x i -e i ) ≤ t i ; (ii) i∈I xi = i∈I e i ; (iii) ImW (p, q) ⊂ Ē .
The following proposition shows the relationship between equilibria and pseudoequilibria.

Proposition 2.1. (a) Let (x, z, p, q) be an equilibrium of the economy E such that rank V (p) = J, then (x, p, q, ImW (p, q)) is a pseudo-equilibrium of E.

(b) Let (x, p, q, Ē) be a pseudo-equilibrium of E such that Ē = ImW (p, q), hen there exist z = (z i ) i∈I ∈ R JI such that (x, z, p, q) is an equilibrium of E if we additionally assume that Non-Satiation: for every i ∈ I, for every s 0 ∈ {0, 1, . . . , S}, there exists x i ∈ P i (x) such that x i (s) = xi (s) for every s = s 0 .

We will use the following lemma to prove Proposition 2.1.

Lemma 2.1. For every pseudo-equilibrium (x, p, q, Ē) of E satisfying the above Non-Satiation Assumption, one has Ē ∩ R 1+S + = {0}, or equivalently there exists λ ∈

R 1+S

++ such that Ē ⊂ λ ⊥ . Proof. First notice that the two conclusions of Lemma 2.1 are equivalent from a standard separation theorem. Let (x, p, q, Ē) be a pseudo-equilibrium and suppose that Ē∩R 1+S + = {0}. Then there exists t = (t(s)) s=0,...,S in Ē such that t(s) ≥ 0 for every s = 0, . . . , S and t(s 0 ) > 0 for some s 0 , 0 ≤ s 0 ≤ S. Under the Non-Satiation Assumption for consumer 1, there exists x 1 ∈ P 1 (x) such that x 1 (s) = x1 (s) for every s = s 0 . But, from the Pseudo-equilibrium Condition (i), there exists t1 ∈ Ē such that p (x 1 -e 1 ) ≤ t1 . Hence, for every integer n large enough, Document de Travail du Centre d'Economie de la Sorbonne -2009.60 p (x 1 -e 1 ) ≤ t1 + nt and t1 + nt ∈ Ē. Thus x 1 ∈ P 1 (x) ∩ B 1 (p, Ē) which is empty from the Pseudo-equilibrium Condition of Agent 1.

Proof of Proposition 2.1. Part (a). Let (x, z, p, q) be an equilibrium of the economy E such that rank V (p) = J. Clearly, one has rankV (p) =rankW (p, q) = J, which guarantees that ImW (p, q) ∈ G J (R 1+S ) and that (x, p, q,ImW (p, q)) is a pseudoequilibrium of E. Part (b). Let (x, p, q, Ē) be a pseudo-equilibrium of E such that Ē =ImW (p, q). For every i ∈ I, there exists ti ∈ Ē such that p (x i -e i ) ≤ ti . Summing up over i these inequalities, we obtain

p i∈I (x i -e i ) ≤ i∈I ti .
But i∈I (x i -e i ) = 0, from the Market Clearing Condition (ii), thus i∈I ti ≥ 0, and we also have i∈I ti ∈ Ē. From Lemma 2.1, one has Ē ∩ R 1+S + = {0} and one deduces that i∈I ti = 0. Finally, i∈I (p (x i -e i ) -ti ) = 0, and each term of this sum belongs to -R 1+S + . Hence, each of these terms is null, that is: p (x i -e i ) = ti , for every i ∈ I.

Since, for every i ∈ I, ti ∈ Ē =ImW (p, q), there exists z i ∈ R J such that ti = W (p, q)z i . So we obtain

0 = p i∈I (x i -e i ) = i∈I ti = W (p, q)( i∈I z i ).
We now let zi = z i -(1/I)( i∈I z i ) for every i ∈ I. Then, i∈I zi = 0, that is, the Market Clearing Condition (iii) holds. Furthermore, we have

p (x i -e i ) = ti = W (p, q)z i = W (p, q)z i .
This shows that (x, z, p, q) is an equilibrium of E.

The main existence theorem

We now set the basic assumptions on the economy E we will consider hereafter. We first define the set of attainable allocations of the economy, that is,

A(E) := {(x 1 , . . . , x I ) ∈ X 1 × . . . × X I : i∈I x i = i∈I e i }.
Consumption Assumption C: For every i ∈ I:

(i) [Consumption sets] X i is closed, convex and bounded below, in the sense that there exists

x i ∈ R L such that X i ⊂ x i + R L + ; (ii) [Continuity] the correspondence P i is lower semicontinuous 3 ;
(iii) [Openess-type Assumption]4 : for all x ∈ X, for all x i ∈ P i (x), for all

x i ∈ X i , x i = x i then [x i , x i ) ∩ P i (x) = ∅; (iv) [Convexity] for every x ∈ X, P i (x) is convex; (v) [Irreflexivity] for every x ∈ X, x i / ∈ P i (x);
(vi) [Non-Satiation] for every x ∈ A(E), for every s 0 ∈ {0, 1, . . . , S}, there exists x i ∈ X i such that x i (s) = xi (s) for every s = s 0 and x i ∈ P i (x);

(vii) [Strong Survival Assumption] e i ∈ intX i .
Financial Assumption F: The mapping p → V (p) is continuous on R L .

We now state the main existence theorem which extends the existence result of Duffie-Shafer [START_REF] Duffie | Equilibrium in incomplete markets I: basic model of generic existence[END_REF] to the case of agents with nonordered preferences.

Theorem 2.1. Assume the economy E satisfy Assumptions C and F. For every λ ∈ R 1+S ++ , there exists a pseudo-equilibrium (x, p, q, Ē) of E such that Ē ⊂ λ ⊥ . The proof of Theorem 2.1 is given in Section 3.

When the payoff matrix V (p) does not depend upon the price p, the financial structure is said to be nominal. We can now deduce the existence result of Cass [START_REF] Cass | Competitive equilibrium with incomplete financial markets[END_REF], Duffie [START_REF] Duffie | Stochastic equilibria with incomplete financial markets[END_REF], Werner [START_REF] Werner | Equilibrium in economies with incomplete financial markets[END_REF] in the case of a nominal financial structure.

Corollary 2.1. Let E = (X i , P i , e i ) i=1....,I , V satisfy Assumptions C and F, where V is a nominal asset structure. For every λ ∈ R 1+S ++ such that λ 0 = 1, there exists an equilibrium (x, z, p, q) of E such that q j = S s=1 λ s V j s for every j = 1, . . . , J.

Proof. For every λ ∈ R 1+S ++ such that λ 0 = 1, define q j = S s=1 λ s V j s for every j. Let J =rank V and consider an economy E = (X i , P i , e i ) i=1....,I , V , where V is a S×J full-rank submatrix of V . From Theorem 2.1 and Proposition 2.1 we deduce the existence of an equilibrium (x, z , p, q ) of E . Then, one can easily modify the agents portfolios z and the asset prices q to get an equilibrium (x, z, p, q) of E.

3. Proof of Theorem 2.1

Proof of Theorem 2.1 under additional assumptions

In this section, we first prove Theorem 2.1 under the following additional assumptions. The next Section 3.2 will then provide a proof of Theorem 2.1 in the general case.

Assumptions A: (i) [Local Non-Satiation] For every x = (x i ) i∈I ∈ A(E), for every i ∈ I, for every x i ∈ P i (x), then (x i , x i ] ⊂ P i (x).

(ii) [Compactness] For every i ∈ I, X i is a compact subset of R L .

Definition of the reaction correspondences.

