Elina Thibeau-Sutre

Olivier Colliot

Didier Dormont

Ninon Burgos

Visualization approach to assess the robustness of neural networks for medical image classification

INTRODUCTION

Deep learning methods have shown a high performance potential for medical image analysis [START_REF] Litjens | A survey on deep learning in medical image analysis[END_REF], particularly classification for computer-aided diagnosis. However, explaining their decisions is not trivial and could be helpful to achieve better results and know how far they can be trusted. Many methods have been developed in order to explain the decisions of classifiers [START_REF] Simonyan | Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps[END_REF]- [START_REF] Ribeiro | Why Should I Trust You?": Explaining the Predictions of Any Classifier[END_REF], but their outputs are not always meaningful and remain difficult to interpret. In this paper, we adapted the method of [START_REF] Fong | Interpretable Explanations of Black Boxes by Meaningful Perturbation[END_REF] to 3D medical images to find on which basis a network classifies quantitative data. Indeed, quantitative data can be obtained from different medical imaging modalities, for example binding potential maps obtained with positron emission tomography (PET) or gray matter (GM) probability maps extracted from structural magnetic resonance imaging (MRI). Our application focuses on the detection of Alzheimer's disease (AD), a neurodegenerative syndrome that induces GM atrophy. We used as inputs GM probability maps, a proxy for atrophy, extracted from T1-weighted (T1w) MRI. The process includes two distinct parts: first a convolutional neural network (CNN) is trained to classify AD from control subjects, then the weights of the network are fixed and a mask is trained to prevent the network from classifying correctly all the subjects it has correctly classified after training. The goals of this work are to assess whether the visualization method initially developed for natural images is suitable for 3D medical images and to exploit it to better understand the decisions taken by classification networks. This work is an original work and has not been submitted elsewhere.

MATERIALS AND METHODS

Data description and preprocessing

Data used in the preparation of this article were obtained from two public datasets: the Alzheimer's Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu) and the Australian Imaging, Biomarkers and Lifestyle (AIBL) study. We used the T1w MRI available in each of these studies. Two diagnosis groups were considered:

• CN: sessions of subjects who were cognitively normal (CN) at baseline and stayed stable during the follow-up;

• AD: sessions of subjects who were diagnosed as AD at baseline and stayed stable during the follow-up.

The population of ADNI and AIBL are described in Table 1.

Values are presented as mean ± SD [range]. M: male, F: female

Preprocessing of T1w MR images was performed with the Clinica software platform (www.clinica.run) [START_REF] Routier | Clinica: an open source software platform for reproducible clinical neuroscience studies[END_REF]. First the datasets were converted to the BIDS format, then the t1-volume preprocessing pipeline of Clinica was applied [START_REF] Samper-González | Reproducible evaluation of classification methods in Alzheimer's disease: Framework and application to MRI and PET data[END_REF]. This pipeline performs bias field correction, non-linear registration and tissue segmentation using the Unified Segmentation approach [START_REF] Ashburner | Unified segmentation[END_REF] available in SPM12. The GM maps in MNI space were retrieved for the image analysis.

CNN classification

The following sections describe the evaluation procedure, the hyperparameters selection and implementation details that are linked to the classification of AD vs CN subjects with CNNs. During training, the weights and biases of the network 𝑤 are optimized to maximize the score function 𝑓 # on a set of images 𝑋 as follows: 𝑤 * = argmax # 𝑓 # (𝑋).

Evaluation procedure

The

Hyperparameter selection

We performed a random search [START_REF] Bergstra | Random Search for Hyper-Parameter Optimization[END_REF] to select the architecture and optimization hyperparameters of our CNN. The hyperparameters explored for the architecture were the number of convolutional blocks, of filters in the first layer and of convolutional layers in a block, the dimension reduction strategy (by using a max pooling layer or by setting the stride of the last convolutional layer of the convolutional block to 2), the number of fully-connected layers and the dropout rate.

Other hyperparameters such as the learning rate, the weight decay, the batch size, the data preprocessing and the intensity normalization were also part of the search.

Only one experiment was performed per architecture tested using the first split of the CV due to the computational cost of the random search. The chosen architecture was the one that obtained the best balanced accuracy on the validation set. This architecture (displayed in Figure 1) is composed of 7 convolutional blocks followed by a dropout layer and a fully-connected layer. Each convolutional block (C1, C2 or C3) is made of 1 to 3 sub-blocks and a max pooling layer with a kernel size and a stride of 2. Each sub-block is composed of a convolutional layer with kernel size of 3, a batchnormalization layer and a leaky ReLU activation. The predicted label of the input image is the class with the highest output probability.

Figure 1: Architecture of the CNN classifier determined following to a random search procedure.

CNN training

The weights of the convolutional and fully connected layers were initialized as described in [START_REF] He | Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification[END_REF], which corresponds to the default initialization method in PyTorch. We applied the following early stopping strategy for all the classification experiments: the training procedure does not stop until the validation loss is continuously higher than the lowest validation loss for N epochs (N=5); otherwise, the training continues to the end of a pre-defined number of epochs (30). The training and validation loss were computed with the cross-entropy loss. For each experiment, the final model was the one that obtained the highest validation balanced accuracy during training. The balanced accuracy of the model was evaluated at the end of each epoch.

Visualization method

The proposed visualization method extends the framework of [START_REF] Fong | Interpretable Explanations of Black Boxes by Meaningful Perturbation[END_REF]. Once the classification network has been trained, its parameters are fixed to the best value found, denoted as 𝑤 * . The method consists in computing a mask that will overlay the most meaningful parts of an image to prevent the network from classifying it correctly. In the following, the goal is to mask AD images that were correctly classified by the CNN so that it systematically classifies them with the CN label. The mask 𝑚 is a 3D volume of the same size as the input image and hide parts of the image in a voxel-wise manner. In this application, each voxel 𝑢 of the input image 𝑋 will be masked by a constant value 𝜇 according to the value of the mask for this voxel. The mask values are included in [0, 1]. The masked input image 𝑋 2 3 at voxel 𝑢 is defined as:

𝑋 2 3 𝑢 = 𝑚 𝑢 𝑋 𝑢 + 1 -𝑚 𝑢 𝜇 ( 1 )
As AD patients suffer from gray matter atrophy, the goal of the masking method would be to artificially simulate gray matter restoration in a minimal number of brain regions to make them look like CN subjects. By setting 𝜇 = 1, the mask was trained to artificially increase the probability of GM for the minimum set of voxels which will lead to the maximum decrease of the performance of the CNN. The optimal mask 𝑚 * is the mask for which the following loss function is minimized:

𝑚 * = argmin 2 𝜆 : 1 -𝑚 ; < ; < + 𝜆 = ∇𝑚 𝑢 ; ? ; ? @ + 𝑓 # * 𝑋 2 3 ( 2 )
The first regularization term ensures that a minimum set of voxels is selected while the second ensures that the mask is smooth enough and is not made of scattered voxels.

Once the mask training is finished, values above 0.95 are set to 1. This ensures that the CNN is only perturbed by the zones identified by the mask, and not by the small gradients that can be found on all the surface of the mask.

Quality check procedure

As the visualization method is very sensitive to outliers when applied to a group of images, a quality check procedure was performed before mask optimization. Figure 2 displays images that passed or failed this procedure. This quality check includes two steps:

1. The GM maps were sorted in increasing order by their maximal value. Images with a maximal value lower than 0.95 were automatically rejected. 8 sessions were removed during this procedure. 2. One image was removed after training a group mask. During the training of the first group mask this session led to a significant increase of the loss. This image was removed as it suffered from defects (the eyes were segmented as gray matter).

Grid search on visualization hyperparameters

A grid search was performed to choose the set of hyperparameters linked to the computation of the mask: the coefficients for the regularization 𝜆 : , 𝜆 = , 𝛽 : , 𝛽 = in equation ( 2). The learning rate was arbitrarily fixed to 0.1. The grid search was only performed on the group level masking for AD label.

