R Destrez

B Albouy-Kissi

S Treuillet
email: sylvie.treuillet@univ-orleans.fr

Y Lucas
email: yves.lucas@bourges.univ-orleans.fr

A Marchadier
email: arnaud.marchadier@gmail.com

Comparison of Visual Registration Approaches of 3D Models for Orthodontics

Keywords: Visual registration, 3D models, projection matrix, orthodontics

We propose to apply vision techniques to develop a main tool for orthodontics: the virtual occlusion of two dental casts. For that purpose, we process photos of the patient mouth and match points between these photos and the dental 3D models. From a set of 2D/3D matches of the two arcades, we calculate the projection matrix, before the mandible registration under the maxillary through a rigid transformation. We perform the mandible registration minimizing the reprojection errors. Two computation methods, depending on the knowledge of camera intrinsic parameters, are compared. Tests are carried out both on virtual and real images. In the virtual case, assumed as perfect, we evaluate the robustness against noise and the increase of performance using several views. Projection matrices and registration efficiency are evaluated respectively by reprojection errors and the differences between the rigid transformation and the reference pose, recorded on the six degrees of freedom.

Introduction

Orthodontics is a dental specialty concerned with the correction of bad positions of jaws and teeth to optimize dental occlusion for functional and esthetical purposes. The first classification of malocclusions was established by E.H Angle in 1898 [START_REF] Angel | Treatment of malocclusion of the teeth and fractures of the maxillae: Angle's system[END_REF]. Today, the treatment can involve a chirurgical intervention or dental device implantation.

Commonly, the orthodontist uses dental casts of patient's arcades to plan the treatment and manually achieve an attempt of occlusion. As in many other medical applications, this manual practice can be replaced by new planning and diagnostic imaging tools. Dental casts can be completed by dentition photos [START_REF] Ettorre | Standards for digital photography in cranio-maxillo-facial surgery -Part I: Basic views and guidelines[END_REF] or radiographies [START_REF] Broadbent | Holly: A new X-ray technique and its application to orthodontia[END_REF]. Other imaging tools can be used like CT-scan [START_REF] Chung | Analyse comparative des valeurs céphalométriques de céphalogrammes latéraux générés par CBCT versus céphalogrammes latéraux conventionnels[END_REF] [START_REF] Plooij | Digital three-dimensional image fusion processes for planning and evaluating orthodontics and orthognathic surgery. A systematic review[END_REF] or face scan devices [START_REF] Terajima | A 3-dimensional method for analyzing facial soft-tissue morphology of patients with jaw deformities[END_REF][7] [START_REF] Schendel | 3D Orthognathic Surgery Simulation Using Image Fusion[END_REF][9] that provide additional information about soft tissues and face appearance, but still requiring dental casting. Anyway these techniques are not widespread because there are expensive and bulky.

A real contribution of imaging is the possibility to create 3D numerical models of dental casts from surface laser scans [START_REF] Asquith | Three-dimensional imaging of orthodontic models: a pilot study[END_REF] or 100µm resolution CT-scan [16]. These techniques differ on speed and cost but none enables an adjustment of the dental arcades corresponding to the real occlusion of patient. This information can be obtained with a silicone device used during the teeth printing [START_REF] Alfano | Using the neutral zone to obtain maxillomandibular relationship records for complete denture patients[END_REF] or another reference [START_REF] Sohmura | High-resolution 3-D shape integration of dentition and face measured by new laser scanner[END_REF] but it requires additional operation for the practitioner. CT-scan or heavy devices [START_REF] Enciso | The virtual craniofacial patient: 3D jaw modeling and animation[END_REF] can be used to obtain the pose of the mandible relative to the maxillary but these techniques are not widespread. In practice, after the scan of dental casts, the search for the optimal occlusion is computerized [16]. A technician moves manually the mandible to modify the occlusion of the dental arcades 3D models. For this task, he relies on dedicated software and photos of mouth patient (Fig. 2a above). This process is very tedious and the result is operator dependent. So, our goal is to perform automatically most of the numerical occlusion setting. Thereby, it will be faster and independent of the operator. We chose to use real photos of the dentition to capture the natural occlusion of the arcades and to map it on the virtual model. This strategy avoids adding another step to the current orthodontist practice. In this paper, we carry out a comparative study of two methods to evaluate the relevance and precision of this approach. These studies are carried out for different configurations of viewpoints and 2D/3D correspondences.

Model Registration from Mouth Images

The 3D models of the two dental arcades are obtained separately, preventing any expression in a shared work coordinate system with the original fitting of the arcades. Consequently, it is necessary to determine a rigid transformation setting in occlusion the mandible against the maxillary. We propose to use colors photos of mouth patient that display the real "in vivo" occlusion for several viewpoints (Fig. 2a). We choose a set of correspondences between 3D points on the surface model and their 2D projections in images to estimate this rigid transformation. The projection matrix obtained from homogenous coordinates can be expressed by the relation between 3D point, M j, and its projection in the image i, m ij :

∝ . ( 1
)
where ∝ means « by a scaling factor ». The 3 4 projection matrix P i can be decomposed into a product of two matrices.

These contain intrinsic parameters of the camera and extrinsic parameters:

| . ( 2
)
where t i and R i are respectively the translation and the rotation defining the pose of the camera and K is the camera calibration matrix where corresponds to the pixel slope, and are the frame principal point coordinates, and horizontal and vertical sensor resolutions and is the focal camera distance:

0 0 0 1 . ( 3
)
For current cameras, pixels are commonly considered as squares and the principal point at the center of pictures, leading to the relations , , ,

with little impact on the precision. Moreover, it allows the calculation of the matrix K from the picture size and the focal distance given in pixels. From a set of 2D/3D correspondences for the two arcades, we propose a two step resolution of the problem: firstly we estimate the projection matrix for each view from maxillary correspondences, before the registration of the mandible with analytical (DeMenthon) [START_REF] Dementhon | Model-based object pose in 25 lines of code[END_REF] or minimization [START_REF] Marquardt | An algorithm for least-squares estimation of nonlinear parameters[END_REF] methods. Two situations can be encountered, depending on the knowledge of the focal distance. If the camera is calibrated, we must only determinate its pose (position and orientation), embedding six degrees of freedom (dof in the following). Conversely, if we don't know the focal length, we calculate the complete projection matrix with 11 dof. In addition, the rigid registration of the mandible provides 6 dof of more. Several views can be used jointly during the registration.

Estimation of P i With K Knowledge (WiK)

In this method called « WiK », the determination of the projection matrix of the image i amounts to the extrinsic parameters estimation: three Euler's angles defining the orientation R i and three translation components of the t i vector. Firstly, we use the POSIT algorithm of DeMenthon [START_REF] Dementhon | Model-based object pose in 25 lines of code[END_REF] for a rough estimation of the six parameters, from correspondences established on the maxillary. Then, from this initialization, the six parameters are optimized by the Levenberg-Marquardt algorithm minimizing the sum of the squared reprojection error:

∑ ∑ ² . ( 4
)
where i, j, and k are respectively picture, point and coordinate indexes, m ijk refers to a reference point coordinate in the picture and refers to a the reprojection of the corresponding 3D point by the P i matrix.

For evaluation in experiments presented in section 3, we use the reprojection mean squared error (MSE):

∑ ∑ ² . ( 5
)

Etimation of P i Without K Knowledge (WoK)

When K is unknown, the projection matrix P i is estimated by least square minimization on the maxillary 2D/3D correspondences. The last element of P i is fixed to 1. Then, the 11 parameters of the P i matrix are optimized with the Levenberg-Marquardt algorithm by minimizing se i (equation 4). This approach is called « WoK ».

Registration of the Mandible

One projection matrix estimated, the mandible is registrated under the maxillary in occlusion position. The reference coordinate system of the maxillary is illustrated figure 1: the origin is the gravity center of the point cloud of the maxillary, the Y axis is facing up and the Z axis is parallel to the rows of molars. The reference coordinate system of the mandible reference is similar but placed on the gravity center of the mandible point cloud. The registration is carried out by calculating the rigid transformation T according to the fact that 3D point projections of the mandible are superimposed with the projections selected in the pictures. Projections are obtained using the matrix P i estimated from the maxillary points where a 3D point M mand on the mandible and its projection m i,mand on the photo is linked by:

, ∝ . (6)
The transformation T embeds 6 dof: three Euler's angles and three translations estimated by minimizing the reprojection error (equation 4) using Levenberg-Marquardt algorithm with an identity transform as initial conditions.