Let B be the closed unit ball of R L , let 1l be the vector in R 1+S whose coordinates are all equal to 1, and let λ ∈ R 1+S ++ such that λ(0) = 1. We now introduce some definitions in which (x, p, E) is given in Π i∈I X i × B × G J (λ ⊥ ):

• for i = 2, . . . , I, we consider the "augmented" budget sets:

β i (p, E) = {x i ∈ X i : ∃t i ∈ E, p (x i -e i ) ≤ t i + (1 -p )1l}, βi (p, E) = {x i ∈ X i : ∃t i ∈ E, p (x i -e i ) t i + (1 -p )1l};
• for i = 1, following the so-called "Cass trick," we consider the "augmented" Walrasian budget set (which does not depend on E):

β 1 (p, E) = {x 1 ∈ X 1 : (λ p) • (x 1 -e 1 ) ≤ 1 -p }, β1 (p, E) = {x 1 ∈ X 1 : (λ p) • (x 1 -e 1 )
1 -p };

• for i = 1, . . . , I, we let:

Φ i (x, p, E) =    {e i } if x i / ∈ β i (p, E) and βi (p, E) = ∅, β i (p, E) ifx i / ∈ β i (p, E) and βi (p, E) = ∅, βi (p, E) ∩ P i (x) if x i ∈ β i (p, E);
• for i = 0, the revision of prices is done according to the standard rule:

Φ 0 (x, p, E) = {p ∈ B : p • I i=1 (x i -e i ) > p • I i=1 (x i -e i )};
• for j = 1, . . . , J, we let:

ψ j (x, p, E) = (- S s=1
λ(s)V j s (p), V j 1 (p), . . . , V j S (p)).

The properties of the above correspondences and mappings are summarized in the following lemma. Throughout this paper, for every Euclidean space V and every integer k ≤dimV , we will consider on the Grassmann manifold G k (V ) the standard topology defined as follows. First, define the Stiefel manifold V k (V ) as the collection of all k-tuples of linearly independent vectors of V . Clearly, V k (V ) is an open subset of V k and it is endowed with the topology induced by the one of V k . Then, define the mapping f : V k (V ) → G k (V ) by f (e 1 , . . . , e k ) =span{e 1 , . . . , e k }, for every (e 1 , . . . , e k ) ∈ V k (V ), and define on G k (V ) the topology as follows; a subset [START_REF] Milnor | Characteristic classes[END_REF]). Lemma 3.1. For every i = 0, . . . , I, the correspondence Φ i is lower semicontinuous and has convex (possibly empty) values. For every j = 1, . . . , J, ψ j is a continuous mapping. Proof. The assertions on Φ 0 and the ψ j , j = 1, . . . , J, are straightforward. We notice that, for every i = 1, . . . , I, Φ i has convex values and we now prove that, for every i ∈ {2, . . . , I}, Φ i is lower semicontinuous, the proof being similar for Φ 1 .

U ⊂ G k (V ) is open if and only if its inverse image f -1 (U ) is open in V k (V ) (see, for example,

Document de

We will use the following claim, the proof of which relies on the definition of the topology of G J (R 1+S ) and is given in [START_REF] Bich | Fixed-point-like theorems on subspaces[END_REF]. Claim 3.1. For every Euclidean space V and every integer k ≤ dim V , the set

{(t, E) ∈ V × G k (V ) : t ∈ E} is closed.
Step (i). First remark that βi is an open-graph correspondence, so the set

A i = {(x, p, E) ∈ Π k∈I X k × B × G J (λ ⊥ ) : βi (p, E) = ∅} is open.
Step (ii). We now prove that β i is a closed-graph correspondence for i = 2, . . . , I. Indeed, consider a sequence (

x n i , p n , E n ) in X i × B × G J (λ ⊥ ) such that x n i ∈ β i (p n , E n )
for every integer n, and suppose that (

x n i , p n , E n ) converges to (x i , p, E) ∈ X i × B × G J (λ ⊥ ).
Then there exists a sequence (t n ) in R 1+S , such that t n ∈ E n for every integer n, and

p n (x n i -e i ) ≤ t n + (1 -p n )1l
. Now, remark that the sequence (t n ) is bounded. Indeed, if it is not the case, without any loss of generality one can suppose that lim n→∞ t n = ∞ and that t n / t n converges to some t ∈ R 1+S such that t = 1. For n large enough, we have

p n (x n i -e i ) t n ≤ t n t n + (1 -p n )1l t n .
Hence, passing to limit in the above inequality, we would obtain 0 ≤ t (since the sequences (p n ) and (x n i ) belong to the bounded sets B and X i ), t = 0 (since t = 1) and t • λ = 0 (since t n ∈ E n ⊂ λ ⊥ for every n). This contradicts λ 0 and ends the proof that (t n ) is bounded. Now, since the sequence (t n ) is bounded, without any loss of generality one can suppose that (t n ) converges to some t ∈ λ ⊥ . From the above inequality, we obtain p (x i -e i ) ≤ t + (1 -p )1l.

Besides, recalling that t n ∈ E n for every n, from Claim 3.1, one gets t ∈ E. This proves that β i is a closed-graph correspondence for i = 2, . . . , I. Consequently, the set

A i = {(x, p, E) ∈ Π k∈I X k × B × G J (λ ⊥ ) : x i / ∈ β i (p, E)} is open.
Step (iii). Now, notice that on the set A i , β i is the closure of the open-graph correspondence βi ; so β i is lower semicontinuous on A i .

Step (iv). Thus, using Steps (i), (ii), (iii), the facts that e i ∈ β i (p, E) and βi (p, E)∩ P i (x) ⊂ β i (p, E), one easily proves that Φ i is lower semicontinuous.

The fixed-point argument.

The existence proof relies on the following fixedpoint-like theorem by Bich and Cornet (see [START_REF] Bich | Fixed-point-like theorems on subspaces[END_REF]).

Document de Travail du Centre d'Economie de la Sorbonne -2009.60

Theorem 3.1. Let V be a finite dimensional Euclidean space, for i = 1, . . . , n, let X i be a nonempty, compact, convex subset of some Euclidean space, let J be a positive integer such that J ≤ dim V , and let X = Π n i=1 X i × G J (V ). For i = 1, . . . , n, let Φ i be a correspondence from X to X i , which is lower semicontinuous and convex-valued (possibly empty-valued), and for j = 1, . . . , J, let ψ j : X → V be a continuous mapping.

Then, there exists

x = (x 1 , . . . , xn , Ē) ∈ X 1 × .. × X n × G J (V ) such that (a) for every i = 1, . . . , n, [either xi ∈ Φ i (x) or Φ i (x) = ∅];
(b) for every j = 1, . . . , J, ψ j (x) ∈ Ē.

The proof of Theorem 3.1 can be found in [START_REF] Bich | Fixed-point-like theorems on subspaces[END_REF]. Remark that this theorem is a generalization first of Gale and Mas-Colell Theorem (see [START_REF] Gale | An equilibrium existence theorem for a general model without ordered preferences[END_REF] and [START_REF] Gale | Corrections to an equilibrium existence theorem for a general model without ordered preferences[END_REF]), and also of Hirsch, Magill and Mas-Colell [START_REF] Hirsch | A geometric approach to a class of equilibrium existence theorems[END_REF], Husseini, Lasry and Magill [START_REF] Husseini | Existence of equilibrium with incomplete markets[END_REF].

