Sessi Tokpavi
email: sessi.tokpavi@u-paris10.fr

Asset Allocation with Aversion to Parameter Uncertainty: A Minimax Regression Approach

Keywords: Asset allocation, estimation error, aversion to uncertainty, minimax regression, Bayesian mean-variance portfolios, least favorable prior

This paper takes a minimax regression approach to incorporate aversion to parameter uncertainty into the mean-variance model. The uncertainty-averse minimax mean-variance portfolio is obtained by minimizing with respect to the unknown weights the upper bound of the usual quadratic risk function over a fuzzy ellipsoidal set. Beyond the existing approaches, our methodology o¤ers three main advantages: …rst, the resulting optimal portfolio can be interpreted as a Bayesian mean-variance portfolio with the least favorable prior density, and this result allows for a comprehensive comparison with traditional uncertaintyneutral Bayesian mean-variance portfolios. Second, the minimax mean-variance portfolio has a shrinkage expression, but its performance does not necessarily lie within those of the two reference portfolios. Third, we provide closed form expressions for the standard errors of the minimax mean-variance portfolio weights and statistical signi…cance of the optimal portfolio weights can be easily conducted. Empirical applications show that incorporating aversion to parameter uncertainty leads to more stable optimal portfolios that outperform traditional uncertainty-neutral Bayesian mean-variance portfolios.

Introduction

The mean-variance model of [START_REF] Markowitz | Mean-Variance Analysis in Portfolio Choice and Capital Markets[END_REF] is one of the most prominent advance in guiding the practice of portfolio selection. This model provides a rigorous framework within which the investor maximizes his expected utility. Yet, the model is unfeasible in practice because the true parameters characterizing the expected utility are not known. A common remedy to this problem is to adopt a plug-in approach which consists in replacing the true unkown parameters by their sample analogues.

However, sample estimates are di¤erent from the true parameters and the resulting estimation error in most cases leads to optimized mean-variance portfolios that perform poorly out-of-sample [START_REF] Michaud | The Markowitz optimization enigma: Is optimized optimal?[END_REF], [START_REF] Best | On the sensitivity of mean-variance e¢ cient portfolios to changes in asset means: Some analytical and computational results[END_REF], [START_REF] Black | Global Portfolio Optimization[END_REF], DeMiguel et al. (2009)). To take care of the problem of estimation error in the mean-variance model, an investor can maximize a Bayesian expected utility function de…ned with respect to a prior density of the unknown parameters. This approach pionnered by [START_REF] Zellner | Prediction and decision problems in regression models from the Bayesian point of view[END_REF] and [START_REF] Bawa | Estimation Risk and Optimal Portfolio Choice[END_REF] was further investigated in the literature with proven empirical success [START_REF] Frost | An Empirical Bayes Approach to E¢ cient Portfolio Selection[END_REF], [START_REF] Jorion | International Portfolio Diversi…cation with Estimation Risk[END_REF][START_REF] Jorion | Bayes-Stein Estimation for Portfolio Analysis[END_REF], [START_REF] Pastor | Comparing Asset Pricing Models: An Investment Perspective[END_REF]).

The Bayesian mean-variance analysis supposes that investors have full information and are certain that the speci…ed prior density is perfectly identical to the true density. Nevertheless investors may not have perfect con…dence on any prior density due to incomplete information and should manifest this by considering multiple prior densities, each with unknown plausibility. In the more general context of decision making under uncertainty, this situation referred as Knightian uncertainty (or ambiguity) describes the uncertainty to the underlying probabilities. As [START_REF] Ellsberg | Risk, Ambiguity and the Savage Axioms[END_REF] shows in several thought experiments, decision-makers are averse to Knightian uncertainty because they prefer known to unknown and ambiguous probabilities of the states of the world.1 Moreover, [START_REF] Gilboa | Maxmin Expected Utility Theory with Non-Unique Prior[END_REF] demonstrate that the maxmin expected utility decision rule, that is, the decision rule which maximizes the minimum expected utility over the set of prior densities is compatible with utility maximization under aversion to Knightian uncertainty. Applications of the maxmin principle in the context of portfolio selection are nested to the general framework of robust optimisation [START_REF] Goldfarb | Robust Portfolio Selection Problems[END_REF], Tütücün and Koenig (2004), [START_REF] Garlappi | Portfolio Selection with Parameter and Model Uncertainty: A Multi-Prior Approach[END_REF]). The main principle is to maximise the investor's expected utility in the worst case scenario regarding the estimation of the unknown parameters. Our paper is related to these works and attempts to provide within a regression framework a methodology to incorporate the investors's aversion to parameter uncertainty when computing the mean-variance portfolio. 2The development of our methodology follows three key steps: …rst, we build on [START_REF] Britten-Jones | The Sampling Error in Estimates of Mean-Variance E¢ cient Portfolio Weights[END_REF] and demonstrate that the computation of the feasible meanvariance portfolio can be recast within a regression framework by minimizing the usual quadratic risk function. Second, we introduce parameter uncertainty into the mean-variance model through a constrained regression problem. In the constrained problem, we minimize the quadratic risk function under the constraint that the true weights of the mean-variance portfolio lie within a fuzzy ellipsoidal set. This set summarizes the investor's incomplete information about the true weights of the meanvariance portfolio. Third we rely on the minimax principle [START_REF] Wald | Statistical decision functions which minimize the maximum risk[END_REF]) to incorporate the investor's aversion to parameter uncertainty. Formally, among all possible allocations that re ‡ect his incomplete information given by the fuzzy ellipsoidal set, he chooses the best allocation in the worst case, that is, the allocation which achieves the smallest maximum quadratic risk.

It is worth noticing that our methodology contrasts with alternative existing approaches [START_REF] Goldfarb | Robust Portfolio Selection Problems[END_REF], Tütücün and Koenig (2004), [START_REF] Wang | A Shrinkage Approach to Model Uncertainty and Asset Allocation[END_REF], [START_REF] Garlappi | Portfolio Selection with Parameter and Model Uncertainty: A Multi-Prior Approach[END_REF]) in three major points: …rst, we make a clear connection between our methodology and the Bayesian mean-variance models. For-mally we demonstrate following [START_REF] Pilz | Minimax linear regression estimation with symmetric parameter restrictions[END_REF] that our minimax mean-variance portfolio can be interpreted as a Bayesian mean-variance portfolio with the least favorable prior density over the fuzzy ellipsoidal set. This result that gives more insights on the ability of our methodology to model aversion to parameter uncertainty allows for a clear comparison with traditional Bayesian mean-variance models. Second, it turns out from our analytic expressions that the minimax mean-variance portfolio shrinks the weights of the feasible mean-variance portfolio towards the weights of the minimum-variance portfolio. However and importantly, the performance of the minimax mean-variance portfolio does not necessarily lie within the performances of these target portfolios. Third, closed form expressions for the standard errors of the minimax mean-variance portfolio weights are available. This is possible because of the regression method we adopt. Hence, statistical signi…cance of the optimal portfolio weights can be easily conducted.

Two empirical applications are conducted to illustrate the relevance of the minimax mean-variance portfolio. In the …rst application we use excess returns on the Fama-French's 6 size and book-to-market assets and compare the accuracy and the stability of the weights of the minimax mean-variance portfolio with the weights of the feasible mean-variance portfolio. The results show that allowing for parameter uncertainty aversion radically decreases the imprecision and the instability of meanvariance portfolios weights. The second application evaluates the economic bene…t of incorporating aversion to parameter uncertainty into the mean-variance model, comparing the out-of-sample performance of the minimax mean-variance portfolio with traditional Bayesian mean-variance portfolios. Using the Fama-French's 25 size and book-to-market assets, the obtained results show that relying on the least favorable prior density leads to optimized Bayesian mean-variance portfolios that exhibit higher out-of-sample performances compared to the Bayes-Stein shrinkage portfolio of [START_REF] Jorion | Bayes-Stein Estimation for Portfolio Analysis[END_REF] and the Bayesian data-and-model approach of [START_REF] Pastor | Comparing Asset Pricing Models: An Investment Perspective[END_REF].

The rest of the paper is organized as follows. In section 2 we brie ‡y review the standard mean-variance model of asset allocation along with the associated problem of parameter uncertainty aversion. In section 3 we introduce our methodology to model aversion to parameter uncertainty and present the minimax mean-variance portfolio. Section 4 illustrates numerically the gain arising from taking into account aversion to parameter uncertainty notably in term of portfolio weights accuracy and stability, while in Section 5 we compare our minimax mean-variance portfolio with traditional Bayesian mean-variance portfolios. Further extensions are considered in section 6 and the last section concludes the paper.

Mean-Variance Model and Aversion to Parameter Uncertainty

Consider an investor who faces the choice of a portfolio among the universe of k risky investable assets. At time t, let r t be the k 1 vector of excess (over the risk-free rate) returns on the k risky assets. We denote t and t respectively the mean and the covariance matrix of r t . In the mean-variance model, the investor chooses the portfolio that maximizes the expected return for a given level of risk. It is well-known that this strategy is equivalent to select at time T the portfolio x = (x 1 ; :::; x k ) 0 to maximize the next-period mean-variance utility

U T +1 (x) = E (r T +1 ) 0 x 2 x 0 V ar (r T +1 ) x (1) = 0 T +1 x 2 x 0 T +1 x;
where the coe¢ cient is the degree of relative risk aversion of the investor. The solution to the above problem is

w = 1 1 T +1 T +1 : (2)
This solution, when scaled to meet the restriction that the asset weights must sum to one, leads to the so-called tangency portfolio

w = x 0 x 1 = 1 T +1 T +1 0 1 T +1 T +1 1 ;
(3) with a vector k 1 of ones. The tangency portfolio is independent of the risk aversion parameter and corresponds to the portfolio of risky assets that maximizes the Sharpe ratio. Theoretically, (3) illustrates that the mean-variance optimal rule is computationally unfeasible since the true parameters characterizing the tangency portfolio are unknown. The usual plug-in solution consists in replacing the unknown parameters by corresponding estimators. Under the assumption that r t is independent and identically distributed, and follows a multivariate normal distribution, natural estimators (maximum likelihood) for the two moments correspond respectively to the empirical mean and covariance matrix, that is

b T +1 = n 1 P t=T t=T n+1 r t ; (4) b T +1 = n 1 P t=T t=T n+1 r t b T +1 r t b T +1 0 ; (5)
with n the available sample size. As a result, the feasible tangency portfolio is computed by replacing the two unknown moments in (3) by their empirical counterparts, leading to the following allocation

b w = b 1 T +1 b T +1 0 b 1 T +1 b T +1 1 : (6)
However, the estimator in (6) are di¤erent from the true weights in (3) and from the di¤erence arises the problem of estimation error. The cost of ignoring this error has been widely documented in the literature. Indeed, relying on the empirical moments generally leads to instable sub-optimal mean-variance portfolios with extremely large positive and negative weights which are not meaningful economically.