Experiments

To validate the two previous steps, tests have been carried out both on virtual and real images. From a "perfect" virtual case, we evaluate the robustness against noise and the increase of performance using several views. The virtual pictures are obtained by screenshots from the VTK rendering of 3D models (Fig. 2a). So, several points of view can be simulated controlling the acquisition settings (focal, view angle, etc…). Such virtual pictures are distortion free but we can raise and control the noise level on the point coordinates to simulate real watching process. Real pictures are taken with a numerical 8 mega pixels camera (Canon EOS 350D), a 60 mm lens and an annular flash (Fig. 2a). We use OpenCV for the calculation functions like POSIT [START_REF] Dementhon | Model-based object pose in 25 lines of code[END_REF] and the levmar functions [START_REF] Lourakis | levmar: Levenberg-Marquardt nonlinear least squares algorithms in {C}/{C}++[END_REF] for the implementation of the Levenberg-Marquardt algorithm. (b) Initial location of the 3D models before the registration.

Projection Matrix Estimation

To evaluate the estimation of projection matrix, the tests are carried out on 3D virtual models registrated in a good occlusion position decided by orthodontics specialist.

Ten features are manually pointed on the 3D models of the maxillary, and ten others on the mandible one. The two previous methods are compared on virtual data and real pictures. Estimated projection matrices are evaluated with the reprojection errors (equation 5). We have done this test for two different points of view: a front view and a side view. This second point of view is one of the most lateral views that we can obtain. The figure 2a represents a typical configuration of this point of view.

Virtual Data

While perfect projections are simulated by using the VTK screen, 2D/3D correspondences can be considered as perfect (Fig. 3). From this "perfect" case, we add an uniform noise on the 2D point coordinates to compare the robustness of WiK and WoK methods against noise. The tables 1 and 2 present the average ± the standard deviation, and the maximal values of the reprojection error (equation 5) respectively for maxillary and mandible points, after the optimization step. For each noise level, we average a hundred random realizations. To evaluate the error introduced by a human operator, we also test a manual 2D point picking.

Regarding the mandible reprojection errors, we can observe that the WiK method surpasses globally the WoK method for the estimation of the projection matrix. It remains that the non-perfect estimation of K used in the WiK method could explain the opposite observation for the maxillary errors. As only 2D/3D maxillary correspondences are used to estimate the projection matrix, the maxillary errors are lower than mandible errors. The errors increase linearly with the noise level (from zero in the perfect case). It should be noticed that these errors remain small compared to the size of pictures expressed in pixels. Concerning the manual selection, mandible MSE for the side view reaches 21.6 pixels (0.93 mm) for the WoK method while it is 10.3 pixels for the WiK method. The manual picking errors are equivalent to a noise around 6 pixels. This demonstrates that capturing manually the 2D reprojection with a high precision is really a tricky task.

Real Case

The estimation of the projection matrix has been tested on two real images (Fig. 2a) with a manual point selection. Five tests have been carried out to display a tendency.

In table 3, we can observe, that mandible reprojection errors are still more important than maxillary ones. In this case, 1 mm corresponds to 11,5 pixels. We note again the superiority of the WiK method. Especially in the manual case, we can see that reprojection errors are higher than on virtual images for the WoK method. This can be explained by a higher complexity for the manual picking in the real case, while the color picture had not the same appearance (colors, texture, different lightings…) than the 3D model (shadows, relief…). In the virtual case, the simulated images are really similar to the 3D model so that ease the selection of correspondences.

Mandible Registration

To test the registration both for virtual and real images, the 3D models of arcades are positioned in the same initial configuration, corresponding to a practical configuration. The registration has been done separately with each viewpoint and then using both images together. From the reference pose, the rigid transformation between mandible and maxillary is defined by the Euler's angles ,  and  and by the translation components X = 5 mm, Y = 24 mm and Z = 11 mm. This configuration is presented in the figure 2b. The quality of the final registration is evaluated by the differences on the six parameters of the rigid transformation. The reference pose is the one obtained by the orthodontist expert with the current technique. We present only the best results, obtained for a registration involving two viewpoints together. It includes also the reprojection error occurring after the registration step because this is the actual criteria minimized by the LM algorithm during the registration. As previously, a uniform noise is added on the perfect data to evaluate algorithm robustness and the errors are averaged on 100 random tests. Moreover, we carry out a set of manual selection tests. Currently, in orthodontic practice, there are no quantitative criteria to estimate the quality of a registration. We considered that a position error of 0.1 mm and an orientation error of 0.5 mm are relevant bounds compared to teeth dimensions.

Virtual Data

Concerning the reprojection errors observed after the registration, they are lower than in table 3. In the manual selection case with the WiK method using only one picture, reprojection errors are 3.24 pixels for the front view and 3.80 for the side view. In the same configuration, for the WoK method, errors are 4.50 pixels for the front view and 4.65 for the side view. We can explain that by the fact that the minimized criterion is the mandible reprojection error. Moreover, the errors are slightly lower if the registration is done with one view rather than with two (in the same configuration for a registration using two points of view, reprojection error is 4.27 pixels for the front view). In this latter case, optimization is a compromise between the two sets of points. This difference is more important for the WoK method.

The figures 4, 5 and table 4 present the 3D differences after the registration relative to the reference positioning provided by the orthodontist expert. Globally, the errors with the WiK method are less important than with the WoK one with a linear dependence to noise. In the "perfect" case, the registration using two viewpoints provide very good results. In the graphs, horizontal straight lines correspond to the manual selection case (MS). Firstly, we have examined the positioning of the gravity center (Fig. 4 and table 4).The Z translation is the less precise because this direction corresponds to the depth in the pictures. The Y translation is the more precise because this axis is perpendicular to the camera axis. For the WiK method, this tendency is not respected, but all errors are low. For the manual selection the maximal error is 0.17 mm. For the WoK method, this tendency is respected but errors can be more important with a maximum of 0.38 mm for the manual selection. Secondly, we focused on the angle errors (Fig. 5 and table 4). The and  angles seem to be the most difficult to evaluate with respectively 0.48° and 0.33° errors for the manual selection with the WiK method. Whatever the method, the angle is the easiest to estimate. Another time, WoK method has higher errors than WiK method. Table 4. 3D errors on the virtual data for the WoK method (in pixels for the gravity center errors and in degrees for the angle errors).

Real Case

For the 3D registration errors (table 5),WiK method produces again globally better results than WoK method. The errors are not very high, but they cannot always afford the precision required for a real dental occlusion. The tests carried out with a unique view reveal sometimes some aberrations. In that case, the registration is not realistic and some 3D errors can be high (until Z = 73 mm for the WoK method). By combining the two viewpoints, the registration is clearly improved and generally the aberrations almost disappear. Nevertheless, 3D errors are higher than with the virtual case. In average, the translation error is maintained below 0.8 mm and the rotation error below 1.2 degrees for the WiK method (respectively 2 mm and 4° for the WoK method).

As in the virtual case, we observe an increase of the reprojection error by using the both picture. Nevertheless, these errors are higher than for the virtual case and the WiK method is still more effective (average of 6.86 pixels) against the WoK method (average of 12.

Conclusion

These first experiments demonstrate that the automatic registration of two dental arcades can be achieved from images of mouth patient. The proposed methods are effective in the « perfect » virtual case, both for the projection matrix estimation and model registration. Nevertheless, the performance of these tests decreases when adding noise, uncertainties or with real shot constrains. It will be interesting to complete this study observing the impact of distortions on virtual cases. Globally, WiK method provides the better results than WoK method for all tested aspects even if in the manual cases, reprojection errors and 3D errors remain significant after registration. For the registration, reprojection error doesn't seem the unique and the best criteria to minimize: relative distance between several critical points or 3D distances obtained from the calculation of triangulated 3D points could be more relevant.

