Vinicius Fiocco Sciuti

Rafael Vargas

Guerrero

María Eugenia Marante

Hild

Franc ¸ois Hild
email: francois.hild@ens-paris-saclay.fr

Keywords: Adapted mesh, crack opening displacement, digital image correlation (DIC), full-scale experiment, mechanical regularization

Masonry walls are subjected to cyclic lateral loads to study damage caused by earthquakes. Imaging techniques are useful to quantify crack networks in such tests, where preferential locations for their initiation are not present. However, detecting small cracks in large structures is challenging. It is shown that FE-based digital image correlation (DIC) can detect and quantify cracks by combining optical and mechanical information of a cyclic shear experiment performed on a full-size masonry wall. Pixel-wise gray level residuals and elementary crack opening displacement fields are the key quantities of the proposed framework. Detection criteria based on standard uncertainties guided the application of new DIC strategies (i.e., mechanical regularization, mesh adaption, and damage). Two damage regimes were quantified.

Zigzagged cracks were first formed for which their opening displacements were on average less than 0.5 mm with very limited damage. They were followed by sliding shear cracks whose mean opening displacements varied between 1 and 3 mm, and damage developed in a more gradual and extended way. Such rich full-field dataset may be used for validating damage models up to full scale simulations.

Introduction

Masonry infilled frames are the structural system most used in low-income housing in Latin American countries due to the accessibility of the materials and simplicity of construction. Most of these countries lie in areas with moderate to high seismic hazards. Almost always, these constructions were built without any structural analysis or design, thus the structural performance under different types of load histories they may be subjected remains uncertain.

It is thus desirable to have models to perform numerical simulations of masonry structures, for instance macromodels accounting for damage initiation and growth. Using these models, essential features observed in experiments should be reproduced (e.g., shear strength, strain levels, and cracking patterns of confined and unconfined masonry panels). [START_REF] Janaraj | Finite element analysis of the in-plane shear behaviour of masonry panels confined with reinforced grouted cores[END_REF][START_REF] Yacila | Pushover analysis of confined masonry walls using a 3d macro-modelling approach[END_REF][START_REF] Shahzada | Seismic vulnerability assessment of confined masonry structures by macro-modeling approach[END_REF][START_REF] Korswagen | High-resolution monitoring of the initial development of cracks in experimental masonry shear walls and their reproduction in finite element models[END_REF] Different Latin American countries have adopted in their analysis and design codes, criteria mainly based on ultimate The approaches for modeling the behavior of infilled frames are divided into two groups, namely, microand macro-models. [START_REF] Asteris | The state-of-theart in infilled frames numerical models[END_REF] In micromodels, the masonry is represented as a continuum discretized into numerous finite elements. This approach gives rich descriptions of the phenomena taking place in the infill under overloads including the interface between frames and infill. In macromodels, the infill is usually substituted by a single, or a set of diagonal struts. The micromodels intend to consider all possible failure modes, but their use is limited to very simple structures due to the complexity of the analysis and the large amount of data that are collected. For large structures such as buildings, macromodels are usually chosen. Very simple and schematic results are obtained yet they are considered accurate enough for engineering purposes.

Petry and Beyer [START_REF] Petry | Limit states of modern unreinforced clay brick masonry walls subjected to in-plane loading[END_REF] indicated that to improve the performance-based evaluation of buildings with unreinforced masonry, future research should focus on the development of models that relate the global force-displacement response to local deformations of the considered elements. The authors proposed two sets of limit states that link localized damage to characteristic points in the global forcedisplacement response of unconfined masonry. Deformation measurements at the brick level were performed with an LED-based optical system. An alternative route consists in utilizing Digital Image Correlation (DIC). [START_REF] Sutton | Computer vision-based, noncontacting deformation measurements in mechanics: A generational transformation[END_REF] DIC was used to measure motions of large scale structures such as bridges. [START_REF] Küntz | Digital image correlation analysis of crack behavior in a reinforced concrete beam during a load test[END_REF][START_REF] Mousa | Application of digital image correlation in structural health monitoring of bridge infrastructures: A review[END_REF] In the following, it will be used to detect and quantify two damage mechanisms at the microscale of a masonry infilled frame subjected to confined shear.

The feasibility of DIC analyses for the analysis of brick walls was shown by Tung et al. [START_REF] Tung | Development of digital image correlation method to analyse crack variations of masonry wall[END_REF] by reporting displacement and strain fields. A lot of emphasis was put on the use of the equivalent von Mises strain. Later on, it was also suggested that major eigen strain fields were more suited for the visualization of damage [START_REF] Korswagen | High-resolution monitoring of the initial development of cracks in experimental masonry shear walls and their reproduction in finite element models[END_REF][START_REF] Bui | Out-of-plane behaviour of hollow concrete block masonry walls unstrengthened and strengthened with cfrp composite[END_REF][START_REF] Herbert | Uniform lateral load capacity of small-scale masonry wall panels[END_REF][START_REF] Vanniamparambil | A data fusion approach for progressive damage quantification in reinforced concrete masonry walls[END_REF][START_REF] Salmanpour | Displacement capacity of contemporary unreinforced masonry walls: An experimental study[END_REF][START_REF] Casolo | Testing masonry blockwork panels for orthotropic shear strength[END_REF][START_REF] Stazi | Experimental study of the mechanical behaviour of a new extruded earth block masonry[END_REF][START_REF] Torres | Study on retrofitted masonry elements under shear using digital image correlation[END_REF][START_REF] Elghazouli | In-plane lateral cyclic behaviour of lime-mortar and clay-brick masonry walls in dry and wet conditions[END_REF][START_REF] Howlader | Digital image correlation for the analysis of in-plane tested unreinforced masonry walls[END_REF] and the analysis of various damage mechanisms in masonry. [START_REF] Guerrero | Experimental analysis of masonry infilled frames using digital image correlation[END_REF][START_REF] Ghorbani | Full-field deformation measurement and crack mapping on confined masonry walls using digital image correlation[END_REF][START_REF] Calderón | Quasistatic testing of concrete masonry shear walls with different horizontal reinforcement schemes[END_REF] Other strain components were also considered depending on the investigated experiment. [START_REF] Tung | Development of digital image correlation method to analyse crack variations of masonry wall[END_REF][START_REF] Serpilli | Earthen claddings in lightweight timber framed buildings: An experimental study on the influence of fir boards sheathing and gfrp jacketing[END_REF] Bolhassani et al. [START_REF] Bolhassani | Failure analysis and damage detection of partially grouted masonry walls by enhancing deformation measurement using dic[END_REF] defined structural cracks as zones in which the strain levels were greater than 0.3%. It was also shown DIC was crucial in other aspects associated with boundary conditions and calibration of mechanical parameters. [START_REF] Fedele | Delamination tests on cfrp-reinforced masonry pillars: Optical monitoring and mechanical modeling[END_REF] Guerrero et al. [START_REF] Guerrero | Experimental analysis of masonry infilled frames using digital image correlation[END_REF] used DIC for evaluations of simplified models of infilled framed structures. Two different deformation mechanisms were found, namely, diagonal cracking (Figure 1(d)) during hardening phases and sliding shear during softening stages (Figure 1(b)). The inclination of the bands depended on the brick dimensions and arrangement.

Didier et al. [START_REF] Didier | Damage quantification in plastered unreinforced masonry walls using digital image correlation[END_REF] quantified two proposed damage scores using DIC. Korswagen et al. [START_REF] Korswagen | Crack initiation and propagation in unreinforced masonry specimens subjected to repeated in-plane loading during light damage[END_REF] used DIC to detect crack formation and the evolution of the crack pattern, to calibrate and validate nonlinear finite element models. The authors defined a damage parameter as a function of the number, length and width of cracks in masonry walls. In the following analyses, two other damage indicators will be introduced.

Ghorbani et al. [START_REF] Ghorbani | Full-field deformation measurement and crack mapping on confined masonry walls using digital image correlation[END_REF] performed uncertainty quantifications and validated DIC readings against point-wise sensors for experiments on full-scale masonry walls. Attempts were made to evaluate crack widths from displacement fields as previously proposed by Fedele et al. [START_REF] Fedele | Delamination tests on cfrp-reinforced masonry pillars: Optical monitoring and mechanical modeling[END_REF] Other studies followed along the same direction. [START_REF] Korswagen | Crack initiation and propagation in unreinforced masonry specimens subjected to repeated in-plane loading during light damage[END_REF][START_REF] Calderón | Influence of a window-type opening on the shear response of partiallygrouted masonry shear walls[END_REF][START_REF] Meriggi | Shear-compression tests on stone masonry walls strengthened with basalt textile reinforced mortar (trm)[END_REF] More recently, systematic damage quantification was performed when analyzing eigen strain fields. Detection threshold were introduced with arbitrary levels [START_REF] Rezaie | Comparison of crack segmentation using digital image correlation measurements and deep learning[END_REF] or based on measurement uncertainties. [START_REF] Gehri | Automated crack detection and measurement based on digital image correlation[END_REF] Uncertainty levels will be utilized hereafter to adapt the mesh in FE-based DIC, and to detect damaged elements.

Prepared using sagej.cls

The experiment studied herein was part of the test campaign reported by Guerrero et al. [START_REF] Guerrero | Experimental analysis of masonry infilled frames using digital image correlation[END_REF] It was selected since it led to two different damage mechanisms (as explained above). The damage patterns, which were the main focus of the study, were revealed thanks to maximum eigen strain fields measured via Q4-DIC (i.e., FE-based DIC with 4noded quadrilaterals [START_REF] Besnard | Finite-element" displacement fields analysis from digital images: Application to Portevin-Le Chatelier bands[END_REF] ). The cyclic nature of the experiment, which could have made DIC analyses demanding, could be performed with a standard code. Conversely, part of the cladding fell off during the test, which made this case challenging. The following analyses are devoted to the quantification of crack opening displacements, and their detection based on measurement uncertainties. Contrary to common practice, [START_REF] Gehri | Automated crack detection and measurement based on digital image correlation[END_REF] the detection criterion used herein is based on the Lagrangian gray level residuals of DIC analyses. [START_REF] Hild | Damage Measurements via DIC[END_REF] In that case, the speckle pattern essentially disappears (except in areas where the cladding was damaged) and its segmentation would have been made easier in comparison with images of the deformed configurations. [START_REF] Rezaie | Comparison of crack segmentation using digital image correlation measurements and deep learning[END_REF] The outline of paper is as follows. First, the investigated experiment is briefly recalled. A recently introduced mesh adaption strategy [START_REF] Sciuti | Pyramidal Adaptive Meshing for Digital Image Correlation Dealing With Cracks. Prepared using sagej[END_REF] was considered, which leads to non conforming meshes with hanging nodes. The latter ones were reconnected to edges of adjacent elements thanks to Lagrange multipliers. For any considered discretization, one key quantity is the standard uncertainty of any measurand. [START_REF] Hild | Comparison of local and global approaches to digital image correlation[END_REF][START_REF]Standardization, Good Practices, and Uncertainty Qunatification Committee. A Good Practices Guide for Digital Image Correlation[END_REF][START_REF]Aerospace series -Metrological assessment procedure for kinematic fields measured by digital image correlation[END_REF] Standard displacement, strain and crack opening displacement [START_REF] Sciuti | Digital image correlation applied to in situ evaluation of surface cracks upon curing of MgO-containing refractory castables[END_REF] uncertainties were first assessed for uniform meshes. A first series of DIC results are then reported for a coarse mesh, which served as initialization for analyses with adapted meshes. Three different criteria were then compared for the detection of damaged elements. Second, from the previous studies, one adapted mesh was considered and the DIC results are analyzed in terms of damage chronology, damage pattern, damage deactivation/reactivation and finally on the distribution of MCODs and surface damage density for the two damage mechanisms.