The Bayesian mean-variance analysis o¤ers a convenient frame within which the problem of estimation error can be treated. Formally, Bayes estimators of the tangency portfolio weights have the following expression

b w b = b b T +1 1 b b T +1 0 b b T +1 1 b b T +1 1 ; (7)
where b b T +1 and b b T +1 correspond to the …rst two moments of the predictive density p (r T +1 jF T ) of asset returns, with F T the set of information available. The predictive density is obtained by integrating with respect to T +1 and T +1 the a posteriori density p r T +1 ; T +1 ; T +1 jF T , that is

p (r T +1 jF T ) = Z T +1 Z T +1 p r T +1 ; T +1 ; T +1 jF T d T +1 d T +1 : (8)
The a posteriori density is derived by updating an a priori density p T +1 ; T +1

of the unknown parameters with the sampling information. The theoretical contributions in this branch of the literature di¤er from the choice of the prior density. Earlier

Bayesian methods [START_REF] Brown | Optimal portfolio choice under uncertainty[END_REF][START_REF] Brown | The portfolio choice problem: comparison of certainty equivalence and optimal Bayes portfolios[END_REF], [START_REF] Klein | The E¤ect of Estimation Risk on Optimal Portfolio Choice[END_REF]) rely on di¤use-prior while Bayes-Stein shrinkage [START_REF] Frost | An Empirical Bayes Approach to E¢ cient Portfolio Selection[END_REF], [START_REF] Jorion | International Portfolio Diversi…cation with Estimation Risk[END_REF][START_REF] Jorion | Bayes-Stein Estimation for Portfolio Analysis[END_REF])) are built with conjugate or hyperparameter priors. Priors that re ‡ect an investor's degree of belief in a given asset pricing model lead to the so-called Bayesian data-and-model method [START_REF] Pastor | Portfolio Selection and Asset Pricing Models[END_REF], [START_REF] Pastor | Comparing Asset Pricing Models: An Investment Perspective[END_REF], [START_REF] Wang | A Shrinkage Approach to Model Uncertainty and Asset Allocation[END_REF]).

A central hypothesis in the Bayesian analysis is that uncertainty is mesurable and can be summarized by a single prior density. This kind of uncertainty is called risk by [START_REF] Knight | Risk, Uncertainty and Pro…t[END_REF] and should be di¤erentiated from the non mesurable uncertainty or ambiguity describing the situation where decision-makers fail to assess with accuracy the probability distribution of the relevant parameters. This distinction is at the core of our paper and we provide in the sequel an econometric methodology which treats the problem of parameter uncertainty along with the reported evidence that investors are averse to such ambiguous situations [START_REF] Ellsberg | Risk, Ambiguity and the Savage Axioms[END_REF]).

Aversion to Parameter Uncertainty and the Minimax Mean-Variance Portfolio

This section is divided into two parts. In the …rst part we extend the results in [START_REF] Britten-Jones | The Sampling Error in Estimates of Mean-Variance E¢ cient Portfolio Weights[END_REF] and introduce an ellipsoid-constrained regression model that solves the computation of the mean-variance portfolio when parameter uncertainty is of concern. In the second part, we rely on the minimax principle to solve the constrained regression model and show that the resulting allocation rule deals with the issue of parameter uncertainty aversion.

Incorporating Parameter Uncertainty into the Mean-Variance Model

Our starting point is the regression-based approach for the computation of the meanvariance portfolio (Britten-Jones (1999)). Formally, consider the following regression

model Y = Xw + u; E (u) = 0; E (uu 0 ) = 2 u ; ( 9
)
where X is the n k matrix of excess returns on the k risky assets, w the k 1 vector of parameters, u the noise term, and Y the n 1 constant vector with all entries equal to with Proposition 1 With the regression model ( 9) and the class of homogeneous linear solutions e w = CY , the optimal unbiased estimator of w under the risk function R ( e w; w; A) is identical to the weights of the feasible tangency portfolio in (6), that is

= 0 b 1 T +1 b T +1 1 1 + b 0 T +1 b 1 T +1 b T +1 : (10
arg min e w2D R ( e w; w; A) = X 0 X 1 X 0 Y = b w: (11)
The proof is straightforward using …rst the equivalence between the least-squares estimation of ( 9) and the minimization of the quadratic risk function R ( e w; w; A) (see

E p R (w; e w; A) = R R k R (w; e w; A) p (dw) : (12)
As already stressed, the uncertainty that arises from the arbitrariness of the choice of the prior density p (w) is at the core of this paper. An approach we take here to incorporate this uncertainty into the mean-variance model is to solve the problem ( 11) by searching the optimal weights over a fuzzy set which re ‡ects the incompleteness of the investor's information, and hence his inability to form the prior density p (w).

The corresponding minimization problem can be written as follows

T +1 0 b 1 T +1 1 . ( 14
)
As a consequence, the distance between the true tangency portfolio weights w and the weights b w min of the global minimum-variance portfolio can serve as an indicator of the level of uncertainty in estimating the expected asset returns. This distance can be formulated in the form of the following ellipsoidal constraint

= w : (w b w min ) 0 H (w b w min ) ; ( 15
)
where H is a k k positive de…nite matrix and a positive real number. The ellipsoid de…nes a region where the true asset weights are believed to lie and the volume of this region is controlled by the parameter . For convenience, we will use the following re-parameterization for

= n 1 (1 =2) 2 ; ( 16
)
with 2 (0; 1) and 1 (:) the inverse of the standard normal cumulative distribution function. This is useful because we summarize the in…nite possible values of 2 R + via the bounded parameter . Note that can be viewed as the investor's degree of skepticism about the estimation of the expected asset returns. Indeed, when the investor assigns a value for near one ( near zero), he has high uncertainty about the estimation of the expected asset returns, and materializes this by locating the true unknown tangency portfolio weights near the weights of the global minimum-variance portfolio. In the opposite case where diverges from one, the investor's skepticism about the precision of the empirical mean b T +1 diminishes. It is worth noting that the role of the scaling factor n in ( 16) is to relax the ellipsoidal constraint when larger sample sizes are available, because in these cases and independently of the investor's degree of skepticism , the precision of the empirical mean b T +1 becomes larger.

A simple choice for the matrix H would be the identity matrix. But this is a rather naive choice, since it measures the distance between w and b w min by placing equal importance on the deviation of each component of w from that of b w min . A better alternative would be to weight the importance of the deviation by a matrix that re ‡ects the precision of the estimated weights of the global minimum-variance portfolio. More precisely, we set the matrix H to the k k diagonal matrix with the i-th element being the inverse of the variance of the i-th weight of the global minimum-variance portfolio, that is, H = (h ij ) with

h ij = ( 1 V ar( b w min;i) if i = j 0 else : (17)
The expression of the variance of the i-th weight of the global minimum-variance portfolio is given by Bodnar and Schmid (2008) and corresponds to

V ar ( b w min;i ) = b 1 T +1 0 ( ) ii P k m=1 ( ) im 2 (n k) b 1 T +1 0 2 ; (18) with b 1 T +1 = ( ) im
the inverse of the sample covariance matrix.

Minimax Principle and Aversion to Parameter Uncertainty

In the last part of this section, we show how to solve the constrained minimization program (13) taking care of the investor's aversion to parameter uncertainty. 5Precisely, we rely on the minimax principle [START_REF] Wald | Statistical decision functions which minimize the maximum risk[END_REF]) to solve this program.

An investor that allocates assets using the minimax estimator b w in ( 19) is averse to the uncertainty (or ambiguity) arising from having multiple prior densities. To show this, let P be the set of arbitrary prior distributions p on and consider the Bayes risk See Theorem 1 in [START_REF] Pilz | Minimax linear regression estimation with symmetric parameter restrictions[END_REF] for the proof. The prior density p 0 is termed the least favorable prior density, that is the prior density which leads to the maximum Bayes risk. Our minimax estimator of the mean-variance portfolio can thus be viewed as a Bayesian mean-variance portfolio for an investor who has multiple prior densities (over the fuzzy set ) and chooses the one which corresponds to the least favorable (or worst-case) scenario. Following [START_REF] Gilboa | Maxmin Expected Utility Theory with Non-Unique Prior[END_REF], this investment strategy is evidently compatible with aversion to uncertainty.

