Andreea Cosnita-Langlais

Jean-Philippe Tropeano
email: tropeano@univ-paris1.fr.

Learning by litigating: An Application to Antitrust Commitments

Keywords: antitrust, commitments, deterrence, legal learning JEL classi…cation: L41, K21, D82

This paper examines the impact of commitment decisions on the e¢ ciency of antitrust enforcement. We discuss the optimal use of commitments considering past rulings as a source of knowledge to better assess future similar antitrust cases. Our framework combines two key e¤ects: the deterrence of the anticompetitive behavior by the di¤erent enforcement regimes, and the dynamic perspective through litigation as a source of learning. We show that if the level of penalty is high enough, the antitrust authorities undervalue the dynamic informational bene…t of litigation and tend to over-use commitments.

Introduction

Currently, two options are available for most competition agencies to address an antitrust violation from unilateral anticompetitive conduct: either rule on an infringement through formal litigation, or instead reach a negotiated settlement with the …rm, also called commitments decisions (Europe) or consent decrees (US). [START_REF] Anderlini | Why Stare Decisis?[END_REF] These two enforcement tools lead to di¤erent outcomes. Litigation results in a formal prohibition of the practice together with the payment of a …ne if the …rm is eventually convicted. But litigation requires a robust theory of harm based on strong evidence. The alternative is to reach a settlement and …x the anticompetitive e¤ect of the practice by negotiating commitments with the …rm. Each procedure involves costs and bene…ts. The key social bene…t of a commitments procedure is an earlier restoration of market competition for lower administrative costs. In contrast, the main bene…ts from a prohibition decision are the deterrence through monetary sanctions and the creation of precedents that may improve future enforcement. Indeed, when using commitments, the competition agency does not formally identify an infringement, and the …rms admit no wrongdoing, [START_REF] Baker | A Theory of Rational Jurisprudence[END_REF] which impedes the creation of precedents. Following Landes and Posner (1976, p. 250 and 251), legal precedents may be viewed as an investment to increase the stock of knowledge useful for the assessment of future cases: competition agencies and courts of justice learn from litigating. Several recent antitrust cases epitomize the implications of the lack of learning from commitment decisions. After …ling a case against the major US studios and Sky UK for geoblocking agreements in 2015, [START_REF] Bebchuk | Litigation and Settlement under Imperfect Information[END_REF] the EC …nally closed it in 2019 after all investigated parties had [START_REF] Anderlini | Why Stare Decisis?[END_REF] See OECD (2016). [START_REF] Baker | A Theory of Rational Jurisprudence[END_REF] "Commitment decisions should …nd that there are no longer grounds for action by the Commission without concluding whether or not there has been or still is an infringement". (Recital 13 of Council Regulation (EC) No 1/2003 of December 2002). [START_REF] Bebchuk | Litigation and Settlement under Imperfect Information[END_REF] Case AT.40023 -Cross-border access to Pay-TV.

o¤ered commitments. However, the main legal question raised by the case (copyright infringement and parallel trade within the common European market) was left unresolved. The latest decision of the European Court of Justice in this case emphasizes that a commitments decision only requires a preliminary assessment of the practice, whereas establishing the existence of an infringement must follow from a thorough examination. [START_REF] Chen | Learning while setting precedents[END_REF] Formal rulings need a solid theory of harm, that may be used in future similar circumstances. After many years where the EC dealt with the abusive royalties in standard essential patent cases by means of commitments, a formal decision was eventually reached in the Motorola case that clari…ed the law and set a precedent. [START_REF] Choné | On the optimal use of commitment decisions under European competition law[END_REF] The European Commission openly acknowledged this potential for learning from formal rulings: it recently …ned Google e2.42 billion for breaching EU antitrust rules and stated that "Google has abused its market dominance as a search engine by giving an illegal advantage to another Google product, its comparison shopping service. [...]Today's decision is a precedent which establishes the framework for the assessment of the legality of this type of conduct". [START_REF] Daughety | Settlements[END_REF] Surprisingly, albeit planning to enforce prohibitions whenever there is signi…cant need for "deterrence, punishment and legal precedent", [START_REF] Gautier | Optimal Enforcement of Competition Policy: The Commitments Procedure Under Uncertainty[END_REF] the European Commission issued only very few prohibition decisions in novel areas of intervention for which legal guidance is much needed. [START_REF] Gennaioli | The Evolution of Common Law[END_REF] Since the entering into force on May 1 2004 of the Council Regulation No 1/2003 making room for commitments, the European Commission has heavily relied on such decisions to deal with antitrust violations: more than 60% of the antitrust cases (excluding cartels) did not formally sanction a violation, and more than 70% of the abuse-of-dominance cases were resolved with commitments (Mariniello, 2014). [START_REF] Landes | Legal Precedent: A Theoretical and Empirical Analysis[END_REF] In this paper we discuss the optimal use of antitrust commitments. Our primary goal is to understand how a forward-looking and benevolent competition authority (CA henceforth) may balance the present and future costs and bene…ts (saving of litigation costs vs lower deterrence and weaker learning process) of using commitments rather than pronouncing a formal prohibition decision.

For this purpose we develop a theoretical model to contrast two enforcement regimes:

"the strict enforcement" where the antitrust authority cannot propose commitments to …rms in order to …x the alleged anticompetitive behavior, but instead commits to always litigate the case, and the " ‡exible enforcement" which allows the CA to choose between proposing commitments and going to trial. Our broad objective is to identify the social cost and bene…t associated with each enforcement regime. To do so we consider alternatively two cases: …rst a purely static analysis, to grasp the basic trade-o¤ between litigation and commitments. Then we turn to a dynamic setting, where present rulings a¤ord better knowledge on future similar cases: the CA learns to correctly determine case facts in future cases by litigating present ones.

We start with the purely static analysis and show that the ‡exible enforcement leads the CA to propose commitments too often. Being able to choose between commitments and litigation leads the CA to neglect the higher deterrence e¤ect of litigation, and instead focus only on the saving of trial costs following a commitments o¤er. We show that the ability of the CA to assess accurately cases before possible litigation has an ambiguous impact on this static bias towards commitments. Indeed, if anticompetitive practices are quite accurately detected by the CA, then commitments are e¢ cient in saving on litigation costs. However, the accurate detection of anticompetitive practices also makes litigation a valuable enforcement tool by imposing …nes and thus e¢ ciently deterring the anticompetitive practices. In the end, if the deterring e¤ect of litigation is high enough, a better assessment of cases reinforces the e¢ ciency of the strict enforcement.

We then go on to examine a dynamic setting where the procedural choice is made at each period, and for which we assume that present litigation increases the accuracy of future detection. The question we address is to what extent the systematic static bias in favor of commitments may cause a dynamic ine¢ ciency by distorting the CA's assessment of the future situations. We show that the static bias is either neutral or leads the CA to undervalue the future informational bene…t from present litigation. Again, the deterrence achieved by the CA's detection activity is critical for this outcome: as explained before, a better future detection accuracy tends to make the litigation procedure optimal if deterrence is strong. This implies that in this case, the future CA's decision to favor the commitments is a source of an even stronger such bias at the present period. The policy implication of our results is that the ‡exible enforcement fosters excessive incentives to use the commitments procedure when the antitrust enforcement produces substantial deterrence of anticompetitive practices. This is true in a static context, but taking into account the dynamic learning e¤ect of litigation reinforces this result. Instead, a weaker deterrence, due to lower …nes in case of conviction for instance, will lower the opportunity cost of using the commitments procedure.

The paper unfolds as follows: next we review the related literature and our contribution to it. Then we present our model and examine …rst the static choice between litigation and commitments, before looking into this trade-o¤ in a dynamic setting.

Contribution to the related literature

This paper examines the optimal use by a Competition Agency of the two options available to dispose of an antitrust violation case, namely prohibition or commitment decisions. Since the latter allow to avoid the costly trial, our paper is connected to the law-and-economics literature on pre-trial settlement (see Daughety and Reinganum (2011) for an in-depth survey). In particular, our static analysis closely mirrors the screening model of Bebchuk (1984), since the uninformed party makes the take-itor-leave settlement o¤er, [START_REF] Kaplow | On the optimal burden of proof[END_REF] but in contrast, we endogenize the primary conduct of the defendant. Our paper actually belongs to a speci…c strand of the literature on pre-trial settlement that considers the context of antitrust with a public authority (the CA) as the plainti¤. The paper closest to ours is the preliminary work of Polo and Rey (2016), which uses the signalling/informed-party-making-the-settlement-o¤er framework to show that commitment decisions are appropriate only when the practice is socially harmful, when it is particularly damaging, and when gathering information is costly enough for the agency. Their paper endogenizes the …rms'choice of practice, as well as the enforcement regime (litigation or commitments), but does not consider the CA's ability to detect the true nature of the practice undertaken by …rms. This in turn is a crucial feature of our model, driving the optimal enforcement decision. In addition,

Polo and Rey (2016) assume a static, one-shot interaction between parties. [START_REF] Maggi | Learning by ruling and trade disputes[END_REF] . We [START_REF] Kaplow | On the optimal burden of proof[END_REF] Spier (1992) considers a …nite-horizon ultimatum game with uninformed party making the settlement o¤er, to show that there is a high likelihood of settlement in the last period. Reinganum and Wilde (1986) examine the opposite, signalling framework, where the informed party makes the settlement o¤er. Nalebu¤ (1987) relaxes the commitment to litigate if the settlement negotiation fails.

In the Appendix we argue the credibility of going to trial in our setting. [START_REF] Maggi | Learning by ruling and trade disputes[END_REF] consider instead a dynamic framework, making room for information arrival through trial, and thereby improved detection of other similar cases.