We will now use Theorem 3.1 to prove the existence of a pseudo-equilibrium. Let V = λ ⊥ and n = I+1. For every i = 1, . . . , I, X i is taken to be the consumption set of the i-th consumer which is convex, compact and nonempty [from Assumption C], and for i = 0, X 0 = B, the closed unit ball of R L . From Lemma 3.1, the correspondences Φ i (i = 0, 1, . . . , I) and the mappings ψ j (j = 1, . . . , J), defined in the previous section, satisfy the assumptions of Theorem 3.1. Consequently, from Theorem 3.1, there exists (x 1 , . . . , xI , p, Ē) ∈ X 1 × . . . × X I × B × G J (V ) satisfying the above conditions (a) and (b) and we let x := (x 1 , . . . , xI ). We first notice that, by construction and from the Irreflexivity Assumption in C, xi / ∈ Φ i (x, p, Ē) for every i = 1, . . . , I, and p / ∈ Φ 0 (x, p, Ē). Consequently (a ) for every i = 0, . . . , I, Φ i (x, p, Ē) = ∅;

(b ) for every j = 1, . . . , J, ψ j (x, p, Ē) ∈ Ē.

We easily deduce the following conditions (using the fact that e i ∈ β i (p, Ē)):

We now let q = (q 1 , . . . , qJ ) by qj = S s=1 λ(s)V j s (p) for every j = 1, . . . , J.

3.1.3. The vector (x, p, q, Ē) is a pseudo-equilibrium of E. This is a consequence of the claims proved hereafter.

Claim 3.2. Im W (p, q) ⊂ Ē.

Proof. From our choice of q we notice that for every j = 1, . . . , J, ψ j (x, p, Ē) = (-q j , V j 1 (p), . . . , V j S (p)) = W j (p, q), Document de Travail du Centre d'Economie de la Sorbonne -2009.60

where W j (p, q) denotes the j-th column of the matrix W (p, q). Hence, Claim 3.2 follows from the Fixed-Point Condition FP3.

The Fixed-Point Condition FP1 implies that xi ∈ β i (p, Ē) for every i ∈ I, hence, from the definition of the sets β i (p, Ē), there exists ti ∈ Ē such that: p (x i -e i ) ≤ ti + (1 -p )1l, for every i = 2, . . . , I;

(3.1)

(λ p) • (x 1 -e 1 ) ≤ 1 -p . (3.2)
Since ti ∈ Ē ⊂ λ ⊥ for every i ∈ I, taking the scalar product of each side of (3.1) with the vector λ ∈ R S+1 ++ , we obtain: Proof.

(λ p) • (x i -e i ) ≤ (1 -p )
If I i=1 (x i -e i ) = 0, it follows from Condition FP2 that p = i∈I (x i -e i ) i∈I (x i -e i )
.

Hence, p =1 and

(λ p)• I i=1 (x i -e i ) = S s=0 λ(s)p(s)•( i (x i (s)-e i (s))) = S s=0 λ(s) i (x i (s) -e i (s)) 2 i (x i -e i ) and consequently (λ p) • I i=1 (x i -e i ) ≥ (min s λ(s)) i∈I (x i -e i ) > 0.
But summing up over i the above Inequalities (3.2) and (3.3), and recalling that p =1, we get

(λ p) • I i=1 (x i -e i ) ≤ (1 -p )(1 + S s=0
λ(s)) = 0, which contradicts the above inequality.

Claim 3.4. For every s = 0, 1, . . . , S, p(s) = 0.

Proof. We first notice that p = 0. Indeed, if p = 0, then βi (p, Ē) = β i (p, Ē) = X i , for every i ∈ I. But, from Claim 3. Proof. We first prove that each consumer budget constraint is binding, i.e.:

p (x i -e i ) = ti + (1 -p )1l for every i = 2, . . . , I.

(λ p) • (x 1 -e 1 ) = 1 -p . (3.4)
Indeed, suppose that (3.4) does not hold, then there exists i = 2, . . . , I such that p (x i -e i ) ≤ ti + (1 -p )1l, with a strict inequality for some component, say the s-th component. But x ∈ A(E) (Claim 3.3) and from the Non-Satiation Assumption in C and the additional Local Non-Satiation Assumption in A, there exists x i ∈ X i such that x i ∈ P i (x), x i (s ) = xi (s ) for every s = s, and [x i , xi ) ⊂ P i (x). Besides, we can choose x i ∈ [x i , xi ) close enough to xi so that x i ∈ β i (p, Ē). Consequently, x i ∈ P i (x) ∩ β i (p, Ē), which contradicts the fact that P i (x) ∩ β i (p, Ē) = ∅ (by Claim 3.5). This ends the proof of (3.4) and the proof of (3.5) is similar. Now, taking the scalar product of both sides of the equalities (3.4) with λ ∈ R S+1 ++ , recalling that ti ∈ Ē ⊂ λ ⊥ , we obtain Consider the set X = co{X 0 ∪ {S, N }}, which is the convex hull of the set X 0 and the points N and S defined as follows: N = (0, 0, 1), S = (0, 0, -1), X 0 = {(x, y, z) ∈ R 3 : z = 0, (x -1) 2 + y 2 ≤ 1}, and a correspondence P from X to X such that P (N ) := {(x, y, z) ∈ X : z < 0}. Then P (N ) is open for the relative topology of X and P (N ) is not open in X.

(λ p) • (x i -e i ) = (1 -p ) S s=0 λ (
3.2.2. Truncating the economy. Consider an economy E = (X i , P i , e i ) i∈I , V which satisfies Assumption C. Since the consumption sets X i are bounded below, we deduce that the attainable set A(E) is compact. Denoting Xi the projection of A(E) on X i , we deduce that the set Xi is bounded for every i ∈ I. Consequently, one can choose r > 0 large enough such that Xi ⊂ int B(0, r) for every i ∈ I.

For every i ∈ I, we let X r i = X i ∩ B(0, r), P r i (x) = P i (x) ∩ X r i , and we define a new economy E r which only differs from E by the fact the consumption sets X i have been replaced by the above sets X r i and the correspondences P i by P r i . To summarize, we let the truncated economy of E be E r := (X r i , P r i , e i ) i∈I , V .

Then, one can easily prove the following lemma:

Lemma 3.3. Let E satisfy Assumptions C, F, and the Local Non-Satiation Assumption in A and let r > 0 be chosen as above. Then E r satisfies Assumptions C, F and A and every pseudo-equilibrium of E r is a pseudo-equilibrium of E.

3.2.3. Proof of Theorem 2.1 in the general case. Let E = (X i , P i , e i ) i∈I , V satisfy Assumptions C and F. Applying Lemma 3.2 and Lemma 3.3, one can define the enlarged economy Ê which satisfies Assumptions C, F and the local Non Satiation Assumption in A and one can choose r > 0 large enough so that the truncated enlarged economy Êr satisfies Assumptions C, F and A. From Section 3.1 (providing the proof of Theorem 2.1 with the additional Assumptions A), there exists a pseudo-equilibrium (x, p, q, Ē) of Êr such that Ē ⊂ λ ⊥ . From Lemma 3.2 and Lemma 3.3, (x, p, q, Ē) is also a pseudo-equilibrium of the economy E. This ends the proof of Theorem 2.1.

Document de

  Travail du Centre d'Economie de la Sorbonne -2009.60

  xi ∈ β i (p, Ē) and P i (x)∩ βi (p, Ē) = ∅ for every i ∈ I; i -e i ) ≤ p• I i=1 (x i -e i ) for every p ∈ B;(FP2) ψ j (x, p, Ē) ∈ Ē for every j = 1, . . . , J.