In the future, the most challenging problem is to realize an automatic matching between singular points or lines on 3D models of dental arcades and their corresponding features in the color pictures of the mouth. Our strategy will consist in bringing together the 2D and 3D representations to simplify the matching process. For this goal, we will carry out a 3D reconstruction from two or more views to extract and describe 3D features (bending radius) or reciprocally take advantage of 2D features (colorimetric gradient, texture). Moreover, we will enhance the 3D model with texture mapping.

Fig. 1 .

 1 Fig. 1. Reference coordinate system of the registration.

Fig. 2 .

 2 Fig. 2.(a)Front view (left) and side view (right) (top) real image (1300 867) (down) virtual views (1300 867). (b) Initial location of the 3D models before the registration.

Fig. 3 .

 3 Fig. 3. (left) Virtual 2D picture obtained by 3D model projection on the screen with 3D points projections (right) dentition 3D model with manually selected 3D points.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Gravity center errors on the virtual data (WiK method).

Table 1 .

 1 Reprojection MSE (in pixels) for the WiK method and two virtual viewpoints.

			Front view
	Noise	MSE	Max. error	MSE	Max. error
	(pixels)	maxillary	maxillary	mandible	mandible
	2	1.37±0.19	2.17	1.92±0.31	3.06
	4	2.73±0.39	4.34	3.83±0.61	6.12
	Manual	3.48±1.53	5.81	5.66±2.96	10.7
			Side view
	2	1.36±0.18	2.18	1.98±0.32	3.17
	4	2.72±0.35	4.37	3.96±0.63	6.34
	Manual	4.00±1.93	7.50	6.35±2.86	10.3

Table 2 .

 2 Reprojection MSE (in pixels) for the WoK method and two virtual viewpoints.

			Front view
	Noise	MSE	Max. error	MSE	Max. error
	(pixels)	maxillary	maxillary	mandible	mandible
	2	1.08±0.22	1.81	5.22±2.16	8.45
	4	2.16±0.44	3.61	10.45±4.36	16.94
	Manual	2.90±1.36	5.20	14.8±4.90	22.1
			Side view
	2	1.09±0.20	1.92	3.69±1.30	6.04
	4	2.18±0.39	3.84	7.37±2.62	12.09
	Manual	3.00±1.57	6.14	12.9±6.00	21.6

Table 3 .

 3 Reprojection MSE (in pixels) for the WiK and WoK methods on two real pictures.

			Front view
		MSE on the	Max. error	MSE on the	Max.error
	Method	maxillary	maxillary	mandible	mandible
	WiK	5.24±2.72	9.57	8.38±2.90	13.6
	WoK	3.77±1.29	5.93	36.6±11.1	53.5
			Side view
	WiK	5.44±2.68	10.6	7.70±3.08	12.2
	WoK	4.54±2.19	8.70	16.8±4.50	24.0

Table 5 .

 5 3D errors relative to the translation (mm) and rotation (degrees) after registration in the real case.

Acknowledgements

We thank the Ortho-Concept laboratory (http://www.ortho-concept.com/) and more especially Daniel Julie for providing us most of numerical data.


Antonio Cabrales

Olivier Gossner

Roberto Serrano

Bob Aumann

Mark Dean

Itay Fainmesser

Juanjo Ganuza

Johannes Gierlinger

Ehud Lehrer

José Penalva

Nicola Persico

Debraj Ray

Larry Samuelson

David Wolpert

Entropy and the value of information for investors

Keywords: C43, D00, D80, D81, G00, G11 Informativeness, Information structures ; Entropy, Decision under uncertainty, investment, Blackwell ordering JEL classification numbers: C00, C43, D00, D80, D81, G00, G11 informativeness, information structures, entropy, decision under uncertainty, investment, Blackwell ordering

Consider any investor who fears ruin when facing any set of investments that satisfy no-arbitrage. Before investing, he can purchase information about the state of nature in the form of an information structure. Given his prior, information structure α is more informative than information structure β if, whenever he is willing to buy β at some price, he is also willing to buy α at that price. We show that this informativeness ordering is complete and is represented by the decrease in entropy of his beliefs, regardless of his preferences, initial wealth, or investment problem. We also show that no prior-independent informativeness ordering based on similar premises exists.

Introduction

Consider a decision maker operating under uncertainty. When can one say that a new piece of information is more valuable to this agent than another? This question is in general hard to answer, because the ranking of information value typically depends upon at least three considerations. (i) The agent's priors matter. (An agent who is almost convinced that a serious crisis in the strength of the dollar is forthcoming may rank the appearance of bad financial news in China or in Europe very differently than an agent who is less convinced about such a crisis.) (ii) The preferences and wealth of the agent matter. (For the same prior, two agents with different degrees of risk aversion may rank in distinct ways a piece of news that almost eliminates uncertainty about an unlikely financial loss versus a less precise piece of news about more likely events.) (iii) The decision problem to which information is applied matters. (The value of being informed about the likelihood of different risks depends on the availability of insurance markets for these risks.)

The agent possesses some initial prior on a set of payoff-relevant states of nature. An information structure specifies, for every state of nature, a probability distribution over the agent's set of signals, so that each signal leads to an update of the agent's beliefs on the state of nature. The question we address is when one information structure provides more information than another.

The first answer to this fundamental question is provided in the seminal work of [START_REF] Blackwell | Equivalent comparison of experiments[END_REF]. According to Blackwell's ordering, an information structure is more informative than another whenever the latter is a garbling of the former, i.e., when there exists a stochastic matrix -interpreted as noise-such that the matrix of conditional probabilities of each signal for the less informative structure is obtained by multiplying the matrix for the more informative structure by the stochastic matrix. Blackwell's Theorem states that this is the case if and only if a decision maker with any utility function would prefer to use the former information structure over the latter when facing any decision problem. This important result provides a decision-theoretic foundation for Blackwell's informativeness ordering. Of course, the requirement that any decision maker would prefer one information structure to another is very strong, and most pairs of information structures cannot be compared according to Blackwell's ordering. In other words, Blackwell's ordering is very incomplete.

Following recent developments in the theory of riskiness, we attempt here to formulate an approach based on decision-theoretic principles in order to complete Blackwell's ordering. 1 Restricting our attention to a class of no-arbitrage investment decisions first studied in Arrow (1971a) and to a specific class of ruin-averse utility functions that like to avoid bankruptcy, we postulate the following informativeness ordering. Fixing a prior over the states, we say that one information structure is more informative than another if, over the allowed class of problems and preferences, whenever the first one is rejected at some price, so is the second. This seems a plausible minimum desideratum for a notion of informativeness.

Our main result is that this informativeness ordering is complete and is represented by the decrease in entropy of the agent's beliefs. 2 Specifically, if one considers the prior and the collection of posteriors generated by the information structure, we show that the informativeness of the information structure is represented by the difference between the entropy of the prior distribution and the expected entropy of the conditional posterior distributions. 3 More precisely, an information structure uniformly investment dominates another one in our sense if and only if the entropy 1 In particular, we follow closely a recent paper by [START_REF] Hart | Comparing risks by acceptance and rejection[END_REF], in which two orderings are proposed to justify the [START_REF] Aumann | An economic index of riskiness[END_REF] index of riskiness and the [START_REF] Foster | An operational measure of riskiness[END_REF] measure of riskiness.

2 Unlike the riskiness papers mentioned in the previous footnote, our decision-theoretic considerations here do not uncover a new index, but provide a new support to the classic concept of entropy.

3 This difference is referred to as "rate of transmission" by Arrow (1971a), who works with a set of Arrow securities and a logarithmic utility function.

decrease resulting from the first is larger than that resulting from the second. Our ordering is complete, since informativeness is characterized by a real number. It is also compatible with Blackwell's ordering, since more information according to Blackwell's ordering necessarily corresponds to decrease in entropy.

