Vincent Daien
email: vincent.daien@gmail.com

A Le Pape

D Heve

I Carriere

Incidence, risk factors and impact of age on retinal detachment following cataract surgery in France: a national population study

Objective: To assess the incidence, risk factors and impact of age on retinal detachment (RD) following cataract surgery.

Design: Cohort study

Subjects: All patients > 40 years old who underwent a primary cataract surgery in France between

Introduction

Cataract surgery is the most commonly performed procedure in people older than 65 years in the developed world. [START_REF] Bellan | The Evolution of Cataract Surgery: The Most Common Eye Procedure in Older Adults[END_REF] Additionally, as the population ages in Western countries, the number of cataract operations is increasing. [START_REF] Congdon | Prevalence of cataract and pseudophakia/aphakia among adults in the United States[END_REF] Retinal detachment (RD) after cataract extraction is a surgical vision-threatening event, with approximately half of patients not recovering better than 20/40 acuity. [START_REF] Haddad | Retinal detachment after phacoemulsification: a study of 114 cases[END_REF] The risk of RD after cataract surgery has been estimated at 0.7% [START_REF] Haug | Risk factors for retinal detachment following cataract surgery[END_REF] and increases up to 20 years after surgery. [START_REF] Erie | Risk of retinal detachment after cataract extraction, 1980-2004: a population-based study[END_REF] As compared with the rate of 0.0065% to 0.079% for rhegmatogenous RD in the general population, [START_REF] Haug | Risk factors for retinal detachment following cataract surgery[END_REF][START_REF] Algvere | The Swedish Retinal Detachment Register. I. A database for epidemiological and clinical studies[END_REF] cataract surgery increases the risk of RD at least 4-fold [START_REF] Haug | Risk factors for retinal detachment following cataract surgery[END_REF] and the absolute incidence of RD is substantial because of the large number of cataract surgeries performed. [START_REF] Baratz | Cataract extraction rates in Olmsted County, Minnesota, 1980 through 1994[END_REF] Cataract surgery continues to improve, with reduced incision size and multifocal lens technologies.

The surgery is increasingly performed in relatively young patients, [START_REF] Baratz | Cataract extraction rates in Olmsted County, Minnesota, 1980 through 1994[END_REF] and clear lens extraction for correction of refractive errors or presbyopia is gaining in popularity. [START_REF] Cochener | Retinal detachment after clear lens extraction for high myopia: seven-year follow-up[END_REF][START_REF] Mato | Retinal detachment after clear lens extraction for high myopia[END_REF] In previous studies, young age was suggested to be a risk factor of RD after cataract surgery. [START_REF] Haug | Risk factors for retinal detachment following cataract surgery[END_REF] However, we need data on the accurate rate of RD by age group for complete patient information and consent before cataract surgery.

In the present study, we aimed to determine the incidence of RD after cataract surgery in France and potential risk factors. A secondary objective was to estimate the risk reduction of RD by year of delay in cataract surgery.

Methods

The study protocol was approved by the national health authority in France.

Data source

The data for all patients > 40 years old who underwent cataract surgery in France between January 2009 and December 2012 were collected from the Programme de Médicalisation des Systèmes d'information (PMSI), a national database in France similar to the US Medicare system. Since 2004, each hospital's budget depends on the medical activity described in the PMSI, which compiles discharge abstracts related to all admissions in the 1,546 French healthcare facilities, public or private, and each patient included in this study was followed until December 2012 by use of an identification number. Information in these abstracts covers both medical and administrative data, including identification number, date of birth, and gender of patients. These data are rendered anonymous, and discharge abstracts related to a given patient can be linked. Routinely collected medical data include the principal diagnosis, secondary diagnoses, and procedures performed.

Data extraction

For each patient, the cataract surgery was identified by the PMSI code BFGA004 corresponding to a cataract extraction, performed by phacoemulsification with intraocular lens implantation in a capsular bag, or by BFGA002, BFGA006, BFGA008 or BFGA009, corresponding to a manual extracapsular extraction. The peri-operative vitrectomy procedure used for posterior capsular rupture was recorded if the code BGFA008 was associated with the surgical care.

Socio-demographic variables including age and gender were recorded. Eye characteristics, including high myopia and history of eye trauma, were collected with ICD-10 diagnostic codes H44.2 or H52.1 and S05, respectively.

Post-operative RD was identified with the ICD-10 code H33 or H35.7 associated with medical care after cataract surgery. Patients with RD recorded before cataract extraction were not included because the aim of the study was to assess the incidence of pseudophakic RD.

We did not include subjects with a history of vitrectomy (done for all causes) to avoid an overestimation of the incidence of pseudophakic RD.

The cumulative risk of RD over the 4 year period in the French population was extracted from the database and compared to the risk of RD after cataract extraction.

Before extraction of the cohort, 0.26% of people were not included because of presumed coding errors. Excluded people were younger (p<0.01) and were more often men (p<0.01) than in the study population.

Statistical analysis

Data are described with meanSD and number (%). Kaplan-Meier analysis was used to assess the cumulative probability of RD after cataract surgery for the overall population and for patients without any potential other risk factors' of RD (patients who underwent phacoemulsification without a peri-operative complication [capsular lens rupture] or local condition that could increase the risk of RD [diabetes, history of eye trauma or high myopia]). The risk of cataract extraction itself was assessed by logistic regression. Cox proportional hazard regression was used to analyze the risk factors associated with RD, including age, gender, diabetes, high myopia, eye trauma, and anterior vitrectomy for peri-operative capsular rupture. A crude hazard ratio (HR) was calculated for each factor, and an adjusted HR was then used to detect the independent relationship between each factor and the occurrence of RD after cataract surgery. The effect of joint exposure to young age and myopia was assessed in the Cox model testing age*myopia interaction.

The duration of follow-up after cataract extraction was based on the patient's last registration in the database, which was December 2012 for all patients. Differences between age groups were investigated by the log-rank test. Significance was set at P < 0.01. All analyses involved use of SAS 9.3 (SAS Inst., Cary, NC).

Results

Over the 4 years, 2,680,167 eyes in 1,787,021 patients > 40 years old (59.4% women, mean age 73.9 ±9.5 years) underwent cataract surgery; 11,424 patients had RD after cataract surgery, for an estimated risk of 0.99% at 4 years after cataract surgery. The cumulative probability of RD after cataract surgery increased in a near-linear manner over the study: 0.43%, 0.66%, 0.84%, 0.99%, at 1, 2, 3 and 4 years after cataract surgery, respectively (figure 1, total population). The median (interquartile range) number of days between cataract surgery and RD onset was 237 (75-738).

In the same period, 2009-2012, among adults > 40 years old, 62,064 underwent RD in France (with or without a history of cataract surgery), giving a 4 year cumulative risk of RD of 0.19% as compared with 0.99% in pseudophakic patients (P<0.001). The odds ratio associated with increased risk of RD was 3.87 (95% confidence interval [95% CI] 3.79 to 3.95) for cataract surgery itself. Pseudophakic RD represented 18.4% of the total rate of RD in this 4-year study. The 4 year risk of RD in the French population was higher for older than younger adults (0.09%, 0.24%, 0.27% and 0.29% for patients 40-54, 55-64, 65-74 and ≥75 years old, respectively, p<0.001).

Risk factors of RD after cataract surgery

After cataract surgery, the mean age was lower for patients with than without RD (67.10.017 vs. 74.00.014 years, p<0.001; table 1). The risk of RD after cataract surgery was higher for younger than older adults (3.64%, 2.20%, 1.06%, and 0.49% for patients 40-54, 55-64, 65-74 and ≥75 years old, respectively, p<0.001), men than women (1.59% vs 0.59%, p<0.001) and patients with extracapsular extraction versus phacoemulsification (3.87% vs 0.96%, p<0.001). The frequency of RD onset was higher with posterior capsular rupture (6.39%), myopia (9.21%) and a history of eye trauma (7.58%) or diabetes (1.21%). The cumulative probabilities of each risk factor of RD during the 4 years are in figures 1 and 2.

The multi-adjusted HR associated with increased risk of RD was 5.22 (95% confidence interval [95% CI] 5.05 to 5.39) for patients aged 40-54 years, 3.69 (95% CI 3.60 to 3.79) for those 55-64 years and 1.98 (95% CI 1.93 to 2.03) for patients 65 to 74 years as compared with those ≥ 75 years old (P<0.001) (Table 2). Adjusted HR pairwise comparisons between age groups 40-54 versus 65-74, 55-64 versus 65-74 and 40-54 versus 55-64 were 2.65 (95% CI 2.48 to 2.81), 1.89 (95% CI 1.78 to 1.96) and 1.41 (95% CI 1.33 to 1.50), respectively.

RD was associated with high myopia (HR 6.12; 95% CI 5.84 to 6.41), vitrectomy for peri-operative capsular rupture (HR 4.36; 95% CI 4.07 to 4.68), history of eye trauma (HR 3.98; 95% CI 3.69 to 4.30), extracapsular extraction (HR 3.11; 95% CI 2.94 to 3.30), male gender (HR 2.39; 95% CI 2.35 to 2.44) and a history of diabetes (HR 1.18; 95% CI 1.15 to 1.21).

