Moez Bennouri

C Robert Clark

Jacques Robert

Information provision in nancial markets

We set up a rational expectations model in which investors trade a risky asset based on a private signal they receive about the quality of the asset, and a public signal that represents a noisy aggregation of the private signals of all investors. Our model allows us to examine what happens to market performance (market depth, price e¢ciency, volume of trade, and expected welfare) when regulators can induce improved information provision in one of two ways. Regulations can be designed that either provide investors with more accurate prior information by improving the quality of prior information, or that enhance the transparency of the market by improving the quality of the public signal. In our rational expectations equilibrium, improving the quality of the public signal can be interpreted as a way of providing information about the anticipations and trading motives of all market participants. We nd that both alternatives improve market depth. However, in the limit, we show that improving the precision of prior information is a more e¢cient way to do so. More accurate prior information decreases asymmetric information problems and consequently reduces the informativeness of prices, while a more accurate public signal increases price informativeness. The volume of trade is independent of the quality of prior information and is increasing in the quality of the public signal. Finally, expected welfare can sometimes fall as prior information or the public signal become more precise.

Introduction

In recent years, regulations have been implemented in nancial markets throughout the world designed to induce improved information provision in order to reduce existing uncertainty about the fundamentals of publicly traded rms. There are two alternative means available to regulators trying to stimulate better information provision. The rst involves tightening the standards of information dissemination of publicly traded rms. This is achieved by improving the precision of the prior information available to all investors about asset value. The Sarbanes-Oxley Act of 2002 is an example of a regulation designed to provide investors with more accurate accounting information.

The second way to improve information provision is to enhance the transparency of the market.

The main argument for this alternative is that uncertainty is more about the actions of competing investors than about fundamentals, and so enhancing transparency can be achieved by providing all market participants better access to information about the information held by their competitors. This can involve giving access to more information about the order book, order imbalances, or about all the transactions that have taken place in the market. For example, in April of 1990 the Toronto Stock Exchange (TSE) established a computerized system, Market by Price, to disseminate real-time detailed information on the limit order book to the public. In particular it provided information on the depth and quotes for the current inside market and also depth and limit order prices for up to four price levels above and below the current market, and it automatically displayed all depth (see [START_REF] Madhavan | Should Securities Markets be Transparent?[END_REF] for further details). The London Stock Exchange has experimented with di¤erent regimes for the publication of last-traded prices (see [START_REF] Gemmill | Transparency and Liquidity: A Study of Block Trades on the London Stock Exchange Under Di¤erent Publication Rules[END_REF].

The objective of this paper is to determine the ways in which these two means of improving information provision a¤ect market performance. We set up a rational expectations model in which investors trade a risky asset based on a private signal they receive about the quality of the asset, and a public signal that represents a noisy aggregation of the private signals of all investors. Our model allows us to examine what happens to market performance when regulators can decrease uncertainty about fundamentals by improving information provision in one of the two ways described above. In our setup, regulators can provide investors with more accurate prior information about the assets value by improving the precision of all private signals, and/or can enhance the transparency of the market by improving the quality of the public signal. In our model improving the quality of the public signal can be interpreted as a way of providing information about the information of others.

Moreover, because expectations are rational in our set up, pre-trade information, for instance from the order book or order imbalances, contains no additional information than would be revealed in equilibrium. We therefore focus on the provision of post-trade information about past transactions such as last-traded prices.

We consider the e¤ect of these alternatives on di¤erent measures of market performance. We focus on market depth, price e¢ciency, volume of trade, and ex-ante expected welfare. Improving the precision of prior information or increasing the level of transparency in the market always increases market depth. However, the e¤ect of increasing transparency is less important than the e¤ect of increasing the precision of prior information. In the limit, while very precise prior information results in an innitely deep market, increasing market transparency results in a level of market depth that is bounded from above. The reason is that the informational content of each of the alternatives is di¤erent. Improving information provision by increasing the precision of prior information directly a¤ects the trading strategies of agents, while enhancing market transparency does so only indirectly by a¤ecting the precision of their information about their competitors signals and the information they infer from the equilibrium price.

Turning to the e¤ect of improved information provision on price e¢ciency we nd that price efciency is decreasing in improvements in the accuracy of prior information. More accurate prior information reduces price e¢ciency since the availability of higher quality information implies that the information transmitted by prices is less important. The opposite e¤ect occurs when the market is made more transparent. The provision of the public signal compensates for the reduction of information asymmetries.

Next we examine the e¤ect of improving the precision of prior information or increasing the level of transparency on the volume of trade that takes place in equilibrium. We nd that the volume of trade is independent of the quality of prior information and is increasing in the quality of the public signal.

Finally, to study the e¤ect on expected welfare we turn to numerical analysis since analytical analysis is intractable. Our simulations show that improving information provision is not always welfare increasing. In particular, we show that starting with very imprecise prior information and/or a very opaque market, improving the precision of prior information, and/or enhancing the transparency of the market will cause expected welfare to fall up to a point, after which expected welfare will increase. The reason is that in our framework expectations are rational and investors are imperfectly competitive and so improving information can eliminate potential informational rents.

We are not the rst to study these issues. In the market microstructure literature, researchers have examined the impact of increasing nancial market transparency on market performance. Theoretical (Pagano and Röell 1996, Madhavan 1995and 1996[START_REF] Baruch | Who Benets from an Open Limit-Order Book?[END_REF], empirical [START_REF] Gemmill | Transparency and Liquidity: A Study of Block Trades on the London Stock Exchange Under Di¤erent Publication Rules[END_REF][START_REF] Porter | Post-trade transparency on Nasdaqs National Market System[END_REF][START_REF] Madhavan | Should Securities Markets be Transparent?[END_REF]) and experimental (Bloomeld and OHara 1999, and [START_REF] Flood | Quote disclosure and price discovery in multiple dealer nancial markets[END_REF]) studies reached mixed conclusions about the impact of transparency on nancial markets. In all of these studies transparency is dened as the information that the market or some participants should disclose in order to increase the information shared between agents. This transparency can be either pre-trade (for instance information about the book) or post-trade (for instance information on past trades). All of this information will a¤ect the trading behavior of agents because it allows information about the trading strategies of competitors to be shared. We are not interested in modeling transparency, but simply in determining its ultimate impact on market performance. Therefore, in our model, we capture the impact of transparency by simply allowing for the existence of a public signal that is a noisy aggregation of the signals held by all the players in the market and which, given our rational expectations equilibrium, provides information about the anticipations of all market participants. In contrast with the market microstructure literature on market transparency, since the precision of the public signal can be varied continuously, our model allows us to study the marginal impact of decreasing uncertainty by increasing transparency.

There have also been a number of papers that study the impact of reducing uncertainty directly by announcing a public signal that is independent of the private signals held by investors. [START_REF] Morris | Social value of public information[END_REF] show that public information may be harmful for expected social welfare in a beauty-contest set up where the payo¤ of an agent decreases in the distance between his action and the actions of the others. The main intuition for their result is that in the beauty-contest set up, public information helps investors achieve coordination, but greater coordination between agents is assumed to be socially irrelevant. 1 [START_REF] Clark | Information and Crowding Externalities[END_REF] focus on situations where individuals care about making the correct choice, but also about the fraction of the population that make the same choice that they do and in which the preferred level of coordination lies between 1=2 and the whole population. They focus on two types of pure strategy equilibriaone in which agents ignore their private information in favor of the public information, and one in which every individual chooses the action that they believe is most likely to be correct. Like Morris and Shin they nd that better public information need not increase players expected utility, but, to the contrary, may even decrease it. This can occur when in the initial situation both types of equilibria exist, and the one with the higher expected utility is the one in which people heed their private information; in this situation, increases in public signal quality can eliminate this equilibrium type, and the remaining equilibrium in which all players follow the public signal may yield a lower expected utility for all players. The same is true for the private signal, as it may cause overcrowding in situations where agents choose the action they believe is most likely to be correct.

There are two important di¤erences between our model and [START_REF] Morris | Social value of public information[END_REF] and [START_REF] Clark | Information and Crowding Externalities[END_REF]. First, there is no direct coordination e¤ect in our model. As a result, the positive e¤ect of decreasing uncertainty would be related to a decrease in the adverse selection problem which would a¤ect their trading behavior. Second, we consider a rational expectations equilibrium set up in which a small number of investors engage in imperfect competition. We model the strategic behavior of investors and explore the way it is a¤ected by the existence of a more valuable information.

The rest of the paper proceeds as follows. In the next section we describe the model. Section 3 characterizes the equilibrium. In Section 4 we analyse the equilibrium e¤ects of improving information provision on market performance. We conclude in Section 5. All proofs are in the Appendix.

The model

Consider a market where Q units of a risky asset are traded. There are n investors who participate in this market. The value of the risky asset is denoted by and is unobserved by the participants.

However, each investor has some information about . Based on the information available to them, each submits a demand function and in equilibrium total demand will equal the quantity supplied, Q.

As in Kyle (1989), we consider a rational expectations equilibrium with imperfect competition among agents in which investors submit downward sloping demand curves. The main di¤erence between our model and Kyles is that, in our model, investors not only observe a private signal about the value of , but also observe a common public signal, which is itself an imperfect signal of the average of individual private signals.

Information structure

We assume that prior to trading each investor i receives two imperfect signals about the value of the risky asset, . The rst signal is private for investor i, and is denoted by s i . The second signal, denoted by S; is a public signal observed by all investors. Both signals are noisy and all random variables are assumed to be normally distributed. So, we suppose that

s i = + t i where N ( ; 2
) and t i N (0;2 t );

and

S = X i s i n + = + X j t i n + where N (0; 2 ): (2)
From these expressions we can see the two means available to regulators for reducing uncertainty about the value of a nancial asset. 2 One option is for standards on information dissemination of publicly traded assets to be tightened. In our setup this is captured by reducing 2 . The second alternative is to enhance the transparency of nancial markets by giving access to more information about the average private signal in the market. This can be done by providing information on past transactions in the market. In our model this is captured by a decrease in 2 . 3Since S is a noisy signal about the average of all private signals received by investors, if 2 ! 0; it is as if participants can observe the average of the signals received by all investors. This would correspond to the case where investors have access to all of the existing valuable information. Alternatively, if

2

! 1; we have E (jS; s i ) = E (js i ) ; so that investors can infer no additional information about the nal value of the asset from observing S: By changing the value of 2 we will explore the impact of information aggregation and market transparency on the market equilibrium and the trading strategies of agents.

In practice, even complete access to information about past trades may not provide investors with all existing information. This is because information on past trades is a compound of private signals and idiosyncratic liquidity shocks. Investors can be motivated to trade either to capitalize on valuable trading opportunities they perceive given their fundamental information or for liquidity needs. As a result it is probably most appropriate to think of as the sum of two variables: one that captures information-based trading and one that captures liquidity-based trading. We are of course interested in the incentive to trade that is marginal relative to the liquidity-based motivation.