  3, x ∈ A(E) and from the Non-Satiation Assumption in C, there exists x i ∈ P i (x) = P i (x) ∩ βi (p, Ē). This contradicts the Fixed-Point Condition FP1. Document de Travail du Centre d'Economie de la Sorbonne -2009.60

  Travail du Centre d'Economie de la Sorbonne -2009.60

  Remark 3.1. The following counterexample shows that Pi may not have open values (relatively to X i ) when P i has open values (relatively to X i ). This is the main reason why we use the Openess-type Assumption C-(iii) (instead of assuming that P i has open values) in both Lemma 3.2 and Theorem 2.1.

s) for every i = 2, . . . , I. (3.6) Document de Travail du Centre d'Economie de la Sorbonne -2009.60

A correspondence F , from X to Y , is said to be lower semicontinuous (l.s.c.) [resp. upper semicontinuous (u.s.c.)] if the set {x ∈ X :F (x) ∩ U = ∅} [resp. {x ∈ X : F (x) ⊂ U }] is open in X for every open set U ⊂ Y . We define the graph of F by G(F ) := {(x, y) ∈ X × Y : y ∈ F (x)}. The correspondence F is said to be open-graph (resp. closed-graph) if G(F ) is open (resp.closed) in X × Y (for its relative topology). Document de Travail du Centre d'Economie de la Sorbonne -2009.60

This assumption, which is satisfied, in particular, when P i (x) is open in X i (for its relative topology) is discussed in Remark 3.1 in Section 3.2.1. Document de Travail du Centre d'Economie de la Sorbonne -2009.60 Existence of pseudo-equilibria in a financial economy

Kansas, USA e-mail: cornet@univ-paris1.fr, cornet@ku.edu Document de Travail du Centre d'Economie de la Sorbonne -2009.60

Philippe Bich and Bernard Cornet

Summing up the equalities (3.6) (for i = 2, . . . , I) and the equality (3.5) (for i = 1), using the fact that

Consequently, p = 1. Claim 3.7. For every i ∈ I, xi ∈ B i (p, Ē) and

Proof. Since p = 1 (from Claim 3.6), we have B i (p, Ē) = β i (p, Ē) for every i = 2, . . . , I. Hence, Claim 3.5 implies the result for every consumer i = 2, . . . , I.

Let us now consider the first consumer. Since Ē ⊂ λ ⊥ , one easily obtains that B 1 (p, Ē) ⊂ β 1 (p, Ē). From Claim 3.5, we get x1 ∈ β 1 (p, Ē) and

The proof of Claim 3.7 will thus be complete if we show that x1 ∈ B 1 (p, Ē), or the stronger assertion that p (x 1 -e 1 ) ∈ Ē. Since i=1 (x i -e i ) = 0 (Claim 3.3) and since the budget constraint of every consumer is binding (cf. the proof of Claim 3.6), for every i = 1, there exists ti ∈ Ē such that:

which ends the proof of the claim.

Proof of Theorem 2.1 in the general case

We now give the proof of Theorem 2.1 without considering the additional assumptions A. First, given an economy E which satisfies Assumptions C and F, we will consider a new economy Ê with enlarged strictly preferred sets as in Gale-Mas Colell ([14], [START_REF] Gale | Corrections to an equilibrium existence theorem for a general model without ordered preferences[END_REF]). Second, we will truncate the economy Ê, to define a new economy Êr , which satisfies Assumptions C, F and A. Thus, Theorem 2.1 (with the additional Assumption A) will provide the existence of a pseudo-equilibrium of Êr , and we will check that it is also a pseudo-equilibrium of E.

3.2.1. Enlarging the preferences. Consider an economy E = (X i , P i , e i ) i∈I , V . Following Gale and Mas-Colell ([14], [START_REF] Gale | Corrections to an equilibrium existence theorem for a general model without ordered preferences[END_REF]), for every x ∈ Π i∈I X i , we define the "augmented preferences" Pi by: Pi

We will need the following lemma, the proof of which is similar to the one given by Gale and Mas-Colell ( [START_REF] Gale | An equilibrium existence theorem for a general model without ordered preferences[END_REF], [START_REF] Gale | Corrections to an equilibrium existence theorem for a general model without ordered preferences[END_REF])) (see also Angeloni-Cornet [START_REF] Angeloni | Existence of financial equilibria in a multi-period stochastic economy[END_REF]). Existence of pseudo-equilibria in a financial economy


Bich Philippe
email: bich@univ-paris1.fr.

Existence of pure Nash equilibria in discontinuous and non quasiconcave games

Keywords: Nash equilibrium, discontinuity, quasiconcavity

In a recent but well known paper, Reny has proved the existence of Nash equilibria for compact and quasiconcave games, with possibly discontinuous payoff functions. In this paper, we prove that the quasiconcavity assumption in Reny's theorem can be weakened: we introduce a measure allowing to localize the lack of quasiconcavity, which allows to refine the analysis of equilibrium existence.

Introduction

The purpose of this paper is to relax the quasiconcavity assumption in the standard Nash equilibrium existence results. Several papers have weakened the continuity assumption of payoff functions (see, for example, [START_REF] Topkis | Equilibrium Points in Nonzero-Sum n-Person Submodular Games[END_REF], [START_REF] Dasgupta | The Existence of an Equilibrium in Discontinuous Economics Games I: Theory[END_REF], [START_REF] Baye | Characterization of the Existence of Equilibria in Games with Discontinuous and Non-quasiconcave Payoffs[END_REF], [START_REF] Reny | On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games[END_REF] or more recently Carmona (in Press)), with various applications, for example to Hotelling's model of price competition or to patent races. Yet, only a few papers have tried to weaken the quasiconcavity assumption, although many games in the economic literature have non quasiconcave payoff functions. Such papers could be classified in several categories, observing the method used to circumvent the non quasiconcavity:

-a first possible method is to relax directly the convexity assumption of the best reply correspondences (see, for example, [START_REF] Friedman | Existence of Nash Equilibrium in n Person Games without Quasi-Concavity[END_REF], [START_REF] Mclennan | Fixed points of contractible valued correspondences[END_REF] or Mc-Clendon (2005)). Unfortunately, the properties assumed on the best reply correspondences are generally not derived from hypotheses on the payoff functions. Thus, such technique may be difficult to use in practice;

-a second method is to use the convexification of preferences when the number of players becomes sufficiently large (see, for example, [START_REF] Starr | Quasi-Equilibria in Markets with Non-Convex Preferences[END_REF]). A drawback of this approach is that it depends on the number of players;

-a third approach is to enlarge the definition of a pure Nash equilibrium, for instance by considering mixed-strategy equilibria, or generalized equilibria (see [START_REF] Kostreva | Nonconvexity in Noncooperative Game Theory[END_REF]);

-a fourth way is to remark that the standard quasiconcavity assumption can be weakened by requiring it to hold only along the diagonal of payoffs (see, for more details, Baye, Tian and Zhou (1993) or [START_REF] Reny | On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games[END_REF]);

-last, another answer to the nonconvexity issue is to look at particular classes of games, as supermodular games (see, for example, [START_REF] Topkis | Equilibrium Points in Nonzero-Sum n-Person Submodular Games[END_REF]), for which the standard topological fixed point theorems can be avoided, using lattice-theoritical techniques.

In this paper, we propose another approach to obtain the existence of a (standard) Nash equilibrium in pure strategies, without assumptions on the best reply correspondences or on the number of agents, and we allow non quasiconcavity of payoff functions.

First, for every player i, we introduce a function ρ i : X → IR (where X is the product of the pure strategy sets of the agents) which measures the non quasiconcavity of the payoff function of player i, and which is easy to compute for many games.