Interaction between each risk factors of RD was performed owing to statistical and clinical significance only between myopia and age groups (p<0.0001). We thus stratified the analysis by age.

In myopic patients, the multi-adjusted HR associated with increased risk of RD was 25.02 (95% CI 3).

Age is a major marker of RD risk after cataract surgery

In the whole population, the younger the patient, the greater the risk of RD after cataract extraction performed with phacoemulsification (p logrank<.001; figure 1). We also assessed the effect of age on RD onset in a subgroup of 1,504,637 patients who underwent surgery with phacoemulsification without any other risk factors of RD such as peri-operative capsular rupture, diabetes, high myopia, or eye trauma. Young age remained significantly associated with RD (p logrank <.001; figure 1).

Relationship between age and risk of retinal detachment with a previous cataract surgery and in the French population is presented in figure 3.

Discussion

By using data from the PMSI program, an exhaustive national database from the 1,546 healthcare facilities in France, public or private, we determined an overall RD risk of 0.99% after cataract surgery during a recent 4-year period. In this work, including a wide range of risk factors, we provide a hierarchy of risk factors for RD onset: high myopia, young age, extracapsular extraction technique, eye trauma, peri-operative capsular rupture, male gender and diabetes.

Incidence of RD

Population-based studies of RD incidence after cataract surgery for patients who underwent surgery between 1980 and 2004 estimated RD based on sometimes a high proportion of extracapsular extraction. [START_REF] Haug | Risk factors for retinal detachment following cataract surgery[END_REF] Phacoemulsification is now generally used for cataract extraction and continues to evolve, so recent data of the RD incidence after cataract surgery are of importance. A retrospective study in Western Australia of 129,982 cataract surgery patients from 46 health facilities over 21 years found an overall pseudophakic RD rate of 0.7% [START_REF] Clark | Whole population trends in complications of cataract surgery over 22 years in Western Australia[END_REF] and the rate of RD after cataract surgery decreased greatly from 1980 to 2001. A recent large retrospective case-control study in Singapore of 24,846 cataract operations performed between 2001 and 2003, with follow-up through 2008, found a pseudophakic RD rate of 0.16%, [START_REF] Quek | Pseudophakic rhegmatogenous retinal detachment in a large Asian tertiary eye centre: a cohort study[END_REF] lower than rates previously found. [START_REF] Haug | Risk factors for retinal detachment following cataract surgery[END_REF] The authors included only cases with RD surgery in the same hospital as the cataract surgery, so the rate could be underestimated. In this 4-year study among the general population in France, we found an estimated risk of RD of 0.99%.

In the present study, cataract surgery itself increased the risk of RD by 3-fold. This finding was close to the study by Bjerrum et al., who reported a 4-fold increase in risk associated with cataract surgery itself. [START_REF] Bjerrum | Risk of pseudophakic retinal detachment in 202,226 patients using the fellow nonoperated eye as reference[END_REF]

Delay of RD onset

The study from Moorfields Eye Hospital found 75% of RD within the first 2 postoperative years, whereas Sheu et al. [START_REF] Sheu | Late increased risk of retinal detachment after cataract extraction[END_REF] found that the mean time between cataract surgery to diagnosis was 40 months. [START_REF] Tuft | Outcomes of pseudophakic retinal detachment[END_REF] In the present study, the median time for RD onset after cataract surgery was 237 (interquartile range 75-738) days and the risk of RD after cataract extraction increased in a nearly linear manner over time in the overall cohort. Regarding the shape of the slopes of RD after cataract surgery, in patients with vitrectomy for peri-operative capsular rupture a history of high myopia, or eye trauma, the RD incidence reached 4.5% to 5.2% during the first year.

Risk factors of RD after cataract surgery

Intraoperative posterior capsular rupture is one of the most significant risk factors for pseudophakic RD, leading to poor final visual acuity in most cases. The anterior movement of the vitreous due to posterior capsular rupture results in dynamic traction on the vitreous, with consequent retinal tear.

In a case-control study performed in Sweden in 2003, among 324 patients with posterior capsular rupture, the risk of RD after cataract surgery was 10-fold increased in patients with capsule complications. [START_REF] Jakobsson | Capsule complication during cataract surgery: Retinal detachment after cataract surgery with capsule complication: Swedish Capsule Rupture Study Group report 4[END_REF] In the present study, the risk of RD after cataract surgery was 4-fold increased with posterior capsular rupture. However, our HR was adjusted for multiple factors to take into account many other potential other risk factors of RD.

A study of patients who had undergone cataract surgery from 1980 through 2004 in Olmsted County, Minnesota, USA, found no difference in probability of RD with extracapsular extraction and phacoemulsification. [START_REF] Erie | Risk of retinal detachment after cataract extraction, 1980-2004: a population-based study[END_REF] We found extracapsular extraction as a major risk factor for RD after cataract extraction, with 3-fold increased risk of RD. This finding may be due to the progress in the technique of cataract surgery, which is now less traumatic than extracapsular extraction. Another hypothesis could be that patients undergoing extracapsular extraction had denser and harder crystalline lens.

A study from Moorefields Eye Hospital found a male predominance with pseudophakic RD: 67.5% males with RD versus 38.2% of controls. [START_REF] Tuft | Outcomes of pseudophakic retinal detachment[END_REF] We found an estimated Cox proportion risk of 2.39 for men as compared with women. This increased risk of RD in men is not entirely understood. Sheu et al.

hypothesized an increased and underreported history of trauma in men versus women. [START_REF] Sheu | Late increased risk of retinal detachment after cataract extraction[END_REF] However, this suggestion has not been verified and in our analysis, we included a history of eye trauma in the multivariate analysis. Myopic eyes have a much higher risk of RD, whether pseudophakic or phakic. High myopia, defined as axial length ≥ 26 mm, is also an established risk factor of pseudophakic RD. [START_REF] Haug | Risk factors for retinal detachment following cataract surgery[END_REF] The retrospective series by Sheu et al. [START_REF] Sheu | Late increased risk of retinal detachment after cataract extraction[END_REF] found an adjusted relative risk of 4.19 with high myopia as compared with axial length ≤ 23 mm. In the present study, the 4-year probability of RD after cataract surgery was 9.21% with high myopia, and the multi-adjusted HR was 6.12.

The retrospective study by Alio JL. et al. of patients with high myopia and a mean follow-up of 61.5  29.6 months found a higher risk of pseudohakic RD in myopic patients < 50 than > 50 years old (4.46% vs. 2.96%, respectively). [START_REF] Alio | The risk of retinal detachment in high myopia after small incision coaxial phacoemulsification[END_REF] Young age and high myopia were found as risk factors of RD. Thus, cataract surgery or refractive lens exchange in patients with high myopia appears to be associated with high risk of RD, and surgical indications may be carefully discussed. [START_REF] Alió | Lens surgery (cataract and refractive lens exchange) and retinal detachment risk in myopes: still an issue?[END_REF] In the present analysis, young age was also a risk factor in myopic patients. Among myopic patients, the risk of RD increased by 25 in age group 40-54 years and by 20 in age group 55-64 years as compared with non-myopic ≥ 75 years old.

Young age is known to increase the risk of RD after cataract surgery. In a large retrospective casecontrol series of 63,298 cataracts between 1994 and 2003 at Moorfields Eye Hospital, young age was a significant risk factor for RD after surgery. [START_REF] Tuft | Outcomes of pseudophakic retinal detachment[END_REF] The mean age with pseudophakic RD was 63.5 years and 71.9 years for controls. The odds ratio was 3.1 for patients ≤ 64 than > 64 years old [START_REF] Tuft | Outcomes of pseudophakic retinal detachment[END_REF] A series from Taiwan found 8-year RD rates of 6.65%, 2.57%, and 2.01% for patients ≤ 50, 50-60 and ≥ 60 years old, respectively. [START_REF] Sheu | Late increased risk of retinal detachment after cataract extraction[END_REF] Our 4-year RD risk estimates were 3.6%, 2.2%, 1.1, and 0.5% for patients 40-54, 55-64, 65-74 and ≥75 years old, respectively. The incidence of RD was not significantly affected after adjusting for other risk factors (gender, history of diabetes, high myopia, eye trauma, and anterior vitrectomy for peri-operative capsular rupture and surgical technique) but was increased by 5-fold for patients 40-54 years old, 4-fold for those 55-64 and 2-fold for those 65-74 as compared with those ≥ 75. Of interest, the 4 year risk of RD (with or without a history of cataract extraction) in the French population was higher for older than younger adults.