Agents

We suppose that agents have CARA preferences with r denoting their risk aversion coe¢cient. Given the assumption of normally distributed random variables, this implies that preferences can be denoted by the mean-variance representation. An agent who trades a quantity x i has expected utility of:

E [W + (w i + x i + F i ) 0 px i jI] 0 r 2 V ar [jI] (w i + x i + F i ) 2 ; ( 3
)
where W is some initial wealth, w i is agent is initial endowment of the stock, and F i corresponds to an idiosyncratic liquidity shock that the agent receives before participating in the market. I denotes all information available to the investor. The initial endowment w i allows us to model the supply of shares in the market. We assume that P i w i = Q: We assume that F i is private information for the agent and that it follows distribution N (0; 2 F ). The introduction of F i adds a second motivation for trading as in [START_REF] Glosten | Insider Trading, Liquidity, and the Role of the Monopolist Specialist[END_REF]. Without these shocks, the rational expectations equilibrium would be fully revealing and so traders would have no incentive to gather information. The result would be a disclose in order to increase the information shared between agents. Essentially this boils down to providing a public signal about the signal of others.

no-trade equilibrium as in [START_REF] Grossman | On the Impossibility of Informationally Ecient Markets?[END_REF]. 4 With the inclusion of F i ; higher demand by a particular agent may be interpreted as being the result either of good information or a large negative liquidity shock.

The precision of the liquidity shock will a¤ect the existence of the linear equilibrium. We prove in what follows that unless this shock is su¢ciently noisy, agents may refuse to take part in the market.

Equilibria with linear downward sloping demand

As in Kyle (1989) we focus on linear equilibria by assuming that investors submit linear demand functions. Under simplifying assumptions of normal and independent idiosyncratic shocks and a CARA utility function, we are able to compute a symmetric rational expectations equilibrium with downward sloping demand curves. The equilibrium is derived by maximizing each agents expected utility against the residual demand curve. Indeed, in the spirit of rational expectations equilibria, after making conjectures about the optimal demand functions of his competitors, each agent will choose his optimal strategy by acting as a monopsonist with respect to a residual demand curve conditional on these conjectures.

Let us suppose that agent i conjectures that each agent j 6 = i has the following inverse demand function :

P (s j ; F j ; S; w j ; x j ) = C 0 + C 1 s j 0 C 2 F j + C 3 S 0 C 4 w j 0 Ex j (4)
Since, the equilibrium price is the same for all agents, summing the demand functions for all j 6 = i, dividing by (n 0 1), and then using the market clearing condition, P j6 =i x j = 0x i ; yields the inverse residual demand function for agent i:

P (x i ; w i ; S; F 0i ; s 0i ) = C 0 + C 1 P j6 =i s j (n 0 1) 0 C 2 P j6 =i F j (n 0 1) + C 3 S 0 C 4 (Q 0 w i ) (n 0 1) + Ex i (n 0 1) ; (5)
where F 0i and s 0i represent the liquidity shocks and private signals for all agents other than i respectively. This can be rewritten as 4 The alternative is to introduce noise traders into the model.

P (x i ; w i ; S; y i ) = C 0 + C 1 y i + C 3 S 0 C 4 (Q 0 w i ) (n 0 1) + Ex i (n 0 1) ; (6)
where

y i h 2 j6 =i sj (n01) 0 C 2 C 1 2 j6 =i Fj (n01) i
. y i represents the indirect information that agent i can infer about the other investors strategies given his demand function x i (1) and the equilibrium price p. When the variance of F j , 2 F ; goes to zero,

y i = h 1 (n01) P j6 =i s j i
and the market equilibrium is fully informative. y i is a su¢cient statistic for the trading motivations of agents. Its two components represent the di¤erent trading motivations of agents alluded to above. The rst term captures the information-based trading motive, while the second represents the liquidity-based motive. If an agent only observes y i , he has no way to distinguish between these two motives. The role of the public signal, S, as we have dened it here is to provide a noisy signal about the rst term of y i which allows agents to better distinguish between the two trading motives. 5 Under the rational expectation equilibrium, we will assume that investor i selects his optimal demand function as if he were observing the y i from the residual demand function. This is illustrated in Figure 1. Suppose that agent i faces the residual demand curve associated with some y i0 . He can select the pair (p 3 0 ; x 3 0 ) lying along this curve which maximizes his expected utility given s i ; S; and y i0 . Now suppose instead that y i = y i1 . The agent can select a price and quantity pair (p 3 1 ; x 3 1 ) that lies along the new residual demand curve associated with y i = y i1 and that maximizes his expected utility given s i ; S; and y i1 . The optimal demand curve for agent i can, therefore, be constructed by connecting the optimal price-quantity pairs for each value of y i . The optimal demand curve provides an optimal response for all values of s i ; S; and y i even if y i is not directly observed by the investor i.

Formally, for some S; y i ; and s i , agent is problem consists of selecting a demand function x i (p)

that solves the following problem

max xi(p) W + E [jy i ; S; s i ] (w i + x i + F i ) 0 px i 0 r 2 V ar [jy i ; S; s i ] (w i + x i + F i ) 2 subject to p = C 0 + C 1 y i + C 3 S 0 C 4 (Q 0 w i ) (n 0 1) + Ex i (n 0 1) :
The rst-order condition is 5 Alternatively, we could model the public signal as being a noisy signal about the liquidity-based trading motives. That is, a noisy signal about the sum of the F j . In this case, enhancing transparency can be achieved by providing market participants better access to information about liquidity-based trading motives of their competitors. This could be achieved for instance by implementing sunshine trading which identies informationless trading (see for instance Admati and Peiderer 1991). and the second-order condition is 2E (n 0 1) + rV ar(jy i ; S; s i ) > 0:

E [jy i ; S; s i ] 0 C 0 + C 1 y i + C 3 S 0 C 4 (Q 0 w i ) (n 0 1) + 2Ex i (n 0 1) 0 rV ar [jy i ; S; s i ] (w i + x i + F i ) = 0; (7)
In order to further characterize the structure of the demand function, we state the following lemma:

Lemma 1 Given the denitions of y i ; S and s i , and the assumption that all random variables are distributed normally, we have the following

E [jy i ; S; s i ] = k 0 k + k 1 k y i + k 2 k (nS 0 s i ) (n 0 1) + k 3 k s i (9)
and V ar(jy i ; S;

s i ) = 2 t k 3 =k; ( 10
)
with k 0 = 2 t ( 11
)
k 1 = (n 0 1) 2 2 t 2 [X 2 2 + 2 t 2 + (n01) n 2 X 2 2 t ] (12)
k 2 = (n01) 2 n 2 X 2 2 2 t [X 2 2 + 2 t 2 + (n01) n 2 X 2 2 t ] (13)
k 3 = 2 (14
)

and k = [k 0 + k 1 + k 2 + k 3 ] (15)
where

X = C 2 C 1 F :
Solving the rst-order condition yields the optimal demand function which is of the form conjectured in ( 4) and which we characterize in the following proposition:

Lemma 2 For strictly positive r; 2 " ; 2 ; 2 ;and 2 t , and n > 2, if a (linear) rational expectations equilibrium exists, then each agent must submit a strictly decreasing inverse demand curve of the form:

P (x i ; S; F i ; s i ) = C 0 + C 1 s i 0 C 2 F i + C 3 S 0 C 4 w i 0 Ex i ( 16
)
where:

E = (n 0 1) k 3 0 k2 (n01) + k 1 r 2 t k 3 =k (n 0 2) k 3 0 (n02)k2 (n01) 0 2k 1 (17) C 0 = k 0 k + k 1 r 2 t k3 k ((n 0 1)k 3 0 k 2 0 k 1 ) Q (18) C 1 = k 3 k 0 k 2 k(n 0 1) + k 1 k (19)
C 2 = r 2 t k 3 k 2 4 k 3 0 k2 (n01) + k 1 k 3 0 k2 (n01) 3 5 (20)
C 3 = nk 2 k(n 0 1) (21)
C 4 = r 2 t k 3 k (n 0 1)k 3 0 k 2 + (n 0 1)k 1 (n 0 1)k 3 0 k 2 0 k 1 ( 22
)
and k; k 0 ; k 1 ; k 2 and k 3 are dened above.

In order to show that the equilibrium inverse demand function is well-dened, we must write the unknown values of the equilibrium inverse demand function (C 0 , C 1 ; C 2 ; C 3 , C 4 , and E) in terms of the parameters of the model (n, r; 2 " ; 2 ; 2 ;and 2 t ) and conrm that the conjectures on prices are satised. Since C 0 , C 1 ; C 2 ; C 3 , C 4 , and E depend on k; k 0 ; k 1 ; k 2 ; and k 3 which in turn depend on C 1 and C 2 through the value of

X = C 2 C 1
F , the problem is somewhat complicated. Using [START_REF] Morris | Social value of public information[END_REF] and [START_REF] Pagano | Transparency and Liquidity: A Comparison of Auction and Dealership Markets With Informed Trading[END_REF],

we can write X = C 2 C 1
F in terms of the parameters of the model (n, r; 2 " ; 2 ; 2 ;and 2 t ). We have:

X = C 2 C 1 F = r 2 t " " [X 2 2 + 2 t 2 + (n01) n 2 X 2 2 t ] 2 [X 2 + 2 t ] # ; (23)
or alternatively,

2 X 0 r 2 t " 3 = r " (n 0 1) X 2 [X 2 + 2 t ] 4 t n 2 2 : ( 24
)
So if we can show that there exists a (unique) X that solves the above equation and that satises the second-order condition (8), then we can characterize the (unique) equilibrium with linear demand.

The following lemma describes when such an X exists and is unique.

Lemma 3 When r 2 2 " n (n02) 2 t
, the equilibrium dened in Lemma 2 is unique and is fully charac-

terized by X = C 2 C 1
F : There is a unique positive X that solves (23) and that satises the second-order condition given by ( 8). When

r 2 2 " < n (n02) 2 t
, there are values of 2 for which no equilibrium exists.

The equilibrium existence condition is su¢cient but not necessary. When this condition is violated, we can only show that an equilibrium does not exist for certain values of 2 : Note that the condition does not depend on 2 or on 2 which are the variables dening the alternatives available to regulators for reducing uncertainty in our model. This fact allows us to perform comparative statics without being constrained by bounded values for 2 and 2 :

The equilibrium existence condition stems from the second-order condition of the optimality of the trading strategies of agents. Roughly speaking, it imposes that the lower bound on X given by (24)

(which is equal to r 2 t " ) is higher than the lower bound given by the second-order condition (which is equal to

n (n02) 1=2
t ): Therefore changing the values of di¤erent parameters does not violate the second-order condition.

Intuitively, the equilibrium existence condition states that a linear equilibrium exists when (i)

there is enough noise caused by liquidity shocks, which would make private signals more valuable and increase the likelihood of trading; (ii) agents are su¢ciently risk averse and therefore willing to trade in the market in order to share risk; and/or (iii) the variance of private signals is su¢ciently high that adverse selection problems are minimized.

In the following, we summarize the results about existence of equilibrium that will be helpful when we turn to equilibrium analysis in the following section.

Proposition 1 For all values of 2 and 2 , a rational expectations equilibrium with linear demand functions exists if and only if r 2 2 " n (n02) 2 t :

Equilibrium analysis

As mentioned above, we are interested in determining the e¤ect of improved information provision on market performance. In particular, we would like to investigate the two alternatives available to regulators for reducing uncertainty about the value of nancial assets. In this section we consider what happens when standards on information dissemination of publicly traded assets are tightened (ie. when 2 is reduced), and what happens when more information about past transactions in the market are provided (ie. when 2 is reduced).