Blackwell showed that there is no ordering of information structures applicable to all decision problems and all possible utility functions. It is thus generally impossible to unanimously rank all information structures. Our approach aims at identifying a wide class of utility functions and investment problems where some version of "unanimity" obtains. More precisely, our approach tells us that in an economy populated with various agents, differing in their ruin-averse utility functions and in their no-arbitrage investment opportunities, if some agent is willing to pay more for information structure α than anyone is willing to pay for β, then we say that α dominates β. 4 The characterization of this ordering indicates that entropy is the only objective way to speak of the informativeness of information structures: by construction, our ordering is independent both of the agent's preferences and of the set of available choices, withing the classes considered. On the other hand, it is prior-dependent, which may seem to be a limitation. We show, however, that this is unavoidable, since any ordering based on the same postulate as ours is necessarily prior-dependent.5 Therefore, in regard to the difficulties described in the first paragraph, we provide an environment in which our complete informativeness ordering takes care of considerations (ii) and (iii), although it cannot possibly do the same with (i).

Section 2 introduces the investment problems that we study, the basic assumptions about the investor, and the notions of an information structure and of valuable information. Section 3 introduces ruin-averse utility functions, no-arbitrage invest-ment sets, and our informativeness ordering, and then proves the main result. Section 4 offers several points of discussion concerning prior independence and wealth uniformity, presents an axiomatization of our assumptions on preferences and investment sets, and analyzes several examples. Section 5 is a review of the literature, and Section 6 concludes. The Appendix provides the proofs.

Investments and Uncertainty

We measure the value of information according to its relevance to investment choices.

To this end, we rely on a standard model of investment under uncertainty à la Arrow (1971a).6

The Investor

We consider an agent with initial wealth w, and with an increasing and twice differentiable monetary utility function u : R + → R. The coefficient of relative risk aversion at wealth z > 0 is:

ρ(z) = - u ′′ (z)z u ′ (z) .
We assume that the agent has weakly increasing relative risk aversion (IRRA), namely, that ρ is nondecreasing on R + . This standard assumption is defended from the theoretical point of view by [START_REF] Arrow | Essays in the Theory of Risk Bearing[END_REF]. It is also consistent with observed behavior both in the field [START_REF] Binswanger | Attitudes toward risk: Theoretical implications of an experiment in rural India[END_REF][START_REF] Post | Deal or no deal? Decision making under risk in a large-payoff game show[END_REF] and in laboratory experiments [START_REF] Holt | Risk aversion and incentive effects[END_REF]. The class of IRRA utility functions includes the widely used constant absolute risk aversion (CARA) and constant relative risk aversion (CRRA) classes.

We denote by U 0 the set of such monetary utility functions. For u ∈ U 0 , we let

u(0) = lim z→0 u(z) ∈ R ∪ {-∞}.
Let K be the finite set of states of nature. The investor has a prior belief p with full support, fixed throughout the paper.7 An investment opportunity or asset is b ∈ R K , with the interpretation that if b is taken, the agent's wealth, once uncertainty is realized, is w + b k in state k. We do not allow for bankruptcy (the possibility of negative wealth), and say that an asset b is feasible at wealth w when w + b k ≥ 0 in every state k ∈ K.8

The Investments

The investor has the opportunity to choose from an investment set B of assets, from which he must take one. One possibility among the available choices is to opt out, namely, to keep his wealth w in a safe asset. We formulate this assumption as 0 K ∈ B, where 0 K is the null vector of R K . Thus, an investment set consists of a subset of R K containing 0 K . For instance, the set B can consist of a set of Arrow securities, or of the asset structure of a complete or incomplete market. Elements in B can be either divisible (for every b ∈ B, λ ∈ [0, 1], λb ∈ B) or indivisible. We say that an investment set B is feasible at w when all its elements are feasible at w.

Information Structures

An information structure α is given by a finite set of signals S α , together with transition probabilities α k ∈ ∆(S α ) for every k. When the state of nature is k, α k (s) is the probability that the signal observed by the agent is s. It is standard practice to represent any such information structure by a stochastic matrix, with as many rows as states and as many columns as signals; in the matrix, row k is the probability distribution (α k (s)) s∈Sα . We assume that every signal s has positive probability under at least one state k. This is without loss of generality, since zero probability signals can be deleted from the set S α .

It is useful to think of α in terms of a distribution over posterior probabilities.

Signal s has a total probability p α (s) = k p(k)α k (s), and the agent's posterior probability on K given s is q s α , derived using Bayes' formula:

q s α (k) = p(k)α k (s) p α (s) .
Information structures are ranked according to the partial [START_REF] Blackwell | Equivalent comparison of experiments[END_REF]

Valuable Information

Given a utility function u, initial wealth w, a feasible investment set B, and a belief q ∈ ∆(K), the maximal expected utility that can be reached by choosing an

investment opportunity b ∈ B is v(u, w, B, q) = sup b∈B, b feasible k q(k)u(w + b k )
with the convention that 0 • (-∞) = 0.

The ex-ante expected payoff before receiving signal s from α is

π(α, u, w, B) = s p α (s)v(u, w, B, q s α ).
The possibility of opting out ensures that both v(u, w, B, q) and π(α, u, w, B) are always larger than or equal to u(w).

The gain V (α, u, w, B) from investment opportunities in B and information α is the difference:

V (α, u, w, B) = π(α, u, w, B) -u(w).
It is often useful for our purposes to rewrite the above expression in this way:

V (α, u, w, B) = s p α (s)(v(u, w, B, q s α ) -u(w)).
This last expression shows that V (α, u, w, B) > 0 if and only if there exists an s such that v(u, w, B, q s α ) > u(w).

3 Entropy as an Ordering of Information for Investment Problems

Ruin-Averse Utility and No-Arbitrage Investments

We make two assumptions in our basic framework. One concerns the agent's utility function u and the other, the set B of available assets. These two assumptions, taken together, ensure that the class of utility functions and investment sets are suitable to rank informativeness.

We call an asset

b ∈ R k belief-supported (given initial belief p) if k p(k)b k ≤ 0,
and we let B * be the set of all belief-supported assets. An investment set B is belief-supported if it contains only belief-supported assets (B ⊆ B * ), and we let B * be the class of belief-supported investment sets. The belief-supported assets are characterized by the property that no weakly risk-averse or risk-neutral agent with belief p would prefer to select such an asset over opting out. We also refer to them as no-arbitrage assets, as they are also characterized by the absence of arbitrage opportunities (see, e.g., [START_REF] Duffie | Dynamic asset pricing theory[END_REF], theorem 1 on page 4 and the later discussion in section 1.B. of risk-neutral probabilities). For instance, for an investor with a uniform prior over three states, the asset with payoffs (-7, 2, 3) offers no arbitrage opportunities ex-ante, but it may be a reasonable investment after acquiring appropriate information, e.g., that the true state of nature is not state 1.

We call a monetary utility u ruin-averse whenever u(0) = -∞. Thus, a ruinaverse agent is one who prefers to opt out rather than making any investment that leads to ruin with positive probability. Let U * ⊂ U 0 be the set of all ruin-averse utility functions in our domain. The next lemma, following the analysis of Hart ( 2010), characterizes ruin aversion by means of coefficients of risk aversion.

Information Purchasing

In order to understand the value of information for the agent, we consider a situation in which the agent has the possibility of purchasing information structure α before making an investment decision in regard to B. Decisions whether to purchase information or not are based on the comparison between the expected payoff under the new information and the sure payoff u(w). The agent with utility function u and wealth w purchases information α at price µ < w, given an investment set B, when:

π(α, u, w -µ, B) ≥ u(w).
Otherwise, the agent rejects information α at price µ.

Our information ordering is defined as follows:

Definition 1 Information structure α uniformly investment-dominates (or investmentdominates, for short) information structure β whenever, for every wealth w and price µ < w such that α is rejected by all agents with utility u ∈ U * at wealth w for every opportunity set B ∈ B * , β is also rejected by all those agents.

The preceding definition seems plausible as a minimum desideratum in order to speak of informativeness. The next lemma identifies the important role played in the definition by an agent with a logarithmic utility function:

Lemma 2 Given an information structure α, a price µ, and a wealth level w > µ, α is rejected by all agents with utility u ∈ U * at wealth level w given every opportunity set B ∈ B * if and only if α is rejected by an agent with ln utility at wealth w for the opportunity set B * .