The pathophysiology of increased risk of RD after cataract surgery in young patients remains speculative. In our study, the main causes of early cataract, such as diabetes, high myopia and a history of eye trauma, were included in the multivariate analyses and the effect of age remained statistically significant. Another possible reason may be the status of the vitreous at the time of cataract extraction. Cataract extraction increases the risk of posterior vitreous detachment, [START_REF] Sheard | Posterior vitreous detachment after neodymium:YAG laser posterior capsulotomy[END_REF][START_REF] Hilford | Posterior vitreous detachment following cataract surgery[END_REF] and pseudophakic eyes have alterations in the vitreous humor structure that are not seen in phakic eyes. [START_REF] Neal | Alterations in human vitreous humour following cataract extraction[END_REF] Changes in the vitreous induced by removal of the crystalline lens may underlie an increased risk of RD. [START_REF] Mirshahi | Incidence of posterior vitreous detachment after cataract surgery[END_REF] The presence of a posterior vitreous detachment in older patients before cataract extraction may protect against subsequent RD by limiting the force to the retina produced during cataract surgery. Consequently, cataract extraction in relatively young patients without previous posterior vitreous detachment may alter the microenvironment of the vitreous and retina, thereby increasing the risk of retinal break and RD. [START_REF] Coppé | Posterior vitreous detachment and retinal detachment following cataract extraction[END_REF] An alternative explanation is that eyes in some young patients are abnormal in their tendency to develop cataracts, which may predispose to pseudophakic RD.

Also people ≤ 65 years old who require cataract surgery may represent those with another pathology that could affect their risk of RD. Among these factors, we could include a history of diabetes, high myopia or ocular trauma. To assess the specific effect of age on the risk of pseudophakic RD, we performed a Kaplan-Meier analysis for a subgroup of patients without any history of diabetes, high myopia, or eye trauma (Figure 1). We also included these 3 covariates in the multivariate Cox model (Table 2). Young age remained significantly associated with RD in both analyses.

Optimal timing for cataract surgery

The optimal timing for cataract surgery is important for public health policies. In Western countries cataract surgery has achieved marked improvements in vision-related activity limitations [START_REF] Harrer | Impact of bilateral cataract surgery on vision-related activity limitations[END_REF] and may not be delayed beyond 20/40 or 20/50 of visual acuity so that older adults can maintain autonomy. [START_REF] Daien | Visual impairment, optical correction, and their impact on activity limitations in elderly persons: the POLA study[END_REF][START_REF] Daien | Visual acuity thresholds associated with activity limitations in the elderly. The Pathologies Oculaires Liées à l'Age study[END_REF] Cataract surgery on eyes with poor preoperative visual acuity is related to surgical complications, and cataract surgery on eyes with excellent preoperative visual acuity is related to adverse visual results, including RD. [START_REF] Lundström | The Changing Pattern of Cataract Surgery Indications: A 5-Year Study of 2 Cataract Surgery Databases[END_REF] As surgical techniques have improved, the demand for surgery at an early stage has increased. The observed strong association between age and pseudophakic RD may have implications when considering clear lens extraction as a refractive procedure in young patients. If refractive lens exchange becomes more commonly accepted, this procedure may add to the potential RD burden in younger eyes.

However, for young subjects with cataract associated with substantial vision loss, cataract surgery cannot be delayed. In both situations of refractive lens exchange or early cataract, we recommend accurate screening for symptomatic flap tears and prophylactic laser treatment before the cataract extraction, and a close follow-up of the fundus in the weeks and months after surgery. [START_REF] Wilkinson | Evidence-based analysis of prophylactic treatment of asymptomatic retinal breaks and lattice degeneration[END_REF]

Strengths and limitations

Strengths of the study are the population size, the national recruitment, the importance of its findings and potential impact on public health. Given the reliance on PMSI codes for the selection of patients and the ascertainment of outcomes, there was a potential for misclassification-or underdetection-related biases. However, this situation may have only a minor impact on the findings.

One possible reason for the increased risk of RD after cataract surgery in young patients may be the status of the posterior vitreous detachment at the time of cataract extraction. However, the administrative database did not include this information, so we could not assess its impact on the risk of pseudophakic RD. We could not exclude patients undergoing any other surgery that of vitrectomy (for example a trabeculectomy), which could have slightly increased the estimation of the risk of pseudophakic RD.

Conclusions

RD is one of the most serious complications following cataract surgery, with an overall risk of 0.99% in our 4-year study in the general population in France. We provide a hierarchy of risk factors for RD onset: high myopia, young age, vitrectomy for peri-operative capsular rupture, history of eye trauma, extracapsular extraction technique, male gender and diabetes. Screening and treatment for symptomatic flap tears before cataract surgery in young patients may help reduce the rate of RD after cataract surgery. Optimal timing of cataract surgery is an important issue for public health policies to obtain the best balance between maintaining autonomy in older adults and avoiding serious adverse events such as RD.

  Diagnoses identified during the hospital stay are coded according to the International Classification of Diseases, Tenth Revision (ICD-10). The PMSI database has provided accurate population-based data in cataract surgery in France between 2009 and 2012.[START_REF] Daien | Incidence and Characteristics of Cataract Surgery in France from 2009 to 2012: A National Population Study[END_REF]

  24.76 to 25.18) for patients 40-54 years old, 20.37 (95% CI 20.21 to 20.53) for those 55-64 years old and 17.05 (95% CI 16.85 to 17.25) for those 65 to 74 years old as compared with non-myopic patients ≥ 75 years old. In non-myopic patients, the multi-adjusted HR associated with increased risk of RD was 5.46 (95% CI 5.40 to 5.52) for patients aged 40-54 years, 3.76 (95% CI 3.70 to 3.82) for those 55-64 years and 1.98 (95% CI 1.92 to 2.04) for those 65 to 74 years as compared with non-myopic patients ≥ 75 years old (P<0.001) (Table

Figures legendsFigure 1 .

 1 Figures legends

Figure 2 .

 2 Figure 2. Cumulative probability of retinal detachment after cataract extraction in sub-groups of risk including high myopia, history of eye trauma, vitrectomy for peri-operative capsular rupture or history of diabetes.

Figure 3 .

 3 Figure 3. Relationship between age and risk of retinal detachment with a previous cataract surgery and in the French population.

Table 1 . Characteristics and estimated 4-year risk of retinal detachment (RD) among 1,787,021 patients who underwent cataract extraction in France between 2009 and 2012

 1

		N=1,787,021	Estimated 4-year	logrank
	Characteristics	No RD	RD	risk of RD by Kaplan Meier analysis	P value
		n=1,775,597	n=11,424
	Age, years, meanSD	74.00.014	67.10.019		P <0.0001
	40-54 (%)	64,374	1,493	3.64%
	55-64 (%)	228,034	3,151	2.20%
	65-74 (%)	527,859	3,596	1.06%
	≥ 75 (%)	955,330	3,184	0.49%
	Gender (%)				P <0.0001
	Men	718,822	7,404	1.59%
	Women	1,056,775	4,020	0.59%
	History of diabetes	255,438	2,101	1.21%	P <0.0001
	Eye characteristics				P <0.0001
	High myopia	7,311	507	9.21%
	Eye trauma	3,269	203	7.58%
	Surgical technique				P <0.0001
	≥ 1 phacoemulsification	1 759,588	10,812	0.96%
	≥ 1 extracapsular extraction	15,948	373	3.87%
	Vitrectomy for peri-operative	5,095	279	6.39%	P <0.0001
	capsular rupture

Table 2 . Crude and adjusted hazard ratio (HR) of retinal detachment after cataract extraction

 2 Adjusted for age, gender, history of diabetes, high myopia, eye trauma, and anterior vitrectomy for peri-operative capsular rupture and surgical technique (extra capsular cataract extraction vs. phacoemulsification).

	Crude HR	95% CI	Adjusted HR*	95% CI

*

Table 3 . Multi-adjusted hazard ratio (HR) of retinal detachment after cataract extraction among myopic and non-myopic patients.

 3 Adjusted for age, gender, history of diabetes, eye trauma, and anterior vitrectomy for perioperative capsular rupture and surgical technique (extra capsular cataract extraction vs. phacoemulsification).

	Myopic patients	Non-myopic patients
	Adjusted HR*	95% CI	Adjusted HR*	95% CI

*


Benjamin Girault

STATIONARY GRAPH SIGNALS USING AN ISOMETRIC GRAPH TRANSLATION

Keywords:

In this communication, we extend the concept of stationary temporal signals to stationary graph signals. We introduce the concept of strict sense stationary and wide sense stationary graph signals as a statistical invariance through an isometric graph translation. Using these definitions, we propose a spectral characterisation of WSS graph signals allowing to study stationarity using only the spectral components of a graph signal.

Finally, we apply this characterisation a synthetic graph in order to study a few important stochastic graph signals. Also, using geographic data, we study weather readings on the graph of weather stations and show evidence of stationarity in the temperature readings.

INTRODUCTION

The evolution of computing capabilities or techniques have produced an avalanche of data, and more importantly structured data. Social networks connecting people, technological networks connecting computers or smart devices, or geographic data are a few examples among many.

The emerging field of signal processing aims at tackling the problem of studying these data using the powerful approach of signal processing. The question at hand is then how to extend successes obtained by signal processing on temporal data or images to the less ordered, more complex, yet very rich graph structures. Recent successes include, but are not limited to, filtering [START_REF] Sandryhaila | Discrete Signal Processing on Graphs[END_REF], short-time Fourier transform and wavelets [START_REF] Shuman | The Emerging Field of Signal Processing on Graphs: Extending High-Dimensional Data Analysis to Networks and Other Irregular Domains[END_REF], empirical mode decomposition [START_REF] Tremblay | Graph empirical mode decomposition[END_REF], or incertitude principle [START_REF] Agaskar | A Spectral Graph Uncertainty Principle[END_REF].