Judicial learning is typically associated with the evolution of legal rules. Although we do not model setting precedents, our framework does exhibit a decision in a current case a¤ecting future decisions. So, to a certain extent, our analysis is close to some contributions dealing with the evolution of law through judicial decision-making. In particular, our dynamic setting in which the agency's decision to litigate improves its own ability to detect future similar cases parallels Parameswaran (2018). This paper builds on Baker and Mezzetti (2017), where the court of law decides whether to summarily dispose of the case, or instead conduct a costly investigation to learn the ideal outcome, thus balancing between adjudication costs, potentially wrongful rulings, and the dynamic e¤ect of better information on future court decisions. [START_REF] Mariniello | Commitments or prohibition? The EU antitrust dilemma[END_REF] While Baker and Mezzetti (2017) assume a random arrival of cases to be tried, Parameswaran (2018) extends their analysis by considering an endogenous distribution of cases to be disposed of, as we also do. Another feature our paper shares with Parameswaran (2018) is that present litigation provides information on the true nature of the practice, thereby reducing future uncertainty. In his paper this will feedback into …rm's behavior, since the adoption of the harmful conduct depends on the court's precedent-setting policy. In our paper we address the agency's potential under-incentives to reduce future uncertainty through judicial learning because of future sub-optimal enforcement. In this sense, the two papers complement each other. Chen and Eraslan (2020) is another related paper studying the impact of learning for the evolution of legal rules, which do not allow for di¤erent types of practice, nor examine the role of improved detection, as we do. [START_REF] Mariniello | Commitments or prohibition? The EU antitrust dilemma[END_REF] A similar trade-o¤ is present in Anderlini et al. (2014), where the court may be either constrained by precedents (which evolve according to a dynamic process) or unconstrained. Gennaioli and Shleifer (2007) provided another dynamic model of (common) law evolution, with possibly biased judges endorsing the existing precedent. also builds on Baker and Mezzetti (2017). They consider a decision-maker ruling on a random case at each period, possibly bound by precedent but improving the quality of that decision by conducting a costly investigation. The purpose is to examine the dynamic consequences of binding or non binding precedents on the decision maker's incentive to acquire information. We also examine in a dynamic framework a decisionmaker's (the CA in our case) incentives to acquire information, but focus instead on the impact of future enforcement on these incentives at the current period. While

Chen and Eraslan (2020) study the role of binding precedent on the future incentives to acquire information, in our model it is the choice of litigation at the present period that represents a costly investment for future detection accuracy. In that sense, our paper is also related to the literature on learning through litigating. Maggi and Staiger (2020) in particular discuss how judicial learning may lead to litigation. For this, they consider a court whose quality of decisions rises as it learns from past rulings, and show that a lower frequency of disputes and rulings over time is consistent with learning from cumulative judicial decisions. Our dynamic model also exhibits this endogenous accumulation of knowledge, but resulting from the CA's decision to litigate, whereas

Maggi and Staiger (2020) relies on the parties'(countries engaged in a trade dispute) decision to go to trial. Moreover, we endogenize the enforcement regime, allowing or not for settlements, as well as the …rms'underlying behavior, in contrast to Maggi and Staiger (2020).

The model

The players and their information A …rm may adopt or not a given practice, which can be either pro-or anticompetitive. The true nature of the practice (i.e anti-or pro-competitive) is observed by the …rm. We denote by A the type of the …rm that may adopt an anticompetitive practice, and P the type of the …rm that may adopt a pro-competitive practice. [START_REF] Nalebu¤ | Credible Pretrial Negotiation[END_REF] Each type has the same a priori probability 1 2 : Adopting the practice, whatever its type, requires a …xed cost of k. This cost is uniformly distributed on the interval [0; K] according to the cdf F (x) = x K : The cost of the practice is …rm's private information. Whatever the type of practice, the …rm bene…ts from it: the resulting pro…t increase is equal to i > 0 for type i = A; P: The adoption of the practice of type A has a negative welfare impact equal to W A < 0 while the practice of type P has a positive welfare impact of W P > 0: The welfare impact is the welfare change with respect to the non adoption of the practice. It includes the entire external e¤ect on competitors and consumers.

The antitrust enforcement

The agency gathers pieces of evidence on the possible anticompetitive nature of the practice with probability e: This probability depends on the true type of the practice (A or P ) :

e = 8 < : if i = A 1 if i = P ; with 1 2 :
Parameter captures the detection accuracy of the CA.

The CA may start a procedure against the practice only if it has gathered pieces of evidence. Two di¤erent procedures are available to dispose of a case:

The litigation procedure (labelled Lit).

In this case, the CA aims to ban the practice. The prohibition is eventually enforced after a formal process of confrontation of evidence, possibly in a court of law. To capture this formal process, we assume that litigation occurs between the …rm and the CA. We denote T the social cost of litigation proceedings. The cost of litigation for the …rm is denoted c. We assume c is distributed on [0; c] according to the cdf G(x) and the density function g(x). The trial allows to discover the true nature of the practice with probability > 0. With probability 1 ; the trial outcome is inconclusive and does not allow to discover the true type, in which case the …rm incurs no liability. When convicted, the …rm must stop the practice and pays a …ne. We denote f the monetary penalty incurred upon conviction. We assume that P > c, meaning that litigation is always worthwhile for type P . This setting captures, in a nutshell, the infringement proceedings speci…ed by Article 7 of the European antitrust regulation. [START_REF]Commitment decision in antitrust cases -DAF/COMP/M[END_REF] The Commitments procedure (labelled Com).

This procedure allows the CA to negotiate commitments with the …rm. We summarize this settlement process by a commitments o¤er made by the CA that the …rm may accept or not. This mirrors the proceedings of article 9 of the European antitrust regulation. In case of refusal by the …rm, the CA commits to engage the formal litigation process described above and the trial occurs [START_REF] Parameswaran | Endogenous Cases and the Evolution of the Common Law[END_REF] . We assume that the only possible commitments consist in abandoning the practice: as a result, they will fully …x the anticompetitive concern if the practice is of type A; but will also cancel any welfare bene…t if the practice is of type P .

The key element of our analysis is the intertemporal role of the litigation procedure.

We consider that the trial has a direct impact on the future ability of both the CA and the judge upon trial to assess the actual nature of the practice. The trial consists of the confrontation of evidence and arguments regarding the practice, which improves the understanding of the latter whenever it leads to the discovery of the true nature of the …rm's conduct. Thus, when faced next with a similar practice, the CA as well as the judge will be better informed: in case of conclusive trial at date t; and of similar practice at date t + 1; the probabilities and increase. In short, present litigated cases are a source of knowledge and thus allows both the CA and the judge to be more accurate in the future. We adopt here the informational view of legal precedents following Landes and Posner (1976) (see also Spier (2007), p. 298) who explicitly consider that each litigated case increases the stock of knowledge for assessing future similar cases. In brief, the CA and the judge learn from successfully litigated cases.

We assume that the CA maximizes the net expected welfare. [START_REF] Polo | Antitrust Enforcement, Commitments and Remedies[END_REF] We contrast two types of enforcement. The …rst enforcement regime consists in not allowing the CA to o¤er commitments to …rms. This "strict" enforcement has the CA impose litigation (procedure Lit only) and forbids commitment negotiation. The other type of enforcement is " ‡exible", to the extent that the CA chooses between both proceedings: either propose commitments to …rms or directly go to trial. These two enforcement regimes lead to two di¤erent games that we describe below.

The "strict enforcement" game:

Stage 1 -The …rm observes its type and the cost of the practice. The …rm undertakes the practice or not.

Stage 2 -If the practice was adopted, the CA gathers pieces of evidence with probability e.

Stage 3 -If the CA obtained evidence, it decides to enforce or not the procedure Lit.

The " ‡exible enforcement" game:

Stage 1 -The …rm observes its type and the cost of the practice. The …rm undertakes the practice or not.

Stage 2 -If the practice was adopted, the CA gathers pieces of evidence with probability e. We represent in …gure 1 and 2 each enforcement game. The ‡exible enforcement game allows the CA to choose between commitments and litigation while with the strict enforcement game, the CA commits to impose litigation (the dotted line does not exist). [START_REF] Spier | The Dynamics of Pretrial Negotiation[END_REF] Insert Figure 1 In what follows, we …rst take a static perspective and determine the Perfect Bayesian Equilibria (PBE) of each game, so as to derive the optimal enforcement. Then we go on to consider a dynamic, two-period, enforcement game. When describing this twoperiod enforcement game later on, we shall also explain the CA's dynamic objective function.

4 The litigation-vs-commitment trade-o¤ and the learning e¤ect

The static choice

Our objective here is to derive the optimal enforcement and discuss to which extent the strict enforcement ought to be imposed.

The strict enforcement

We …rst examine the …rm behavior under "strict" enforcement, i.e. when litigation is imposed. To avoid trivial cases, we assume that if the CA gathers evidence, it litigates the case [START_REF] Spier | Litigation[END_REF] .

At stage 1, the …rm of type i = A; P adopts the practice i¤ the expected bene…t covers the cost of the practice. The expected gain for type A is equal to:

(1

) A f R c 0 cg(c)dc + (1 ) A :
Therefore, the probability for type A to adopt the practice is given by:

a Lit ( ; ) = F h (1 ) A f R c 0 cg(c)dc + (1 ) A i :
By the same token, the probability for type P to adopt the practice is:

p Lit ( ; ) = F h P (1 ) R c 0 cg(c)dc i .
We deduce that the expected welfare net of social litigation cost under "strict" enforcement is given by:

W Lit ( ; ) = 1 2 p Lit ( ; ) W P (1 )T 1 2 a Lit ( ; ) (1 ) W A + ((1 )W A + T ) :
The expected welfare is given by the sum of the welfare gain net of the social litigation cost in case of adoption of the pro-competitive practice (with probability p Lit ( ; )); and the welfare loss net of the social litigation cost in case of adoption of the anticompetitive practice (with probability a Lit ( ; )):

The ‡exible enforcement

We now consider the " ‡exible" enforcement game where the CA chooses between procedures Com and Lit. Below we determine the Perfect Bayesian Equilibrium of this game.

When the CA chooses which procedure to enforce, her choice is based on beliefs regarding the probability to face either a type A or a type P …rm. Let the CA believe that the probability for type A to have adopted the practice is a; and respectively p for type P .

If the CA chooses to impose litigation, the resulting expected welfare is equal to:

T + (1 )p a+(1 )p W P (1 )W A a a+(1 )p .
The expected welfare is composed of three terms: the social litigation cost that will always be incurred, the welfare gain of the pro-competitive practice (W P ) a¤ected by the conditional probability to face a P -type, and the expected welfare loss from the anti-competitive practice ( (1 )W A ) a¤ected by the conditional probability to face an A-type.

If instead the CA chooses to o¤er commitments, the expected welfare depends on the …rm's decision to accept or refuse the commitments proposal. A type A …rm accepts the commitments i¤ the expected pro…t in case of litigation, given by (1 ) A c f;

is lower than the zero pro…t earned if commitments were adopted. As a result, there is a Here, the expected welfare is composed, on the one hand, of the welfare gain of the pro-competitive practice net of the social litigation cost, (W P T ); a¤ected by the conditional probability to face a P -type, and the welfare loss from the anti-competitive practice adjusted for the social litigation cost, (1 )W A +T; a¤ected by the conditional probability to face an A type.

As a result, comparing both values of expected welfare, it is clear that the CA always proposes commitments: they allow some screening of types, albeit imperfect, which enables the CA to …x the behavior of a fraction of type A …rms at no cost. This highlights the two main bene…ts of the commitments proposal: by inducing type A to accept remedies, not only does the CA's o¤er …x the anticompetitive concern of the practice, but it also avoids costly litigation to some extent.

We summarize this result as follows (see proof in the Appendix): [START_REF] Wright | The economic analysis of antitrust consents[END_REF] Lemma 1 At the Perfect Bayesian Equilibrium of the ‡exible enforcement game, the CA always o¤ers commitments.

Lemma 1 makes clear the extent to which the CA always has incentives to o¤er commitments. As before mentioned, this is due to the screening outcome, together with the fact that the commitments save on litigation costs.

The optimal enforcement in the static case

We have previously spelled out the expected welfare expression in the case of strict enforcement, and we can now do the same for the ‡exible one. This will enable us to compare both enforcement options.