Entropy Ordering

With the assumptions made about assets and utility functions, our next step is to arrive at a representation of the ordering just defined. In effect, achieving this representation will provide an index of informativeness for information structures, i.e., an objective way to talk about an information structure being more informative than another, based on the investment framework described. The result below characterizes entropy as such an index, which is independent of the specific utility function of the decision maker, of his wealth, and of the specific investment decision considered.

In contrast, such an index cannot be independent of the decision maker's prior, as we also show in the next section.

Following [START_REF] Shannon | A mathematical theory of communication[END_REF], the entropy of a probability distribution q ∈ ∆(K) is the quantity:

H(q) = - k∈K q(k) log 2 q(k),
where 0 log 2 (0) = 0 by continuity.9 The entropy of p is a measure of the level of uncertainty about the state of nature held by the investor with belief p. The entropy is always nonnegative, and is equal to zero only in the case of certainty, i.e., when q puts weight 1 on some state k. It is concave, representing the fact that distributions that are closer to the extreme points in ∆(K) correspond to a lower level of uncertainty. On the other hand, entropy achieves its global maximum at the uniform distribution, a situation of "maximal uncertainty."

Recall that following information structure α, (i) the agent's signal is s with probability p α (s); and (ii) the posterior probability on K following s is q s α . The entropy informativeness of information structure α is the expected reduction of entropy of the investor's beliefs due to his observation of s. It is this quantity:

I(α) = H(p) - s p α (s)H(q s α ).
As shown in Subsection 4.1, I(α) depends on p as well as on α. (For notational simplicity, we do not include p as an argument of I. Only in Subsection 4.1 do we make this dependence explicit.) The informativeness is minimal when α is α with no informational content, and I(α) = 0. It is maximal when α is α that fully reveals the state of nature k, and value I(α) = H(p). Note that given a prior p, I is a numeric index, which hence defines a complete ordering of information structures.

Main Result

Our main result establishes that the ordering of information structures given by investment dominance coincides with the ordering according to entropy informativeness. Hence this ordering is complete.

Theorem 1 Information structure α investment-dominates information structure

β if and only if I(α) ≥ I(β).
The proof of this result hinges on two crucial steps. First, we establish that an agent with logarithmic utility values information about our class of investments using entropy. Then, since by Lemma 2 all agents in the class U * reject an information structure α at a price if and only if so does the logarithmic utility, entropy must order information structures in our investment-domination sense for all agents in U * .

Notice that although the proof in Lemma 2 borrows from the techniques in Hart (2010), we establish an important additional step, namely, that the logarithmic agent, the universally less risk-averse agent in our class, is also the one who is the most willing to pay for information.

Discussion

This section discusses each of the assumptions used in our approach.

Prior-Independent Ordering

For a given prior p, the informativeness ordering we have suggested is represented by a decrease in entropy. Making the dependence of I(α) on p explicit, we denote here I(α) by I(α, p). An information structure α is more informative than another β, I(α, p) ≥ I(β, p), if and only if α causes a larger reduction in entropy (from the entropy of the prior p to the expected entropy of the generated posteriors) than does β.

We now prove that there can be no index that orders information structures that is both compatible with uniform investment dominance and independent of the agent's prior. In order to do this, let us define the following.

Definition 2 An information structure α investment-dominates independently of the prior β, whenever α investment-dominates β for all priors p.

This definition turns out to be too strong a requirement, and leads to the following impossibility result:

Theorem 2 There exists no numerical representation that orders information structures according to the ordering of investment dominance independently of the prior.

Uniformity in Wealth

After the discussion opened in the previous subsection, we now return to fixing a prior p, which will remain fixed for the rest of the paper.

We have defined information dominance as wealth-independent, but this is not really a restriction. To see this, consider the following alternative definition:

Definition 3 Information structure α investment-dominates information structure β for wealth w if, for every price µ < w such that α is rejected by all agents with utility u ∈ U * for every opportunity set B ∈ B * , β is also rejected by all those agents.

This definition leads to the following theorem:

Theorem 3 Information structure α investment-dominates information structure β for wealth w if and only if I(α) ≥ I(β).

The result clearly follows because Lemma 2 holds for either Definition 2 or Definition 3 and the ordering I(•) induced by logarithmic preferences is independent of wealth.

In a similar vein, we have made so far comparisons for agents with the same level of wealth. This can be avoided provided the pricing is done as a proportion of wealth.

Definition 4 Information structure α proportionally investment-dominates information structure β if, for any 0 < λ < 1, whenever α is rejected by all agents with utility u ∈ U * for every opportunity set B ∈ B * at every wealth w for price λw, β is also rejected by all those agents.

Before getting to the next result, we prove the following lemma:

Lemma 3 Let 0 < λ < 1. An information structure α is rejected by all agents with utility u ∈ U * for every opportunity set B ∈ B * at every wealth w for the price λw if and only if α is rejected by an agent with ln utility at every wealth level for the opportunity set B * .

Using Lemma 3 we show:

Theorem 4 Information structure α proportionally investment-dominates information structure β if and only if I(α) ≥ I(β). According to this property, in the absence of information beyond the prior p, the agent prefers to opt out rather than to invest in risky elements of B. As stated, it is a joint assumption on the possible investment set B and the agent's utility function u. It can be viewed as a normalization: for a decision maker who is considering improving his information before investing, we define his initial position as "not being ready to invest" if he gets no new information.

On the Ruin-Aversion and No-Arbitrage Assumptions

The NINI property expresses the idea that B is such that V is null unless α has some informational content; more precisely: NINI: B is the class of investment sets B such that for V (α, u, w, B) = 0 for every u ∈ U 0 , w ∈ R + .

To motivate the second property, we now discuss the circumstances under which information is valuable to the agent. First, note that if B does not contain feasible elements b such that b k > 0 in some state k, the agent always weakly prefers to opt out. More generally, an agent who fully learns that k is the state of nature cannot take advantage of such information, unless there exists a feasible b offering a gain in state k. We say that an investment set B is investment-prone if, for every k ∈ K,

there exists b ∈ B such that b k > 0.
What quality of information is needed to ensure that every investor takes advantage of investment-prone sets? We say that information structure α is sometimes certain if q s α (k) = 1 for some k and s, that is, when there is a signal s which, if received, reveals that the state of nature is k for sure. If α is not sometimes certain, we call it always uncertain.

The next lemma shows that sometimes-certain information structures are always advantageous, provided B is feasible and investment-prone.

Lemma 4 If B is investment-prone and feasible at wealth level w, then V (u, w, α, B) > 0 for every α that is sometimes certain and for every u ∈ U 0 .

The second joint property of utility functions and investment sets is that only investors with access to sometimes-certain information structures are always inclined to invest. This property, (SCAI for short) is expressed as SCAI: U consists of the elements u of U 0 satisfying the condition that there exists a wealth level w and an investment-prone set B of feasible investment opportunities such that V (α, u, w, B) = 0 for every always-uncertain α.

According to SCAI, whenever α is always uncertain, then there exists a feasible and investment-prone set of investment opportunities such that the agent weakly prefers to opt out. The idea is that not every piece of information is always valuable, i.e., valuable for every agent in every circumstance. In particular, risk-averse agents, like ours, may not use investment opportunities with a positive expected profit if the associated risk is too high. On the other hand, our agents surely can take advantage of being fully informed. The SCAI property establishes a restriction on the class of agents, requiring that only sometimes-certain information structures be always valuable to them.

As noted above, both the classes U * of ruin-averse utility functions and B * of no-arbitrage investment sets satisfy NINI and SCAI. That they satisfy NINI is clear:

no-arbitrage assets offer no profitable investment opportunity to risk-averse agents if there is no new information. To see that they also satisfy SCAI, think of a typical investment-prone asset with large negative payoffs in all states but one; with such no-arbitrage assets around, a risk-averse investor will want to opt out unless the new information completely reveals one state. Taken together, NINI and SCAI therefore depict situations in which a risk-averse investor is cautious in utilizing new information, given that the available investments out there may include very risky deals.

What is perhaps more surprising is that these two properties uniquely define a set U of utility functions and a class B of investment sets, and that both U = U * and B = B * :

Theorem 5 U and B satisfy NINI and SCAI if and only if U is the class U * of ruin-averse utility functions, and B is the class B * of no-arbitrage investment sets.