E p R (w; e w; A) = R R (
To derive explicit solutions for the estimator b w , additional structural assumptions are needed. In the case where the matrix A is of rank one, the explicit solution (Kuks andOlman (1971, 1972)) is given by b

w = X 0 X + 1 2 u H 1 X 0 Y + 1 2 u H b w min : (21
To …nish, remark that b w is not properly an estimator of the tangency portfolio weights because its components do not sum to one except for the limiting cases b w and b w min . Therefore, we follow [START_REF] Toutenburg | On the Combination of Equality and Inequality Restrictions on Regression Coe¢ cients[END_REF] de…ning the equality restricted analogue of the estimator b w in ( 21), that is

w = b w D 1 0 D 1 1 0 b w 1 ; (24) with D = X 0 X + 1 b 2 u H.
A salient feature of our regression approach is that the bias and the variance of w can be derived [START_REF] Toutenburg | On the Combination of Equality and Inequality Restrictions on Regression Coe¢ cients[END_REF])

yielding bias w = 1 b 2 u h I k + D 1 0 D 1 1 0 i D 1 Hw; b 2 u V ar ( w ) = D 1 X 0 XD 1 +D 1 0 D 1 1 0 D 1 X 0 XD 1 0 X 0 X 1 1 D 1 0 D 1 1 0 D 1 X 0 XD 1 D 1 X 0 XD 1 0 D 1 1 0 D 1 ; (25)
The bias (resp. the variance) is a decreasing (resp. increasing) function of . This implies that when the investor degree of skepticism in estimating the expected returns is high ( ! 1) the parameter ! 0 and the bias (resp. the variance) increases (resp.

decreases). Hence, allowing for uncertainty aversion has a shrinkage-like e¤ect on the estimated portfolio weights which are biased but more precise according to the usual bias-variance trade-o¤ for regression parameters. Notice that for the limiting case ! 0 ( ! 1), we have bias w = 0 and

V ar ( w ) = b 2 u h S 1 S 1 0 S 1 0 1 S 1 i ; (26)
which correspond to the bias and the variance of the feasible tangency portfolio.

Stability and Accuracy of the Minimax Mean-Variance Portfolio

In this section, we evaluate the behaviour (from the viewpoint of stability and accuracy) of the minimax mean-variance portfolio, using the monthly excess returns of the Fama-French's 6 size and book-to-market assets from July 1963 to September for uncertainty aversion with a small amount of uncertainty equal to 10% lower the variances which lie between 0:02 and 3:2.

To illustrate the e¤ect of aversion to parameter uncertainty on the out-of-sample performance of the mean-variance model, we compute from the N n optimal portfolios returns three statistics: the out-of-sample means, standard-deviations and sharpe ratios. Table 1 in Appendix displays these statistics for di¤erent values of the investor's degree of skepticism . First, one can see that for the limiting case = 0%

(resp. = 100%) the statistics are identical to those of the feasible tangency portfolio (resp. the global minimum-variance portfolio). Second, the minimax mean-variance portfolio w that allows for parameter uncertainty aversion exhibits higher means and lower volatilities. For instance, while the out-of-sample mean (resp. standarddeviation) of the feasible tangency portfolio is equal to -0:1732 (resp. 3:9550) the same statistic when the investor's degree of skepticism corresponds to = 10% is equal to 0:0175 (resp. 0:1937). Third, for a given value of 2 ]0; 1[ the out-of-sample Sharpe ratio of our minimax mean-variance portfolio w does not lie necessarily within the sharpe ratios of the two limiting cases ( = 0% and = 100%). Indeed, for 20%

the out-of-sample Sharpe ratios are higher than the ones of the feasible tangency and the global minimum-variance portfolios. Figure 3 in Appendix gives a complete description of the evolution of the Sharpe ratio with respect to . The …gure depicts a quadratic evolution, with an increase in Sharpe ratio up to a given value of and a decrease over this value. Relatively to this evolution our methodology to deal with the issue of uncertainty aversion contrasts with the multi-prior approach of Garlappi, Uppal and [START_REF] Garlappi | Portfolio Selection with Parameter and Model Uncertainty: A Multi-Prior Approach[END_REF]. In the multi-prior approach, uncertainty is introduced in the mean-variance model by specifying con…dence intervals (around the true expected asset returns) in the form of ellipsoid constraint, that is This ellipsoid re ‡ects the investor's a priori information about the true expected asset returns for a given level of uncertainty . Therefore, there is a close connection between the max-min problem (28) and the minimax regression approach we follow in this paper. The methodological di¤erence arises from the fact that in our framework, the uncertainty is instead about the true unknown portfolio weights. [START_REF] Garlappi | Portfolio Selection with Parameter and Model Uncertainty: A Multi-Prior Approach[END_REF] show that the solution of the max-min problem ( 28) is a weighted average of the classical mean-variance portfolio that ignores parameter uncertainty and the global minimum-variance portfolio

G T +1 ; b T +1 = T +1 : n (n k) (n 1) k T +1 b T +1 0 T +1 T +1 b T +1 ; (27)
w = b w min + (1 ) b w (29)
with a scalar that depends on the level of uncertainty . Thus, the performance of this asset allocation rule lies between the ones of the two reference portfolios. As already stressed, our minimax mean-variance portfolio w di¤ers from w with regards to this characteristic.

Comparison with Uncertainty Neutral Bayesian Mean-Variance Portfolios

From proposition 3, we have shown that the minimax mean-variance portfolio corresponds to a Bayesian mean-variance portfolio under the least favorable prior density. Hence, we compare in this section the out-of-sample performances of the minimax mean-variance portfolio w with traditional uncertainty neutral Bayesian meanvariance portfolios.

Description of the portfolios Considered

The …rst mean-variance portfolio we consider is the Bayes-Stein shrinkage portfolio developed by [START_REF] Jorion | Bayes-Stein Estimation for Portfolio Analysis[END_REF], which exploits the idea of shrinkage estimation [START_REF] Stein | Inadmissibility of the Usual Estimator for the Mean of a Multivariate Normal Distribution[END_REF], [START_REF] James | Estimation with Quadratic Loss[END_REF]). Under this model, the weights of the tangency

b bs T +1 = (1 b v) b T +1 + b vb g T +1 ; (31) b bs T +1 = 1 + 1 n + b V T +1 + b n n + 1 + b 0 0 V 1 T +1 ; ( 32
) b v = k + 2 (k + 2) + n b T +1 b g T +1 0 V 1 T +1 b T +1 b g T +1 ; b = k + 2 b T +1 b g T +1 0 V 1 T +1 b T +1 b g T +1 ; V T +1 = n b T +1 n k 2 ; b g T +1 = 0 V 1 T +1 b T +1 0 V 1 T +1 :
Thus, the empirical mean b T +1 are shrunk towards the target value b g T +1 with b v the bounded shrinkage parameter. This operation while introducing a small amount of bias in the estimation procedure of the expected asset returns signi…cantly reduces the variance, leading to more stable out-of-sample portfolio weights (see [START_REF] Jorion | Bayes-Stein Estimation for Portfolio Analysis[END_REF] and DeMiguel et al. (2009aDeMiguel et al. ( , 2009b) ) for empirical applications).

The second mean-variance portfolio is derived from the Bayesian data-and-model approach in [START_REF] Pastor | Comparing Asset Pricing Models: An Investment Perspective[END_REF]. For a brief description, suppose the existence of m benchmark portfolios related to a given asset-pricing model. The key issue in the Bayesian data-and-model approach is to make a balance between the asset-pricing model and the sampling information. In this line, the two extreme views for an investor is to believe or not to believe in the asset-pricing model. In the former case, estimators of the two unknown moments T +1 and T +1 are computed from the asset-pricing model r t = r f;t + e t ; 8t = T n + 1; :::; T;

where r f;t is the m 1 vector of excess returns on the benchmark portfolios, is the k m matrix of the betas, and e t the k 1 vector of residuals. In the latter case, empirical counterparts of the two moments are used instead, and this is equivalent to estimate the moments via the asset-pricing model ( 33) by allowing for a mispricing vector . A middle approach that is more relevant than the extreme views is to update the sampling information with the prior degree of con…dence about the assetpricing model. As suggested by [START_REF] Pastor | Comparing Asset Pricing Models: An Investment Perspective[END_REF], one way of doing this is to specify a prior distribution for the asset's mispricing vector , that is

p ( j T +1 ) = N 0; 2 s 2 T +1 ; ( 34
)
where 2 is a positive parameter that controls the variance of the prior distribution of , and s 2 a …xed parameter. Note that when 2 = 0 the investor believes dogmatically in the asset-pricing model and for 2 ! 1 the investor believes that the asset-pricing model is not useful. [START_REF] Wang | A Shrinkage Approach to Model Uncertainty and Asset Allocation[END_REF] shows that with the prior in (34), the mean of the predictive distribution of r T +1 has a shrinkage expression

E (r T +1 j ) = b f T +1 + (1 ) b T +1 ; (35)
with b f T +1 the estimator of the expected returns from the asset-pricing model, the shrinkage parameter that depends on 2 the variance of the mispricing. 6 Hence, the weights of the tangency portfolio in the Bayesian data-and-model approach are computed by replacing T +1 and T +1 in (3) by E (r T +1 j ) and V ar (r T +1 j ).

Data and Results

The comparison is conducted using k = 25 risky assets that correspond to the 25 Fama-French size and book-to-market portfolios. The data set contains monthly excess returns (from Kenneth French's Web site) for these assets over the period July 1963-September 2009. We set the estimation sample size to n = 60 and use the rolling-window methodology described in the last section to compute the outof-sample portfolio weights respectively for our minimax mean-variance portfolio w and the two Bayesian mean-variance portfolios described above.

Table 2 in Appendix displays the out-of-sample means, standard deviations and Sharpe ratios for the three portfolios. For the minimax mean-variance portfolio w , we report the results for di¤erent values of the investor's degree of skepticism .

To implement the Bayesian data-and-model approach we consider the Fama-French three-factor model and set the shrinkage parameter respectively to 10%, 25%, 50%, 75%, 90% and 100%. From this table, we see that the out-of-sample mean of the Bayes-Stein shrinkage portfolio is negative and equal to -0:0225. Thus, it seems, at least for the data set considered here, that the shrinkage intensity of this model is not su¢ cient enough to move the out-of-sample mean of the feasible tangency portfolio from a negative value (-0:0455) to a positive one. Compared to the feasible tangency portfolio that ignores parameter uncertainty, both the Bayesian data-andmodel portfolios of [START_REF] Pastor | Comparing Asset Pricing Models: An Investment Perspective[END_REF] and our minimax portfolios exhibit higher means in all cases, that is for the di¤erent values of and . However, the out-of-sample means are uniformly higher for the former portfolios for 50%. To

give more insights about this result, note that the Bayesian data-and-model (resp.

the minimax) approach makes a compromise between the feasible tangency portfolio and the asset-pricing (resp. the minimum-variance) portfolio. Now, it turns out in the case of the data set used that the out-of-sample mean of the Fama-French three-factor portfolio ( = 100%) is higher than the one of the minimum-variance portfolio. The converse case is obtained with regards to the out-of-sample standarddeviations, where the values respectively for the Fama-French three-factor portfolio and the minimum-variance portfolio are equal to 0:4582 and 0:0437. The minimax mean-variance portfolios bene…t from this characteristic by exhibiting lower standarddeviations compared to the Bayesian data-and-model portfolios. Lastly, the out-ofsample Sharpe ratios of the minimax mean-variance portfolios are higher than the ones of the Bayesian portfolios. Summarizing our …ndings, we can conclude from this empirical exercise that relying on the least favorable prior density via our minimax regression approach is economically bene…cial, because the corresponding portfolio leads to lower (resp. higher) out-of-sample standard-deviations (resp. Sharpe ratios).

