Mehdi Boroumand

Rémi Cogranne

Jessica Fridrich
email: fridrich@binghamton.edu

Remi Cogranne
email: remi.cogranne@utt.fr

Are we there yet?

Motivation

Steganography is the art of covert communication in which messages are hidden in cover objects so that the very existence of the secrets cannot be established. The objective of steganalysis is to detect the usage of steganography and do so as reliably as possible. A popular choice for cover objects today are digital multi media files, such as digital images, audio, and video. Such objects are ideal for covert communication for two reasons. They contain an indeterministic component, the acquisition noise, that helps mask the presence of steganographic embedding changes. Additionally, the inherent complexity of these objects is hard to capture using tractable and estimable statistical models, which further complicates detection. Steganographers fine-tune their embedding algorithms to locally adapt to content complexity since complicated textures and small-scale details are extraordinarily difficult to model statistically. This forced steganalysts to use complex high-dimensional (rich) media models [START_REF] Fridrich | Rich models for steganalysis of digital images[END_REF][START_REF] Kodovský | Steganalysis of JPEG images using rich models[END_REF][START_REF] Chen | Variable multi-dimensional co-occurrence for steganalysis[END_REF][START_REF] Goljan | Rich model for steganalysis of color images[END_REF][START_REF] Holub | Low-complexity features for JPEG steganalysis using undecimated DCT[END_REF][START_REF] Song | Steganalysis of adaptive JPEG steganography using 2D Gabor filters[END_REF][START_REF] Denemark | Steganalysis features for content-adaptive JPEG steganography[END_REF][START_REF] Denemark | Improving selectionchannel-aware steganalysis features[END_REF][START_REF] Abdulrahman | Color image steganalysis based on steerable Gaussian filters bank[END_REF] and, recently, non-linear hierarchical models with a large number of parameters, deep neural networks [START_REF] Qian | Deep learning for steganalysis via convolutional neural networks[END_REF][START_REF]Learning and transferring representations for image steganalysis using convolutional neural network[END_REF][START_REF] Xu | Structural design of convolutional neural networks for steganalysis[END_REF][START_REF] Xu | Ensemble of CNNs for steganalysis: An empirical study[END_REF][START_REF] Yang | Steganalysis based on awareness of selection-channel and deep learning[END_REF][START_REF] Yang | JPEG steganalysis based on densenet[END_REF][START_REF] Zeng | Large-scale JPEG image steganalysis using hybrid deep-learning framework[END_REF][START_REF] Boroumand | Deep residual network for steganalysis of digital images[END_REF][START_REF] Xu | Deep convolutional neural network to detect J-UNIWARD[END_REF][START_REF] Yedroudj | Yedroudjnet: An efficient CNN for spatial steganalysis[END_REF].

It should be stressed that, fundamentally, it is the unavailability of statistical models for natural images that is responsible for this seemingly never ending spiral develop-ment. Steganography in artificial sources (sources with a known statistical model) can be perfectly secure 1 as covers can be synthesized [START_REF] Anderson | On the limits of steganography[END_REF] to communicate at a positive rate (payload whose size is linear w.r.t. the number of cover elements) [START_REF] Moulin | New results on steganographic capacity[END_REF][START_REF] Wang | Perfectly secure steganography: Capacity, error exponents, and code constructions[END_REF]. Likewise, optimal detectors of imperfect steganography methods in artificial sources can be constructed and their performance computed.

The situation is quite different for empirical sources that lack description using tractable and estimable statistical models. All steganographic methods inevitably become imperfect and the size of their secure payload sublinear in the number of pixels due to the so-called square root law [START_REF] Ker | The Square Root Law of steganographic capacity[END_REF][START_REF] Filler | The Square Root Law of steganographic capacity for Markov covers[END_REF][START_REF] Ker | On the relationship between embedding costs and steganographic capacity[END_REF][START_REF] Ker | The square root law of steganography: Bringing theory closer to practice[END_REF]. Detectors can be built that can distinguish between cover and stego objects better than randomly guessing. Without a cover model, however, we are unable to assess how good our steganography methods are and how well our detectors perform.

This paper is an attempt to address this problem by forming an artificial source of realisticly looking images while forcing a known statistical model on pixels to allow derivation of optimal statistical tests for benchmarking empirical detectors built using machine learning. While it is entirely possible to synthesize artificial images for this purpose, the authors believe that it is valuable to keep a more realistic dataset with images visually similar to popular sources, such as BOSSbase 1.01 [START_REF] Bas | Break our steganographic system -the ins and outs of organizing BOSS[END_REF], in which content adaptive schemes execute changes with a similar selection channel as in the original source. We also need to avoid sources in which steganography would be too easy or too hard to detect while making sure that an optimal detector can be derived. Since these requirements are in conflict, preparing a suitable source of both cover and stego images is quite challenging

The idea for the cover source proposed in this paper was inspired by the experiment reported in Fig. 5 of [START_REF] Sedighi | Contentadaptive steganography by minimizing statistical detectability[END_REF]. The authors selected one BOSSbase image, denoised it, and then created 10, 000 different versions of the same image by adding to it 10, 000 independent realizations of sensor acquisition noise. Steganalysis in such a homogeneous cover source with the spatial rich model (SRM) [START_REF] Fridrich | Rich models for steganalysis of digital images[END_REF] and MiPOD embedding algorithm [START_REF] Sedighi | Contentadaptive steganography by minimizing statistical detectability[END_REF] was reported to be rather close in terms of the Receiver Operating Characteristic (ROC) to the optimal statistical test designed for the noise component. However, for a heterogeneous source with images of diverse content, the SRM detector lagged behind the optimal Likelihood Ratio Test (LRT) quite a bit most likely due to the inability of the empirical detector to deal with 1 In Cachin's sense [START_REF] Cachin | An information-theoretic model for steganography[END_REF].

1 the diversity of natural images (Fig. 6 in [START_REF] Sedighi | Contentadaptive steganography by minimizing statistical detectability[END_REF]).

The strategy adopted in this paper is to start with an existing dataset, apply a denoising filter to all images to remove complex noise introduced during acquisition and the subsequent development of the image from the raw sensor capture to a viewable form. Then, independent realizations of a Gaussian noise whose variance was estimated per pixel from the original images is reintroduced to force a known and tractable noise model in the cover source. This needs to be executed with care to prevent introducing dependencies among stego pixels. In particular, the pixels costs cannot be computed from the cover itself as the stego pixels would be dependent, which would prevent derivation of a closed-form LRT.

The proposed cover source dataset is described in the next section. TBD

Cover source

In this section, we first describe in detail the cover source preparation and then discuss the specific choices that were made.

The cover source was generated the from the union of BOSSbase 1.01 [START_REF] Bas | Break our steganographic system -the ins and outs of organizing BOSS[END_REF] and BOWS2 [START_REF] Bas | BOWS-2[END_REF] grayscale images resized from their original 512 × 512 size in Matlab (using 'imresize.m' with default parameters) to 20, 000 256 × 256 grayscale images. The smaller image size was selected in anticipation that the best empirical detectors will be deep neural networks, which typically require smaller images for effective training to fit reasonable size minibatches to the memory of current GPUs. We note that the leading performers in the spatial domain, the YeNet [START_REF] Ye | Deep learning hierarchical representations for image steganalysis[END_REF], Yedroujd-Net [START_REF] Yedroudj | Yedroudjnet: An efficient CNN for spatial steganalysis[END_REF], and the SRNet [START_REF] Boroumand | Deep residual network for steganalysis of digital images[END_REF], were trained and benchmarked on this same database. The union of both databases with 20, 000 256 × 256 grayscale images will be denoted B. We note that should future deep architectures require more than 20, 000 images, cropping instead of resizing the original images would produce four times as many, 80, 000 images, for a bigger dataset.