Under ‡exible enforcement, given that at the PBE the CA always proposes commitments, the ex ante expected welfare is given by

W Com ( ; ) = 1 2 p Com ( ; ) W P (1 )T ) 1 2 a Com ( ; ) (1 ) W A + G(b c)(( 1
)W A + T ) ; (1)
with the probability for type A and P to have adopted the practice under the ‡exible enforcement (Com) denoted a Com and p Com respectively, where:

a Com ( ; ) = F h (1 ) A + R b c 0 ((1 ) A c f )g(c)dc i and p Com ( ; ) = p Lit ( ; ).
The optimal enforcement is determined by comparing the two expected welfare expressions, hence by the sign of W Lit ( ; ) W Com ( ; ).

The strict enforcement is more e¢ cient than the ‡exible enforcement i¤ (see the Appendix):

W Lit ( ; ) W Com ( ; ) > 0 (2) i¤ T + (1 )W A W A | {z } Litigation cost (denoted ) < 1 | {z } Detection e¤ect (1 G(b c)) a Com ( ; ) a Com ( ; ) a Lit ( ; ) 1 1 | {z } Deterrence e¤ect S ( ; )
Condition (2) highlights the three di¤erent e¤ects driving the choice of optimal enforcement: a litigation cost e¤ect, a detection e¤ect and a deterrence e¤ect.

First, the litigation costs e¤ect comprises both the direct cost of litigation, T; and the associated welfare loss, since with probability 1 the trial may be inconclusive.

This e¤ect goes against the strict enforcement, because the ‡exible enforcement allows the CA to save on litigation costs, given that type A may be induced to accept commitments rather than go to trial.

The second e¤ect in condition (2) is the "detection e¤ect": the CA has ability to detect, although imperfectly, the anticompetitive practices ( > 1 2 ) and thus, it faces a higher share of A-types than P -types. Because of the litigation costs, it is then more e¢ cient to o¤er commitments, and thus this e¤ect goes against the strict enforcement

( 1 < 1).
Finally, we call the third e¤ect in condition (2) the "deterrence e¤ect": it is the change in incentives for type A. Explicitly, imposing litigation lowers the incentives for type A to adopt the practice (a Lit ( ; ) < a Com ( ; )) and leaves those of type P unchanged (type P prefers litigation even when commitments are o¤ered: p Com ( ; ) = p Lit ( ; )). This deterrence e¤ect stems from the higher cost incurred by the …rm, since it will have to pay a penalty f in case of conviction upon trial, which cannot be avoided following the initial detection. This e¤ect goes against the ‡exible enforcement.

In particular the better deterrence of the penalty f on type A (the lower a L ( ; )); the lower the deterrence e¤ect term making more likely the optimality of the strict enforcement.

The trade-o¤ between these three e¤ects explains the optimal choice. Imposing litigation, i.e. the strict enforcement, will be optimal as long as the relative total cost of trial (denoted ) is low enough. Note that the critical threshold level, denoted (S for Static) S ( ; ); depends on both accuracy parameters: the one for the quality of detection, , and the one for the quality of trial, .

It is straightforward to see that a higher increases the threshold S ( ; ): the accuracy of the trial lowers the cost of litigation, thus making the strict enforcement optimal more often.

In contrast, the accuracy of detection ( ) has an ambiguous impact in terms of optimal procedural choice. On the one hand, a more accurate detection allows the CA to better screen from the beginning the true type of the practice. This reduces decision errors and increases the bene…t of proposing commitments, since they will avoid costly litigation. This will strengthen the detection e¤ect, and thereby favor the ‡exible enforcement. But on the other hand, a more accurate detection lowers the expected pro…t of type A and thus decreases the share of this type facing the CA. But this decline in the share of type A is relatively larger under strict rather than under ‡exible enforcement, because type A would more often pay the …ne if litigation were imposed. This strengthens the deterrence e¤ect of the strict enforcement, and thus increases its expected bene…t. Finally, note that a higher …ne, hence a stronger deterrence through litigation, ampli…es the impact of a more accurate detection on the deterrence e¤ect. As a result, depending on the level of monetary sanction incurred in case of conviction (the …ne f ), two regimes must be considered for the impact of the accuracy of detection ( ).

Either the monetary penalty is high, and then the better detection (higher ) will favor the strict enforcement thanks to the increased deterrence -we call this "strong deterrence" regime. Or, on the contrary, if the …ne is low, the resulting deterrence e¤ect of litigation is quite low too, and hence the more accurate detection will favor instead the commitments o¤er (i.e. the ‡exible enforcement). We call this "weak deterrence" regime. 20 Next we summarize the outcome of the welfare comparison between the strict and ‡exible enforcement (see proof in the Appendix):

Proposition 1 The ‡exible enforcement is optimal i¤ the litigation cost is high enough ( > S ( ; )). The critical threshold S ( ; ) is always increasing with , but increasing with i¤ the …ne f is high enough.

Proposition 1 clari…es that the choice between the strict and ‡exible enforcement depends on the litigation cost. The critical value of the latter depends in its turn on the accuracy of both detection and trial. A higher trial accuracy will naturally favor the strict enforcement (i.e. imposing litigation). In contrast, the impact of the detection accuracy on the optimal enforcement, strict or ‡exible, depends on the deterrence achieved when imposing litigation. The monetary penalty in case of conviction upon trial is crucial for the deterrent e¤ect of litigation, and eventually signs the impact of the accuracy of detection on the choice of optimal enforcement. This result is important because it allows to compare the informational bene…t of a more accurate detection between the two enforcement regimes: a more accurate detection is more valuable with strict enforcement only if the deterrence derived from the monetary penalty is high enough. This result will play a key role when we consider the dynamic impact of the choice between litigation and commitments, which we do next.

The dynamic case and the learning e¤ect

We now examine each type of enforcement when the CA's procedural choice takes into account the learning e¤ect of present litigation on future enforcement. For this, we consider a two-period setting (t = 1 and t = 2) where the previous game is repeated twice. It has been established (Proposition 1) that in a static context, there is a strong bias toward commitments in case of ‡exible enforcement. The pending question is whether such a future bias at period 2 is likely to provide additional ine¢ cient incentives to adopt commitments at the …rst period, if the ‡exible enforcement is adopted. To answer this, we contrast the procedural choice made by the CA at t = 1 in two di¤erent cases:

-the ‡exible enforcement in the dynamic case: the ‡exible enforcement is used at both periods.

-the optimal enforcement in the dynamic case: the CA determines the optimal enforcement for each period.

The ‡exible enforcement in the dynamic case

The game is displayed in Figure 3 below. Insert Figure 3 here

We allow the values of parameters and to be di¤erent at t = 2 from t = 1, depending on the procedure chosen at t = 1. The following table explains this, and captures the learning e¤ect of litigation at t = 1 :

21 t = 1 t = 2
and > and > if conclusive trial at t = 1 and if no trial or inconclusive trial at t = 1

The CA maximizes the sum of the expected welfare net of the social litigation cost at each period. 22 This objective function preserves the agency's choice from any exogenous bias, since the CA will perfectly internalize the e¤ect of a present decision on future welfare.

We now determine the PBE of the whole game with two periods in the following lemma:

Lemma 2 With ‡exible enforcement in the dynamic case, litigation is the equilibrium procedural choice at the …rst period i¤

< D W Com W A ; where W Com = W Com ( ; ) W Com ( ; ) :
Explicitly, for litigation to be chosen at the …rst period, the total relative cost of litigation incurred by the CA at the time of the procedural choice ( ), must now be compensated by the future relative bene…t of present-period litigation, given by W Com ( ; ) W Com ( ; ) . The …rst period procedural choice may be interpreted as an investment decision that requires a cost at period 1, and yields a future bene…t through an increase in the future detection accuracy. Note that the future bene…t is measured taking into account the second period procedural choice, i.e. commitments.

We then compare that threshold D (D for dynamics), which dictates the adoption of litigation at t = 1 if ‡exible enforcement is used at t = 2; with the corresponding threshold in case of optimal enforcement: In doing so, we examine whether the learning dynamic e¤ect magni…es or lessens the previously identi…ed over-incentives toward commitments under ‡exible enforcement.

The optimal enforcement in the dynamic case

We consider the following game (see Figure 4 below), where: at t = 1 : the CA chooses between the ‡exible enforcement and the strict enforcement at t = 2 : the CA observes the values of and at t = 2 and chooses between the ‡exible enforcement and the strict enforcement.

Insert Figure 4 here

The objective of the CA is to maximize the sum of the expected welfare net of the social litigation cost.

We solve this game backwards 23 by determining …rst the optimal enforcement at 23 At this point there is an important remark to make on the resolution of the optimal enforcement game. We consider here that the CA determines the optimal enforcement at each period. We could have instead allowed the CA to choose the optimal enforcement for both periods at t = 1. This would have led to the same result because there is no bene…t to distort the enforcement choice at t = 2 in t = 2 according to ( 2 ; 2 ): We then identify the optimal enforcement at the …rst period taking into account the optimal enforcement chosen at t = 2: We derive the following result (see proof in the Appendix):

Lemma 3 If the optimal enforcement is adopted at the second period, then strict enforcement is optimal at the …rst period i¤:

< D = (1 ) (1 G(b c)) a Com (a Com a Lit ) 1 1 | {z } static e¤ ect= S ( ; ) + W A W Opt ( ; ) W Opt ( ; ) | {z } dynamic e¤ ect= W Opt ;
where W Opt ( ; ) is the expected welfare from the optimal procedure at t = 2 depending on the level of and at t = 2:

Lemma 3 indicates that the optimal enforcement choice at t = 1 is driven by the level of litigation cost, ; relative to a critical threshold denoted D :

The threshold D has two components: one due to the "static e¤ect" of strict enforcement, already present in expression (2). The second component is due to the "dynamic e¤ect" of the present procedural choice on future expected welfare. This dynamic impact is driven by the application of future optimal enforcement and not by the commitments procedure as in the case of ‡exible enforcement. It is easy to

show that W Opt = W Opt ( ; ) W Opt ( ; ) > 0; because the increase in and improves the expected welfare (see the Appendix). This means that the dynamic cost threshold for optimally choosing litigation ( D ) exceeds the critical threshold identi…ed order to in ‡uence the CA strategy at t = 1: The only decision that may be in ‡uenced at t = 1 is the choice between commitments and litigation (under ‡exible enforcement) towards commitments, since the CA can always impose litigation. But, we showed that there is a static over-incentive to choose commitments. Thus, there is no bene…t to further increase this incentive by distorting the enforcement choice at t = 2 in order to favor the commitments choice at t = 1 under the ‡exible enforcement. As a result, the optimal enforcement for t = 2 is the same whatever the period at which this enforcement is chosen (t = 1 or t = 2):

in Proposition 1, S ( ; ), if the dynamic e¤ect of litigation is ignored. The reason is that the litigation procedure yields future bene…ts: the improved future detection and litigation accuracy. These bene…ts logically lead the CA to optimally prefer the strict enforcement for a higher litigation cost. It implies that for < S ( ; ); the strict enforcement is optimal at the …rst period. We then focus on values of such that > S ( ; ):