Material and Methods

Experiment

The experiment reported hereafter was carried out at the A servo-controlled hydraulic actuator with a capacity of 500 kN was used to apply the desired displacement history and to measure the corresponding applied force. The actuator was mounted on a reaction frame and connected to the specimen through a steel device (Figure 3). In order to apply a uniform vertical load to the frame, an additional structure was built on top of it, in which blocks and steel plates were placed until reaching a total weight of 50 kN. A cyclic horizontal displacement was applied on the top part by a hydraulic actuator to mimic seismic motions.

DIC framework

FE-based registrations All the DIC analyses were performed with the Correli 3.0 framework, [START_REF] Leclerc | [END_REF] and the selected parameters are summarized in Table 2. Brightness conservation was considered

f (x) = g(x + u(x)) (1)
for each pixel x in the reference image f (of the unloaded state, see Figure 4(a)) and in deformed (state) corresponding to image g where a displacement u has occurred. The displacement fields are discretized using 3-noded (T3) elements whose shape functions are Ψ T3 i . The corresponding nodal displacements υ i , which are gathered in the column vector {υ υ υ} are determined via non-linear least squares Prepared using sagej.cls minimization of the global gray level residual ϱ

ϱ 2 ({υ υ υ}) = ROI [f (x) -g(x + i υ i Ψ T3 i (x))] 2 (2)
by iteratively solving linear systems of the form [H]{δυ υ υ} = {h}, in which {δυ υ υ} is the vector of nodal displacement corrections, [H] the Hessian matrix

H ij = ROI (Ψ Ψ Ψ T3 i • ∇ ∇ ∇f )(x)(Ψ Ψ Ψ T3 j • ∇ ∇ ∇f )(x) (3)
and {h} the right-hand member

h i = ROI [f (x) -g(x)](Ψ Ψ Ψ T3 i • ∇ ∇ ∇f )(x) (4)
with g the corrected image g by the current estimate of the displacement field ũ, g(x) = g(x + ũ(x)). The normalized pixel-wise gray level residuals are thus defined as

ρ(x) = f (x) -g(x) ∆f ( 5
)
where ∆f is the dynamic range of the reference picture (i.e., max(f ) -min(f )).

Once convergence is achieved, the elementary strains are obtained by exact differentiation of the shape functions.

Usually, the maximum eigen strain ϵ 1 is used to quantify crack networks [START_REF] Guerrero | Experimental analysis of masonry infilled frames using digital image correlation[END_REF][START_REF] Rezaie | Comparison of crack segmentation using digital image correlation measurements and deep learning[END_REF][START_REF] Gehri | Automated crack detection and measurement based on digital image correlation[END_REF] independently of their orientation or cracking mode. The Mean Crack Opening Displacement (MCOD) per element is calculated from the mean strain tensor estimated for each element e covering a domain Ω e . [START_REF] Sciuti | Digital Image Correlation applied to in situ evaluation of surface cracks upon curing of MgO-containing refractory castables[END_REF] In the presence of cracks, the mean strain tensor is expressed as 49

ϵ ϵ ϵ = 1 |Ω e | Ωe ϵ ϵ ϵ(x)dx + 1 |Ω e | Γe u(x) ⊠ n(x) ds (6)
where u denotes the displacement jump, n the normal to the cracked surface Γ e , ⊠ the symmetrized tensorial product, ϵ ϵ ϵ the strain tensor of the uncracked matrix, and s the curvilinear abscissa along the crack path. In the following equations, it is assumed that the elastic strain levels are negligible in comparison to the singular contribution provided by cracks. Consequently, the mean strain tensor becomes

ϵ ϵ ϵ ≈ 1 |Ω e | Γe u(x) ⊠ n(x) ds (7)
The MCOD u per finite element is defined as

u ⊠ n = 1 |Γ e | Γe u ⊠ n ds ( 8
)
where n is the mean crack normal. The average maximum principal strain ϵ 1 is then related to the equivalent MCOD

|Ω e | ϵ 1 = |Γ e | u (9)
that accounts for both mode I and II contributions. It is worth noting that |Ω e |/|Γ e | scales with the element size ℓ, provided only one crack is present in each element. This is likely to be true for small element sizes. Consequently, the MCOD is approximated by

u = ℓ ϵ 1 (10)
Mechanical regularization Mechanical regularization, [START_REF] Réthoré | An extended and integrated digital image correlation technique applied to the analysis fractured samples[END_REF][START_REF] Tomičević | Mechanics-aided digital image correlation[END_REF] which is based on the equilibrium gap, [START_REF] Claire | A finite element formulation to identify damage fields: The equilibrium gap method[END_REF] was used to help convergence in the present case. Considering that no internal forces should be present if the sample obeys linear elasticity, the regularization scheme changes the system to be solved to

[H t ]{δυ υ υ} = {h t }, in which [H t ] = [H] + ω m {v} ⊤ [H]{v} {v} ⊤ [K] ⊤ [K]{v} [K] ⊤ [K] (11)
and

{h t } = {h} -ω m {v} ⊤ [H]{v} {v} ⊤ [K] ⊤ [K]{v} [K] ⊤ [K]{ υ υ υ} (12)
where [K] is the stiffness matrix, {v} the nodal displacements associated with a trial displacement field described by a plane wave, { υ υ υ} the vector containing the current estimate of the nodal displacements, and ω m the weight, which is related to regularization length L reg by ω m ∝ L 4 reg . In this paper, L reg = 0 and L reg = 50 px were selected and compared for highlighting the interest of the discussed methodology. The 50-px length was chosen as a compromise between measurement uncertainty and not filtering out too much high frequency displacements. The interested reader is referred to Refs. [START_REF] Tomičević | Mechanics-aided digital image correlation[END_REF][START_REF] Mendoza | Complete mechanical regularization applied to digital image and volume correlation[END_REF] for additional details on the implementation of such regularization schemes.

Hanging nodes

The mesh considered hereafter will not be uniform, contrary to classical practice. [START_REF] Besnard | Finite-element" displacement fields analysis from digital images: Application to Portevin-Le Chatelier bands[END_REF][START_REF] Réthoré | An extended and integrated digital image correlation technique applied to the analysis fractured samples[END_REF] It will be adapted according to a criterion based on the mean crack opening displacement [START_REF] Sciuti | Pyramidal Adaptive Meshing for Digital Image Correlation Dealing With Cracks. Prepared using sagej[END_REF] and its corresponding uncertainty that is assessed for each element. Nodes with a reduced number of connectivities, also called hanging nodes, [START_REF] Sciuti | Pyramidal Adaptive Meshing for Digital Image Correlation Dealing With Cracks. Prepared using sagej[END_REF][START_REF] Fries | Hanging nodes and xfem[END_REF] were generated during the refinement steps. Lagrange multipliers were used to prescribe the displacement of each hanging node to that of its adjacent nodes. For instance, the displacement of a hanging node in the middle of an edge is the mean value of the displacement of the two adjacent nodes. This approach was implemented in the DIC code by concatenating an auxiliary Lagrange matrix [L] to the Hessian matrix [H t ], thereby generating an augmented

Hessian matrix [H a ] [H a ] = [H t ] h[L] ⊤ h[L] [0] ( 13
)
where h is the ratio of the Frobenius norms of [H t ] and [L]

to avoid degrading the conditioning of the full system

[H a ]{δβ β β} = {h a } (14)
composed of a vector of unknowns {β β β} containing nodal displacements and Lagrange multipliers, and the augmented

residual vector {h a } {h a } ⊤ = {{h t } ⊤ {0} ⊤ } (15)
A detailed discussion of this approach is given in Ref. [START_REF] Sciuti | Pyramidal Adaptive Meshing for Digital Image Correlation Dealing With Cracks. Prepared using sagej[END_REF]

Uncertainty quantification with uniform meshes

Only one reference image (i.e., taken in the unloaded state) was available (Figure 4(a)). Since the gray level residual from the DIC analysis of the first pair of images was essentially associated with white Gaussian noise, 100 new reference images were created by adding Gaussian white noise (i.e., standard deviation equal to 4.2 gray levels) to the reference image. Then, the uncertainties for displacements, strains, and MCODs were assessed by taking the standard deviation of the values measured in this set of images. If several images of the unloaded state had been available, the uncertainty analysis would have been directly conducted on them as is current practice [START_REF] Hild | Comparison of local and global approaches to digital image correlation[END_REF][START_REF]Standardization, Good Practices, and Uncertainty Qunatification Committee. A Good Practices Guide for Digital Image Correlation[END_REF][START_REF]Aerospace series -Metrological assessment procedure for kinematic fields measured by digital image correlation[END_REF] .

First, uniform meshes were considered, starting with an element size ℓ = 560 px, which was divided by two each time as shown in Figure 5. This analysis allows the thresholds to be set for adapting the mesh to better capture the crack pattern. as functions of the element size. When no mechanical regularization was used (i.e., L reg = 0), the uncertainties decreased with an increase in element length. This trend is the well-known compromise between measurement uncertainty and spatial resolution. [START_REF] Hild | Comparison of local and global approaches to digital image correlation[END_REF] Conversely, when mechanical regularization was used (i.e., L reg = 50 px) the standard displacement uncertainties decreased for elements whose size was greater than the regularization length. For small elements, the displacement uncertainty was controlled by the regularization length and no longer by the element size.