Group level masking

To find the regions most related to Alzheimer's disease according to the CNN, we computed a mask based on the subset of images of AD subjects that were in the training set and validation set and all correctly classified by the network. For AIBL, the subset of all AD sessions correctly classified by the network was used. We exploit the fact that a voxel-wise correspondence exists between the GM maps, thanks to the non-linear registration, to iteratively build a group mask: the mask is initialized with all its values set to 1 and is then updated each time with a different image. The subset of wellclassified AD of the validation set used for the CNN training was again used as a validation set for mask optimization. At the end of each epoch, the masking loss was evaluated on this set to save the best mask according to the validation masking loss. To assess the robustness of the CNN training, the masks obtained for different folds (i.e. different input images and initializations) and different runs of the same folds (i.e. same input images but different initializations) were compared by pairs. The mean value of all pairs gave the similarity between folds or runs of the same fold.

Session level masking

Masks were also produced at the session level based on a single image. To avoid the overfitting risk due to the use of only one image instead of a set of images as in the previous section, the regularization terms 𝜆 : and 𝜆 = were multiplied by 100.

No validation set was used for these experiments as the goal is precisely to overfit the individual pattern of one image instead of finding a general pattern that may correspond to a group of images. Session level experiments include a longitudinal and cross-sectional analysis. In the longitudinal analysis, all the sessions of one subject were compared by pairs and the mean value of these comparisons gave the intra-subject similarity for this subject. The mean intra-subject similarity is then the mean value of the intra-subject similarity of all AD subjects. For the cross-sectional analysis, the mean value of all pairwise comparisons of baseline sessions of all AD subjects gave the inter-subject similarity measure. These analyses are performed to assess the stability of the visualization method and provide a baseline value by using the inter-subject similarity for the different metrics.

Visualization method training

The mask was initialized with a matrix of the same size than the input images (121x145x121) full of ones. We applied a similar early stopping strategy than for the classification experiments: the training procedure for group level masking on ADNI does not stop until the relative difference between the validation loss and the lowest validation loss is superior to a tolerance of 0.05 for a patience of N epochs (N=5); otherwise, the training continues to the end of a pre-defined maximum number of epochs (150). For the group level masking on AIBL, the patience was increased to 25 and the maximum number of epochs to 300 as the number of AD subjects is smaller than for ADNI. For the session level masking, the patience was increased to 200 and the maximum number of epochs to 5,000, while the tolerance was decreased to 0.01. The loss corresponds to the argmin argument of Equation (2). For each experiment, the final mask was the one that obtained the lowest validation loss during training. The loss of the mask was evaluated at the end of each epoch.

Metrics of evaluation

The similarity between masks was evaluated in two ways. The output probabilities of the CNN for the true class (prob CNN ) for an input masked by two masks optimized in two different contexts (e.g. different runs for the group level masking, different sessions of the same subject for the session level masking) are used to establish a comparison based on the CNN perception of the input. A mean output probability close to 1 means that the first model is not perturbed by the mask optimized for the second model, meaning that the two models are dissimilar. A ROI-based similarity was also computed to assess the similarity of two masks according to the 120 regions-of-interest (ROIs) of the AAL2 atlas [START_REF] Rolls | Implementation of a new parcellation of the orbitofrontal cortex in the automated anatomical labeling atlas[END_REF]. For each ROI, 1 minus the sum of the values in the ROI is computed, resulting in a ROI-vector of size 120 for each mask. Each value in the ROI-vector represents the density of the mask in the associated ROI. The ROI-based similarity between two masks is then the cosine similarity of two ROI-vectors. A value close to 1 means that the densities of the masks are the same between the ROIs, a value close to 0 means that the locations of the masks have no intersection.

RESULTS

Once the architecture was chosen and the CNN was trained on all folds, the classification performance was evaluated on the independent test set to ensure the absence of overfitting. The validation balanced accuracies on the five folds were 0.95, 0.82, 0.96, 0.85 and 0.87, giving an average of 0.89. The test balanced accuracies on the five folds were 0.89, 0.87, 0.90, 0.86 and 0.87, giving an average of 0.88. Moreover, the balanced accuracies obtained on the independent test set AIBL were 0.85, 0.92, 0.91, 0.92 and 0.92, giving an average value of 0.90. We could thus conclude that the network was not overfitting and we could use it for the visualization task. First the hyperparameters 𝛽 : and 𝛽 = were chosen with fixed 𝜆 : = 0.0001 and 𝜆 = = 0.001. The choice of the values 𝛽 : = 0.1 and 𝛽 = = 1 was made based on visual inspection. We observe on Figure 3 that when 𝛽 : decreases, the minimal value of the mask decreases and this prevents from producing a mask with a large set of values close but different from 1. When 𝛽 = increases, the value of the second term becomes negligible before the first term of equation ( 2). This leads to a very scattered mask as it is dominated by the first term of the regularization.

Grid search on visualization hyperparameters

The hyperparameters 𝜆 : and 𝜆 = were then chosen with fixed 𝛽 : = 0.1 and 𝛽 = = 1. The choice of the values 𝜆 : = 0.0001 and 𝜆 = = 0.01 was made based on visual inspection and the stability of the loss during mask training. We observe on Figure 4 that when 𝜆 : increases, the surface covered by the mask decreases until it only becomes scattered points. When 𝜆 = increases, the surface covered by the mask increases.

Robustness of the visualization method

Different experiments were conducted to assess whether the method was robust enough to help interpret the results of the CNN. Indeed [START_REF] Adebayo | Sanity checks for saliency maps[END_REF] highlighted that some visualization methods developed to interpret the results of neural networks (for example guided back-propagation and guided grad-CAM) did not depend on model parameters, as they gave the same result with a pretrained network or a randomized one. Hence we need to check if the visualization method gives coherent results based on the CNN training. Note that for 𝜆 : = 0.0001 and 𝜆 = = 0.1, the learning rate was fixed to 0.01 as the mask optimization did not converge with a learning rate of 0.1.

Group level masking

This experiment aims to assess the coherence of the proposed visualization approach with the a priori knowledge of the disease. One mask was optimized for each of the five models trained on the five folds of the cross-validation. Though these masks do not always overlap, they focus on a set of ROIs known to be particularly affected during AD progression. To confirm this visual observation, the list of the 5 ROIs in which the mask has the lowest values was extracted for each fold. All masks include in this list at least one hippocampus and parahippocampal gyrus. Moreover, the fusiform gyri (4 masks out of 5) and the amygdalae (3 masks out of 5) are frequently highlighted by the masks. Other regions such as the putamen, the pallidum, the inferior temporal gyrus and the thalamus appear only once in these lists.

Moreover, to assess the robustness of the method towards data used for mask optimization we compared the masks obtained by applying the visualization method on ADNI or AIBL data using the five networks trained on the five folds of the CV on ADNI training/validation set. The corresponding masks are displayed on Figure 5. The ROI-based similarities between the pairs of masks were 0.92, 0.99, 0.93, 0.89 and 0.97. These are comparable to the intra-subject ROI-based similarity (0.94). The prob CNN dissimilarities were very small as all the dissimilarities were smaller than 10 CD . This indicates that for a given pretrained network, a mask optimized for images of ADNI (resp. AIBL) correctly occludes the images of AIBL (resp. ADNI). However, the comparison of the masks in this way may not be completely fair. Even though the number of epochs and the patience of the early stopping procedure were increased for AIBL masking, the masks on ADNI and AIBL did not benefit from the same number of iterations. This factor leads to masks that comprise a different number of points, as the effect of the regularization terms is correlated to the number of iterations. This means that though the masks highlight the same locations in the brain, the difference in regularization makes the masks more dissimilar that they would be if we could find an equivalence for the hyperparameters that control the number of epochs (patience, tolerance and maximum number of epochs). Hence the dissimilarity here may not be due to the difference of datasets, but to the different number of iterations done during the mask optimization.

Session level masking

The inter-subject similarity and dissimilarity were evaluated to 0.80 and 0.58 for the ROI-based and the prob CNN metrics respectively. The intra-subject similarity and dissimilarity were evaluated to 0.94 and 0.11 for the ROI-based and the prob CNN metrics respectively. The higher intra-subject similarity compared to the inter-subject similarity ensures that the visualization metric is robust as the same pattern is generated for different sessions of the same subject.

Similarity across hyperparameters

With the ROI-based similarity, we can assess whether the masks produced by varying one hyperparameter value are similar. To observe this similarity, we reused the same masks as those produced for the grid search (see Figure 3 and4).