Then, using the measures ρ i , we exhibit a condition called strong better-reply security, which is a reinforcement of Reny's better-reply security condition (Reny (1999)), and which provides the existence of a Nash equilibrium in pure strategies without quasiconcavity assumption. More precisely, our main condition says that for every non equilibrium strategy profile x * and every payoff vector u * resulting from strategies approaching x * , some player i has a strategy yielding a payoff strictly above u * i + ρ i (x * ) even if the others deviate slightly from x * . Since for quasiconcave games we obtain ρ i (x * ) = 0 for every player, in this case the last condition is exactly the better-reply security assumption of Reny.

Roughly speaking, the principle of the Nash equilibrium existence proof is to associate to each game G a quasiconcavified game G as follows: for every player i and every strategy profile x -i of the other players, player i's payoff function u i (., x -i ) is replaced by its quasiconcave hull. Then, assuming G is strongly better-reply secure, it is proved that the game G is better-reply secure, and also that the set of equilibria of G is equal to the set of equilibria of G, which finally yields the existence of an equilibrium from Reny's theorem.

The motivation for the weakening of the quasiconcavity assumption, aside from the fact that many games exhibit non quasiconcavity, could be also a better understanding of the existence or non existence issue of equilibria: up to now, most attention in the literature has been concentrated on the continuity problem, and one of the aim of this paper is to offer a method to refine the analysis of equilibrium existence in game theory, in particular to be able to localize the non quasiconcavity issues.

The remainder of this paper is organized as follows: in Section 2, we describe the non quasiconcavity measure and its main properties. In Section 3, we use the idea introduced in Section 2 to define our class of games, strongly better-reply secure games, which strictly contains the class of quasiconcave and better-reply secure games. Then, our main pure strategy equilibrium existence result is stated and proved. In Section 4 are described two simple conditions (payoff security and weak reciprocal upper semicontinuity), which together imply strong better-reply security. Finally, in Section 5, the previous results are extended to quasisymmetric games, for which the non quasiconcavity measure can be restricted along the diagonal of payoffs. This permits to extend some standard equilibrium existence results for quasisymmetric games, as Reny's one (1999) or Baye et al.'s one (1993), to a nonconvex framework.

Measure of lack of quasiconcavity

In this section, we define a measure ρ f of lack of quasiconcavity for every real-valued function f defined on a nonempty convex subset of a topological vector space. The idea we introduce will be used in the next section to measure the lack of quasiconcavity of payoff functions. Roughly, we want to overcome the dichotomy "to be quasiconcave or not to be quasiconcave", by defining a local index of non quasiconcavity.

For every n ∈ IN, let ∆ n-1 be the simplex of R n , defined by ∆ n-1 = {(t 1 , ..., t n ) ∈ R n + : t 1 + t 2 + ... + t n = 1}. Let X be a topological vector space. For every n ∈ IN, t ∈ ∆ n-1 and (x 1 , ..., x n ) ∈ X n , we denote t • x = t 1 x 1 + t 2 x 2 + ... + t n x n . Let Y be a convex subset of X, and consider a function f : Y → R. In the following, coY denotes the convex hull of Y .

We recall that f is said to be quasiconcave if for every n ∈ IN and for every (t, y) ∈ ∆ n-1 × Y n , one has f (t • y) ≥ min{f (y 1 ), ..., f (y n )}. We would like to measure how much the previous inequality can be false at x ∈ Y . For this purpose, we introduce the mapping π f (x) defined as the following supremum

π f (x) = sup{min{f (y 1 ), ..., f (y n )} -f (x)}
over all n ∈ N and all families {y 1 , ..., y n } of Y such that x ∈co{y 1 , ..., y n }. Our final measure of lack of quasiconcavity of f is the upper semicontinuous regularization of the previous mapping:

∀x ∈ Y, ρ f (x) = lim sup x →x π f (x ).
Definition 2.1 The mapping ρ f defined above is called the measure of lack of quasiconcavity of f Figure 1 gives an example of non quasiconcave mapping and of its measure of lack of quasiconcavity. The following lemma gathers some straightforward properties of ρ f .

Lemma 2.1 i) One has ρ f ≥ π f ≥ 0. ii) If f is bounded, then so is ρ f . iii) f is quasiconcave if and only if for every x ∈ Y, ρ f (x) = 0.
Now, the following proposition links ρ f to the quasiconcave envelope of f :

d d d d d f (x) x c a b x 0 x 1 x 2 d x d d d d d ρ(x) b -a x 0 x 1 x 2
Figure 1: graph of a non quasiconcave mapping f and of its measure of lack of quasiconcavity.

Proposition 2.2 Let f be the quasiconcave hull of f (see [START_REF] Frenk | Chapter 1 in the Handbook of Generalized Convexity and Generalized Monotonicity[END_REF], p.33), defined by

∀x ∈ Y, f (x) = inf{h(x) : h : Y → R quasiconcave, f ≤ h}.
Then, one has f (x) = sup{min{f (y 1 ), ..., f (y n )}} over all n ∈ N and all families {y 1 , ..., y n } of Y such that x ∈co{y 1 , ..., y n }. Thus,

π f (x) = f (x) -f (x).
Proof. Define f1 (x) for every x ∈ Y , as the following supremum f1 (x) = sup{min{f (y 1 ), ..., f (y n )}} over all n ∈ N and all families {y 1 , ..., y n } of Y such that x ∈co{y 1 , ..., y n }. Proposition 2.2 is equivalent to

∀x ∈ Y, inf{h(x) : h : Y → R quasiconcave, f ≤ h} = f1 (x).
To prove this equality, first notice that one has clearly f ≤ f1 from the definition of f1 . Second, if h : Y → R is quasiconcave with f ≤ h, then for every x ∈ Y , every n ∈ N and every family {y 1 , ..., y n } of Y such that x ∈ co{y 1 , ..., y n }, one has min{f (y 1 ), ..., f (y n )} ≤ min{h(y 1 ), ..., h(y n )} ≤ h(x). Passing to the supremum in the above inequality, one obtains f1 (x) ≤ h(x). Third, to finish, we prove that f1 is quasiconcave.

Let (x, y) ∈ Y 2 , λ ∈ [0, 1]
and ε > 0. From the definition of f1 , there exists n ∈ N and (x 1 , ..., x n , y 1 , ..., y n ) ∈ Y 2n such that x ∈ co{x 1 , ..., x n }, y ∈ co{y 1 , ..., y n }, and such that min{f (x 1 ), ..., f

(x n )} ≥ f1 (x) -ε and min{f (y 1 ), ..., f (y n )} ≥ f1 (y) -ε. Now, since λx + (1 -λ)y ∈ co{x 1 , ..., x n , y 1 , ..., y n }, one has f1 (λx + (1 -λ)y) ≥ min{f (x 1 ), ..., f (x n ), f (y 1 ), ..., f (y n )},
thus one finally gets

f1 (λx + (1 -λ)y) ≥ min{ f1 (x), f1 (y)} -ε.
This finishes the proof of Proposition 2.2.

Remark 2.2 Proposition 2.2 provides another possible definition of ρ f : it is the upper semicontinuous regularization of the distance between f and its quasiconcave hull. For a practical purpose, it is the definition we shall often use.

3 The class of strongly better-reply secure games

The aim of this section is to define a class of non quasiconcave games for which a Nash equilibrium exists. First, in the following subsection, we extend the definition of the measure of lack of quasiconcavity to payoff functions.

Definition of a game and measure of lack of quasiconcavity of payoff functions

Consider a game with N players. The pure strategy set of each player i, denoted by X i , is a non-empty, compact and convex subset of a topological vector space. Each agent i has a bounded payoff function

u i : X = N i=1 X i → R. A game G is a couple G = ((X i ) N i=1 , (u i ) N i=1
). Throughout this paper, a game G satisfying the above assumptions will be called a compact game.