The standard strain uncertainty had a dual decrease depending on the element size. This trend was due to the fact that the standard strain uncertainty was proportional to the standard displacement uncertainty divided by the element size. For MCODs, which again considered the element length in its calculation, the uncertainty was smaller than for small elements, further highlighting the interest of using such regularization scheme.

Prepared using sagej.cls

Initial DIC results

The present case was very challenging for two main reasons.

First, very high displacement amplitudes (i.e., more than 100 px) were observed for some images of the cyclic test. Second, the wall cladding was also damaged as a result of the loading history (i.e., significant deviations from gray level conservation). Therefore, some care had to be taken for the first DIC analysis that was used to initialize the subsequent procedures. A very coarse mesh with only 15 nodes (Figure 5(a)) was first used with no regularization. Moreover, an incremental approach was first run, meaning that the reference image and the nodal positions were updated for each new image to mitigate the influence of the changes in speckle pattern. This calculation was also initialized with a rigid body translation estimate using the cross-correlation product computed via fast Fourier transform. [START_REF] Hild | Multi-scale displacement field measurements of compressed mineral wool samples by digital image correlation[END_REF] It took at most 30 iterations (apart from the last frame) to converge for the norm of displacement increment to become less than 10 -3 px. In a direct analysis, it would have taken more than 200 iterations (or even diverge) if no proper initialization was performed.

To illustrate the result of this first analysis and to compare with the following ones, the final gray level residual for the most loaded state (with positive horizontal displacement) is shown in Figure 7

Mesh adaption

Starting from the coarse discretization shown in Figure 5(a), the mesh was locally refined following the procedure described in Ref. [START_REF] Sciuti | Pyramidal Adaptive Meshing for Digital Image Correlation Dealing With Cracks. Prepared using sagej[END_REF] Each element is refined if the MCOD is greater than k times the standard uncertainty. For each step, the considered uncertainty was changed; it followed the estimates obtained for a uniform mesh (L reg = 0 px in Figure 6(c)). Six refinement steps were performed, which led to element sizes ranging from 9 px up to 560 px. Moreover, instead of using only one image for the deformed state as in Ref. [START_REF] Sciuti | Pyramidal Adaptive Meshing for Digital Image Correlation Dealing With Cracks. Prepared using sagej[END_REF] , two images were considered (i.e., #67 and #70, see Figure 8), to account for both positive and negative prescribed displacements (Figure 4(b)). DIC analyses were run for each step using such images, and the elements were divided if the measured MCOD surpassed the threshold on any of these two results. This type of mesh adaption was tested for monotonically increasing crack openings and propagation. [START_REF] Sciuti | Pyramidal adaptive meshing for Digital Image Correlation dealing with cracks[END_REF] In this work, due to the cyclic nature of the loading history, two systems of cracks were created, and therefore the use of two images together for such procedure was tested (see Section "Crack network development"). The gray level residual image using the adapted mesh is shown in Figure 11 (to be compared with the results displayed in Figure 7). The areas with higher residuals are more localized in the cracked region. Far from them, the residual was considerably lower with no special pattern (i.e., signature of noise). In some specific regions, it is possible to read the initial pattern in the residual since part of the wall cladding fell off and the background became white (see Figure 8).

Detection of Damage

Even though mesh adaption per se already gave a good description of the cracked regions, the use of mechanical regularization spread the strains around them. [START_REF] Hild | Damage Measurements via DIC[END_REF] Therefore, damage was added, leading to a relaxation of mechanical regularization locally in order to allow for higher displacement gradients. To choose which elements to damage (by multiplying its rigidity by (1 -D 0 ) with D 0 = 0.9), three criteria were tested, namely, related to the MCOD, gray level residual, and both coupled. If, for a given image, the quantity of interest is higher than K times the uncertainty, the element is labeled damaged and its elementary stiffness matrix is degraded. All the following calculations were initialized with the one fully converged that used the adaptive mesh with no damage. The parameter D 0 = 0.9 was selected to lower the regularization weight in damaged elements. This choice was motivated by two conflicting effects. If D is not high enough, it will be difficult to relocalize the strain and MCOD fields. [START_REF] Hild | Damage Measurements via DIC[END_REF] Conversely, given the fact that cladding loss was also observed (Figure 8), the damaged areas extended over many neighboring elements. If D was to high, it would have been impossible to achieve convergence of the DIC code.

For the sake of simplicity, the first choice was to keep K = k = 3 from the mesh adaption procedure. In the gray level residuals, the three approaches (i.e., MCOD, gray level or coupled criterion) were indistinguishable.

However, the damage fields were studied very carefully for deformed image #58 (Figure 4 When testing K = 6 for the same image #58, using MCODs for the detection criterion, undamaged areas were detected (Figure 13(a)). If only the gray level residuals were considered, a fine description of the cracked state was observed in the damage field (Figure 13(b)). Contrary to the cases with K = 3, the coupled criterion now seems to erase some small aspects of the cracked network seen in the gray level residuals, when comparing Figures 13(b,c), yet both led to very similar damage fields. (c, d) Corresponding damage fields.

Prepared using sagej.cls

Results and Discussion

The following section mostly focuses on MCOD data. The way they were computed, namely, from the maximum eigen strains averaged over each element, they will inform on locations where the cracks were open or sliding (i.e., only positive MCODs are shown).

Comparisons with simpler DIC implementations

If no adaptive mesh or damage was considered, an analysis with a very fine mesh with mechanical regularization would provide the best description of the displacement field in such challenging case. The results obtained with a uniform mesh with an element size of 9 px, which was the smallest in the adapted mesh, are first discussed. This mesh was created using the same element subdivision routine, but choosing all elements to be subdivided at each step. The MCOD field for image #58 is shown in Figure 15 Similar results are reported in Figure 16 with the adapted mesh (with k = 3). The DIC analyses were conducted with damage based on the gray level residual (with K = 6). In the MCOD field (Figure 16(a)), the localization of the cracks is visible in single elements. Even though the dynamic range of the color map was saturated at openings greater than 2 mm (for comparison purposes with the uniform mesh), local openings of up to 10 mm were captured for some elements.

This better description is also perceptible in the gray level residuals (Figure 16(b)) in which, for smaller openings, they are concentrated in a single pixel line. It is worth noting that for this small ROI, the adapted mesh had about 30% fewer elements than the uniform mesh, while still yielding a better description of the underlying phenomenon.

Crack network development

The following procedure aims to have a spatial and temporal description of damage development, which is summarized in The same type of map was drawn using only the elements that were damaged at the peak loads for positive (Figure 18 For a finer description of crack initiation, it is proposed to further investigate both series of images highlighted in the loading history displayed in Figure 19, for maximum positive or negative strokes where softening occurred. The range for the color map was set small in order to highlight the damage pattern (i.e., MCOD levels less than 1 px or 0.8 mm). The same analysis was performed for the fourth cycle.

Two intermediate MCOD fields are also shown in Figure 23 (i.e., images #20 and 21, see green triangles in Figure 19) in Contrary to the previous results (Figure 22), the horizontal crack experienced much higher MCOD levels. Prepared using sagej.cls

Macroscopic damage quantification

One first quantification of macroscopic damage is shown in Figure 24, by considering the mean shear strain of the whole ROI and comparing to its evolution with the applied forces.

The mean shear strain was very low for the first images, and as it increased, the force started to decay (in absolute value). The red circles depict the first 19 images (up to the peak load), which are the same highlighted inside the red box in Figure 25. It is interesting to note that from the previous results, it was concluded that damage had already initiated before reaching the ultimate force. To further investigate the oscillations observed in (among which the test investigated herein), which focused on damage mechanisms and on the final failure mode.

To quantify crack opening displacement fields, the maximum principal strain was utilized at the element level (i.e., microscale). Such indicator does not distinguish simultaneous mode I and II contributions. In the present case, it is believed that such situation did not occur. Had it been the case, directional indicators associated with the block arrangement would have been needed.

For the DIC analyses, the chosen regularization length and thresholds for adapting the mesh or setting damage (i.e., parameters k and K) may be further investigated for a broader range of combinations. Brightness and contrast corrections [START_REF] Sciuti | On the benefits of correcting brightness and contrast in global digital image correlation: Monitoring cracks during curing and drying of a refractory castable[END_REF] could also be applied to improve the description in the regions where cladding fell off and for describing changes in illumination. Different experimental setups and loading protocols could be tested using such methodology. More cameras may be added for stereocorrelation purposes, [START_REF] Sutton | Computer vision-based, noncontacting deformation measurements in mechanics: A generational transformation[END_REF] in which the out-of-plane motions would also be measurable. Last, these full-field analyses provide very rich data sets that can be used for comparisons to numerical simulations of full scale structural elements to validate damage models at various scales.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Figure 1 .

 1 Figure 1. Different failure modes of masonry-infilled frames subjected to simple shear. (a) Corner crushing mode; (b) sliding shear mode; (c) diagonal compression mode; (d) diagonal cracking mode; (e) frame failure mode (adapted from Ref. 12 ).

  Laboratory of Structural Mechanics (LME) in theUniversidad Centroccidental Lisandro Alvarado (Barquisimeto-Venezuela). The compressive strength of concrete (f ′ c ) was 25.0 MPa. The steel bars used in the panels were of ASTM designation A615, with a yield strength of 420 MPa. The size of the concrete masonry blocks was 40 × 20 cmwith a thickness of 15 cm , and their compressive strength was 1.5 MPa. The infill was built in standard overlapped way. Specifications of the frame are shown in Figure2. The interested reader will find more details on the geometry of the infill frame in Refs.[START_REF] Guerrero | Experimental analysis of masonry infilled frames using digital image correlation[END_REF][START_REF] Guerrero | Análisis Teórico-Experimental del Daño y del Pandeo Local en Estructuras de Ingeniería Civil[END_REF]

Figure 2 .

 2 Figure 2. Dimension and internal steel reinforcement details of the studied RC frame.

Figure 3 .

 3 Figure 3. Masonry infilled frame before the experiment. The hydraulic actuator is shown in the dashed red box. The 50 kN dead load is visible above the frame.

Figure 4 (

 4 Figure 4(a) shows the acquired image of the reference configuration with the Region Of Interest (ROI) for DIC purposes depicted as a yellow box. To achieve good contrast for DIC purposes, the speckle pattern (Figure3) was made first with a thin layer of black paint to emphasize the texture of the cladding. Black dots were then painted, and last yellow dots over the former. The hardware parameters are gathered in Table1.

Figure 4 .