Further Extensions

It is well documented in the literature that sample means are more a¤ected by estimation error than the sample covariance matrix [START_REF] Merton | On Estimating the Expected Return on the Market: An Exploratory Investigation[END_REF], [START_REF] Chopra | The E¤ects of Errors in Means, Variances, and Covariances on Optimal Portfolio Choice[END_REF]). Yet the uncertainty in estimating the covariance matrix by its empirical counterpart is not negligible for high dimensional problem, that is when the asset universe is large [START_REF] Chan | On Portfolio Optimization: Forecasting Covariances and Choosing the Risk Model[END_REF], [START_REF] Jagannathan | Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps[END_REF], [START_REF] Ledoit | Improved estimation of the covariance matrix of stock returns with an application to portfolio selection[END_REF], 2004a, 2004b)). This is the case because when k=n ! c > 0 with k the number of assets and n the sample size, the eigenvalues of the sample covariance matrix spread out more than the true unobservable ones [START_REF] Marcenko | Distributions of eigenvalues of some sets of random matrices[END_REF]), and the eigenvectors are not consistent [START_REF] Johnstone | Sparse principal components analysis[END_REF]). Therefore, unless k=n ! 0, errors in the sample covariance matrix can a¤ect the mean-variance allocation through the inconsistency of eigenvalues and eigenvectors. Needless to say that investors should be averse to this type of uncertainty. Hence, our objective in this section is to extend our methodology to incorporate into the mean-variance model the investor's aversion in estimating both the expected asset returns and the covariance matrix.

If we denote w 0 the vector weights of the equally-weighted portfolio, it is easy to see that the fuzzy ellipsoidal set de…ned as follows

= n w : w w 0 0 H w w 0 o ; (36)
can summarize the incompleteness of the investor's information. Indeed, for ! 0 the investor allocates his wealth equally between the assets because he believes that sampling information is completely useless. In the opposite case where ! 1, he relax his belief and search the weights of the tangency portfolio in a larger space. As previously, we use for convenience a re-parameterization of the parameter , that is

= n 1 (1 =2) 2 ; ( 37
)
with the level of uncertainty in estimating the two unknown moments T +1 and T +1 . For the choice of the matrix H, we use the diagonal matrix with elements the inverse of the empirical variances of the asset returns. Hence, the more risky is an asset, the less important is its contribution to the construction of the ellipsoid.

Using once again the minimax principle, the optimal portfolio weights e w for an uncertainty averse investor will be identical to w in (24) except that the weights of the target minimum-variance portfolio is replaced by the weights of the noise-free equally-weighted portfolio

e w = ! w D 1 0 D 1 1 0 ! w 1 ; (38) with D = X 0 X + 1 b 2 u H and ! w = D 1 X 0 Y + 1 b 2 u Hw 0 : (39)
Thus, our methodology does not change radically. One can expect that for large dimensional problems where the minimum-variance portfolio performs poorly, shrinking (using our minimax method) the feasible tangency portfolio towards the equallyweighted portfolio would be more bene…cial. This is all the more true as it is shown by DeMiguel et al. (2009a) that the equally-weighted portfolio constitutes a relevant benchmark investment rule as many others investment strategies, even sophisticated, cannot beat it.

To illustrate all these statements, we consider the excess returns for the Fama-French's 100 size and book-to-market assets over the period July 1963 -September 2009. The estimation sample size is set to n = 120 and a rolling-window procedure is use to compute w and e w which are considered as competitive investment strategies.

Recall that w (resp. e w ) refers to the weights of the minimax mean-variance portfolio for an investor's who worries about the estimation of the expected asset returns (resp.

the expected asset returns and the covariance matrix). Table 3 in Appendix reports for di¤erent values of the investor's degree of skepticism the out-of-sample means, standard errors and Sharpe ratios of the two investment strategies. The table also displays the same statistics for the target minimum-variance and equally-weighted portfolios. As expected, the out-of-sample standard deviations (resp. the Sharpe ratios) of the equally-weighted portfolio are lower (resp. higher) than those of the minimum-variance portfolio. As a consequence, for the di¤erent values of the minimax mean-variance allocation rule e w is uniformly better than w at least for the data set considered here.

Conclusion

Standard models of asset allocation would be very easy to implement if the true parameters characterizing the distribution of asset returns are perfectly known. In practice, this is not the case and the knowledges of investors are generally too limited to form a single prior density for these parameters. The resulting Knightian uncertainty (or ambiguity) renders the usual uncertainty-neutral Bayesian solutions restrictive because investors are averse to the uncertainty arising from having multiple prior densities.

For an investor allocating assets using the mean-variance model, this article provides an econometric framework to incorporate his aversion to parameter uncertainty.

The starting point of our methodology is the regression-based approach for the computation of the mean-variance portfolio in [START_REF] Britten-Jones | The Sampling Error in Estimates of Mean-Variance E¢ cient Portfolio Weights[END_REF]. Extending this approach using the minimax principle [START_REF] Wald | Statistical decision functions which minimize the maximum risk[END_REF]) we provide an estimator for the optimal portfolio weights that solves the mean-variance problem under aversion to parameter uncertainty. The uncertainty-averse mean-variance portfolio weights correspond to the minimax estimator for regression parameters, that is the estimator that minimizes the upper bound of the usual quadratic risk function. This upper bound is derived by searching over a fuzzy ellipsoidal set that summarizes the incompleteness of the investor's information about the unknown parameters. The new estimator has a shrinkage expression and makes a compromise between the usual plug-in estimator of the mean-variance portfolio and the global minimum-variance portfolio. An empir-ical application using the Fama-French 6 size and book-to-market assets shows that the minimax estimator by introducing a small amount of bias improves substantially the accuracy and the stability of the optimal portfolio weights. Moreover, the minimax mean-variance portfolio corresponds to a Bayesian mean-variance portfolio with the least favorable prior density. Using the Fama-French 25 size and book-to-market assets we show that relying on the least favorable prior density leads to better outof-sample performances compared to traditional hyperparameter or pricing-model priors.

Appendix : Tables and …gures Notes: This table provides for di¤erent values of the investor's degree of skepticism in estimating the expected asset returns, the monthly out-of-sample means, standard-deviations and Sharpe ratios of the minimax mean-variance portfolios. For comparison, the table also displays the same statistics for the feasible tangency portfolio and the global minimumvariance portfolio. The statistics are computed using a rolling-window procedure over the period July 1963 to September 2009. Notes: This table compares the monthly out-of-sample performances (means, standarddeviations and Sharpe ratios) of the minimax mean-variance Portfolios for di¤erent values of the investor's degree of skepticism about the estimation of the expected returns with the Bayes-Stein shrinkage portfolio of [START_REF] Jorion | Bayes-Stein Estimation for Portfolio Analysis[END_REF] and the Bayesian data-and-model portfolio of [START_REF] Pastor | Comparing Asset Pricing Models: An Investment Perspective[END_REF]. For the latter portfolio, we display the performances for di¤erent values of the shrinkage parameter in equation ( 35). The statistics are computed using excess returns of the Fama-French's 25 size and book-to-market assets over the period July 1963 to September 2009. Notes: This table compares the monthly out-of-sample performances (means, standard-deviations and Sharpe ratios) of the two types of minimax mean-variance portfolios. The …rst (resp. second) type includes portfolios that take into account the investor's aversion in estimating the expected asset returns (resp. the expected asset returns and the covariance matrix). The statistics are computed using a rolling-window procedure over the period July 1963 to September 2009 with an estimation sample size equal to n = 120.

  ) Let D = f e w : e w = CY g be the class of homogeneous linear solutions for the regression model (9) with C a k n unknown matrix and consider the quadratic risk function R ( e w; w) R ( e w; w; A) = E ( e w w) 0 A ( e w w) ; with A a k k positive de…nite matrix. The following proposition recasts the computation of the feasible tangency portfolio within a regression framework.

  Theorem 4.1 in[START_REF] Rao | Linear Models: Least Squares and Alternatives[END_REF]), and second the results of Theorem 1 in[START_REF] Britten-Jones | The Sampling Error in Estimates of Mean-Variance E¢ cient Portfolio Weights[END_REF]. The proposition states that the traditional plug-in method to compute the weights of the tangency portfolio and the minimization of the quadratic risk function R ( e w; w; A) lead to the same solution. Now, the question of interest is how one can exploit this result to deal with the issue of estimation error. First let us recall that from a Bayesian perspective, the problem of estimation error in the mean-variance model is usually treated by specifying a prior density for the unknown parameters T +1 and T +1 . The direct equivalence in our regression framework would be to form a prior density p (w) directly for the unknown weights of the tangency portfolio. The resulting Bayesian meanvariance portfolio can be obtained by minimizing (with respect to e w 2 D) the Bayes risk given by

  of the weights of the tangency portfolio and the fuzzy set.3 To construct the fuzzy set , consider an investor who believes that the empirical mean b T +1 provides a good approximation to the true expected asset returns T +1 . This investor should rely on the feasible tangency portfolio by replacing the two unknown moments in (3) by their empirical counterparts. In the converse case where the investor assumes that the uncertainty in approximating the expected asset returns by its empirical counterpart is too high to be economically valuable, he should allocate the assets by cancelling the contribution of the expected returns in (3). This task can be achieved by setting the expected returns to the constant vector. The corresponding allocation is identical to the weights of the global minimum-variance portfolio given by 4 b w min = b 1

De…nition 2 R

 2 The minimax estimator b w 2 D = f e w : e w = CY g solution of the constrained problem (13) is de…ned as follows sup w2

  w; e w; A) p (dw) : (20) Proposition 3 There exists a prior density p 0 2 P such that the Bayes estimator that minimizes the Bayes risk in (20) corresponds to the minimax estimator b relation E p 0 R (w; e w; A) E p R (w; e w; A) 8p 2 P:

)

  An operational version of this estimator is obtained by replacing 2 u by its leastsquares estimator b 2 u . It can be seen easily that b w behaves like a shrinkage estimator and makes -with respect to the parameter or equivalently -a trade-o¤ between b w the weights of the feasible mean-variance portfolio and the weights b w min of the global minimum-variance portfolio. The corresponding limiting behaviours are lim

  2009. Thus the number of assets is equal to k = 6 and the total number of time series observations is equal to N = 555 months. We set the estimation sample size to n = 60 months and use a "rolling-window" procedure to compute the out-of-sample mean-variance portfolio weights and returns. More precisely, the asset returns for the …rst n = 60 months are used to compute w and the corresponding portfolio return for the next month. This step is repeated by moving each time the estimation window (including the data for a new month and dropping the data for the earliest month) until the end of the data set is reached. Note that at the end of the procedure, we have computed N n portfolio weights and returns.