First, the similarities using different 𝛽 : and 𝛽 = values were computed with fixed 𝜆 : = 0.0001 and 𝜆 = = 0.001 and a learning rate of 0.1. The similarities between masks produced with 𝛽 : values in {0.1, 0.5, 1, 2} and fixed 𝛽 = = 1 are given in Table 2.A. As expected when looking at the masks obtained in Figure 3, the masks are highly similar except for the value 𝛽 : = 2 for which the first regulation term became negligible in front of the second regulation term in equation [START_REF] Simonyan | Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps[END_REF]. It resulted in a very smooth mask which is dense in all regions of the brain as 97.5% of values are below 0.95. This explains why the ROI-based similarity is so low between this mask and the others, though the regions identified seem similar at visual inspection. Other masks have a high similarity (> 0.95 in all cases). The similarities between masks produced with 𝛽 = values in {1, 2, 3} and fixed 𝛽 = = 0.1 are given in Table 2.B. There is more variability for this hyperparameter, though the similarity between two consecutive values is still high (> 0.90). With the prob CNN metric using the validation set, the dissimilarity between folds and 5 runs of the same fold are equivalent with respectively 0.78 and 0.82. The similarities computed with the ROI-based metric are also equivalent with respectively 0.65 and 0.69 between folds and between runs of the same fold. This indicates that the impact of the distribution of the data between training and validation is minimal compared to the initialization of the CNN and the training process. Moreover, we observe that the dissimilarity between folds / runs of the same fold is higher that the intersubject dissimilarity obtained with session level masking. This could mean that the regions on which the CNN relies on to identify the diagnosis mainly depend on the initialization and the training process and that the CNN training is not robust towards the regions identified.

DISCUSSION

We extended a visualization method to 3D medical imaging data and used it to better understand the decisions made by a classification network.

We first assessed the robustness of the proposed visualization approach. We showed that it gave coherent results as the regions identified by the mask are representative of AD [START_REF] Whitwell | 3D maps from multiple MRI illustrate changing atrophy patterns as subjects progress from mild cognitive impairment to Alzheimer's disease[END_REF]. This coherence is also confirmed by the fact that the intra-subject similarity is higher than the inter-subject similarity. Moreover, the high similarity across neighbouring values of the hyperparameters of the masking method indicates that this method is stable towards hyperparameter selection. Finally, we assessed that the method appears robust towards the data used for the construction of group masks by comparing masks computed using ADNI and AIBL datasets.

We then applied the visualization approach to assess the robustness of the CNN training. We demonstrated that even if the classification performances on the test set are very similar between the different folds, the training of the CNN for our application is not robust as the inter-subject similarity for one training is higher than the similarity between two retrainings or two folds of the network. This problem of robustness in CNN training may exists for many medical applications in which the number of samples is not sufficient for the network to learn stable meaningful features. This means that it may not be possible to study individual variations using visualization methods on deep learning applied to imaging data. This problem might be resolved by using more samples and with a better initialization, given for example by an autoencoder pretraining. Moreover, we found that the regions identified by the networks were very small (restricted to the hippocampus, amygdala and part of the temporal lobe) and that most of the image is not exploited by the CNN to find the diagnosis. This confirms the findings of [START_REF] Wen | Convolutional Neural Networks for Classification of Alzheimer's Disease: Overview and Reproducible Evaluation[END_REF] in which an equal performance was found by learning AD vs CN classification with a CNN on the whole MRI or the hippocampus only. This focus of the CNN on the hippocampi only may be partly due to the dataset: ADNI is a research cohort from which patients with multiple phenotypes are excluded, leading to a very homogeneous cohort in which the main symptom of the patients is memory loss. It does not fully represent the diversity of AD phenotype, and they may be biased towards hippocampus atrophy.

The visualization method we used has several limitations. First, the quality check of the data is crucial otherwise the training of the group level mask is not stable. Second, our method is only meaningful for quantitative data: for T1-MRI it would not have been sensible to increase the value of the voxels as it would have deformed the image in a non-meaningful way. This is an issue as the advantage of deep learning is precisely to be able to adapt to the rawest data possible. Finally, though we explored the effect of four hyperparameters of the visualization method (𝛽 : , 𝛽 = 𝜆 : , 𝜆 = ) we did not conduct an exhaustive study on the impact of the learning rate and the number of epochs performed (correlated to the patience, the tolerance and the maximum number of epochs). As we have seen when comparing masks trained on ADNI and AIBL, these parameters impact the amount of regularization of the masks. Future work will consist of developing a mask for non-quantitative images by creating another version of the input that is sensible. For example, for AD classification with T1-MRI, we would need the healthy version of an AD subject to mask the image with the healthy version. This kind of transformation from AD to normality has been explored in [START_REF] Bowles | Modelling the progression of Alzheimer's disease in MRI using generative adversarial networks[END_REF]. Moreover, we would like to study whether the visualization method is able to differentiate between overfitting, random and correctly trained networks. Finally, we will explore how masks can be used to enhance the training performance, for example by following an adversarial strategy to force the network to base its decision on more various regions.

Figure 2 :

 2 Figure 2: A. Example of image that passed the quality check. B. Example of image removed during the first step of the quality check. C. Image removed during the second step of the quality check.

Figure 3 :

 3 Figure 3: Comparison of masks obtained for different values of the visualization hyperparameters 𝛽 : and 𝛽 = .

Figure 4 :

 4 Figure 4: Comparison of masks obtained for different values of the visualization hyperparameters 𝜆 : and 𝜆 = .Note that for 𝜆 : = 0.0001 and 𝜆 = = 0.1, the learning rate was fixed to 0.01 as the mask optimization did not converge with a learning rate of 0.1.

Figure 5 :

 5 Figure 5: Coronal view of the group masks trained on ADNI (first line) and AIBL (second line). Each column corresponds to a model trained on one fold of the CV on training/validation ADNI set.

Figure 6 :

 6 Figure 6: Coronal view of the group masks obtained for the five folds of the CV on the first run (first line) and of the group masks obtained for five runs of the first fold (second line)

Table 1 .

 1 Summary of ADNI and AIBL participant demographics, mini-mental state examination (MMSE) and global clinical dementia rating (CDR) scores at baseline.

	Dataset Label Subjects Sessions Age	Gender	MMSE	CDR
		CN	330	1 830	74.4±5.8 [59.8, 89.6] 160 M / 170 F 29.1±1.1 [24, 30] 0: 330
	ADNI	AD	336	1 106	75.0±7.8 [55.1, 90.9] 185 M / 151 F 23.2±2.1 [18, 27] 0.5: 160; 1: 175; 2: 1
		CN	429	730	72.5±6.2 [60, 92]	183 M / 246 F 28.8±1.2 [25, 30] 0: 406, 0.5: 22, 1: 1
	AIBL	AD	76	108	73.9±8.0 [55, 93]	33 M / 43 F	20.6±5.5 [6, 29]	0.5: 31; 1: 36; 2: 7, 3: 2

  ADNI dataset was split into training/validation and test sets. The ADNI test set consisted of 100 randomly chosen age-and sex-matched subjects for each diagnostic class (i.e. 100 CN subjects, 100 AD patients). The rest of the ADNI dataset was used as training/validation set. We ensured that age and sex distributions between training/validation and test sets were not significantly different. The model selection procedure, including model architecture selection and training hyperparameter fine-tuning, was performed using only the training/validation dataset. For that purpose, a 5-fold crossvalidation (CV) was performed, which resulted in one fold (20%) of the data for validation and the rest for training. Note that the 5-fold data split was performed only once for all the experiments with a fixed seed number (random_state = 2), thus guaranteeing that all the experiments used exactly the same subjects during CV. The AIBL dataset was used as an independent test set to assess the CNN generalization ability. Test and validation sets included only one session per subject.

ACKNOWLEDGEMENTS

The research leading to these results has received funding from the French government under management of Agence Nationale de la Recherche as part of the "Investissements d'avenir" program, reference ANR-19-P3IA-0001 (PRAIRIE 3IA Institute) and reference ANR-10-IAIHU-06 (Agence Nationale de la Recherche-10-IA Institut Hospitalo-Universitaire-6).

All experiments were performed on the cluster of the Brain and Spine Institute in Paris, which is equipped with 4 NVIDIA P100 GPU cards (64 GB shared memory) and 24 CPUs (120 GB shared memory).