Examples

We now compute some examples. The first one illustrates how our framework serves to complete Blackwell's ordering.

Example 1 Let K = {1, 2, 3} and fix a uniform prior. Consider two information structures that are not ordered in the Blackwell sense. For instance, let each of the two information structures have two signals:

α 1 =   1 0 1 0 0 1   , α 2 =   1 0 0.1 0.9 0 1  
To see that they are not ranked according to Blackwell, we exhibit two decision problems where a decision maker would rank them differently. For instance, in Problem 1 the agent must choose one of two actions: action 1 gives a utility of 1 only in the first two states, and 0 otherwise, while action 2 gives a utility of 1 only in the third state, and 0 otherwise. Problem 2, in contrast, has action 1 pay a utility of 1 only in the first state, and 0 otherwise, while action 2 gives a utility of 1 only in states 2 or 3, and 0 otherwise. Facing Problem 1, the decision maker would value α 1 more than α 2 : following the first signal in α 1 , he would choose the first action and following the second signal in α 1 , he would choose the second action, thereby securing a utility of 1. This would be strictly greater than his utility after α 2 . On the other hand, facing Problem 2, he would under α 2 choose action 1 after the first signal and action 2 after the second, yielding a utility of 29/30, which is greater than his optimal utility after α 1 .

But by calculating their entropy reduction from the uniform prior, we know that I(α 1 ) > I(α 2 ). Thus, for every investment problem we consider and every utility function in our allowable class, the first information structure is more valuable -more informative-than the second when starting from a uniform prior. The difficulty in the two problems of this example is that, if one specifies an economic environment like ours to make sense of the action-utility pairs provided, the resulting investment set fails to be belief-supported, for at least some wealth levels.

The second example illustrates the optimization over investment sets.

Example 2 Let the prior belief p and the investment opportunity b i be such that for a unit of investment the return is:

b i k = 1 p(k) -1 if k = i -1 if k = i .
Then, one can easily see that

K k=1 p(k)b i k = 0
and hence, b i is belief-supported given initial belief p. Suppose that B is composed of a set of K perfectly scalable investment opportunities b 1 through b K , one for each state of nature. Then, suppose w i is the amount an investor invests in b i . This implies that:

v(ln, w, B, q) = sup b∈B, b feasible K k=1 q(k) ln w k p (k)
so that the optimal w * k is equal to q(k)w. Hence,

v(ln, w, B, q) = K k=1 q(k) ln q (k) p (k) + ln w.
For other CRRA utility functions with parameter φ we can write,

v(φ, w, B, q) = sup b∈B, b feasible K k=1 q(k) 1 -φ w k p (k)
1-φ so that the optimal w * k can be expressed as:

w * k = λq(k) 1/φ p (k) -(1-φ)/φ w with λ = 1/ K k=1 q(k) 1/φ p (k) -(1-φ)/φ . Hence, v(φ, w, B, q) = K k=1 q(k) 1 -φ q (k) p (k) λw 1-φ φ .
The next example illustrates how entropy ranks lotteries over two information structures compared to an "expected information structure" which combines the two outcomes of the lottery.

Example 3 Assume a prior p of K equiprobable states. Then

H (p) = - K k=1 1 K log 2 (K) = -log 2 (K).
Consider the following information structure:

α 1 =   1 0 0.5 0.5 0 1   .
Given the uniform prior, each signal is equally likely. The corresponding posteriors are (2/3, 1/3, 0) and (0, 1/3, 2/3). The expected entropy of the posteriors is

-(2/3) ln 2+ln 3 ln 2
, which is approximately 0.9182.

On the other hand, consider a situation in which the agent is offered two information structures α 1 and α 2 , each with one-half probability, as follows:

α 1 =   1 0 1 0 0 1   , α 2 =   1 0 0 1 0 1   .
Note how the "average" of these two information structures is the original information structure α.

In α 1 , the first signal has a probability of 2/3 and the second signal has a probability of 1/3. The corresponding posteriors are (1/2, 1/2, 0) and (0, 0, 1). Following α 2 , the probability of the first signal is 1/3 and that of the second is 2/3; the corresponding posteriors are (1, 0, 0) and (0, 1/2, 1/2). The corresponding expected entropy is

(2/3) ln 2 ln 2 , which is 2/3. Thus, the experiment consisting of purchasing the lottery of the two information structures is more informative than its expected information structure. More generally, this follows from the convexity of thelog 2 function.

Finally, although we have worked with finitely many states to avoid measuretheoretic technicalities, the results in this paper are easy to extend to distributions with a continuum of states. This is useful because many applications assume such a continuum.

Example 4

The normal distribution has a particularly simple entropy. Suppose s is an n × 1 vector of normally distributed variables, where s ∼ N (µ, Σ) . Then, if we let |Σ| be the determinant of the variance-covariance matrix, we find that:

H (s) = - 1 2 log 2 ((2πe) n |Σ|) .
When s follows a univariate uniform distribution in the interval [A, B], then H(s) can be written as:

H (s) = - B A 1 B -A log 2 1 B -A du = log 2 (B -A) .
It is clear that information structures can be ordered by the expected reduction on |Σ| in the first case and by the expected reduction of B -A in the second.

Related Literature

Other authors have justified the use of the entropy index based on information theoretic considerations, and have showed that it arises naturally in a variety of dynamic setups. A salient feature of our work is that it shows that entropy is also rooted in economic and decision-theoretic arguments in static setups. [START_REF] Shannon | A mathematical theory of communication[END_REF], who introduced entropy as a measure of information, characterizes it as the only measure that jointly satisfies these three properties: continuity, monotonicity, and decomposability. [START_REF] Marschak | Remarks on the economics of information[END_REF] presents formal arguments in favor of using entropy in the study of the demand for information and its cost. After these classic contributions, the concept has arisen separately in several fields of economics, and we provide only a brief partial survey here. [START_REF] Gossner | Optimal use of communication resources[END_REF] study repeated games in which one of the players, who can forecast the realization of future states of nature, can transmit information to others through his choice of actions. They provide a closed-form characterization, based on entropy, of the set of distributions which the players can achieve. [START_REF] Gossner | Empirical distributions of beliefs under imperfect observation[END_REF] analyze games in which a team of players uses private signals in a repeated game to secretly coordinate their actions. They show that entropy is an adequate measure of informativeness to study the trade-off between the generation of signals for future coordination and the use of acquired secret information.

The Theil coefficient for economic inequality is based on the entropy of observed data. [START_REF] Bourguignon | Decomposable income inequality measures[END_REF] axiomatizes this measure by showing that it is the only one that is consistent with a property of income-weighted decomposability.

In [START_REF] Sims | Implications of rational inattention[END_REF]'s model of rational inattention, entropy is used to measure information acquisition by agents with bounded information-processing capabilities.

This approach has been applied to different economic problems. For instance, Peng (2005) explores its implications for asset-price dynamics and consumption behavior.

Sims ( 2005) offers a summary of other contributions in this area. Arrow (1971a) considers an investor who has access to a set of securities that pay a positive amount in only one state of nature. He shows that, if the value of information about the state is independent of the returns, then this value is given by the entropy of this information.

Entropy is also the basis for the relative-entropy measure of proximity of probability distributions. [START_REF] Blume | Evolution and market behavior[END_REF] and [START_REF] Sandroni | Do markets favor agents able to make accurate predictions?[END_REF] show that in dynamic exchange economies, markets favor agents who make the most accurate predictions when accuracy is measured according to relative entropy. Other applications of relative entropy include ambiguity aversion [START_REF] Maccheroni | Ambiguity aversion, robustness, and the variational representation of preferences[END_REF] and reputation models [START_REF] Gossner | Simple bounds on the value of a reputation[END_REF].