The formation of the dataset is explained in five steps and an additional, sixth, step needed for creating the stego images in a way that allows computation of an optimal statistical test.

Step 1: Estimate pixel variance

MiPOD's variance estimator (Section V in [START_REF] Sedighi | Contentadaptive steganography by minimizing statistical detectability[END_REF]) was applied to all 20,000 images from B to estimate the local variance of pixels' noise residual. For a given image and its pixel (i, j), 1 ≤ i, j ≤ 256, we denote its estimated variance σ 2 ij . Note that the output of MiPOD's estimator is lower bounded: σ 2 ij ≥ 0.01 for all (i, j).

Step 2: Denoising

All images from B were first denoised to remove complex dependencies among pixels introduced by the RAW developer and subsequent processing. We used the wavelet denoising method described in [START_REF] Mihcak | Low-complexity image denoising based on statistical modeling of wavelet coefficients[END_REF] with Daubechies 8-tap wavelets and standard deviation of the removed Gaussian i.i.d. noise σ den = 10. The pixel values in the denoised image were left in their non-rounded form but were clipped to the interval corresponding to 8-bit grayscale images [0, 255].

Step 3: Narrowing dynamic range

As the third step, the dynamic range of each denoised and clipped image was narrowed to the range [START_REF] Goljan | Rich model for steganalysis of color images[END_REF]240] by linearly mapping the interval [0, 255] to [START_REF] Goljan | Rich model for steganalysis of color images[END_REF]240] using :

g(x) = 15 + 225 255 x. ( 1
)
The scaled values were also rounded to integers, which we will denote µ ij ∈ {15, . . . , 240}. The resulting 8-bit grayscale image with a narrower dynamic range, which we denote µ ij , will next be noisified with the variances estimated in Step 1 and further adjusted in Step 4.

Step 4: Adjusting the variance

The estimated pixel variances σ 2 ij were adjusted so that the probability of a pixel getting out of the 8bit dynamic range [0, 255] after noisification is at most 2.87 × 10 -7 , the probability of a one-sided 5σ-outlier. This was done by making sure that σ ij is smaller or equal to one fifth of the distance between the pixel mean µ ij to the dynamic range boundary (0 or 255) :

σ ij = min 1 5 min{µ ij , 255 -µ ij }, σ ij . (2)
Additionally, we also introduced a floor for the variance to be able to simplify the optimal test using the fine quantization limit approximation :

σ ij = max{σ ij , σ 0 }. (3)
In this paper, experimented with σ 0 ∈ {0.5, 1}.

Step 5: Noisifying

The noisified pixel c ij is obtained by adding to µ ij a sample ξ ij from N (0, σ 2 ij ), rounding to an integer and clipping to [START_REF]Learning and transferring representations for image steganalysis using convolutional neural network[END_REF]254] to make sure the embedding will be free to modify all pixels by ±1 without getting out of the dynamic range. In other words, in our dataset we impose the cover image model from MiPOD -pixels are realizations of independent Gaussian variables N (µ ij , σ 2 ij ) that are rounded to integers (denoted with the square bracket [•]), and then clipped to a finite dynamic range. Symbolically,

c ij = [µ ij + ξ ij ] ( 4
)
c ij =    c ij if 1 ≤ c ij ≤ 254 1 if c ij ≤ 0 254 if c ij ≥ 255. (5)
The cover image pixels thus follow a p.m.f.

p ij , c ij ∼ p ij : p ij (m) =              0 m = 0 Q ij m -1 2 m = 254 Q ij m -1 2 -Q ij m + 1 2 1 < m < 254 1 -Q ij m + 1 2 m = 1 0 m = 255 2 (6)
with Q ij (x) defined as the tail probability of N (µ ij , σ 2 ij ) :

Q ij (x) Pr{N (µ ij , σ 2 ij ) > x}. ( 7
)
The values c ij form the data source used in our experiments. This cover source will be denoted B(σ 0 ).

Stego images

The stego methods used in this study are all ternary embedding algorithms that change the (i, j)th pixel by ±1 with equal probability β ij . Since we curbed the cover values in Step 5 to the interval [START_REF]Learning and transferring representations for image steganalysis using convolutional neural network[END_REF]254], the embedding does not need to be constrained in any way -all pixels can be changed either way. For content adaptive steganography, the probabilities (change rates) β ij are determined from pixel costs, which are typically computed from a local neighborhood of pixel (i, j). This dependence is quite complicated as the costs are usually computed in a non-linear fashion from outputs of several high-pass filters. Thus, computing β ij from the noisified cover c ij would create complex dependencies among stego pixels, preventing thus a closed-form expression for the distribution of stego pixels and tractable evaluation of the associated likelihood ratio test.

We resolved this problem by computing the pixel costs (change rates β ij ) from the corresponding original image from B. Another possibility is to compute the costs from a different independent noisification of the image. Both versions gave similar results in our experiments. The former was used as default for the rest of this paper.

Since β ij now does not depend on the specific noisification of the image and since the embedding changes are executed independently, the stego pixel p.m.f. is factorizable. In particular, it is a product of the following Gaussian mixtures q ij over all pixels :

q ij (m) = (1 -2β ij )p ij (m) + β ij p ij (m -1) + β ij p ij (m + 1). ( 8
)
For S-UNIWARD, HILL, and WOW, the change rates were obtained from an embedding simulator (e.g., assuming optimal source coding).

Optimal test

Given an image with pixels s ij , the steganalyst is facing the following statistical hypothesis test for all (i, j) :

H 0 :s ij ∼ p ij H 1 :s ij ∼ q ij . ( 9
)
We will assume that the parameters of the added MVG noise, the mean µ ij , and the variance σ 2 ij , are known. We also assume that the change rates β ij are known. Under these assumptions, the test is simple, and the optimal statistic is the log-likelihood ratio

Λ(s) = i,j Λ ij = i,j log q ij (s ij ) p ij (s ij ) (10)
by the statistical independence of pixels. For convenience, we will use the following normalized form of the log-LRT :

Λ (s) = i,j Λ ij -E H0 [Λ ij ] i,j V ar H0 [Λ ij ] , (11)
where

E H0 [Λ ij ] = i,j p ij Λ ij (12) V ar H0 [Λ ij ] = i,j p ij Λ 2 ij -E H0 [Λ ij ] 2 . ( 13
)
Under the fine quantization limit, σ 0 ≤ σ ij for all i, j, and as the number of pixels approaches infinity, the Lindeberg's version of the Central Limit Theorem implies

Λ (s) N (0, 1) under H 0 N ( , 1) under H 1 , ( 14
)
where means convergence in distribution and = i,j σ -4 ij β 2 ij > 0 is the deflection coefficient.

Discussion

The denoising step in the preparation of the cover source is essential because we give the means of the MVG (the denoised signal) to the optimal test while the network would have to deal with the complexity of numerous noise sources that occur in natural images. By adding the MVG to the denoised image, we force the cover complexity to be primarily in the noise component. This way, both the optimal test and the network need to deal with the content complexity due to indeterminism, the added noise. Of course, the network still needs to learn a model for the (heterogeneous) denoised content.