The similarities using different 𝜆 : and 𝜆 = values were then computed with fixed 𝛽 : = 0.1 and 𝛽 = = 1 and a learning rate of 0.1. For both hyperparameters the similarity is high between two consecutive values (>0.90), as can be seen in Table 3. These results highlight the stability of the method toward the hyperparameters choice, as two consecutive hyperparameter values led to two masks with a ROI-based similarity superior to the inter-subject similarity (0.80). Moreover, all the masks involved in this section analysis correctly occlude the CNN perception: for all masks, the mean output probability of the AD class on the validation dataset is below 10 CH .

Robustness of the CNN training

After having assessed the robustness of the visualization method, we applied it to better understand the factors influencing the training process of the CNN classifier based on several scenarios: for different folds (different initialization, different training/validation split) and different runs (different initialization, same training/validation split). Figure 6 displays the masks obtained for the five folds and five runs of the first fold.


Nada Belhadj
email: nadabelhadj@gmail.com

Didier Laussel
email: didier.laussel@univ-amu.fr

Joana Resende

Marketplace or reselling? A signalling model

Keywords: JEL classification: D4, L1 Marketplace, Reselling, Asymmetric information, Platform, Demand uncertainty

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In the last years, many economic sectors have experienced considerable changes. While in some cases there has been a significant disintermediation process, in other cases new intermediation players have emerged. Digital platforms are a good example of this last phenomenon (e.g. Amazon, Ebay or Alibaba). These platforms have adopted different types of business models: some of them act as resellers (retailers), whereas others are pure marketplaces, which bring together sellers and buyers (without making any sales decisions). Finally, in some other cases, digital platforms act both as a marketplace (for some products) and as a reseller (for other products). For example, [START_REF] Rysman | The economics of two-sided markets[END_REF] reports that the Internet giant Amazon sometimes acts as a conventional reseller (Amazon Retail), while on other occasions it offers a marketplace to independent buyers and sellers (Amazon Marketplace). For some particular jresende@fep.up.pt (J. Resende).

products (e.g. specific books), Amazon basically buys the product at a wholesale price and then resells it. However, for many other products,1 Amazon adopts a two-sided business model, acting as a platform which connects buyers and sellers, with sellers choosing both the sales level and the price of the product. [START_REF] Jiang | Firm strategies in the 'mid tail' of platform based retailing[END_REF] provided significant empirical evidence that platforms like Amazon tend to resell high-demand products while letting third-party independent sellers sell low-demand goods (with the platform opting for the marketplace mode). For example, in the category of electronics, they noticed that Amazon was reselling only 7% of the products but that those products included 64% of the 100 best sellers. In the categories of Tools and Home improvement, Sports and Outdoors, Jewelry, Toys and Games and Shoes the respective figures were 5.8% and 88%, 3.1% and 76%, 3.2% and 34%, 5.9% and 66%, 16.7% and 72%. 2 These findings have been confirmed by [START_REF] Zhu | Competing with complementors: an emprical look at amazon[END_REF] who concluded from a large scale empirical study using data from Amazon.com that "the likelihood of Amazon's entry is positively correlated with the popularity and customer ratings of third-party sellers' products".

The present paper proposes a very stylized model to examine whether asymmetric information on demand may constitute one possible determinant explaining why the platform may prefer to resell a high-value good, while acting as a marketplace for lowvalue goods.

To this end, we analyze the signalling game played between an intermediary platform and a representative seller (thereafter "she"), who may be interested in using the platform to distribute its product. More precisely, we assume that the seller may use (either separately or together) two distribution channels3 to reach its final consumers: (i) its own direct sales channel, such as a brick and mortar shop or an online website; (ii) an independent platform. One may find plethoric evidence of the combination of these business practices. For example, HP computers are for sale on HP online store as well as on Amazon and other platforms, Nike shoes may be purchased on Nike official site as well as on Amazon, Dior perfumes are available on Dior's official site as well as on many intermediaries sites including Amazon.

In our set-up, the platform has private information about demand. This modelling option intends to account (in a very simple way) for the digital platforms' ability to collect, gather and process enormous volumes of data4 on their customers (e.g. demographic data, data on consumers' online behavior, or data on consumers' previous online searches and purchases). In this respect, [START_REF] Petro | Amazon's acquisition of whole foods is about two things: data and product[END_REF] actually argues that "Amazon has a better understanding of the customer than any other retailer. The Motley Fool estimates that over 80 million people are Amazon Prime members. With this data, it is capable of building analytic models which can predict what these consumers will want, how much they will want, and when they will want it." In a completely different set-up, [START_REF] Heiko | De-targeting: advertising an assortment of products to loss-averse consumers[END_REF] find that an advertising agency or an Internet portal can make use of a tracking technology which allows it to perfectly identify a consumer's preferred product category, giving support to our assumption that the platform may be better informed about demand than the sellers.

The strategic interaction between the representative seller and the privately informed platform is here examined within a sequential game. In the first stage of this game, the platform first (privately) learns the true value of the good for consumers and then proposes a trading contract to the seller. This contract may be either a marketplace contract (in which all the commercial decisions are transferred to the seller) or a reselling contract (in which the platform itself defines the platform's sales level). In the second stage, after revising her beliefs about the consumers' tastes using the Bayes' Rule, the seller decides to accept or reject the platform's proposition. Finally, in the last stage, the seller takes her sales decisions: in the case of a marketplace contract, she chooses both the level of direct sales and the level of sales intermediated by the platform, whereas in the case of a reselling contract, she only chooses the level of direct sales. In this context, the choice of the platform sales is a "transferable action"5 : it is made by the platform under the reselling mode and by the seller under the marketplace mode. Differently, the choice of the direct sales level is a "non-transferable action": it is determined by the seller in the reselling mode as well as in the marketplace one.

We find that asymmetric information (with the platform being better informed than the seller about final demand) may indeed lead the platform to prefer to resell a high-value good rather than acting as a pure marketplace. More precisely, we find that there is a separating equilibrium in which, all other things equal, the platform is willing to act as a reseller for high demand products but not for low demand products. This separating equilibrium is more likely to occur when: (i) the consumers' willingness to pay for the high-value good is sufficiently larger than their willingness to pay for a low demand product (which opens the door to potentially large information rents); (ii) the negative spillover between the two vending sites (due to some degree of substitution between them) is not too large.

In the literature, it is well established that the marketplace mode allows for the internalization of the spillovers between two vending sites since this business mode transfers to the seller the sales' choice in both channels (direct sales and platform sales), thereby eliminating any double marginalization issues (Tirole, 1988). For this very reason, under complete information, we find that, as expected, the marketplace business model generates, at the same time, greater aggregate profits and greater platform's profits (in comparison with the reselling business model).

When the platform has private information on final demand, the business model choice is no longer exclusively determined by the need to eliminate double marginalization issues. Indeed, the platform shall also consider the possibility of obtaining informational rents (everything else the same). More precisely, the platform faces a trade-off between: (i) letting the informed party (the platform) adjust its own sales level to the state of demand (by proposing a reselling contract), at the cost of not internalizing the externalities between direct sales and platform sales, or (ii) letting the uninformed party (the seller) choose the level of sales in the two sites (by offering a marketplace contract) at the cost of not adapting them to the true state of demand.

We find that the platform will never be interested in reselling a low demand product. In this case, there is not much room for informational rent and therefore it prefers to offer a marketplace contract. Now, what about a high-value good? In the context of our model, the platform may prefer to offer a reselling contract, depending on two parameters: (i) the degree of substitutability between the direct vending site and the platform's vending site and (ii) the ratio between the consumers' willingness to pay for a high-value good and their willingness to pay for a low-value good. When the former is not too important and the latter is large enough, the only Perfect Bayesian Equilibrium satisfying the [START_REF] Cho | Signaling games and stable equilibria[END_REF] Intuitive Criterion is a separating equilibrium in which the platform always resells a high quality good. The rational behind this result is the following: the strategic cost of reselling under these circumstances is small (as the spillover between the two vending sites is limited) while the informational benefit is important. When the sites are strong substitutes and/or there is a small gap in the consumers' willingness to pay for a high-value versus a low-value good, the only Perfect Bayesian Equilibrium is a pooling equilibrium where the platform always acts as marketplace for both types of goods (high-value and low-value goods). Finally, for intermediate parameter values, there exist three Perfect Bayesian Equilibrium: a separating one, a pooling one and an hybrid equilibrium where the platform resells the high value good with a certain probability, which is strictly positive and lower than 1.