Figure 1 in

 1 Figure 1 in Appendix reports for four di¤erent values of the investor's degree of skepticism the boxplot of the estimated portfolio weights w over the out-ofsample period. The considered values are 0%, 10%, 75% and 100%. Recall that the …rst value = 0% corresponds to the feasible tangency portfolio that ignores parameter uncertainty, while the last value = 100% corresponds to the global minimum-variance portfolio. Compared to the feasible tangency portfolio in which parameter uncertainty is not of concern, the minimax mean-variance portfolios have more stable weights. Indeed, while the weights of the feasible tangency portfolio

T

  with the level of uncertainty in estimating expected asset returns. Aversion to uncertainty is modelled by referring to the following constrained mean-variance op-+1 w 2 w 0 T +1 w; s.t. T +1 2 G T +1 ; b T +1 and 0 w = 1:(28)In this setting, the standard problem of utility maximization over the unknown portfolio weights is modi…ed, by …rst minimizing the investor's utility with respect to the expected returns which are constrained to lie within the ellipsoid G T +1 ; b T +1 .

  T +1 and b bs T +1 the shrinkage estimators of T +1 and T +1 equal to

Figure 1 :Figure 2 :Figure 3 :

 123 Figure1: Boxplots of the out-of-sample weights of the minimax mean-variance portfolio using the Fama-French's 6 size and book-to-market assets

Table 2 :

 2 Comparison of the Minimax Mean-Variance Portfolios with Bayesian Mean-Variance Portfolios (Fama-French's 25 size and book-to-market assets)

Table 3 :

 3 Comparison of Minimax Mean-Variance Portfolios using the Fama-French's 100 size and book-to-market assets

	Strategy	Mean Standard Deviation Sharpe ratio
	Minimum-Variance	0.0039	0.0879	0.0442
	Equally-Weighted	0.0055	0.0527	0.1047
	Panel A: Uncertainty about only the …rst moment
	= 5%	0.0097	0.3658	0.0265
	= 10%	0.0093	0.3285	0.0284
	= 50%	0.0092	0.2024	0.0454
	= 75%	0.0108	0.1454	0.0744
	= 99%	0.0038	0.0889	0.0426
	Panel B: Uncertainty about the …rst two moments
	= 5%	0.0136	0.1478	0.0919
	= 10%	0.0131	0.1361	0.0961
	= 50%	0.0101	0.0966	0.1048
	= 75%	0.0072	0.0790	0.0915
	= 99%	0.0046	0.0561	0.0827

See also[START_REF] Becker | What Price Ambiguity? Or the Role of Ambiguity in Decision-Making[END_REF] and[START_REF] Camerer | Recent Developments in Modeling Preferences: Uncertainty and Ambiguity[END_REF] for more designed experiments which con…rm Ellsberg's intuition.

In the rest of the paper, the term "uncertainty" will thus refer to the "Knightian uncertainty".

The notion of fuzzy sets is introduced by[START_REF] Zadeh | Fuzzy sets[END_REF] and provides an interesting alternative (in representing uncertainty) to the conventional approaches using probabilistic modelling. For comprehensive descriptions and interpretations of the notion of fuzzy sets, see for e.g.[START_REF] Bandemer | Fuzzy Sets, Fuzzy Logic, Fuzzy Methods with Applications[END_REF] or[START_REF] Zimmermann | Fuzzy Set Theory and its Applications[END_REF].

Note that we focus here on the uncertainty in estimating the expected asset returns since they are more a¤ected by estimation error than the covariance matrix. For empirical evidences see[START_REF] Merton | On Estimating the Expected Return on the Market: An Exploratory Investigation[END_REF] and[START_REF] Chopra | The E¤ects of Errors in Means, Variances, and Covariances on Optimal Portfolio Choice[END_REF]. In Section 6 we will extend our framework by considering uncertainty in estimating both the expected asset returns and the covariance matrix.

We focus in a …rst time on the constrained least squares problem (13) without the last restriction that the asset weights must sum to one. We will consider this restriction later.

[START_REF] Wang | A Shrinkage Approach to Model Uncertainty and Asset Allocation[END_REF] also derived the formula for the second moment V ar (rT +1 jFT ; ) of the predictive distribution which also has a shrinkage expression.


Francesca Busetto

Giulio Codognato

Sayantan Ghosal

Noncooperative Oligopoly in Markets with a Continuum of Traders: A Limit Theorem

Keywords:

In this paper, in an exchange economy with atoms and an atomless part, we analyze the relationship between the set of the Cournot-Nash equilibrium allocations of a strategic market game and the set of the Walras equilibrium allocations of the exchange economy with which it is associated. In an example, we show that, even when atoms are countably infinite, Cournot-Nash equilibria yield different allocations from the Walras equilibrium allocations of the underlying exchange economy. We partially replicate the exchange economy by increasing the number of atoms without affecting the atomless part while ensuring that the measure space of agents remains finite. We show that any sequence of Cournot-Nash equilibrium allocations of the strategic market game associated with the partially replicated exchange economies approximates a Walras equilibrium allocation of the original exchange economy.

1 Introduction [START_REF] Okuno | Oligopoly and competition in large markets[END_REF] proposed an approach to modeling oligopoly in general equilibrium where "[...] either perfectly or imperfectly competitive behavior may emerge endogenously [...], depending on the characteristics of the used in this paper ensures that the mixed measure space remains finite: without this feature required, the model of oligopoly in general equilibrium we study would not be well-defined. Then, we show that any sequence of Cournot-Nash equilibrium allocations of the strategic market game associated with the partially replicated exchange economies approximates a Walras equilibrium allocation of the original exchange economy. [START_REF] Sahi | The noncooperative equilibria of a trading economy with complete markets and consistent prices[END_REF] showed the convergence of sequences of Cournot-Nash equilibrium allocations to a Walras equilibrium allocation starting from a finite exchange economy by replicating traders. [START_REF] Codognato | Cournot-Nash equilibria in limit exchange economies with complete markets and consistent prices[END_REF] showed the equivalence between Cournot-Nash and Walras equilibrium allocations in exchange economies with an atomless continuum of traders. While our convergence result synthesizes elements of both these papers, there are two new elements. First, the replication of atoms we use is different form the one used by [START_REF] Sahi | The noncooperative equilibria of a trading economy with complete markets and consistent prices[END_REF] for a finite number of traders: our replication ensures that the mixed measure space remains finite and, further, the convergence proof relies in an essential way on it. Second, we have to solve a new technical issue: in order to ensure that a sequence of Cournot-Nash equilibrium allocations approximates a Walras equilibrium allocation, we have to use a version of Fatou's Lemma in several dimensions proved by [START_REF] Artstein | A note on Fatou's lemma in several dimensions[END_REF]. Some limit results have already been proved in the approach adopted by [START_REF] Gabszewicz | Oligopoly 'à la Cournot-Walras' in a general equilibrium analysis[END_REF], where competitive or noncompetitive traders' behavior is explicitly assumed (see [START_REF] Roberts | The limit points of monopolistic competition[END_REF], [START_REF] Mas-Colell | Walrasian equilibria as limits of noncooperative equilbria. Part I: mixed strategies[END_REF], Novshek and Sonnenschein ((1983), (1987)), among others). Here, consistently with the Okuno et al. (1980)'s approach we have adopted, we prove a limit result à la Cournot (see [START_REF] Cournot | Recherches sur les principes mathématiques de la théorie des richesses[END_REF]) without making any further behavioral assumption and preserving the feature that the mixed measure space remains finite. atoms and T 0 = T \ T 1 the atomless part of T . A null set of traders is a set of measure 0. Null sets of traders are systematically ignored throughout the paper. Thus, a statement asserted for "all" traders, or "each" trader, or "each" trader in a certain set is to be understood to hold for all such traders except possibly for a null set of traders. The word "integrable" is to be understood in the sense of Lebesgue.

In the exchange economy, there are l different commodities. A commodity bundle is a point in R l

+ . An assignment (of commodity bundles to traders) is an integrable function x:

T → R l + .
There is a fixed initial assignment w, satisfying the following assumption.

Assumption 1. w(t) > 0, for each t ∈ T .
An allocation is an assignment x for which T x(t) dµ = T w(t) dµ. The preferences of each trader t ∈ T are described by a utility function u t : R l + → R, satisfying the following assumptions. Assumption 2. u t : R l + → R is continuous, strongly monotone, and quasiconcave, for each t ∈ T .

Let B(R l + ) denote the Borel σ-algebra of R l + . Moreover, let T B denote the σ-algebra generated by the sets E × F such that E ∈ T and F ∈ B. Assumption 3. u : T × R l + → R, given by u(t, x) = u t (x), for each t ∈ T and for each x ∈ R l + , is T B-measurable.
We also need the following assumption (see [START_REF] Sahi | The noncooperative equilibria of a trading economy with complete markets and consistent prices[END_REF]).