The ‡exible vs optimal enforcement in the dynamic case

If the dynamic e¤ect is ignored, the commitments are ine¢ ciently proposed under ‡exible enforcement for in [0; S ( ; s)]. Taking into account the dynamic e¤ect will magnify this bias i¤ this ine¢ ciency-prone range of expands:

[ D D ] > [ S ( ; s)
] : Hence, to identify this possible additional bias toward commitments at the …rst period, we need to compare D and D : For that, it is necessary to weigh the dynamic consequences of increasing both detection and litigation accuracy, so as to determine whether the future ‡exible enforcement is likely to strengthen the static bias against litigation in case of ‡exible enforcement at the present period, or if instead, the future ‡exible enforcement alleviates this static bias. The following lemma provides the outcome of this comparison: Lemma 4 Under strong deterrence, we have:

(i) If > S ( ; ); then W Opt = W Com ; (ii) If S ( ; ) < < S ( ; ); then W Opt = W Lit ( ; ) W Com ( ; ) > W Com :
Under weak deterrence, for all > S ( ; ) we have W Opt = W Com :

Starting with the strong deterrence case, we …rst (i) consider the high litigation cost (i.e. when > S ( ; )), where commitments are optimal at the second period. In this case, the source of ine¢ ciency is purely static, and was already identi…ed before. It is only the static bias that leads the CA to undervalue the bene…t of imposing litigation:

D D = S ( ; ).
There is no dynamic ine¢ ciency, because at t = 2 commitments (which are chosen in equilibrium under the ‡exible enforcement) are also optimal.

The second possible subcase with strong deterrence, (ii), is that of an intermediate level of litigation cost: S ( ; ) < < S ( ; ). Here, the optimal choice at t = 2 depends on the accuracy of detection: if the latter is high following conclusive litigation at t = 1, then, under strong deterrence, it is optimal to impose litigation at t = 2, whereas if it remains low, it is better to allow for commitments at t = 2. Hence, the CA's under-incentives to impose litigation in subcase (ii) are partly due to the ine¢ cient choice at t = 2: with strong deterrence, a better detection makes litigation relatively more e¢ cient, therefore the sub-optimal choice at t = 2 leads the CA to undervalue the bene…t of litigation at t = 1 (since W Opt > W Com ). Here the dynamic bene…t of litigation strengthens the CA's under-incentives to impose litigation: D D > S ( ; ). Turning now to the weak deterrence regime, the same outcome prevails as in subcase (i) above, where the ine¢ ciency is only static, since at t = 2 commitments are optimal and also chosen in equilibrium under ‡exible enforcement. This does not come as a surprise given that with weak deterrence, the improved detection comforts the optimality of commitments.

To sum up, and in line with the interpretation of the choice of enforcement as an investment in future detection accuracy, we …nd that the ‡exible enforcement leads to a miscalculation of this investment cost, which always gets to be over-estimated because the deterrence e¤ect of the choice made at the present period is not considered.

In addition, there is also an under-estimation of the future bene…t from the higher detection accuracy whenever the CA can easily deter the anticompetitive behavior (i.e. when the strong deterrence regime holds).

Below we summarize these result in terms of the incentives for the CA to adopt litigation at the …rst period: Proposition 2 (1) In the dynamic case, the ‡exible enforcement leads to an overincentive to propose commitments at the …rst period as compared with the optimal enforcement (for 2 [ D ; D ]).

(2) The CA undervalues the learning bene…t of the Litigation procedure i¤ the monetary penalty upon trial conviction is deterrent enough, i.e. when the strong deterrence regime applies.

Proposition 2 has two implications.

First, for an intermediate level of litigation costs, the CA does not choose to enforce the Litigation procedure in equilibrium, so it should be imposed because it is the optimal option under optimal enforcement in the dynamic case.

Second, the ine¢ cient use of the commitment procedure typically occurs when the anticompetitive practice is strongly deterred. In this case, the CA's sub-optimal equilibrium choice at t = 2 to not impose litigation will comfort her sub-optimal choice at t = 1 to not litigate: the future static ine¢ ciency triggers dynamic ine¢ ciency. More precisely, the fact that the CA will not litigate tomorrow reduces her incentives to impose litigation today. To see this, recall that with strong deterrence, a better detection accuracy makes welfare bene…t of litigation higher than o¤ering commitments. As a result, given that at t = 2 the CA will always choose commitments in equilibrium, the future bene…t of a trial at the present period is lower than if the CA imposed litigation at t = 2. In other words, the CA's over-incentives to propose commitments tomorrow makes her undervalue the bene…t to impose litigation today: tomorrow's static inef-…ciency creates present dynamic ine¢ ciency. In short, a given period's bias toward commitments gives rise to a dynamic bias toward commitments.

Conclusion and …nal remarks

This paper discussed the optimal use of commitments in antitrust. We consider two alternative proceedings that the agency can use to dispose of a case, either by litigation or by o¤ering commitments, and conduct a comparison between two di¤erent enforcement regimes, one where the agency always imposes litigation and one where the agency chooses between litigation and commitments. In a purely static setting we identify a …rst ine¢ ciency a¤ecting the optimal enforcement choice: the CA undervalues the bene…t of litigation and therefore imposes trial in equilibrium less often than it is optimal. In a dynamic setting, allowing for increased accuracy of detection and adjudication following a conclusive trial, we …nd that the above-mentioned ine¢ ciency may lead the CA to overlook the dynamic bene…t of litigation, depending on the strength of deterrence e¤ect associated with the better detection of the practice. This shows the need to factor in the deterrence exerted by the antitrust enforcement when assessing the true bene…t of a commitments procedure.

Our analysis suggests that a ‡exible antitrust enforcement that allows for commitments proceedings has potentially a cumulative and self-reinforcing e¤ect over time, since the future CA's decision to favor commitments is a source of an even stronger such bias at the present period. In short, the agency has under-incentives to reduce future uncertainty via judicial learning due to future sub-optimal enforcement. This outcome hinges on the strength of the deterrence e¤ect associated with the future trial and detection of anticompetitive practices. So the direct policy implication is that commitments are not optimal when the antitrust enforcement achieves substantial deterrence of anticompetitive practices, and even more so when the dynamic bene…t of litigation is considered.

Appendix

Proofs

Proof of Lemma 1. In order to determine the Perfect Bayesian Equilibrium, we start …rst with the …rm's choice when o¤ered commitments: Type P chooses litigation over commitments i¤ P c > 0: This is always the case, by assumption, so type P always chooses litigation and refuses commitments.

Type A chooses litigation over commitments i¤ (1 We determine here the optimal procedural choice between the strict and the ‡exible enforcement (we omit ( ; ) in the notation of a Lit ; a Com ; p Lit and p Com ). For this we need to compare the corresponding ex ante expected welfare levels:

) A c f > 0, i.e. c < (1 )
-under strict enforcement:

W Lit ( ; ) = 1 2 p Lit W P (1 )T 1 2 a Lit (1 ) W A + ((1 )W A + T ) ; -under ‡exible enforcement: W Com ( ; ) = 1 2 p Com W P (1 )T 1 2 a Com (1 ) W A + G(b c)((1 )W A + T :
The di¤erence between the two writes:

W Com ( ; ) W Lit ( ; ) = 1 2 T + (1 )W A (G(b c)a Com a Lit ) 1 2 (1 )(a Com a Lit )W A :
We then consider only the case where (G(b c)a Com a Lit ) < 0: Otherwise, litigation is always optimal (that may be the case, in particular, if f is high enough):

we have that W Com ( ;

) W Lit ( ; ) > 0 i¤: T +(1 )W A W A = > 1 : R c b c ( f +c (1 ) A )g(c)dc+(1 G(b c)) R b c 0 ((1 ) A c f )g(c)dc R c b c ( f +c (1 ) A )g(c)dc + (1 G(b c)) A R c b c ( f +c (1 ) A )g(c)dc 1 = 1 A + B 1 S ( ; ); with B > 0:
The sign of A depends in particular on the level of f: We have @A @f < 0: It follows that if f is high enough, then it is possible to have A < 0: Thus the critical threshold S ( ; ) increases with i¤ A < 0:

In addition, if increases, 1 A + B 1 increases too, therefore S ( ; ) increases with :

Proof of Lemma 2. The ‡exible enforcement in the dynamic case

The procedural choice at t = 2 is always commitments, as described by Lemma 1.

The procedural choice at t = 1 is driven by a welfare comparison at stage 2 of period 1:

If the CA imposes litigation, the expected welfare is equal to

p(1 ) p(1 )+a W P a p(1 )+a (1 )W A T + W Com ( ; ) + (1 )W Com ( ; ) ,
If the CA decides to o¤er commitments, the expected welfare becomes where we denote by a and p the beliefs that type A and type P respectively have adopted the practice at t = 1.

The comparison of the two expected welfare expressions shows that litigation is the equilibrium procedural choice at the …rst period i¤ the following condition is satis…ed: Proof of Lemma 3. The optimal enforcement in the dynamic case Strict enforcement at t = 1 (litigation imposed at t = 1) :

(1 )W A +T W A < W Com
The expected welfare is given by: 1 2 pW P 1 2 W A a Lit (( 1) + ( 1)) 1 2 (p(1 ) + a Lit )T +

Q Lit W Opt ( ; ) + (1 Q Lit ) W Opt ( ; ) ;
where Q Lit denotes the probability of conclusive litigation at t = 1 under procedure

Lit and where W Opt ( ; ) is the expected welfare when the CA determines the optimal enforcement at t = 2 according to the level of parameters and : We have

Q Lit = 1 2
a Lit + 1 2 (1 ) p and also p = p Lit = p Com :

Flexible enforcement at t = 1 :

Following Lemma 2, commitments are o¤ered i¤ > W Opt W A : Then, in this case (i.e. when commitments are o¤ered), the expected welfare is given by: Optimal enforcement in the dynamic case

The strict enforcement (litigation imposed) is optimal at t = 1 i¤:

< (1 ) h (1 G(b c)) a Com (a Com a Lit ) 1 i 1 + 1 W A W Opt ( ; ) W Opt ( ; ) = D with D > W Opt W A .
6.2.2 The CA' s decision to impose trial in case of …rms' refusal of commitments

In case of ‡exible enforcement, we assume that the CA commits to go to trial if the …rm does not accept commitments. We justify below this assumption.

First, as long as the expected welfare is positive after Bayesian revision, then the decision to impose litigation is credible even after the …rm's decision to refuse commitments. This is the case i¤ In the other cases, we show that the CA commitment is optimal from the CA perspective: if we allow the CA to choose between "commit" or "no commit", it is always better for the CA to commit.