Related literature. A growing number of papers has investigated the factors explaining why platforms like Amazon act sometimes as resellers and sometimes as a pure marketplace. As far as we know, the only paper which considers the role of interactions between direct sales and platform sales is [START_REF] Abhishek | To platform-sell or resell? Channel structures in electronic retailing[END_REF] . Under complete information, they analyze the platform's choice between the reselling and the marketplace modes. They propose a model in which a good is distributed both through a traditional channel (brick and mortar retailing) and an electronic channel (platform). Under the reselling mode,6 the retailer buys the product at a fixed wholesale price and sets the retail price. Under the marketplace mode, the seller is the one choosing the retail price and the platform retains a fixed fraction of her revenues. The basic result is that the reselling mode is preferred to the marketplace mode when electronic sales stimulate traditional sales. Otherwise, the authors find that the platform prefers to work as a marketplace between buyers and sellers. [START_REF] Jiang | Firm strategies in the 'mid tail' of platform based retailing[END_REF] try, as we do here, to explain why retail platforms, like Amazon, resell high-demand products while letting third-party independent sellers sell low-demand products (marketplace mode). They also build a signalling model. However, in their two-period model, it is the seller who has initially private information about demand (high or low). In the first-period, the product is sold directly by the seller, who pays a fixed unit fee to the platform. However, the latter revises its priors on the seller's type from observing the price-quantity pair selected by the independent seller. Anticipating this possibility, the seller may try to hide her type by reducing the level of her first-period services to consumers. Based on its observation, the platform chooses the reselling mode or sticks to the marketplace one in the second period. The authors show that their model has a separating equilibrium (in which the platform identifies the independent seller's type and resells the high-demand good) when the ex-ante probability of high demand is above a given threshold. The game has a pooling equilibrium (in which the seller's type is not identified and the platform chooses the marketplace mode) when such probability is below this critical value.

Here, we look at a different case in which the platform is actually better informed than the sellers. Hence, we are looking at a different problem, trying to account for the evidence that Internet giants, such as Amazon, may have "a better understanding of the customer than any other retailer" [START_REF] Petro | Amazon's acquisition of whole foods is about two things: data and product[END_REF].

Interesting, but more loosely related works, are the papers by [START_REF] Hagiu | Merchant or two-sided platform[END_REF] ; [START_REF] Hagiu | Do you really want to be an Ebay?[END_REF], 2015a, 2015b) ; Hagiu (2007) 's paper is "a first pass" to compare the reselling and the "two-sided platform" modes. It is mostly focused on the analysis of "chicken and egg" problems for two-sided platforms (the more severe this problem is, the more often the reselling mode will be selected by the platform). [START_REF] Hagiu | Merchant or two-sided platform[END_REF] also surveys a lot of other factors affecting the platforms' business modus operandi (such as the complementarity or substitutability between sellers' products that are not internalized in the prices, indirect network effects between buyers and sellers, as well as asymmetric information between sellers and the platform). However, this work "rulesout direct sales by sellers to consumers",7 which constitute a key element in our set-up.

The main interest of Hagiu and Wright (2015a) is to model the choice between the reseller and the marketplace modes "as a decision between whether control rights over a non-contractible decision variable (the choice of some marketing activity) are better held by suppliers (in the marketplace mode) or by the intermediary (in the reseller mode)". This is similar to our paper in which, however, the non-contractible decision variable is the level of direct seller's sales (instead of a marketing activity). In their approach, the choice leans toward the reselling mode (resp. the marketplace mode) when the platform (resp. the seller) is better in-formed about "the optimal tailoring of marketing activities ". As in [START_REF] Hagiu | Merchant or two-sided platform[END_REF] , network effects between sellers and unfavorable expectations are shown to tilt the choice toward the reselling mode.

Hagiu and Wright (2015b) generalize the previous paper by analyzing the factors which favor the adoption of the multi-sided platform (MSP) mode versus vertical integration (VI) or the reseller mode. They focus on the first trade-off (MSP vs VI) while referring to Hagiu and Wright (2015a) for the second one.

Even, more loosely related are the works of [START_REF] Balasubramanian | Mail versus mall: a strategic analysis of competition between direct marketers and conventional retailers[END_REF] or [START_REF] Yoo | Internet channel entry: a strategic analysis of mixed channel structures[END_REF] . Those authors deal with the introduction of an Internet distribution channel in a market where there already existed a traditional physical channel (brick-and-mortar). They highlight the important role played by the different features of the Internet channel vis-á-vis the traditional vending site. In particular, the authors are more interested in understanding the impact of introducing the new online channel rather than looking at the platforms' business choice.

Finally, notice also a paper by [START_REF] Foros | Apple's agency model and the role of most-favored-nation clauses[END_REF] who study the marketplace model adopted by Apple and Google. 8 They set up a single period model of perfect and symmetric information with imperfect competition both at the upstream (sellers) and the downstream (platform) level, investigating the optimal pricing strategies for the platforms and the sellers. They prove that "the agency9 model may not be universally adopted even if adoption would mean higher profits for all firms."

The rest of the paper is organized as follows. In Section 2 , we set up the model. In Section 3 , we study the full information benchmark. We examine the Perfect Bayesian Equilibria of our game in Section 4 , providing an analytical characterization for the Least Cost Separating Equilibrium, in which the platform chooses to resell o nly high-value goods, preferring to work as a marketplace for low-demand goods. Section 5 concludes the paper.

The model

A good is produced by a given seller, who may sell the good directly to consumers and/or via a platform. The latter may act as a reseller who buys the product from the seller and resells it to final consumers (i.e. a pure reseller) or as a simple marketplace. In the former case, the platform fixes the quantities it sells itself while the seller determines the level of her direct sales, after observing the platform's sales decisions (which are known to the seller as we assume that the platform sells all the units of the good it buys from the seller). In the marketplace case, the seller is in charge of choosing her sales, both in the direct vending site and in the marketplace (i.e. the sales mediated by the platform are decided by the seller). For the sake of simplicity, and without loss of generality, we normalize unit production and distribution costs to zero.

In each distribution channel, the following inverse demand functions are assumed:

p = θ -y -δx, (1) v = θ -x -δy,
where p denotes the price of the good when it is sold through the platform; v is the price of the good charged by the seller when she directly sells her good to the final consumers; θ is the consumers' maximum willingness to pay for the good (or, equivalently, a measure of the width of the potential market) and δ ∈ (0, 1) is the parameter that formally measures the degree of substitutability between direct sales x and platform sales y .

The demand specification is based on the underlying assumption that opening a second vending site allows to increase the aggregate sales of the good (measured at an identical price). However, the aggregate sales, when the second distribution channel is used, are never more than twice the sales at a single site.10 According to (1) , at identical selling prices, the ratio between aggregate sales when the two distribution sites are used and the sales when only one vending site is active is11 equal to 2 1+ δ , which, as expected, is decreasing in δ.

In our set-up, θ is assumed to be private information of the platform. The seller does not know the true value of θ . Her priors are such that θ = θ H (the product is a high-value one) with probability μ and θ = θ L (the product is a low-value one) with probability 1 -μ. We denote by k = θ H θ L the ratio between the consumers' maximum willingness to pay for a high-value good and the consumers' maximum willingness to pay for a low-value good. Hence, k provides information on the demand expansion when the product is a high-value rather than a low-value one. This parameter will play an important role in our analysis since it allows us to assess whether there is a large potential for information rents (which amounts to a large k -value).

The strategic interaction between the seller and the platform is examined within a sequential game with asymmetric information, whose timing is as follows:

1. The platform privately learns whether the good is a high-value good (θ = θ H ) or a low-value one (θ = θ L ) ; 2. The platform decides its business model strategy ( S ), proposing to the seller either a reselling contract (S = R ) or a marketplace contract ( S = M). A reselling contract specifies the level of platform sales y X and a transfer T X to the seller. In the case of a marketplace contract, the platform only defines the fixed fee F that it charges the seller in order to give her access to the platform (with all decisions on the level of platform sales being made by the seller, afterwards); 3. The seller observes the contract offered by the platform and revises her beliefs using Bayes' Rule. Then, she decides whether to accept or reject the platform's proposition. 4. If the seller has accepted a reselling proposition, given her expost beliefs about θ , she selects a level x of direct sales as a best reply to y X . If, instead, she has accepted a marketplace contract, at this stage, the seller simultaneously chooses the level of direct sales and the level of platform sales ( x and y , respectively) in order to maximize her expected profits, given her expost beliefs about θ .