Measuring the amount of information is a common problem in economics and decision theory.10 Most of the work in this area follows the seminal work of [START_REF] Blackwell | Equivalent comparison of experiments[END_REF]. For Blackwell, an information structure α is more informative than an information structure β if every decision maker prefers α to β in any decision problem. As noted in the introduction, the main drawback of this approach is that this criterion does not provide a complete ordering. Researchers have made progress by focusing on decision makers who have preferences in a particular class. [START_REF] Lehmann | Comparing location experiments[END_REF], for instance, restricts the analysis to problems that generate monotone decision rules (and hence satisfy single-crossing conditions). [START_REF] Persico | Information acquisition in auctions[END_REF], [START_REF] Levin | The value of information in monotone decision problems[END_REF], and Jewitt (2007) extend Lehmann's analysis to more general classes of monotone problems. We follow this tradition with two main differences. First, unlike the measures in those papers, our measure of informativeness provides a complete order of all information structures. Second, we achieve this through a different kind of restriction on admissible preference orderings, and we characterize decision problems in terms of investment opportunities, thereby restricting the framework. [START_REF] Gilboa | The value of information -an axiomatic approach[END_REF] and [START_REF] Azrieli | The value of a stochastic information structure[END_REF] take an approach that differs markedly with respect to the one used in papers cited in the previous para- A recent paper by [START_REF] Ganuza | Signal orderings based on dispersion and the supply of private information in auctions[END_REF] provides a different way to order information structures (also a partial order) which is based not on decision-theoretic considerations, but rather on various measures of dispersion of distributions. (Many of those measures are presented in [START_REF] Shaked | Stochastic orders[END_REF]. They show that while some of their measures are implied by notions of informativeness based on the value of information, the strongest of their criteria, supermodular precision, is strictly different: it neither implies nor is implied by those notions of informativeness.

They then proceed to study the implications of greater informativeness (in their sense) for auction problems, and show that while greater informativeness improves allocational efficiency, the auction organizers are not always interested in increasing informativeness since that may increase the buyers' informational rents.

Conclusion

In the classic framework of information structures proposed by Blackwell, we have found that, for a given prior, a natural informativeness ordering (namely, that if a decision maker is willing to pay a price for an informative information structure, he is willing to pay that price for a one that dominates it) is complete when considered over the class of ruin-averse utility functions and no-arbitrage investment sets. Furthermore, this ordering is represented by the expected decrease of entropy from the prior to the posteriors, and this ordering is complete. We have also found that no such ordering can be made independent of the decision maker's prior.

u(z)u(1) ≤ u ′ (0) ln(z), and hence that u(0) = lim z→0 u(z) = -∞.

Now assume that there exists z 0 > 0 where ρ(z 0 ) < 1. Since u is IRRA, then for every z ≤ z 0 , ρ(z) ≤ ρ(z 0 ) < 1. Integrating shows that for every z ≤ z 0 ,

ln u ′ (z) -ln u ′ (z 0 ) ≤ -ρ(z 0 )(ln(z) -ln(z 0 )),
which can be expressed as

u ′ (z) ≤ u ′ (z 0 ) z z 0 -ρ(z 0 )
.

A second integration between z < z 0 and z 0 shows:

u(z) -u(z 0 ) ≥ z 0 u ′ (z 0 ) 1 -ρ(z 0 ) z z 0 1-ρ(z 0 ) -1 .
Since 1ρ(z 0 ) > 0, the limits of the right-hand side, and hence of the left-hand side, are finite. This shows that u(0) > -∞.

A.2 Proof of Lemma 2

First, note that the only if condition is satisfied since the ln utility function belongs to B * and B * ∈ B * .

We now prove the if part. Assume that α is rejected at price µ given the investment set B * by an agent with ln utility and with wealth w. For u ∈ * , Lemma 1 shows that ρ(z) ≥ 1 for z > 0; hence:

u ′′ (z) u ′ (z) ≤ - 1 z .
By integration between w and z:

ln u ′ (z) -ln u ′ (w) ≤ -ln(z) + ln(w) if z ≥ w; ln u ′ (z) -ln u ′ (w) ≥ -ln(z) + ln(w) if z ≤ w.
Once w is fixed, a second integration with respect to z between w and z ′ shows that for every z ′ , u(z ′ )u(w) ≤ wu ′ (w)(ln(z ′ )ln(w)).

Hence, given any belief q, B ∈ R * and µ < w, we can write:

v(u, wµ, B, q)u(w) ≤ wu ′ (w)(v(ln, wµ, B, q)ln(w));

and by summation over q α , for every B ∈ * and µ < w, we obtain:

π(α, u, w -µ, B) -u(w) ≤ u ′ (w)w(π(α, ln, w -µ, B) -ln(w)).
Since π(α, u, wµ, B) is nondecreasing in B, and B * is the maximal element of B * , then for every B ∈ B * and µ < w we have:

π(α, u, w -µ, B) -u(w) ≤ wu ′ (w)(π(α, ln, w -µ, B * ) -ln(w)) < 0,
which is the desired conclusion.

A.3 Proof of Theorem 1

Lemma 2 shows that α uniformly investment-dominates β if and only if, for every w and µ < w, an agent with ln (or, equivalently, log 2 ) utility function who rejects α for the opportunity set B * also rejects β. The following lemma characterizes the value of information for an agent with log 2 utility function and opportunity set B * .

Lemma A.1 For every w > 0 and belief q, 1.

v(log 2 , w, B * , q) = log 2 (w) -H(q)k q(k) log 2 p(k).

2.

π(α, log 2 , w, B * ) = I(α) + log 2 (w).

Proof. For the first point, v(log 2 , w, B * , q) is the maximum of k q(k) log 2 (w + b k ) over (b k ) such that k p(k)b k ≤ 0. The first-order condition shows that w + b k is proportional to q k p k , and hence equal to w q k p k . We then obtain:

v(log 2 , w, B * , q) = log 2 (w) + k q(k) log 2 q(k) - k q(k) log 2 p(k) = log 2 (w) -H(q) - k q(k) log 2 p(k).
For the second point, we rely on the previous expression to deduce:

π(α, log 2 , w, B * ) = s p α (s)v(log 2 , w, B * , q α (s)) = log 2 (w) - s p α (s)H(q s α ) - k,s p α (s)q s α (k) log 2 p(k) = I(α) + log 2 (w) since s p α (s)q s α (k) = p(k).
We now complete the proof of Theorem 1. Recall that by Lemma A.1, an agent with utility function log 2 rejects α at price µ < w for the opportunity set B * if and only if:

I(α) < log 2 w w -µ .
If I(α) ≥ I(β), then β is rejected whenever α is. If, on the contrary, I(α) < I(β), let µ be such that:

I(α) < log 2 w w -µ < I(β).
At this price µ, α is rejected whereas β is accepted. Hence, α does not investmentdominate β.

A.4 Proof of Theorem 2

By Theorem 1 and a fixed prior p, the only possible index is I(α, p) = H(p)s p α (s)H(q s α ). Therefore, it suffices to construct an example to show that this index orders two information structures in different ways for two different priors.

The example follows.

Let K = {1, 2, 3}. Let p 1 = (1/2, 1/2, 0) and p 2 = (1/3, 1/3, 1/3) and an agent with u(x) = ln(x).11

Let information structures α 1 and α 2 be described by these two-signal three-state matrices:

α 1 =   1 0 0 1 0.5 0.5   , α 2 =   0.3 0.7 0 1 1 0   .
Clearly, the expected utility for the agent with logarithmic utility under α 1 is larger than that for α 2 when priors are p 1 as the former gives her full information while the latter does not. It thus follows that:

I(α 1 , p 1 ) = H(p 1 ) - s p α 1 (s)H(q s α 1 ) > H(p 1 ) - s p α 2 (s)H(q s α 2 ) = I(α 2 , p 1 ).
What is the expected entropy of the posteriors generated by α 1 and α 2 under p 2 ? First, the utility for a ln agent of prior p 2 is ln(1/3). Then for α 1 the expected utility is: 2 3 ln 2 3 + 1 3 ln 1 3 = -0.63651.

Therefore, H(p 2 )s p α 1 (s)H(q s α 1 ) =

(2/3) ln 2 ln 2 = 0.462 10 ln 2 = 2/3.

As for α 2 , the (conditional on p 2 ) probability of either signal is 13/30 and 17/30.

After she observes each signal, her posteriors are (3/13, 0, 10/13) and (7/17, 10/17, 0), respectively. Thus, her expected ln utility from α 2 is: Hence, whereas for prior p 1 information structure α 1 is more informative than α 2 , the opposite is true for prior p 2 .