Assumption 4. There are at least two traders in T 1 for whom w(t) 0;

u t is continuously differentiable in R l ++ ; {x ∈ R l + : u t (x) = u t (w(t))} ⊂ R l ++ . A price vector is a vector p ∈ R l + .
A Walras equilibrium of E is a pair (p * , x * ), consisting of a price vector p * and an allocation x * , such that, for each t ∈ T , u t (x * (t)) ≥ u t (y), for all y ∈ {x ∈ R l + : p * x = p * w(t)}. We introduce now the strategic market game, Γ, associated with E.

Let b ∈ R l 2 + be a vector such that b = (b 11 , b 12 , . . . , b ll-1 , b ll ). A strategy correspondence is a correspondence B : T → P(R l 2 + ) such that, for each t ∈ T , B(t) = {b ∈ R l 2 + : l j=1 b ij ≤ w i (t), i = 1, . . . , l}. A strategy selection is an integrable function b : T → R l 2 , such that, for each t ∈ T , b(t) ∈ B(t). For each t ∈ T , b ij (t), i, j = 1, . . . , l
, represents the amount of commodity i that trader t offers in exchange for commodity j. Given a strategy selection b, we define the aggregate matrix B = ( T b ij (t) dµ). Then, we introduce two further definitions (see [START_REF] Sahi | The noncooperative equilibria of a trading economy with complete markets and consistent prices[END_REF]).

Definition 1. A nonnegative square matrix A is said to be irreducible if, for every pair (i, j), with i = j, there is a positive integer k = k(i, j) such that a

(k) ij > 0, where a (k) ij denotes the ij-th entry of the k-th power A k of A.
Definition 2. Given a strategy selection b, a price vector p is market clearing if

p ∈ R l ++ , l i=1 p i bij = p j ( l i=1
bji ), j = 1, . . . , l.

(

By Lemma 1 in Sahi and Yao (1989), there is a unique, up to a scalar multiple, price vector p satisfying (1) if and only if B is irreducible. Then, we denote by p : R l 2 + → R l + a function such that, for each strategy selection b, p(b) is the unique, up to a scalar multiple, price vector satisfying (1), if B is irreducible, and p(b) = 0, otherwise.

Given a strategy selection b and a price vector p, consider the assignment determined as follows:

x j (t, b(t), p) = w j (t) - l i=1 b ji (t) + l i=1 b ij (t) p i p j , if p ∈ R l ++ ,
x j (t, b(t), p) = w j (t), otherwise, j = 1, . . . , l, for each t ∈ T . According to this rule, given a strategy selection b and the function p(•), the traders' final holdings are determined as follows:

x(t) = x(t, b(t), p(b)),
for each t ∈ T . It is straightforward to show that the assignment corresponding to the final holdings is an allocation. This reformulation of the Shapley's model allows us to define the following concept of Cournot-Nash equilibrium for exchange economies with an atomless part (see [START_REF] Codognato | Cournot-Nash equilibria in limit exchange economies with complete markets and consistent prices[END_REF]).

Definition 3. A strategy selection b such that B is irreducible is a Cournot- Nash equilibrium of Γ if u t (x(t, b(t), p( b))) ≥ u t (x(t, b \ b(t), p( b \ b(t)))),
for all b ∈ B(t) and for each t ∈ T .

3 The existence of a δ-positive Cournot-Nash equilibrium of Γ

We introduce now the notion of a δ-positive strategy correspondence which was used by Sahi and Yao (1989) to prove their existence and limit theorems and which we shall use here for the same purposes. Let T1 ⊂ T 1 be a set consisting of two traders in T 1 for whom Assumption 4 holds. Moreover, let δ = min t∈ T1 { 1 l min{w 1 (t), . . . , w l (t)}}. We say that the correspondence

B δ : T → R l 2 is a δ-positive strategy correspondence if B δ (t) = B(t) ∩ {b ∈ R l 2 : i ∈J j∈J (b ij + b ji ) ≥ δ, for each J ⊆ {1, . . . , l}}, for each t ∈ T1 ; B δ (t) = B(t), for the remaining traders t ∈ T . We say that a strategy selection b is δ-positive if b(t) ∈ B δ (t), for each t ∈ T . Moreover, we say that a Cournot-Nash equilibrium b of Γ is δ-positive if b is a δ-

An example

Codognato and Ghosal (2000) analyzed the Sahi and Yao (1989)'s model in exchange economies with an atomless continuum of traders. In this framework, they showed an equivalence result à la Aumann (see [START_REF] Aumann | Markets with a continuum of traders[END_REF]) between the set of the Cournot-Nash equilibrium allocations of Γ and the set of the Walras equilibrium allocations of E. The mixed measure space we are considering here may contain countably infinite atoms. This raises the question whether an equivalence result à la Aumann could hold in this case.

The following example considers an exchange economy E with countably infinite atoms and it shows that any Cournot-Nash equilibrium allocation of the strategic market game Γ associated with E is not a Walras equilibrium allocation of E.

Example. Consider an exchange economy E where l = 2,

T 1 = T 1 ∪ T 1 , T 1 = {2, 3}, T 1 = {4, 5, . . .}, T 0 = [0, 1], w(2) = w(3) 0, w(t) = (0, 1), for each t ∈ T 1 ∪ T 0 , u t (•) satisfies Assumptions 2 and 3, for each t ∈ T , u 2 (•) and u 3 (•) satisfy Assumption 4, u 2 (x) = u 3 (x), u t (x) > u t (y), whenever x ∈ R l ++ and y ∈ (R l + \ R l ++ ), for each t ∈ T 1 ∪ T 0 ,
dx 2 dx 1 = -p 2 b12 (t) b21 (t) + h = -p,
for each t ∈ T 1 . But then, we must also have

b21 (2) + h b12 (2) = b21 (2) + b21 (3) + h b12 (2) + b12 (3) = b21 (3) + h b12 (3) . ( 2
)
The last equality in (2) holds if and only if b21 (2) = k( b21 (3) + h) and b12 (2) = k b12 (3), with k > 0. But then, the first and the last members of (2) cannot be equal because

k( b21 (3) + h) + h k b12 (3) = b21 (3) + h b12 (3) .
This implies that the pair (p, x) such that p = p( b) and x(t) = x(t, b(t), p), for each t ∈ T , cannot be a Walras equilibrium of E.

The example shows that the condition that E contains a countably infinite number of atoms is not sufficient to guarantee that the set of the Cournot-Nash equilibrium allocations of Γ coincides with the set of the Walras equilibrium allocations of E. This leads us to consider the question whether partially replicating E à la Cournot (see Cournot (1838))that is increasing the number of atoms, while making them asymptotically negligible, without affecting the atomless part -would generate sequences of Cournot-Nash equilibrium allocations which approximate, in some way, a Walras equilibrium allocation of E. We shall address this question in the following sections.

The replication à la Cournot of E

We consider the replication à la Cournot of E which, by analogy with the replication proposed by Cournot (1838) in a partial equilibrium framework, consists in replicating only the atoms of E, while making them asymptotically negligible. Let E n be an exchange economy characterized as in Section 2 where each atom is replicated n times. For each t ∈ T 1 , let tr denote r-th element of the n-fold replication of t. We assume that w(tr) = w(ts) = w(t),

u tr (•) = u ts (•) = u t (•), r, s = 1, . . . , n, µ(tr) = µ(t) n , r = 1, . . . , n, for each t ∈ T 1 . Clearly, E coincides with E 1 .
The strategic market game Γ n associated with E n can then be characterized, mutatis mutandis, as in Section 2. Clearly, Γ coincides with Γ The interpretation is that an outside agency places fixed bids of for each pair of commodities (i, j). Given > 0, we denote by p (b) the function which associates, with each strategy selection b, the unique, up to a scalar multiple, price vector which satisfies

l i=1 p i ( bij + ) = p j ( l i=1 ( bji + ), j = 1, . . . , l. (3)
u t (x(t, b (t), p ( b ))) ≥ u t (t, b \ b(t), p ( b \ b(t)))),
for all b ∈ B(t) and for each t ∈ T 0 .

We neglect, as usual, the distinction between integrable functions and equivalence classes of such functions and denote by L 1 (µ, R l 2 ) the set of integrable functions taking values in R l 2 , by L 1 (µ, B(•)) the set of strategy selections, and by L 1 (µ, B * (•)) the set of atom-type-symmetric strategy selections.

Note that the locally convex Hausdorff space we shall be working in is L 1 (µ, R l 2 ), endowed with its weak topology. The first lemma has been proved by [START_REF] Busetto | Noncooperative oligopoly in markets with a continuum of traders[END_REF].

Lemma 1. The set L 1 (µ, B(•)) is nonempty, convex and weakly compact.

The next lemma provides us with the properties of L 1 (µ, B * (•)) required for the application of the Kakutani-Fan-Glicksberg Theorem (see Theorem 17.55 in [START_REF] Aliprantis | Infinite dimensional analysis[END_REF], p. 583). We say that an atom-type-symmetric

t : L 1 (µ, B(•)) → B(t) be a correspondence such that α t (b) = argmax{u t (x(t, b \ b(t), p (b \ b(t)))) : b ∈ B(t)}, for each t ∈ T 0 . Let α : L 1 (µ, B * (•)) → L 1 (µ, B(•))
-Cournot-Nash equilibrium b of Γ n ( ) is δ-positive if b is a δ-positive strategy selection. Now, given > 0, let α δ tr : L 1 (µ, B * (•)) → B(tr) be a correspon- dence such that α δ tr (b) = α tr (b) ∩ B δ (tr), r = 1, . . . , n, for each t ∈ T 1 , and let α t : L 1 (µ, B(•)) → B(t) be a correspondence such that α δ t (b) = α t (b) ∩ B δ (t), for each t ∈ T 0 . Let α δ : L 1 (µ, B * (•)) → L 1 (µ, B(•)) be a correspondence such that α δ (b) = {b ∈ L 1 (µ, B(•)) : b(tr) ∈ α δ tr (b), r = 1, . . . , n, for each t ∈ T 1 , and b(t) ∈ α δ t (b), for each t ∈ T 0 }. Moreover, given > 0, let α δ * : L 1 (µ, B * (•)) → L 1 (µ, B * (•)) be a cor- respondence such that α δ * (b) = α δ (b) ∩ L 1 (µ, B * (•))
. Now, we are ready to prove the existence of a δ-positive atom-type-symmetric -Cournot-Nash equilibrium of Γ n ( ).

{b ij ∈ R l 2 : b ij ≤ n T 1 w i (t) dµ + T 0 w i (t) dµ, i, j = 1, . . . , l, i ∈J j∈J (b ij + b ji ) ≥ n T1 δ
dµ, for each J ⊆ {1, . . . , l}}, the sequence { b m (tr)} belongs to the compact set B δ (tr), r = 1, . . . , n, for each t ∈ T 1 , and the sequence {p m }, where p m = p( b m ), for each m = 1, 2, . . ., belongs, by Lemma 9 in [START_REF] Sahi | The noncooperative equilibria of a trading economy with complete markets and consistent prices[END_REF], to a compact set P, implies that there is a subsequence { B km } of the sequence { B m } which converges to an element of the set

{b ij ∈ R l 2 : b ij ≤ n T 1 w i (t) dµ + T 0 w i (t) dµ, i, j = 1, . . . , l, i ∈J j∈J (b ij + b ji ) ≥ n T1 δ

The limit theorem

The following theorem shows that the sequences of Cournot-Nash equilibrium allocations generated by the replication à la Cournot of E approximate a Walras equilibrium allocation of E. In particular, the next theorem shows that given a sequence of atom-type-symmetric Cournot-Nash equilibrium allocations of Γ n , there exists a Walras equilibrium allocation of E such that, for each trader, his final holding at this Walras equilibrium is a limit point of the sequence of his final holdings at the sequence of atom-type-symmetric Cournot-Nash equilibrium allocations of Γ n .

We now state and prove the limit theorem. T1 δ dµ, for each J ⊆ {1, . . . , l}}, the sequence { bn (t)} belongs to the compact set B δ (t), for each t ∈ T 1 , and the sequence {p n }, belongs, by Lemma 9 in Sahi and Yao, to a compact set P, implies that there is a subsequence { Bk n } of the sequence { Bn } which converges to an element of the set {b ij ∈ R l 2 : b ij ≤ T w i (t) dµ, i, j = 1, . . . , l, i ∈J j∈J (b ij + b ji ) ≥ T1 δ dµ, for each J ⊆ {1, . . . , l}}, a subsequence { bk n (t)} of the sequence { bn (t)} which converges to an element of the set B δ (t), for each t ∈ T 1 , and a subsequence {p k n } of the sequence {p n } which converges to an element p of the set P . Moreover, by Lemma 9 in Sahi and Yao, p 0. Since the sequence { bk n } satisfies the assumptions of Theorem A in Artstein (1979), there is a function b such that b(t) is the limit of the sequence { bk n (t)}, for each t ∈ T 1 , b(t) is a limit point of the sequence { bk n (t)}, for each t ∈ T 0 , and such that the sequence { Bk n } converges to B. (ii) Let x(t) = x(t, b(t), p) for each t ∈ T , xk n (t) = x(t, bk n (t), pk n ), for each t ∈ T , and for n = 1, 2, . . .. Then, x(t) is the limit of the sequence {x kn (t)}, for each t ∈ T 1 , as b(t) is the limit of the sequence { bk n (t)}, for each t ∈ T 1 , and the sequence {p kn } converges to p, x(t) is a limit point of the sequence {x k n (t)}, for each t ∈ T 0 , as b(t) is a limit point of the sequence { bkn (t)}, for each t ∈ T 0 , and the sequence {p kn } converges to p. (iii) BΓ n = Bn as bΓ n ij = t∈T 1 n r=1

bΓ n ij (tr)µ(tr) + t∈T 0 bΓ n ij (t) dµ = t∈T 1 n bn ij (t) µ(t) n + t∈T 0 bn ij (t) dµ = t∈T 1 bn ij (t)µ(t)+ t∈T 0 bn ij (t) dµ = bn ij , i, j = 1, . . . , l, for n = 1, 2, . . .. Then, pn = p( bn ) as pn and bn satisfy (1), for n = 1, 2, . . .. But then, by continuity, p and b must satisfy [START_REF] Aliprantis | Infinite dimensional analysis[END_REF]. Therefore,

  Moreover, we denote by b \ b(t) a strategy selection obtained by replacing b(t) in b with b ∈ B(t). With a slight abuse of notation, b \ b(t) will also represent the value of the strategy selection b \ b(t) at t.

  positive strategy selection. The following theorem, which follows straightforwardly from the existence theorem in Busetto et al. (2011), shows the existence of a δ-positive Cournot-Nash equilibrium of Γ. Theorem 1. Under Assumptions 1, 2, 3, and 4, there exists a δ-positive Cournot-Nash equilibrium of Γ, b Proof. Busetto et al. (2011) showed that, under Assumptions 1, 2, 3, and 4, there exists a Cournot-Nash equilibrium of Γ, b, such that, for each t ∈ T , b(t) ∈ B δ (t). This implies that b is a δ-positive Cournot-Nash equilibrium of Γ.

Theorem 2 .

 2 1 . A strategy selection b of Γ n is atom-type-symmetric if b n (tr) = b n (ts), r, s = 1, . . . , n, for each t ∈ T 1 . We can now provide the definition of an atom-type-symmetric Cournot-Nash equilibrium of Γ n . Definition 4. A strategy selection b such that B is irreducible is an atomtype-symmetric Cournot-Nash equilibrium of Γ n if b is atom-type-symmetric and u tr (x(tr, b(tr), p( b))) ≥ u tr (x(tr, b \ b(tr), p( b \ b(tr)))), for all b ∈ B(tr), r = 1, . . . , n, and for each t ∈ T 1 ;u t (x(t, b(t), p( b))) ≥ u t (x(t, b \ b(t), p( b \ b(t)))),for all b ∈ B(t) and for each t ∈ T 0 .6 The existence of a δ-positive atom-type-symmetric Cournot-Nash equilibrium of Γ n Let δ be determined as in Section 3 and define the δ-positive strategy correspondence B δ , mutatis mutandis, as in Section 3. Notice that B δ (tr) = B δ (ts), r, s = 1, . . . , n, for each t ∈ T 1 . We say that a strategy selection b is δ-positive if b(tr) ∈ B δ (tr), r = 1, . . . , n, for each t ∈ T 1 , b(t) ∈ B δ (t), for each t ∈ T 0 . Moreover, we say that an atom-type-symmetric Cournot-Nash equilibrium b of Γ n is δ-positive if b is a δ-positive strategy selection. The following theorem shows the existence of an atom-type-symmetric δ-positive Cournot-Nash equilibrium of Γ n . Under Assumptions 1, 2, 3, and 4, there exists a δ-positive atom-type-symmetric Cournot-Nash equilibrium of Γ n , b. Proof. Following Sahi and Yao (1989) and Busetto et al. (2011), we define the game Γ n ( ). Given > 0 and a strategy selection b, we define the aggregate bid matrix B = ( bij + ). Clearly, the matrix B is irreducible.

Definition 5 .

 5 Given > 0, a strategy selection b is an atom-type-symmetric -Cournot-Nash equilibrium of Γ n ( ) if b is atom-type-symmetric and u tr (x(tr, b (tr), p ( b ))) ≥ u tr (tr, b \ b(tr), p ( b \ b(tr)))), for all b ∈ B(tr), r = 1, . . . , n, and for each t ∈ T 1 ;

Lemma 2 .

 2 The set L 1 (µ, B * (•)) is nonempty, convex and weakly compact.Proof. L 1 (µ, B * (•)) is nonempty, convex and it has a weakly compact closure by the same argument used by[START_REF] Busetto | Noncooperative oligopoly in markets with a continuum of traders[END_REF]. Now, let {b m } be a convergent sequence ofL 1 (µ, B * (•)). Since L 1 (µ, R l 2 ) is complete, {b m } con-verges in the mean to an integrable function b. But then, there exists a subsequence {b km } of {b m } such that b km (tr) converges to b(tr), r = 1, . . . , n, for each t ∈ T 1 , and b k m (t) converges to b(t), for each t ∈ T 0 (see Theorem 25.5 in Aliprantis and Burkinshaw (1998), p. 203). The compactness of B(t), for each t ∈ T , and the fact that b k m (tr) = b k m (ts), r, s = 1, . . . , n, for each t ∈ T 1 , implies that b ∈ L 1 (µ, B * (•)). Hence L 1 (µ, B * (•)) is norm closed and, since it is also convex, it is weakly closed (see Corollary 4 in Diestel (1984), p. 12). Now, given > 0, let α tr : L 1 (µ, B * (•)) → B(tr) be a correspondence such that α tr (b) = argmax{u tr (x(t, b\b(tr), p (b\b(tr)))) : b ∈ B(tr)}, r = 1, . . . , n, for each t ∈ T 1 , and let α

  be a correspondence such that α (b) = {b ∈ L 1 (µ, B(•)) : b(tr) ∈ α tr (b), r = 1, . . . , n, for each t ∈ T 1 , and b(t) ∈ α t (b), for each t ∈ T 0 }. The following lemma can be proved by the same argument used to show Lemma 2 in Busetto et al. (2011).Lemma 3. Given > 0, the correspondence α :L 1 (µ, B * (•)) → L 1 (µ, B(•)) is such that the set α (b) is nonempty and convex, for all b ∈ L 1 (µ, B * (•)),and it has a weakly closed graph.

Lemma 4 .