In order to find the equilibrium of the model, we proceed by backwards induction, first solving the last stage of the game and then finding the solution of the previous stages.

The full information benchmark

Let us first analyze the full information benchmark. Given the structure of our game, we need to separately examine two different subgames, defined on the basis of the platforms' business model choice: the marketplace subgame; and the reselling subgame.

When the two sites are open, the aggregate direct demand is 2 1+ δ (θp) . Thus, opening a second site expands demand by a coefficient 2 1+ δ .

Marketplace

Under a marketplace contract, the seller, who knows the realized value of the good, θ X , X = H, L, chooses the level of sales at both sites (platform and direct sales) in order to maximize her profits: max

x,y [ x (θ X -x -δy ) + y (θ X -y -δx ) -F ] .
The equilibrium level of direct and platform sales are given by

y A = x A = θ X 2(1 + δ)
, and aggregate profits are:

θ 2 X 2(1 + δ) .
(2)

Hence, the platform will set the transfer F in such a way that, for the selected transfer F a , the seller's participation constraint is binding. Put otherwise, the seller's equilibrium profits should not be smaller than the level of profits obtained when she only uses its direct distribution channel to sell the good. Hence, to obtain the seller's stand-alone profit, we need to look for the solution of the seller's profit maximization problem w.r.t x , when only the direct sales channel is active. This yields to a level of equilibrium (direct) sales equal to θ X 2 , with the corresponding stand-alone profit being

θ 2 X 4 .
Hence,

F a = θ 2 X 2(1 + δ) - θ 2 X 4 = (1 -δ) 4(1 + δ) θ 2 X . (3)
Summing up, in a full information with a marketplace contract, the seller's profit is

θ 2 X 4
, whereas the platform's profit is equal to the fixed fee F a . Let us now look at the reselling subgame.

Reselling

In the reselling contract, the seller chooses the direct sales level x in order to maximize her profits max

x [ x (θ X -x -δy ) + T ] ,
given the level y of platform's sales and the fixed transfer T received from the platform.

Thus, when solving the seller's profit maximization problem, one obtains the following best reply function

x (θ X , y ) = max { θ X -δy 2 , 0 } ,
defining the optimal level of direct sales ( x ), as a function of the platform's sales levels (and the demand parameter θ X , which under full information is also known by the seller).

The platform acts like a Stackelberg leader, maximizing its profit max y [ y (θ Xy -δx (θ X , y )) -T ] subject to the seller's participation constraint. Again, this constraint simply requires that the seller's profit should at least equal to her stand-alone profit, θ 2 X 4 , which is obtained when the seller rejects the contract proposal12 , opting for not doing any sales through the platform, i.e;

( x (θ X , y ) ) 2 + T ≥ θ 2 X 4 . ( 4
)
Since the platform benefits from choosing T as low as possible, y is then set to maximize the following function (which already incorporates the seller's participation constraint).

y (θ X -y -δ θ X -δy 2 ) + ( x (θ X , y ) ) 2 - θ 2 X 4 .
Accordingly, in the reselling subgame, the platform's full information sales ( y f X ), transfer ( T f X ) and payoff ( f X ) are given by 13 :

y f X = 2(1 -δ) θ X 4 -3 δ 2 ,
(5)

T f X = δ(4 -δ + δ 2 ) θ 2 X (4 + δ 2 ) 2 , f X = θ 2 X (1 -δ) 2 4 -3 δ 2 .
In order to solve the full game, we now need to compare the platform's equilibrium profit under a marketplace contract and a reselling contract.

Comparison of the platform's strategies under full information

It is illuminating to compare the full information platform's profits when it selects the marketplace business model versus the reselling contract. The respective profits are given by (3) in the former case and by the last equation in ( 5) in the reselling one. The difference is

θ 2 X (1 -δ) 4(1 + δ) - θ 2 X (1 -δ) 2 4 -3 δ 2 = θ 2 X (1 -δ) δ 2 4(1 + δ)(4 -3 δ 2 ) . ( 6
)
This difference equals 0 when δ = 0 (no externality between the two vending sites) and when δ = 1 (sales at the two sites are perfect substitutes so that selling at the platform does not bring any additional profits, whatever the distribution mode). It is strictly positive for all other values of δ, showing that the marketplace business model gives higher profits to the platform (in comparison to the reselling business model). The marketplace model also necessarily yields greater aggregate profits than the reselling model, since in both cases the seller's profit is equal to θ 2 X 4 , which corresponds to the maximum profit direct sales only.

The rationale behind the superiority of the marketplace mode under full information is straightforward. In the context of a marketplace model, we have that the same agent, the seller, chooses the level of direct sales as well as the level of platform sales. Hence, when doing so, the seller internalizes all the externalities between the two vending sites. In contrast, in the reselling business strategy, the seller only maximizes her profits from direct sales, ignoring the external effects of the direct sales on the platform's profits. It's only when there are no externalities ( δ = 0 ) or when selling through the platform does not possibly increase aggregate profits ( δ = 1 ), that the two modes are equivalent.

We now turn our attention to the model with asymmetric information, which allows us to show that, all other things equal, the platform may no longer always prefer to propose a marketplace business model to the seller.

Perfect Bayesian equilibria

In a full information set-up, the possibility to internalize the externalities between the two sales channels always gives a strategic advantage to the marketplace model over the reselling contract. When, as we suppose here, the platform has private information about the true demand state (high or low), this may no longer 13 It is easy to check that x ( θ X , y X ) > 0.

be the case. In particular, the platform may actually prefer the reselling mode over the marketplace model to obtain greater informational rents. This is more likely to occur when the consumers' willingness to pay for the good in the high-demand scenario is sufficiently high (leading to potentially large informational rents). The platform's business model choice may signal the true demand state and therefore, we examine the strategic interaction between the platform and the seller within a signalling game framework.

We start by proving an important result, which shows that reselling a low-value good is always a dominated business strategy for the platform.

Proposition 1. In equilibrium, the platform never resells a low-value good.

Proof. If the platform were to resell the low-value good, its maximum possible profits would be equal to max

y y (θ L -y -δ θ e -δy 2 ) + ( x (θ e , y ) ) 2 - θ e 2 2 = (θ L -δθ e ) 2 4 -3 δ 2 ,
where θ e = ρθ H + (1 -ρ) θ L , ρ being the ex-post probability that the good is a high-value one (conditional on the platform's choosing a reselling contract, R ). Since a marketplace mode would, in the low-demand scenario, guarantee to the platform profits at least equal to

(1 -δ) θ 2 L 4(1+ δ) > (θ L -δθ e ) 2
4 -3 δ 2 , it follows that the platform will always prefer to act as a marketplace when the true state of θ is θ L .

From Proposition 1 follows that indeed the platform's business model choice may be a signal of the true demand state. In particular, if the platform chooses the reselling mode, the seller immediately concludes that the good is a high value ( θ = θ H ) one:

ρ(θ = θ H | S = R ) = 1 . (7)
Now, what about the probability that the consumers' willingness to pay for the good is high when the platform chooses the marketplace mode? Given (7) , when the H -type platform chooses R with probability λ ∈ [0, 1] and M with complementary probability, the Bayes' Rule implies that the ex-post probability that the product is a high value one conditional on the choice of a marketplace system must be

ρ(θ = θ H | S = M) = μ(1 -λ) μ(1 -λ) + (1 -μ) . ( 8
) Notice that, when λ = 1 , ρ(θ = θ H | S = M) = 0 and, when λ = 0 , ρ(θ = θ H | S = M) = μ. For any value λ ∈ [0 , 1] , ρ(θ = θ H | S = M) ∈ [0 , μ] .
It is well-known that there may be three possible types of equilibria of a signalling game: separating equilibria, pooling equilibria and hybrid equilibria. In the paper, we focus on the separating equilibria, in which the platform's business model actually depends on the true demand state: in the case of low-demand products, the platform always acts as a marketplace, whereas in the case of high-demand goods, the platform opts for a reselling strategy. As, will be shown below, the separating equilibria will emerge when k is large enough, meaning that the consumers' willingness to pay for a high-value good is sufficiently higher than their willingness to pay for a low-value good (i.e. the platform's informational rents are potentially high).

In what follows, we provide an analytical characterization of the separating equilibria. The analysis of the hybrid and the pooling equilibria is left to the Appendix.