Because the added noise mimics the content complexity of the original image (a combination of the indeterminism in the original image and texture), when the change rates β ij are computed from the noisified cover c ij , the costs profile (change rates) of the tested content-adaptive algorithms looks similar to the costs (change rates) computed from the original image from B (see Figure 1). This justifies our choice of computing the change rates from the original image.

Note that since p ij (0) = p ij (255) = 0, the boundary values do not occeur in covers from B(σ 0 ), and whenever the embedding produces these "forbidden" values, the LRT becomes infinity, arranging for a perfect detection in this case. Fortunately, due to our choice of the standard deviation σ ij (2)-(3), this occurs with very small probability. In our tests, we typically saw at most one such image.

The scaling and clipping of the dynamic range in Step 3 was also necessary because otherwise the embedding would have to be adjusted not to change the boundary value 255 up and 0 down, which would reintroduce a dependence between a specific noisification c ij and the change rates β ij , preventing the derivation of a closed-form expression for the LRT.

Experiments

Figure 2 shows the ROCs of three detectors -the optimal LRT [START_REF] Denemark | Improving selectionchannel-aware steganalysis features[END_REF], the convolutional network SRNet [START_REF] Boroumand | Deep residual network for steganalysis of digital images[END_REF], and the maxSRM [START_REF] Denemark | Selection-channel-aware rich model for steganalysis of digital images[END_REF] with the low-complexity linear classifier [START_REF] Cogranne | Is ensemble classifier needed for steganalysis in highdimensional feature spaces[END_REF]. The SRNet and the maxSRM were trained on 2 × 15, 000 images (out of which 2 × 1000 were used for validation for the SRNet) randomly selected from our dataset and tested on the remaining 2 × 5, 000 images. The SRNet was first trained on the "easier" dataset B(0.5) from a random initialization. Then, the trained SRNet was used as a seed for training on B(1). The LRT's performance was computed on the same testing set. Table 1 shows the comparison in terms of three scalar performance descriptorsthe minimal total error probability P E under equal priors

P E = min F FA 1 2 (P FA + P MD ), ( 15
)
the false-alarm rate at 50% detection, P FA (0.5), and the missed-detection rate for 5% false alarm, P MD (0.05). The SRNet is markedly better than the detector with maxSRM. For S-UNIWARD, the gap between the bound and the ROC of the empirical detector was roughly cut by half. For HILL, there seems more room for improvement. In both cases, our analysis shows space for improvement for both embedding algorithms especially for low false alarms.

Justification of design choices

In this section, we provide a justification for the choices made when creating the dataset in Section "Cover source." First, we demonstrate the impact of narrowing the dynamic range of cover images and the effect of flooring the noise variance with σ 2 0 for noisification (the failure to comply with the fine quantization assumption).

4 left shows the distribution of the normalized LRT Λ [START_REF] Denemark | Improving selectionchannel-aware steganalysis features[END_REF] under H 0 on a dataset created by skipping both Step 3, narrowing the dynamic range, and Step 4, adjusting the variance for noisification, for the alternative hypothesis with stego images embedded with S-UNIWARD at payload 0.4 bpp. The right figure shows the same distribution when including Step 3 but skipping Step 4. The values below the figures are the mean, variance, skewness, and kurtosis. The normalized LRT is much closer to N (0, 1) when the dynamic range is narrowed but its skewness and kurtosis still indicates deviations from a Gaussian distribution.

This deviation is due to the failure of complying with the fine quantization limit needed for the asymptotic result [START_REF] Fridrich | Rich models for steganalysis of digital images[END_REF] to hold. To further investigate this issue, we selected the BOSSbase image BOSSbase '559.pgm', which contains many pixels with a small noise variance σ 2 ij , and executed the following experiment. The image was independently processed 10, 000 times as described in Section "Cover model" with and without flooring the variance with σ 2 0 in Step 4. Figure 5 shows the distribution of Λ for the 10, 000 noisifications for both the cover and stego versions of this image with S-UNIWARD at 0.4 bpp. The top two graphs are for the dataset B(∞), i.e., when not flooring the variance with σ 2 0 while the bottom two graphs show the effect of enforcing the fine quantization limit with σ 0 = 1. Notice that when not enforcing the fine quantization limit, under both hypotheses (9) the distribution of the LRT fails to follow ( 14) -the distribution is asymmetrical with a thicker right tail, which is pronounced especially for the alternative hypothesis. After flooring the variance, however, both distributions become much closer to the expected limit N (0, 1).

Finally, we comment on an alternative to scaling down the dynamic range of the images to resolve the problem with boundary conditions, and that is to skip the scaling and allow the embedding to change pixel values to -1 and 256. Of course, the existence of a single pixel with one of these two values is 100% indicative of embedding. The LRT will "see" this since for such stego pixels the LR will be infinite. However, an empirical detector that forms the test statistic by computing noise residuals or convolutions of the input image may not.

To test this approach, we prepared a different version of our dataset in which the scaling (1) as well as the variance adjustment ( 2)-(3) were skipped. The detection performance of the LRT for S-UNIWARD at 0.4 bpp is shown in Figure 6 together with the ROC for the SRNet. The third ROC (SRNet-Adjusted) is for the performance of the SRNet with its output augmented by the fact that whenever pixels -1 and 256 are present in the image, it is automatically labeled as stego. The performance of this "informed" SRNet is very close to the theoretical upper bound. Figure 7 shows the distribution of the normalized LRT across cover images. Notice that the distribution indeed matches N (0, 1) even though it is still slightly asymmetric. This is because we do not have the fine quantization assumption satisfied. This measure was not selected for our experiments, however, because of the rather artificial setup as the stego images are not really images.

I am not sure about this experiment as we ignore the fine quantization limit condition here. It would make sense if we floored the variance. Shall we redo?

Conclusions

TBD All code used to produce the results in this paper, including the network configuration files are available from http://dde.binghamton.edu/download/.

Figure 1 .

 1 Figure 1. Sorted total change rate computed for ten images (above) from B(1) for HILL at 0.4 bpp. Blue: change rates computed from the original images, Red: from noisified covers cij .

Figure 3 .

 3 Figure 3. Contrasting the performance of the LRT and SRNet for payloads 0.2, 0.4, and 0.6 bpp for S-UNIWARD on B(0.5) and B(1). Place holder only.

Figure 4 .

 4 Figure 4. Left: Distribution of the LRT under H0 on a dataset created by skipping both Step 3 and 4 for S-UNIWARD 0.4 bpp. Right: The same distribution with Step 3 included but not Step 4. The values below the figures are the mean, variance, skewness, and kurtosis.

( 4 .Figure 5 .

 45 Figure 5. Distribution of Λ and its mean, variance, skewness, and kurtosis for 10,000 different noisifications of BOSSbase image '559.pgm'. Left: covers, Right: stego images for S-UNIWARD 0.4 bpp. Top: dataset B(∞), Bottom: B(1).

Figure 7 .

 7 Figure 7. Distribution of the normalized LRT on cover images across the dataset when permitting changes to -1 and 256. S-UNIWARD at 0.4 bpp.