 4 Figure 4. (a) Reference image of the unloaded configuration with the ROI highlighted in the yellow box. The smaller cyan box is discussed in Figure 14. (b) Loading history in which the blue dots depict image acquisitions. The red circles highlight the images used for adapting the meshes and the green diamonds those utilized for setting the damage parameter.

Figure 5 .

 5 Figure 5. Uniform meshes with element lengths of 560 px (a) and 280 px (b). Each triangular element is divided into four new triangles. The physical size of one pixel is 790 µm.

Figure 6 .

 6 Figure 6. Standard uncertainties in terms of horizontal displacement (a), major eigen strain (b), and MCOD (c) for uniform meshes and two regularization lengths. The physical size of one pixel is 790 µm.

Figure 7 .

 7 Figure 7. (a) Gray level residual for the initial DIC analysis using a coarse mesh (image #67). (b) Normalized standard deviation of gray level residuals (see Equation (5)). The green diamonds and red circles highlight images #19, 58, 67, and 70 further discussed in the text.

Figure 8 .Figure 9 )Figure 9 .

 899 Figure 8. Images #67 and #70 of two deformed states that were used in the mesh adaption scheme (Figure 4(b)), respectively for the most loaded state with positive (a) or negative (b) displacements in the x-direction.

Figure 10 .

 10 Figure 10. Standard MCOD uncertainties for the adapted mesh (AM, with k = 3, see Figure 9(b)) by groups of element size with no mechanical regularization or with Lreg = 50 px. The mean standard uncertainties of each group are compared to the uncertainty of the uniform mesh (UM) with same element size. The error bar length depicts the standard deviations associated with the adapted meshes. The physical size of one pixel is

Figure 11 .

 11 Figure 11. Gray level residual for the DIC analysis using the adaptive mesh (with k = 3, see Figure 9(b))

Figure 12 .

 12 Figure 12. Damaged elements for deformed image #58 using as detection criterion only MCODs (a), gray level residuals (b), or both of them (c), all with K = 3

Figure 13 . 6 AFigure 14 .

 13614 Figure 13. Damaged elements for deformed image #58 using as detection criterion only MCODs (a), gray level residuals (b), or both (c), all with K = 6

  (a) for a zoomed area. The field was blurred due to the mechanical regularization, which spread the strains over few elements. Its corresponding gray level residual map is shown in Figure 15(b). Although a proper description of the crack network is observed, there are interpolation errors close to the horizontal cracks, which were created by enforcing the continuity and mechanical admissibility of displacements.

Figure 15 .

 15 Figure 15. Details (see Figure 4(a)) of (a) MCOD and (b) gray level residual fields for image #58 using a fine uniform mesh.

Figure 16 .

 16 Figure 16. Details (see Figure 4(a)) of (a) MCOD and (b) gray level residual maps for image #58 using the adapted mesh.

Figure 17 .

 17 Figure 17. The color rendering of each element represents the image when damage initiated (i.e., when the threshold was surpassed). If the element is white, it means that it remained undamaged for the whole analysis. Many undamaged regions are observed between cracked regions. Damage initiated in the central horizontal region and along the main diagonals up to image #19. By reaching the maximum load, more horizontal cracked regions were formed, and damage grew toward the outside frame. The bigger damaged regions were related to a sudden fall of the cladding.

Figure 17 .

 17 Figure 17. Map of first image for which elements were damaged. If damage never occurred, the element remains white.

  (a)) or negative (Figure 18(b)) applied displacements. Apart from the previous conclusion pointing out that damage started in the center and grew outward, a first diagonal band is seen in both cases, which was aligned along the loading direction applied at the top part of the wall. Then, beyond the maximum force, other horizontal bands appeared, some for the very last images (e.g., see top of Figure 18(b)). These maps provide interesting data to validate damage models. Prepared using sagej.cls

Figure 18 .

 18 Figure 18. Map of first image for which elements were damaged in the peak loads for positive (a) and negative (b) applied displacements (see Figure 4(b)). If damage never occurred, the element remains white.

Figure 19 .

 19 Figure 19. Stroke and force histories. The blue dots depict image acquisitions, and the triangles the series of images studied for crack initiation for positive (in purple, images #13, 19, 25, and 37) or negative (in orange, images #16, 22, 28, and 40) prescribed displacements. To analyze the activation / deactivation of crack systems, additional frames were considered (in green, images #14, 15, 20, and 21).

Figure 20 .Figure 21 . 13 (

 202113 Figure 20. MCOD fields for images #13 (a), 19 (b), 25 (c), and 37 (d), at peak loads of positive displacement (Figure 19)

Figure 22 .

 22 Figure 22. MCOD fields for images #13 (a), 14 (b), 15 (c), and 16 (d). It is worth noting that the dynamic range was lowered in comparison to Figure 20

  addition to images #19 and 22 for comparison purposes. The result of the first image (Figure 23(a)) was already reported (Figure 20(b)) but with a different dynamic range. The highest MCOD levels were significantly higher in the central part of the wall. Figure 23(b) corresponds to the unloading stage with zero applied force. Many of the zigzagged cracks had essentially closed (as in the previous case, see Figure 22). However, the horizontal crack remained open, presumably due to friction. The zigzagged cracks further closed when the applied displacement was equal to zero (Figure 19(c)) and the horizontal crack was still substantially open. Conversely, the second system of zigzagged cracks that was fully open for image #22 (Figure 23(c)) was still mostly closed. The second system of zigzagged cracks opened again for image #22 (Figure 23(d)) but most of the MCOD took place in the central part of the wall and for the horizontal crack.

Figure 23 .

 23 Figure 23. MCOD fields for images #19 (a), 20 (b), 21 (c), and 22 (d). Sub-figures (a) and (d) were already reported with different dynamic ranges in Figures 20(b) and 21(b), respectively.

Figure 24 .Figure 25 .

 2425 Figure 24. Measured force as a function of mean shear strain calculated over the whole ROI. The red circles depicts all the images up to the peak load (image #19).

Figure 25 ,Figure 26 .

 2526 Figure 25, both mean MCOD and SCD are shown as functions of the minimum (Figure 26(a)) or maximum (Figure 26(b)) shear strain per cycle, depending whether the prescribed displacement was negative or positive. The increasing trend in both cases highlights the fact that damage mostly developed on the envelope of the loading history, while the intermediate points on loading reversal were related to closure of one crack system and opening of the other one.

Prepared using sagej.cls

Journal Name XX(X) (a)

Funding

The experiment discussed in this paper was conducted in the course of an investigation sponsored by FONACIT-ECOS NORD and CDCHT-UCLA. The present study was supported in part by the Coordenac ¸ão de Aperfeic ¸oamento de Pessoal de Nível Superior -Brasil (CAPES) -Finance Code 001, and grants #2018/23081-0 and #2021/09238-6, São Paulo Research Foundation (FAPESP).

Appendix: DIC parameters


Emilie Dargaud
email: dargaud@gate.cnrs.fr.

Armel Jacques
email: armel.jacques@univ-reunion.fr.

Slowdown antitrust investigations by decentralizationA Review

Keywords: Collusion, antitrust policy, leniency programs, multimarket contact, organizational form K42, L22, L41 Collusion, antitrust policy, leniency programs, multimarket contact, organizational form. JEL: K42, L22, L41

When multi-product firms make simultaneous price-fixing agreements in different markets, the introduction of leniency programs may induce firms to compartmentalize their activities. Doing so results in slowdown antitrust investigations and decentralized firm can easily request leniency for a second cartel after the detection of an other. We study how variation of fine reduction may produce procompetitive but also procollusive effects.

Introduction

The fight against cartels is the priority of antitrust authority and leniency program is widely used to facilitate their detection. 1 Despite a consensus regarding its utility, the design and effects of this program are yet a matter of many debates. 2 This article analyzes the interactions between leniency program and internal organization of multiproduct collusive firms. Theoretical works have often been produced on the premise that inter-firm contact may facilitate collusion. 3 Antitrust authorities have incorporated this into their procedures and investigate all anticompetitive conducts inside a firm when they detect a collusive agreement. In 1999 the US adopted amnesty plus programs which consist of reduced fines for a convicted firm if it reports another, undetected, cartel. 4 Multiproduct firms may adopt appropriate collusive strategies. In [START_REF] Jp | Multi-market collusion with demand linkages and antitrust enforcement[END_REF], firms collude in a first market and subsequently in another once the first cartel is dissolved, if the substitutability of products is low. [START_REF] Dargaud E | Hidden collusion by decentralization: firm organization and antitrust policy[END_REF] [ DJ (2015)] show that the firm's choice of internal structure affects the detection probability of a cartel inside a firm. Particularly the probability that the antitrust authority uncovers inculpatory evidence of several infringements when investigating only a single market decreases if firms compartmentalize collusive agreements. [START_REF] Dargaud E | Leniency programs and cartel organization of multiproduct firms[END_REF] [DJ (2020)] address the role of leniency programs in the previous model and describe how they can defeat the compartmentalization strategy.

The antitrust authority seeks to substitute for its own investigations by inducing CEOs to launch internal investigations so as to collect evidence and apply for leniency. However, by defeating the compartmentalization strategy, leniency may promote centralization with higher collusive prices.

Assumptions of this article are in a large part those used in DJ (2015DJ ( , 2020) ) but we study another effect of the organizational structure. In DJ (2015,2020) firms may compartmentalize collusive agreements to avoid the contagion of antitrust authority investigations. In this paper we eliminate this effect by assuming that the contagion probability is the same for each organization but we focus on different durations for the investigations. Decentralized firms, by compartmentalizing activities, make investigations slower. In this case these firms are much more likely to seek leniency for the second cartel (before its detection) than centralized firms.

When a cartel is either denounced by whistle-blowers or suspected by antitrust authorities, firms are not immediately informed. Authorities can wiretap firm's line 5 or raid the head office in order to obtain hard evidence. If a second cartel is detected during the first wave of investigations then firms lack the time to apply for leniency. Otherwise they may do so before a deeper investigation. 6 The probability that the second cartel is discovered during one-market investigations is higher for centralized firms since both cartels are managed by the same people. 7

This article analyzes the impact of leniency on the choice of compartmentalization. In the main part, we assume that the authority always finds hard evidence of a second cartel when it investigates the other market and both cartels are dissolved. This reduced the number of collusive strategies and firms, where possible, always apply for leniency for the second cartel since its conviction is inescapable. Firms face a trade-off: competition between divisions of decentralized firms results in low collusive prices (firms do not internalize the effects of the price on the other product demand) but expected fine is lower since they are much more likely to seek leniency for the second collusive agreement. Without leniency, firms never select decentralized structure. Leniency can affect this result if goods are weak substitutes and may cause collusive prices to fall.