Separating equilibria

In a separating equilibrium, the platform chooses different business trading strategies when the product is a high-value or a lowvalue one. The seller' beliefs are then revised accordingly, so that the seller is actually able to learn the true state of demand i.e. θ H or θ L .

From Proposition 1 , there cannot exist an equilibrium in which the platform would resell the low value good and distribute the high value one as a marketplace. Thus, the only possible separating equilibria are such that:

• Upon observing privately that the product is a low value one, the platform chooses to act as a marketplace, i.e. S(L ) = M, and asks F L from the seller;

• Upon observing privately that the product is a high value one, it chooses to act as a reseller, i.e. S(H) = R, or, equivalently, λ = 1 , selling quantities y H and paying T H to the seller.

From ( 8) , upon observing that the platform chooses a marketplace the seller infers that the product is low value one, i.e.

ρ(θ

= θ H | S = M) = 0 .
At equilibrium, the H -type platform should not benefit from pretending that the product is a low value one, leading to the following condition:

(θ H -y H -δ θ H -δy H 2 ) y H - θ 2 H 4 - θ H -δy H 2 2 ≥ F L = (1 -δ) θ 2 L 4(1 + δ) . ( 9
)
The LHS of ( 9) equals the platform's (direct) operating profits when selling a quantity y H minus the price T H it pays to its supplier to guarantee her a profit just equal to its stand-alone value. 14 The RHS is the platform's profit when it pretends that the product is a low-value one. It just equals to the maximum transfer which it can ask in that case from the seller, who, upon observing S = M, would anticipate that consumers' willingness to pay is low (since we are looking here at a separating equilibrium).

Notice that Proposition 1 guarantees that the L -type platform never benefits from choosing a business strategy R , which would amount to pretend that the product is a high-value one. Accordingly, the L -type platform's incentive constraint is automatically satisfied.

When there exist several separating equilibria, i.e. several values of y H which satisfy the incentive constraints (9) , 15 it is natural to focus on the ones which maximize the H -type platform's profits, i.e. the Least Cost Separating Equilibria (LCSE). Proposition 2. A Separating Equilibrium, in which the platform resells the good when θ = θ H and distributes it as a marketplace when

θ = θ L , exists iff k ≥ 1 G = 4 -3 δ 2 4(1 -δ 2 ) .
Proof. Since the type L-platform's incentive constraint is always satisfied and the participation constraint is satisfied by construction, there exists a separating equilibrium iff the type H-platform's incentive constraint (9) is satisfied. This is obviously the case iff the maximum of the LHS of (9) w.r.t y H , i.e.

(1 -δ) 2 θ 2 H 4 -3 δ 2 is not smaller than

(1 -δ) 2 θ 2 L 4(1 -δ 2 ) , i.e. iff k ≥ 4 -3 δ 2 4 1 -δ 2 .
(10) 14 T H is the minimum price which satisfies the supplier's participation constraint. 15 It is implicitly assumed that the participation constraints are satisfied.

Notice that the threshold 4 -3 δ 2 4 ( 1 -δ 2 ) in condition (10) corresponds to the square root of the ratio between marketplace and reselling platform's full information profits, for any θ . This threshold is greater than 1 for all δ = 0 and it is increasing in δ.

Condition (10) simply states that the ratio between the consumers' maximum willingness to pay under different realizations of the demand parameter ( k = θ H θ L

) must be greater than the square root of the ratio between the marketplace and the reselling platform's full information profits, for a given θ . The LHS ( k ) provides a measure of the platform's informational rents on good demand states, 16 whereas the RHS provides a measure of the strategic advantage of marketplace over reselling. 17Hence, the results of Proposition 2 are quite consistent with intuition. The existence of a (least-cost) separating equilibrium in which a low-value good is distributed via a marketplace while a high-value good is resold by the platform is more likely when k is large and when the externalities between the two sites, as represented by δ 2 , are small enough. Indeed, the smaller the externalities between the sites, the less important the strategic advantage of a marketplace mode (which allows the seller to internalize the externalities between the two vending sites) over a reselling contract (in which decisions on direct sales and platform sales are taken by different firms).

Corollary 1. In the LCSE, when θ = θ H , the platform: (i) optimally resells the quantity y Proof. The best separating equilibrium from the H-platform's point of view, i.e. the LCSE, is the one which maximizes its profits, with

{ y f H = 2(1 -δ) θ H 4 -3 δ 2 , T H = θ 2 H 4 - θ H -δy f H 2 2 } .
Corollary 1 provides a full description of the LCSE. It shows that, in this equilibrium, the level of platform sales with asymmetric information actually coincides with the level of platform sales in a full information scenario (with θ = θ H ). However, in this case, the platform takes advantage on its private information, to pay a lower fee to the seller, i.e., T H < T f X . The separating equilibria studied above are not the only equilibria in our signalling game. As shown in Appendix, the game also displays pooling and hybrid equilibria. In a pooling equilibrium, the platform's business model is not affected by the true realization of demand. From Proposition 1 , the only possible pooling equilibrium is such that S(H) = S(L ) = M and ρ(X = H| S = M) = μ. In this equilibrium, the platform chooses the marketplace system whatever the value of the good and therefore the seller does not update her beliefs upon observing the marketplace proposal.

In a hybrid equilibrium the L -type platform offers a marketplace contract, while an H -type platform randomizes, i.e, it chooses R with probability λ and M with complementary probability. According to the Bayes' Rule, the ex-post probability that the product is a high value one conditional on the choice of a marketplace system is then given by (8) . The next subsection analyses in more detail the equilibrium configuration in the ( δ, k )-space, leaving to the Appendix the analytical characterization of the pooling and the hybrid equilibria.

Equilibrium configuration in the ( δ, k )-space

The common point between the three possible types of equilibria is that the platform always prefers to use the marketplace mode to distribute a low value good. At a separating equilibrium, it always chooses the reselling mode to distribute a high value good. At an hybrid equilibrium, it randomizes between marketplace and reselling. At a pooling equilibrium, it distributes it as a marketplace.

The Fig. 1 below (which is drawn for μ = 0 . 845 ) pictures the equilibrium configuration in the ( δ, k ) -space. 18 There are three areas:

• S , where the unique equilibrium that meets [START_REF] Cho | Signaling games and stable equilibria[END_REF] Intuitive Criterion is the least cost separating equilibrium; • P , where the pooling equilibrium is the unique Perfect Bayesian Equilibrium (PBE for short); • SHP , where the equilibria exist: separating, pooling and hybrid.

As Fig. 1 shows, in the area S , the only PBE meeting the Intuitive Criterion is the (least cost) separating equilibrium, where the high value good is distributed by the platform under a reselling contract, whereas the low-value good is distributed within a marketplace system. This equilibrium occurs when (i) the externalities between the vending sites are weak, (ii) the ratio between the willingness to pay for the high-value and for the low-value goods is high and (iii) the ex-ante probability of a high-value good is small enough.

In contrast, in the area P , the only PBE meeting the Intuitive Criterion implies a pooling strategy, for which the platform prefers to distribute both types of goods as a marketplace. In this subdomain, the externalities between direct sales and platform sales are large ( δ is large) and/or the difference in consumers' willingness to pay for the two demand realizations is not too large (i.e. k is small enough). On the one hand, the strong externalities favor the marketplace system (which allows for internalization of those externalities by assuring that the same player takes decision s on the level of direct sales and the level of platform sales). On the other hand, the small difference between the goods' value limits the informational advantage of using a reselling strategy.

Finally, for the subdomain of parameters in the area SHP , three possible PBE exist: pooling, separating and hybrid. In this area, both externalities between the two vending sites ( δ) and the ratio between θ H and θ L are relatively mild. The intuition behind this equilibrium multiplicity lies on the existence of a self-reinforcing mechanism which makes the hybrid equilibrium look like an un-18 Remember that δ is the degree of substitutability between the two sites and k the ratio between the willingness to pay for the high and the low-value good.

stable equilibrium and the separating as well as the pooling equilibria look like (locally) stable ones. Indeed, when the frequency with which the platform chooses to resell the H -good ( λ) increases, the Bayes' Rule (8) leads to a reduction in the ex-post probability that θ = θ H , conditional on the platform having chosen a marketplace mode. This way, the platform's marketplace profits,19 go down, thus increasing the relative profitability of Reselling versus Marketplace. On the contrary, any reduction in λ leads to a reduction of the relative profitability of a marketplace contract vis-à-vis a reselling contract.