Table 1 . Performance of three empirical detectors in terms of

 1 P E , P FA (0.5), and P MD (0.05).

		1				SUNI -0.4 bpp
		0.9
		0.8
		0.7
		0.6
	PD	0.5
		0.4
		0.3
		0.2							SRNet
		0.1							SRNet-Adjusted LRT
		0
		0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
							PFA
	Figure 6.	ROCs when permitting changes to -1 and 256, "SRNet-
	Adjusted": helping SRNet by pronouncing the test image 'stego' if it contains
	-1 or 256.
	5

SRNet LRT

Figure 2. ROCs for (top down) S-UNIWARD, HILL, WOW at 0.4 bpp, and LSBM for total change rate β = 0.03 for the optimal test (LRT), SRNet, and maxSRM with the low-complexity linear classifier with the left and right part columns corresponding to datasets B(0.5) and B(1), respectively. Place holder only.

Acknowledgments

The work on this paper was supported by NSF grant No. 1561446 and by DARPA under agreement number FA8750-16-2-0173. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation there on. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied of DARPA or the U.S. Government.

Author Biography

Mehdi Boroumand


Nicolas Aubert
email: niaubert@gmail.com

Hachmi Ben Ameur
email: hbenameur@groupeinseec.com

Guillaume Garnotel
email: ggarnotel@groupeinseec.com.

Jean-Luc Prigent
email: jean-luc.prigent@u-cergy.fr

H G E C L G E C H B E C L B

Optimal Employee Ownership Contracts under Ambiguity Aversion

Keywords: JEL classification: G11, G32, G34, J33

, we prove that if aversion to ambiguity is sufficiently high, then we find the same behavior as in the case of no ambiguity.

Introduction

Employee stock ownership is when employees own shares of the company for whom they work. Employee ownership is a major part of shared capitalism, with 14 million employee owners in the US in 2014. 6 Shared capitalism refers to "schemes that tie worker pay or wealth to their own workplace performance, whether at the level of the workgroup, institution, or company" (Kruse et al, 2010;p. 5). Shared capitalism was extended to a wider public in 2015 by former members of the US government 7 who suggested promoting inclusive prosperity. The main argument of shared capitalism is that sharing wealth helps to stimulate growth. By participating in the wealth of the company they work for, employees work more and better, which subsequently improves corporate performance and stimulates economic growth. The idea was initially promoted by Martin Weiztman to fight stagflation in the 1980s [START_REF] Weitzman | The Share Economy: Conquering Stagflation[END_REF]. Shared capitalism resurfaced after 2008 as a solution to share wealth, and fight poverty and inequality (Kruse et al, 2010). [START_REF] Blasi | The citizen's share: Putting ownership back into democracy[END_REF] even argue that shared capitalism has always been a major component of the US economy, promoted by the founding fathers, and one of the reasons for America's economic growth since the 19 th century. However, employee stock ownership outcomes remain debatable.

The decision to introduce and develop employee ownership generally lies with management. When offered as an investment option by the employer, the employees can decide whether or not they want to buy the company stocks on offer. Consequently, employee ownership can be analyzed as a contract between the employer and the employee. This paper investigates the characteristics of an employee ownership contract when the employee is ambiguity averse. Ambiguity aversion or uncertainty aversion refers to the preference for known risks over unknown risks. More technically, the presence of ambiguity reflects the non-uniqueness of prior probabilities over sets of states. Rather than having in mind a unique probability to determine expected utility, the agent faces a whole set of possible probabilities, or so-called prior probabilities.

The present paper also converges the findings of three different fields of economics, since labor, financial and behavioral economics have investigated employee ownership separately to date. Labor and financial economics have focused on the consequences of employee ownership, while behavioral economics has documented the causes of employee ownership. Labor economics and industrial relations establish a link between employee ownership and enhanced corporate performance. Financial economics on the other hand, tends to show that employee ownership damages corporate governance. Behavioral economics documents many cognitive biases, leading employees to overinvest in their company stocks, but has never included ambiguity aversion as one of them. The presence of positive and negative effects suggests an optimal level of employee ownership that stimulates corporate performance without harming corporate governance. This level suits the behavioral preferences of the employee in terms of ambiguity aversion. Unlike much of the prior literature, our model does not consider employees' and employers' behavior as being separate from their environment. Instead, we consider interactions within the workplace by illustrating strategic interactions between the employer and the employee.

The main conclusions of labor economics and industrial relations' regarding employee ownership is that it improves corporate performance (called 'performance motive' hereafter). Performance motive relies on the positive effects of employee ownership within the workplace in terms of cooperation, mutual monitoring, staff turnover and absenteeism. The meta-analysis by O'Boyle et al (2016) reviewed 102 samples representing 56,984 firms in different countries. They reported an overall positive and significant relationship between employee ownership and firm performance measured in terms of efficiency or growth. Several chapters of the book by Kruse et al (2010) also point to evidence of the positive relationship between employee ownership workplace outcomes and corporate performance based on survey and archival data collected on a sample of 40,000 employees representative of the US workforce. The shared capitalism survey items are now included in the General social survey of the National opinion research center. 8Financial economics often regards employee ownership as a management entrenchment tool ('entrenchment motive' hereafter). Schleifer and Vishny (1989) claim that managers entrench themselves in their position when they counter the financial market disciplinary forces "by making themselves valuable to shareholders and costly to replace" (p. 123). Since employees and managers are natural allies against hostile takeovers [START_REF] Pagano | Workers, managers, and corporate control[END_REF]p. 841), managers prefer to place shares in the friendly hands of employees [START_REF] Benartzi | The law and economics of company stock in 401(k) plans[END_REF]. The negative effect of employee ownership on corporate governance is well documented empirically [START_REF] Beatty | The cash flow and informational effects of ESOP[END_REF][START_REF] Chaplinsky | The role of ESOPs in takeover contests[END_REF][START_REF] Park | Employee stock ownership plans, firm performance and monitoring by outside blockholders[END_REF][START_REF] Chang | ESOPs and shareholder wealth: an empirical investigation[END_REF][START_REF] Chang | Managerial vote ownership and shareholder wealth: evidence from ESOP[END_REF][START_REF] Conte | Financial returns of public ESOP companies: investors effects vs. manager effects[END_REF]. [START_REF] Rauh | Own company stock in defined contribution pension plans: a takeover defense[END_REF] and [START_REF] Brown | 401(k) matching contributions in company stock: costs and benefits for firms and workers[END_REF] also concluded more recently that employee ownership has a deterrent effect on the likelihood of a takeover. Behavioral economics began investigating why employees invested in their company's stocks for their pension plan following the dotcom crisis when thousands of employees lost their jobs and savings. Most of the literature on household finance [START_REF] Guiso | Household finance: an emerging field[END_REF] and behavioral finance [START_REF] Barber | The behavior of individual investors[END_REF] examined retail investment through the data on pension plan participants. As reported by [START_REF] Guiso | Household finance: an emerging field[END_REF], the concentration of US citizens' pension wealth in employer's stocks is noteworthy and of some concern to the public authorities. According to [START_REF] Benartzi | The law and economics of company stock in 401(k) plans[END_REF], investment in employer's stock comes at a high cost for both employees and employers. Previous research suggests that employees' investment in their company stock is a consequence of cognitive biases such as excessive extrapolation of past returns and endorsement [START_REF] Benartzi | Excessive extrapolation and the allocation of company stock to retirement accounts[END_REF], loyalty and familiarity [START_REF] Cohen | Loyalty-based portfolio choice[END_REF][START_REF] Huberman | Familiarity breeds investment[END_REF], framing effect [START_REF] Benartzi | Excessive extrapolation and the allocation of company stock to retirement accounts[END_REF]Thaler, 1999, 2002), risk myopia [START_REF] Mitchell | Company Stock and Retirement Plan Diversification[END_REF], disposition effect [START_REF] Choi | Employees' investment decisions about company stock[END_REF], and default heuristics [START_REF] Madrian | The power of suggestion: Inertia in 401(k) participation and savings behavior[END_REF].