For every x ∈ X and every i ∈ {1, ..., N }, we denote x -i = (x j ) j =i and X -i = Π j =i X j . We say that the game G is quasiconcave if for every player i and every strategy x -i ∈ X -i , the mapping u i (., x -i ), defined on X i , is quasiconcave.

Recall that x * = (x * 1 , ..., x * N ) ∈ X is a Nash equilibrium if for every player i, one has

u i (x * i , x * -i ) ≥ u i (x i , x * -i ) for every x i ∈ X i .
For instance, it is well known that for every compact and quasiconcave game, if the payoff functions are continuous then there exists a Nash equilibrium.

To weaken the standard quasiconcavity assumption, we introduce the measure of lack of quasiconcavity of payoff functions as follows, using the previous section: in the following definition, for every player i and every x = (x i , x -i ) ∈ X, u i (., x -i ) denotes the mapping defined from X i to R by u i (., x -i )(x i ) = u i (x) for every x i ∈ X i . Definition 3.1 For every i = 1, ..., N and every x ∈ X, we define the measure ρ i : X → R of lack of quasiconcavity of player i's payoff function as follows:

ρ i (x) = lim sup x →x π u i (.,x -i ) (x i ),
where the definition of π is given in Section 2.

Thus, from Proposition 2.2, the measure ρ i : X → R of lack of quasiconcavity of player i's payoff function at x = (x i , x -i ) is the upper semicontinuous regularization (with respect to the strategy profile x) at x of the distance between the quasiconcave hull ũi of u i (with respect to the action of player i) and u i . Clearly, for every x ∈ X, ρ i (x) ≥ 0, and a compact game G is quasiconcave if and only if one has ρ i = 0 for every player i. Besides, by definition, ρ i is upper semicontinuous.

The class of better-reply secure games

Before defining our class of games, we recall the definition of better-reply secure games. This notion was introduced by Reny (1999), who has proved that every quasiconcave, compact and better-reply secure game has a Nash equilibrium, thus extending most of the previous Nash equilibrium existence results. In the following, for every profile of payoff functions (u i ) N i=1 , Gr(u) = {(x, (u 1 (x), ..., u N (x)) : x ∈ X} denotes the graph of the payoff functions, and Γ(u) = Gr(u) denotes the closure of Gr(u).

Definition 3.2 Player i can secure a payoff of α ∈ R at x = (x i , x -i ) ∈ X if there exists x i ∈ X i and V x -i , an open neighborhood of x -i , such that u i (x i , x -i ) ≥ α for every x -i in V x -i .
Definition 3.3 A game G is better-reply secure if for every (x * , u * ) ∈ Γ(u) such that x * is not a Nash equilibrium, some player i can secure a payoff strictly above u * i .

The class of strongly better-reply secure games

In this subsection, we define our class of games, call strongly better-reply secure games:

Definition 3.4 A game G = ((X i ) N i=1 , (u i ) N i=1
) is said to be strongly better-reply secure if for every (x * , u * ) ∈ Γ(u) such that x * is not a Nash equilibrium, some player i can secure a payoff strictly above u * i + ρ i (x * ).

Remark 3.1 Clearly, our definition strengthens Reny's Definition: every strongly betterreply secure game is better-reply secure. But the class of compact and strongly better-reply secure games strictly generalizes the class of compact, quasiconcave and better-reply secure games, as stated in the following proposition.

Proposition 3.5 If a game G is quasiconcave, then it is strongly better-reply secure if and only if it is better-reply secure. Moreover, there exists some compact games which are strongly better-reply secure and which are not quasiconcave.

Proof. The first assertion is clear, since one has ρ i = 0 for every quasiconcave game and every player i. To prove the second assertion, see Example 1 and Example 2 of Section 3 or Example 3 of Section 4, where are defined compact games which are strongly better-reply secure and which are not quasiconcave.

Existence of Nash equilibria in compact and strongly better-reply secure games

The purpose of this subsection is to prove our main equilibrium existence result:

Theorem 3.2
If G is a compact and strongly better-reply secure game, then it admits a pure strategy Nash equilibrium.

Proof. The proof2 rests on Reny's main existence result, and is a clear consequence of the two following lemma. Hereafter, G is a strongly better-reply secure game; for every player i and every strategy x = (x i , x -i ) ∈ X, ũi (x i , x -i ) denotes the quasiconcave envelope of u i (., x -i ) (with respect to x i ) at x i , and G = ((X i ) N i=1 , (ũ i ) N i=1 ).

Lemma 3.3

The set of equilibria of G is equal to the set of equilibria of G.

Proof. First suppose that x ∈ X is an equilibrium of G. Then, for every player i, one has ũi (x) ≥ u i (x) ≥ u i (y i , x -i ) for every strategy y i ∈ X i . Now, let y i ∈ X i , n ∈ N and

(t, z) ∈ ∆ n-1 × (X i ) n such that y i = n k=1 t k z k . From the previous inequality, one has ũi (x) ≥ min{u i (z 1 , x -i ), ..., u i (z n , x -i )}, which entails ũi (x) ≥ sup n∈N, (t,z)∈∆ n-1 ×X n i , t•z=y i min{u i (z 1 , x -i ), ..., u i (z n , x -i )} = ũi (y i , x -i ),
which proves that x is an equilibrium of G. Now suppose that x ∈ X is an equilibrium of G, and suppose that it is not an equilibrium of G. From the strong better-reply security assumption, and since (x, u(x)) ∈ Γ(u), there exists a player i, a neighborhood V x -i of x -i and a strategy x i ∈ X i of player i such that one has

∀x -i ∈ V x -i , u i (x i , x -i ) > u i (x) + ρ i (x).
Recalling that ρ i (x) is the u.s.c. regularization of ũi (x) -u i (x) at x, one has ρ i (x) ≥ ũi (x) -u i (x). From this inequality and from the above inequality for

x -i = x -i , one obtains ũi (x i , x -i ) ≥ u i (x i , x -i ) > u i (x) + (ũ i (x) -u i (x)) = ũi (x)
which is a contradiction with the fact that x ∈ X is an equilibrium of G. Thus, x is an equilibrium of G Lemma 3.4 If G is strongly better-reply secure, then the game G is better-reply secure.

Proof. To prove Lemma 3.4, suppose that G is strongly better-reply secure; let (x, ũ) ∈ Γ(ũ) such that x is not an equilibrium of G. Let Π

1 : X × R N → X be defined by Π 1 (x, u) = x for every (x, u) ∈ X × R N . Let u ∈ (x,ũ)∈V u(Π 1 (V ∩ Gr(ũ))),
where the intersection is taken over all open neighborhoods V , in Γ(ũ), of (x, ũ). Note that this intersection is nonempty, because u is bounded and (u(Π 1 (V ∩ Gr(ũ)))) (x,ũ)∈V is a family of compact subsets of R N that has the finite intersection property. In a metric space context, u can be seen as the limit of a sequence (u(x n )) n∈N , where (x n , ũ(x n )) n∈N is a sequence converging to (x, ũ).