We consider the case where the CA does not want to go to trial if the …rm does not accept the commitments. In the absence of the CA's commitment to go to trial, the only possible equilibrium is one where the CA mixes between trial with probability and no trial with probability 1 :

In equilibrium, this mixed strategy imposes the same expected welfare in both cases, trial and no trial:

Stage 3 -

 3 If the CA obtained evidence, it chooses the procedure to enforce: either Com or Lit. Stage 4 -If procedure Com was chosen, the CA proposes commitments. The …rm observes its litigation cost c and accepts or not the proposition. If the …rm refuses commitments, procedure Lit is applied.

here Insert Figure 2 here

 2 Before turning to the formal analysis, we sum up below the parameters a¤ecting the …rm's and the CA's choices (to undertake the practice, to accept commitments or to litigate, and to propose commitments or to litigate), which are displayed in the game trees above. Industry parameters type and a priori probability of conduct A, P ; 1=2 cost of adopting practice and its cdf k 2 [0; K] ; F (x) = x K type-wise pro…t change i , i = A; P litigation cost and its cdf c 2 [0; c] ; G(x) monetary …ne if conviction f Antitrust parameters type-wise detection probability if A; 1 if P type-wise welfare change W i , i = A; P social cost of trial T trial accuracy probability

probability 1 G

 1 (b c) for type A to accept the commitments, with b c = (1 ) A f: In contrast, type P always refuses the commitments o¤er (since c + P > 0 for all c c by assumption): Therefore the commitment proposal triggers an imperfect screening of types, since type P always prefers to decline the o¤er and go to trial, whereas the decision of type A depends on her expected pro…t. The resulting expected welfare when the CA o¤ers commitments is equal to: (1 )p a+(1 )p (W P T ) aG(b c) a+(1 )p ((1 )W A + T ).

1 )

 1 A f = b c: Thus type A refuses commitments i¤ c < b c: There exists a unique equilibrium where the CA chooses the commitments procedure. To show this, we compare the expected welfare if the CA o¤ers commitments, which is (1 )p a+(1 )p (W P T ) aG(b c) a+(1 )p ((1 )W A + T ); with the expected welfare if the CA imposes litigation, which is T + we always have a 0 and thus, it always holds that (1 )p a+(1 )p (W P T ) a+(1 )p W A : Proof of Proposition 1.

  p+ a W P T + W Com ( ; ) + (1 )W Com ( ; T + W Com ( ; ) + (1 )W Com ( ; ) +(1 G(b c))W Com ( ; )

D

  ; with W Com = W Com ( ; ) W Com ( ; ) : This condition does not depend either on a or on p. Thus the CA chooses the commitments procedure i¤ > D :

1 2 pW P 1 2 W 2 G(b c)a Com + 1 2 ( 1 )

 1221 A a Com ((1 ) G(b c) + (1)) 1 2 (p(1 ) + a Com G(b c) )T + Q Com: W Opt ( ; ) + (1 Q Com ) W Opt ( ; ) ;where Q Com denotes the probability of conclusive litigation at t = 1 under procedureCom. We have that Q Com = 1 p and also p = p Lit = p Com :

( 1 )

 1 a G(b c) a G(b c)+p(1 ) W A + p(1 ) a G(b c)+p(1 ) W P T > a G(b c) a G(b c)+p(1 ) W A + p(1 )a G(b c)+p(1 ) W P : This is true as long as G(b c) is high enough or the trial e¢ cient enough with low T and high :

( 1 )

 1 Figure 1

Figure 3 t=1

 3 Figure 3

Figure 4

 4 Figure 4

See case C-132/19 P Groupe Canal + v. Commission -December 9th, 2020.

Case 2014/C 344/06.

Brussels, 27 June 2017 -IP 17/1784, at https://ec.europa.eu/commission/presscorner/detail/en/IP_17_1784

See Alexander Italianer, Director General of the Commission's Competition Directorate-General, speech of 11 December 2013 at the CRA Competition Conference, http://ec.europa.eu/competition/speeches/text/sp2013_11_en.pdf.

See Marinielllo (2014) for examples on patent abuse in standard setting, air transport, energy or new media/ebooks.

The same trend is noted in the US, where the FTC and DoJ have "resolved nearly their entire civil enforcement docket by consent decree" -Wright and Ginsburg (2018).

We follow[START_REF] Kaplow | On the optimal burden of proof[END_REF] in making this modelling assumption.

Council Regulation (EC) No 1/2003 of 16 December 2002.

We discuss in the appendix the CA's engagement to litigate if the …rm declines to adopt commitments.

The net expected welfare is further explained below. It includes all the external e¤ects i.e. W i and the social trial cost, T . All private gains and costs are considered as mere transfers within the industry. In particular, the expected welfare does not include c; which is assumed to be a purely private transfer between the …rm and the providers of legal services.

In the Appendix we brie ‡y justify the CA's choice to start litigation proceedings when it gathers evidence that the …rm's conduct is anticompetitive.

If the CA does not start a procedure even if it gathers evidence, this amounts to pure laissezfaire. We provide in the Appendix (see the Discussion section) the condition for laissez-faire to be dominated.

We show in the Appendix (section 6.2) that it is optimal for the CA to commit to litigation if the commitments proposal is not accepted. Thus, to the extent that the CA determines the optimal enforcement design, the ‡exible enforcement with CA's commitment to impose litigation if commitments are not accepted dominates the ‡exible enforcement without such a commitment from the CA.

In particular, we should note that a …ne equal to the expected extra pro…t from an anticompetitive practice is su¢ cient to ensure a strong deterrence regime. But if the probability is low, such a …ne may need to be very high.

Learning requires knowledge on the case. The mere fact that a type A accepts commitments is not su¢ cient to increase the accuracy of future assessment because there is no production of knowledge.

This amounts to assuming that the discount factor is equal to 1.

Proof of Lemma 4 and Proposition 2. From Proposition 1, we have:

Strong deterrence ( S ( ; ) < S ( ; ))

In this case we have D D = S ( ; ) (ii) for S ( ; ) < < S ( ; ); W Opt ( ;

In this case we have W Opt ( ;

In this case we have D D = S ( ; ): For D < < D ; commitments are proposed at t = 1 in case of ‡exible enforcement at t = 2; while it is optimal to impose litigation at t = 1 if the optimal enforcement is chosen at t = 2:

Discussion

The CA' s decision to start litigation proceedings if it gathers evidence

We brie ‡y discuss here the CA's options when it gathers evidence that the …rm's conduct is anticompetitive: the CA may either start litigation or not.

In case of pure laissez faire, the expected welfare is equal to

Laissez faire is dominated i¤ M ax(W Lit ( ; ); W Com ( ; )) W A 1 2 F ( A )+ 1 2 W P F ( P ): This inequality is satis…ed as long as the legal costs c and T are low enough.


Baptiste 1 Brac

Charles Sabbagh

Professor Jean-Marc M Regimbeau
email: regimbeau.jean-marc@chu-amiens.frlesauteursdéclarentnepasavoirde

Sabbagh

Regimbeau Jm

laparoscopy after left colectomy with end colostomy (Hartmann procedure)

Keywords: Reversal colostomy, laparoscopy, Hartmann procedure

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

Hartmann's procedure is a simplified surgical procedure for left colonic resection used mainly in emergency situations in order to reduce post-operative morbidity and mortality. Depending on the underlying pathology and the general condition of the patient, the Hartmann procedure is carried out with the intention to restore intestinal at a later date (which is feasible in 32 to 85% of cases) with postoperative morbidity of 3-50% and mortality of 0-7.1%, respectively [START_REF] Dumont | Morbi-mortality after Hartmann procedure for peritonitis complicating sigmoid diverticulitis. A retrospective analysis of 85 cases[END_REF][START_REF] Van De Wall | Conventional and laparoscopic reversal of the Hartmann procedure: a review of literature[END_REF].

Although in the majority of cases, the initial intervention is performed by laparotomy, it has been shown that Hartmann reversal (HR), the procedure to restore intestinal continuity, can be performed laparoscopically in 78 to 100% of cases (3). This approach has been associated with benefits in terms of reduced post-operative morbidity, duration of hospital stay, post-operative ileus and parietal infections [START_REF] Celentano | Laparoscopic versus open Hartmann's reversal: a systematic review and meta-analysis[END_REF][START_REF] De'angelis | Comparison between open and laparoscopic reversal of Hartmann's procedure for diverticulitis[END_REF][START_REF] Gustafsson | Guidelines for Perioperative Care in Elective Colorectal Surgery: Enhanced Recovery After Surgery (ERAS®) Society Recommendations: 2018[END_REF].

Figure 1: Patient setup

The patient is positioned supine as for a two-team laparoscopic left colectomy.

The two arms lie alongside the body, with shoulder pads and a bladder catheter that may lie within the operative field (optional). The operator starts the operation on the patient's left side (stoma removal and laparoscopic viscerolysis), and then moves to the right for the rest of the operation.

Figure 2: Global strategy for laparoscopic Hartmann reversal

We propose here a technique for an all laparoscopic HR with a strategy of dissection in two stages: a first stage of abdominal viscerolysis (abdominal wall and enterolysis) and, if possible, a stage of pelvic viscerolysis (enterolysis and freeing of the rectal stump ). This makes it possible to limit the size of the incision (in the event that conversion becomes necessary) to an infra-umbilical incision for the pelvic stage, without necessarily increasing the operating time. It is therefore necessary to operate "with the intention of an all-laparoscopic HR" on all patients. The indications for laparoscopy, on the other hand, must be reasonable so that the operative approach does not become deleterious for the patient (for example, HR in a patient with a ventral hernia).

Figure 3: Feasibility of optical trocar insertion to enable placement of first trocars

The procedure begins with dissection to free the stoma from the abdominal wall.

The dissection of the colostomy is made in direct contact with the intestine down to the rectus fascia, avoiding injury to the mesocolon so as not to damage the arcade of Drummond. Viscerolysis (B) is then performed as far into the abdomen as possible through the ostomy opening, dissecting toward the old midline scar. Dissection is carried up to the umbilicus (A) in order to facilitate the laparoscopic phase by enabling insertion of the umbilical optical trocar (a) under visual control through the ostomy opening (prevention of a trocar-associated intestinal wound). A plastic skirt with two wall-retracting rings (b) is inserted through the ostomy wound, which can be made air-tight around a trocar (c). If this step proves unfeasible, the surgeon may choose to proceed to a midline laparotomy. If the laparoscopic approach is to be pursued, the anvil of the endto-end mechanical stapler is inserted into the proximal colon for an eventual stapled colorectal anastomosis. The rest of the operation is performed according to the strategy described in Figure 2, with two distinct dissection phases: an initial abdominal phase followed by a pelvic phase. A 12mm trocar is placed through the double ring skirt and then a 12mmHg pneumoperitoneum is insufflated. Complete release of the parietal adhesions of the submesocolic area is necessary in order to be able to continue the operation without difficulty. However, complete small intestinal viscerolysis is not necessary in most cases and could lead to intestinal injury. The mobility of the descending colon should be promptly assessed to see whether it can be brought down into the pelvis for a tension-free anastomosis. If mobilization of the colon by the laparoscopic route proves impossible, the HR should be converted to a midline laparotomy. On the other hand, if the gesture is feasible, a complete liberation of the descending colon, and omental/small intestinal adhesions is performed. If there is tension on the liberated colon in the pelvis, several options may be considered: if the splenic flexure was not freed during the first operation, it must now be freed. If the colon still lacks sufficient mobility, the inferior mesenteric vein can be divided.