Our main focus is on the impact of changes in these variables on di¤erent measures of market performances. We concentrate our analysis on the e¤ect of reducing 2 and/or 2 on market depth, market e¢ciency, and expected welfare. As in [START_REF] Kyle | [END_REF] we dene market depth as the inverse of sensitivity of prices to changes in quantities; that is, the size of an order ow innovation required to change prices a given amount. So market depth provides a measure of the liquidity of the market.

By market e¢ciency we mean the way prices aggregate the available information in the market. We study the impact on these two measures since the objective of regulators is to ensure high quality fair, orderly, and e¢cient 6

Market depth and price e¢ciency

In order to study the e¤ect of reducing 2 and 2 on market depth and price e¢ciency we start by characterizing the way in which these variables a¤ect X and the ex-post variance of . Using (24), we can sate the following:

Lemma 4 X has the following properties:

(i) X does not depend on 2

(ii) X is a decreasing function of 2 : X converges to r 2 t " as 2 goes to innity and to innity as 2 converges to 0.

Using [START_REF] Gemmill | Transparency and Liquidity: A Study of Block Trades on the London Stock Exchange Under Di¤erent Publication Rules[END_REF] and ( 23), we obtain an expression for the ex-post variance of :

V ar(jy i ; S;

s i ) = 2 t k 3 k 0 + k 1 + k 2 + k 3 = 2 t 2 2 t + n 2 0 2 (n01)X 2 2 [X 2 2 + 2 t 2 + (n01) n 2 X 2 2 t ] = 2 t 2 ( 2 t + n 2 ) 0 2 r" 2 t (n01)X [X 2 + 2 t ] (25)
Note that the expression for V ar(jy i ; S; s i ) is written so that it varies with 2 only indirectly through its dependence on X; and that X does not depend at all on 2 : Hence, we can use (25) to compute the impact of changes in 2 and 2 . V ar(jy i ; S; s i ) is increasing in

X [X 2 + 2 t ]
, which in turn is decreasing 6 See, for example, http://www.sec.gov/about/whatwedo.shtml. 13 in X whenever X 2 2 t . But this is always true under the assumption of Lemma (3), indeed we have X r . Hence, V ar(jy i ; S; s i ) is decreasing in X and increasing in 2 . Also V ar(jy i ; S; s i ) is increasing in 2 . We summarize these results with the following lemma:

Lemma 5 The ex-post variance of , V ar(jy i ; S; s i ), is (i) decreasing in X and increasing in 2

(ii) increasing in 2 .

Market depth analysis

In our rational expectations equilibrium set-up market depth is measure by 1=E. The value E measures the way prices change in response to quantity changes and is dened in Proposition 2. The lower is E; the deeper is the market. 7 Using (17), we have:

E = rV ar(jy i ; S; s i ) (n 0 1) k 3 0 k2 (n01) + k 1 (n 0 2) k 3 0 (n02)k2
(n01) 0 2k 1 = rV ar(jy i ; S; s i )

" (n 0 1) 0 X 2 + n 2 t 1 (n 0 2)X 2 0 n 2 t # = 2 4 2 t 2 ( 2 t + n 2 ) 0 2 r" 2 t (n01)X [X 2 + 2 t ] 3 5 " (n 0 1) 0 X 2 + n 2 t 1 (n 0 2)X 2 0 n 2 t # (26)
Note that E is increasing in 2 , and that both its terms are decreasing in X. Hence, E is also increasing in 2 .

We summarize the results for market depth with the following proposition:

Proposition 2 (i) Market depth, is decreasing in both 2 and 2 .

(ii) As 2 converges to zero, market depth converges to innity. However, as 2 converges to zero, market depth converges to a nite value. In the limit, we have:

lim 2 !0 1 E = (n 0 2) n (n 0 1)r 2 t :
(iii) The marginal e¤ect of 2 on market depth is increasing in 2 ; i.e.

@ 2 ( 1 E ) @ 2 @ 2
> 0:

7 Note also that in our set-up E is a measure of the e¤ective bid-ask spread.

Making the market more transparent increases market depth. Increasing the precision of the prior information available to all investors about asset value also increases market depth, and the marginal e¤ect is increasing in market transparency. This suggests that regulators should somehow link market transparency to an increase in the quality of prior information.

More interestingly, (ii) states that in the limit, as prior information becomes extremely precise, the market becomes innitely deep, while an extremely transparent market would yield a level of market depth bounded from above. This demonstrates the di¤erence between the two alternative ways of improving information provision. While, both generate an increase in market depth, they do so with di¤erent levels of e¢ciency. Increasing the precision of prior information is more e¢cient since doing so would have a direct impact on the trading behavior of agents by a¤ecting the accuracy of all available information. On the other hand, enhancing transparency only inuences agents strategies by a¤ecting the precision of their information about their competitors signals and the information they infer from the equilibrium price. Therefore, we argue that a further increase in the quality of prior information or of private signals has a more important impact on market depth than does a marginal increase in market transparency. Henceforth, working on rules that allow for an improvement in the quality of the information about rms (whether by increasing prior information or even the private signals) would be a more e¢cient way of increasing market depth than would rules that increase the quality of the information of the traders behavior reecting a higher market transparency.

Note, however, that (iii) implies that the two e¤ects reinforce one another. This nding is particularly interesting since almost all of the literature focuses on one or the other of these types of information, whereas our setup considers both. It suggests that a regulator should encourage both types of informational improvements if it wants to achieve greater market depth.

Price e¢ciency analysis

We now consider the e¤ect of changes of 2 and 2 on price e¢ciency. By price e¢ciency we mean the quality of information about the asset value transmitted by equilibrium price. We consider the following measure of price e¢ciency: . As 2 goes to innity, X goes to r 2 t " and e converges to lim

e = 1 0 V ar(jp) var() = 1 0 V ar(jy i ; s i ; S) 2 = 1 0 2 t ( 2 t + n 2 ) 0 2 r" 2 t (n01)X [X 2 + 2 t ] = n[X 2 + 2 t ] 0 r " 2 t (n 0 1)X 2 t 2 + n [X 2 + 2 t ] 0 r " 2 t (n 0
X!r 2 t " e = [ 0 r 2 t " 1 2 + n 2 t ] 2 t 2 [(r 2 t " ) 2 + 2 t ] + [(r 2 t " ) 2 + n 2 t ]
:

The rst part of (i) may seem counter-intuitive in the sense that it states that increasing the quality of prior information reduces price e¢ciency. However, since price e¢ciency measures the ability of prices to transmit signals about the information available in the market, it makes sense that if the available information is of higher quality, prices are less e¢cient as information aggregators. Furthermore, a marginal increase in the quality of information would negatively a¤ect the informativeness of prices. On the other hand, (ii) states that enhancing market transparency by increasing the quality of the aggregate signal (decreasing 2 ) positively a¤ects the informativeness of the equilibrium price.

However, the positive e¤ect of transparency on market e¢ciency is bounded since, in the limit, as 2 converges to zero or to innity, price e¢ciency converges to nite values. As 2 is lowered, it becomes easier to distinguish the informational and liquidity motivations for trading. This allows traders to better assess the information-motivations for trading and so the resulting price is more informative.

Equilibrium trade

From the derived equilibrium we have that

x i = E [jy i ; S; s i ] 0 h C 0 + C 1 2 j6 =i sj (n01) 0 C 2 2 j6 =i Fj (n01) + C 3 S 0 C 4 (Q0wi) (n01) i 0 rV ar [jy i ; S; s i ] (w i + F i ) 2 E (n01) + r 2 V ar [jy i ; S; s i ] :
If we sum the above expression for all x i , then since P j x j = 0, we have that x i 0 P j x j = x i , which can be written as

x i = C 1 (n01) s i 0 2 j sj n 0 C 4 (n01) + rV ar [jy i ; S; s i ] w i 0 Q n + C 2 (
x i = [(n 0 2) X 2 0 n 2 t ] (n 0 1)[X 2 + 2 t ] " X t i 0 P j t j n + [X 2 + 2 t ] [X 2 ] Q n 0 w i + P j F j n 0 F i : (28)
Note here that, interestingly, x i does not depend on S: So, the trading quantity of the agent is not a¤ected directly by the public signal. The agents net trade will depend on the di¤erence between his private signal the average signal in the market, the di¤erence between his endowment and the average endowment in the market, and the di¤erence between his liquidity shock and the average liquidity shock. Furthermore, recall that X 2 is independent of 2 and therefore so is an agents net trade. Now, consider the impact of 2 on net trade. The variance 2 a¤ects net trade indirectly through its impact on X, which is decreasing in 2 . Because t i 0

2 j tj n , Q n 0 w i ; and
2 j Fj n 0 F i can be negative or positive, the impact of 2 is ambiguous.

However, we can derive unambiguous results on the dispersion of net trades among agents, E(x 2 i ).

Doing so allows us to say something about the impact of 2 on the volume of trade (since E(x 2 i ) is related to P jx i j ): For tractability, we assume that for i, we have

w i = Q n :
Proposition 4 The dispersion of net trades among agents, E(x 2 i ); is given by

E(x 2 i ) = [(n 0 2) X 2 0 n 2 t ] 2 (n 0 1)[X 2 + 2 t ] 2 " nX 2 ;
and is increasing in X 2 and decreasing in 2 :

The above result may seem paradoxical. An increase in 2 lowers the value of the public signal which reduces the relative importance of shared information. It follows that individual demand will rely more on the idiosyncratic information available. This e¤ect tends to increase the amount of trade. However, despite this, there is in fact less trade since there is another more dominant e¤ect.

An increase in 2 increases the adverse selection e¤ect and therefore increases the absolute value of the slope of the demand curve (or equivalently decreases market depth). This reduces trade in the market.

Expected welfare

Finally we analyse the e¤ect of 2 and 2 on expected welfare. We start by dening an agents welfare conditional on all of the available information

V (s i ; y i ; S; F i ) = (E(js i ; y i ; S; F i ) (w i + x + F i ) 0 px 0 r 2 var(js i ; y i ; S; F i )(w i + x + F i ) 2 :
By substituting the rst order condition ( 7) we obtain

V (s i ; y i ; S; F i ) = (E(js i ; y i ; S; F i ) (w i + F i ) 0 r 2 V ar(js i ; y i ; S; F i )(w i + F i ) 2 + r 2 V ar(js i ; y i ; S; F i ) + E (n 0 1) x 2 i ; (29)
where x i is given by (28).

When there is no uncertainty (ie. var(js i ; y i ; S; F i ) = 0), V (s i ; y i ; S; F i ) = E(js i ; y i ; S; F i ) (w i + F i ) = (w i + F i ). That is, when there is no uncertainty the equilibrium price is equal to the value of the asset; also because there is no risk there are no gains from trade. When there is uncertainty, the second and third terms of (29) are non-zero. The second term is negative and it reects the disutility associated with risk. The third term is positive and it measures to what extent trade can mitigate risk. The impact of an increase in variance depends on an agents initial endowment and liquidity shock (w i + F i ). For instance if w i + F i = 0, then agent i is initially not exposed to any risk and so can trade purely in order to take advantage of the uncertainty faced by rival agents.