 4 Given > 0, there exists a δ-positive atom-type-symmetric -Cournot-Nash equilibrium of Γ n ( ), b . Proof. Let > 0 be given. By Lemma 6 in Sahi and Yao (1989), we know that, for each b ∈ L 1 (µ, B * (•)), α δ tr (b) is nonempty, r = 1, . . . , n, for each t ∈ T1 . Moreover, for each b ∈ L 1 (µ, B * (•)) and for each t ∈ T 1 , there exists b ∈ B(t) such that b ∈ α δ tr (b), r = 1, . . . , n as b is an atom-typesymmetric strategy profile. But then, by the same argument of Lemma 2 in Busetto et al. (2011), α δ * (b) is nonempty. The convexity of α δ (b), for each b ∈ L 1 (µ, B * (•)), is a straightforward consequence of the convexity of α tr (b) and B δ (t), r = 1, . . . , n, for each t ∈ T 1 , and of α t (b) and B δ (t), for each t ∈ T 0 . But then, α δ * is convex valued as L 1 (µ, B * (•)) is convex. α δ tr is upper hemicontinuous and compact valued, r = 1, . . . , n, for each t ∈ T 1 , as it is the intersection of the correspondence α tr , which is upper hemicontinuous and compact valued by Lemma 2 in Busetto et al. (2011), and the continuous and compact valued correspondence which assigns to each strategy selection b ∈ L 1 (µ, B * (•)) the strategy set B δ (tr) (see Theorem 17.25 in Aliprantis and Border (2006), p. 567). Moreover, α δ t is upper hemicontinuous and compact valued, for each t ∈ T 0 , using the same argument. Therefore, α δ has a weakly closed graph, by the same argument used in the proof of Lemma 3. Finally, α δ * has a weakly closed graph as it is the intersection of the correspondence α δ and the continuous correspondence which assigns to each strategy selection b ∈ L 1 (µ, B * (•)) the weakly closed set L 1 (µ, B * (•)) which, by the Closed Graph Theorem (see Theorem 17.11 in Aliprantis and Border (2006), p. 561), has a weakly closed graph (see Theorem 17.25 in Aliprantis and Border (2006), p. 567). But then, by the Kakutani-Fan-Glicksberg Theorem (see Theorem 17.55 in Aliprantis and Border (2006), p. 583), there exists a fixed point b of the correspondence α δ * and hence a δ-positive atom-type-symmetric -Cournot-Nash equilibrium of Γ n ( ). Let m = 1 m , m = 1, 2, . . .. By Lemma 4, for each m = 1, 2, . . ., there is a δ-positive atom-type-symmetric -Cournot-Nash equilibrium b m . The fact that the sequence { B m } belongs to the compact set

  dµ, for each J ⊆ {1, . . . , l}}, a subsequence { b km (tr)} of the sequence { b m (tr)} which converges to an element of the set B δ (tr), r = 1, . . . , n, for each t ∈ T 1 , and a subsequence {p km } of the sequence {p m } which converges to an element of the set P . Since the sequence { b k m } satisfies the assumptions of Theorem A in Artstein (1979), there is a function b such that b(tr) is the limit of the sequence { b k m (tr)}, r = 1, . . . , n, for each t ∈ T 1 , b(t) is a limit point of the sequence { b km (t)}, for each t ∈ T 0 , and such that the sequence { B k m } converges to B. Then, b(tr) = b(ts) as { b km (tr)} = { b km (ts)}, r, s = 1, . . . , n, for each t ∈ T 1 , and b(tr) is the limit of the sequence { b k m (tr)}, r = 1, . . . , n, for each t ∈ T 1 . Hence, it can be proved, by the same argument used by Busetto et al. (2011) to show their existence theorem, that b is a δ-positive atom-type-symmetric Cournot-Nash equilibrium of Γ n ( ).

  µ is the Lebesgue measure, when restricted to T 0 , and µ(t) = (12 )

t , for each t ∈ T 1 . Then, if b is a Cournot-Nash equilibrium of Γ, the pair (p, x) such that p = p( b) and x(t) = x(t, b(t), p), for each t ∈ T , is not a Walras equilibrium of E.

Proof. Suppose that b is a Cournot-Nash equilibrium of Γ and that the pair (p, x) such that p = p( b) and x(t) = x(t, b(t), p), for each t ∈ T , is a Walras equilibrium of E. Clearly, b21 (t) > 0, for each t ∈ T 1 ∪ T 0 . Let h = T 1 ∪T 0 b21 (t) dµ. Since, for each t ∈ T 1 , at a Cournot-Nash equilibrium, the marginal price (see

[START_REF] Okuno | Oligopoly and competition in large markets[END_REF]

) must be equal to the marginal rate of substitution which, in turn, at a Walras equilibrium, must be equal to the relative price of commodity 1 in terms of commodity 2, we must have

  Theorem 3. Under Assumptions 1, 2, 3, and 4, let { bn } be a sequence of strategy selections of Γ and let {p n } be a sequence of prices such that bn (t) = bΓ n (tr), r = 1, . . . , n, for each t ∈ T 1 , bn (t) = bΓ n (t), for each t ∈ T 0 , and pn = p( bΓ n ), where bΓ n is a δ-positive atom-type-symmetric Cournot-Nash equilibrium of Γ n , for n = 1, 2, . . .. Then, (i) there exists a subsequence { bk n } of the sequence { bn }, a subsequence {p k n } of the sequence {p n }, a strategy selection b of Γ, and a price vector p, with p 0, such that b(t)isthe limit of the sequence { bk n (t)}, for each t ∈ T 1 , b(t) is a limit point of the sequence { bk n (t)}, for each t ∈ T 0 , the sequence { Bk n } converges to B, and the sequence {p kn } converges to p; (ii) x(t) is the limit of the sequence {x k n (t)}, for each t ∈ T 1 , and x(t) is a limit point of the sequence {x kn (t)}, for each t ∈ T 0 , where x(t) = x(t, b(t), p) for each t ∈ T , xkn (t) = x(t, bk n (t), pk n ), for each t ∈ T , and for n = 1, 2, . . .; (iii) The pair (p, x) is a Walras equilibrium of E. Proof (i) Let { bn } be a sequence of strategy selections of Γ and let {p n } be a sequence of prices such that bn (t) = bΓ n (tr), r = 1, . . . , n, for each t ∈ T 1 , bn (t) = bΓ n (t), for each t ∈ T 0 , and pn = p( bΓ n ), where bΓ n is a δ-positive atom-type-symmetric Cournot-Nash equilibrium of Γ n , for n = 1, 2, . . .. The fact that the sequence { Bn } belongs to the compact set {b ij ∈ R l 2 : b ij ≤ T w i (t) dµ, i, j = 1, . . . , l, i ∈J j∈J (b ij + b ji ) ≥

The mathematical modelWe consider a pure exchange economy, E, with large traders, represented as atoms, and small traders, represented by an atomless part. The space of traders is denoted by the measure space (T, T , µ), where T is the set of traders, T is the σ-algebra of all µ-measurable subsets of T , and µ is a real valued, non-negative, countably additive measure defined on T . We assume that (T, T , µ) is finite, i.e., µ(T ) < ∞. This implies that the measure space (T, T , µ) contains at most countably many atoms. Let T 1 denote the set of

Lemma 1 in Sahi and Yao implies that B is completely reducible as p 0. Moreover, b(t) ∈ B δ (t) since b(t) is a limit point of the sequence { bkn (t)}, for all t ∈ T . Then, b is δ-positive. But then, by Remark 3 in Sahi and Yao, B must be irreducible. Consider the pair (p, x). It is straightforward to show that the assignment x is an allocation as p and b satisfy [START_REF] Aliprantis | Infinite dimensional analysis[END_REF] and that x(t) ∈ {x ∈ R l + : px = pw(t)}, for all t ∈ T . Suppose that (p, x) is not a Walras equilibrium of E. Then, there exists a trader τ ∈ T and a commodity bundle x ∈ {x ∈ R l + : px = pw(τ )} such that u τ (x) > u τ (x(τ )). By Lemma 5 in [START_REF] Codognato | Cournot-Nash equilibria in limit exchange economies with complete markets and consistent prices[END_REF], there exist λj ≥ 0,

Let bij = w i (τ ) λj , i, j = 1, . . . , l. Then, it is straightforward to verify that xj = w j (τ ) -

for each j = 1, . . . , l. Consider the following cases. Bh n , for n = 1, 2, . . .. Moreover, let q Γ h n , q Γ h n τ ρ , and q hn denote the vectors of the cofactors of the first column of, respectively, ∆ BΓ hn -BΓ hn , ∆ BΓ h n \ b(τ ρ) -BΓ h n \ b(τ ρ), and ∆ Bh n -Bh n , for n = 1, 2, . . .. Clearly, q Γ hn = q hn as BΓ hn = Bhn , for n = 1, 2, . . .. Let ∆ B be the matrix of row sums of B and q be the cofactors of the first column of ∆ B -B. The sequences {q Γ hn } and {q hn } converge to q as the sequence Bh n converges to B and q Γ h n = q hn , for n = 1, 2, . . .. Let w = max{w 1 (τ ), . . . , w l (τ )}. Consider the matrix

and the sequences { BΓ h n -BΓ h n \ b(τ ρ) } and { Bh n -B } converge to 0. Then, the sequence {q Γ h n τ ρ } converges to q. u τ ρ (x(τ ρ, bΓ h n (τ ρ), p( bΓ

be a strategy selection obtained by replacing bhn (τ ) in bhn with b, for

Sahi and Yao, bΓ hn \ b(τ ρ) = bh n \ b(τ ), and p( bΓ hn \ b(τ ρ)) = β hn q Γ hn τ ρ , with β h n > 0, by Lemma 2 in Sahi and Yao, for n = 1, 2, . . .. But then, u τ (x(τ )) ≥ u τ (x), by Assumption 2, as the sequence { bh n (τ )} converges to b(τ ), the sequence {q Γ h n } converges to q, the sequence {q Γ h n τ ρ } converges to q, and p = θq, with θ > 0, by Lemma 2 in Sahi and Yao, a contradiction. Case 2. τ ∈ T 0 . Let { bh k n (τ )} be a subsequence of the sequence { bk n (τ )} which converges to b(τ ). Moreover, let bΓ h kn \ b(τ ) be a strategy selection obtained by replacing bh k n (τ ) in bΓ h k n with b, for n Hence, the pair (p, x) is a Walras equilibrium of E.