Figure 6: Feasibility of pelvic viscerolysis

At this stage, it is important to assess the extent of the small bowel adhesions in the pelvis and in the left iliac fossa to evaluate whether pelvic adhesions can be lysed laparoscopically. If not, conversion to open laparotomy may be necessary.

Since the colon has already been mobilized, a conversion may require only a limited infra-umbilical laparotomy incision. The pelvis is completely freed from adhesions and the rectal stump is identified by the non-absorbable monofilament sutures left at the ends of the running hemostatic suture that overran the stapled closure of the rectal stump during the initial intervention [START_REF] Brac | Colectomie gauche en urgence avec colostomie terminale (intervention de Hartmann)[END_REF].

Figure 8: Performing colorectal anastomosis

A standard colorectal stapled anastomosis is then performed. The anastomosis should be air-tested for leaks. We do not recommend the placement of a perianastomotic drain, nor the routine performance of a protective ileostomy.

Conclusion

The reversal of a Hartmann procedure by laparoscopy is a safe and feasible procedure in more than 70% of cases and offers a benefit to the patient in terms of the post-operative course.

Figure 4 :

 4 Figure 4: Feasibility of parietal viscerolysis and introduction of the working trocars

Figure 5 :

 5 Figure 5: Feasibility of laparoscopic mobilization of the splenic flexure and/or colon (if necessary)

Figure 7 :

 7 Figure 7: Dissection of the rectal stump

  E c r a n d e r a p p e l Co l o n n e Feasibility of insertion of optical trocar and first working trocars Feasibility of parietal viscerolysis and of insertion of the other trocars No Feasibility of pelvic viscerolysis Yes No Splenic flexure already mobilized, left colon mobilization if feasible and necessary Feasibility of splenic flexure take down by laparoscopy (if necessary) and left colon


Emilie Hucteau
email: emilie.hucteau@u-bordeaux.fr

Pernelle Noize
email: pernelle.noize@u-bordeaux.fr

Antoine Pariente
email: antoine.pariente@u-bordeaux.fr

Catherine Helmer
email: catherine.helmer@u-bordeaux.fr

Karine Peres
email: karine.peres@u-bordeaux.fr

Karine Pérès

ADL-dependent older adults were identified in medico-administrative databases

Keywords: Dependency, Activities of daily living, Aged, Pharmacoepidemiology, Health insurance data, Cohort study

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

INTRODUCTION

The worldwide intensification of population aging led age-related diseases and their consequences to become a major public health concern [START_REF]World population ageing 2017[END_REF]. According to the United Nations Highlights on World Population Ageing 2017, the over-80s will more than triple until 2050, then representing around 425 million people [START_REF]World population ageing 2017[END_REF]. This will have major consequences in terms of healthcare and social care needs, relating to both the burden of the multiple chronic conditions it implies, and to their consequences in terms of dependency. Dependency is defined as a need for human assistance in activities of daily living (ADL) [START_REF] Edjolo | Natural History of Dependency in the Elderly: A 24-Year Population-Based Study Using a Longitudinal Item Response Theory Model[END_REF] and this is obviously the most severe stage of dependency that particularly mobilise the health, social and informal resources.

Whilst this dimension of health appears very important to monitor when considering population of older adults, most of the existing medico-administrative databases that are now routinely used for the assessment of health do not capture directly such information. Indeed, the detection of dependency ideally requires individual assessments performed by specially trained interviewers using standardised procedures and validated scales. If such expensive procedures have been deployed in the context of dedicated cohort studies, it cannot be implemented at the general population level. Despite massive amount of information, medico-administrative databases poorly contribute nowadays to the study of dependency. Yet, dependency is a relevant indirect indicator of health and its assessment in quasiexhaustive samples of the older population would be crucial for public health to estimate the current and future needs of social and health care, which are expected to dramatically increase in the future.

Such assessment could also allow to screen large populations of older adults in order to identify dependent people who could benefit from specific interventions. Finally, the identification of dependency in large unbiased databases would have applications for research purposes, for example to describe the path of dependent people, study the determinants of dependency, or controlling other analyses in administrative healthcare claims using dependency as a proxy of global health. The detection of dependency has previously been performed through the developing of dedicated algorithms from US claims data, but in a population and health system limiting their applicability to the very different European setting [START_REF] Davidoff | A novel approach to improve health status measurement in observational claims-based studies of cancer treatment and outcomes[END_REF][START_REF] Faurot | Using Claims Data to Predict Dependency in Activities of Daily Living as a Proxy for Frailty[END_REF][START_REF] Kinosian | Validation of the JEN frailty index in the National Long-Term Care Survey community population: identifying functionally impaired older adults from claims data[END_REF].

The objective of this study was to develop an algorithm allowing to identify older ADL-dependent adults in French Health Insurance databases using population-based cohort combining individual assessment of ADL-dependency and health insurance data.

METHODS

Study population and data collection

The AMI (Aging Multidisciplinary Investigation) study is a prospective population-based cohort on health and aging conducted in former farmers living in rural environment in Gironde department (South-Western France) [START_REF] Pérès | Health and aging in elderly farmers: the AMI cohort[END_REF]. A total of 1002 individuals aged 65 or older, retired from agriculture and affiliated to the French Farmer Health Insurance System ("Mutualité Sociale Agricole", MSA) were included in 2007-2009. At baseline (T0), trained neuropsychologists used standardized questionnaires at the participant's home to collect data (socio-demographics characteristics, living conditions, lifestyle, health status, drugs use) and to assess dependency. Participants were interviewed in the same conditions two years later in 2010-2011 (T2). In addition to these data, the MSA provided health insurance data for the cohort participants.

Definition and assessment of ADL-dependency

Among the different domains of activity limitation, we chose to focus on the most severe one, the basic activities of daily living (ADL). The neuropsychologists used the Katz's scale [START_REF] Katz | Progress in development of the index of ADL[END_REF] to assess participants' ADL-dependency level during the home interviews. This scale includes five different activities: dressing, bathing, toileting, transferring and eating. A participant was considered as dependent if he/she could not perform at least one activity without a given level of human assistance (personal assistance, directive assistance or supervision). This assessment was used as gold standard. To increase the power of our analysis, we considered dependency either at baseline (T0) or 2 years later (T2) (Supplementary Figure 1). Accordingly, in the analysis, we used: i) T0 data for participants who were independent at T0 and remained independent at T2 and for those who were already dependent at T0); ii) and T2 data for participants with incident dependency at T2 and for those who were dependent at T2 with missing information on dependency at T0. Date of the assessment was considered as the inclusion date for this study.

Health-related reimbursements from the merged French Farmer Health Insurance System

For each participant, all health-related reimbursements were extracted from the French Health Insurance System database and linked to the cohort data. For this analysis we considered healthrelated reimbursements that were concomitant with the dependency status. The information provided included: out-hospital medications data (all reimbursements performed), out-hospital medical and paramedical acts, medical equipment and devices and the registered Long-Term Diseases (LTD) (30 specific LTDs allowing 100% health expenditures coverage by the Insurance System, supplementary Table 1). The data included in the analysis were selected based on expert's opinion (co-authors of this paper: experts in epidemiology of aging and dependency (CH, KP) and in pharmacoepidemiology with a high experience of health insurance databases (PN, AP)) and the exposure period considered for each factor was defined according to the type of the data (supplementary Table 2). The LTD being mainly irreversible, we considered that a person identified as suffering from a LTD at a time remained diseased for the rest of the follow-up.

Drugs and LTD that concerned at least 20 participants and medical acts and devices that concerned at least 5 participants were included in our analyses. After adding age (< 75 / 75-<85 / ≥ 85) and sex, 70 factors were thus included in the analysis for the algorithm development: 39 drugs, 10 medical and paramedical acts, 9 medical equipment and devices and 10 LTD (Figure 1).

Data analysis

Creation of the algorithm of ADL-dependency

Participants' main characteristics (sociodemographic, drugs consumption, medical and paramedical acts, medical equipment and devices and LTD) were described according to the ADL-dependency status using bivariate analyses with Pearson's chi-square and Fisher exact test.

To develop the algorithm, we first identified the factors associated with ADL-dependency. Because of a high number of factors, we used a logistic regression model with LASSO regularization repeated for 1000 bootstrap samples [START_REF] Pavlou | Review and evaluation of penalised regression methods for risk prediction in low-dimensional data with few events[END_REF][START_REF] Tibshirani | Regression Shrinkage and Selection via the Lasso[END_REF][START_REF] Guo | SparseLearner: Sparse Learning Algorithms Using a LASSO-Type Penalty for Coefficient Estimation and Model Prediction[END_REF]. Over these 1000 iterations, the frequency of selection was recorded for each factor. Controlling for age and sex, three provisional algorithms were built as the sum of each variable multiplied by its beta coefficient from the variables selected in 90% (algorithm90), 80% (algorithm80), and 70% (algorithm70) of the bootstrap samples and different cut-offs were considered for each algorithm. Using estimated prevalence, the area under the receiver operating characteristics (ROC) curve (AUC), sensitivity (Se), specificity (Sp), positive and negative predictive value (PPV and NPV respectively), and Youden index, we identified the most accurate algorithm out of the three; for this, we then determined the most accurate cut-off for ADL-dependency identification.

Application and external validation on the French General Health Insurance database

The "Echantillon Généraliste de Bénéficiaires (EGB)" is an anonymous 1/97 th sample of the French health insured population [START_REF] Tuppin | French national health insurance information system and the permanent beneficiaries sample[END_REF]. Established in 2005, it is representative of the general population in terms of sex and age and contains for each affiliate an exhaustive recording of all out-hospital health expenditures (including drug deliveries, medical and paramedical acts, and reimbursements for medical equipment and devices), hospitalisation data, and information on LTD.

For the validation phase, we selected persons aged 65 and over and affiliated to the Health Insurance General Scheme (covering around 85% of the population) in January 2010 (n=74,652, Supplementary Figure 2). In this sample, we applied the developed algorithm quarterly from 2010 to 2015. Thus, the same population being followed over the 6 years, we obtained for a person followed until December 2015, 24 dependency status, one by quarter. The end of the follow-up was defined as the date of death, the date of end of the affiliation, or December 2015 the 31 st , whichever came first. Yet, applying the algorithm by quarter may lead to an unstable status due to a transitory increased or decreased consumption of care. However, the level being severe (with the Katz's scale) [START_REF] Barberger-Gateau | A hierarchical model of domains of disablement in the elderly: a longitudinal approach[END_REF] and ADLdependency generally consisting in stable dependency, we assumed its reversibility to be very low.