A.5 Proof of Lemma 3 and Theorem 4

For a fixed w, it follows from Lemma 2 that α is rejected by all agents with utility u ∈ U * at wealth w and price λµ if and only if it is rejected by an agent with ln utility for the opportunity set B * at the same wealth level and price. Hence Lemma 3.

Note that the property that an agent with ln utility purchases some information at price λw for wealth level w is independent of w.

From Lemma 3, α proportionally investment dominates β if and only if, for every λ, whenever an agent with utility ln and opportunity set B * purchases β at some wealth level w for the price λw, the same agent purchases α. This is equivalent to the fact that given some w > 0 and for every µ, whenever an agent with utility Hence, b ′ ∈ B.

We observe that if B is not investment-prone, neither is any element of B, a subset of B. In this case, SCAI becomes trivially equivalent to U = ∅. In contrast, the following proposition characterizes SCAI when B is investment-prone.

Proposition A.4 If B is investment-prone, then U satisfies SCAI if and only if

U = U * .
Proof. We divide the proof into a series of lemmata.

Lemma A.5 Let u ∈ U 0 . If u(0) > -∞, then for every w and for every B that is investment-prone and feasible, there exists an always-uncertain α such that V (u, w, α, B) > 0.

Proof. Fix u, w, and the set B that is investment-prone and feasible.

For 1 > ε > 0, let α ε be defined by S α ε = K, α ε k (s) = 1ε if k = s, and α k (s) = ε K-1 otherwise. It can easily be werified that α ε is always uncertain for every ε > 0, and that as ε → 0, q k α ε (k) → 1 for every s.

  This subsection is intended to shed additional light on the role of the ruin-averse utility functions -the class U * -and the class of no-arbitrage or belief-supported investment sets -B * . So far, the assumptions underlying our main result, theorem 1, were that (1) the decision makers we study strongly dislike situations in which their wealth approaches zero (ruin aversion), and (2) the investments they consider do not offer profitable opportunities in the absence of new additional information (no arbitrage). Now we offer a joint axiomatization of such economic circumstances, i.e., of such preference-investment pairs. Thus, consider in general any class of utility functions included in our original IRRA class U 0 and any class B of feasible investment sets. First, it is worthwhile to point out that the classes U * of ruin-averse functions and B * of no-arbitrage sets jointly have two properties that we now turn to discuss.The first property is No Investment under No Information, or NINI for short.

  graphs. Rather than choosing a class of decision problems, and then providing an ordering of information structures, they characterize the orderings that are possible for any prespecified class of decision problems. The 1991 paper considers deterministic information structures, and the 2008 paper extends the analysis to stochastic ones. The entropy function satisfies the axioms of the first paper, and hence it is a "value of information" function over partitions of the set of states. The 2008 paper shows that reducibility, weak order, independence, continuity, and convexity characterize all binary relations on information structures induced by decision problems, entropy being one of them.

  ln and opportunity set B * purchases β at wealth level w for the price µ, the same agent purchases α. From point 2 of Lemma A.1, this statement is equivalent toI(α) ≥ I(β). Hence Theorem 4. A.6 Proof of Lemma 4 Let k, s be such that q s α (k) = 1, and let b ∈ B be such that b k > 0. If b k is feasible, then we have v(u, w, B, q s α ) ≥ u(w + b k ) > u(w). Hence, V (α, u, w, B) ≥ p α (s)(u(w + b k )u(w)) > 0.Lemma A.3 B is comprehensive. Proof. Assume that b ∈ B and that b ′ is such that b ′ k ≤ b k for every k. Then, b is feasible at wealth w whenever b ′ is; and for every u ∈ U, w ∈ R + , k p(k)u(w + b ′ k ) ≤ k p(k)u(w + b k ) ≤ u(w).

Since

  B is investment-prone, there exist k * and b * ∈ B such that b * k * > 0. We now have v(u, w, B, q k * α ε ) = sup b∈B k q k * α ε (k)u(w + b k ) ≥ k q k * α ε (k)u(w + b * k ) ≥ q k * α ε (k * )u(w + b * k * ) + (1q k * α ε (k * ))u(0).We now show that U * and B * satisfy NINI and SCAI. With the assumption that B = B * , U * satisfies SCAI from Proposition A.4. With U = U * , B * satisfies NINI from Lemma A.7.

  Noting that ln(1/3) is the expected utility from the prior, we can derive:H(p 2 )p α 2 (s)H(q s α 2 ) = I(α 2 , p 2 ).

				s	p α 2 (s)H(q s α 2 ) =	0.480 61 ln 2	.
	That is,
	I(α 1 , p 2 ) = H(p 2 ) -	p α 1 (s)H(q s α 1 ) < H(p 2 ) -
		s
		13 30		3 13	ln	3 13	+	10 13	ln	10 13
	+	17 30	7 17	ln	7 17	+	10 17	ln	10 17	= -0.618 .

s

We can accomodate agents with heterogeneous wealth if we require the agents to pay not a price, but a proportion of their wealth.

There are of course other limitations, e.g., our postulate of an objective prior.

This basic asset-investment model is used often, e.g., in Mas-Colell, Whinston, and Green (1995), Section 19.E, where it is integrated into a general-equilibrium economy.

Except for Subsection 4.1, where we discuss the impossibility of a prior-independent ordering.

Notice that this is an ex-ante notion of feasibility, which does not take into account the payment of any amount to purchase information. The dominance relation defined later accounts for this.

The specific function log 2 (•) stems from the normalization that the amount of information carried by the observation of a Bernoulli random variable with parameter 1/2 is exactly one bit. For us, any log function would work, including, for example, ln(•).

[START_REF] Veldkamp | Information choice in macroeconomics and finance[END_REF] provides a good summary of ways in which economists have measured informativeness and its applications.

We make our computations below on the basis of this utility function, but recall that for all x, log 2 x = ln x/ ln 2, a positive transformation.

A Proofs

A.1 Proof of Lemma 1

We follow [START_REF] Hart | Comparing risks by acceptance and rejection[END_REF]. Assume that for every z > 0,

By integrating between z < 1 and 1 we obtain, ln u ′ (z)ln u ′ (0) ≥ln(z), which can be rewritten as:

A.7 Proof of Theorem 5

Given a class U ⊆ U 0 of utility functions, we say that an investment b individually satisfies NINI if for every u ∈ U and w ∈ R + such that b is feasible,

Thus, b individually satisfies NINI when, under no information, the agent does not prefer b to opting out. We denote as B the set of investments that individually satisfy the NINI property. Since 0 K satisfies NINI, the NINI investment set is a nonempty investment set.

Lemma A.2 Given U, B satisfies NINI if and only if B is the class of investment sets contained in B.

Proof. B satisfies NINI if and only if it contains all the investment sets B such that for every w > 0 and u ∈ U,

An equivalent way to write the previous statement is: for every w > 0, u ∈ U and b ∈ B feasible, then we have:

Therefore, we assume from this point on that B is the NINI investment set corresponding to a set of utility functions U, and that B is the class of investment sets contained in B.

We say that a set A ⊆ R K is comprehensive if, for every feasible b ′ and for every

which shows that for ε small enough, v(u, w, B, q k * α ε ) > 0 and therefore V (u, w, α ε , B) > 0.

Lemma A.6 Let u ∈ U 0 and assume that B is investment-prone. If u(0) = -∞, then there exist w and an investment-prone set B that is feasible at w such that V (u, w, α, B) = 0 for every always-uncertain α.

Proof. Since B is investment-prone, for every k ∈ K there exists b ′k such that

Let α be always uncertain and assume u(0) = -∞. For every s ∈ S α and for every b k ∈ B, the expected utility from investing in b k conditional on s is

Thus, for every s ∈ S α , v(u, w, B, q s α ) = u(w), which implies that:

V (u, w, α, B) = 0.

Lemmata A.5 and A.6 provide the proof of Proposition A.4.

belief-supported investment sets.

Proof. We need to show that the NINI investment set B coincides with the set B * of belief-supported assets.

For any b ∈ B * , and for any u ∈ U * and w such that b is feasible, u is concave and increasing. This implies: