Frédéric Palomino
email: eric.palomino@edhec.edu

Anil Peyrache
email: peyrache@hec.fr

Psychological Bias and Gender Wage Gap *

Keywords: bias, gender, optimal contract, wage gap. JEL Classification: J16, J33, J71

Basing our work on the literature on psychology that highlights differences in confidencelevels between men and women, we provide a rationale for the simultaneous existence of the two dimensions of the gender wage gap: (i) on average, men earn more than women and (ii) the difference in wages depends on the gender structure. We also argue that although an equal compensation policy does not eliminate the gender wage gap, it nonetheless increases social welfare. Finally, endogenizing the choice of task by agents, we show how women, in a vast majority, can self-select on low paying jobs.

A c c e p t e d M a n u s c r i p t 1 Introduction

The existence and persistence of a gender wage gap is certainly an open secret. A large number of papers have well documented the pay difference based on gender characteristics. A recent release from the Census Bureau (CENSR-15, May 2004) states that "the female-to-male earnings ratio at the median for year-round, full time workers was 77 percent in 2002". This gender wage gap is mainly bi-dimensional. The first dimension is well-known and accounts for the fact that, on average, women earn less than men (see, for example, [START_REF] Fields | Interindustry wage differentials and the gender wage gap[END_REF][START_REF] Blau | Gender difference in pay[END_REF][START_REF] Graddy | Wage differences by gender: evidence from recently graduated MBAs[END_REF]. The second dimension is a group effect: the larger the fraction of woman in an industry or a job, the lower the average wage (see [START_REF] Johnson | Estimates of the direct effect of comparable worth policy[END_REF][START_REF] Reilly | Does more mean less? The male/female wage gap and the proportion of females at the establishment level[END_REF][START_REF] Boraas | How does gender play a role in earnings gap? An update[END_REF]. According to [START_REF] Boraas | How does gender play a role in earnings gap? An update[END_REF], in 1999, taking into account personal characteristics, women who are employed in occupations in which they represent 90 percent of the workforce earn 19 percent less than women employed in occupations where men represent 90 percent of the workforce. The disadvantage falls to 13 and 7 percent in occupations where female represent 50 percent and 30 percent of the workforce, respectively. This suggests that not only demographic characteristics (gender, age...) are at stake but, additionally, that the composition of the group to which the employee belongs is also important.

Various explanations have been offered to justify the existence of such a gender wage gap. It is widely advocated that women's career breaks and their participation in the labor market are relevant explanation for the existence of a gender wage gap (see [START_REF] Corcoran | Work history, labor forces attachment and earning difference between the sexes[END_REF], Mincer and Ofek1982, Cox 1984[START_REF] Mincer | Family investments in human capital: earning of women[END_REF][START_REF] Polachek | Occupational self-selection: a human capital approach to sex differences in occupational structure[END_REF]. However, many other rationales have been provided. [START_REF] Rothschild | A model of employment outcomes illustrating the effect of the structure of information on the level of distribution of income[END_REF] stress differences in the noise of productivity signals across genders whereas [START_REF] Milgrom | Job market, market forces, and the invisibility hypothesis[END_REF] underline poor observability of women's quality on the external market. Other sources of the heterogeneity among genders have been analyzed such as difference in tastes [START_REF] Killingsworth | Heterogeneous preferences, compensating wage differentials, and comparable worth[END_REF], differences in the degree of risk aversion [START_REF] Eckel | Sex and risk: experimental evidence[END_REF], the existence of non-market benefits for women [START_REF] Lazear | Male-female wage differentials in job ladders[END_REF], and a difference in commitment power in the presence of firm specific human capital accumulation [START_REF] Kuhn | Demographic group and personnel policy[END_REF]. Many authors have also pointed out some sociological explanations such as the desire by men to maintain their occupational status [START_REF] Goldin | A pollution theory of discrimination: male and female differences in occupations and earnings[END_REF], the mechanics of interactions within households (François 1997 andFrançois and[START_REF] Francois | Gender wage differentials in a competitive labor market: the household interaction effect[END_REF] and differences in expectation regarding future wages [START_REF] Filippin | Gender wage gap in expectations and realizations[END_REF]. However, all these models focus on the equal occupation dimension of the gender wage gap and do not consider the group effect In this article, we propose a new approach which, endogenously, generates that in a cohort of equally talented agents, both aspects of the gender wage gap can emerge in equilibrium. This approach is based on two sets of empirical findings. First, following the work of Tomaskovic-Devey and Skaggs (1999), we assume that there are no real productivity differences between men and women, hence that the gender wage gap cannot be attributed to a difference of innate ability.

Second, we borrow from the literature in psychology according to which it is now a well-established empirical fact that men and women differ noticeably in their level of self-confidence in many areas of achievement (see Section 2). Our guiding line is that if such differences in self-confidence have proven to be tangible in many areas, then it is of interest to study whether it could generate some gender wage gap. Illustrating our model with some data analysis would be of great value, but unfortunately having access to a data base that could match some psychological parameters of agents, the proportion of men/women on the occupation and finally their salary is a very difficult task. In this paper, we then concentrate on a first theoretical step aiming at building a bridge between empirical evidence in psychology and some stylized facts in labor markets.

We consider a standard principal-agent model in which a rational principal delegates the execution of a task to either a woman or a man. Traditionally, the main source of productivity-difference between men and women is considered to be due to the smaller amount of time and energy that women devote to their career. We depart from such an analysis and, in line with the literature in psychology, we assume that men and women are biased regarding the perception of their own productivity of effort. Whereas men over-estimate such productivity, women under-estimate it. As mentioned above, we assume that both genders have the same innate ability and, additionally, the same cost of effort. We proceed in two steps. First, we let the principal discriminate between the genders and demonstrate that only the first dimension of the wage gap can be explained. Second, we consider the case in which discrimination is banned 1 . We show that the unique equilibrium has the feature that the greater the probability of men applying for the job, the larger the average wage. Furthermore, because of the over-estimation of their own productivity, men exert more effort than women and then, on average, receive a higher financial compensation. The same results hold true A c c e p t e d M a n u s c r i p t when the choice of tasks is endogenous.

The implications of our results are twofold. First, whenever discrimination is not allowed, the optimal contract proposed by the principal is such that both dimensions of the gender wage gap are obtained. An equal compensation policy is then not sufficient to eliminate the gender wage gap. Second, conducting a welfare analysis, we show that the difference in effective average utility between women and men depends on the difference in the magnitude of the bias between men and women. Stated differently, despite a lower revenue, on average, women's surplus can be higher than men's if, for example, they assess their marginal productivity of effort accurately whereas men over-estimate it. Additionally, we show that an equal compensation policy increases social welfare.

Finally, building upon our base model and endogenizing the allocation of agents on various projects, we show how, in a vast majority of cases, women self-select on low paying jobs whereas men choose high-paying jobs.

Before proceeding, we should mention that we do not pretend to capture the "full picture" about the gender wage gap, but we aim at building a bridge between the standard economic approach based on incentives and research outcomes in psychology highlighting behavioral difference between women and men. Moreover, this paper provides a first theoretical step towards highlighting group effects in the decomposition of the gender wage gap.

The organization of the paper is as follows. Section 2 reviews the literature on gender-related psychological biases. Section 3 describes the model. In Section 4 the optimal contracts in cases where discrimination is successively allowed and not allowed are derived, and welfare implications are analyzed. In Section 6 the choice of projects is endogenized. Finally, Section 7 offers a conclusion.

Gender and Psychological Bias

Gender differences in self-confidence have been extensively documented in the academic literature in psychology. Testing for the perception of competences on various tasks, [START_REF] Beyer | Gender differences in the accuracy of self-evaluations of performance[END_REF] shows that men tend either to be accurate or to over-estimate their ability, whereas women tend either to be accurate or to underestimate their ability. Similarly, testing differences in confidence judgement, [START_REF] Lundeberg | Highly confident, but wrong: Gender differences and similarities in confidence judgments[END_REF] conclude that "the problem may not be that women necessarily lack confidence but that, in some cases, men have too much confidence".

In a different vein, [START_REF] Babcock | Women don't ask: negotiation and the gender divide[END_REF] advocate that, as they don't use negotiation to get what they want, women just don't ask for pay raises, promotions and better job opportu-A c c e p t e d M a n u s c r i p t nities. [START_REF] Deaux | Attributing causes for one's own performance: the effect of sex, norms and outcome[END_REF], [START_REF] Meehan | Gender differences in expectancies for success and performance on Piagetian spatial tasks[END_REF] and Beyer (1999a) find that men have a tendency to take too much credit for their successes and, consequently, suffer from a greater self-serving attribution bias than women do. In addition, Beyer (1999b) also finds that female undergraduate students overestimate their grades less than men do. Finally, Gneezy, Niederle and Rustichini (2003) provide experimental evidence that women may be less effective in competitive environments even if they are able to perform similarly in non-competitive environments. Additionally, Gneezy, Niederle and Rustichini (2003) provide evidence that, when asked to choose the difficulty of tasks, men choose more difficult tasks relative to women. [START_REF] Deaux | Attributing causes for one's own performance: the effect of sex, norms and outcome[END_REF] also advocate that the gender bias is highly task-dependent by claiming that "men claim more ability than do women, but this difference emerges more strongly on masculine tasks". This is confirmed by several other studies. [START_REF] Prince | Women, men and money styles[END_REF] finds that men feel more competent than women in financial matters and, as observed by [START_REF] Barber | Boys will be boys: gender, overconfidence and common stock investment[END_REF], men are disproportionately represented in the financial industry. [START_REF] Beyer | The effect of gender, dysphoria, and performance feedback on the accuracy of self-evaluation[END_REF] finds differences in self-valuation between men and women in a mathematics test (considered as a masculine task) but no difference in an English test (feminine task) and or in history and geography tests (neutral tasks). Similarly, De Keuster, Rynes and De Heers (2004) provide evidence that female undergraduate students majoring in Management Information System (a supposedly "masculine" field) under-estimate their ability to a greater extent than do male students, while [START_REF] Beyer | Deterrents to women taking computer science courses[END_REF] find that women have lower confidence-levels than men in their ability in computer sciences (another supposedly "masculine" field).

The Basic Moral Hazard Model

We propose a standard moral hazard type of model where an employer (the principal) delegates the realization of a project to one of his employees (the agent), who can either be a man (m) or a woman (w). There is a proportion θ of women and, consequently, (1 -θ) of men. Both types of agents are risk neutral and subject to a limited liability constraint normalized to zero. The project generates a net return V h > 0 whenever the agent is successful and V l < 0 otherwise. We denote by ∆V the difference V h -V l . The effective probability of success on the project depends on the innate ability of the agent (ρ) and an effort e he/she exerts. To make the analysis as general as possible, A c c e p t e d M a n u s c r i p t we model the true probability of success as follows Probability of success = e(a 1 + a 2 ρ) + a 3 ρ i = m, w where a 1 , a 2 and a 3 are positive parameters weighting the impact of effort and ability in the probability of success. 2 In the rest of the paper, it is assumed that a 1 + a 2 > 0. Otherwise, the probability of success would be independent of the effort exerted.

Agents are uniquely differentiated according to their gender characteristics. Following a large body of literature in psychology, men are assumed to over-estimate their probability of being successful on a project whereas women are assumed to under-estimate it. Formally, we account for such psychological effect by defining bias parameters k m ≥ 1 and k w ≤ 1, for men and women respectively, and modelling their perceived probability of success as follows

p(e i , k i ) = k i e i (a 1 + a 2 ρ) + a 3 ρ i = m, w
That is, we focus on a setting where the bias impacts agents' marginal productivity of effort.

Effort is costly and is given by C(e) = c e 2 2 with c > 0 for both types of agents. 3 As is shown in the next section, as a result of this modelling of overconfidence, men exert more effort than women in equilibrium and, under the conditions provided in footnote 3, the equilibrium probability of success is strictly smaller than 1. One could certainly have a different interpretation of the psychology literature on overconfidence, namely that men overestimate their innate ability rather than their productivity of effort. Such analysis would suggest a scenario of the type "I am so good that I can succeed without exerting much effort". However, for such a scenario to lead to men exerting less effort than women in equilibrium, it must be the case that men believe they will succeed with probability 1 at the optimal (i.e., utility maximizing) effort level. This form of overconfidence appears very extreme and does not fit in a richer (and more realistic setup) in which agents tackle several project successively and learning takes place (as will be discussed in Section 7) since at their first failure, agents would realize that they are overconfident. As a consequence, we have chosen to rule this extreme type of overconfidence out. This does not mean that we discard 2 Adding a white noise, as is often modelled in the literature, would not alter our result and would not bring any additional insight.

3 Assuming that ∆V c ∈ a 3 ρ (a 1 +a 2 ρ)kw , 2(1-a 3 ρ) (a 1 +a 2 ρ) 2 km
ensures that the principal is willing to undertake the project, and that in all cases the equilibrium probability of success is strictly smaller than 1. The above interval is non empty as long as ρ is small enough.

A c c e p t e d M a n u s c r i p t the case in which men overestimate both the innate ability and their productivity of effort, this case being addressed is Section 4.2. However, we will still focus on sets of parameters such that the equilibrium probability of success is strictly smaller than 1.

Finally, as it is standard in contract theory, we assume that effort is not observable and, therefore, cannot be contracted upon. Finally, the principal has all the bargaining power and makes a takeit-or-leave-it offer to the agent.

To summarize, we consider a game with the following timing. The principal first makes a takeit-or-leave-it offer to the agent. The latter then accepts or refuses the contract. If he/she refuses, the game ends and all parties get 0. If he/she accepts the contract, he/she chooses a corresponding level of effort. Then, the output is realized and the agent receives transfers accordingly.

Contract Design

We start by describing the optimal contract in a setting where discrimination is either allowed or banned. We then consider welfare aspects.

Optimal Contract

As is standard in models of moral hazard, the agent must be given incentives to exert effort. In the previous section, we have described the simplest model that allows for such a structure. Given that effort is not contractible, the principal provides incentives by conditioning the reward scheme on the outcome of the project which is both observable and enforceable. Denote q such an outcome. Given the two-point distribution of the project value, we define a compensation contract conditioning the reward on the realized project value as follows.

h(α, β, q) =    α if q = V l β if q = V h with α < β.
One can naturally think of bonus reward schemes as being described by such a binary outcome model but it also accounts for standard "promotion" incentive schemes or "up or out contracts" settings. 4 We solve the model by backward induction. Taking the contract offered by the principal as given, 4 We discuss the robustness of our approach to other incentive schemes later in the paper.

8

A c c e p t e d M a n u s c r i p t an agent i with a bias parameter of magnitude k i performs an effort

e(k i , α i , β i ) = k i c (a 1 + a 2 ρ)(β i -α i ) (1)
It directly follows from the difference in perception of marginal productivity between men and women (i.e., k m > k w ) that, for a given contract, men exert more effort than women.

We can distinguish two (legal) environments under which the principal offers his contract. In the first one, the principal is free to discriminate among agents, while in the second the principal is constrained to offer the same (menu of) contract(s). The next propositions derive the optimal contract in each setting.

Proposition 1 If the principal can discriminate among gender, then the optimal contract is

   α o (k i ) = 0 if q = V l β o (k i ) = ∆V 2 - a 3 ρc 2(a 1 +a 2 ρ) 2 k i if q = V h (2)
Proof. The proof is straightforward. Rationally anticipating the effort of the agent in the continuation game, the objective of the principal is, for each gender, to maximize

F (α, β, k) = V l -α + p(e i (k i , α, β), 1)(∆V -β + α)
subject to α ≥ 0 and β ≥ 0. The solution then follows directly.

Since k m > k w , we have

β o (k m ) > β o (k w ).
As a result, men exert more effort than women, succeed more often and get a higher prize in case of success. However, the group effect dimension of the wage gap is left unexplained.

Let us now turn to the optimal contract when compensation discrimination is not allowed. In the rest of the paper, we will denote by X(θ) ≡ θ k w + (1 -θ)k m . We have the following result.

Proposition 2 When discrimination is not allowed, the unique equilibrium is pooling and, for all

i, is such that          α * (θ, k w , k m ) = 0 if q = V l β * (θ, k w , k m ) = ∆V 2 - a 3 ρ c 2(a 1 +a 2 ρ) 2 X(θ) if q = V h
The fact that the equilibrium is pooling stems from two opposite forces. On the one hand, the principal would like to tailor contract to agents. However, to do so, he would have to pay some fixed fee (α w ) to women and choose an incentive component larger for men than for women (i.e.,

9

A c c e p t e d M a n u s c r i p t β m -α m > β w -α w ), so that each gender self-select on each contract. There are then two costs associated to a separating equilibrium: first, rewarding women in case of failure and, second, a lower effort level chosen by women hence a lower probability of success. The proposition states that these costs exceed the benefits of a separating equilibrium (i.e., contracts tailored to agents).

Note also that the assumption of limited liability plays an important role in the derivation of this result. It implies that the participation constrain of the agents are slack. In a setting where the principal could set arbitrarily large negative wages in the event of a poor output realization, then there would be scope for separating contracts in which participation constraints are binding.

In addition, a necessary condition for our result to hold is that the probability of success can be positive even in the absence of effort (namely, a 3 > 0). This is a usual assumption that can be defended on different grounds. First, innate ability is a component that does not only affect the marginal productivity of effort. Second, one can also reinterpret our model as a setting where some minimum effort (given by a 3 ρ) can be monitored by the principal and that the contract is design to provide incentives for additional effort exertion. In the rest of the analysis, we will assume that a 3 > 0. A remaining question then concerns the impact of outside opportunities on the viability of our result. This is discussed in the following section on robustness.

A closer look at the optimal contract helps us to deepen our analysis of the difference in expected revenue between men and women. The reward in case of success, i.e., β * (θ, k w , k m ) can be rewritten as

β * (θ, k w , k m ) = θ k w X(θ) β o (k w ) + (1 -θ) k m X(θ) β o (k m ) (3) = β o (k w ) + (1 -θ) k m X(θ) (β o (k m ) -β o (k w )) (4)
where β o (k i ) (i = m, w) is the payment in case of success when discrimination is allowed (See proposition 1). We then obtain the following result

Corollary 1 When discrimination is banned, the optimal contract is such that:

(i) men's average revenue is larger than women's one.

(ii) the larger the fraction of men (i.e., the lower θ), the higher women's average revenue.

An equal opportunity regulation implies that men and women obtain the same compensation contract. However, this does not necessarily imply that they have the same chance of receiving a bonus or being promoted. First, due to their bias toward over-estimating their productivity, men 2000). Second, the larger the proportion of women in the industry, the lower is β. This effect drives the result in point (ii).

Finally, measuring the absolute wage gap by the difference in average wages received by men and women and the relative wage gap by the ratio of average wage of men to average wage of women, additional testable implications can be derived from our model:

Corollary 2 When discriminatory conducts are banned, both absolute and relative expected wage gap increase with the proportion of men in the considered occupation.

The proof is straighforward once noting that both the absolute and the relative expected wage gap between men and women are given by (a 1 + a 2 ρ) 2 β 2 c (k m -k w ) and a 3 ρc+(a 1 +a 2 ρ) 2 kmβ a 3 ρc+(a 1 +a 2 ρ) 2 kwβ , respectively.

Robustness of the results

We now aim at analyzing the impact on our results of several assumptions made in the core of the paper, namely the fact that reservation utilities are normalized to zero for both genders, agents are risk neutral and, finally, the fact that psychological biases are associated to effort rather than ability.

First, our results remain valid in a framework where genders have different outside options as long as we keep on assuming that women accept the contract offered when discrimination is allowed.

Suppose indeed that, for various sociological or economical reasons, women have a higher value than outside work (see Gronau 1973aGronau , 1973b)), i.e., u o w > u o m = 0 and, additionally, that u(β 0 (k w )) > u o w . Given that, (i) for all θ > 0, β * (θ, k w , k m ) > β 0 (k w ) and (ii) women's utility increases with the magnitude of the transfer β, then women's participation constraint is necessarily slack and the contract described in Proposition 2 remains optimal.

Second, we have assumed that men overestimate their marginal productivity whereas women underestimate it. However, some studies in the field of psychology suggest that both men and women tend to overestimate their probability of success but men to a larger extent than women.

11

A c c e p t e d M a n u s c r i p t

Our results carry over under this assumption, since our results rely on men being more overconfident than women. The gender wage gap is generated by the fact that there exists a gap between these biases. The fact that the equilibrium is uniquely pooling whenever equal treatment is imposed is key to explaining the second result of corollary 1.

An interesting argument sometimes advocated when comparing men and women's performances is that the women are more risk averse (see [START_REF] Eckel | Sex and risk: experimental evidence[END_REF]). We could account for differences in preferences in our model and show that if the difference in risk-aversion between genders is not too large, then our results hold. Otherwise, agents could self-select on different contracts but the group effect of wage inequality would not emerge at equilibrium. Finally, let us discuss the way psychological biases influence the probability of success in our model. As already mentioned, one could certainly reinterpret the results derived in the psychology literature as evidence that men overestimate their innate ability, from ρ to k m ρ, rather than their marginal productivity of effort. For example, it can be assumed that the probability of success is

p(e, k i ) = e(a 1 + a 2 k i ρ) + a 3 k i ρ (5)
In such a case, if discrimination is not allowed, we still obtain that the unique equilibrium is pooling and

β(θ, k w , k m ) = ∆V 2 - a 3 ρcX(θ) 2[a 2 1 + 2a 1 a 2 ρX(θ) + a 2 2 ρ 2 (θk 2 w + (1 -θ)k 2 m )] (6)
It can be shown that if a 1 is small enough, then β(θ, k w , k m ) is decreasing in θ. In such a case, the results obtained in the case in which biases are related to effort only, still hold in the case in which biases are related to the innate ability.

Welfare implications

The implementation of an equal compensation regulation influences both workers' welfare and the level of social welfare. This influence is twofold. First, following an increase in revenue in case of good performances, women's effort and surplus increase. Second, men's revenue in case of good performances decrease. As a consequence, their effort also decreases. However, the total impact on their average realized welfare level is ambiguous. To see this, note that for a contract paying β in A c c e p t e d M a n u s c r i p t case of success, women's and men's average realized utility 5 is given by

U i (β) = β a 3 ρ + (a 1 + a 2 ρ) 2 k i β c 1 - k i 2 i = m, w.
It is a routine matter to check that U w is an increasing function of β. Given that β * (θ, k w , k m ) > β o (k w ), women's surplus then increases with the implementation of an equal compensation policy.

Regarding men, a direct effect is of an equal compensation policy regulation is to decrease β.

However, as a positive implication of a downgraded bonus, the over-optimal effort level decreases.

Formally, there exists k > 1 such that ∂U m (β)/∂β > 0 (resp. < 0) whenever k m < k (resp. > k).

The larger the salary in the case of good performance the more men exert effort. If the magnitude of the bias is large, then a decrease in β generates a large decrease of the effort level, and the effort cost reduction dominates the expected salary reduction. On the contrary, if the bias is small, the decrease in effort level generated by a decrease in β is small, and the expected salary reduction dominates the effort cost reduction.

In the absence of discrimination, men and women receive the same salary β * (θ, k w , k m ) in the case of good performance. The workers' average surplus is given by

U ND = θU w (β * (θ, k w , k m )) + (1 -θ)U m (β * (θ, k w , k m ))
We can then provide the following results

Proposition 3 If compensation discrimination is banned, the difference in average realized utility between women and men depends on the difference in the magnitude of the bias, i.e., (k m -1) -

(1 -k w ).
The proof is straightforward once observing that the difference in average realized utility between women and men is given by

U w (β * (θ, k w , k m )) -U m (β * (θ, k w , k m )) = β * 2 (θ, k w , k m ) c (a 1 + a 2 ρ) 2 (k m -k w ) (k m -1) -(1 -k w ) 2 (7)
which sign is that of the difference (k m -1) -(1 -k w ). That is, the agent with the higher average realized utility is the one endowed with the smaller bias.

5 If we consider a large population, the average realized utility is equal to the expected revenue net of effort cost given the true probability distributions.

13

A c c e p t e d M a n u s c r i p t

Let us turn now to the level of social welfare which we measure by the wealth created net of effort cost. Under discriminatory conduct, the level of social welfare corresponds to

SW D = V l +a 3 ρ∆V +(a 1 +a 2 ρ) 2 θk w β o (k w ) c ∆V - k w β o (k w ) 2 + (1 -θ)k m β o (k m ) c ∆V - k m β o (k m ) 2
In absence of compensation discrimination, the level of social welfare is

SW ND = V l + a 3 ρ∆V + (a 1 + a 2 ρ) 2 X(θ) β * (θ, k w , k m ) c ∆V - β * 2 (θ, k w , k m ) c θk 2 w + (1 -θ)k 2 m
We then get that

SW ND -SW D = (a 1 + a 2 ρ) 2 a 3 ρc 2X(θ) θ(1 -θ)(k w -k m ) 2 ∆V - a 3 ρc 2X(θ) > 0. ( 8
)
This result is summarized in the next proposition.

Proposition 4 An equal compensation policy increases the level of social welfare.

This result is not perfectly intuitive in the first place. Relying on a single contract when facing heterogenous agents should decrease the principal's profit. However, given that

β * (θ, k w , k m ) ∈ (β * (k w ), β * (k m ))
, an equal opportunity treatment lowers the effort exerted by men (who have a tendency to overwork) whereas it increases effort by women (who have a tendency to under-work).

Globally, social welfare increases.

Endogenous Choice of Projects

So far, we have assumed that the fraction of men and women for a given job was exogenous. However, as mentioned in Section 2, [START_REF] Deaux | Attributing causes for one's own performance: the effect of sex, norms and outcome[END_REF] advocate that the gender bias is highly task dependent. Some later contributions such as [START_REF] Lundeberg | Highly confident, but wrong: Gender differences and similarities in confidence judgments[END_REF] and [START_REF] Beyer | Gender differences in self-perceptions: convergent evidence from three measures of accuracy and bias[END_REF] confirm such an analysis. Following these pieces of evidence, we consider now the case in which there are tasks for which men and women accurately perceive their productivity and others in which they differ in their perceived productivity. Furthermore, agents non-cooperatively choose which type of task they dedicate to.

Formally, we assume that there are two projects (A and B). Each project generates a value V h > 0 in case of success and V l < 0 in case of failure. On project A, men and women accurately perceive their productivity of effort:

k A w = k A m = 1.
Conversely, on project B, men over-estimate their productivity (i.e., k B m > 1) while women under-estimate it (i.e., k B w < 1). Finally, without loss A c c e p t e d M a n u s c r i p t of generality, we assume that there is a continuum of length 1 of agents, half of them being women and the rest being men.

The sequence of events is the following. First, agents choose a project. Then, the owner of project j (j = A, B) makes a take-it-or-leave-it offer to each agent that has chosen project j. If an agent rejects the offer, he/she receives his/her reservation utility. He/she does not have the possibility to apply for a job on the other type of project.

Suppose that discrimination is not allowed, and denote θ j (j = A, B) the anticipated fraction of women among agents working on project j. Furthermore, let β j (θ j ) be the payment received in case of success on a project j if women represent a fraction θ j of agents anticipated to work on this project. Then, from Proposition 2, we deduce that the principals choose

α j = 0 β * (θ j , k j m , k j w ) = ∆V 2 - a 3 ρc 2(a 1 +a 2 ρ) 2 (θj k j w +(1-θ j )k j m) (9)
Denote Û ND w (θ, k m , k w ) and Û ND m (θ, k m , k w ) women's and men's expected perceived surplus, respectively, for a project in which a fraction θ of agents are women, and the perceived probabilities of success of men and women are p(e, k m ) and p(e, k w ), respectively. Hence,

ÛND w (θ, k m , k w ) = β * (θ, k m , k w ) a 3 ρ + k 2 w 2c (a 1 + a 2 ρ) 2 β * (θ, k m , k w )
and

ÛND m (θ, k m , k w ) = β * (θ, k m , k w ) a 3 ρ + k 2 m 2c (a 1 + a 2 ρ) 2 β * (θ, k m , k w )
An equilibrium is then a pair (θ * A , θ * B ) such that the contract proposed are (0,

β * (θ * A , k A m , k A w )) and (0, β(θ * B , k B m , k B w )
) and, for (j, j ′ = A, B, j ′ = j), we have

• If θ * j ∈ (0, 1) and If θ * j ′ ∈ (0, 1) then    Û ND w (θ * j , k j m , k j w ) = Û ND w (θ * j ′ , k j ′ m , k j ′ w ) Û ND w (θ * j , k j m , k j w ) = Û ND w (θ * j ′ , k j ′ m , k j ′ w ) • If θ * j = 1 and θ * j ′ = 0 then    Û ND w (1, k j m , k j w ) ≥ Û ND w (0, k j ′ m , k j ′ w ) Û ND m (1, k j m , k j w ) ≤ Û ND m (0, k j ′ m , k j ′ w ) • If θ * j ∈ (0, 1) and θ * j ′ = 0 then    Û ND w (θ * j , k j m , k j w ) ≥ Û N D w (0, k j ′ m , k j ′ w ) ÛND m (θ * j , k j m , k j w ) = Û N D m (0, k j ′ m , k j ′ w ) • If θ * j ∈ (0, 1) and θ * j ′ = 1 then    Û ND w (θ * j , k j m , k j w ) = Û N D w (1, k j ′ m , k j ′ w ) ÛND m (θ * j , k j m , k j w ) ≥ Û N D m (1, k j ′ m , k j ′ w )
Page 17 of 25

A c c e p t e d M a n u s c r i p t a task (i.e., k B w < 1), they believe that their probability of success is low. If this bias is sufficiently high (i.e., k B w < G(k B m )), the second effect is stronger than the first one and men and women choose different projects. As the bias weakens (i.e., k B w ∈ (G(k B m ), H(k B m ))) women's incentives to choose project B increase and women split on the two projects. Still, they tend to work in occupation where their gender is highly represented. This is in line with empirical finding since, as stated by [START_REF] O'neill | The Gender wage gap, circa 2000[END_REF], "In 2001 women, on average, worked in occupations in which the percentage of female employees was close to 68 percent; men worked in occupations that were only 30 percent female."

The previous analysis assumes that employees choose their job. Consider now the case in which a principal has to find a worker for a project B (i.e., a project about which women underestimate their productivity and men overestimate theirs) and that there are two applicants, a man and a woman. If men and women have the same ability then the man is the one hired by the principal since his profit is increasing in self-perceived productivity. Furthermore, allowing for different levels of ability, a woman must be endowed with greater ability than a man in order to be chosen by the principal.

Discussion and Concluding Remarks

It is now well established that men and women differ in their confidence levels although such empirical results have not been used in a personnel economics perspective. Building upon this observation and using a mechanism design approach, we have shown that contract theory provides a useful framework to highlight how the two dimensions of the gender wage gap, namely the traditional dimension and the group-effect dimension, can emerge at equilibrium. Furthermore, we have derived additional testable implications, such as the increase of both absolute and relative wage gap with the proportion of men in an occupation.

A consequence of our results is that an equal-opportunity policy will not eliminate the gender wage gap. In our framework, as long as biases are effectively at stake, women always earn less than men. However, the sign of the difference in surplus is ambiguous and stems from the fact that both genders do not necessarily choose the optimal amount of effort. Although women have a tendency to under-work, men have a tendency to over-work. The fact that an equal-compensation regulation does not eliminate the gender wage gap does not imply that such regulation should be re-evaluated. The forces underlying our results easily extend to different incentives schemes and production technologies. Consider for example that agents effectively produce goods in quantity (or quality) q i = (a 1 + a 2 ρ ′ )e i + a 3 ρ ′ + ǫ i , where ǫ i is a white noise. Each unit has a value ∆V for the principal and he offers w i for each unit produced by gender i. It can be shown that the contract offered by the principal is exactly the same as the one described earlier in the sense that the reward scheme is such that w i = β o (k i ) whenever the principal can discriminate, and w(θ) = β * (θ, k w , k m ), otherwise.

The wage gap highlighted in this setup is then not based on the average revenue but, in contrast, on the effective revenue.

Our behavioural-economics approach can also provide a useful theoretical framework compatible with a growing literature stating that women are less competitive than men [START_REF] Niederle | Do women shy away from competition? Do men compete too much?[END_REF]Vesterlund 2005 or Gneezy, Niederle andRustichini 2004). An important difference between these studies and our model, apart from the fact that these papers are experimental, is that they consider contests with exogenously given prizes and test behavioral differences among genders whereas we start from the stylized fact that there exist behavioral differences among men and women and we endogenously determine the optimal reward scheme. Even in our framework where agents are not rewarded on the relative performance, we show that the expected utility derived by each party depends on his identity and the proportion of the respective genders in the pool.

Formally, consider that agents are not paid solely according to their own performance but rather enter a contest. Whenever facing a man, the expected probability of success of a woman is denoted P w m and is given by

P w m = Pr (a 3 ρ + (a 1 + a 2 ρ)k w e w + ǫ w > a 3 ρ + (a 1 + a 2 ρ)k m e m + ǫ m ) = Pr (k w e w -k m e m > η) ≡ G (k w e w -k m e m )
where η = (ǫ m -ǫ w )/(a 1 + a 2 ρ) and G(.) is the distribution function of η. To keep things simple, assume that G(.) is twice differentiable and denote g(.) the density function associated to G(.).

Furthermore, assume that g(.) is symmetric and single-peaked at 0. Denote B > 0 the prize for the winner. Women then choose their effort level such that ). Hence, men choose a higher effort level than women (e * m = km k w e * w ) and, consequently, have a higher performance level. Their chances to succeed in the contest are then higher.

To summarize, in the specific framework of a contest, the assumption that men and women overestimate and underestimate their productivity, respectively, provides then a rationale for the fact that women under-perform men in a contest (see Gneezy, Niederle and Rustichini 2003) and, relatedly, women prefer individual rather than relative pay schemes (when men are in the pool) (see [START_REF] Niederle | Do women shy away from competition? Do men compete too much?[END_REF].

Finally, we based our analysis on the fact that agents misperceive their true marginal productivity of effort and, consequently, take sub-optimal decisions. Such a statement can however become problematic in the long run since, whenever endowed with a minimum of rationality, agents should revise their perceived marginal impact of effort based on their performance record. To see that learning does not eliminate the gender wage gap, consider a model of promotion in which agents are promoted based on the number of successes on several projects they tackle, and wage increases are associated to promotions. In such a framework, updating their beliefs in a Bayesian fashion, agents will realize that for a given effort level e i , their frequency of success is p(e i , 1) and not p(e i , k i ).

Hence, agents should revise their effort level accordingly. However, if early in their career men and women have beliefs about their productivity such that k m > k w , this has two implications. First, early in their career, men exert more effort than women. As a consequence, their performance is greater than that of women (see Corollary 1) and their success rate is higher. Consequently, men get "promoted" faster, implying that they work on projects of higher value (and higher difference in value between success and failure, i.e., ∆V ). Hence men's revenues increase faster than those of women. Second, through their record of successes and failures men and women learn their true marginal productivity. As a consequence, after learning has taken place, men and women are not biased anymore. Everything else being equal, this implies that they would have the same rate of success. Still, a wage gap remains through out the working life cycle due to differences in performance, which implied faster promotion pace, early in the career.

A c c e p t e d M a n u s c r i p t

In such a setting, it is straightforward to see that men will always self select on the contract designed for them. That is

α w + p(e w m , k m )(β w -α w ) -c (e w m ) 2 2 < p(e m m , k m )β m -c (e m m ) 2 2 . ( 13
)
The principal will then aim at maximizing his profit under constraints (12), α w ≥ 0, and β w ≥ α w .

Denote λ, µ, and ν, the Lagrange multipliers associated to the constraints (12), α w ≥ 0, and β w ≥ α w , respectively. Formally, writing the Lagrangian and solving the program while maximizing with respect to α w and β w , we get the following first order conditions

∂L ∂α w = -1 - k w c (a 1 + a 2 ρ) 2 (∆V -(β w -α w )) + a 3 ρ + k w c (a 1 + a 2 ρ) 2 (β w -α w )) +λ[1 -a 3 ρ + k 2 w c (a 1 + a 2 ρ) 2 (β w -α w )] + µ -ν = 0 ∂L ∂β w = k w c (a 1 + a 2 ρ) 2 [∆V -(β w -α w )] -a 3 ρ - k w c (a 1 + a 2 ρ) 2 (β w -α w ) +λ[a 3 ρ + k 2 w c (a 1 + a 2 ρ) 2 (β w -α w )] + ν = 0
which directly implies that λ + µ = 1. First, note that it must be the case that ν = 0, otherwise β w = α w so that women do not exert effort.

Now, two cases have to be considered. First, if µ > 0, then α w = 0. Second, if µ = 0, then λ = 1.

In such a case, the first-order conditions ∂L ∂αw = 0 and ∂L ∂β w = 0 are equivalent to β w -α w = ∆V 2 + k w Substituting β w -α w by its expression in the profit of the principal, it is straightforward that this profit is a decreasing function of α w . Hence, the principal chooses α w = 0.

It follows that in all cases, the principal chooses α w = 0, implying that there is no separating equilibrium in which both types of agents are employed.

It is finally obvious that the principal can never exclude one type by offering a single contract since both genders are better off exerting some effort than remaining unemployed. The optimal contract is then pooling.

It now remains to derive the structure of payments in the optimal contract. In a pooling equilibrium, the principal maximizes Finally, we need to verify that each gender always has an incentive to exert effort. This can be deduced from the fact that, when the agent does not exert effort, his/her utility is ρβ whereas if he/she exerts effort, his/her perceived expected utility is β a 3 ρ + (a 1 + a 2 ρ) 2 (k i ) 2 2c β > a 3 ρβ for all i and triplet (k i , ρ, c).

Proof of Proposition 5

As mentioned in section 6, since k A m = k A w = 1, then for all θ A ∈ [0, 1], β * (θ A , 1, 1) = β A = ∆V 2 -a 3 ρc 2(a 1 +a 2 ρ) 2 . It follows that, if choosing project A, men and women exert the same effort level and their expected revenue net of effort cost is

Û ND m (θ, 1, 1) = Û ND w (θ, 1, 1) = U A = β A a 3 ρ + β A 2c (a 1 + a 2 ρ) 2 (14)
Conversely, if choosing project B, men and women do not exert the same effort level. The effort level chosen by an agent of type i is ). This completes the proof.

  3

  than women (see equation 1) and, therefore, receive the payment β more often (point (i)). This is in line with the empirical observation inPuri and Robinson (2005) who present a large-scale micro level evidence linking optimism to a series of significant work related choices and show that more optimistic agents (regardless of their employment status) work harder. Moreover, it is acknowledged that women work fewer hours (BLS Bulletin 2340; US Bureau of the Census,

  As we also show, an equal-compensation policy increases social welfare.

  g(k w e * w -k m e * m ) k w B = ce * w (10) whereas men act as follows g(k m e * m -k w e * w ) k m B = ce * m Given that the function g(.) is symmetric and single-peaked in 0, we have g(k m e * m -k w e * w ) = g(k w e * w -k m e * m

V

  l -α + (∆V -β + α)[a 3 ρ + (a 1 + a 2 ρ)(θe * (α, β, k w ) + (1 -θ)e * (α, β, k m ))]

  s c r i p t for which the unique solution is α* (θ, k w , k m ) = 0 and β * (θ, k w , k m ) = ∆V 2 -a 3 ρc 2X(θ)(a 1 +a 2 ρ) 2

  1 + a 2 ρ)β * (θ B , k B w , k B m ).It follows that the expected revenue net of effort cost of an agent of type i isÛND i (θ B , k B m , k B w ) = β * (θ B , k B w , k B m ) a 3 ρ + + a 2 ρ) 2 β * (θ B , k B w , k B m )A necessary condition for women to work on both projects, in equilibrium, is thatU A = ÛND w (θ B , k B m , k B w ).It implies that if women work on both projects, men always choose project B sinceÛ ND m (θ B , k B m , k B w ) > Û ND w (θ B , k B m , k B w ).Assume now that all women work on project A (i.e θ B = 0). Then, it can be shown thatÛ ND m (0, k B m , k B w ) > U A . Therefore, if women choose project A, then men choose project B. Stated differently, if U A > Û ND w (0, k B m , k B w ), the unique equilibrium is fully separating, i.e, θ * A = 1 and θ * B = 0. Such a condition can be rewrittenk B w < G(k B m ), we have θ * A = 1 and θ * B = 0. Now, β * (θ, k B w , k B m ) is a decreasing function of θ which implies that ÛND w (θ, k B m , k B w ) is a decreasing function of θ. Therefore, if U A ∈ ( Û ND w (1/2, k B m , k B w ), Û ND w (0, k B m , k B w )) , there exists θ * B such that U A = Û ND w (θ * B , k B m , k B w ). Let F (k B m , k B w ) = Û ND w (1/2, k B m , k B w ) The equation F (k B m , k B w ) = U Ahas then a unique positive solution in k w which, of course, is a function of k B m . Denote H(k B m ) such a solution. Given that A c c e p t e d M a n u s c r i p t ∂F/∂k B w > 0, then whenever k B w > H(k B m ), we have F (k B m , k B w ) > U A , and women self-select on project B, i.e., θ * B = 1/2. In contrast, if k B w ∈ (G(k B m ), H(k B m )), θ * A = 1 and θ * B ∈ (0, 1/2). Finally, given that β * (0, k B w , k B m ) > β * (1/2, k B w , k B m ) and Û ND m (θ B , k B m , k B w ) is increasing in k B m , we have that H(k B m ) > G(k B m

As empirical evidence shows, discrimination in wages and bonuses may end up being very costly for firms. As an illustration, Merril Lynch has already paid out more than USD 100 millions to past and present female employees. See, The Economist, "Sexism and the City", June 10,

2004, for more evidence.

A c c e p t e d M a n u s c r i p t Therefore, an equilibrium is a situation in which 1) principals choose contracts according to their beliefs θ * j (j = A, B), 2) agents choose their task so as to maximize their expected perceived surplus given the contracts chosen by principals, and 3) the fraction of women among agents working on project j (j = A, B) is effectively θ * j . The following proposition derives the fraction of women involved on projects A and B.

Proposition 5 Men only work on project B. There exist functions G(k m ) and H(k m ) (with

, then women only work on project B, i.e., θ * B = 1/2 and nobody works on project A.

The proposition characterizes the equilibrium for all sets of parameters. In part (i), women and men self-select on two different jobs and men earn, on average, more than women. However, the group effect of the gender wage gap is absent. Part (ii) and (iii) are certainly the most interesting ones since they establish that the two dimensions of the gender wage gap may arise endogenously when agents choose between several tasks for which they have different perceived productivity. In part (ii), the proportion of women choosing project B, in equilibrium, is determined such that women are indifferent between both projects. Their perceived utilities are then identical on both projects. Nevertheless, it is a routine matter to show that the average revenue of agents working on project B is larger than that of agents working on project A and, additionally that the average revenue of men (working on project B) is larger than that of women working on project B. Finally, part (iii) is very similar to the analysis of the base model and the same conclusions carry on.

The proposition also states that men always choose the job for which they feel more competent. This is consistent, for example, with results concerning the finance industry: men generally feel more competent than women do in financial matters (See Prince 1993) and, as stated by [START_REF] Barber | Boys will be boys: gender, overconfidence and common stock investment[END_REF] "casual observation reveals that men are disproportionately represented in the finance industry". Another interesting feature of Proposition 5 is that women can dedicate themselves to project A which, in our model, can de denominated as the low paying job (β A < β B ). Two effects explain why women may self-select on low pay jobs, rather than choosing high pay jobs. On the one hand, both genders would like to choose project B since the transfer received in case of success is larger for project B than for project A. On the other hand, if women feel less able to perform such To derive the proof, we start by showing that the equilibrium is pooling. To do so, we proceed by contradiction, and assume that the principal offers two different contracts (α w , β w ) and (α m , β m ), with α w = α m and β w = β m , that are chosen by women and men respectively. We proceed in different steps.

Lemma 1 If a separating equilibrium exists, it has to be such that only α m is set to 0.

Assume that both α m and α w are strictly positive, then the principal could gain by lowering all transfers without affecting incentives. Moreover, there cannot exist a separating equilibrium where α w = α m = 0 and β m = β w since both genders would always choose the contract with the higher β.

Suppose now that α w = 0 and α m > 0. Then β w > β m otherwise both agents choose (α m , β m ).

But then if women indeed choose (α w , β w ) over (α m , β m ), so will men since their chance to reach the high state is higher. Therefore, if there exists a separating equilibrium, it is such that α w > α m = 0 and β m > β w .

Let us now proceed one step further by showing that there is no separating equilibrium in this game, implying that α m = α w = 0.

Lemma 2 The equilibrium is pooling and α m = α w = 0. Suppose, still, that a separating equilibrium satisfying lemma 1 exists and that men are indifferent between both contracts whereas women prefer their contract. Then the principal could decrease transfers to women such that both genders still separate on two different contracts and incentives are kept unchanged. Such a reasoning does not apply when women are indifferent since α m is already set at 0. Therefore, a separating equilibrium must be such that women are indifferent to both types of contract whereas men (weakly) prefer the contract (0, β m ).


Thierry Bastogne

Pierre Vallois

An extension of the target theory in biology applied to system reliability

Keywords: Reliability, repairing procedure, target theory, treatment of cancer by radiotherapy, damaged cell, Markov chain, large deviations AMS 2000 subject classifications : 60 F 10, 60

We consider rough products produced by a factory. Each product coming from the plant has m vital elements and some elements can be damaged. To obtain a perfect product (i.e. all the constitutive m elements are safe) all the damaged elements are repaired and a test phase follows. The result of this two-steps procedure is random. We suppose that the number Z k of non-damaged elements is a Markov chain valued in the set {0, 1, • • • , m}, where k is the number of applied repairingtest phases. We have a qualitative result which says that if the repair phase is efficient then P (Z k = m) is close to 1. As for production of a large number n of products, the former result allows us to give conditions under which either the n elements or a fraction of these n elements are (is) safe after the application of k previous maintenance phases.

Introduction

Most engineering systems endure degradations with time from wear, manufacturing defects, shocks and damages, and ultimately fail when the total damage exceeds a failure critical level [START_REF] Nakagawa | Shock and Damage Models in Reliability Theory[END_REF]. Such events are usually mathematically described as stochastic processes and different models have been developed to analyze reliability properties and improve maintenance policies. In this chapter, we propose a bio-inspired modeling of engineering systems reliability, based on the target theory and hit-modeling paradigm introduced in the 1920s when biologists were beginning to develop quantum approaches to inactivation phenomena in irradiated biological tissue [START_REF] Dessauer | Uber einige Wirkungen von Strahlen[END_REF][START_REF] Blau | Uber einige Wirkungen von Strahlen[END_REF][START_REF] Crowther | Some considerations relative to the action of x-rays on tissue cells[END_REF][START_REF] Atwood | On the interpretation of multi-hit survival curves[END_REF][START_REF] Lea | Actions of Radiations on Living Cells[END_REF][START_REF] Pollard | The direct action of ionizing radiation on enzymes and antigen[END_REF][START_REF] Pollard | Radiation inactivation of enzymes, nucleic acids and phage particles[END_REF].

1. [START_REF] Atwood | On the interpretation of multi-hit survival curves[END_REF] In target theory, a cell is assumed to get different vital sites called targets which must be all inactivated to kill the cell. For instance, it is accepted that the chromosomes are sensitive targets but there is additional evidence that the nuclear membrane, or some cell organelles close to the nuclear membrane can also be regarded as targets. Each target is deactivated when it is hit by a number of radiation particles. There are several classes of hit models classified by the number of targets and the number of hits. In these models, it is generally assumed that cells have an homogeneous behavior. In practice, there are at least three main reasons to put this hypothesis into question. The first cause of heterogeneity comes from the nonuniform spatial distribution of the radiation dose. The second cause is due to the differences between the cell types (necrotic, quiescent, proliferating, stem cells, etc.) and the nonuniform concentration of oxygen and nutrients. A third factor corresponds to a cell-to-cell variability of damages and to the variation of the cell sensitivity to radiation. In [START_REF] Bastogne | Multinomial model-based formulations of tcp and ntcp for radiotherapy treatment planning[END_REF], a model of tumor growth based on the target and hit modeling paradigm has been proposed. It is based on a Markov chain formalism able to describe both the target reparation between two consecutive dose fractions of the radiation schedule and the heterogeneity of damages induced by radiations. 1. [START_REF] Bastogne | Multinomial model-based formulations of tcp and ntcp for radiotherapy treatment planning[END_REF] We claim that this model may be applied to other fields than cell biology, for instance reliability of engineering systems. For instance, let us consider a space launcher composed of numerous subsystems such as the payload structure, the guidance part, and the propulsion system, which are themselves made up of thousands of constitutive elements. The production of such sophisticated systems is not perfect and the products can have different manufacturing faults. If the m components of a given product have no failure, the product can be directly used or sold. Otherwise, if at least one of its m constitutive elements is out of order, the product has to be repaired. As illustrated in Figure 1, the maintenance procedure is iteratively repeated until each product is completely safe. We assume that the time period of the maintenance cycle is constant and the number of cycles for each product is noted k, which is equivalent to a discrete time. k = 0 corresponds to the time instant when the product is coming out from manufacturing and has undergo no maintenance. During the maintenance cycle, two successive random phenomena can occur.

1. Reparation phase. At the beginning of the k-th cycle, all the m elements of a product are checked. If i ≥ 1 elements are detected as damaged then each element is repaired. The repair mechanism only concerns these i elements. However the result of a repair of a given element is not certain, we suppose that it is successful with probability r and the damaged element remains in the same state with probability 1 -r. The results concerning the maintenance of these i elements are supposed to be independent. If we adopt the convention that a disk with (without) a cross denotes a damaged (resp. safe) element, the previous procedure can be summarized as

⊗ r -→ ⃝ ⊗ 1-r -→ ⊗ ⃝ 1 -→ ⃝. (1.1)
2. Damage occurence. The reparation phase is always risky. Indeed, the maintenance operator can make mistakes, forget connexions or introduce undesirable elements in the product, which therefore leads to cause new damages. To account for this risk, we denote q (resp. 1 -q) the probability for any constitutive element to be damaged (resp. to remain undamaged). The associated scheme is :

⃝ q -→ ⊗ ⃝ 1-q -→ ⃝ ⊗ 1 -→ ⊗. (1.2)
These assumptions of heterogeneous damages caused to a complex system come from [START_REF] Bastogne | Multinomial model-based formulations of tcp and ntcp for radiotherapy treatment planning[END_REF] : the cell and its inward targets are herein replaced by the manufactured product and its constitutive elements respectively.

The parameters q and m are considered as fixed and only depend on the own feature of the product. However r is allowed to vary freely : an active maintenance strategy consists in carefully repairing the failed elements so that after a few repairing phases, they will be without any default.

1.3

Let Z k the random variable denoting the number of safe elements in the manufactured product at time k. We suppose that (Z k ) is a discrete-time Markov chain, i.e. the unit state at time k + 1 only depends on the current state at time k.

Let Π be the corresponding transition matrix of (Z k ). We briefly define Π, interested readers can refer to [START_REF] Bastogne | Multinomial model-based formulations of tcp and ntcp for radiotherapy treatment planning[END_REF] for details. The dynamics of (Z k ) takes into account first repair mechanisms and second the phase of testing as follows :

Π = RQ (1.3)
where R models repairing and Q describes the procedure of tests. According to our assumptions, the matrices R and Q take the form :

R(i, j) = { ( m-i j-i )r j-i (1 -r) m-j i ≤ j 0 j < i.
(1.4)

Q(i, j) = { ( i j )q i-j (1 -q) j j ≤ i < m 0 i < j. (1.5) When i = m, Q(m, m) = 1, Q(m, j) = 0, for any 0 ≤ j < m.
If we assume that the product is initially in state i 0 (i 0 active component(s)), i.e. P (Z 0 = i 0 ) = 1, then

P (Z k = i) = Π k (i 0 , i) i ∈ {0, . . . , m}. (1.6)
The first question which is addressed herein is the efficiency measure of the repairing procedure at the element scale. Suppose that k is fixed and Z 0 = i 0 . According to (1.6), Π k (i 0 , m) is the probability that all the m elements are safe after k periods of time. It is clear that the design of repairing procedure is successful if Π k (i 0 , m) is close to 1. The strategy is the following : choose r near 1 so that Π k (i 0 , m) ≈ 1. Note that if r = 1 (all the repaired elements are safe) then Z k = m and Π k (i 0 , m) = 1. We provide a qualitative result (see Theorem 2.1 below), which gives the exact rate of convergence of 1 -Π k (i 0 , m) in terms of 1 -r. This actually means that if the repairing procedure is efficient, i.e. 1 -r ≈ 0, then with a high level of confidence, the m elements have no default after k periods. Theorem 2.1 has an interesting interpretation initially developed in a biological setting and more precisely for anti-cancer treatments [START_REF] Bastogne | Tumor growth modeling based on cell and tumor lifespans[END_REF]. Let us introduce:

T := inf{i, Z i = m}. (1.7)
Since m is an absorbing state for (Z k ),

Π k (i 0 , m) = P (T ≤ k). (1.8)
Note that T is the minimal random number of periods necessary to repair and deliver a product without any default. As a result, for r sufficiently close to 1, the probability that the product is completely repaired before k periods is near 1.

The parameter E(m -Z k ) is a second pertinent qualitative indicator evaluating the quality of the procedure. Indeed, since Z k converges in distribution as r → 1 then lim

r→1 E(Z k ) = m. Using Theorem 2.
1, the rate of convergence of E(m -Z k ) in terms of 1 -r → 0 is given in Proposition 2.3.

1.4

Our second goal is to measure the efficiency of the repairing procedure applied to the production of a large number n of elements. We propose to take into account the performance of the whole production with the following control probability :

CP k,n := P (all the n products have no default at time k).

(1.9)

Suppose that the repairing and test phases are independent for any product and i 0 is the common initial state of each product, then

CP k,n = ( P (Z k = m) ) n = ( Π k (i 0 , m) ) n . (1.10)
It is clear that CP k,n ≈ 1 actually means that the ability of the system to repair defective products is good.

In the setting of cancer cells exposed to a treatment of radiotherapy, the analog of the parameter CP k,n is the tumor control probability (TCP), see [START_REF] Bastogne | Tumor growth modeling based on cell and tumor lifespans[END_REF], [START_REF] Bastogne | Multinomial model-based formulations of tcp and ntcp for radiotherapy treatment planning[END_REF] and Section 3. Suppose that q and k are fixed. The number n of products being large, CP k,n is close to zero. This is obviously a non desirable answer. To solve this issue, we propose to take r as an explicit function of n such that r ≈ 1 and the CP k,n is larger that a given threshold (for instance 95%). This goal is easy to perform (see Theorem 2.5), it is a direct consequence of Theorem 2.1.

Obviously, the requirement that all the n products are ready after k periods, is a very stringent condition. It seems interesting to define a weaker criterium based on the proportion of products presenting a default at time k :

CP * k,n,α := P ( N k n ≤ α ) (1.11)
where α ∈ [0, 1[ and N k denotes the number of non-damaged products at time k.

It is easily seen that N k = 0 means all the n elements have non-default. Consequently,

CP k,n ≤ CP * k,n,α
for any α and CP * k,n,α = CP k,n when α = 0. We prove in Theorem 2.8 that it is actually possible to tune the parameters r and k such that probability CP * k,n,α ≈ 1. The proposed index CP k,n and CP * k,n,α which measure the quality of the repairing procedure have to be compared. Suppose that θ 0 is a given level close to 1, for instance θ 0 = 95%. If 1 -r is chosen as a decreasing function of n of the type C n 1/k where C is a constant, then CP k,n ≈ θ 0 . As for the second index, the answer is very different. Suppose that α is small. It can be shown that if n is larger than n 1 (which only depends on α and θ 0 ) and 1 -r is proportional to

α 1/k , then CP * k,n,α ≈ θ 0 .
In particular, it is not necessary that r depends on n. A more complete result can be found in Theorem 2.8 and Proposition 2.9 below. As for the organization of the paper, the mathematical results are given in Section 2. In Section 3 we interpret our results presented in terms of reliability associated with the treatment of cancer cells by radiations. Finally the proofs of Theorems and Propositions are postponed in Section 4.

The main results

2.1

Our first goal is to study the behavior of r → Π k (i 0 , m) when r → 1. It is intuitive that more carefully a technician (or a machine) repairs the elements (i.e. more r is close to 1), more the probability of having m non-damaged elements after the test phase is high. It is clear from definitions that if r = 1 then Z k = m for any k ≥ 1 and r → Π k (i 0 , m) is continuous. Note that it has been proved in [START_REF] Bastogne | Tumor growth modeling based on cell and tumor lifespans[END_REF] that r → Π k (i 0 , m) is increasing. Therefore Z k converges in distribution to m as r → 1. The issue is to measure this convergence. It is actually possible determining the rate of decay of 1 -Π k (i 0 , m) in terms of 1 -r. This result is important since it will be applied to prove Proposition 2.3 and Theorem 2.5.

Theorem 2.1 Let 0 ≤ i 0 < m. Set ε := 1 -r. Then 1 -Π k (i 0 , m) ∼ (m -i 0 ) ( 1 + (m -1)q ) k-1 ε k , ε → 0. ( 2
R = R 0 + m ∑ l=1 R l ϵ l , (ϵ → 0).
Since Π = RQ we have :

Π = Π 0 + m ∑ l=1 Π l ϵ l , (ϵ → 0) (2.2)
where

Π l = R l Q. Therefore it is expected that Π k (i 0 , m) = 1 + aϵ + ϵo(ϵ), where lim ϵ→0 o(ϵ) = 0. Theorem 2.1 says
there are non-trivial cancelations.

2. One interesting property in (2.1) is the fact that the coefficient in front of ϵ k is explicitly given in terms of i 0 , m, q and k. This permits interpretations, see item below.

3. In [START_REF] Bastogne | Tumor growth modeling based on cell and tumor lifespans[END_REF], it has been observed that i → Π k (i, m) and r → Π k (i 0 , m) are increasing and q → Π k (i 0 , m) is decreasing. Note that the coefficient φ(i, r, q

) := -(m -i) ( 1 + (m -1)q ) k-1 (1 -r) k
inherits the same properties : i → φ(i, r, q) and r → φ(i, r, q) are increasing and q → φ(i, r, q) is decreasing. These properties corresponds to intuition.

Since Z k converges in distribution to m, as r goes to 1 and [START_REF] Blau | Uber einige Wirkungen von Strahlen[END_REF] It is interesting to introduce the time T r devoted to repair one element. It seems reasonable to suppose that r is increasing function of T r , for instance :

Z k takes its values in {0, 1, • • • , m}, then lim r→1 E(Z k ) = m. Theorem 2.1 permits to give the exact rate of convergence of m -E(Z k ). Proposition 2.3 Suppose that Z 0 = i 0 and k ≥ 1. Then m -E(Z k ) ∼ (m -i 0 ) ( 1 + (m -1)q ) k ε k , ε → 0. (2.3) Remark 2.
r := 1 - α T β r (2.4)
where α, β > 0.

Assume that the cost generated by repairing one element is proportional to T r . Therefore the mean cost for repairing at time k all the damaged elements is

E ( (m -Z k )γT r ) = γT r ( m -E(Z k ) )
, (where γ > 0).

Therefore, under (2.4), Proposition 2.3 implies that the above quantity is closed to zero if T r is large and kβ > 1. In practice, as soon as β is known, we have an effective way to choose k.

2.2

We now consider the production of n elements. We suppose that before to be delivered each element produced by the factory is repaired and tested as explained in details in Section 1.2 of Introduction. First we consider the index CP k,n defined by (1.10). It measures the global capability to produce non-defective elements after k repair-test phases. More CP k,n is a close to 1, more the maintenance may be considered as successful. To reach this objective, the parameter r is taken as an increasing function of n such that CP k,n ≈ 1. One second main result of this study is the following.

Theorem 2.5 Let θ 0 ∈]0, 1[. Let r such that 1 -r := ( -ln(θ 0 ) ( m -i 0 ) ( 1 + (m -1)q ) k-1 ) 1/k 1 n 1/k (2.5) Then lim n→∞ CP k,n = θ 0 .
Theorem 2.5 is a direct consequence of Theorem 2.1.

Remark 2.6

1. In practice θ 0 is often chosen to be equal to 95%.

Let us give a practical consequence of Theorem 2.5. Following the analysis developed in Remark 2.4, the mean cost of the quality control applied to the n elements is proportional to

ρ := n ( m -E(Z k ) )
. Using (2.5) and Proposition 2.3, we have

ρ ≈ n(m -i 0 ) ( 1 + (m -1)q ) k (1 -r) k ≈ -ln(θ 0 ) ( 1 + (m -1)q ) .
Consequently ρ does not depend on n. Moreover, as expected, the function θ 0 → ρ (resp. the functions m → ρ and q → ρ) is (resp. are) decreasing (resp. increasing).

3. In the spirit of Remark 2.2, let us introduce :

Ψ(i, q, n, k, θ 0 ) := - ( -ln(θ 0 ) ( m -i) ( 1 + (m -1)q ) k-1 ) 1/k 1 n 1/k
It is immediate to see that Ψ(i, q, n, k, θ 0 ) is an increasing (resp. decreasing) function of either q or n (resp. i). Moreover it is decreasing (resp. increasing) with respect to

k if n is large i.e. n > -ln(θ 0 ) ( 1 + (m -1)q ) m -i (resp.

otherwise). It is clear that r inherits the same behavior and its variations correspond to what is expected.

We now study the second parameter CP * k,n,α introduced in subsection 1.4 of the Introduction. Recall that CP * k,n,α measures efficiency of the whole system and CP * k,n,α near 1 means that the system operates successfully. To get CP * k,n,α ≈ 1 or equivalently 1 -CP * k,n,α ≈ 0, first we determine in Proposition 2.7 below an upper bound of 1 -CP * k,n,α in terms of 1 -Π k (i 0 , m). Second combining this result with Theorem 2.1 permits to prove that, under some conditions, the parameter CP * k,n,α is near 1 (see Theorem 2.8 below).

Proposition 2.7 Let 0 ≤ i 0 < m and α ∈]0, 1[. Suppose that Π k (i 0 , m) > 1 -α. Then 1 -CP * k,n,α ≤ { ( 1 -Π k (i 0 , m) α ) α ( Π k (i 0 , m) 1 -α ) 1-α } n , ∀ n ≥ 1. (2.6)
Recall that we have already observed that if r ≈ 1, then Π k (i 0 , m) is close to 1. Therefore if r is in the neighborhood of 1, then inequality (2.6) implies CP * k,n,α ≈ 1.

Theorem 2.8 Let 0 ≤ i 0 < m, α and θ 0 ∈]0, 1[. Suppose

1 -(1 -θ 0 ) 1/n ≤ α(1 -α) α 2 .
(2.7)

1. If we have

1 -Π k (i 0 , m) ≤ α - √ 2α [ 1 -(1 -θ 0 ) 1/n ] (1 -α) α (2.8) then CP * k,n,α ≥ θ 0 . 2. Suppose that α ≈ 0. If r is taken so that 1 -r ≃    1 (m -i 0 ) ( 1 + (m -1)q ) k-1 ( α - √ 2α [ 1 -(1 -θ 0 ) 1/n ] (1 -α) α )    1/k
(2.9)

then CP * k,n,α ≽ θ 0 .
Let us give a sufficient and easy condition under which relation (2.7) and (2.9) holds resp. Before enouncing the result let us introduce a useful notation :

n 0 := 1 + ⌊ - 2 ln(1 -θ 0 ) α(1 -α) α ⌋ (2.10)
where ⌊a⌋ stands for the integer part of a.

Proposition 2.9 Let 0 ≤ i 0 < m, α and θ 0 ∈]0, 1[.

1.

If n ≥ n 0 , then condition (2.7) is satisfied.

2. Suppose that α ≈ 0 and n ≥ 4n 0 and r is given as

1 -r = { α 2(m -i 0 ) ( 1 + (m -1)q ) k-1 } 1/k . (2.11) then CP * k,n,α ≥ θ 0 .
Remark 2. [START_REF] Lea | Actions of Radiations on Living Cells[END_REF] We would like to compare the two methods leading to CP k,n and CP * k,n,α close to θ 0 respectively. As for the first one, Theorem 2.5 says that if 1 -r is a decreasing function of n (namely (2.5) holds) then the goal is achieved, i.e. CP k,n ≈ θ 0 . We emphasize that the objective CP * k,n,α ≈ θ 0 may be obtained under a weaker condition. This result is expected since CP * k,n,α is bigger than CP k,n . Suppose that assumptions of Proposition 2.9 are satisfied. Relation (2.11) tells us that it is not necessary to take r depending on n (as for the previous case), it suffices that 1 -r is proportional to α1/k . That shows that for n large this condition is weaker than the first one.

Interpretation in terms of cancer cells and tumor

3.1

The goal of this Section is to interpret the main results given in Section 2 in the setting of an anticancer treatment by radiotherapy. The usual treatment planning in radiotherapy consists in applying a radiation dose fraction everyday k to the tumor. Let us begin by briefly recalling the model introduced in [START_REF] Bastogne | Multinomial model-based formulations of tcp and ntcp for radiotherapy treatment planning[END_REF] to take into account effects of the treatment on a cell and a tumor. At the level of a cancer cell, it is supposed : 2. Each target may be made inactive after the application of a fraction dose of radiation with a probability q 0 .

3. The cell death occurs when all the m targets are deactivated.

4. Between two consecutive doses, if the cell is still alive, then an inactive target can be repaired with probability r 0 . All the targets behave independently.

The aim of radiotherapy is to deliver enough radiation to the tumor to control it without irradiating normal tissue to a dose leading to serious complications (morbidity). Since radiation delivery at a precise point of cells is generally described as a random variable, the effects of the radioactive treatments on cancer and healthy cells are characterized by two probabilities: (i) the tumor Control Probability (T CP ) and (ii) the Normal Tissue Complication Probability (N T CP ) [START_REF] Zaider | Tumour control probability: a formulation applicable to any temporal protocol of dose delivery[END_REF][START_REF] Dawson | Derivation of the tumour control probability (tcp) from a cell cycle model[END_REF][START_REF] Gay | A free program for calculating eud-based ntcp and tcp in external beam radiotherapy[END_REF]. T CP k is defined as the probability that no cancer cell remains in a tumor after applying k dose fractions of radiation. N V CP k,v , is defined as the probability that a complication appears in a normal tissue. Its expression depends on the tissue architecture. Niemierko and Goitein [START_REF] Niemierko | Calculation of normal tissue complication probability and dosevolume histogram reduction schemes for tissues with a critical element architecture[END_REF] proposed a division of normal tissues into three different architectures: serial architecture (e.g. nerves or spinal cord), parallel architecture (e.g. kidney, liver or lung) and graduated response (e.g. skin or mucous membranes). The optimum choice of radiation dose delivery technique in the treatment of a given tumor has to maximize the TCP so that at the same time the NTCP must be lower than an acceptable level (see for details [START_REF] Bastogne | Tumor growth modeling based on cell and tumor lifespans[END_REF]). In clinical radiotherapy a typical choice is T CP ≥ 0.5 and N T CP ≤ 0.05 [START_REF] Suntharaligam | Radiation oncology physics : a handbook for teachers and students, chapter Basic Radiobiology[END_REF]. Let Z k be the (random) number of damaged target(s) in a cell at day k, i.e. after application of k fraction dose. It is clear that this model is the analog of the one given in the Introduction and related to reliability of engineering systems, with r := q 0 , q := r 0 .

(3.1)

In practice, the probability q 0 should be an increasing function of the applied fraction dose u 0 . In the linear-quadratric model of Target Theory (see Section 2.3 in [START_REF] Bastogne | Multinomial model-based formulations of tcp and ntcp for radiotherapy treatment planning[END_REF] for details) it is supposed that

q 0 = ( 1 -e -γu0-δu 2 0 ) 1/m (3.2)
where γ > 0 and δ ≥ 0 are given parameters scaled to be adapted to the range of the possible values of u 0 . In particular if u 0 = 0, then q 0 = 0, this actually means that there is no effect coming from radiation. However, as for contrary effect, it is important to note that if u 0 is large then q 0 is close to 1. In other words, there exists a natural way to force q 0 to be near 1.

3.2 A tumor is a population of n cells. We suppose that all the cells have the same independent behavior given above. In particular, the parameters q 0 and r 0 are common for all the cancer cells. The efficiency of the treatment applied to a tumor is measured by the Tumor Control Probability. This quantity is the probability that all the cancer cells are killed after application of k radiation dose fractions. Due to previous assumptions of independent behaviors of constitutive cells of the tumor and (1.9)

T CP k,n := CP k,n = ( P (Z k = m) ) m (3.3)
where CP k,n is defined with parameters q and r given by (3.1). Suppose that the tumor has a large number n of cancer cells and the parameters q 0 and r 0 are fixed. Then, relation (3.3) implies that T CP k,n goes to 0 since P (Z k = m) < 1 and n is large. This obviously means that the treatment is not efficient at all. One way to remove this undesirable conclusion is to strengthen the radiation, i.e. to choose u 0 large so that q given by (3.2) is close to 1. This corresponds exactly to the problematic introduced in the above subsection 1.4. As a result, the approach developed in Section 2 may be applied. Following subsection 1.4, let us introduce the second interesting parameter which also permits to measure efficiency of the treatment :

T CP * k,n,α := CP * k,n,α = P ( N t n ≤ α ) (3.4)
where N t n is the fraction of malign cells which have not been destroyed by the treatment. Recall that T CP k,n ≤ T CP * k,n and the goal is to have either T CP k,n or T CP * k,n close to 1. Here, this objective can be achieved choosing q 0 ≈ 1 or equivalently u 0 large.

3.3

It is interesting to interpret the main results given in Section 2 and their consequences on the radiotherapy response model. We keep notation given in Section 2.

Interpretation of Proposition 2.3 and Remark 2.4

Relation (2.3) can be written as :

m -E(Z k ) ∼ (m -i 0 ) ( 1 + (m -1)r 0 ) k (1 -q 0 ) k , q 0 → 1. (3.5)
where

i 0 ∈ {0, 1, • • • , m -1} is the initial state of the cell, Z k its state at time k. Using moreover (3.2) we get m -E(Z k ) ∼ (m -i 0 ) ( 1 + (m -1)r 0 ) k [ 1 - ( 1 -e -γu0-δu 2 0 ) 1/m ] k , u 0 → ∞. ( 3.6)
In particular relation (3.6) provides a quantitative way to choose the dose fraction u 0 large so that the mean of Z k is close to 1.

Interpretation of Theorem 2.5 and Remark 2.6

T CP k,n converges to a given real number θ 0 as soon as

1 -q 0 := ( -ln(θ 0 ) ( m -i 0 ) ( 1 + (m -1)r 0 ) k-1 ) 1/k 1 n 1/k . ( 3.7)
This formula can be applied in practice once we have an estimate of the number of cancer cells. Indeed, recall that q 0 is expressed via the dose fraction u 0 by (3.2). Therefore u 0 and k can be determined so that the treatment has a good chance to be successful, i.e. the probability to kill the whole tumor equals .95% for instance.

The comments related to the variation of the coefficient ψ(i, r 0 , n, k, θ 0 ) given in item 3 of Remark 2.6 remain valid in the biological context.

Interpretation of Theorem 2.8 and Proposition 2.9

We now focus on the second indicator T CP * k,n,α and the goal is again to force T CP * k,n,α ≈ 1. Let us start with a small α, for instance α = .001. Suppose that the size of the tumor before the beginning of the treatment is important. If u 0 is chosen such that

γu 0 + δu 2 0 = -ln   1 -   1 - { α 2(m -i 0 ) ( 1 + (m -1)r 0 ) k-1 } 1/k   m   (3.8) then T CP * k,n,α ≈ θ 0 . Introducing ψ 0 := 1 2(m -i 0 ) ( 1 + (m -1)r 0 ) k-1 .
and using

[ 1 - { α ψ 0 } 1/k ] m = 1 - m ψ 1/k 0 α 1/k + o ( α 1/k ) , α → 0
we finally obtain :

γu 0 + δu 2 0 ≈ - 1 k ln(α). (3.9)
Note that u 0 implicitly depends on θ 0 and α since it is supposed that the initial number of cancer cells constituting the tumor is larger than the threshold n 0 given by (2.10). Moreover (3.9) corresponds to intuition : the smaller α is, the larger u 0 has to be.

Developing (1 -ϵ) j-i we get

R(i, j) = j-i ∑ r=0 ( m -i j -i ) ( j -i r ) (-1) r ϵ m+r-j .
Setting l := m + r -j leads to (4.4) where

R l (i, j) := (-1) l+j-m ( m -i j -i ) ( j -i l + j -m ) 1 {m-j≤l≤m-i} . Since : ( m -i j -i ) ( j -i l + j -m ) = ( m -i l ) ( l m -j ) the result follows. Lemma 4.3 1. R 0 Q = R 0 .
2. For any 1 ≤ l ≤ m, we have :

R l 0 = R 0 , R 0 R l = 0, R 0 R l Q = 0.
Proof. The proof is straightforward and is left to the reader.

Definition 4.4 A multi-index i is an element of ∪ k≥1 {1, 2, • • • , m} k . For any multi-index i = (i 1 , • • • , i k ) we set : |i| := k, ∥i∥ := i 1 + • • • + i k , Π i := Π i1 × • • • × Π i k .
where Π l := R l Q.

Lemma 4. [START_REF] Crowther | Some considerations relative to the action of x-rays on tissue cells[END_REF] The matrix Π k admits the following asymptotic expansion :

Π k = R 0 + ∑ |i|=k ϵ ∥i∥ Π i + ∑ |i|≤k ϵ ∥i∥ Π i R 0 . ( 4.5)
Proof. Since Π = RQ, then relation (4.4) implies

Π k = ( m ∑ l=0 ϵ l Π l ) k = ∑ i1,••• ,i k Π i1 × • • • × Π i k ϵ i1+•••+i k where i 1 , • • • , i k belong to {0, 1, • • • , m}. We now consider three different cases. a) If i 1 = i 2 = • • • = i k = 0 then the corresponding term is R k 0 = R 0 . b) If each i j belongs to {1, 2, • • • , m}, set i := (i 1 , • • • , i k ). Then |i| = k and ∥i∥ = i 1 + • • • + i k ≥ k.
This gives the second term in the right hand-side of (4.5). c) The third and last case is the one where (i 1 , • • • , i k ) ̸ = (0, • • • , 0) and there exists j such that i j = 0. There are two possibilities :

1. (i 1 , i 2 , • • • , i k ) = (i 1 , i 2 , • • • , i k ′ , i k ′ +1 , • • • , i k ) where i k ′ = 0 and i k ′ +1 ≥ 1, • • • , i k ≥ 1, 2. (i 1 , i 2 , • • • , i k ) = (i 1 , i 2 , • • • , i k ′ , i k ′ +1 , • • • , i k ) where i k ′ ≥ 1 and i k ′ +1 = • • • = i k = 0.
In the first case,

Π i1 × • • • × Π i k = Π i1 × • • • × Π i k ′ Π i k ′ +1 × • • • × Π i k .
The above product vanishes since Lemma 4.3 implies that

Π i k ′ Π i k ′ +1 = R 0 QR i k ′ +1 Q = R 0 R i k ′ +1 Q = 0.
As for the second case, we have :

Π i k ′ +1 × • • • × Π i k = ( R 0 Q ) × • • • × ( R 0 Q ) = R 0 × • • • × R 0 = R 0 .
If there exists 1 ≤ j < k ′ such that i j = 0, proceeding similarly leads to

Π i1 × • • • × Π i k ′ = 0. Finally it can be supposed that (i 1 , • • • , i k ′ ) is a multi-index with length k ′ .
Remark 4.6 Lemma 4.5 is crucial in the proof of Theorem 2.1. Indeed, we will see in Lemma 4.8 below that the terms in the second sum of the right hand-side of (4.5) do not contribute to the equivalent of 1 -Π k (i 0 , m), as ϵ → 0.

Lemma 4.7

We have the following identities :

m ∑ j=0 R l (i, j) = 0, ∀ i ∈ {0, 1, • • • , m}, l ≥ 1.
Proof. 1) If i = m then according to the definition of the matrix R l , we have R l (m, j) = 0 for any 0 ≤ j ≤ m. From now on suppose that i < m. Using Lemma 4.2 we get

m ∑ j=0 R l (i, j) = ( m -i l ) m ∑ j=0 ( l m -j ) (-1) l+j-m 1 {i≤j, m-j≤l≤m-i} .
Setting s = j -m + l we obtain :

m ∑ j=0 R l (i, j) = ( m -i l ) 1 {l≤m-i} l ∑ s=0 ( l l -s ) (-1) s . Since l ∑ s=0 ( l l -s ) ( -1
) s = (1 -1) l = 0, then Lemma 4.7 follows.

Lemma 4.8

Let i be a multi-index, then :

Π i R 0 (k, m) = 0, for any k ∈ {0, 1, • • • , m}. (4.6)
Proof. We prove (4.6), reasoning by induction on the length n of i. 1) Let us start with n = 1. Then i = i ≥ 1 and

Π i R 0 (k, m) = Π i R 0 (k, m) = R i QR 0 (k, m).
We claim that Lemma 4.1 may be applied. If i < m, then using (1.5), we get

m ∑ j=0 Q(i, j) = i ∑ j=0 ( i j ) q i-j (1 -q) j = (1 + 1 -q) i = 1. ( 4.7)
Since Q(m, j) = 1 {m=j} , it is clear that (4.7) holds when i = m.

As a result we may apply Lemma 4.1 with

B 1 := R i , B 2 = Q and a = 1 : R i QR 0 (k, m) = R i R 0 (k, m).
Lemma 4.7 implies that we can apply twice Lemma 4.1 with B 1 := I, B 2 = R i and a = 0 :

R i R 0 (k, m) = 0. Consequently Π i R 0 (k, m) = 0.
2) It remains to prove n → n + 1. Let i be a multi-index of length n + 1. It can be written as i = (j, i ′ ) where i ′ is a multi-index with length n and j ≥ 1. Consequently, for any i 0 ∈ {1, 2, • • • , m}, we have :

Π i R 0 (k, m) = Π j Π i ′ R 0 (k, m) = m ∑ s=0 Π j (k, s) ( Π i ′ R 0 ) (s, m) = 0 since i ′ being a multi-index with length n, then ( Π i ′ R 0 ) (s, m) = 0 .
Lemma 4.9 For any 0 ≤ i < m we have

Π 1 (i, k) =    -(m -i) if k = m (m -i) ( m -1 k ) q m-1-k (1 -q) k , if 0 ≤ k < m (4.8)
Moreover Π s 1 (m, j) = 0 for any 0 ≤ j ≤ m and s ≥ 1.

Proof. Suppose that 0 ≤ i < m and 0 ≤ j ≤ m. Using the definition of R 1 (cf Lemma 4.2) we easily prove that R

1 (i, j) = 0 if j < m -1, R 1 (i, m -1) = m -i and R 1 (i, m) = -(m -i).
As a result :

Π 1 (i, k) = R 1 Q(i, k) = m ∑ j=0 R 1 (i, j)Q(j, k) = (m -i) ( Q(m -1, k) -Q(m, k)
) .

Using (1.5), we get (4.8).

It remains to study the case i = m. Du to the fact that R 1 (m, j) = 0 for any j, we deduce :

Π 1 (m, k) = R 1 Q(m, k) = m ∑ j=0 R 1 (m, j)Q(j, k) = 0, ∀ k ∈ {0, 1, • • • , m}.
Let s ≥ 1 and 0 ≤ j ≤ m. We have :

Π s+1 1 (m, j) = m ∑ i=0 Π 1 (m, i)Π s 1 (i, j) = 0.
Lemma 4.10 For any 0 ≤ i < m we have

Π k 1 (i, m) = -(m -i) ( 1 + (m -1)q ) k-1 . (4.9)
Proof. 1) When k = 1, identity (4.9) is a direct consequence of Lemma 4.9.

Let us deal with k = 2. Let 0 ≤ i < m. Using Lemma 4.9 we get :

Π 2 1 (i, m) = m-1 ∑ j=0 Π 1 (i, j)Π 1 (j, m) = - m-1 ∑ j=0 (m -i) ( m -1 j ) q m-1-j (1 -q) j (m -j).
It is convenient to introduce

S(x) := m-1 ∑ j=0 (m -j) ( m -1 j ) x m-1-j , x ∈ R.
Let y ∈ [0, 1]. Taylor formula leads to :

g(y) = g(0) + g ′ (0)y + g ′′ (z) 2 y 2 = 1 - α 1 -α (1 -z) α-2 ( 1 + α 1 -α z ) -α-1 y 2 2
where 0 < z < y. Since, α -2 < 0 and 1 -z belongs to ]0, 1[, then (1 -z) α-2 > 1.

Similarly :

z < 1 ⇒ 1 + α 1 -α z ≤ 1 + α 1 -α = 1 1 -α . Therefore ( 1 + α 1 -α z ) -α-1 ≥ (1 -α) α+1
and

g(y) ≤ 1 - α 1 -α (1 -α) α+1 y 2 2 = 1 -α(1 -α) α y 2 2 .
Proof of Theorem 2.8 Suppose that (2.7) holds, i.e.

1 -

(1 -θ 0 ) 1/n ≤ α(1 -α) α 2 . ( 4
y ≥ √ 2 [ 1 -(1 -θ 0 ) 1/n ] α(1 -α) α ⇔ 1 -Π k (i 0 , m) ≤ α - √ 2α [ 1 -(1 -θ 0 ) 1/n ] (1 -α) α
Consequently, (2.8) implies that

y ≥ √ 2 [ 1 -(1 -θ 0 ) 1/n ] α(1 -α) α . ( 4

.20)

According to (4.16),(4.17), (4.18) and Proposition 2.7 it follows that :

( 1 -CP * k,n,α ) 1/n ≤ f ( Π k (i 0 , m) ) = g(y) ≤ 1 - α(1 -α) α 2 y 2 .
Using (4.20), we have successively :

1 - α(1 -α) α 2 y 2 ≤ ( 1 -θ 0 ) 1/n 1 -CP * k,n,α ≤ 1 -θ 0 ⇒ CP * k,n,α ≥ θ 0 .
2) Suppose that α ≈ 0. Here we suppose that (2.9) holds, i.e.

1 -r ≃    1 (m -i 0 ) ( 1 + (m -1)q ) k-1 ( α - √ 2α [ 1 -(1 -θ 0 ) 1/n ] (1 -α) α )    1/k . Since α- √ 2α [ 1 -(1 -θ 0 ) 1/n ]
(1 -α) α ≤ α, the former relation implies that r is in the vicinity of 1. Theorem 2.1 may be applied :

1 -Π k (i 0 , m) ≃ (m -i 0 ) ( 1 + (m -1)q ) k-1 (1 -r) k ≃ α - √ 2α [ 1 -(1 -θ 0 ) 1/n ] (1 -α) α .
This actually means that (2.8) holds.

Step one of Theorem 2.8 gives CP * k,n,α ≽ θ 0 .

4.6 Proof of Proposition 2.9

1) Recall that 1 -e -x ≤ x for any positive x. Therefore 2) Suppose that n ≥ 4n 0 . We claim that We thus get :

α - √ 2α [ 1 -(1 -θ 0 ) 1/n ] (1 -α) α ≥ α 2 . ( 4
√ 2α [ 1 -(1 -θ 0 ) 1/n ] (1 -α) α ≤ α 2 .
A straightforward calculation shows that (4.22) follows.

Recall that r has been defined by (2.11), i.e.

1 -r = { α 2(m -i 0 ) ( 1 + (m -1)q ) k-1 } 1/k .
Since α ≈ 0, then 1 -r → 0. Consequently, Theorem 2.1 may be applied, using moreover the previous identity we obtain :

1 -Π k (i 0 , m) ∼ (m -i 0 ) ( 1 + (m -1)q ) k-1 (1 -r) k = α 2 .
As a result, we deduce from (4.22) that (2.8) holds. According to item 1 of Theorem 2.8 we can conclude that CP * k,n,α ≥ θ 0 .

Figure 1 :

 1 Figure 1: Reliability test schedule of a manufacturing system based on a loop composed of a checking phase and a maintenance step. k is the number of maintenance cycles in the loop.

.19) 1 )

 1 We claim that under(2.8) then CP * k,n,α ≥ θ 0 where θ 0 ∈]0, 1[. We set Π k (i 0 , m) = 1 -α + αy where y ∈]0, 1]. Obviously, y = Π k (i 0 , m) + α -1 α y ≤ 1 and

1 - 1 n

 11 (1 -θ 0 ) 1/n = 1 -e ln(1-θ0) ≤ -ln(1 -θ 0 ) n . (4.21) Since n 0 := 1+ ⌊ -2 ln(1 -θ 0 ) α(1 -α) α ⌋, we deduce immediately that if n ≥ n 0 then condition (2.7) is satisfied.

  .22) Indeed (4.21), n ≥ 4n 0 and n 0 ≥ -2 ln(1 -θ 0 ) α(1 -α) α yield to : 1 -(1 -θ 0 ) 1/n ≤ -ln(1 -θ 0 ) n ≤ -ln(1 -θ 0 ) 4n 0 ≤ α(1 -α) α 8 .

a cell has m vital targets.

Proofs

Proof of Theorem 2.1

We keep notation introduced in Sections 1 and 2.We adopt the convention that for any matrix the first line and the first column are labeled 0. We begin with a preliminary result which will be used several times. Suppose :

Then

Proof. According to the definition of R 0 and (4.1), we have :

In Lemma 4.2 below, we give the behavior of R as a function of

when ϵ → 0. This allows us to determine the asymptotic expansion of Π in Lemma 4.5.

Lemma 4.2

The matrix R admits the following asymptotic expansion

where for

Proof. Using (1.4) and (4.3) we have :

where 0 ≤ i ≤ j ≤ m and R(i, j) = 0 otherwise. Indeed,

Let us calculate S(x) :

We easily deduce that

3) We now prove (4.9) by induction on k. We have already proved it for k = 1 and k = 2. Let us detail k → (k + 1). Observe that

Let 0 ≤ i < m. We know from Lemma 4.9 that Π k 1 (m, m) = 0, consequently

, therefore Lemmas 4.5 and 4.8 imply :

It is clear that

Proof of Proposition 2.3

First we focus on the calculation of E(Z k ). Although it is not possible to give an explicit value of the expection of Z k we are however able to provide a recursive relation between E(Z k+1 ) and E(Z k ).

Lemma 4.11 Suppose that Z 0 = i 0 and k ≥ 0. Then

Proof. Denote Z ′ the number of non-damaged elements after the repairing procedure applied to one element. According to the scheme (1.2), conditionally on Z ′ = j < m, the distribution of Z k+1 is binomial with parameters j and 1 -q. Therefore

Consequently :

We deduce from (1.1) that conditionally on

). That implies

Proposition 2.3 will proved by induction on k. First we study the case k = 1 in Lemma 4.12 below.

Lemma 4.12 Suppose that Z 0 = i 0 , then

Proof. Using Lemma 4.11 with k = 0 we obtain :

, we deduce :

Proof of Proposition 2.3

Let us prove by induction on k :

Obviously the case k = 1 corresponds to Lemma 4.12. Suppose that (4.11) holds and prove

Using Lemma 4.11 and (4.11) we get :

Starting with the classical inequality :

Using (4.11) we conclude that R(ϵ) = o(ϵ k+1 ) and

This implies (4.12).

Proof of Theorem 2.5

Let 0 ≤ i 0 < m and k ≥ 1. Suppose that identity (2.5) holds. Therefore r is a function of n and moreover ϵ := 1 -r goes to 0 as n → ∞. According to the definition (1.10) of CP k,n , we have :

Theorem 2.1 and (2.5) imply that :

Proof of Proposition 2.7

Let ( (Z

) i≥1 a collection of independent Markov chains with common transition matrix Π given by (1.3) and having the same initial point i 0 . Let us introduce :

Recall that from (1.11) we have :

) .

The technique developed below is classical in the theory of large deviations. In our specific setting (2.6) is a particular case of Cramér's theorem (see Section 2.2 in [START_REF] Dembo | Large deviations techniques and applications[END_REF] and more specifically Exercise 2.2.23).

Note that

} for any λ > 0. Using the Markov inequality and the fact that r.v.'s ξ i are iid, we get :

where a(λ) := -λα + ln

and p := P (ξ 1 = 0) = P (Z

. Using classical analysis it can be proved that a(λ) ≥ a(λ * ) for any λ > 0 where

It is clear that (2.6) is a direct consequence of (4.13) (with λ = λ * ) and (4.15).

Proof of Theorem 2.8

Let θ 0 ∈]0, 1[ and α in ]0, 1[. Let us introduce the function :

Obviously, Proof. 1) Taking the second derivative, we get :

where h(y) = α(α -1)

This identity implies that g ′′ (y) < 0 for all y in ]0, 1[ and therefore g is concave over [0, 1].

2) Since g ′ (y) = (1 -y) α-1

] then g ′ (0) = 0