Using V (s i ; y i ; S; F i ) we can dene the ex-ante expected welfare of an agent as

EW = 0e 0rW E (si;yi;S;Fi) [0 exp [0rV (s i ; y i ; S; F i )]] (30)
where the variables, s i , y i , S, and F i have means equal to ; ; ; and 0 respectively, and have variance-covariance matrix given by

A = 0 B B B @ 2 + 2 t 2 2 + 2 t n 0 2 2 + 1 (n01) 2 t + X 2 (n01) 2 + 2 t n 0 2 + 2 t n 2 + 2 t n 2 + 2 t n + 2 0 0 0 0 2 F 1 C C C A : (31)
Note that V (s i ; y i ; S; F i ) is a quadratic function and so by applying moment generating techniques we can rewrite (30). For tractability we assume that w i = Q n . We summarize the result in the following proposition.

Proposition 5 Assuming that w i = Q n , expected welfare can be written in the following form

EW = 0 j3j 1=2 jAj 1=2 exp ( 0r W + Q n 0 r 2 t k 3 2k Q n 0 r 2 t03t !) = 0 exp 8 < : 0r 0 @ 0 @ W + Q n 0 r 2 t k 3 2k Q n 2 0 log( j3j jAj ) 2r 1 A 0 r 2 t03t 1 A 9 = ; (32)
where 3 and t are dened in the appendix.

Solving for the impact on expected welfare of reducing2 and 2 is analytically intractable and so we turn to simulation analysis. Figure 2 plots expected welfare as a function of 2 and/or 2 . From the picture it can be seen that starting at high values of 2 and/or 2 ; expected welfare is initially decreasing as 2 and/or 2 are reduced, before nally increasing as 2 and/or 2 become very small.

Figure 3a plots ex-ante expected welfare as a function of 2 for di¤erent values of 2 , while gure 3b plots ex-ante expected welfare as a function of 2 for di¤erent values of 2 . In each we see the same pattern as in the the three-dimensional plot. Expected welfare is initially decreasing as 2 or 2 are reduced, before nally increasing as 2 or 2 become very small.

To explain what is going on in these gures, we turn back to equation (32) from which it can be seen that expected welfare is made up of mean and variance components. The second term in the exponent of (32), r 2 t03t, represents the variance component. Expected welfare is decreasing in this component. The rst term, W + Q n 0

r 2 t k3 2k Q n 2 0 log( j3j jAj ) 2r
, therefore represents the mean component, which is positively related to expected welfare.

The variance component is always increasing in 2 and/or 2 since more precise information (more precise prior information and/or greater transparency) will always reduce uncertainty (see Figure 3).

The relationship between the mean component and 2 / 2 is more complicated (see Figure 4). Starting from perfect information ( 2 = 2 = 0) where there is no trading, we consider a small increase in 2 and/or 2 . This reduces the mean component since for these values trading will occur only for liquidity purposes. This is only true up to a certain point, after which increasing 2 and/or 2 increases the mean component since less precise information provides the opportunity for informational rents.

So what is the overall e¤ect of more precise information on expected welfare? Starting from high 2 and 2 we consider the impact of a decrease in 2 and/or 2 on expected welfare. As 2 and/or 2 falls both the variance component and the mean component decrease leading to conicting e¤ects on welfare since welfare is decreasing in the former and increasing in the latter. Because of the greater e¤ect of informational rents, the mean-component e¤ect will dominate leading to an overall reduction in expected welfare. This is true up until information becomes relatively precise (low 2 and/or 2 ).

At this point trading becomes more liquidity motivated and the uncertainty e¤ect comes to dominate causing expected welfare to increase as information becomes more precise. 9As mentioned in the Introduction, other articles have also found that improved information may lower welfare, but these mostly focus on a class of economies that feature externalities, strategic complementarity or substitutability (see for instance Morris and Shin 2002, and Clark and Polborn 2006, and see Angeletos and Pavan 2007 for a summary). We focus instead on a rational expectations set up with imperfect competition and improving information can lower welfare in this case since doing so eliminates potential informational rents.

Concluding remarks

In this paper we have studied the impact on market performance of regulations that would improve the level of information provision in nancial markets. In our rational expectations model we are able to examine what happens to market performance when regulators can provide investors with more accurate prior information by improving the precision of all private signals, and/or can enhance the transparency of the market. We show that the two alternative ways of decreasing uncertainty have the same e¤ect on market depth, but opposite e¤ects on market e¢ciency. Providing more precise

Appendix

Proof of Lemma 1. To simplify notation we dene the following:

Z 4 = N (; 2 ); Z 3 = s i = + t i N (; 2 + 2 t ); Z 2 = nS 0 s i (n 0 1) = n + 2 i ti n + 0 s i (n 0 1) = + P j6 =i t j n 0 1 + n n 0 1 N ; 2 + 1 (n 0 1) 2 t + n 2 2 (n 0 1) 2 ! ; Z 1 = y i = 1 (n 0 1) X j6 =i s j 0 C 2 C 1 (n 0 1) X j6 =i F j N ; 2 + 1 (n 0 1) 2 t + C 2 2 C 2 1 (n 0 1) 2 " :
We then let

X = C 2 C 1 " ; so that Z 1 N ; 2 + 1 (n01) 2 t + X 2 (n01)
: Our objective is to compute E(Z 4 jZ 1 ; Z 2 ; Z 3 ) and V ar(Z 4 jZ 1 ; Z 2 ; Z 3 ):

Note that all variables are normally distributed, so the joint distribution is

(Z 1 ; Z 2 ; Z 3 ; Z 4 ) N (m; 6) ;
where m = (E(Z 1 ); E(Z 2 ); E(Z 3 ); E(Z 4 )) = (; ; ; ) and 6 is the variance-covariance matrix dened as follows For notation purposes, let 6 = ( ij ) for i; j = 1; :::4 and 6 01 = ( ij ) for i; j = 1; ::; 4: Moreover, let 6 (04) be the upper left 3 by 3 corner of 6; so

6 = 0 B B B @ 2 + 1 (n01) 2 t + X 2 (n01) 2 + 2 t (n01) 2 2 2 + 2 t (n01) 2 + 2 t (n01) + n 2 2 (n01) 2
6 (04) = 0 B @ 2 + 1 (n01) 2 t + X 2 (n01) 2 + 2 t (n01) 2 2 + 2 t (n01) 2 + 2 t (n01) + n 2 2 (n01) 2 2 2 2 2 + 2 t 1 C A :
Finally, let 6 01 (04) = ( pq ) for p; q = 1; ::; 3. Since all variables are normally distributed, we have

(Z 4 jZ 1 = z 1 ; Z 2 = z 2 ; Z 3 = z 3 ) N 0 m 3 ; 2 1
; where 2 = det(6) det(6 (04) ) , 10 and

m 3 = E(Z 4 ) + 3 X l=1 l (z l 0 E(Z l )); (33)
with l = 3 X q=1 lq q;4 ; pour l = 1; 2; et 3:

Similar manipulations allow us to write

2 = (n 0 1) 2 X 2 2 2 t k[n 2 2 2 t + n 2 2 X 2 + (n 0 1) 2 t X 2 ]
; and

3 = 2 k :
Substitution of the values of z i in (33) gives

m 3 = + 3 X l=1 l (z l 0 ) = 1 0 3 X l=1 l ! + 1 y + 2 nS 0 s i (n 0 1) + 3 s i : Note that 1 0 3 X l=1 l = 2 t k ;
then substitution of l (for l = 1; 2; 3) gives

m 3 = E [jy i ; S; s i ] = 2 t k + n 2 (n01) 2 2 t 2 [n 2 2 2 t +n 2 2 X 2 +(n01) 2 t X 2 ] k y i + (n01) 2 X 2 2 2 t [n 2 2 2 t +n 2 2 X 2 +(n01) 2 t X 2 ] k nS 0 s i (n 0 1) + 2 k s i : Finally let: k 0 = 2 t ; k 1 = n 2 (n01) 2 2 t 2 [n 2 2 2 t +n 2 2 X 2 +(n01) 2 t X 2 ] ; k 2 = (n01) 2 X 2 2 2 t [n 2 2 2 t +n 2 2 X 2 +(n01) 2 t X 2 ] and k 3 = 2 :
Note that from these denitions and from (36

) that k = [k 0 + k 1 + k 2 + k 3 ] :
This establishes the results of Lemma 1:

Proof of Lemma 2. Recall that we have:

y i = p 0 C 3 S + C 4 (Q0wi) (n01) 0 Ex (n01) 0 C 0 C 1 and (38)
E [jy i ; S; s i ] = k 0 k + k 1 k y i + k 2 k (nS 0 s i ) (n 0 1) + k 3 k s i (39)
Hence, we can rewrite the rst-order condition in [START_REF] Clark | Information and Crowding Externalities[END_REF] in order to nd a relationship between x, a demand quantity, and, p, the equilibrium price which should not depend on y i : Since V ar [jy i ; S; s i ] = 2 t k3 k , we obtain the following relationship:

0 = 2 4 k 0 k + k 2 k (nS 0 s i ) (n 0 1) + k 3 k s i + k 1 k 0 @ p 0 C 0 0 C 3 S + C 4 (Q0wi) (n01) 0 Ex (n01) C 1 1 A 3 5 0p 0 E (n 0 1) + r 2 t k 3 k x 0 r 2 t k 3 k (w i + F i ) (40)
Isolating p, we obtain the following inverse demand function:

p 1 0 k 1 C 1 k = k 0 k + k 1 C 1 k C 4 Q (n 0 1) 0 k 1 C 0 kC 1 + k 3 k 0 k 2 (n 0 1)k s i 0 r 2 t k 3 k F i + nk 2 (n 0 1)k 0 k 1 k C 3 C 1 S 0 k 1 C 1 k C 4 (n 0 1) + r 2 t k 3 k w i (41) 0 E (n 0 1) + r 2 t k 3 k + k 1 C 1 k E (n 0 1) x (42)
Since by assumption:

P (x i ; S; F i ; s i ) = C 0 + C 1 s i 0 C 2 F i + C 3 S + C 4 w i 0 Ex i
matching the arguments of the two above equations, we get

C 0 = 2 4 0 k0 k 1 + k1 C 1 k C 4 Q (n01) 0 k1C 0 kC 1 1 0 k1 C 1 k 3 5 C 1 = 2 4 k3 k 0 k2 (n01)k h 1 0 k1 C 1 k i 3 5 = k 3 k 0 k 2 (n 0 1)k + k 1 k C 2 = 0 r 2 t k3 k 1 h 1 0 k1 C 1 k i C 3 = h nk2 (n01)k 0 k1 k C 3 C 1 i h 1 + k1 C 1 k i C 4 = h k1 C 1 k C 4 (n01) + r 2 t k3 k i h 1 0 k1 C 1 k i E = h E (n01) + r 2 t k3 k + k1 C 1 k E (n01) i h 1 0 k1 C 1 k i
The expression for C 1 is as in the Lemma. Subbing this into the other expressions, solving, and simplifying, we obtain the expressions in Lemma 2.

Proof of Lemma 3. From (24), the right side of this equation is positive. So, the left side must also be positive and consequently X must be strictly bigger than r 2 t " : Now in order to show the existence of a solution to (24), we only need to show that the left-and right-hand sides dened as functions of X cross at least once. To do this note that at X = r 2 t " , the left-hand side of ( 24) is 0 and is lower than the right-hand side which is strictly positive. On the other hand, when X goes to innity, the left-hand side goes to innity and is bigger than the right side which is nite.