Since G is strongly better-reply secure, one has u i (x i , x -i ) > u i + ρ i (x) + ε for some ε > 0, some player i, some x i ∈ X i and for all x -i in some neighborhood

V x -i of x -i . Fix now x -i ∈ V x -i .
From the definition of u, and since ρ i (x) is the u.s.c. regularization of ũi (x) -u i (x) at x, there exists (y, ũ(y)) ∈ V , where V is some open neighborhood in Γ(ũ) of (x, ũ), such that one has

u i (x i , x -i ) > u i + (ũ i (y) -u i (y)) + ε 2 , (1)
d d d d d d d d d d d d u i (x i , x -i ) x i V i -x -i -x -i x -i V i d d d d d d d d d ũi (x i , x -i ) x i V i -x -i x -i V i Figure 2: Graph of u i (., x -i ) and ũi (., x -i ) in Example 1. d d ρ i (x i , x -i ) x i x -i x -i V i
( 1 2 -x)(1 -x) ( 1 2 -x)x x 2x 1 d d ũ2 (x, y) Graph of ũ2 (x, .) when x < 1 2 y ( 1 2 -x)(1 -x) ( 1 2 -x)x x 2x 1
Figure 4: Graph of u 2 (x, .) and of ũ2 (x, .) when x < 1 2 .

In this game, player 1 would like to choose the same location as player 2, whereas the behaviour of player 2 depends on the location of player 1: he would like to be far from player 1 if x < 1 2 , would like to be close to player 1 if x > 1 2 , and does not care for x = 1 2 . The game G is not quasiconcave, because u 2 (x, .) is not quasiconcave for x < 1 2 (see figure 4). More precisely, since u 1 (., y) is quasiconcave for every y ∈ Y , one has ρ 1 = 0, and we now compute ρ 2 to measure the lack of quasiconcavity of this game. Figure 4 represents the graph of u 2 (x, .) and of its quasiconcave envelop ũ2 (x, .) for x < 1 2 fixed; from Proposition 2.2, and since the payoff functions are continuous, one has ρ 2 (x, y) = ũ2 (x, y) -u 2 (x, y). Moreover, for x ≥ 1 2 , u 2 (x, .) is quasiconcave, thus ρ 2 (x, .) = 0 in this case. Now, to prove that G is strongly better-reply secure, let (x * , y * , u * 1 , u * 2 ) ∈ Γ(u) such that (x * , y * ) is not an equilibrium. First notice that if x * = y * , then player 1, whose payoff function is continuous and quasiconcave with respect to x, can strictly secure a payoff of u * 1 + ρ 1 (x * ) = u 1 (x * ) = -| x * -y * |, by playing y * . Thus, now suppose that x * = y * . Since (a, a) is an equilibrium for every a ∈ [ 1 2 , 1], one has x * < 1 2 . This implies ρ 2 (x * , x * ) = ( 12 -x * )x * . Consequently, player 2 can strictly secure u * 2 + ρ 2 (x * , x * ) = ( 1 2 -x * )x * by playing 2x * + ε ∈]0, 1[ for ε > 0 small enough: indeed, it gives him a payoff of ( 12 -x * )(x * + ε). Thus, G is strongly better-reply secure.

The class of weakly reciprocal upper semicontinuous games

Reny (1999) has introduced two simple conditions, called payoff security and reciprocal upper semicontinuity (rusc), which together imply better-reply security. An advantage of these conditions is that they can be checked without any reference to the set of Nash equilibria, contrarily to better-reply security. Following the idea of the last section, we propose3 a natural generalization of reciprocal upper semicontinuity in a non quasiconcave framework.

First recall the definitions of weak reciprocal upper semicontinuity. 4 In the following, if A and B are two sets, we let

A \ B = {x ∈ A : x / ∈ B}. Definition 4.1 A game G = ((X i ) N i=1 , (u i ) N i=1
) is weakly reciprocal upper semicontinuous (wrusc) if for any (x * , u * ) ∈ Γ(u) \ Gr(u), there is a player i and xi ∈ X i such that u

i (x i , x * -i ) > u * i . Definition 4.2 A game G = ((X i ) N i=1 , (u i ) N i=1
) is payoff secure if for every player i, for every x ∈ X and for every ε > 0, player i can secure a payoff of u i (x) -ε, which means that there exists xi ∈ X i such that u i (x i , x -i ) ≥ u i (x) -ε for all x -i in some neighborhood of x -i .

We now generalize wrusc to (possibly) non quasiconcave games:

Definition 4.3 A (possibly non quasiconcave) game G = ((X i ) N i=1 , (u i ) N i=1 ) is weakly recip- rocal upper semicontinuous (wrusc) if for any (x * , u * ) ∈ Γ(u) \ Gr(u -ρ), there is a player i and xi ∈ X i such that u i (x i , x * -i ) > u * i + ρ i (x * ).
For example, let G be a game whose payoff functions are continuous and satisfying the following property: for every x * ∈ X with ρ(x * ) = 0, there is a player i and xi ∈ X i such that u i (x i , x * -i ) > u i (x * ) + ρ i (x * ). Then, it is payoff secure and wrusc.

Proposition 4.4 If G is payoff secure and weakly reciprocal upper semicontinuous, then it is strongly better-reply secure.

Proof. The proof is a simple adaptation of the proof of Proposition 1 in [START_REF] Bagh | Reciprocal Upper Semicontinuity and Better Reply Secure Games: A Comment[END_REF]: take (x * , u * ) ∈ Γ(u), where x * is not an equilibrium. If (x * , u * ) ∈ Γ(u) \ Gr(u -ρ), then wrusc implies that there is a player i and xi ∈ X i such that u i (x i , x * -i ) > u * i + ρ i (x * ); clearly, payoff security at (x i , x * -i ) implies that player i can secure a payoff strictly above u * i + ρ i (x * ). Now, suppose that (x * , u * ) ∈ Gr(u-ρ). Then, for all i, u * i = u i (x * )-ρ i (x * ). Since x * is not an equilibrium, there is a player i and xi ∈ X i such that u i (x i , x * -i ) > u i (x * ) = u * i + ρ i (x * ), and again payoff security implies that player i can secure a payoff strictly above u * i + ρ i (x * ).

Note that there exist some strongly better-reply secure games which are not weakly reciprocal upper semicontinuous: indeed, in Example 1, for a ∈]0, min{V 1 , V 2 }[, one has ((a, a), (V 1 -a, V 2 -a)) ∈ Γ(u) \ Gr(u -ρ), but for every player i = 1, 2, there is no xi ∈ X i such that u i (x i , a) > V i -a + ρ i (a, a) = V i .

Example 3. Consider the following game G: there are two players i = 1, 2; the strategy sets of each player are X = Y = [-1, 1]; given the strategy x ∈ X of player 1 and the stratey y ∈ Y of player 2, the payoffs are defined as follows:

u 1 (x, y) = -x u 2 (x, y) = xy 2
This game is continuous, so it is payoff secure, but not quasiconcave. We now prove that it is wrusc. Let (x * , y * ) ∈ X × Y such that ρ(x * ) = 0. Clearly, it implies x * > 0, and one has u 1 (-1, y * ) = 1 > u 1 (x * , y * ) + ρ 1 (x * , y * ) = -x * . It finally proves that G is wrusc (see the remark following Definition 4.3).

3

  Document de Travail du Centre d'Economie de la Sorbonne -2009.61

Figure 3 : 9 y

 39 Figure 3: Graph of ρ i (., x -i ) in Example 1.

Document de Travail du Centre d'Economie de la Sorbonne -2009.61

Document de Travail du Centre d'Economie de la Sorbonne -2009.61

The principle of the following proof was suggested by Philip Reny. 6 Document de Travail du Centre d'Economie de la Sorbonne -2009.61

I am grateful to two anonymous referees for suggesting me to adapt the results of the previous section for the class of payoff secure and reciprocal upper semicontinuous games.