To the best of our knowledge, no study to date has investigated ambiguity aversion, either theoretically or empirically. However, ambiguity aversion affects less sophisticated individual investors like employees who are not always able to infer the probability distribution of stock returns when they are offered an investment in their employer's stock. Ambiguity was first documented by [START_REF] Ellsberg | Risk, ambiguity, and the Savage axioms[END_REF]. [START_REF] Charness | Portfolio choice and risk attitudes: an experiment[END_REF] conducted an experimental comparison that considers three biases well known in the behavioral finance literature: ambiguity aversion, the illusion of control and myopic loss aversion. [START_REF] Gilboa | Maxmin expected utility with a non-unique prior[END_REF] suggest accommodating ambiguity within economic decision-making, and assume that in the presence of ambiguity, individuals cannot identify a single probability distribution over states of nature. They thus consider multiple probability distributions and then evaluate their choices according to the worst probability distribution for that choice. This approach is based on multiple priors. The decision model is called "maxmin expected utility" (MEU hereafter) whose beliefs are given by a set of probabilities, with decisions made to maximize the minimal expected utility of the act chosen. Our model adopts this approach.

The model postulates that employers' financial contributions are a means to stimulate employee ownership9 and reveal the manager's type. From this standpoint, the prevailing management type is supposedly revealed by the amount of company stock that management decides to grant to its employees. [START_REF] Rauh | Own company stock in defined contribution pension plans: a takeover defense[END_REF] claims that strategic corporate control is one of the main reasons why managers encourage employees to hold company stock in their defined contribution pension accounts. The monetary contribution given by managers to their employees plays a major role. The forms of employer monetary contributions depend on the context in which they are granted to the employee: an Employee stock purchase plan (ESPP), defined contribution plans like the 401(k) or Employee stock ownership plans (ESOP). Within an ESPP, a discount is offered on the stock price. Managers can also help to boost employee ownership through an ESOP. ESOPs frequently give employees company stocks, often with no up-front cost to the employees. In a 401(k) style company-based savings plan with several investment options, matching contributions in company stocks are offered by the employer. This means that additional company stocks are granted to employees when they choose to invest in their employer's stocks. We extend the conclusions of the [START_REF] Aubert | Employee ownership: a theoretical and empirical investigation on reward management vs. management entrenchment[END_REF] model by introducing ambiguity aversion. In the context of standard utility theory, [START_REF] Aubert | Employee ownership: a theoretical and empirical investigation on reward management vs. management entrenchment[END_REF] considered that a firm's management type is revealed by the way management contributes financially to the development of employee ownership. Good managers stimulate employee ownership because they want to boost corporate performance (performance motive), whereas bad managers encourage it because it can help them to retain their position (entrenchment motive). In the ambiguity theory framework, we extend the previous contribution by providing a general finding about the existence of a perfect subgame Nash equilibrium, taking ambiguity into account. We propose a sequential game where a riskneutral manager grants company stock to risk-averse employees, while both face ambiguity. 10 In the framework of [START_REF] Gilboa | Maxmin expected utility with a non-unique prior[END_REF], we demonstrate that the optimal ownership contract is respectively increasing with respect to the lower bound of the return expectation in the case of a high level of effort, and decreasing with respect to the upper bound of the return expectation in the case of a low level of effort. In the framework of [START_REF] Maccheroni | Ambiguity aversion, robustness, and the variational representation of preferences[END_REF], we prove that if the ambiguity aversion is sufficiently high, then we find the same behavior as in the scenario without ambiguity. The remainder of the paper is organized as follows. Section 2 presents the model setup and the optimal employee ownership contract under ambiguity aversion. Section 3 concludes. Proof is provided in the appendix

Optimal employee ownership under ambiguity

In this section, we present the model setup (subsection 2.1) and we derive the optimal employee ownership contract characteristics under ambiguity aversion (section 2.2). In the second subsection, we develop the conditions under which an equilibrium exists that satisfies the employees' and the managers' preferences.

Model setup

Our model relies on the MEU approach, which is more flexible than the standard expected utility criterion. Furthermore, it provides a distinction between risk and ambiguity, and can capture preference with ambiguity aversion. [START_REF] Maccheroni | Ambiguity aversion, robustness, and the variational representation of preferences[END_REF] characterize preferences under ambiguity by introducing both a utility function U defined on outcomes and an ambiguity index C defined on a set  of probabilities regarding the random events. The latter set is viewed as the individual's set of priors. Ambiguity is associated with the multiplicity of priors. Thus, they consider the following representation of preferences: For all random variables X and Y which represent results or consequences, and with values in [ , ] MM  , we have:

        pp XY Min U X dP C P Min U Y dP C P        (1)
The function U corresponds to decision risk attitude. Index C represents individual attitude towards ambiguity. This representation of preferences includes both the multiple priors' preferences of [START_REF] Gilboa | Maxmin expected utility with a non-unique prior[END_REF] and the multiplier preferences of Hansen andSargent (2000, 2001). The MEU criterion of [START_REF] Gilboa | Maxmin expected utility with a non-unique prior[END_REF] corresponds to the case where 0 C  and  is a convex set. Hansen andSargent (2000, 2001) propose a robust preference approach where individuals are uncertain about their modeling of random events. The ranking of decisions is based on:

        ,, pp XY Min U X dP R P Q Min U Y dP R P Q         (2)
where   , R P Q denotes relative entropy with respect to the given probability distribution Q . 11

11 Recall that the relative entropy of the probability distribution P with respect to the probability distribution Q is defined by  

, P dP R P Q E Log dQ        . Note that   ,0 R Q Q  .
Relative entropy has been introduced in both the probability and information theories to measure the difference between probability distributions P and

Q . It
This kind of preference is known as a multiplier preference. Parameter  represents the weight individuals give to the possibility that due to the lack of information, P is not the appropriate probability distribution. [START_REF] Maccheroni | Ambiguity aversion, robustness, and the variational representation of preferences[END_REF] establish theoretical connections between the two approaches. For a given set  , the lower C , the higher the ambiguity aversion. MEU captures an extreme form of ambiguity aversion in the sense that the decision-maker bases his or her decisions on worst-case scenarios.

We consider three stages involving a risk-neutral manager and a risk-averse employee. The manager's wealth is fully invested in company stock but the resulting stake is insufficient to ward off a takeover attempt. We assume the manager holds a given amount of company stock at the beginning of the game. These features allow us to analyze the game without explicitly including the other shareholders as players. The employee is representative of all company employees.

There are two types of managers: good ( G ) managers and bad managers (

B ) with { , } k G B  . At 0 t  ,
the manager's type is determined. Only the manager knows this information. The employee does not observe the manager's type and has an a priori probability distribution 0 P with regard to perception of the manager's type. Thus, 00 () P G p  is the probability that the manager is good and 00 ( ) 1 P B p  is the probability that the manager is bad with 0 0 P  1  . The manager's type affects the rate of return on company stock. Good managers are more likely to perform better than bad managers. This is denoted by index k of the rate of return.

At 1  t
, the manager designs a compensation scheme aimed at motivating the employee to make more effort. The compensation scheme we investigate only consists of company stock.

c is the level of employee ownership. We thus consider a setup where the manager has to decide whether to offer company stock to the employee and the amount of stock to offer.