Conclusion

In this paper, we investigate optimal managerial strategies in markets where product sellers may distribute their goods using two distinct channels: direct vending sites and intermediation platforms, which are not necessarily perfect substitute channels. We assume that, contrary to the level of direct sales, the level of the platform's sales may be a transferable decision of the seller to the platform, who has private information on the true demand realization.

We propose to analyze the strategic interaction between the seller and the platform within a sequential signalling game. Everything else the same, this stylized model allows us to examine how the platform's optimal business choice (marketplace mode versus reselling mode) may be affected by the market features such as the degree of substitutability between direct sales and platform sales; or the consumers' additional willingness to pay for a high-value good (vis-à-vis their willingness to pay for a low-demand good).

While it is clearly the case that, under full information, the marketplace mode leads to the highest-profits for all the market players (the marketplace mode allows for the internalization of all the profit effects originated by the intermediated sales), in an imperfect information set-up, that may not necessarily be the case.

Accordingly, our results provide one possible explanation (not excluding others) for the observation that the platform tends to resell more high-value goods than low-value ones. This finding is consistent with previous empirical evidence pointed by [START_REF] Jiang | Firm strategies in the 'mid tail' of platform based retailing[END_REF] on the distribution strategy of important digital platforms such as Amazon. It is also aligned with the spirit of the Chief Executive Jeff Bezos's bons mots"Your margin is my opportunity". 20Given that the platform never benefits from reselling a lowvalue good, the adoption of the reselling mode signals that the good is a high-value one. A separating equilibrium, in which the platform always resells the high-value good, occurs when the two vending sites are not strong substitutes and/or when the highvalue good is substantially more valued than the low-value one. A pooling equilibrium, in which the platform always acts as a marketplace, arises in the opposite case. Finally, an area of multiple equilibria (separating, pooling and hybrid) exists for intermediate parameters values.

In the future, several interesting extensions of this model may be analyzed. In particular, it may be interesting to extend the present approach to account for competition between platforms and for the existence of several sellers, incorporating network effects. Another interesting extension would be to look at the platform's incentives to act in the upstream market as well (as a producer), in line with the recent developments in the strategy of firms like Amazon. Indeed, this platform has recently bought "Whole Foods", it is already involved in media content production (e.g. the movie "Manchester by the Sea") and it also produces several private brand products.

  -δ) θ H 4 -3 δ 2 ; and (ii) it pays the transfer T H =

Fig. 1 .

 1 Fig. 1. Equilibrium configuration in the ( δ, k )-space.

Many of these products may have low individual sales but, collectively, if the distribution channel is large enough, they achieve a market share which exceeds the relatively few best sellers. This corresponds to the "long tail" of Internet sales, according to the term coined by[START_REF] Anderson | The Long Tail why the Future of Business is Selling Less of More[END_REF] .

Table 1, page 758 in Jiang et al. (2011) .

This modelling option is aligned with the concept of "augmented retail" (corresponding to a blend of digital and physical sale channels), which is becoming more and more prevalent. A number of papers[START_REF] Brynjolfsson | Battle of the retail channels: how product selection and geography drive cross-channel competition[END_REF][START_REF] Goolsbee | Competition in the computer industry: online versus retail[END_REF][START_REF] Smith | Piracy or promotion? The impact of broadband internet penetration on DVD sales[END_REF][START_REF] Hilton | The short-term influence of free digital versions of books on print sales[END_REF] investigated the impact of electronic sales on traditional brick-and-mortar sales which is a related but different question (since here we are more interested in looking at the business model of the platform).

 4 According to[START_REF] Marr | Want to use big data? Why not start via google, facebook, amazon[END_REF] , "Amazon compares products we browse and buy with millions of other customers around the world. By building a profile of our habits, it is able to match us with products and recommendations from others, which will most likely fit our needs. The Big Data technology at work here is known as a recommendation engine and Amazon's was one of the first, and most

sophisticated".5 We use here the distinction introduced by Hagiu and Wright (2015b) .

It is assumed that the traditional channel always uses it.

p.117

For example, the retail price of the popular game Angry Birds is controlled by its inventor, Rovio Entertainment, and e-book retail prices are determined by the publishers. Apple and Google keep 30% of the revenue created when a sale is made through its platform.

The agency model mentioned here refers to what we call "Marketplace Model" in the present paper.

The basic assumption is that, at any given price, the aggregate sales of the two vending sites (when they are both active) are greater than the sum of their sales when each one of them is the only distribution channel. For the sake of simplicity, we suppose that the two sites are symmetrical. A generalization is available from the authors upon demand.

Let p = v . If only one site is open, the direct demand for the good is θp.

In which case y = 0 .

The informational advantage of the platform under reselling is the greater, the higher is the ratio between the consumers' willingness to pay for the high and the low-value good: it is precisely measured by the ratio θH θL = k .

This advantage is precisely measured by the ratio between the full information profits of the two business strategies, 1 G .

As given by expressions (11) and (12) in the Appendix

Citation from an article in the Wall Street Journal, written by[START_REF] Mims | The limits of amazon[END_REF] .

R This article is a result of the project NORTE-01-0145-FEDER-028540, supported by Norte Portugal Regional Operational Programme (NORTE 2020), under the POR-TUGAL 2020 Partnership Agreement, through the European Regional Development Fund (ERDF) and through national funds by the FCT -Fundação para a Ciência e a Tecnologia. This research also benefited from the support of COMPETE 2020 (POCI) and FCT through the project POCI-01-0145-FEDER-006890.

Appendix A. Pooling equilibria

In a pooling equilibria, the platform chooses the same strategy regardless of the state of demand. Proposition 1 allows us to pin down out-of-equilibrium beliefs in order to evaluate possible deviations. Indeed, as Proposition 1 establishes that the choice of a reselling strategy when the good is a low value one is always a dominated strategy, we can naturally suppose that the seller, upon observing a deviation towards the reselling mode, must conclude that the good is a high value one. The pooling equilibrium then satisfies the [START_REF] Cho | Signaling games and stable equilibria[END_REF] "Intuitive Criterion" iff the Htype platform does not benefit from choosing the reselling mode (thus revealing its type to the seller).

Lemma 1. A pooling equilibrium, in which the platform always chooses the marketplace mode satisfies [START_REF] Cho | Signaling games and stable equilibria[END_REF] ) "Intuitive Criterion" if either:

Proof. The Intuitive Criterion requires that there should be no feasible deviation from the marketplace equilibrium yielding greater profits to the H -type platform, when we account for the fact that the platform is automatically recognized as a H -type platform when it deviates from marketplace to reselling, since S = R is a dominated strategy for a L -type platform as a result of Proposition 1 and expression ( 7) .

The H -type platform's profits, when recognized as such upon deviating, are equal to its full information profits. They should not be greater than its profits at the pooling equilibrium, i.e.

(1

, where θ = μθ H + (1 -μ) θ L .

This condition holds always true

4 -3 δ 2 . When μ < G , the inequality above only holds for 1

G , the Intuitive Criterion eliminates all pooling equilibria which do not satisfy these conditions.

Appendix B. Hybrid equilibria

Lemma 2. There exists an hybrid equilibrium when either

In this hybrid equilibrium:

, where ρ is given by ( 14), the platform opts for a reselling strategy, setting in that case y

4 -3 δ 2 ; with the complementary probability, the platform opts for the marketplace business model, (ii) When the platform privately knows that the good is a low value one, it always prefers to act as a marketplace.

Proof. When θ = θ H and the platform chooses to act as a reseller, it obtains profits equal to

Differently, when θ = θ H and the platform decides to become a marketplace, it obtains profits equal to

where

For the type H -platform to have interest in randomizing its strategies, the expected profit under a marketplace contract and the expected profit under a reselling contract must be the same. Then, the following necessary conditions for the existence of an hybrid equilibrium follow:

For this to be an equilibrium, it must be that λ ∈ [0, 1] and ρ ∈ [0 , μ] . Accordingly, necessary conditions for the existence of an hybrid PBE are:

G > 1 which guarantees that ρ ≥ 0;

(ii) k ≤ 1 -μ G -μ which guarantees that ρ ≤ μ when G > μ.

When G < μ, we check that ρ ≤ μ for all k > 1.