We thus established a strategy for the use of the algorithm in the database considering both the potentiality of an unstable status and the limitations of the data. First, we applied several rules to smooth the results: i) if, while considered dependent over the preceding and following quarters, a person was non-dependent in one quarter owing to the algorithm results, the person's status was changed for this quarter into "dependent"; ii) similarly, if a person was considered dependent for a quarter but not the two preceding and following ones, the person's status for this quarter was changed for "non-dependent" (Figure 2). Second, as the reimbursement data (drugs, medical acts, equipment…) are not available during a period of hospitalization, if a person was dependent the quarter before a long period of hospitalization (30 days or more), we considered that the person remained dependent during this hospitalization. Third, we classified some people as dependent regardless of the algorithm: 1) people who were admitted to hospital from long-term care or from a medico-social housing structure (including nursing homes for which information on care received is lacking from the database) or who left hospital towards these structures; and 2) people hospitalized for at least 3 months. For these people, the beginning of the dependency period was the quarter where the hospitalization started.

Finally, to confirm the relevance of the algorithm: i) we estimated the prevalence of dependency according to age and sex to check for consistency with literature; ii) we evaluated the reversibility rates to unsure that this rate was low as expected for ADL-dependency; and iii) we calculated death rate at the end of the follow-up period according to the dependency status (among dependent in 2010 vs nondependent in 2010) and compared it to AMI death rate at the 6-year follow-up (among dependent at T0 or T2 vs non-dependent at these times).

Analyses were performed using SAS 9.4 (SAS Institute, Inc., Cary NC) and R 3.4.1.

RESULTS

Description of the AMI cohort

Among the 1002 participants, 995 were included in our analyses after exclusion of 7 individuals with missing data on ADL scale: 75 were dependent at T0, 39 were incident dependent at T2 and 881 were independent at T0 and T2, leading to 11.5% of dependent participants in our sample. Mean age of participants was 76.4 (standard deviation: 6.7) and 37.5% were female. Main characteristics of the participants are shown in Table 1 according to the ADL-dependency status.

Development of the algorithm

Aside of age and sex that were systematically considered, among the 70 variables initially included, [START_REF] Cuthbertson | Controlling for Frailty in Pharmacoepidemiologic Studies of Older Adults: Validation of an Existing Medicare Claims-based Algorithm[END_REF] were selected in at least 70% of the bootstrap samples, 15 in at least 80% and 10 in at least 90% (Figure 3). For the three provisional algorithms created, the AUC was excellent: 0.92 for algorithm90 and 0.94 for algorithm70 and algorithm80. Different cut-offs were tested for each algorithm; the most performant ones are shown in Table 2. Although algorithm70 at 2.5 cut-off had a better Youden Index, its VPP was quite small and its estimated prevalence quite high compared to the expected prevalence.

Thus, the retained algorithm, showing the best compromise between estimated prevalence and performances, was algorithm80 at a 3.1 cut-off with a Se at 75.4%, a Sp at 95.9%, a PPV at 70.5% and a NPV at 96.8%. The average of this algorithm was 0.8 (SD=1.1) for non-dependent and 5.7 (SD=3.6) for dependent persons. The final algorithm corresponding to algorithm80 included 17 variables (Figure 2).

The prevalence of ADL-dependency based on the algorithm with the defined 3.1 cut-off was estimated at 12.3% from the French Farmer Health Insurance System merged to the AMI cohort vs. 11.5% based on the individual ADL assessment of cohort participants.

External validation on the EGB sample

In the EGB, 74,652 persons met the inclusion criteria (Supplementary Figure 2). Among them, 59.0% were female, and mean age was 76.3 (SD=7.7). The average follow-up was 5.3 years (SD=1.4) for a maximum of 6 years.

Among the 74,652 persons, 9.5% were classified as dependent during 2010 first quarter. This prevalence increased with age: 2.4% among people aged less than 75, 10.4% among those aged 75-84 and 30.7% among those aged 85 or more. With the aging of the studied sample between 2010 (first quarter) and 2015 (last quarter) (same population followed-up over the 6 years), the prevalence of dependency increased as expected to reach 16.3%. Over the 6-year follow-up period, 27.9% were considered ADL-dependent at least once, with a higher frequency in females (31.5% vs 22.6% in males). Among persons classified as dependent at least once, 80.0% remained dependent after the first dependency period; this stable status was higher in males than in females (82.7% and 78.7% respectively). On average, men became dependent at younger ages than women (81.6 years (SD=6.8) vs. 84.0 years (SD=6.7) respectively) probably due to a lower life expectancy of about 6 years in men than in women in France [START_REF] Pérès | Recent Trends in Disability-Free Life Expectancy in the French Elderly. Twenty Years Follow-Up of the Paquid Cohort[END_REF]. Finally, death rate was 73.7% among dependent EGB people (vs 22.7% among non-dependent) and 77.2% among dependent AMI participants (vs 22.1% among non-dependent).

DISCUSSION

Using a population-based cohort combining individual face-to-face assessment of ADL-dependency (as gold standard) and health insurance data, we developed an algorithm to identify ADL-dependency in administrative healthcare claims. In addition to age and sex, this algorithm retained 15 variables related to health care (drugs, medical and paramedical acts and medical devices) and LTD. In the AMI cohort, the developed algorithm presented excellent performances, with an AUC of 0.94; the identification of ADL-dependency using the developed algorithm allowed obtaining a prevalence similar to that observed in participants (12.3% vs. 11.5%). Within the external validation in a large sample of the French older adults insured population aged 65 and over, the obtained prevalence was lower, of 9.5%.

In the context of intense aging of the populations [START_REF]World population ageing 2017[END_REF], this algorithm, easy to apply in administrative healthcare claims, may be valuable in terms of public health and population health monitoring, allowing to provide key indicators of prevalence, incidence, transition probabilities, time spent in dependency, or DALYs. It also offers excellent opportunity to estimate the costs of dependency in medico-economic analyses, to follow trends over time and to perform projections; crucial data to anticipate and organise the society to face future health care and social care needs associated to the intensive aging of the population. Moreover, it could also allow studying the determinants of dependency, including health events and drug consumption, and identifying individuals at-risk of dependency who could benefit from targeted preventive strategies.

In the dependency process, different levels of increasing severity exist according to the domains of activity limitation hierarchically affected (first mobility, then instrumental ADL and finally basic ADL) [START_REF] Edjolo | Natural History of Dependency in the Elderly: A 24-Year Population-Based Study Using a Longitudinal Item Response Theory Model[END_REF][START_REF] Barberger-Gateau | A hierarchical model of domains of disablement in the elderly: a longitudinal approach[END_REF]. For the development of this algorithm, we chose to focus exclusively on basic activities for several reasons. First, this is the level of dependency associated to the highest levels of medical, social, informal care, caregiver's burden [START_REF] Pérès | Recent Trends in Disability-Free Life Expectancy in the French Elderly. Twenty Years Follow-Up of the Paquid Cohort[END_REF][START_REF] Coe | A Comprehensive Measure of the Costs of Caring for a Parent: Differences According to Functional Status[END_REF] and costs [START_REF] Andrieu | Burden experienced by informal caregivers assisting Alzheimer's patients in the REAL.FR study[END_REF][START_REF] Van Lier | Predictors of Societal Costs of Older Care-Dependent Adults Living in the Community in 11 European Countries[END_REF] and poorer quality of life of the elderly [START_REF] Lindholm | Costs explained by function rather than diagnosis--results from the SNAC Nordanstig elderly cohort in Sweden[END_REF]. Second, we assumed that less severe stages of dependency were less likely to be identified from health expenditures.

To our knowledge, all the algorithms existing to date for the identification or prediction of dependency have been developed in the United-States using Medicare [START_REF] Davidoff | A novel approach to improve health status measurement in observational claims-based studies of cancer treatment and outcomes[END_REF][START_REF] Faurot | Using Claims Data to Predict Dependency in Activities of Daily Living as a Proxy for Frailty[END_REF][START_REF] Kinosian | Validation of the JEN frailty index in the National Long-Term Care Survey community population: identifying functionally impaired older adults from claims data[END_REF]. Davidoff et al. developed and validated an algorithm aiming to detect a summary measure of dependency considering globally all stages of dependency (IADL and ADL) [START_REF] Davidoff | A novel approach to improve health status measurement in observational claims-based studies of cancer treatment and outcomes[END_REF]. Combining different levels of dependency in a single indicator leads to a heterogeneous group with different associated risks, costs and needs. As in the present work, Faurot et al. developed an algorithm to detect basic ADL-dependency among older adults, used as a proxy of frailty in order to improve unmeasured confounding in the analyses conducted using healthcare claims [START_REF] Faurot | Using Claims Data to Predict Dependency in Activities of Daily Living as a Proxy for Frailty[END_REF]; this algorithm has recently been validated within the ARIC population-based study [START_REF] Christiansen | Health-related quality of life and related factors among a sample of older people with cognitive impairment[END_REF]. Finally, Kinosian et al. recently tested an already existing commercialized algorithm (whose development has not been published), to evaluate its performances for ADL dependency detection within the National Long-Term Care Survey [START_REF] Kinosian | Validation of the JEN frailty index in the National Long-Term Care Survey community population: identifying functionally impaired older adults from claims data[END_REF]. With an AUC at 0.94 in the developing sample, our algorithm slightly outperforms these previous algorithms (AUC of 0.80 to 0.92 in the estimation samples, and of 0.71 to 0.92 in the validation samples) [START_REF] Davidoff | A novel approach to improve health status measurement in observational claims-based studies of cancer treatment and outcomes[END_REF][START_REF] Kinosian | Validation of the JEN frailty index in the National Long-Term Care Survey community population: identifying functionally impaired older adults from claims data[END_REF][START_REF] Christiansen | Health-related quality of life and related factors among a sample of older people with cognitive impairment[END_REF]. Additionally, the use of LASSO logistic regression model allowed us obtaining a lighter algorithm including only 17 variables, thus theoretically easier to apply in health insurance databases.

Contrary to the other algorithms, we chose to include the drugs used, instead of only including diagnoses, similarly to what was done in adaptations of comorbidity score to health insurance databases [START_REF] Cuthbertson | Controlling for Frailty in Pharmacoepidemiologic Studies of Older Adults: Validation of an Existing Medicare Claims-based Algorithm[END_REF][START_REF] Bannay | The Best Use of the Charlson Comorbidity Index With Electronic Health Care Database to Predict Mortality[END_REF]. The LASSO logistic regression model allowed us to include all the drug classes in the model, without having any a priori hypothesis on drugs that could improve the detection of dependency. At the end, four drug classes were retained in the final algorithm. Several of the variables identified in previous algorithms, such as nursing visit, home visit, ambulance, hospital bed, dementia, stroke, heart failure were similarly retained in our algorithm. In addition, we also retained dopaminergic antiparkinson drugs, antidiabetic agents and lipid modifying agents, as well as incontinence equipment, whereas Faurot et al. retained diagnosis of Parkinson's disease, diabetic complications, lipid abnormalities, and bladder continence [START_REF] Faurot | Using Claims Data to Predict Dependency in Activities of Daily Living as a Proxy for Frailty[END_REF]. Contrary to other algorithms [START_REF] Davidoff | A novel approach to improve health status measurement in observational claims-based studies of cancer treatment and outcomes[END_REF][START_REF] Faurot | Using Claims Data to Predict Dependency in Activities of Daily Living as a Proxy for Frailty[END_REF], wheelchair was not retained in the final algorithm, even it would have been if we had selected algorithm70. However, our statistical approach probably led to retain other variables in the model, highly correlated with wheelchair use.