At 2  t
, the employee evaluates the compensation scheme and chooses the level of effort. The employee's effort can either be high ( H ) or low ( L ) with { , } j H L 

. The level of effort affects the rate of return on company stock.

The rate of return on company stock is given by

, jk RM 
, where R is a random variable with a zero mean, volatility  and density () fr. It represents the stochastic part of the rate of return. The random variable Ceteris paribus, we assume that a better manager or more employee effort increases the rate of corresponds to the Kullback-Leibler divergence from Q to P and quantifies the information gain when we revise beliefs from the prior probability distribution Q to the posterior probability distribution P .

12 The independence assumption can be relaxed for the manager since the latter is risk-neutral, which implies that the manager's expected utility can be computed without any assumption about the joint distribution of R and , jk M . However, we must use this joint law to compute the employee's expected utility. Thus, given R , we have to introduce specific multipriors to describe ambiguity with respect to the conditional distributions of , jk M . To avoid this additional and cumbersome complexity, we assume that R and , jk M are independent. return on the company stock. We then consider the rational conditions

  , , , , L B L G H B Min     and   , , ,
,

H G L G H B Max     .
In what follows, we model ambiguity for both the manager and the employee as in [START_REF] Maccheroni | Ambiguity aversion, robustness, and the variational representation of preferences[END_REF]. This means that there are two sets of multipriors, . The latter functions are assumed to satisfy the conditions defined in [START_REF] Maccheroni | Ambiguity aversion, robustness, and the variational representation of preferences[END_REF].

We make the following assumption:

ASSUMPTION 1 Both sets of multipriors ,, , j k m m R     and ,, , j k e e R     are compact.
REMARK 1 The previous assumption is satisfied when, for example, the minimization of expected utilities on the given set m  (or e  ) is equivalent to the minimization of a given set of parameters. This is true when the distribution of

,, j k l M is parametrized with parameters belonging to a compact subset of R d . For instance, ,, j k l M is characterized by its expectation ,, j k l  , with ,, , ,. ,, [ , ] j k l j k l j k l    
, where , ,. The risk-neutral manager's expected utility is ,,

i j k V
, where i denotes the level of company stock c with 0, ic  . The manager seeks to maximize the expected value of his/her wealth given the uncertainty and ambiguity regarding some parameters that we define in the following. The manager's initial wealth d W is positive and wholly invested in company stock. When the manager offers employee ownership, it takes the form of a contribution in company stock denoted by c . We assume c to be a proportion of s W , which is the positive initial employee's wealth. While the manager cannot directly observe the employees' wealth, (many employee characteristics that correlate with overall wealth can be observed such as age, gender, tenure, rank, level of education, fixed wage and bonus. Altogether, these variables make it possible to commensurate the employee ownership policy with the employees' wealth. s cW is therefore the monetary value of company stock granted to the employee that decreases the manager's initial wealth d W . Indeed, the monetary value of the contribution granted to the employee is often commensurate with the amount invested by the employee. This may take the form of a discount or a matching contribution for instance. [START_REF] Degeorge | Selling company shares to reluctant employees: France Telecom's experience[END_REF] provide empirical evidence that wealthier employees are more willing to invest in their company stocks. If the manager grants a contribution in company stock, the manager's expected wealth is given by equation (4); otherwise, the manager's expected wealth is given by equation (3).

    ,, ,, , ,, ,, , 0, , , , , ,, , 1 ( ) 1 (
)

j k m j k m m RM j k m j k m m RM j k j k m P d m RM j k m P d m RM V Min W E R M C P Min W C P               (3)         ,, ,, , ,, ,, , , , , , , ,, , ( ) E 1
( ) ( ) 1 ( ) . j k m j k m RM m j k m j k m RM m c j k j k m P d s m RM j k m P d s m RM V Min W cW R M C P Min W cW C P               (4) LEMMA 1 Since the set m  of multipriors is compact, the utility 0, , jk V corresponds to 0, , , , 0, , 0, , , , , 1 ( ),
j k m j k j k m dm RM V W C P         for a given 0, , , , , j k m RM P  in m 
, and ,,

c j k V corresponds to   , , , , , , , , , , , ( ) 1 ,
c j k m c j k c j k m d s m RM V W cW C P          for another given , , , , , c j k m RM P  in . m 
The risk-averse employee's expected utility is considered to be additively separable between the gains and the disutility of effort. The employee's positive initial wealth s W is wholly invested in a risk-free asset whose rate of return is 0 r . Effort e is costly to the employee who endures () e  , the disutility that increases with effort. If company stock is granted, the initial employee's wealth s W increases by the amount s cW . If the employee receives s cW , his or her expected utility is given by equation ( 6), otherwise, it is given by equation ( 5):

0, 0 [ (1 )] ( ) jj s U u W r e    
(5) and:

    ,, , ,, ,, , ,, 00 , ,, 00 , [ (1 )
(1 )] ( ) (1 ) [ (1 ) (1 ) ( ) ]
( ).

p u W r cW R M C P E p u W r cW R M C P e                      (6)
where (.) u is a risk-averse von Neumann-Morgenstern utility function.

LEMMA 2 Since the set e  of multipriors is compact, the utility , cj U corresponds to

    , , , , , , , , , , , 00 , , , , 00 , [ (1 ) (1 ) ( )]
(1 j k e j k e

) [ (1 ) (1 )] ( ) ( ).



We make the following assumptions about the utility function of the employee and the parameters:

ASSUMPTION 2 1) The employee is risk-averse and cautious [START_REF] Gollier | Understanding saving and portfolio choices with predictable changes in assets returns[END_REF].

''( ) : '( ) 0, ''( ) 0, ( ) 1. '( )

ux x u x u x RRA x x ux       ( 8
)
This assumption is equivalent to , [ '( )]' 0 x xu x  , which implies that '( ) x xu x  is an increasing function.

2) The employee's expected utility is strictly positive when company stock is not granted and effort is low:

0, 0 [ (1 )] ( ) 0. LL s U u W r e      (9)
This assumption implies in particular that the disutility for low effort is relatively small (upper bounded by 0

[ (1 )] s u W r 
).

3) The employee's expected utility is higher with employee ownership for a given level of effort:

  , 0, , 0, , .

c j j U U c j H L      (10)
This assumption means that, whatever the level of effort, the employee benefits from receiving a non-zero contribution.

Derivation of the optimal contract

In what follows, we derive the model solutions and detail the conditions under which employee ownership is used either as an entrenchment tool or as a reward mechanism. We describe the employee ownership contract with ambiguity aversion when there is no threat of dismissal for the manager. We define () c  as the difference between the employee's utility of wealth when a high level of effort is exerted and the utility of wealth resulting from a low level of effort (without taking into account the respective disutilities resulting from the effort). The difference is given by:

 ) (c  (11)     , , , , , , , , , , , 00 , , , , 00 , [ (1 ) (1 ) ( )]
(

P E p u W r cW R M C P                          , , , , , , , , , , , 00 , , , , 00 ,
[

(1

) [ (1 ) (1 )] ( ) c L G e c L B e c L G e s s e RM c L B e s s e RM p u W r cW R M C P E p u W r cW R M C P                      
The employee chooses the high level of effort if this difference is greater than the supplementary disutility of effort ( ) ( )

HL ee  
needed to reach this level of effort. First, we consider a case which is a straightforward extension of the framework introduced by [START_REF] Aubert | Employee ownership: a theoretical and empirical investigation on reward management vs. management entrenchment[END_REF]. We get the following result:

PROPOSITION 1 Assume we are dealing with the basic example 1. Recall in particular that both ambiguity indices m C and e C are null (as in the framework of Gilboa and Schmeidler, 1989) and that ambiguity exists only for

;

L B m L B e L G m L G e H B m H B e                    ; , , , , . j H G j H G          Proof:
The result is proved by using the fact that both utilities are increasing functions of all the parameters ,, . jk  Second, we investigate a more general framework. For this purpose, we introduce some specific conditions which allow us to establish the mild properties of () c  , leading to the derivation of the optimal solutions (see Assumption 3).