Since both sides are continuous functions in X; there exists at least one solution to this equation.

Note also that the left-hand side is linear and strictly increasing in X while the right-hand side is strictly increasing and concave in X for all X > r 2 t " , so they must cross only once which guarantees equilibrium uniqueness. Also, we must show that the obtained X satises the second-order condition.

This condition is represented in (8), 2E (n 0 1) + rV ar(jy; s i ; S) > 0:

Substitution of E from (17) and using the fact that V ar(jy i ; S; s

i ) = 2 t k 3 =k yields r 2 t k 3 k 2 4 0 @ 2k 3 0 2k2 (n01) + 2k 1 (n 0 2)k 3 0 (n02)k2 (n01) 0 2k 1 1 A + 1 3 5 > 0 or, since r 2 t k3 k > 0; n k 3 0 k2 (n01) (n 0 2)k 3 0 (n02)k2 (n01) 0 2k 1 > 0:
Then, substitution of the values of k 1 ; k 2 and k 3 from equations ( 12), ( 13) and ( 14) and simplication gives the following condition

n 0 X 2 + 2 t 1 (n 0 2)X 2 0 n 2 t > 0:
Hence, the second-order condition is veried for all X 2 > n 2 t (n02) . So, a su¢cient condition for the equilibrium existence is that the lowest value of X satises the second order condition, i.e.,

0 r 2 t " 1 2 > n 2 t (n 0 2)
or, after simplication

r 2 2 " > n (n 0 2) 2 t :
Under this condition, the equilibrium exists and is unique. Finally, if this condition does not hold, i.e., r 2 2 " n (n02) 2 t ; then we can show that the equilibrium may not exist. In order to see this, note that the right side of (24) converges to 0 when 2 goes to innity. So, from the left side, if X exists it must converge to r 2 t " . Thus, there exists a 2 su¢ciently large so that the solution to (24) is such that X 2 < n 2 t (n02) , and, consequently, the second-order condition is not veried.

Proof of Lemma 4. The result in (i) follows immediately from (24) which does not depend on 2 .

In order to prove the results in (ii), let us rewrite (24) as follows

2 X 0 r 2 t " 3 [X 2 + 2 t ] X 2 = r " 4 t (n 0 1) n 2 1 2 ; ( 43
)
where the right hand side is decreasing in 2 : The left hand side must be consequently decreasing in 2 : So, in terms of di¤erentiation with respect to 2 , this gives

@ @ 2 2 X 0 r 2 t " 3 [X 2 + 2 t ] X 2 ! = @X @ 2 X 3 0 2 t X + 2r 4 t " X 3 0:
We can easily show that the polynomial 0 X 3 0 2 t X + 2r 3 must converge to innity.

Proof of Proposition 2. Part (i) is proven in the text. For part (ii), if we take the limit as 2 converges to 0, 1 E goes to 1. From Lemma 4, when 2 converges to 0; X goes to r 2 t " . Hence, we have:

lim 2 !1 1 E = lim X!r 2 t " 1 E = 2 (n 0 2) 0 r 2 2 t 2 " 1 0 n 3 [(r 2 2 t 2 " ) + n] 0 2 t + n 2 1 (n 0 1)r 2 t 2 0 r 2 " (n 0 1) [(r 2 2 t 2 " ) + 1]

!

This establishes (ii). Using (26) in order to write 1 E and di¤erentiating with respect to 2 and X gives

@ 2 0 1 E 1 @ 2 @X = @ @X 2 (n 0 2) X 2 0 n 2 t 3 [X 2 + n 2 t ] ! @ @ 2 0 @ 2 t 2 + n (n 0 1)r 2 t 1 A < 0:
From Lemma 4 we have @X @ 2 < 0; so

@ 2 0 1 E 1 @ 2 @ 2 > 0 which shows (iii).
Proof of Proposition 3. We can easily see from ( 27) that e is increasing in 2 : When 2 goes to 0, e goes to 0. Conversely, if 2 goes to 1, then 1 e and e go to 1. This establishes (i).

Note that e is decreasing in

X [X 2 + 2 t ] ; and that X [X 2 + 2 t ] is decreasing in X since we have X 2 > 2 t .
It follows that e is increasing in X. Since X decreases in 2 , market e¢ciency, e, must decrease in . When 2 goes to 0, X goes to innity. Hence, we have:

lim 2 !0 e = lim X!1 e = n 2 t 2 + n = 2 n 2 t + 2 n
Also when 2 goes to 1, X goes to r " 2 t , hence the result presented in the Proposition.

Proof of Proposition 4. Since all variables are independently and normally distributed. We have:

E(x 2 i ) = [(n 0 2) X 2 0 n 2 t ] (n 0 1)[X 2 + 2 t ] 2 2 " X 2 E " t i 0 P j t j n 2 # + E " P j F j n 0 F i 2 #! = [(n 0 2) X 2 0 n 2 t ] (n 0 1)[X 2 + 2 t ] 2 2 " X 2 (n 0 1) 2 t n + (n 0 1) 2 " n = [(n 0 2) X 2 0 n 2 t ] (n 0 1)[X 2 + 2 t ] [(n 0 2) X 2 0 n 2 t ] nX 2 2
" which is increasing in X 2 , hence from Lemma (4) is decreasing in 2 :

Analysis of ex ante expected welfare. We need to calculate

EW = E (;si;yi;S;Fi) [0 exp [0r(W + (w i + x + F i ) 0 px]] = 0e 0rW E (si;yi;S;Fi) [E [exp [0r( (w i + x + F i ) 0 px] js i ; y i ; S; F i ]] = 0e 0rW E (si;yi;S;Fi) [0 exp [0rV (s i ; y i ; S; F i )]]
where V (s i ; y i ; S; F i ) is dened in (29).

Substitution of the expression for x i in (29) as well as (E(js i ; y i ; S; F i ); var(js i ; y i ; S; F i ); and E, and assuming a symmetric initial portfolio position, i.e., w i = Q=n for all i, gives

V = k 0 k + k 1 k y i + k 2 k nS (n 0 1) + k 3 k 0 k 2 k(n 0 1) s i Q n + F i 0 r 2 t k 3 2k ( Q n + F i ) 2 + r 2 t k 3 2nk k 3 0 k 2 (n 0 1) (n 0 2) k 3 0 (n 0 2) k 2 (n 0 1) 0 2k 1 2 4 0 @ (s i 0 y i ) r 2 t k 3 0 F i k 3 0 k2 (n01) 1 A 3 5 2 :
In the following, we rewrite V so as to bring out a quadratic expression in the vector of variables Z = (s i ; y i ; S; F i ): In order to simplify notation, let

A = k 3 0 k2 (n01) (n 0 2) k 3 0 (n02)k2 (n01) 0 2k 1 2nk (r 2 t k 3 ) B = 0r 2 t k 3 k 0 @ k 3 0 k2 (n01) + k 1 n k 3 0 k2 (n01) 1 A C = (n 0 2) nk k 3 0 k 2 (n 0 1) + k 1 D = nk 2
(n 0 1)k :

So we can write V as follows

V = k 0 k Q n 0 r 2 t k 3 2k Q n 2 + k 3 k 0 k 2 k(n 0 1) Q n s i + k 1 k Q n y i + k 2 k QS (n 0 1) + k 0 k 0 r 2 t k 3 k Q n F i +A 0 s 2 i + y 2 i 1 + BF 2 i 0 2As i y i + Cy i F i + DSF i + 2C n 0 2 s i F i :
Since we have E(Z) = (; ; ; 0); we can write

V = k 0 k Q n 0 r 2 t k 3 2k Q n 2 + k 3 k 0 k 2 k(n 0 1) Q n s i + k 1 k Q n y i + k 2 k QS (n 0 1) + 0 r 2 t k 3 k Q n F i +A 0 s i 0 1 2 + 0 y i 0 1 2 + BF 2 i 0 2A 0 s i 0 1 0 y i 0 1 + C 0 y i 0 1 F i + D 0 S 0 1 F i + 2C n 0 2 0 s i 0 1 F i
which can be written as follows

V = k 0 k Q n 0 r 2 t k 3 2k Q n 2 + ht; Zi + (Z 0 E(Z)) 0 1 (Z 0 E(Z)) (44)
with the vector t is such that

t 0 = k 3 k 0 k 2 k(n 0 1) Q n ; k 1 k Q n ; k 2 k Q (n 0 1) ; 0 r 2 t k 3 k Q n ( 45
)
h1; 1i is the dot product and 1 is a 4 2 4 matrix dened as follows Thus we have

1 = 0 B B @ A 0A 0 C (n02) 0A A 0 C 2 0 0 0 D 2 C (n02)
EW = 0e 0r W +( k 0 k ) Q n 0 r 2 t k 3 2k ( Q n ) 2 j3j 1=2 jAj 1=2 exp 0rE(Z) 0 t + r 2 2 t03t = 0 j3j 1=2 jAj 1=2 exp ( 0r W + k 0 k Q n 0 r 2 t k 3 2k Q n 2 + E(Z) 0 t 0 r 2 t03t
!)

:

Yet we have

E(Z) 0 t = Q k 1 + k 2 + k 3 nk so that EW = 0 j3j 1=2 jAj 1=2 exp ( 0r W + Q n 0 r 2 t k 3 2k Q n 2 ! 0 r 2 t03t !)
This gives the expression of expected welfare that will be used in the numerical analysis.

  markets, and the microstructure literature studying the impact of transparency on market quality generally uses liquidity as a measure of market quality (see for instancePagano and Röell 1996, Madhavan 1995 and 1996[START_REF] Madhavan | Should Securities Markets be Transparent?[END_REF]).

2

 2

3 Z0 1 2 ( 3 Z 3 :

 3233 value of V in the expression of Expected welfare gives EW = 0e 0rW E (si;yi;S;Fi) (si;yi;S;Fi)exp 0r 0 ht; Zi + (Z 0 E(Z)) 0 1 (Z 0 E(Z)) 1 Z 0 E(Z)) 0 A 01 (Z 0 E(Z)) dZ (si;yi;S;Fi) exp 0rht; Zi 0 1 2 (Z 0 E(Z)) 0 0 A 01 + 2r1 1 (Z 0 E(Z))dZ29where A is the variance covariance matrix of Z and it is equal to E(Z))0 3 01 (Z 0 E(Z)) dZIf the matrix 3 is positive denite then the integral part in this equation is written as the moment generating function of a multi-normal distribution with mean E(Z) and variance-covariance matrix It is such that Z

1 EFigure 1 :

 11 Figures

Figure 2 :Figure 3 :

 23 Figure 2: Expected welfare as function of and

  When e is zero (one) p is completely uninformative (perfectly informative) about the nal value . Note that e varies with2 only indirectly through its dependence on V ar(jy i ; s i ; S):

		;	(27)
	1)X
	which lies in [0; 1]. 8 Hence, we can use (27) to compute the impact of 2	and 2	on e.
	n 2
	2 t +n 2

Proposition 3 (i) Price e¢ciency is increasing in

2

. e converges to 0 as 2 goes to 0, and to 1 as 2 goes to 1.