We show that leniency's overall level of generosity may lead to procollusive as well as procompetitive effects.

Then we check the robustness of the results in a richer model and assume that the detection of the second cartel by serendipity is lower than one, irrespective of the organizational structure. Then firms can sometimes continue to collude on the second market after the condemnation of the first cartel. Equilibria depend on the serendipitous probability and the design of leniency programs. If the contagion probability is high, it is more difficult to pursue collusion in the second market and the major results obtained in the main part of the paper are robust. If leniency is sufficiently generous, then firms seek leniency for the second cartel and prefer a decentralized organization if product substitutability is low. If the contagion probability is low then results depend on the amnesty plus program. Without it, firms should collectively commit not to apply for leniency and adopt a centralized organization. But pursuing collusion in the second market once the first cartel is detected is destabilized if we introduce amnesty plus. Then, with amnesty plus, firms wish to apply for leniency for this second cartel and firms may select decentralized organization to seek leniency early and to reduce the risk of being fined. Thus we obtain similar results as in the main part of the paper if we consider high contagion probability or a lower value together with amnesty plus program.

The paper is organized as follows. In section 2, we describe our model. Section 3 characterizes collusive strategies without leniency programs and puts in evidence that firms never select a decentralized organization.

Section 4 analyzes the role of leniency in facilitating decentralization adoption. In section 5 we study the effects of different levels of fine reduction and show that leniency may lead to pro-competitive and pro-collusive effects. In section 6 we check the robustness of our results. The conclusion follows in section 7. 6 In 1995 SC Johnson blew the whistle on a cartel. Then antitrust authority raided a meeting of the managers and the head offices of the firms involved. Among these was Colgate-Palmolive, which earlier had applied twice for leniency. The first application concerned the cartel under investigations, but the second an undetected cartel. If different people manage cartels then the risk of a fast detection by contagion is lower.

7 A counter-argument is that a centralized firm can quicker collect hard evidences for both cartel and apply for leniency. But it seems reasonable to expect that this counter-argument is not prevailing.

This model largely takes over the assumptions used in DJ (2015DJ ( , 2020)). Two identical firms, 1 and 2, each produce two differentiated products, A and B. Marginal costs are assumed to be identical and equal to zero. The firms determine their organizational form and then they play an infinitely repeated game of price competition. Firms can decide to implement a collusive agreement on one or two products but we introduce a third player, antitrust authority, which seeks to detect and fight collusion.

Organizational design: Before competing in price, firms determine their organizational form cooperatively. Each firm can choose between a centralized (unitary) organizational structure (U-form) and a decentralized (multidivisional) structure (M-form). In the first case each CEO chooses prices in order to maximize overall firm profit, whereas in the second option two divisional managers each determine the price of a single product so as to maximize the profit of their business unit. In this case firms do not internalize the effects of the price on the other product demand.

Price competition: Once organizational structures have been decided, firms play an infinitely repeated game of price competition. In each period, they can decide to implement a collusive agreement. Collusive outcomes are modelled on the basis of grim trigger strategies [START_REF] Friedman | A non-cooperative equilibrium for supergames[END_REF]: as soon as one firm deviates from the agreement, the other plays non-cooperatively forever. Both firms face the same discount factor δ.

The demand function for product i (i = {A, B}, j = {A, B} and i = j) is8 (d is a positive parameter reflecting product substitutability):

Q i p i , p j = max 0, min a -p i + d p j -p i , 1 + 2d 1 + d a -p i
Probabilities of detection and fines: Collusion among firms or their divisions generates hard evidence which can be found by the authority with probability ρ. The authority can still find collusion during a deviation period, but past offences cannot be detected once firms have reverted to competitive pricing. If a cartel is detected, the authority launches an investigation which always leads to successful prosecution, and a fine F is imposed on each cartel member. Cartels are dissolved once they have been convicted and firms cannot collude again on the market in which they have been condemned. During such a one-market investigation, the authority may find, which probability µ, actionable evidence of a collusive agreement in the other market which leads firms to pay a second fine F . In DJ (2015,2020) the probability of this event depends on the organizational structure. In this article we remove this assumption since we focus on the investigations slowdown allowed by the decentralized structure. To simplify presentation of the major results

we assume that µ = 1 until Section 6.

Timing of leniency programs: They are three moments at which firms can apply for leniency. The first is before collusive agreements are detected. When deviating from a cartel, firms simultaneously apply for leniency as the authority can still find collusion during this deviation period. Second, firms could apply for leniency in the interval between the prosecution of the first cartel and the serendipitous detection of the second. This event 9 depends on the duration of authority investigations. Firms can seek leniency for the second cartel with probability q M or q U (the subscripts M and U refer respectively to decentralized and centralized firms) and we assume 10 q U ≤ q M since decentralization slowdowns investigations. Finally, it is possible for the firms to apply for leniency after completion of the investigation if the cartel has not been detected.

Reduced fines with leniency policy: If only one firm applies for leniency before the cartel is detected, then it receives total immunity from fines. If both firms apply simultaneously, the expected fine is 0.5F . 11 Offering total immunity to the first firm applying for leniency is optimal since it weakens collusive sustainability by maximizing the deviation profit of the firm. 12 Firm applying for leniency for the second cartel, during the first market investigation or shortly after this investigation is closed, obtains a reduced fine for the second cartel: τ F , with τ ∈ [-1, 1]. Once again only the first firm benefits from the reduced fine. τ = 0 is the total immunity and τ ∈ ]0; 1] the partial immunity case. In 1999 the US adopted amnesty plus programs which consist of reduced fines for a convicted firm if it reports another, undetected, cartel. This option is introduced in our model for τ ∈ [-1; 0[.

Timing of each period:

The timing of each period is: (1) CEO or managers choose their competitive or collusive prices. Collusion generates hard evidence and firms may stay on collusive path or apply for leniency if they deviate from the cartel agreement.

(2) The authority suspects an active cartel with probability ρ and launches investigations.

(3) During the first-market investigation, firms can apply for leniency for the second cartel with probability q U or q M (depending on the organizational choice). ( 4) If firms do not apply for leniency then the authority detects the second cartel with probability µ. (5) Firms can apply for leniency if this second cartel has not been detected by the authority.

Who applies for leniency? CEOs of centralized structures have clear evidence concerning both cartels and are free to request leniency. Under decentralization collusive agreements are organized by managers who hold proof of their own cartel. DJ (2020) study CEO and manager divergence concerning the revelation to the cartel to the authority. This potential conflict does not appear for µ = 1. In section 6, the case µ < 1 is 9 Firms can never apply for leniency for the first cartel after its detection. A major reason of the leniency application (see [START_REF]Leniency programs and cartel prosecution[END_REF]) is then rejected.

10 DJ (2020) study the following cases: q U = q M = 0 or q U = q M = 1. In this case the organizational structure does not influence the ability to apply for leniency for the second cartel during the first market investigation.

11 [START_REF] Sauvagnat | Are leniency programs too generous?[END_REF] suggests that total immunity from fines be implemented if only one firm applies for leniency. If there is more than one applicant (apply simultaneously) the fine reduction is negligible. Our paper follows the traditional approach.

12 See Spagnolo ( 2004) and Harrington (2008). We do not introduce the possibility for the antitrust authority to reward informed employees reporting evidence as was the case in [START_REF]The impact of leniency and whistle-blowing programs on cartels[END_REF].

already rather complex and we assume that when CEOs launch internal audits in order to gather evidence then audit cost is negligible and CEOs always obtain information about collusive agreements.

No leniency programs

In this section we compute the expected profits and sustainability conditions obtained without any leniency programs. Until section 6, we assume that µ = 1.

Firms can select one of the following four strategies. They can choose not to collude. In this case they obtain zero profit since we assume Bertrand competition with homogeneous products. Firms can collude simultaneously on the two markets. Since µ = 1 the detection of one collusive agreement automatically leads to the second cartel detection and both cartels are dissolved. We denote by U s this strategy for centralized firms or M s in the opposite case. Firms finally can choose to cartelize in only one market and, if the cartel is discovered, they start collusion in the second market: this is the Seq strategy. If the firms select this strategy we assume that they choose a centralized structure.13

Comparison of collusive strategies

Without any leniency programs the M s strategy is dominated by the U s one.14 Indeed detection probabilities and fines are the same but collusive prices and profits are higher if firms play the U s strategy thanks to a coordination of the two prices. We now compute the expected profits and sustainability conditions considering only the U s and Seq strategies.

U s strategy

Solving the individual maximization program for each firm, we obtain the following equilibrium values (i = 1, 2; X = A, B):

Collusion Deviation Punishment Price p X i = a 2 p dX i = a 2 -ε p pX i = c = 0 Quantity q X i = a 4 q dX i = a 2 q pX i = a 2 Profit π cU i = a 2 4 π dU i = a 2 2 π pU i = 0
The present discounted value of a firm from colluding is given by:

Π U s i = π cU i -1 -(1 -ρ) 2 2F + δ (1 -ρ) 2 Π U s i ⇔ Π U s i = π cU i -2ρ (2 -ρ) F 1 -δ (1 -ρ) 2
Deviations are punished with Nash reversion, so the current gain form deviation is:

Π dU s i = 2π cU i -2ρ (2 -ρ) F
The U s strategy is sustainable if and only if:

Π U s i ≥ Π dU s i ⇔ F ≤ F U s ≡ δ (1 -ρ) 2 -1 2 δ (1 -ρ) 2 ρ (2 -ρ) π cU i 3.1.2 Seq strategy
Firms cartelize in only one of the two markets (named market B), maintaining price competition in the second market. If the cartel is discovered, firms start collusion in the market A. We successively determine equilibrium values in the second market and in the first cartels.

Second cartel: In the market B the cartel has been discovered and equilibrium prices equal marginal cost:

p B 1 = p B 2 = c = 0.
In the market A firms act as a monopoly. The best-response function of a firm monopolizing the market A is: p A = a+dp B 2(1+d) . Setting p B = 0, we obtain:

p A i = a 2 (1 + d) , q A i = a 4 and π c i = a 2 8 (1 + d)
The present discounted value of a firm from colluding is given by:

Π i = π c i -ρF + δ (1 -ρ) Π i ⇔ Π i = π c i -ρF 1 -δ (1 -ρ)
The deviation profit is:

Π d i = 2 π c i -ρF .
This second cartel is sustainable if and only if:

Π i ≥ Π d i ⇔ F ≤ F ≡ 2δ (1 -ρ) -1 δρ (1 -ρ) π c i
First cartel: Each firm's expected payoff associated with collusion is defined as:

Π Seq i = π c i -ρF + δ (1 -ρ) Π Seq i + δρ Π i ⇔ Π Seq i = (1 -δ + 2δρ) ( π c i -ρF ) [1 -δ (1 -ρ)] 2
The deviation profit is:

Π dSeq i = 2 π c i -ρF .
Then collusion is sustainable if and only if:

Π Seq i ≥ Π dSeq i ⇔ F ≤ F Seq ≡ -1 + 3δ -2δρ -2δ 2 (1 -ρ) 2 1 -δ (1 -ρ) 2 δρ π c i
Intuitively this second condition is easier to sustain than: F ≤ F since it accounts the expected collusive outcome in the second cartel once the first cartel is detected.