This communication aims at being part of this journey to a powerful signal processing on graphs. We propose a definition of stationary graph signals, a class of signal that proved to be crucial in the context of temporal data [START_REF] Flandrin | Time-frequency/time-scale analysis[END_REF]. A spectral characterisation of those signals completes the definition. Finally we apply those to a few illustrative applications.

Work partially funded by Labex MILYON.

BACKGROUND

Stationary Time Signals

We denote x(t) a stochastic signal. Throughout this communication, variables written with a bold font are stochastic variables. Among the wide class of stochastic signals, one subclass is of particular interest: the stationary signals. The initial definition is that of Strict-Sense Stationary (SSS) signals verifying: ∀t, ∀s, x(t)

d = x(s), (1)
i.e. the probability density functions of x(t) and x(s) are equal. This property is in practice impossible to verify since it involves the computation of all statistical moments. We usually are satisfied with a less strict definition named Wide Sense Stationary (WSS). A signal is WSS if its first two statistical moments η x (t) = E[x(t)] and R x (t, s) = E[x(t)x * (s)] verify:

η x (t) = η x (2) R x (t, s) = γ x (t -s). (3)
The first order property (2) is equivalent to the mean being the Direct Current (DC) component of the signal. In the second order property (3), the function γ is called the autocorrelation function of the signal. In the context of WSS time series, R is a Toeplitz matrix with R ij = γ(t it j ). WSS signals admit a harmonic decomposition, i.e. they can be written as the Fourier-Stieljes integral:

x(t) = ∞ -∞ e -ıωt dX(ω), (4)
with dX(ω) a probability measure on the spectral decomposition of the signal named the spectral increments. Stationarity is then characterised by a double orthogonality [START_REF] Flandrin | Time-frequency/time-scale analysis[END_REF]: The exponential basis is orthogonal, and the spectral increments are orthogonal:

∞ -∞
e ıωt e -ıω t dt = 2πδ(ωω )

E[dX(ω)dX(ω ) * ] = 2πδ(ω -ω )Γ(ω)dωdω , (5)
Γ(ω) being the Fourier transform of γ(t), and called the power spectrum density. Stochastic signal analysis of x of-ten involves the study of γ(τ ) and Γ(ω) making these two functions crucial to the field [START_REF] Papoulis | Probability, random variables, and stochastic processes[END_REF]. Both definitions of stationarity naturally invoke the time shift operator. Let T τ be the time shift operator acting on a signal x as T τ {x}(t) = x(tτ ). The SSS property reduces to a statistical invariance of a stochastic signal with respect to T τ , while WSS is the invariance of the first two moments (η x = η Tτ {x} and R x = R Tτ {x} ).

In other words, given a time instant, the statistical properties of the signal and its translated version are statistically the same at that instant. Also, the samples of the signal at all time instants are statistically the same. For temporal signals those two properties are equivalent, but are worth noticing since only one of them will remain true for graph signals.

Moreover, given e ω (t) = e ıωt the Fourier mode of (angular) frequency ω, the expression of the time shift in the frequency domain is given by:

T τ {e ω }(t) = e ıτ ω e ω (t). (7)
The Fourier modes are eigenvectors of the time shift operator. Therefore, the time shift behaves as a phase shifting operator in the frequency domain.

We refer the interested reader to [START_REF] Flandrin | Time-frequency/time-scale analysis[END_REF] and [START_REF] Papoulis | Probability, random variables, and stochastic processes[END_REF] for details on stationarity.

Signal Processing on Graphs

We now introduce the field of signal processing on graphs. Let G = (V, E) be a graph, with V the set of vertices and E the set of edges ij. We suppose the graphs to be symmetric, i.e. ij ∈ E if and only if ji ∈ E. Let A be the adjacency matrix of G, with a ij the weight of the edge ij. Let D be the diagonal matrix of degrees d ii = j a ij = d i . Since G is symmetric, A is a symmetric matrix. We define then L = D -A the Laplacian matrix of G. L is a semi-definite positive matrix. Let Lχ l = λ l χ l be the eigendecomposition of L, with 0 = λ 0 ≤ λ 1 ≤ . . . [START_REF] Brouwer | Spectra of graphs[END_REF]. We will suppose the graph to be connected such that λ 1 > 0 [START_REF] Brouwer | Spectra of graphs[END_REF].

We call a graph signal a function assigning values to vertices: X : V → R or C. Most often, the Fourier transform is defined as the projection on the basis of eigenvectors of L [START_REF] Shuman | The Emerging Field of Signal Processing on Graphs: Extending High-Dimensional Data Analysis to Networks and Other Irregular Domains[END_REF]. Let F be the matrix of this Fourier transform. We have L = F * ΛF , with Λ = diag(λ 0 , . . . , λ N -1 ). Since L is semi-definite positive, {χ l } l is orthogonal and the matrix F is unitary. We have an equivalence of Parseval's identity:

X 2 = F X 2 , with X 2 2 = i |X i | 2 the l 2 -norm. We denote X = F X the Fourier transform of a signal X.
Let H be a linear operator on graph signals. We use the notation H to denote both the operator and its matrix representation. Its action is that of X → HX, with X represented as a column vector. We denote H = F HF * the expression of H in the Fourier domain.

STATIONARITY FOR GRAPH SIGNALS

Stochastic analysis for temporal signals have produced very interesting results, and we naturally wish to leverage the full potential of the statistics of graph signals to analyse them. Naturally, we need definitions, concepts and associated characterisations for that purpose. As seen in the previous section, stationnarity is one of those important concepts and we propose to define its counterpart for graph signals.

For temporal signals, stationarity can be interpreted as the statistical invariance by time shifting. There exists in the literature two operators defining equivalents of the time shift for graph signals. The generalized translations in [START_REF] Shuman | The Emerging Field of Signal Processing on Graphs: Extending High-Dimensional Data Analysis to Networks and Other Irregular Domains[END_REF] define operators acting as generalized convolution by a delta centred on vertex i whose expression in the Fourier domain is

T i X(l) = √ N δ i (l) X(l).
These operators, when applied to very specific input signals, localise their input around vertex i. The authors of [START_REF] Sandryhaila | Discrete Signal Processing on Graphs[END_REF] proposed the graph shift as the matrix multiplication X → AX. The graph shift is then an operator diffusing a sample from one vertex to its neighbours according to the edge weights.

Unfortunately, neither operator is isometric with respect to the l 2 -norm, i.e. T i X 2 = X 2 and AX 2 = X 2 . As opposed to the time shift, these two operators lack the mathematical comfort of isometry to define statistical invariance. In another publication, we proposed an alternative timeshift-like operator for graph signals which is isometric by design. After recalling its definition, we use it to propose a definition of stationary graph signals.

Graph Translation

In [START_REF] Girault | Translation on Graphs: an Isometric Shift Operator[END_REF] we proposed the graph translation as a new operator defined by analogy to the time shift for temporal signals. To this end, the graph translation is defined as an isometric operator shifting the phase of each of the spectral components of the input, similarly to [START_REF] Brouwer | Spectra of graphs[END_REF]. More precisely, we define graph frequencies as:

ω l = π λ l /ρ G , (8)
where ρ G is an upper bound on the eigenvalues of L [START_REF] Ch | Extremal graph characterization from the bounds of the spectral radius of weighted graphs[END_REF]:

ρ G = max i∈V 2d i (d i + di ) with di = ij∈E w ij d j d i .
The graph frequencies lie in [0, π], and a small ω l corresponds to a Fourier mode χ l of lower frequency. The graph translation is then defined as a phase shifting operator on the Fourier modes, i.e. such that it verifies T G χ l = e ıω l χ l : Definition 1 (Graph Translation). The graph translation operator is defined using its matrix representation as:

T G = exp ıπ L/ρ G .
Details on the construction, justification of the formula, and comparison to the generalized translations and graph shift can be found in [START_REF] Girault | Translation on Graphs: an Isometric Shift Operator[END_REF].

Definition of stationary signals

We now propose to define stationarity for graph signals as the statistical invariance to the graph translation: Definition 2 (Strict-Sense Stationary). A stochastic signal X on the graph G is Strict-Sense Stationary (SSS) if and only if:

X d = T G X. (9)
Def. 2 is tractable thanks to the isometric nature of the graph translation. However, as for temporal signals, SSS is difficult to verify in practice, and we introduce a weaker, more practical definition of stationarity: Definition 3 (Wide-Sense Stationary). A stochastic signal X on the graph G is Wide-Sense Stationary (WSS) if and only if:

E[X] = E[T G X] (10)
E[XX * ] = E[(T G X)(T G X) * ]. (11)
The invariance property is interpreted as follows: Given a vertex i ∈ V , the random variables X i and (T G X) i are statistically equal (SSS) or have identical moments of the first and second orders (WSS). However, in contrast to temporal signals, given two different vertices i and j, there is in general no equality in law between X i and X j .

The definition of WSS signals is introduced in the vertex domain. As for temporal signals, we propose a dual characterisation of WSS signals in the spectral domain. We begin by the first moment. Let η = E[X] be the mean of the graph signal. Eq. 10 is then equivalent to η = T G η. In other words, η is an eigenvector of T G associated to the eigenvalue 1 = e 0 = e ω0 . Since λ 1 > 0, η is collinear to χ 0 : The vector of mean is collinear to the Fourier mode of frequency ω 0 = 0. As for temporal signals, the mean of the signal is the equivalent of its DC component.

Next, we characterise the second moment. Let R = E[XX * ] be the autocorrelation matrix of the signal. Eq. 11 gives then:

R = E[(T G X)(T G X) * ] = T G E[XX * ]T * G = T G RT * G . Let S = E[(F X)(F X) * ]
be the autocorrelation matrix of the Fourier transform of the signal. By linearity of the Fourier transform, we obtain S = F RF * and R = F * SF . The equality above becomes:

S = T G S T G * . ( 12
)
Assuming that all eigenvalues of L are distinct, then (12) is verified if and only if S is diagonal. In other cases, S diagonal is a sufficient, but not necessary, condition. We can now formally write the spectral characterisation of WSS signals:

Proposition 1 (Second Moment Characterisation). The second moment of a graph signal is invariant through graph translation if its spectral components are uncorrelated (sufficient condition). This condition is also necessary if all graph frequencies are distinct.

The eigenvalue uniqueness condition is actually verified on many real world graphs. i.e. when weights depend on measurements. This comes from the fact that a small random perturbation of weights will slightly change the eigenvalues making them easily unequal. On the contrary, synthetic graphs showing high regularity, such as cycles, or regular grids with unit weights have many multiple eigenvalues [START_REF] Brouwer | Spectra of graphs[END_REF].

Prop. 1 is similar to the spectral characterisation of WSS time series (resp. WSS signals) where the spectral components (resp. the spectral increments) are uncorrelated. We have then a doubly orthogonal decomposition.

Discussion

We remark first that the operator T G is in general a complex operator, meaning that if X is real-valued, then T G X is usually complex. For WSS signals, this is not an issue since Prop. 1 shows that only the correlation between spectral components matters: A signal can be both WSS and real, and both sides of (11) are real for WSS signals.

We consider now the SSS property in the particular case of real-valued signals. Let X be such a signal. Def. 2 and the Fourier transform give then:

∀i, ∀x, P[X i = x] = P[(T G X) i = x],
and by linear combination of the equalities above:

∀l, ∀ x, P[ X l = x] = P[( T G X) l = x] ⇔ ∀l, ∀ x, P[ X l = x] = P[e ıω l X l = x].
When l = 0, e ıω l X l is complex and X l is real, such that the probabilities above are non zero only for x = 0 = X. Therefore:

∀l = 0, P[ X l = 0] = 1.
A real SSS signal is reduced to a DC component of random amplitude. This case corresponds to the deterministic relation X = T G X since a DC component is invariant through the graph shift. On the other hand, complex graph signals do not suffer from this constraint. This contrasts with temporal SSS signals that can be real without reducing to a DC component.

Another remark concerns the autocorrelation matrix R. We know that for temporal WSS signals, this matrix is Toeplitz. This is not the case for graph signals. This is to be expected since having such a structure would mean having an autocorrelation function γ which is a function of the difference between vertex indices. In general such a difference does not make sense. For example, shifting all indices by one would leave the matrix R unchanged. However, the graph translation does not perform such a shift such that it is unlikely that the signal is invariant to both the graph translation and this vertex index shifting operation.

Next, the characterisation of the first order moment in the previous section is actually independent of the underlying graph structure. Indeed, χ 0 being always a constant vector, the mean of a WSS signal is constant across vertices. However, several definitions of Fourier transform for graph signals exists, F being one of them. There exists an alternative definition based on the normalized Laplacian L = D -1/2 LD -1/2 [START_REF] Shuman | The Emerging Field of Signal Processing on Graphs: Extending High-Dimensional Data Analysis to Networks and Other Irregular Domains[END_REF]. Both matrices share similar properties. In particular, they are both semi-definite positive. We denote ψ l the eigenvector associated to the eigenvalue µ l of L. Prop. 1 is still valid for the Fourier transform F. However, and as opposed to χ 0 , ψ 0 is not constant and equals

( √ d 1 , . . . , √ d N ) T .
We denote T G the associated graph translation. We can then use this graph translation to obtain an alternative definition of stationarity. A connected graph verifies µ 1 > 0 such that a WSS signal has a mean vector collinear to ψ 0 . Also, SSS real-valued graph signals are now collinear to ψ 0 , and not constant. Finally, results on the second moment can be adapted to T G directly.

The use of F or F carries then a different notion of DC component, adapted or not to the local structural properties of the vertices. This accounts for the heterogeneity between vertices weighing differently in the graph structure. Using the Laplacian matrix or the normalized Laplacian matrix carries then different meanings, and using one or the other depends on the application at hand. The rest of this communication will use F as the Fourier transform.

APPLICATIONS

In this section, we wish to apply the notion of stationary graph signals we introduced to concrete graph signals. First, using synthetic signals, and then to study a dataset and the stationarity of a real world signal.

Synthetic data

In this section, we study several simple stochastic signals. To that end, we use the graph of Fig. 1(a). This graph has 100 vertices randomly sampled in the 2 dimensional plane, each connected to its 8 nearest neighbours. The edge weights are defined by a Gaussian kernel of the Euclidean distance in the plane (a ij = exp(-d(p i , p j ))).

We propose to define white noise on graphs as a stochastic signal having a flat power spectrum, i.e. with S = σ 2 I N , with mean collinear to χ 0 . A realization of white noise with zero mean is shown on Fig. 1(a). Using Prop. 1, a white noise is a WSS signal. Since the Fourier matrix F is unitary, we have also R = σ 2 I N . Therefore, the samples on vertices are uncorrelated, and of equal variance.

We consider now a signal X with samples on the vertices independent and identically distributed (i.i.d.). Then, R = σ 2 I N = S. X is therefore a white noise. The whole class of white noise is actually independent of the underlying graph structure. Indeed, the samples being uncorrelated, the edges play no role in explaining any correlation between them.

The third signal we consider is a WSS signal with nonconstant spectral power, i.e. with S diagonal but of non constant diagonal. Without loss of generality, we suppose the mean of the graph signal to be constant. Fig. 2 shows the empirical correlation matrices R and S obtained using 50k realisations. In general, such a signal has correlated samples, but in contrast to temporal signals, the second moment is in general not constant across vertices, i.e. the diagonal of R is not constant. Therefore, the mere fact that the signal is WSS is not a guarantee that the samples have the same variance. This is an illustration of the property that WSS depends on the underlying graph structure. Also, as soon as the signal shows some correlations between vertices, the edges of the graph contribute to explaining those correlations.

Real dataset

The French national meteorological service published in open access a dataset of hourly weather observations in Brittany for the month of January 2014 1 . From this dataset, we wish to study the temperature readings on the graph of ground weather stations. We define the graph of ground stations to be an image of the spatial distances between stations. Fig. 3(a) shows it. Each vertex (i.e. ground station) is connected to its five nearest neighbours, and the weights are a Gaussian 1 https://www.data.gouv.fr/ kernel (of parameter 10 -5 ) of the spatial distance between ground stations. Note that these arbitrary parameters can be slightly changed without altering the results. We wish to assess the stationarity of the temperature readings. Let X i (t) be the temperature at vertex i at time t. Fig. 3(a) shows the mean temperature of each station. This mean is obviously not constant, hence not collinear to χ 0 . Therefore, we cannot model the readings X i (t) using a stationary graph signal of which the readings are realisations.

Fig. 3(b) shows the temperature variation over time of three ground stations. We observe that these time series have a significant trend, decomposed into seasonal variations and non-deterministic coarse scale variations. To attenuate this trend, we consider the first Intrinsic Mode Function (IMF) of the Empirical Mode Decomposition of each of the time series X i [START_REF] Huang | The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis[END_REF]. The first IMF goal is precisely to remove the trend of a time series. We denote the IMFs Y i . We assume then that the graph signals {Y (t)} t are independent realisations of the stochastic graph signal Y. Its first empirical moment is zero everywhere, since the time series Y i are centered.

Fig. 4 shows the empirical correlation matrices of Y and of its spectral decomposition Y. The matrix R Y shows some significant off-diagonal elements, suggesting significant spatial correlations between vertices. On the other hand, S Y is close to a diagonal matrix, suggesting that Y has uncorrelated spectral component and is indeed a stationary graph signal.

We stress here that this assessment of stationarity is performed with respect to the graph structure we defined. This graph can be defined differently leading to another definition of Fourier transform, and eventually of stationarity.

CONCLUSION

This communication introduces two definitions of stationary graph signals. In particular, our definition of WSS graph signals is simple yet rich enough to allow a tractable spectral characterisation. These definitions and their consequences share many properties with temporal signals, however, the more complex structural properties of graphs introduce fundamental differences with time, accounting for the heterogeneity of the structure.

Facing the still infancy of graph signal processing, our contribution brings in an important ingredient that will serve as the starting point of further development towards the constitution of an entire toolbox for analysing real-world measurements modelled as stochastic signals on graphs. A few examples in that toolbox include the study of more classes of stochastic signals, being stationary or not, optimal filtering, or the development of a stationarity test.