The main strength of our study is its design relying on the combined use in the same study of information on individual assessment of dependency performed by trained neuropsychologists using a validated scale and information on health care obtained from merged health insurance data exhaustive in terms of outpatient care and hospitalisation episodes. Given the major present and future implications of ALD-dependency, this study provides an indicator particularly relevant in a public health perspective. However our study also has some limits. The algorithm has been developed on a specific older population of retired farmers, which explains for instance the specific sex-ratio in favor to men. Medical acts and prescriptions may thus differ from other populations. Also, the sample used to develop the algorithm only included 995 participants, of which 114 were ADL-dependent. However, this did not preclude developing an algorithm with excellent AUC, which application on an external sample allowed obtaining results for prevalence and age at ADL-dependency entry consistent with the literature [START_REF] Pérès | Recent Trends in Disability-Free Life Expectancy in the French Elderly. Twenty Years Follow-Up of the Paquid Cohort[END_REF][START_REF] Harwood | Current and future worldwide prevalence of dependency, its relationship to total population, and dependency ratios[END_REF][START_REF] Duée | La dépendance des personnes âgées : une projection en 2040[END_REF]. Moreover, as a validation step of our algorithm, we found much higher mortality rate for those identified as dependent compared to those nondependent in the EGB, and this higher rate was very close to that found in the AMI cohort among dependent participants. This algorithm based on French health prescriptions and patients medical and paramedical paths, may be not directly transposable to other countries, all having specific health and social care insurances and procedures.

The same would be true for the strategy of use we developed. This was constrained by the lack of inhospital or nursing home care data that might be available in health insurance databases in other countries. The classification rules we retained could be adapted and improved in such settings.

CONCLUSION

Our algorithm allowing the identification of ADL-dependency from health insurance databases might constitute a valuable tool for public health and population health monitoring from electronic healthcare databases. The easy to obtain information its use could provide regarding dependency prevalence, incidence, or determinant identification represents key data for policy makers to anticipate and adapt societies to aging and if possible to develop preventive strategies.

13

Figure 1. Selection of factors included in the analysis

The figure should be read as follows: among all medical and paramedical acts, 12 were selected by experts' opinion and, after exclusion of factors that concerned less than 5 participants, 10 were included in the LASSO analysis. 1 In the ATC (Anatomical Therapeutic Chemical) classification drugs are divided into groups according to the organ or system on which they act or their therapeutic and chemical characteristics; drugs are coding on seven characters (letters and digits). The first level (first letter) defines the anatomical group among 14 different ones. The second level (first two digits) gives the main pharmacological or therapeutic subgroup and the third level (second letter) corresponds to chemical, pharmacological or therapeutic subgroups. In the analysis we used the second level but for drugs of the nervous system where the third level was considered due to their potential high relation with dependency. Algorithms at 90%, 80% and 70% refer to the factors selected in 90%, 80% and 70% of the bootstrap samples for their construction The figure should be read as follows: among all medical and paramedical acts, 12 were selected by experts' opinion and, after exclusion of factors that concerned less than 5 participants, 10 were included in the LASSO analysis. 1 In the ATC (Anatomical Therapeutic Chemical) classification drugs are divided into groups according to the organ or system on which they act or their therapeutic and chemical characteristics; drugs are coding on seven characters (letters and digits). The first level (first letter) defines the anatomical group among 14 different ones. The second level (first two digits) gives the main pharmacological or therapeutic subgroup and the third level (second letter) corresponds to chemical, pharmacological or therapeutic subgroups. In the analysis we used the second level but for drugs of the nervous system where the third level was considered due to their potential high relation with dependency.

Figure 2 .Figure 3 .

 23 Figure 2. Algorithm to identify quarters of ADL-dependency Figure 3. Factors selected by LASSO bootstraps

Figure 4 .

 4 Figure 4. Prevalence of ADL-dependency among the EGB patients in 2010 and 2015 first quarter and 2015 last quarter

Figure 1 .

 1 Figure 1. Selection of factors included in the analysis

Table 1 .

 1 Characteristics of the participants according to their ADL-dependency status. AMI cohort

	2007-2011, N=995
	Only variables significantly associated to ALD-dependency (p<0.05) are presented

Table 2 .

 2 Performance of the scores selected in 90%, 80% and 70% of the bootstrap samples according to different cut-offs

	Se: sensitivity; Sp: specificity; PPV: Positive predictive value; NPV: Negative predictive value
	Youden index is calculated as follows: (Se+Sp)-1

Table 1 .

 1 Characteristics of the participants according to their ADL-dependency status. AMI cohort

	1
	2	2007-2011, N=995.
			Dependent	Independent
			n=114	n=881
		Characteristics	n (%)	n (%)
		Sociodemographic
		Sex
		Men	59	(51.8)	563	(63.9)
		Women	55	(48.2)	318	(36.1)
		Age
		less than 75	13	(11.4)	443	(50.3)
		75 to less than 85	61	(53.5)	370	(42.0)
		85 and over	40	(35.1)	68	(7.7)
		Drugs consumption
		A02: Drugs for acid related disorders	43	(37.7)	191	(21.7)
		A03: Drugs for functional gastrointestinal disorders	17	(14.9)	52	(5.9)
		A06: Drugs for constipation	23	(20.2)	40	(4.5)
		A10: Drugs used in diabetes	21	(18.4)	97	(11.0)
		A12: Mineral supplements	13	(11.4)	34	(3.9)
		B01: Antithrombotic agents	46	(40.4)	251	(28.5)
		B03: Antianemic preparations	8	(7.0)	20	(2.3)
		C03: Diuretics	40	(35.1)	161	(18.3)
		C10: Lipid modifying agents	24	(21.0)	279	(31.7)
		D01: Antifungals for dermatological use	11	(9.7)	11	(1.3)
		H03: Thyroid therapy	12	(10.5)	49	(5.6)
		M02: Topical products for joint and muscular pain	9	(7.9)	29	(3.3)
		N02B: Other analgesics and antipyretics	34	(29.8)	163	(18.5)
		N03A: Antiepileptics	12	(10.5)	30	(3.4)
		N04B: Dopaminergic agents for Parkinson	13	(11.4)	9	(1.0)
		N05A: Antipsychotics	14	(12.3)	11	(1.3)
		N05B: Anxiolytics	22	(19.3)	65	(7.4)
		N05C: Hypnotics and sedatives	19	(16.7)	58	(6.6)
		N06A: Antidepressant drugs	31	(27.2)	52	(5.9)
		N06D: Anti-dementia drugs	21	(18.4)	38	(4.3)
		Medical and paramedical acts
		Daily nursing care	13	(11.4)	1	(0.1)
		Non-daily nursing care	14	(12.3)	3	(0.3)
		Home treatment accessories	25	(21.9)	5	(0.6)
		Ambulance	7	(6.1)	2	(0.2)
		Medical vehicle	5	(4.4)	12	(1.4)
		At-home GP' visits	52	(45.6)	71	(8.1)
		Medical equipment and devices
		Walker	3	(2.6)	4	(0.4)
		Anti-bedsore mattress	13	(11.4)	4	(0.4)
		Wheelchair	12	(10.5)	1	(0.1)

Table 2 .

 2 Performance of the algorithms selected in 90%, 80% and 70% of the bootstrap samples according to different cut-offs Se: sensitivity; Sp: specificity; PPV: Positive predictive value; NPV: Negative predictive value Youden index is calculated as follows: (Se+Sp)-1

	Algorithm	Cut-off	Se (%) Sp (%) PPV (%) NPV (%) Youden (%)	Prevalence (%)
	90%	2.5	75.4	93.9	61.4	96.7	69.3	14.1
		3.0	71.1	96.7	73.6	96.3	67.8	11.1
		3.5	64.9	97.4	76.3	95.5	62.3	9.7
	80%	2.5	79.8	91.6	55.2	97.2	71.4	16.6
		3.0	75.4	95.8	69.9	96.8	71.2	12.4
		3.1	75.4	95.9	70.5	96.8	71.4	12.3
		3.2	73.7	96.1	71.2	96.6	69.8	11.9
		3.3	72.8	96.5	72.8	96.5	69.3	11.5
		3.4	70.2	96.5	72.1	96.2	66.7	11.2
		3.5	70.2	96.7	73.4	96.2	66.9	11.0
	70%	2.5	78.1	95.2	67.9	97.1	73.3	13.2
		3.0	73.7	96.5	73.0	96.6	70.2	11.6
		3.5	69.3	98.0	81.4	96.1	67.3	9.7

ACKNOWLEDGMENTS

This study is part of the DRUGS-SAFE research program, funded by the French Medicines Agency (Agence Nationale de Se ́curite ́ du Me ́dicament et des Produits de Sante ́, ANSM). This publication represents the views of the authors and does not necessarily represent the opinion of the French Medicines Agency. The sponsor has no involvement in any parts of this work. The AMI project was supported by the Association de Gestion pour le Compte des Institutions Complémentaires Agricoles, the Caisse Mutuelle Autonome de Retraités Complémentaires Agricoles, the Caisse de Retraite Complémentaire des Cadres de l'Agriculture, the Caisse Centrale de Prévoyance Mutuelle Agricole Prévoyance, the Caisse de Prévoyance des Cadres d'Entreprises Agricoles, Agri Prévoyance, the Mutualité Sociale Agricole de Gironde, the Caisse Centrale de la Mutualité Sociale Agricole, and the Caisse Nationale de Solidarité pour l'Autonomie.

Algorithms at 90%, 80% and 70% refer to the factors selected in 90%, 80% and 70% of the bootstrap samples for their construction Supplementary Figure 1. Selection of AMI participants included in the analysis Supplementary Figure 2. Flow Chart for the EGB patients, an anonymous 1/97th sample of the French health insured population Supplementary Table 1. List of the thirty long-term diseases Supplementary Table 2. Medical, paramedical acts and medical equipment and devices included in the analysis with exposure period considered N: number of factors included ATC: Anatomical, Therapeutic and Chemical) classification system GP: General practitioner For example, a person was considered exposed to "GP consultation" if he had at least 3 consultations within the three-month period before the inclusion date. 27.9% were dependent at least once (31.5% in women vs. 22.6% in men) → 80% of stable status (82.7% in men vs. 78.7% in women)

Drugs Medical and paramedical acts

Medical equipments and devices

Long-term diseases