(ii) Price e¢ciency is decreasing in 2 . As 2 goes to 0, X goes to innity and e converges to

  n01) + rV ar [jy i ; S; s i ] Substituting for the equilibrium values of C 0 , C 1 , C 2 , C 3 , C 4 , V ar [jy

	(n01) + rV ar [jy i ; S; s i ] 2E	2 n j Fj	0 F i	:

i ; S; s i ], and E and using (23), we obtain:

[START_REF] Svensson | Social value of public information: Morris and Shin (2002) is actually pro transparency not con[END_REF] shows that this result holds only for special cases and that when the parameters of the[START_REF] Morris | Social value of public information[END_REF] model take reasonable values the condition for public information to be welfare decreasing is violated. Other papers have challenged Morris and Shins result by showing that the increase in public information announcements would increase the expected social welfare if sent only to a proportion of agents(Cornand and Heinemann,

2008), in economies with investment complementarity, or in economies featuring monopolistic competition among heterogeneously informed rms[START_REF] Angeletos | Transparency of information and coordination in Economies with Investment complementaries[END_REF][START_REF] Angeletos | Transparency of information and coordination in Economies with Investment complementaries[END_REF][START_REF]E¢cient Use of Information and Social Value of Information[END_REF].

Note that we do not focus on a trade-o¤ between these two types of signals, but rather on their respective impacts on market performance. See[START_REF] Morris | Social value of public information[END_REF],[START_REF] Clark | Information and Crowding Externalities[END_REF] for models in which agents weigh the relative benets of the two types of information.

Modeling transparency in this way allows us to study the marginal impact of decreasing uncertainty. In the marketmicrostructure literature transparency is always dened as the information that the market or some participants should

See Brown and Zhang (1997) for a discussion of the properties of this measure of price e¢ciency.

!

Note that the extent to which expected welfare decreases and the point at which expected welfare begins to increase as information becomes more precise depend on 9 , 3, , and H.

0 See[START_REF] Hoel | Introduction to Mathematical Statistics[END_REF].

prior information increases market depth and decreases market e¢ciency since adverse selection e¤ects are less relevant. Enhancing market transparency will both increase the market depth and market e¢ciency. The e¤ect of increasing transparency is less important for market depth than the e¤ect of increasing the precision of prior information since in the limit, while very precise prior information results in an innitely deep market, increasing market transparency results in a level of market depth that is bounded from above. The volume of trade is independent of the quality of prior information and is increasing in the quality of the public signal. Finally, we show that starting with very imprecise private signals and/or a very opaque market, improving the precision of all private signals, and/or enhancing the transparency of the market will cause expected welfare to fall up to a point, after which expected welfare will increase.

For the conditional variance, 2 ; we have

(n 0 1)

with

In order to derive the conditional expectation of Z 4 ; we need to derive the inverse matrix of 6 (04) :

We present 6 01 (04) as follows:

2 ( 1 ; 2 ; 3 )

where

From (34) and since

for l = 1; 2; 3: For 1 ; substitution of the values of 1 i (for i = 1; 2; 3) in (37) and simplications give

:


Preface

This book is the result of years of collaboration between different teams interacting in the framework of a European Research and Training Network on"Modeling, Mathematical Methods and Computer Simulation of Tumour Growth and Therapy". Throughout this Network, the collaboration between groups has brought new insight and motivated research leading to mechanical, mathematical, physical and biological approaches all related to the simulation of the behavior of cells and, in particular, of tumor cells.

While looking at the phenomena from different points of view, it became soon clear that multiscale problems are ubiquitous and fundamental in cell mechanics as illustrated in figure 1. For instance, such problems naturally emerge from the description of situations involving cell adhesion, migration, invasion, intravasation and extravasation, mitosis, cell-cell interactions, etc. Modeling the role of mechano-transduction in the behavior of the cell, its fate (duplication, survival, or death) or even its differentiation, or how the internal dynamics of actin and microtubules determine the migratory characteristics of the cells, leads also to a combination of intrinsically coupled scales. These aspects are, of course, not only of major interests to describe the growth of tumors, but also in embryogenesis, in the description of physiological and pathological states of all tissues, and nowadays in tissue engineering, in tissue remediation and regeneration. It is our motivation here to present non exhaustive tools which will help graduate students, researchers, and professors in their research and teaching activity.

The book is a pluridisciplinary collective effort which contains reviews or articles interesting for students and researchers from different disciplines such as the ones presented above. i ii Some of the collected works were presented in a European Workshop on "Multiscale approaches in cell mechanics" (January 7-10, 2008) which took place in Autrans (France) and was organized by Arnaud Chauvière (TU Dresden -Germany) and Claude Verdier (CNRS, Grenoble -France). Several other works from experts in the field were also included in order to get a more general overview of the subject. At the moment, it appears that solving such multiscale problems is still a difficult task to achieve. Throughout this textbook, various approaches and methods are presented in order to highlight how phenomena happening at various scales can be modeled and coupled to account for multiscale aspects. It is also our purpose here to show the existing tools available, and iii by default to bring the need to carry on new approaches. In the spirit of the Workshop mentioned above, efforts have been made to propose, in some chapters, both experimental and theoretical aspects of the same topic. In some other ones, experiments and theories are discussed separately in order to provide a deeper insight of, either biological aspects in the first case, or complex modeling in the second one.

The book is divided into four parts. In the first part, some subcellular and cellular aspects, and their links, are presented. Microrheology (Chapter 1) is shown to be a powerful tool for the investigation of cell mechanical properties. Links between length scales are properly identified in a unified concept. Chapter 2 presents multiphysics and multiscale approaches to investigate intra-cellular mechanisms involved in cell motility. Minimal ingredients are used to explain how the actin polymerization can take place and trigger forces to generate motion. This approach is very promising for further investigations and understanding of cell motion.

The last chapter of the first part discusses how subcellular effects involving certain genes can play a role to induce cell motility, in the particular context of cancer. The multiscale approach is explained within the concept of sophisticated experiments.

In the second part, cell migration is considered. The basic ingredients which have been proposed earlier1 are revisited through three different models respectively based on physical (chapter 4), mathematical (chapter 5) and computational (chapter 6) approaches. All of them introduce and develop tools to account for the description of the complex interplay between cell adhesion molecules and the dynamic evolution of the cell cytoskeleton. They underline the ubiquity of multiscale aspects when one aims to understand the complex interaction between molecules at the nanometer scale, and their combination to create motion at the cellular scale (i.e. micron scale).

The third part is devoted to the study of cells interaction with the environment, in particular the role of external mechanical forces and their effects on cell behaviour. In the first chapter (chapter 7), a model approach is considered to see how a flow field (involving shear stresses mainly) can affect the dynamic behavior of a cell (a bead here) in close contact to a wall. Such a motion is very important when considering the rolling motion of leukocytes on the vessel wall, but may also be relevant when cancer cells interact with the endothelium of blood vessels to intravasate or extravasate. The next two chapters (8 and 9) consider the role of a substrate on the adhesion of a cell and the exerted traction forces. In particular, the role of rigidity of the substrate is very important and affects mechano-transduction (chapter 8) as well as the development of stable or unstable focal adhesions. Another related aspect is the corresponding forces which are exerted by cells on a rigid substrate (chapter 9) which depend on its rigidity. Such features are important when considering many cells in their natural environment, for example when in contact with a soft tissue. Mechano-transduction is also reported in chapter 10 in an attempt to see how external mechanical forces can lead to calcium fluxes, and then to biochemical responses at the cell level.

In the last part, models of multicellular systems are presented. A substantial review of innovative models developed in the context of cancer has been presented before2 , but some recent developments are now available and can be considered complementary. This is the aim of this last part to propose such new approaches. Chapter 11 investigates the collective amoeboid motion of cells within the extracellular matrix (i.e. collagen in a real tissue). This problem is important in particular when looking at cancer cell migration3 . The next three chapters present applications of population models to developmental biology and cancer (chapter 12), embryogenesis (chapter 13) and cancer invasion (chapter 14). Finally a new approach based on Delaunay-Object-Dynamics is considered (chapter 15) to study the behaviour of highly motile cells involved in certain tissues.

Of course, there is still a large gap between the acceptance of multiscale models and their effectiveness in predicting accurate pathologies, or their import towards real medical applications. Nevertheless, we have tried here to collect, among others, some new methods to apply to cells and tissues, through a multiscale concept.

The editors wish to thank warmly all the authors who contributed to this book. Special thanks are also given to all participants of the Marie Curie RTN Network (2004-2008) for bringing into play such challenging scientific efforts. We are also indebted to N. Sunil at CRC Press for his helpful advices.

Figure 1 :

 1 Figure 1: Multiscale view of cell mechanics. A biological tissue is made of several components including cells and the extracellular matrix. A zoom allows to identify a single cell embedded in the extracellular matrix and surrounded by neighboring cells. Details illustrate the configuration at the cellular scale. At the microscopic scale, various elements interact to create cell motion (actin filaments, adhesion molecules within focal adhesions). At the nanoscale, cell sub-elements combine to create actin filaments generating forces through the acto-myosin complex.

D. A. Lauffenburger, and A. F. Horwitz (1996) Cell migration: a physically integrated molecular process. Cell 84:359-369.

Cancer Modelling and Simulation, Chapman & Hall/CRC Mathematical and Medicine Series, Ed Luigi Preziosi, 2003.

P. Friedl and Wolf K. (2003) Tumour-cell invasion and migration: diversity and escape mechanisms, Nature, 3:362-374


Khadoudja Ghanem
email: gkhadoudja@yahoo.fralice.caplier@gipsa-lab.inpg.fr

Alice Caplier

An Efficient Method to Classify Positive and Negative Expressions

Keywords: Facial expression, classification, negative expression, positive expression, transient features, permanent features

In this paper, we present a new method to classify expressions into positive, negative or neutral classes.. The pre processing step is based on transient features analysis. By detecting the presence or not of such transient features, into some facial regions, the considered expression is classified as negative or positive. Otherwise, the method is performed, and permanent features are then studied to classify the expression into neutral, positive or negative classes. Expression classification into negative or positive classes is done in only few nanoseconds when the pre processing step is performed , this leads to be used in realtime applications. Obtained results are given to prove the efficiency of the proposed method.

Introduction

Recognizing human facial expression by computer is an interesting and challenging problem.

From the time where researches in this field started, the most common approaches were interested in classifying a given expression as one of the six universal expressions postulated by Ekman namely: Joy, Disgust, Anger, Sadness, Surprise and Fear. Much progress has been made in the last decade, and a thorough survey of the exiting work can be found in [START_REF] Pantic | Automatic analysis of facial expressions: the state of art[END_REF][START_REF] Fasel | Automatic facial expression analysis: a survey[END_REF].