This property, very similar to rusc, and which is implied by rusc, was introduced by Bahg and Jofre (2006), who have noticed that every wrusc and payoff secure game was better-reply secure.

and

Thus, from the above equations, one obtains u i (x i , x -i ) > u i (y)+(ũ i -u i (y))+ ε 4 = ũi + ε 4 , which also implies ũi (x i , x -i ) > ũi + ε 4 . This prove that G is better-reply secure, and finishes the proof of Lemma 3.4.

If G is quasiconcave, Theorem 3.2 is exactly Reny's Theorem. Thus, Theorem 3.2 gives a simple characterization (thanks to strong better-reply security assumption) that guarantees that the quasiconcavified game G is better-reply secure, and has the same equilibria as the game G. Besides, this characterization measures precisely the "cost"(through the measure of non quasiconcavity ρ) that has to be paid to restore equilibrium existence. Two examples illustrate Theorem 3.2:

Example 1. Consider the following game G: there are two players i = 1, 2; the strategy sets of each player are

, where V 1 > 0 and V 2 > 0; the payoff functions are defined as follows, where -i denotes 2 if i = 1 and 1 if i = 2: for every

Clearly, G is not quasiconcave (see Figure 2) but is compact. To compute the measure of lack of quasiconcavity ρ i , from Proposition 2.2, we only have to find ũi (., x -i ), the envelop of u i (., x -i ) (x -i ∈ X -i being fixed). Then, ρ i (x i , x -i ) is the upper semicontinuous regularization (with respect to x = (x i , x -i )) of ũi (x) -u i (., x -i ). See Figure 2 and Figure 3 for a representation of u i (., x -i ), ũi (., x -i ) and ρ i (., x -i ). Now, to prove that G is strongly better-reply secure, let

is not an equilibrium. Thus, x * 1 = x * 2 , because for every a ∈ [0, min{V 1 , V 2 }], (a, a) is a Nash equilibrium of G. Without any loss of generality, one can suppose that x

, it proves that player 2 can secure a payoff strictly above u * 2 + ρ 2 (x * ) by playing x 2 (because the payoff of player 2 moves continuously when the strategy x * 1 = x 2 of player 1 is slightly modified). In the next example, we provide a continuous and compact game which is not quasiconcave, but which is strongly better-reply secure:

Example 2. Consider the following location game G: there are two players i = 1, 2; the strategy sets of each player are X = Y = [0, 1]; for every strategy x of player 1 and every strategy y of player 2, the payoff functions are defined as follows:

Symmetric equilibria

In this section, we improve the conclusion of the previous section, by considering the more restricted class of quasisymmetric games. Recall that a game G = ((

y, y, ..., y) = u 2 (y, x, y, y, ..., y) = ... = u N (y, y, ..., y, x) for every x ∈ X 1 and for every y ∈ X 1 . For N = 2, a quasisymmetric game is called a symmetric game. In the following, we let X = X 1 , and the quasisymmetric game will be denoted G = (X, (u i ) N i=1 ). In such games, one can define the diagonal payoff function v : X → IR by v(x) = u 1 (x, ..., x) for every x ∈ X.

First of all, we define a measure of non quasiconcavity for quasisymmetric games:

) be a quasisymmetric game. For every x ∈ X, we define the measure ρ : X → R of lack of quasiconcavity of G at x as follows:

It is worthwhile to note that in the above definition, one can replace player 1 by any player withouth changing the value of ρ. Moreover, from Proposition 2.2, one can relate the previous measure to the notion of quasiconcave envelop as follows: for every x ∈ X, define the quasiconcave envelop of u 1 (., x , ..., x ) with respect to the first variable, denoted ũ1 (., x , x ..., x ). Then, one has

Now, recall that G is said to be diagonally quasiconcave (see Reny (1999)) if X is convex and if for each x ∈ X and y 1 , ..., y n in X such that x ∈co{y 1 , ..., y n }, one has -v(x ) + min{u 1 (y 1 , x , ..., x ), ..., u 1 (y n , x , ..., x )} ≤ 0.

Thus, if X is convex, then G is diagonally quasiconcave if and only if ρ(x) = 0 for every x ∈ X.

Following Reny (1999), we say that player i secures a payoff of α ∈ R along the diagonal at (x, x, ..., x) ∈ X N if there exists x ∈ X such that u i (x , x , ..., x, x , ..., x ) ≥ α for all x in some neighborhood of x ∈ X. Note that if G is quasisymmetric, then player i secures a payoff of α ∈ R along the diagonal at (x, x, ..., x) ∈ X N if and only if player j secures a payoff of α ∈ R along the diagonal at (x, x, ..., x) ∈ X N , for every j = 1, ..., N .

We now adapt Definition 3.4 to quasisymmetric games:

) is strongly diagonally better-reply secure if whenever (x * , v * ) ∈ X × IR is in the closure of the graph of its diagonal payoff function and (x * , ..., x * ) is not an equilibrium, some player i can secure a payoff strictly above v * + ρ(x * ) along the diagonal at (x * , ..., x * ).

If G is diagonally quasiconcave, we have seen that ρ = 0 : in this case, the previous definition is exactly the definition of diagonally better-reply secure games, introduced by Reny. Note also that since G is quasisymmetric, in the definition above, "some player i" can be replaced by "every player i" without altering this definition.

The following theorem is an extension of Theorem 3.2 to quasisymmetric games, and can be proved similarly by considering the game G associated to G. ) is quasisymmetric, compact and strongly diagonally betterreply secure, then it admits a symmetric pure Nash equilibrium. We now give an example of quasisymmetric, compact, strongly diagonally better-reply secure game which is not diagonally quasiconcave:

Example 4 In their paper, [START_REF] Baye | Characterization of the Existence of Equilibria in Games with Discontinuous and Non-quasiconcave Payoffs[END_REF] introduce the following game G: two duopolists with zero costs set prices (p 1 , p 2 ) ∈ [0, T ] × [0, T ], where T > 0. For i = 1, 2, the payoff functions are

where 0 < c < T , and where -i denotes 1 if i = 2 and 2 if i = 1. The interpretation is that each firm pays brand-loyal consumers a penalty of c if the other firm beats its price. It is easy to prove that this game is symmetric, compact, diagonally quasiconcave and diagonally better-reply secure. Thus, one could apply [START_REF] Reny | On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games[END_REF] or the main result of Baye et al., in order to obtain the existence of a pure Nash equilibrium. Now, let ε ∈]0, min{c, T -c}[, and consider the following modification G ε of the previous game: suppose that the penalty of firm i is reinforced for p -i ≤ T -c: in this case, firm i pays brand-loyal consumers a penalty of c if p -i < p i + ε, and nothing otherwise. Thus, firm i may have to pay the penalty even if p -i is (slightly) larger than p i . Consequently, one can write the modified payoff function as follows:

Besides, clearly, G ε is symmetric and compact. We now prove that it is strongly diagonally better-reply secure. First, we compute ρ, the measure of non quasiconcavity of G ε . For this purpose, recall that for every p 2 ∈ [0, T ], ũ1 (., p 2 ) denotes the quasiconcave envelop of the mapping u 1 (., p 2 ) with respect to the first variable, defined in Section 2. Now, consider the three following cases: i) Suppose p 2 < ε, which implies p 2 < T -c. Then, one has u 1 (p 1 , p 2 ) = p 1 -c for every p 1 ∈ [0, T ]. Thus, u 1 (., p 2 ) is quasiconcave, and ũ1 (p 1 , p 2 ) = p 1 -c for every

From the three previous cases, one has ũ1 (p, p) -u 1 (p, p) = 0 for every p < ε, ũ1 (p, p) -