Strategy choice

If they are sustainable, collusive strategies dominate the non-cooperation since they lead to a strictly positive profit. Firms choose the U s strategy if:

Π U s i ≥ Π Seq i ⇔ F ≤ F 1 ≡ [1 -δ (1 -ρ)] 2 π cU i -1 -δ (1 -ρ) 2 (1 -δ + 2δρ) π c i 2 [1 -δ (1 -ρ)] 2 (2 -ρ) -1 -δ (1 -ρ) 2 (1 -δ + 2δρ) ρ
Proposition 1 Firms choose the U s strategy if and only if F ≤ min (F U s , F 1 ). They choose the Seq strategy if F is high. If the products are weak substitutes the Seq strategy is now available. If the firms choose this strategy then the per period profit is lower but the collusion lasts longer and the second fine is paid later.

if min (F U s , F 1 ) < F ≤ F . They do not collude if F > max F U s , F .
The Seq strategy is selected if F is relatively high. In the other case firms select the U s strategy.15

Leniency Programs

Since µ = 1 firms never reach the stage (5). Firms can apply for leniency only at stage 1 (before the detection of the second cartel) or stage 3 (between the detection of the first cartel and the detection of the second one).

At stage (1), when deviating from the cartel, firm simultaneously applies for leniency. All the threshold values of the sustainability of the collusion are impacted (F U s , F and F Seq ).

The detection of the second cartel once the investigation is launched is inescapable. In this case, firms apply for leniency for the second cartel whenever they can, that is with probability q U if firms are centralized or q M under decentralized structure. Since q M ≥ q U the M s strategy is not always dominated by the U s.

Collusive strategies

Three collusive strategies are now available: the Seq, U s and M s strategies.

Seq strategy

When a cartel is detected firms can not apply for leniency for an other cartel since firms sequentially collude in the two markets. But they can seek leniency when they deviate from an active cartel which has not been yet detected.

The expected collusive profits are not altered by leniency. Nonetheless, sustainability conditions are modified since deviation profits are increased. The new threshold values are indicated below:

F ≡ 2δ (1 -ρ) -1 ρ π c i and F Seq ≡ 3δ -2δρ -2δ 2 (1 -ρ) 2 -1 (1 -δ + 2δρ) ρ π c i 4.1.2 U s strategy
Leniency programs have a double effect on the U s strategy. First firms can apply for leniency before the detection of the first cartel, increasing the deviation profit and making collusive agreements more difficult to sustain. Second firms can apply for leniency between the detections of the first and the second cartel: the expected profit under the U s strategy is increased (sustainability condition is modified). If µ = 1 conviction for the second cartel is inescapable, firms always apply for leniency whenever possible.

Collusive profit is:

Π U s i = π cU i -ρ 2 2F -2ρ (1 -ρ) (1 -q U ) 2F + q U 1 2 2F + 1 2 (F + τ F ) + δ (1 -ρ) 2 Π U s i ⇔ Π U s i = π cU i -[2ρ + (1 -ρ) (4 -q U + τ q U )] ρF 1 -δ (1 -ρ) 2 Deviation profit is : Π dU s i = 2π cU i .
U s strategy is sustainable if and only if:

Π U s i ≥ Π dU s i ⇔ F ≤ F U s ≡ 2δ (1 -ρ) 2 -1 [2ρ + (1 -ρ) (4 -q U + τ q U )] ρ π cU i 4.1.3 M s strategy
The expected per-period payoff of colluding firms playing the M s strategy is lower than with the U s strategy (because of internal competition between substitutable products) but decentralized firms are much more likely to seek leniency for the second collusive agreement. We obtain the same equilibrium prices under the M s strategy as in the duopoly case with differentiated products (each firm producing only one product).

For each period we obtain:

Collusion Deviation Punishment Price p X i = a 2+d p dX i = a 2+d -ε p pX i = c = 0 Quantity q X i = 1+d 2+d a 2 q dX i = 1+d 2+d a
depends on the price in the other market Profit of a division

π cM i = 1+d 2(2+d) 2 a 2 π dM i = 2 1+d 2(2+d) 2 a 2 π pM i = 0
A deviating division reduces its price (p X i -ε) in order to absorb the global market. But this price reduction is not computed in order to absorb the consumers buying the other product since the collusive price is the best response to the other market price. The expected collusive profit is :

Π M s i = π cM i -ρ 2 F -ρ (1 -ρ) F -(1 -ρ) ρ (1 -q M ) F + q M 1 2 F + 1 2 τ F + δ (1 -ρ) 2 Π M s i ⇔ Π M s i = π cM i -1 + 1 2 (1 -ρ) (2 -q M + q M τ ) ρF 1 -δ (1 -ρ) 2
The deviation profit is: Π dM s i = 2π cM i . M s strategy is sustainable if and only if:

Π M s i ≥ Π dM s i ⇔ F ≤ F M s ≡ 2δ (1 -ρ) 2 -1 1 + 1 2 (1 -ρ) (2 -q M + q M τ ) ρ π cM i 4.

Organizational choice

Comparison of the profits is used to endogenize the choice of organizational structure. Firms adopt the U sim strategy instead of the M c strategy if and only if:

Π U s i ≥ 2Π M s i ⇔ F ≤ F 2 ≡ π cU i -2π cM i (q M -q U ) (1 -τ ) (1 -ρ) ρ
Decentralization can appear at equilibrium if q M is higher than q U , if the fine reduction is high (low value of τ ), if the products are weak substitutes (2π cM i close to π cU i ) and for high values of F .

If the products are independent then F 2 = 0 (π cU i = 2π cM i ). M s strategy dominates U s whatever the amount of fine. 16We have to compare Seq strategy with the others (ρ ≡ 1 -ρ):

2Π M s i ≥ Π Seq i ⇔ F ≤ F 3 ≡ (1 -δρ) 2 2π cM i -1 -δρ 2 (1 -δ + 2δρ) π c i (1 -δρ) 2 [2 + ρ (2 -q M + q M τ )] -1 -δρ 2 (1 -δ + 2δρ) ρ Π U s i ≥ Π Seq i ⇔ F ≤ F 1 ≡ (1 -δρ) 2 π cU i -1 -δρ 2 (1 -δ + 2δρ) π c i (1 -δρ) 2 [2ρ + ρ (4 -q U + q U τ )] -1 -δρ 2 (1 -δ + 2δρ) ρ
Firms choose the sequential collusion if F is relatively high, q M and q U are low, τ is high, δ is high (collusion lasts longer with Seq strategy) and, most importantly, if the products are weak substitutes.

The comparisons of the threshold values are reported in the Appendix A. The relative positions of the borderlines F U s , F M s and F 2 do not depend on parameters values (see Appendix A). However the relative position of F 1 and F 3 with respect to the other thresholds depends heavily on δ.

In order to graphically illustrate these results and to compare with the previous graphical representation we consider: a = 10, ρ = 0.01, q M = 0.9, q U = 0.1 and τ = 0. Then we vary δ considering always sufficiently high value to make all the collusive strategies sustainable for each figure (the benchmark boundaries without leniency program are plotted with dots for comparison).

F ≤ min (F 1 , F U s , F 2 ), the M s strategy if min (F 2 , F U s ) ≤ F ≤ min (F M s , F 3 ). They do not collude if F > max( F , F M s , F U s ).
In the other cases, firms choose the Seq strategy.

Impact of leniency programs

Major impact on the organizational choice: The major above mentioned result is that leniency can favor the adoption of decentralized organization. Without leniency firms always select a centralized structure.

For some parameters values firms can switch to the M s strategy when leniency is introduced. Decentralized firms face competition between divisions inside the firms and this impacts the collusive prices. Without leniency, the M s strategy is always dominated by U s (except for independent products). If we introduce leniency, decentralized firms are much more likely to seek leniency for the second collusive agreement than centralized firms and this reduces the expected fine. If competition between products is not too fierce then the M s strategy dominates the U s.

Proposition 3 Leniency programs may favor the adoption of decentralized structure.

Other effects relating to the organizational choice: In the previous graphical representation the traditional destabilizing effect of leniency appears. Deviation profits are increased when firms can apply for leniency before the first cartel is detected. The boundary F U s moves down and F moves to the left. Leniency makes some collusive agreements more difficult to sustain.

In the graphical representation set for δ = 0.8 there is a small area in which firms switch from the U s to the Seq strategy because the former is no longer sustainable.

Moreover there is an area in which firms switch from the Seq strategy to M s. With this latter strategy leniency programs partially protect firms from a double fine, there are fewer incentives to choose the Seq strategy of which one advantage is to protect firms from this risk.

In the graphical representation set for δ = 0.95 there is a collusive area which appears only if leniency programs are introduced. This is due to two reverse effects of leniency on the boundary F M s . Allowing leniency before the first cartel is detected increases deviation profit and makes the M s strategy more difficult to sustain. But the decreased expected fine increases the expected profit of the M s strategy and makes this strategy easier to sustain. If leniency is sufficiently generous (low value of τ ) the second effect may dominate the first one.

Impacts on the prices: In the area in which firms switch from the U s strategy to the M s strategy, leniency programs decrease collusive prices (except in the extreme case d = 0) and do not impact the existence or the duration of the cartel. But when firms switch from the Seq strategy to the M s strategy collusive price increase (except for d = 0) and collusion is shorter-lasting. Finally when firms switch to the U s strategy to the Seq strategy, collusive prices decrease and collusion lasts longer.

Proposition 4 Even when leniency programs do not deter collusive agreements they may impact collusive prices.

Moreover we reach traditional effects of leniency: prices decrease when leniency makes collusive agreements not sustainable and increase if collusive opportunities are strengthened.

Impact of τ : There is a broad consensus regarding the usefulness of leniency programs but their design is still being debated, notably concerning the fine reduction. In our model granting total immunity is optimal if firms apply for leniency before the detection of the first cartel, since the destabilizing effect of leniency is maximized. But we obtain inconsistent results regarding the fine reduction for the second cartel (computed with τ ) if leniency is requested during the first cartel investigation.

The following table sums up the impact of a decreased value of τ on prices, consumer surplus and duration of collusion.

border line shift of the border line potential switch prices surplus duration

F 2 Us ⇒ Ms = F M s Seq ⇒ Ms ambiguous F M s No collusion ⇒ Ms - F 3 Seq ⇒ Ms ambiguous F U s ↑ No collusion ⇒ Us - F 1 Seq ⇒ Us ambiguous
The above effects of the border lines F 1 and F 3 appear if the discount factor is very high (in the two graphical representations below, F 1 and F 3 do not appear since we assume δ = 0.8).

A decreased value of τ can yields to collusive or competitive effects. We reach the same effects concerning the introduction of amnesty plus (τ < 0).

In the short term, a decreased value of τ increases the number of active cartels since firms switch from the Seq strategy to M s strategy (one more cartel) or can decide to start collusion under the M s or U s strategies (two more cartels). In a longer term, the effect of τ on the number of active cartels is quite ambiguous since the switch from the Seq strategy to the M s strategy results in shorter lasting cartel. In any case, leniency applications increase.

The two graphical representations below illustrate these points (for δ = 0.8).

Partial amnesty for the second cartel: Border lines are plotted for τ = 0 (full line) and τ = 0.5 (dotted line). In this section we check the robustness of the results obtained with µ = 1. We obtain similar results considering µ < 1 if µ is sufficiently high or if amnesty plus program is active for lower value of µ.

When the antitrust authority investigates the first cartel, then firms, irrespective of the organizational structure, face a risk of detection for the second agreement that is lower than one µ ∈ ]0; 1[. Another strategy may appear: firms simultaneously collude in the two markets, and when a cartel has been successfully detected they can continue to collude in the other market provided that this second cartel is not detected. This strategy is denoted U c if firms are centralized and M c in the other case.

No leniency programs

Without leniency program, the M s and M c strategies are respectively dominated by the U s and U c strategies.

The expected profit under the U s strategy is:

Π U s i = π cU i -ρ 2 2F -2 (1 -ρ) ρ (F + µF ) + δ (1 -ρ) 2 Π U s i ⇔ Π U s i = π cU i -2 [1 + (1 -ρ) µ] ρF 1 -δ (1 -ρ) 2
The deviation profit is:

Π dU s i = 2π cU i -ρ 2 2F -2 (1 -ρ) ρ (1 + µ) F .
This strategy is sustainable if and only if:

Π U s i ≥ Π dU s i ⇔ F ≤ F U s ≡ δ (1 -ρ) 2 -1 2 δ (1 -ρ) 2 [1 + (1 -ρ) µ] ρ π cU i
The expected profit under the U c strategy is:

Π U c i = π cU i -ρ 2 2F -2 (1 -ρ) ρ (F + µF ) + 2δ (1 -ρ) ρ (1 -µ) Π i + δ (1 -ρ) 2 Π U c i ⇔ Π U c i = π cU i -2 [1 + (1 -ρ) µ] ρF + 2δ (1 -ρ) ρ (1 -µ) Π i 1 -δ (1 -ρ) 2
The deviation profit is:

Π dU c i = 2π cU i -2 [1 + (1 -ρ) µ] ρF .
This strategy is sustainable if and only if:

F ≤ F and Π U c i ≥ Π dU c i ⇔ F ≤ F U c ≡ [1-δ(1-ρ)][δ(1-ρ) 2 -1 2 ]π cU i +δρ(1-ρ)(1-µ) π c i {[1-δ(1-ρ)](1-ρ)[1+(1-ρ)µ]+ρ(1-µ)}δ(1-ρ)ρ .
If we consider F ≤ F then F U c ≥ F U s and U c dominates U s. The only valuable comparison is between U c and Seq (with ρ = 1 -ρ) :

Π U c i ≥ Π Seq i ⇔ F ≤ F 4 ≡ (1 -δρ) 2 π cU i + -1 + δ 1 -2ρ 2 + (1 -δ) ρ 2 -2δρρ (1 -δρ) µ π c i 1 -δρ 2 + 2 (1 -δρ) ρµ (1 -δ) ρ
We obtain the following proposition:

Proposition 5 Firms choose the U c strategy if F ≤ min( F , F U c , F 4 ). They choose the U s strategy if F < F < F U s . They choose the Seq strategy if min(F U c , F 4 ) < F ≤ F . They do not collude if F > max( F , F U s ).
The main difference with the case µ = 1 is that the U c strategy is now an equilibrium structure. Moreover F U s is increased: some cartels are sustainable only for a lower value of µ.

The figure below illustrates these results (a = 10, ρ = 0.01, δ = 0.8 et µ = 0.6). We also plot on the graph the border lines obtained with µ = 1 for comparison.

Leniency programs

To simplify the exposition and highlight the main effects we assume: q M = 1 and q U = 0. The U c strategy dominates the M c strategy: firms prefer the centralized structure when they aim to collude as long as possible.17

Collusive strategies

The Seq strategy does not depend on the value of µ. We compute the expected profits and sustainability conditions for the U s, U c and M s strategies.

U s strategy: If µ < 1 firms can reach the stage (5). In this case the antitrust authority can not detect the second collusive agreement since it is no longer active but firms can denounce it. Without amnesty plus program (τ ≥ 0) two pure strategy equilibria exist: either both firms denounce the cartel or not. We assume that firms can coordinate on the best equilibrium that is the non denunciation. But, if τ < 0, firms denounce the second cartel. We have to distinguish these two cases.

If τ ≥ 0, the expected profit is:

Π U s i = π cU i -ρ 2 2F -2ρ (1 -ρ) [µ2F + (1 -µ) F ] + δ (1 -ρ) 2 Π U s i ⇔ Π U s i = π cU i -2 (1 + µ -ρµ) ρF 1 -δ (1 -ρ) 2
The deviation profit is: Π dU s i = 2π cU i . This strategy is sustainable if:

Π U s i ≥ Π dU s i ⇔ F ≤ F U s ≡ 2δ (1 -ρ) 2 -1 2 (1 + µ -ρµ) ρ π cU i .
If τ < 0, the expected profit is:

Π U s i = π cU i -ρ 2 2F -2ρ (1 -ρ) µ2F + (1 -µ) F + 1 2 F + 1 2 τ F + δ (1 -ρ) 2 Π U s i ⇔ Π U s i = π cU i -{2ρ + (1 -ρ) [3 + µ + τ (1 -µ)]} ρF 1 -δ (1 -ρ) 2
The deviation profit is: Π dU s i = 2π cU i . This strategy is sustainable if:

Π U s i ≥ Π dU s i ⇔ F ≤ F U s ≡ 2δ (1 -ρ) 2 -1 {2ρ + (1 -ρ) [3 + µ + τ (1 -µ)]} ρ π cU i U c strategy:
This strategy is sustainable if firms do not deviate from the collusive agreement at stage

(1) and do not apply for leniency for the second cartel at stage (5). Once the investigation is closed then expected fine is nil for the firms and they do not apply for leniency if τ ≥ 0. But if τ < 0 they may apply for leniency if:

-τ F ≥ δ Π i ⇔ F ≥ F rep ≡ δ δρ -[1 -δ (1 -ρ)] τ π c i
If the firms stay on the collusive path then the expected profit under the U c strategy is the same than without leniency but the deviation profit is altered: Π dU c i = 2π cU i . Firms do not deviate from the cartel at stage (1) if:

Π U c i ≥ Π dU c i ⇔ F ≤ F U c ≡ [1 -δ (1 -ρ)] 2δ (1 -ρ) 2 -1 π cU i + 2 (1 -ρ) ρ (1 -µ) δ π c i 2 1 -δ (1 -ρ) 2 + (1 -δ) (1 -ρ) µ ρ
M s strategy: Firms always apply for leniency at stage (3) since the second cartel disappears once the first one has been detected.

The expected profit is:

Π M s i = π cM i -ρF -(1 -ρ) ρ 1 2 + 1 2 τ F + δ (1 -ρ) 2 Π M s i ⇔ Π M s i = π cM i -1 + 1 2 (1 -ρ) (1 + τ ) ρF 1 -δ (1 -ρ)
The deviation profit of a firm which deviates at stage (1) is: Π dM s i = 2π cM i . Then this strategy is sustainable if:

Π M s i ≥ Π dM s i ⇔ F ≤ F M s ≡ 2δ (1 -ρ) 2 -1 1 + 1 2 (1 -ρ) (1 + τ ) ρ π cM i 6.2.2
Π Seq i ≥ 2Π M s i ⇔ F ≥ F 3 ≡ [1 -δ (1 -ρ)] 2 2π cM i -1 -δ (1 -ρ) 2 (1 -δ + 2δρ) π c i [1 -δ (1 -ρ)] 2 [2 + (1 -ρ) (1 + τ )] -1 -δ (1 -ρ) 2 (1 -δ + 2δρ) ρ Π U c i ≥ 2Π M s i ⇔ [1 -δ (1 -ρ)] π cU i -2π cM i + 2δ (1 -ρ) ρ (1 -µ) π c i (1 -ρ) ρ ≥ [2µ (1 -δ) -(1 -δ) (1 + τ ) + δρ (1 -τ )] F If µ < 1+τ 2 -δρ(1-τ ) 2(1-δ)
, the right hand side is negative whereas the left hand side is positive: if sustainable, U c dominates M s.

If µ > 1+τ 2 -δρ(1-τ ) 2(1-δ) then: Π U c i ≥ 2Π M s i ⇔ F ≤ F 5 ≡ [1-δ(1-ρ)](π cU i -2π cM i )+2δ(1-ρ)ρ(1-µ) π c i [2µ(1-δ)-(1-δ)(1+τ )+δρ(1-τ )](1-ρ)ρ .
Remark: For d = 0, π cU i -2π cM i = 0 and F 5 > 0.

To compare M s and U s we have to distinguish two sub-cases depending on the existence of amnesty plus.

Case 1: if τ ≥ 0, then:

Π U s i ≥ 2Π M s i ⇔ π cU i -2π cM i ≥ [2µ -(1 + τ )] (1 -ρ) ρF .
If µ < 1+τ 2 , the right hand side is negative and the above inequality always applies.

Each decentralized firm is individually better of applying the leniency at stage (3) but it is not the collectively best action: this is a prisoner dilemma situation. With centralized structure, the stage (3) is removed and firms do not cooperate with the antitrust authority.18

If µ ≥ 1+τ 2 , we obtain:

Π U s i ≥ 2Π M s i ⇔ F ≤ F 2a ≡ π cU i -2π cM i (2µ -1 -τ ) (1 -ρ) ρ Case 2: if τ < 0, then: Π U s i ≥ 2Π M s i ⇔ F ≤ F 2b ≡ π cU i -2π cM i (1-ρ)(1-τ )µρ .
Firms tend to prefer decentralization if the product substitutability decreases (2π cM However if τ decreases the expected profits are increased and F U s moves upward. Note that the expression of F U s is exactly the same for τ = -1 and for τ = 0.

The upper border of the U c area decreases from F to F rep . Amnesty plus program incites firms to reveal the second cartel once the first cartel has been detected and then firms switch from U c to U s.

The M s area is increased and some firms can switch from the U s to the M s strategy in order to enjoy amnesty plus with certainty.

Moreover there is an area in which firms switch from the U c to the M s strategy whether because the U c strategy is not sustainable or because the expected profit of the M s strategy is sufficiently increased compared to the U c strategy.

The threshold F M s is increased and firms can switch from a no collusion strategy to the M s strategy for high values of fine: this is a procollusive effect. Firms can also switch from Seq to M s strategy: collusion is shorter-lasting but collusive prices are increased.

In closing, results obtained with µ < 1 are relatively similar than those obtained for µ = 1. First, amnesty plus program favors the decentralization organization. Second it can deter some cartel formations. Finally pro-collusive effects can appear.

The following graphical representation illustrates these points. We set a = 10, ρ = 0.01 , δ = 0.8, µ = 0.6 and τ = -0.5 (the boundaries obtained for τ = 0 are plotted with dots for comparison).

low value of µ

If µ < 1+τ 2 -δρ(1-τ ) 2(1-δ) and τ ≥ 0 then the M s strategy is dominated by the U c and U s strategies and is never selected in equilibrium.

Firms playing the M s strategy have enough time to apply for leniency. But if µ is low and τ relatively high, requesting leniency is not the collectively best action. Firms not applying for leniency pay the second fine with a low probability µ whereas if one firm applies then the second firm pays the global fine F and the first firm obtains a reduced fine. The opportunity for seeking leniency in stage (3) decreases the expected profit of firms for low values of µ. Then firms may prefer centralization since (1) they can not apply for leniency (q U = 0) and (2) the competition between the two products is removed. 21 In a richer model in which firms should select decentralized form without leniency program 22 we could 21 These two explanations support the fact that U s is sustainable if M s is sustainable too. Under the U s strategy, collusive prices are higher and the expected fine is lower.

22 Organizations theory puts in evidence some factors which drive firms to select decentralization structure. Indeed decentralization structure can increase the provisions of incentives inside the firm [START_REF] Aghion | Some implications of growth for organizational form and ownership structure[END_REF][START_REF] Maskin | Incentives, information, and organizational form[END_REF], obtain the opposite effect. If only a slight reduced fine was allowed then firms could select centralized structure in order to decrease their unilateral ability to apply for leniency.

Effects of amnesty plus: For low values of µ, the U c strategy, if sustainable, always dominates the M s strategy. But amnesty plus program decreases the area for which U c is sustainable. Firms now compare the M s structure with U s. If τ < 0 M s dominates the U s strategy if goods are independent. We reach similar comparative static properties as in the case µ = 1, a decreased value of τ may drive firms to adopt a more decentralized structure. The switch from U c to M s induced by amnesty plus reduces the duration of collusion and decreases collusive prices.

Moreover amnesty plus programs may drive firms to switch from U c to U s, U s to M s and U s to no collusion. For all these switches we obtain procompetitive effects.

But procollusive effect can appear (as in the case µ = 1) since F M s may shift upward: some cartels are sustainable only if amnesty plus program is introduced. For some parameter values, firms switch from the Seq to the M s strategy inducing higher collusive prices (but collusion is shorter-lasting).

We illustrate these points with the following graphical representation. We set: a = 10, ρ = 0.01, δ = 0.8, µ = 0.4 and τ = -0.5 (we also plot with dots the previous graphic obtained with τ = 0). can allow experiment of creative ideas on small scale [START_REF] Qian | Coordination and experimentation in M-form and U-form organizations[END_REF] or avoid heavy workload of CEO (Spiegel, 2009).

Conclusion

In the present article, we describe how leniency may induce firms to select decentralized organization and then to compartmentalize hard evidence between divisions inside a firm. Doing so results in slowdown investigations of antitrust authority relating to a cartel once another has been detected. This may facilitate the application for leniency for the undetected cartel. Leniency programs can modify the choice of organizational structure and favor the adoption of a decentralized organization.

DJ (2020) obtain the opposite effect but the two models differ in key respects. 23 This article does not invalidate the results obtained in DJ (2020) but complements them in examining another effect. An interesting topic for future research would be to mix the effects of leniency obtained in this article and in DJ ( 2020) into a same model but the large potential number of cases could make them difficult to interpret.

The major result of this article is that the introduction of leniency may induce firms to adopt a more decentralized structure with decreased collusive prices.

Despite a consensus regarding the utility of leniency, the designs are yet a matter of many debates. The purpose of this research is, in particular, to further knowledge of the implications of reduced fines when leniency is requested during investigations. We reach procompetitive and procollusive effects depending on the values of some parameters. The determination of optimal fine reduction for the second cartel depends on the distribution of these values and goes beyond the framework of this article but can be studied in future research.

The implications that firms may choose to compartmentalize activities to delay investigations are broader than just the analysis of collusion. Other firm behavior can be analyzed in this context, such as tax evasion or standard (environmental or other) infringement. An interesting topic for future research is to introduce the major assumption of this article into static model of self-reported crimes. 24

23 DJ (2015) suggest that firms may prefer decentralized structure in order to decrease the contagion detection probability inside multi-product firms. DJ (2020) show that leniency programs can overcome this strategy. In these two articles, the major assumption relies on the fact that the probability that the antitrust authority uncovers inculpatory evidence of several infringements when investigating only a single market decreases if firms compartmentalize agreements. In this article we eliminate this effect and focus on the investigations slowdown achievable with decentralization.

24 See Kaplow and Shavell (1994), Innes (2000), Feess et Walzl ( 2004) and Landeo and Spier (2020).

Figure 1a :Figure 1b :

 1a1b Figure 1a: Equilibria obtained without leniency program and δ = 0.8

Figure 2c :

 2c Figure 2a: Organizational choice with leniency program (δ = 0.8)

Figure 3 :Figure 4 :

 34 Figure 3: Impacts of τ in the partial amnesty case

Figure 5 :

 5 Figure 5: Equilibria obtained without leniency program (µ = 0.6)

  Strategy choice Four collusive strategies are available: U s, U c, M s and Seq. If sustainable, the U c strategy dominates the U s strategy. The comparison between U c and Seq remains unchanged. So we have to compare the M s strategy with the others.

iFigure 6 :

 6 Figure 6: Equilibria obtained for µ = 0.6 and τ = 0

Figure 7 :

 7 Figure 7: Equilibria obtained for µ = 0.6 and τ = -0.5

Figure 8 :

 8 Figure 8: Equilibria obtained for µ = 0.4 and τ = -0.5

The last terms correspond to the demands for each product when the other is priced above its choke price. We include these terms for completeness, but such prices do not arise in equilibrium.

Centralized firms can easily coordinate themselves on the time to put in place the second agreement.

Except if we consider independent products (d = 0). In this case firms are indifferent between the two organizational structures.

The value of F 1 for d = 0 decreases with δ. It is negative if δ is sufficiently high (δ 0.987 for ρ = 0.01)).

It is not the case if µ < 1 (see section 6).

Electronic copy available at: https://ssrn.com/abstract=3606082

For 0 < q U < 1 centralized firms could apply for leniency for the second cartel at stage (3) but do not and continue to collude in the second market if they reach the stage (5). A similar strategy could appear for decentralized firms if 0 < q M < 1.

If q U > 0 the stage (3) would be partially removed and the trade-off between M s and U s would remain roughly the same.20Electronic copy available at: https://ssrn.com/abstract=3606082

It would be too tedious since all the threshold values should be compared and some comparisons depend on the parameters values.

Similar representation can be obtained for δ = 0.95 (M s dominates Seq for all the values for which M s is sustainable and dominates U c).

Implications of µ < 1 on the previous results

In this section, we do not characterize all the equilibria for each parameter value 19 but we aim to show that some of the previous results are robust to the case µ < 1. Moreover we obtain new results. Equilibrium choices depend to a large extent on the values of µ and τ and on the existence of amnesty plus programs.

The results obtained for µ = 1 are robust to the case µ < 1 if µ is high. For lower value of µ we obtain similar results only when amnesty plus program is active.

high value of µ

If µ > 1+τ 2 and τ ≥ 0, the joint expected profits increase if firms can apply for leniency at stage (3). We reach the same results as obtained for µ = 1. Since decentralization allows firms to apply for leniency for the second cartel once an investigation is opened, firms can select the M s equilibrium strategy. This benefits consumers in two ways: collusive prices are decreased and collusion is shorter lasting when firms switch from the U c strategy to the M s strategy.

Contrary to the case µ = 1 firms can select centralized organization when the products are independent since cartels can be longer sustainable with this type of organization.

Moreover firms may switch from the U c strategy to the U s strategy since F is reduced with the application of leniency. In such cases, prices are not altered when firms simultaneously collude in the two markets but collusion is shorter-lasting.

The following graphical representation illustrates the previous comments.

We set: a = 10, ρ = 0.01, δ = 0.8, µ = 0.6 and τ = 0. The boundaries obtained without leniency program are plotted with dots for comparison. 20

Appendix A

We have to compare the threshold values obtained in the section 4.2. These values are:

Note that F U s does not depend on d.

Independent products If d = 0 then:

and π c i = π cM i . These conditions imply that:

1) F 2 = 0 then F U s > F 2 if δ is sufficiently high for U s to be sustainable.

2)

necessary condition for sustainability of collusion).

We conclude that

Substitutable products (d > 0)

We have to compare the intersection between the border lines: F M s and F U s and between F U s and F 2 .

increases with d since: (2+d) 2 (1+d) 2 decreases with d. If F 3 > F M s for d = 0 then this inequality applies ∀d ≥ 0. This is the case for intermediate values of δ, thus the borderline F 3 does not appear in the graph. However, for high values of δ then F 3 < F M s if d = 0. For even higher values, F 3 goes negative. In this case the Seq strategy is selected if d = 0 whatever the level of the fine.