[……..].However in the 90s, Ekman expanded his list of basic emotions, including a range of positive and negative emotions not all of which are encoded in facial muscles [START_REF] Ekman | Basic Emotions[END_REF]. The newly included emotions are Amusement, Contempt, Contentment, Embarrassment, Excitement, Guilt, Pride in achievement, Relief, Satisfaction, Sensory pleasure, Shame. As it is difficult to distinguish between all these emotions, we propose in this paper, to classify expressed emotions in Positive, Negative or Neutral classes just to know whether the person is in a good mood, or in a bad mood. Many different approaches reported in the psychological literature have led to the proposal of dimensions underlying emotional concepts [START_REF] Schroder | Speech and emotion research: An Overview of Research Frameworks and a Dimensional Approach to Emotional Speech Synthesis[END_REF]. Different researches propose two essential dimensions: activation (from active/aroused to passive/relaxed) and evaluation (from negative/bad to positive/ good), sometimes complemented by a third dimension: power (from powerful/dominant to weak/submissive). One concrete proposal for an emotion dimension model is the activation-evaluation space, proposed by Cowie et al. [START_REF] Cowie | FEELTRACE': An Instrument for Recording Perceived Emotion in Real Time[END_REF].

Fig. 1 The two-dimensional disk-shaped activation-evaluation space proposed by Cowie et al.

In [START_REF] Peters | Positive-Negative Asymmetry in the Identification of Faces in the Basis of Descriptions of Happy and Sad Facial Expressions[END_REF] the positive-negative asymmetry hypothesis is reported, in [START_REF] Eastwood | Negative Facial Expression Captures Attention and Disrupts Performance[END_REF] John counts features of negative and positive expressions to deduce that negative faces are particularly effective at capturing attention to the global face level. Husvar [START_REF] Husvar | The Effects of Gender and Facial Expressions in Human Interaction[END_REF] studied the effects of gender and facial expressions in human interaction and Miura [START_REF] Miura | Discrimination of Facial Expressions using Velocity Estimation from Image Sequence[END_REF] used velocity from image sequences and optical flow to know if an expression is positive, negative or neutral.

In this paper we present a new method to classify facial expressions in three classes namely positive, negative or neutral using two types of features: transient features or wrinkles analyzed in the pre processing step and permanent features deformations in the main core of the process. In section 2, the proposed approach is presented. Obtained results are given in section 3. We also evaluate across-dataset generalization using another database in Section 4. Finally, Section 5 concludes the paper.

Overview of the proposed Method:

In a first step, only transient features are considered. The detection of transient features on three facial regions : nasal root, mouth corners and chin (Fig. 3), is done. If wrinkles appear in one of these regions, our expert system yields to the conclusion that the considered expression is negative. The nasolabial region is also considered. In the case of the presence of transient features on this region, we compute the angle formed by the nasolabial furrows in order to classify the expression as negative or positive.

If no transient features are detected on the corresponding regions of interest, and given that the absence of transient features on the different regions of the face does not necessarily mean that we have a neutral face, a second step is then performed. It is based on permanent features deformations (eyes, mouth, brows). Geometric distances are measured to know if the face is or not in the neutral state. If it is not the case, some of the measured distances are used to classify the face as positive or negative.

The Pre Processing Step:

As it is not the aim of our work, facial features detection is not automatically done. 18 characteristic points of permanent features are manually extracted (Fig: 2). These characteristic points represent mouth corners, bottom and top of leaves, eye corners, bottom and top of eyelids , the centers of irises and finally eye brows corners. Selected points allow calculating the biometric distances as well as automatically determining regions of interests where transient features can appear (Fig: 3). The selection step is then followed by a segmentation of the two facial images, to detect the presence or absence of transient features on the determined regions of the face (Fig: 3). Different kinds of transient features occur with facial expressions. Sometimes on one region and in other times on several regions. Here six regions are considered (see Fig. 3): the nasal root, the two nasolabial regions, the two regions surrounding the mouth, and finally the chin. The Canny edge detector [START_REF] Li Tian | Recognizing Action Units for Facial Expression Analysis[END_REF] associated to a thresholding stage [START_REF] Otsu | A Threshold Selection Method from Gray Level Histograms[END_REF] is used to detect the presence or absence of transient features. We compare the number of edge pixels in a wrinkle area in case of an expressive face with the number of edge pixels in the same area in case of a neutral face (Fig. 3). If the ratio is larger than a threshold, the furrows are determined to be present. Otherwise they are absent. 1: Presence or absence of transient features on each region for each facial expression.

(0) (0) (0) (1 OR 0) (1 OR 0) (1 OR 0) Mouth corners (0) (0) (0) (0) (1 OR 0) (1 OR 0) Nasolabial (1 OR 0) (0) (1 OR 0) (1 OR 0) (1 OR 0) (1 OR 0) Eyes corners (1 OR 0) (0) (1 OR 0) (1 OR 0) (0) (0) Nasal root (0) (0) (1 OR 0) (1 OR 0) (1 OR 0) (1 OR 0) Forehead (0) (1 OR 0) (0) (1 OR 0) (1 OR 0) (1 OR 0) Table
On table 1, columns represent the six universal expressions and the lines represent the different regions where wrinkles can appear. For example with surprise we can get wrinkles only in the front head and with joy we can get wrinkles in nasolabial regions and/or in eye corners regions. The study of Table 1 shows that some transient features are associated with negative expressions only. The considered wrinkles are: on the chin due to AU17, wrinkles can appear with Anger, Sadness or Fear, around mouth corners due to AU15, wrinkles can appear with Sadness or Fear and on the nasal root due to AU4, wrinkles can appear with Disgust, Anger, Sadness or Fear (Fig. 6).

Presence of mouth corner

Presence of wrinkles Presence of wrinkles on the wrinkles due to AU15 on chin due to AU17. nasal root due to AU4. Figure 6: Transient features formed with negative expressions.

To validate these results, we generalize with another publicly available database: the JAFFE database [12]. This database contains (212 images) 10 Asiatic female subjects with seven expressions namely: Joy, Anger, Sadness, Fear, Disgust, Surprise and Neutral, each expression with different intensities. Results are presented in Table 2 Using results on table 2, we can conclude that all images from the studied database with transient features on the chin, or on the mouth corners or on the nasal root correspond to the negative expressions which are Anger, Sadness, Fear and Disgust.

We computed the statistical correlation between the presence of transient features and the 6 expressions in the different used emotion databases as a similarity measure:

where denotes standard deviation and is the covariance of these two variables. This measure achieves the generalization performance of ~1 for the presence of transient features with negative expressions and performance of ~0 for the presence of these features with positive expressions.

Decision: As a generalization, we can say that if wrinkles appear on the chin due to AU17, around mouth corners due to AU15 or in the nasal root due to AU4, the corresponding expression is Negative. These findings are in perfect conformity with those of [START_REF] Michael | Dominic A Psychometric Evaluation of the Facial Action Coding System for Assessing Spontaneous Expression J[END_REF] where Sayette et al. found good to excellent reliability of Emotion-specified expressions, which are defined in terms of specific combinations of AUs, included both those thought to represent discrete emotions and more molar categories of positive and negative emotion. They found that Negative expressions are characterized by the absence of AU 12 and the presence of at least one of the following AUs: 9 (nose wrinkle); 10 (upper lip raise); 14 (unilateral dimpler); 15 (lip corner depress); 20 (lip stretch), and 1+ 4 (pulling the medial portion of the eyebrows upwards and together). These AUs are thought to appear during the expression of negative emotion (Ekman & Friesen, 1982;1986

Classification of positive and negative expressions based on the nasolabial furrows: Learning stage

Nasolabial furrows can appear with positive expressions like Joy as well as with Anger, Disgust and sometimes with Sadness or Fear. But these furrows are not due to the same action units. Joy nasolabial furrows appear by the activation of AU12. In case of Disgust, Anger, Sadness or Fear, nasolabial furrows appear by the activation of AU9 (Nose wrinkle) or AU10 (Upper lip raiser) (Fig. 7). As a result the angle between the line approximating the nasolabial furrow and the horizontal line connecting both mouth corners, formed when AU12 is activated, is lower than the angle formed when AU9 is activated (Fig. 8) [START_REF] Ekman | Facial Action Coding System : A Technique for the Measurement of Facial Movement[END_REF]. To validate this idea, we measure the angle formed by the nasolabial furrows for all images with Disgust, Anger and Joy expressions of the two databases EEbase and Dafex. Calculated angles are normalized with respect to the angle 90° (the highest possible angle reached in case of the activation of AU9 or AU10). Because of the lake of Fear and Sadness images with nasolabial furrows, we did not calculate angles for these two expressions. After that, we calculate thresholds, according to each intensity (Medium or High), for each expression because the intensity criterion allows us to recognize if the expression is positive without any doubt or if it can be a positive or negative expression. The highest thresholds are computed as the average of the highest values corresponding to each expression intensity and the lowest thresholds are the average of the lowest ones. These thresholds are represented on (Fig. 9) where abscissa represents thresholds values and ordinate represents arbitrary values used to separate the different expressions. From this figure we can see that the angle formed by the nasolabial furrows with Joy expression is lower than the angle formed with Disgust or Anger, so we deduce the rules of a is the minimum threshold corresponding to Joy expression with high intensity. d is the minimum threshold corresponding to Disgust expression with medium intensity. e is the minimum threshold corresponding to Joy expression with medium intensity. j is the maximum threshold corresponding to Disgust expression with high intensity.

Decision: In the case of transient features in the nasolabial regions, the calculated angle formed by the nasolabial furrows is compared with thresholds computed in the training step. If the calculated angle is in the interval [a,d], the considered expression is considered as positive, if the angle is in the interval [d,e], there is a doubt between positive and negative expression, and if the angle is in the interval [e,j], the expression is considered as negative.

2.2

The Proposed Method : To classify an expression in positive, negative or neutral classes without being concerned by the presence or no of transient features, automated software measurements of facial actions were supplemented with Facial Action Coding System (FACS) (Ekman & Friesen, 1978) which is widely acknowledged to be the most comprehensive and objective system available. The method is based mainly on permanent features where five distances are measured and normalized with respect to the static distance between the two irises. Computed distances are then compared with thresholds calculated in a precedent work [START_REF] Ghanem | Estimation of Facial Expression Intensity Based on the Belief Theory[END_REF] to determine if the distance increase or decrease. The aim of the use of results obtained in the previous work is that to avoid comparing the expressive face to the neutral one in case of a lake of data.

Recognition of Neutral state:

To know if the face is in the neutral state, we compute the distance between eyes and brows (D2), between lids (D1), between lips (D4), between mouth corners (D3) and between eye and mouth (D5), then compare them with the corresponding distances in the neutral face (Fig. 10). If these distances have not changed , the face is considered as neutral. Otherwise the expression is classified as positive or negative. Since all these Aus are considered as negative Aus, we study the effect of activation of each AU to facial deformations and thus on the calculated distances on the neutral state. With the activation of AU9 and AU10, the distance between mouth corners (D3) does not change, but the face is not no more in the neutral state (D5 : the distance between mouth and eye corners decreases) so the specified expression is negative. With the activation of AU15 or AU16, the distance between mouth corners does not change but the distance between mouth and eye corners increases, so the specified expression is negative too. With the activation of AU20, the distance between mouth corners increases like with joy, but the distance between mouth and eye corners increases, here the specified expression is negative. Another type of activation effect of some other Aus like AU17, AU18, AU22, AU23, AU26 and AU27 shows that the distance between mouth corners decreases and the expressed emotion is negative. Positive emotions are expressed by the activation of AU12, AU13 or AU14. In this case, the distance between mouth corners increases and the distance between mouth and eye corners decreases. In the other cases, we can not be sure that expression is negative or positive so we are in a doubt between positive and negative expressions. Sometimes upper action units express negative emotions and lower action units express positive ones, the reality is that smiles are used to mask or to control negative emotions. Table 4 presents the rules in term of distance modifications according to positive or negative expressions. The Dempster-Shafer [START_REF] Dempster | A generalization of Bayesian inference[END_REF] theory has been used to fuse the data from multiple sensing modalities in various applications. The uncertainty of the measurements is modeled with the probability mass function, which defines a value between zero and one (basic probability assignment) to indicate the degree of support for a proposition. In the framework provided by DS theory, the frame of discernment ( ) consists of 2 singletons, which are based on mutually exclusive and exhaustive propositions. In practical applications, such proposition is expressed as a discrete value or pre-defined class number to describe certain condition or status.