NOTATION 1 Denotes the four kinds of wealth level for the employee:

, , , ,

(1 ), (1 )

(1 ), (1 )

(1 ), (1 )

(1 ).

c H G e ss c L G e ss c H B e ss c L B e ss

X W r cW r M Y W r cW r M Z W r cW r M T W r cW r M                         ASSUMPTION 3 1)
We assume that, despite ambiguity, the employee is more optimistic about the value of his or her wealth when making an effort, whatever the quality of the manager. This means that we have and X Y Z T  . Thus, using assumption (2), we get the following inequalities: for any value r of R,

                ( ) ( ) , ( ) ( ) , ( ) ( ) , ( ) ( ) .
E u Y E u X E u T E u Z E Xu X E Yu Y E Zu Z E Tu T                
2) The sensitivities of the optimal probability distributions , , , c j k e

M

 to the level of company stock c weighted by the employee's marginal utility are higher when the employee makes an effort, whatever the quality of the manager:

        , , , , , , , , , , , , , , , , , , , , , , , , , , / ( )
/ ( ) , / ( ) / ( ) . c H G e c L G e MM c H B e c L B e MM c H G e c L G e c H B e c L B e E M c u X E M c u Y E M c u Z E M c u T                                  
3) When the employee makes an effort, the latter's ambiguity index is more sensitive to the level of employee ownership:

, , , ,

, The previous result means that the greater the contribution of the employee in the company stock, the higher the difference between the satisfaction level when the employee makes an effort and the higher the satisfaction level when the employee does not make this effort. Recall that () c  does not take the effort's disutility into account. Taking ambiguity into account, the equilibrium is obtained if the manager grants a positive amount of company stock and if the employee exerts a high level of effort. Although other solutions may be investigated, we focus on the situation where the employee exerts a high level of effort in the presence of a positive level of employee ownership. We thus illustrate the manager's strategies by examining a scenario that is the most widely identified by the literature on employee ownership.

c H G e c L G e ee R M R M C P C P c       , , , , , , ,, ( ) ( ) 0.

COROLLARY 1 For the basic example 1, with a null ambiguity index

This situation describes a perfect subgame Nash equilibrium where managers' and employees' utilities are, respectively,

        , , , , 0, , , , , , , , , 0, , , , , ( ) 1 1
( ) . c H k m L k m c H k m d s m RM L k m dm RM W cW C P W C P                 (12)
The manager's expected wealth ,, c H k V must decrease with respect to the contribution c in company stock. We thus state the following proposition.

PROPOSITION 3 If ,, c H k V
is strictly decreasing with respect to the contribution c , then threshold k c % is the unique solution of Equation ( 12). The previous result is illustrated by the two following fundamental examples.

EXAMPLE 2 If the ambiguity index is null (i.e. 0 m C  ), then: , , , , , , , , ,
. 1 13) In this case, using assumptions of the basic example 1, the threshold k c % increases with respect to , , , H k m   , meaning that the higher the minimum return expectation when the employee makes an effort, the higher the contribution given to the employee. On the contrary, the threshold k c % decreases with respect to , , , L k m   , meaning that the higher the minimum return expectation when the employee does not make an effort, the smaller the contribution given to the employee. If , , , L k m   is high, the manager does not need to significantly remunerate the employee. Note that, if we compare the above solution to the solution when there is no ambiguity corresponding to a unique value is decreasing, which corresponds exactly to the assumption of Proposition 3. This means that if the aversion to ambiguity is sufficiently high, then we find the same behavior as in the no ambiguity case.

H m L k m d k H k m s W c W        % (
Note that the ambiguity index of the manager

 

, , , ,

, c H k m m RM CP  is itself decreasing, which means that the higher the contribution c given by the manager to the employee, the higher the manager's aversion to ambiguity. Proof: See Appendix.

PROPOSITION 4

Under previous assumptions, for

p u W r cW R M E p u W r cW R M e                  (15)
Given the above assumptions, we have , where N denotes a Gaussian random variable with zero mean and standard deviation equal to 1. Thus, we have:

      2 1 , , /2 , , 00 [ (1 ) (1 )] (1 )
/ 1 j G e N j G e s s s s E u W r cW R M W r cW e                         2 1 , , /2 , , 00 [ (1 ) (1 )] (1 )
/ 1 j B e N j B e s s s s E u W r cW R M W r cW e                  
The two previous terms increase respectively with respect to ,, . Thus, we get:

  1 () 1 c     (16)                 2 2 2 2 1 ,, /2 0 0 1 ,, /2 0 1 ,, /2 0 0 1 ,, /2 0 (1 ) (1 ) (1 ) (1 )
(1 )

H G e N ss L G e N ss H B e N ss L B e N ss W r cW e p W r cW e W r cW e p W r cW e                                                       
j G e ss j B e ss

j p u W r cW R M E p u W r cW R M e               
Due to the above assumptions, we have , , , ,

11 j G e j G e R M N       and , , , , 11 j B e j B e R M N      
, where N denotes a Gaussian random variable with zero mean and standard deviation equal to 1. Thus, we have:

    2 , , , , 00 [ (1 ) (1 )] exp (1 ) 1 / 2 s j G e j G e s s s s a cW E u W r cW R M a W r cW a                              2 , , , , 00 [ (1 ) (1 )] exp (1 ) 1 / 2 s j B e j B e s s s s a cW E u W r cW R M a W r cW a                         
The two previous terms increase respectively with respect to ,, . Therefore, we get: . Thus, we have min c , such that for min cc  , () c

1 () c a     (18)                 2 2 2 2 ,, 0 2 0 ,, 0 2 ,, 0 2 0 ,, 0 2 exp (1 ) 1 exp
 is decreasing with respect to c .

In such a case, we recover the property given in Proposition 2. Therefore, the optimal ownership contract c  is respectively increasing with respect to

Concluding remarks

In this paper, we introduce ambiguity aversion to a model setup previously advanced by [START_REF] Aubert | Employee ownership: a theoretical and empirical investigation on reward management vs. management entrenchment[END_REF]. In this model, the authors consider that managerial monetary contributions in company stocks to the employee reveal the management type. The present paper proposes a sequential game where a risk-neutral manager grants company stocks to an ambiguity-averse employee. We focus on the most common situation in which employee ownership positively affects corporate performance. While previous experimental and empirical literature has documented behavioral cognitive biases affecting employees when their employer's stocks are offered, this paper is the first attempt to take ambiguity into account in a theoretical model. We extend the results of [START_REF] Aubert | Employee ownership: a theoretical and empirical investigation on reward management vs. management entrenchment[END_REF] in the framework of ambiguity using the multiple prior preferences of [START_REF] Gilboa | Maxmin expected utility with a non-unique prior[END_REF] and its extension provided by [START_REF] Maccheroni | Ambiguity aversion, robustness, and the variational representation of preferences[END_REF]. We provide a general outcome of optimal employee ownership under ambiguity. The optimal contribution in company stock is determined in the situation described by a subgame perfect Nash equilibrium. We show that in the framework of [START_REF] Gilboa | Maxmin expected utility with a non-unique prior[END_REF], first, the higher the minimum return expectation when the employee makes an effort, the higher the contribution given to the employee. Second, the higher the minimum return expectation when the employee does not make an effort, the smaller the contribution given to the employee. Third, if the lower bound of the mean of the company excess returns in the case of low effort is high, the manager does not need to significantly remunerate the employee. Concerning the optimal ownership contract, we show that the contribution in company stocks respectively increases with respect to the lower bound of the return expectation in the case of a high level of effort and decreases with respect to the upper bound of the return expectation in the case of a low level of effort. In the framework of [START_REF] Maccheroni | Ambiguity aversion, robustness, and the variational representation of preferences[END_REF], we prove first that, if the aversion to ambiguity is sufficiently high, then we find the same behavior as in the no ambiguity case. Second, the contribution given to the employee by the manager decreases when the manager faces stronger ambiguity.