Fig. 1 :

 1 Fig. 1: White noise on a graph with σ 2 = 1 and zero mean. (a) shows a realisation of this white noise, and (b) the empirical spectral correlation matrix computed using 50k realisations.

Fig. 2 :

 2 Fig.2: Empirical second moments of a WSS signal with zero mean and non uniform spectral power with 50k realizations.

Fig. 3 :

 3 Fig. 3: Spatial temporal variations of temperatures in Brittany in January 2014. (a) mean temperature of each station. temporal variations of three stations circled in (a) on top, and first IMF of these time series below.

  |R Y |.

  |S Y |.

Fig. 4 :

 4 Fig. 4: Empirical spectral correlation matrices of Y.


Delphine Lahet
email: delphine.lahet@u-bordeaux.fr

Anne-Gaël Vaubourg
email: anne-gael.vaubourg@u-bordeaux.fr

Banks' shareholding in multilateral trading facilities: A two-sided market perspective

Keywords: G10 G23 G24 L10 L11 L22 banks, shareholding, multilateral trading facilities, two-sided markets 1

The aim of this paper is to account for the observation that banks are both owners and clients of Multilateral Trading Facilities (MTFs) which were created in Europe after the implementation of the Markets in Financial Instruments Directive (MiFID). Using a duopoly model of two-sided markets, we show that banks' participation in MTFs crucially affects their objective function shape, pricing policy and profit. We show that when brokerage and trading activities are particularly important for banks' revenue compared to their profit as MTF operators, some market outcomes may emerge, whereby both MTFs include banks' interest as clients in their objective function. In these situations, although they earn negative profit as shareholders, banks benefit from lower fees as MTF's clients. This finally results in larger global revenue. This may explain why banks are at the origin of the creation of MTFs and why they maintain their stake despite negative profit.

Introduction

The Markets in Financial Instruments Directive (MiFID), applicable in November 2007, aims to remove the remaining barriers to the supply of cross-border securities-related financial services and create a single securities market in Europe. The objective is to promote cross border competition in secondary securities (primarily equity) markets on the basis of three pillars: increased competition on a level playing field between trading venues, enhanced market efficiency and liquidity and better investor protection through improved transparency about trading venues. Consequently, under this European regulation, newly created trading electronic platforms or multilateral trading facilities (MTFs) are now allowed to compete directly with regulated markets. MTFs, which can be operated by investment firms or market operators, are similar to regulated markets in matching buying and selling orders: they allow the trading of securities that are admitted to trading on regulated markets. However, they are subject to different regulations and have no listing process.

BATS Chi-X Europe and Turquoise provide good examples of such trading platforms. BATS Chi-X Europe represents the 2011 merger of the two leading pan-European MTFs: BATS Europe (established in 2008 by BATS Global Market, a leading US operator of stock and options markets) and Chi-X Europe (created in 2007 by Instinet and a consortium of twelve financial institutions). Turquoise was initially funded by nine investment banks. Since December 2009, it has been mainly owned by LSE (51% since March 2010). Several other MTFs, such as Equiduct and Nyse Arca, still operate in Europe, but BATS Chi-X Europe and Turquoise are the most important. In 2013, BATS Chi-X Europe's and Turquoise's market shares amounted to 9.40% and 4.36% on FTSE, 11.81% and 5.10% on CAC40 and 10.62% and 3.16% on DAX30 respectively. 1This paper focuses on two aspects of MTFs, that few previous studies have examined. First, one observes that some MTFs are mainly owned by banks, which were also involved in their creation (see Table 1, in the appendix). For example, before the merger with BATS, BNP Paribas' ownership in Chi-X Europe was 1.89%. The stake was 8.24% for Credit Suisse, 8.23% for Merrill Lynch, 5.37% for Morgan Stanley and 5.12% for UBS. Similarly, Citigroup's stake in Turquoise has been 5.48% since 2011 whearas the stake has been 7.05% for Deutsche Bank and 3% for Barclays and JP Morgan. 2 Banks' shareholding in MTFs is all the more striking because they also route many orders to these MTFs for their own account or on behalf of their clients (AMF 2009). If banks are shareholders of MTFs, they may be tempted to urge the MTF they own to reduce the level of fees for themselves as clients and to ultimately respect the best execution principal for third parties. 3 Second, it is noteworthy that MTFs in Europe charge lower prices than regulated markets. Although acquiring accurate data about connectivity and trading fees charged by trading venues is very difficult, [START_REF] Fleuriot | La révision de la directive sur les marchés d'instruments financiers (MIF)[END_REF] indicates that, during the first three quarters of 2008, the average execution cost of a round-trip trade was between 0.25 and 0.30 basis points on MTFs whereas it was approximately 0.80 on regulated markets such as Deutsche Börse, LSE and Nyse-Euronext. In the same vein, AMF (2009) concludes that MTFs offer the best prices and the best volumes concerning CAC40 equities one in ten times, which is close to what regulated markets are proposing (one out of four times for Nyse-Euronext). Due to low fees and despite weak fixed costs and increasing trading volume, MTFs regularly incurred losses. For example, Turquoise's losses before tax were approximately 15.7 million pounds in 2008 and BATS Chi-X Europe's losses were 4.67 million dollars in 2012. Before the merge in 2011, BATS Europe never generated profit and Chi-X Europe earned profit in only 2010 (800,000 pounds), followed by huge losses in 2011 (10 million pounds). 4 One possible rationale for this market outcome is that because banks own the MTFs in which they participate as clients, they are able to influence the MTF pricing policy towards lower fees. Consequently, according to the weight banks attribute to their profit as shareholders and operators of MTFs compared to their utility as clients of MTFs (which depends on the relative importance of brokerage and trading activities for them), the banks' profit may be reduced whereas their utility and global revenue (defined as the sum of utility and profit) may increase.

This paper aims to account precisely for the influence of banks, both as shareholders and clients, in MTFs' objective function shape, pricing policy and profitability. We refer to previous studies on two-sided markets, in which two groups of agents interact through an intermediary called a platform [START_REF] Evans | Some Empirical Aspects of Multi-Sided Platform Industries[END_REF][START_REF] Roson | Two-Sided Markets: A Tentative Survey[END_REF][START_REF] Rochet | Platform Competition in Two-Sided Markets[END_REF][START_REF] Armstrong | Competition in Two-Sided Markets[END_REF]. Because the participation of each group contributes value to the other group, two-sided markets are associated with a specific class of network externalities, called cross externalities. Therefore, the attractiveness of the platform for agents of one group largely depends on the participation of agents of the other group. As revealed by [START_REF] Armstrong | Competition in Two-Sided Markets[END_REF] in a duopoly model, participation fees charged to agents when they access the platform are lower than without any externality effects. The author also demonstrates that participation fees charged to agents from one group decrease with the way their participation is estimated by the opposite group. Few articles resort to a two-sided market perspective to analyze financial markets externalities. [START_REF] Wright | One-Sided Logics in Two-Sided Markets[END_REF] notes trading venues as examples of two-sided market that allows security issuers and investors to interact, creating liquidity externalities. Foucault, Kadan and Kendel (2013) and Skjeltorp, Sojli and Wah Tham (2012) focus on interactions between liquidity makers and takers and examine their impact on the speed of liquidity provision and consumption, the intensification of market orders and new trading opportunities. However, this literature does not account for the situations where a participant in a platform is also its owner.

To fill this gap, we introduce the notion that a participant in a platform can also be its shareholder in a two-sided market model. Transposing the [START_REF] Armstrong | Competition in Two-Sided Markets[END_REF] framework to the MTF industry, we demonstrate that if brokerage and trading activities are important for banks' revenue compared to their profit as MTF operators, some market configurations may emerge, where both MTFs include banks' interest as clients in their objective function. In this case, MTFs' profit can be negative. However, because banks benefit from lower fees as MTFs' participants, they eventually earn substantial global revenue. The remainder of the paper is organized as follows. In section 2, we present the model. Section 3 concludes the paper.

We now turn to the theoretical model. We first present assumptions and then focus on the equilibrium of the model.

Assumptions

Following [START_REF] Armstrong | Competition in Two-Sided Markets[END_REF], we consider two MTFs i and j in a duopoly. This assumption accounts for the structure of the European MTF industry which is dominated by two platforms, Turquoise and BATS Chi-X Europe. Each platform incurs a unit cost denoted c.5 Two groups of agents also exist, denoted 1 and 2, that participate in MTFs to route buying or selling orders. For example, type 1 agents are buyers and type 2 agents are sellers. Agents 1 and 2 are uniformly and exogenously located on a unit segment and platforms are located at each extremity. Agents incur a unit transport cost, denoted t. This parameter accounts for the degree of agents' subjective differentiation between both platforms (in terms of ease of access or the order submission process for example), i.e., the degree of platforms' market power.

In this model, we address single homing: agents 1 (resp. agents 2) choose between platforms i and j to connect and place buying (resp. selling) orders on the venue. To do this, they are charged a fee by the platform. We denote p 1,i (resp. p 1,j ) the fee charged to agents 1 by the platform i (resp. j) to connect and place buying orders and p 2,i (resp. p 2,j ) the fee charged to agents 2 by the platform i (resp. j) to connect and place selling orders. Hence, following [START_REF] Armstrong | Competition in Two-Sided Markets[END_REF], we focus on participation or registration fees, i.e., fees charged to agents when they access the platform. For this reason, the fees do not depend on trading volume. The number of agents 1 participating in the platform i (resp. j) is denoted n 1,i (resp. n 1,j ) and the number of agents 2 participating in the platform i (resp. j) is denoted n 2,i (resp. n 2,j ). The total number of agents 1 and 2 is normalized such that n 1,i + n 1,j = 1 and n 2,i + n 2,j = 1.

Trading venues are characterized by the existence of cross liquidity externalities. In each group, agents positively value interacting with agents from the opposite group. Indeed, an increase in the participation of buyers increases the probability that sellers will find a counterpart, and vice versa. It seems reasonable to consider that liquidity externalities are valued similarly by buyers and sellers. To avoid unnecessary complexity, we assume that the benefit enjoyed by agents from each agent on the other side is 1.

Each MTF is assumed to have a majority shareholder, which is a bank. We denote by i the bank that is the main shareholder of platform i and by j the bank that is the main shareholder of platform j. Banks trade securities in financial markets and participate in their own MTF to submit orders. 6 For convenience, we assume that banks are buyers and belong to the group 1. 7 Because they are both owners and clients of the MTF, banks have a special position. As owners of a MTF, banks' interest is to maximize the platform's profit. However, as clients, they are also concerned about the price they are charged and the utility they obtain when submitting orders on the platform. Hence, banks have the choice between two strategies. They can let MTFs maximize profit without taking into account their interest as clients. Conversely, because they are majority shareholders of MTFs, banks have the ability to urge them to include in the objective function not only their profit as shareholders but also their utility as participants (which corresponds to the utility of type 1 agents). We denote U 1,i (resp. U 1,j ) agents 1's utility on the platform i (resp j) and U 2,i (resp. U 2,j ) agents 2's utility on the platform i (resp j). Finally, MTF i' and MTF j's profits are denoted Π i and Π j respectively, and banks' total revenues, defined as the sum of their profit and their utility as clients, are denoted R i and R j respectively. We denote α as the weight of MTF profit in banks' revenue and (1 -α) the weight of their utility as clients of the platform in banks' revenue, with 0 < α < 1. Parameter α accounts for the importance of brokerage and trading activities for banks as clients compared to profit as an operating member of the MTF: the lower α is, the more important brokerage and trading activities are.8 Hence,

R i = αΠ i + (1 -α)U 1,i and R j = αΠ j + (1 -α)U 1,j .
The model has two stages. First, banks choose whether to urge or not the MTF to include their utility as clients into their objective function. Then, platforms compete in prices. We examine Nash equilibria in each pricing subgame and in the full game.

Equilibrium

We now solve the model. We first examine the second-stage subgames. We then consider the first-stage game.

The second-stage subgames

In this section, we consider the three following cases: the subgame where both MTFs only maximize profit, the subgame where both MTFs maximize the global revenue of banks, and the subgame where one MTF maximizes profit and the other maximizes bank revenue.

(i) The subgame where both MTFs maximize profit Agents' utilities can be written as follows:

U 1,i = n 2,i -p 1,i , U 1,j = n 2,j -p 1,j , U 2,i = n 1,i -p 2,i , U 2,j = n 1,j -p 2,j .
(1)

Following [START_REF] Armstrong | Competition in Two-Sided Markets[END_REF], we rely on the Hotelling specification to determine agents' participation:

n 1,i = 1 2 + U 1,i -U 1,j 2t , n 1,j = 1 2 + U 1,j -U 1,i 2t , n 2,i = 1 2 + U 2,i -U 2,j 2t , n 2,j = 1 2 + U 2,j -U 2,i 2t . (2)
Because each MTF's objective function includes only the platform's profit, equilibrium prices p * 1,i , p * 1,j , p * 2,i and p * 2,j are set as follows: 1) and (2), we determine each agent's participation as a function of prices. Substituting for profit expressions, we derive first-order conditions. Thus, we obtain the following lemma.

{p * 1,i , p * 2,i } = ArgMax Π i = ArgMax (p 1,i -c)n 1,i + (p 2,i -c)n 2,i , {p * 1,j , p * 2,j } = ArgMax Π j = ArgMax (p 1,j -c)n 1,j + (p 2,j -c)n 2,j . From (
Lemma 1 For t > 1 (H1) 9 , the subgame where both MTFs maximize profit has a unique equilibrium, given by

p * 1,i = p * 2,i = p * 1,j = p * 2,j = c + t -1, n * 1,i = n * 2,i = n * 1,j = n * 2,j = 1 2 , Π * i = Π * j = t -1, U * 1,i = U * 2,i = U * 1,j = U * 2,j = 3 2 -c -t, R * i = R * j = 3 2 -c -t - 5 2 α + 2αt + αc.
This subgame refers to the equilibrium obtained by [START_REF] Armstrong | Competition in Two-Sided Markets[END_REF]. Prices increase with the agents' transport cost and the platforms' cost. Moreover, consistent with the standard results of the literature on two-sided markets, each group of agents is subsidized: prices charged to the agents of one group is reduced by the amount of externality (i.e., 1) they cause on each agent of the other group. Reducing the price charged to the agents of one group encourages them to participate in the trading venue. This effect increases the liquidity of MTFs and their appeal for the agents of the other group.

(ii) The subgame in which both MTFs maximize the bank's revenue Agents' utilities and agents' participation can be written as in subgame (i). However, because MTFs now maximize banks' revenue, we have:

10 {p * 1,i , p * 2,i } = ArgMax R i = ArgMax α[(p 1,i -c)n 1,i + (p 2,i -c)n 2,i ] + (1 -α)(n 2,i -p 1,i ), {p * 1,j , p * 2,j } = ArgMax R j = ArgMax α[(p 1,j -c)n 1,j + (p 2,j -c)n 2,j ] + (1 -α)(n 2,j -p 1,j
). Proceeding in the same way as in (i), we obtain:

9 Under this condition, the second-order condition is satisfied (the trace of the Hessian matrix is negative and its determinant is positive).

10 When banks are assumed to be both type 1 and type 2 agents, banks' objective functions also include agents 2's utility, i.e. n 1,i -p 2,i and n 1,j -p 2,j respectively.

Lemma 2 Under H1, the subgame in which both MTFs maximize banks' revenue has a unique equilibrium, given by

p * 1,i = p * 1,j = -2t -α + αc + 3αt α , p * 2,i = p * 2,j = 2 -3α + αc + αt α , n * 1,i = n * 2,i = n * 1,j = n * 2,j = 1 2 , U * 1,i = U * 1,j = 4t + 3α -2αc -6αt 2α , U * 2,i = U * 2,j = -4 + 7α -2αc -2αt 2α , Π * i = Π * j = (t -1)(-1 + 2α) α , R * i = R * j = 4t + 5α -2αc -12αt -7α 2 + 2α 2 c + 10α 2 t 2α .
It is interesting to compare this equilibrium with the one obtained in the subgame (i). When both MTFs include the banks' utility (i.e., the type 1 agent's utility), in their objective function, the price charged to agents 1 is lower than when the bank's utility is not taken into account.

Lemma 2 also indicates that cross subsidies exist between both types of agents. Because type 1 agents are charged lower fees, attracting them by encouraging type 2 agents to participate in the platform becomes less necessary. This effect allows platforms to balance lower prices charged to agents 1 with higher fees charged to agents 2 (p * 1,i < p * 2,i and p * 1,j < p * 2,j ).11 Moreover, in contrast with Lemma 1, if α is sufficiently low (i.e., α < 2 3 ), prices p * 1,i and p * 1,j are decreasing in t. The higher the market power of platforms, the lower the fees charged to agents 1. Indeed, when t is large, the pressure to charge high prices is strong, contrary to banks' interest as MTFs' clients. If banks' utility represents a large part of banks' revenue, platforms balance this effect by reducing the fees charged to agents 1.

Finally, consistent with the stylized facts explained in the introduction, the pricing policy described in Lemma 2 induces negative profit for MTFs if α < 1 2 , i.e., if brokerage and trading activities for banks as participants in MTFs represent a large share of their revenue. However, comparing equilibrium revenues in subgames (i) and (ii) respectively indicates that, in this case, due to lower participation fees, banks earn a larger revenue than when their utility as clients is not included in MTFs' objective function.

(iii) The subgame in which MTF j maximizes profit and MTF i maximizes the bank's revenue Agents' utilities and agents' participations can be written as in subgames (i) and (ii). The maximization program is as follows:

{p * 1,i , p * 2,i } = ArgMax R i = ArgMax α[(p 1,i -c)n 1,i + (p 2,i -c)n 2,i ] + (1 -α)(n 2,i -p 1,i ), {p * 1,j , p * 2,j } = ArgMax Π j = ArgMax (p 1,j -c)n 1,j + (p 2,j -c)n 2,j .
Proceeding in the same way as in subgames (i) and (ii), we obtain: Lemma 3 Under H1 and for α > 2 5 ,12 the subgame in which MTF i maximizes profit and MTF j maximizes the bank revenue has a unique equilibrium, given by

p * 1,i = -4t -3α + 3αc + 7αt 3α , p * 2,i = 4 -7α + 3αc + 3αt 3α , p * 1,j = -2t -3α + 3αc + 5αt 3α , p * 2,j = 2 -5α + 3αc + 3αt 3α , n * 1,i = 2 + α 6α , n * 2,i = 1 2 , n * 1,j = -2 + 5α 6α , n * 2,j = 1 2 , U * 1,i = 8t + 9α -6αc -14αt 6α , U * 1,j = 4t + 9α -6αc -10αt 6α , U * 2,i = U * 2,j = -2 + 5α -2αc -2αt 6α , Π * i = -4t + 3α + 5αt -12α 2 + 8α 2 t 9α 2 , Π * j = 2t + 6α -10αt -15α 2 + 17α 2 t 9α 2 , R * i = 16t + 33α -18αc -56αt -51α 2 + 18α 2 c + 58α 2 t 18α , R * j = 16t + 39α -18αc -62αt -57α 2 + 18α 2 c + 64α 2 t 18α .
Because it includes the bank's utility as a client in the objective function, MTF i charges agents 1 less than MTF j. As in Lemma 2, this effect is balanced by a higher price charged to agents 2. It is also noteworthy that the participation of type 1 agents in MTF i is higher than in MTF j (n * 1,i > n * 1,j ), thus making MTF i more attractive for agents 2. Hence, two effects are at play in the participation of agents 2. As explained above, a higher price is charged on MTF i than on MTF j but MTF i is more attractive than MTF j. Both effects balance each other such that the participation of agents 2 is the same on both MTFs. Finally, it can be easily demonstrated that Π j > Π i provided α is sufficiently low (α < 2t 3t-1 ), which indicates that if brokerage and trading incomes represent a large proportion of banks' revenue compared to profit as platform operators, the MTF that includes the bank's utility as a client in the objective function charges less agents 1 and is less profitable than the MTF that only maximizes profit. From elementary calculus, we also have R i > R j . Hence, the MTF that includes the bank's utility as a client in the objective function earns a lower profit but higher global revenue than its rivale.

The equilibrium described in Lemma 3 can be compared to the one obtained in subgame (i). Because MTF i internalizes banks' utility as clients (i.e., agents 1's utility), it charges a lower fee to agents 1 than in subgame (i). This effect is balanced by charging agents 2 a higher price. Hence, because prices on each platform are strategic complements, platform j charges a lower (resp. higher) fee to agents 1 (resp. agents 2) than in subgame (i). In other words, the fact that platform i internalizes the bank i's utility allows type 1 agents to be charged less on platform j.

The same reasoning applies when comparing subgames (iii) and (ii). Because MTF j does not internalize type 1 agents' utility, it charges them a higher price than in subgame (ii). This effect is balanced by a lower fee paid by agents 2. Hence, because prices on MTF i and j are strategic complements, platform i charges a higher (resp. lower) price to agents 1 (resp. agents 2) than in subgame (ii).

The first-stage subgame

We now turn to the first-stage subgame. The first-stage subgame is described in Table 2, in the appendix. Using Table 2, we can consider the three following cases: -case 1:

(Max Π i , Max Π j ) is a subgame-perfect equilibrium if 3 2 -c -t - 5 2 α + 2αt + αc > 16t + 33α -18αc -56αt -51α 2 + 18α 2 c + 58α 2 t 18α , i.e., if α > 8t 11t-3 ≡ α 1 . -case 2: (Max R i , Max R j ) is a subgame-perfect equilibrium if 4t + 5α -2αc -12αt -7α 2 + 2α 2 c + 10α 2 t 2α > 16t + 39α -18αc -62αt -57α 2 + 18α 2 c + 64α 2 t 18α ,
i.e., if α < 10t 13t-3 ≡ α 2 .

-case 3:

(Max Π i , Max R j ) (or (Max R i , Max Π j )) is a subgame-perfect equilibrium if 3 2 -c -t - 5 2 α + 2αt + αc < 16t + 33α -18αc -56αt -51α 2 + 18α 2 c + 58α 2 t 18α and 4t + 5α -2αc -12αt -7α 2 + 2α 2 c + 10α 2 t 2α < 16t + 39α -18αc -62αt -57α 2 + 18α 2 c + 64α 2 t 18α ,
i.e., if α < α 1 and α > α 2 . Because α 2 > α 1 , this situation is not possible.

Noting that 1 2 < α 1 < α 2 < 1, these conditions are summarized in Figure 1.

From Figure 1, we derive the following proposition:

Proposition 1 Two thresholds α 1 and α 2 (with 0 < α 1 < α 2 < 1) exist such that case 2 case 2 case 2 and case 1 case 1

1 2 α 1 α 2 0 1
Figure 1: The first-stage game: equilibria according to the level of α (a) if α < α 1 , the full game has a unique subgame-perfect equilibrium (Max R i , Max R j ) where both MTFs maximize banks' revenue, (b) if α 1 < α < α 2 , the full game has two subgame-perfect equilibria: (Max R i , Max R j ), where both MTFs maximize banks' revenue, and (Max Π i , Max Π j ), where both MTFs maximize profit, (c) if α > α 2 , the full game has a unique subgame-perfect equilibrium (Max Π i , Max Π j ), where both MTFs maximize profit.

Proposition 1 demonstrates that subgame-perfect equilibria crucially depend on the level of α. When α is low, i.e., when brokerage and trading activities are important for banks compared to their profit as MTF's operators, banks' utility as clients is included in the MTF's objective function. When the level of α is intermediate, two equilibria can emerge: one, in which MTFs only maximize profit, and another that include banks' utility as clients in the objective function. When α is large, banks' brokerage and trading activities are not very important for banks. For this reason, the utility linked to these activities is not taken into account in the MTF's objective function. Finally, the main result of Proposition 1 is that some market configurations may exist where MTFs include banks' utility as clients in the objective function.

Moreover, recalling Lemma 2, that in the case of the (Max R i , Max R j ) equilibrium, MTFs' profit is negative if α < 1 2 . Hence, we can conclude that, when α is low, i.e., when the weight of brokerage and trading activities in banks' revenue is sufficiently large, both MTFs include banks' utility as clients in the objective function and incur losses. Interestingly, this result accounts for the observed market outcomes whereby banks that own BATS Chi-X Europe or Turquoise earn negative profit as shareholders while benefiting from lower fees as participants to these platforms.

Conclusion

This paper aimed to account for the observation that European MTFs, such as Turquoise or Chi-X Europe, are owned by banks that also route orders on these MTFs for their own account or on behalf of clients. Considering a two-sided model in a duopoly, we investigate the influence of banks, both as shareholders and clients, in MTFs' objective function shape, pricing policy and profitability. We demonstrate that when brokerage and trading activities are particularly important in banks' revenue compared to their profit as MTF operators, some market outcomes may emerge, in which both MTFs include banks' interest as clients in their objective function. Although they earn negative profit as shareholders, banks benefit from lower fees as MTF's clients. This effect eventually results in larger global revenue. Our model thus explains why banks were behind the creation of MTFs and why they have an incentive to maintain their stake in these MTFs despite negative profit.

Our model could be extended in several interesting ways. First, we could consider that banks take stakes in both platforms and examine the implications of this cross-shareholding assumption on MTFs' price schemes and profitability. Second, we could introduce an historical platform, such as Euronext, to investigate how a third venue, which also allows IPOs and security issuing, affects MTFs' behavior.

Table 1 :

 1 Ownership of Chi-X Europe and Turquoise, in % Source: Authors calculations from firms' documents.

	Chi-X Europe

Table 2 :

 2 The first-stage game: banks' equilibrium revenues (The first entry in each cell corresponds to the bank i's equilibrium total revenue while the second entry corresponds to the bank j's equilibrium global revenue.)

	j
	Max Π
	j
	i

Source Fidesa and Agefi Hebdo 7-13 November

2 Source: Authors' calculation, from MTF's financial statements.3 The best execution principle requires choosing the best venue for the clients in terms of factors such as transaction costs, quality and speed of execution.

Since the acquisition by LSE in 2009, the profit of Turquoise is less obvious to interpret because it is combined with the profit of LSEG (London Stock Exchange Group).

It seems reasonable to assume that the technology and the business model of platforms are very similar such that they incur the same cost. Differences in platforms' unit cost do not qualitatively affect our findings.

Everything happens as if, due to habits, close relationships between the bank and the platform or banks' better knowledge of their own platform's functioning, banks' transportation costs was so large that each MTF had a monopoly power on its shareholder.

The fact that banks are both type 1 and type 2 agents does not qualitatively change our results.

For example, in 2013, Société Générale earned

million euros from brokerage activities while their profit as a shareholder of BATS Chi-X Europe can be approximated to 0.53% x 16.9 million pounds = 9 million pounds (0.53% being the ownership percent indicated in Table1 and 16.9 million pounds the profit of Bats Chi-X Europe in 2013). This can be interpreted as a low level of α.

This effect vanishes when it is assumed that banks submit not only a buying but also a selling order. However, this phenomenon does not challenge our key result.

This condition ensures that n * 1,j > 0.