In our case we define the following: If an expression is positive we associate to it the state E+, if it is negative we associate it E-. Our aim is to characterize each type of expression with a specific combination of state distances. If a distance increases, we associate to a state variable Vi associated to the distance Di the state "C+", and if the distance decreases, we associate to it the state "C-" and we associate "C" if the distance do not change.

A mass of evidence mDi =1 is associated to the distance Di if this distance changes. In another hand, we formulate state distances by type expressions, we get : mD3(C+)=mD3(E+UE-) mD5(C-)=mD5(E+) mD5(C+)=mD5(E-) mD5(C)=mD5(E+UE-) To make a decision about the classification of the studied expression , and as we have in most cases more than one sensor, we use the Dempster-Shafer fusion low for two characteristic distances Di and Dj , the conjunctive combination (orthogonal sum) between these two distances is defined as :

Where A, B and C denote the propositions and B ∩ C denotes the conjunction (intersection) between the propositions B and C.

3

Results: We evaluate our proposed method on different databases which are: Hammal_Caplier database [9], EEbase database [10] Dafex database [11], JAFFE database [12] and [START_REF]Cohen and colleagues :Facial Expression Recognition from Video Sequences[END_REF] positive and negative database.

In our experiments, we evaluated the preprocessing procedure on all images with transient features appearing on facial regions of interest which correspond to negative class (Deduced

∑ = ∩ = ⊕ = A C B C m B m A m m m m ) ( ). ( ) ( 2 1 2 1 (2)
from the training step). In the case of the presence of nasolabial transient features, we cropped the nasolabial region and calculated the formed angle. This angle is compared to thresholds (computed in the training step) to classify the expression into positive or negative classes. Then we evaluated the performance of the proposed method on the remainder images without transient features, and also on images where the result presented a doupt between classes. By summing rates obtained on the pre processing step and on the processing one, we can note that classification rates of negative expressions are higher than classification rates of positive ones, it means that negative faces are particularly effective at capturing attention to the global face level as it has been proved by John in [START_REF] Eastwood | Negative Facial Expression Captures Attention and Disrupts Performance[END_REF]. However both classification rates can be considered as good. Sometimes, classification rates are given to the recognition of doubt between positive and negative expressions, this is due to the fact that Joy which is a positive expression is generally confused with Disgust which is a negative expression. This confusion has been exhibited in different analysis systems such as in [START_REF] Hammal | A Fusion Process Based on Belief Theory Classification of Facial Basic Emotions[END_REF], [START_REF] Ghanem | Estimation of Facial Expression Intensity Based on the Belief Theory[END_REF]. We can see that more than half of images of tested databases are classified in the pre processing step. Finally, we note that the Neutral state can be recognized only by analyzing permanent features, because the absence of transient features does not always mean that it is a neutral face.

Generalization to Cohn and Kanade database [24]

We also evaluated the proposed approach on the Cohn and Kanade database. The Cohn and Kanade database includes 97 university students. They ranged in age from 18 to 30 years. Sixty-five percent are female, 15 percent are African-American, and three percent are Asian or Latino. Subjects were instructed by an experimenter to perform a series of 23 facial displays that included single action units (e.g., AU 12, or lip corners pulled obliquely) and combinations of action units (e.g., AU 1+2, or inner and outer brows raised). Six of the displays are based on descriptions of prototypic basic emotions (i.e., joy, surprise, anger, fear, disgust, and sadness). Image sequences from neutral to target display were digitized into 640 by 480 or 490 pixel arrays with 8-bit precision for grayscale values. Obtained rates with Cohn and Kanade database are better than those with the other databases, this can be explained by the fact that the Cohn and Kanade database is larger than the others. Recognition rates of positive expressions are still lower than those of negative ones. The rates after the preprocessing step with the other datasets are better than those with Cohn and Kanade database. This can be explained by the fact that images of the other databases were collected in a uniformly controlled environment. With Cohn and Kanade, images were collected with variations in image conditions like different lightening. When the lightening is intense, transient features can not be reliably detected even if they exist and the preprocessing procedure can not be performed. So these images are treated by the proposed method.

Conclusion

In this work, we presented a logical process to classify expressions into negative, positive or neutral classes. The main process consists in a preprocessing step based on transient features and on a processing step based on permanent features. The pre processing step which allows classifying more than the half of images of used databases avoid following method's steps.

Our method follows the rules that we defined, however, these rules have common criterions with FACS coding system. The dempster fusion low was used to perform the fusion of all available data. One limitation of this work is that the recognition is performed by using static images without exploiting temporal behaviors of facial expressions. Another limitation of the current work is that we do not consider some variations, such as pose variations and occlusions. In our future work, all these gaps will be addressed. We hope also to refine the classification by proposing other classes like excited positive, calm positive, dangerous negative (like Anger) and neutral negative (like Sadness).

Fig. 2 :

 2 Fig. 2: Selected characteristic points

Fig. 3 :

 3 Fig.3 : (a): Face without transient features (b): Face with transient features in the nasolabial areas

Fig. 7 : 1 )Fig. 8 :

 718 Fig.7: on the left AU9 (Nose wrinkle) and AU10 (Upper lip raiser), on the right AU12 (Lip corner puller)

a b c d e f g h i jFig. 9 :

 9 Fig. 9: Thresholds for Joy, Disgust and Anger expressions

Fig. 10 :

 10 Fig.10 : Computed facial distances 2.2.2 Classification in positive or negative classes: By studing permanent features deformations when expressing an emotion, and by considering the lower face action units table deduced by Eckman et al, we can note that the half bottom part of the face is the most important part to know if the face expresses good mood (AU 12,13,14) or bad mood (AU 15,16,17..)

Figure 10 :

 10 Figure10: Lower face action Units Table[START_REF] Ekman | Facial Action Coding System : A Technique for the Measurement of Facial Movement[END_REF].The most Aus of this part of the face correspond to negative faces. According to Ekman(Ekman & Friesen, 1982; 1986; Ekman et al, 1980; Gosselin, Kirouac, & Dore, 1995; Rozin, Lowery, & Ebert, 1994; Soussignan & Schaal, 1996; Vrana,1993), the following AUs: 9 (nose wrinkle); 10 (upper lip raise); 14 (unilateral dimpler); 15 (lip corner depress); 20 (lip stretch), and 1+ 4 (pulling the medial portion of the eyebrows upwards and together) are supposed to appear during with a negative emotion. Since all these Aus are considered as negative Aus, we study the effect of activation of each AU to facial deformations and thus on the calculated distances on the neutral state. With the activation of AU9 and AU10, the distance between mouth corners (D3) does not change, but the face is not no more in the neutral state (D5 : the distance between mouth and eye corners decreases) so the specified expression is negative. With the activation of AU15 or AU16, the distance between mouth corners does not change but the distance between mouth and eye corners increases, so the specified expression is negative too. With the activation of AU20, the distance between mouth corners increases like with joy, but the distance between mouth and eye corners increases, here the specified expression is negative. Another type of activation effect of some other Aus like AU17, AU18, AU22, AU23, AU26 and AU27 shows that the distance between mouth corners decreases and the expressed emotion is negative. Positive emotions are expressed by the activation of AU12, AU13 or AU14. In this case, the distance between mouth corners increases and the distance between mouth and eye corners decreases. In the other cases, we can not be sure that expression is negative or positive so we are in a doubt between positive and negative expressions. Sometimes upper action units express negative emotions and lower action units express positive ones, the reality is that smiles are used to mask or to control negative emotions. Table4presents the rules in term of distance modifications according to positive or negative expressions.

4 :

 4 between mouth and eyes:D5 / Distance between mouth corners:D3 Increases=C+ Second case Distance between mouth and eyes:D5 Increases=C+ Distance between mouth corners:Rules for each expression classes based on the two facial distances (mouth cornersD3, mouth and eyesD5).

2.1.1 Presence or absence of transient features with different facial expressions: learning stage

	Transient features on four regions of interest which are : The nasal root, mouth corners and on
	the chin are used to determine the negativity of the expression, however transient features of
	nasolabial regions convey another type of information allowing to classify the expression in
	negative or positive classes. For this raison, we study these features separately.
	In this training stage, we investigate the relation between some specific transient features and
	negative expressions or positive expressions.
	To determine which wrinkle corresponds to which class of expressions (Negative or Positive),
	we train our program on 2520 images of Joy, Disgust, Surprise and Neutral expressions
	coming from Hammal_Caplier database [9], on 462 of Joy, Disgust, Anger, Sadness, Fear,
	Surprise and Neutral images coming from EEbase database [10] and on 2400 images of the
	same expressions coming from Dafex database [11]. Using the Canny edge detector, we
	detect all transient facial features appearing with each facial expression. Results are presented
	in

Table 1 .

 1 If transient features appear at least in one of the considered regions of interest (see

	Nasal Root Region
	Nasolabial Region

Fig 3)

for a given expression, the "1" value is associated , otherwise the "0" value is associated. This procedure ensures a validation of the presence or absence of wrinkles with each class of facial expressions across different databases.

Table 2 :

 2 : Wrinkles on chin, around mouth corners and in the nasal root according to each facial expression of the JAFFE database.

		Wrinkles on Chin	Wrinkles around	Wrinkles in the
			mouth corners	nasal root
	Joy	/	/	/
	Disgust	/	/	60%
	Anger	60%	/	30%
	Sadness	20%	20%	40%
	Fear	/	/	50%

  ; Ekman et al, 1980; Gosselin, Kirouac, & Dore, 1995; Rozin, Lowery, & Ebert, 1994; Soussignan & Schaal, 1996; Vrana, 1993). Ekman called them Negative Action Units.

Table 3 :

 3

	Considered	Positive	Positive Exp. OR	Negative
	intervals	Expressions	Negative Exp.	Expressions
	[a,d]	X
	[d,e]		X
	[e,j]			X

Table 3 :

 3 Classification of positive and negative expressions based on the nasolabial furrows.

Table 5

 5 presents obtained classification rates correspending to the pre processing procedure and to the proposed method.

			Neutral	Neg.Exp	Pos.Exp
		Neutral	/	/	/
	Pre processing	Neg.Exp.		58,29%
		Pos. Exp.			72,28%
		Neg.OR Pos.		0,89%	10,87%
	The Processing	Neutral	60,77%	5,70%	1,09%
		Neg.Exp.	39,23%	35,12%
		Pos.Exp.			10,87%
		Neg.OR Pos			4,89%
	Total Pos/Neg			93,41%	83,15%
	Total		100%	100%	100%

Table 5 :

 5 Classification rates obtained on the four databases.