e e e e R M R M R M R M p E u X p E u Z f r dr p E u Y p E u T f r dr p C P C P p C P C P                             
For the second property, we note that:

  ) (c  (20)             1 ( ( ) (1 ) ( ) ( ) 1 ( ) (1 ) ( ) ) ( ) (1 ) ( ( ) (1 ) ( ) ( ) (1 ) c H G e c H B e MM c L B e c L G e M M c H B e c H G e M M cL M s s p E Xu X p E Zu Z f r dr c p E Yu Y p E Tu T f r dr c Wr p E u X p E u Z f r dr c Wr pE c                                                   ( )
E M c u X E M c u Y f r dr p E M c u Z E M c u T f                                                 , , , , , , , , , , , , , , , , 00 ()
( ) ( ) ( ) ( ) (1 ) c H G e c L G e c H B e c L B e e e e e R M R M R M R M r dr C P C P C P C P pp cc                    
Using assertion (1) of Assumption 3, we have:

                ( ) ( ) , ( ) ( ) , ( ) ( ) , ( ) ( ) .
E u Y E u X E u T E u Z E Xu X E Yu Y E Zu Z E Tu T                
Using assertion (2) of Assumption 3, we have:

        , , , , , , , , , , , , , , , , , , , , , , , , , , / ( )
E M c u X E M c u Y E M c u Z E M c u T                                  
Using assertion (3) of Assumption (3), we have:

, , , , , , ,, ( ) ( ) 0
, c H G e c L G e ee RR C P C P c        , , , , , , ,, ( ) ( ) 0.

PROOF OF COROLLARY 1

For the basic example, we get the result directly. Indeed, we have:

 ) (c  (
                              
First, we have (0) 0   . For the second property, we note:

-Condition (1) of Assumption 3 is satisfied since we assume 3) is satisfied since (.) 0.

M c M c M c M c                 -Condition (
( , ) j k j k m j k m j k m j k m j k m m j k m j k m RM M dM dM C P R M M E Log dM dM                   (22) 2 , , , , 0 1 . 2 j k m j k m             , 0 , , , , , , , , 0 , , , , , 00 ( )
Using Lemma 2, the manager's initial expected wealth when the contribution c is null is defined by equation ( 23

):   , ,, , 0, , , , , 1 ( ) jk j k m m RM j k j k m P d m RM V Min W E R M C P        (23) 0, , , 2 0, , , , , , , 0 1 1, 2 j k m j k m j k m d W               
   which leads to (first-order condition):

(24)

The manager's initial expected wealth for any given contribution c is defined from: Both intuitively and in practice, this means for example that the manager increases the contribution given to the employee if the minimal expected return itself increases. is decreasing with respect to c . Thus, using (1), () c  is itself decreasing, which corresponds exactly to the assumption of Proposition 3. This means that, if the aversion to ambiguity is sufficiently high, then we find the same behavior as in the no ambiguity case.

    , , , , , , , , , ( ) 1 (
) jk j k m RM m c j k j k m P d s m RM V Min W cW E R M C P         (25) 2 , , , , , , , , , , 0 1 ( ) 1 , 2 c j k m j k m c j k m ds W cW                    which leads to (first-order condition):   , , , , , , 2 0 /. c j k m j k m ds W cW         (

Examine now the maximum compensation

PROOF OF PROPOSITION 4

A Perfect Subgame Nash Equilibrium  .

Thus, we can conclude that the employee's participation in the firm is optimal. Thus, This characterizes a Perfect Subgame Nash Equilibrium with a positive e c , with equality in the preceding relation.

  to manager's type k and employee's effort j . Random variables R

  evaluate the expectation of respective utilities of both the agents. There are also two ambiguity indices,

  basic example corresponds to straightforward ambiguity about the parameter values of the joint distributions of R and

  Under assumption (3), the difference between the employee's utilities of wealth with high and low levels of effort defined by () c  has the following properties:

  where c  is the equilibrium contribution in company stock. The value k c % is given by the manager's participation condition the maximum compensation that can be granted by the manager. Above k c % , employee ownership becomes too costly for the manager. The two threshold constraints G c and B c differ according to the two types of managers who know their type, but which cannot be observed by the employee. Indeed, GB cc  . Consider the following equation corresponding to the equality of ,,

  If the ambiguity index corresponds to the Log entropy, then if the aversion to ambiguity is sufficiently high, i.e.

.EXAMPLE 5

 5   Assumption 3 is satisfied since all three of its assertions are verified (in particular, depend on c, implying that assertion (2) is satisfied).Therefore, from Proposition 2, we deduce that () a low level of effort). Finally, using the theorem of implicit functions, the optimal ownership contract c  is respectively increasing with respect to ,, This property related to the employee's ambiguity is analogous to the sensitivities of the threshold k c % corresponding to the manager. Consider the assumptions of example 5 and suppose that the utility function is given by

  utility of the employee is strictly positive when they choose the level of effort L j  , and strictly greater than the value obtained when they choose the level of effort

  the manager's exit cannot correspond to an equilibrium since the first point, and it is strictly positive. Thus, condition the maximum compensation that can be granted by the manager. Above k c % , employee ownership becomes too costly for the manager. Indeed, the manager grants company stock if respect to c and that the value of k c % is characterized by

According to the National Center for Employee Ownership (https://www.nceo.org/articles/statistical-profileemployee-ownership accessed on December 2016).

http://www.norc.org/research/projects/pages/general-social-survey.aspx

[START_REF] Benartzi | Excessive extrapolation and the allocation of company stock to retirement accounts[END_REF],[START_REF] Holden | 401(k) plan asset allocation, account balances, and loan activity in 2000[END_REF],[START_REF] Brown | 401(k) matching contributions in company stock: costs and benefits for firms and workers[END_REF] andPurcell (2003) found that concentration in company stock is significantly higher in companies where matching contributions in company stocks are offered.

Risk-neutrality is the standard assumption about managers' behavior in both the microeconomic and corporate finance literature. Thus, we make this assumption while introducing risk aversion for employees. The same setup is followed by[START_REF] Aubert | Employee ownership: a theoretical and empirical investigation on reward management vs. management entrenchment[END_REF].

Appendix PROOF OF PROPOSITION 2

First we have (0) 0   . Indeed, when the contribution to the employee in the company stock is null, the expected utility of the employee, without taking their effort into account, always equals 0, 0

. Thus, the difference (0)  is null.

Let   c  be defined by Equation (11). Recall that we have:

, , , ,

(1

Therefore, we deduce: