A Lachapelle
email: lachapelle@ceremade.dauphine.fr

F Santambrogio

On the strategic use of risk and undesirable goods in multidimensional screening

Keywords: principal-agent problem, adverse selection, calculus of variations, nonlinear pricing, random contracts

A monopolist sells goods with possibly a characteristic consumers dislike (for instance, he sells random goods to risk averse agents), which does not affect the production costs. We investigate the question whether using undesirable goods is profitable to the seller. We prove that in general this may be the case, depending on the correlation between agents types and aversion. This is due to screening effects that outperform this aversion. We analyze, in a continuous framework, both 1D and multidimensional cases.

Introduction

Principal-Agent problems and the theory of multidimensional nonlinear pricing (see for instance [START_REF] Bolton | Contract Theory[END_REF] for an overview of the topic) have known rapid advances at the end of the XX th century with the works of Armstrong [START_REF] Armstrong | Multiproduct nonlinear pricing[END_REF] and Rochet & Choné [START_REF] Choné | Ironing, Sweeping and Multidimensional Screening[END_REF]. This framework considers a partially informed monopolist (or principal), selling multiproducts to a continuum of agents having multidimensional types. This is the classical adverse selection case of informational asymmetries, involving multidimensional screening. The starting point of the paper, and its motivation, is the following very natural and fundamental question. Suppose that a risk neutral monopolist may "randomize" the goods, facing risk averse agents (so that random contracts are undesirable from their viewpoint), then what will happen in the end? Should the seller actually offer randomized goods (i.e. lotteries) or not? Will adverse selection force him to produce those goods preferred by the consumers, or will it be convenient to use this extra possibility in order to make profit? Although these questions are not novel, only few answers can be found in the literature. A simple case where risk aversion is introduced in a scalar problem involving two agents can be found in [START_REF] Maskin | Monopoly with incomplete information[END_REF] or [START_REF] Bolton | Contract Theory[END_REF]. In such cases, it is shown that the monopolist benefits by randomizing contracts as soon as the difference between both agent's aversion is large enough. Another work on n risk averse agents facing a not informed monopolist has been developed by Maskin & Riley in [START_REF] Maskin | Optimal auctions with risk averse buyers[END_REF], where they focus on auctions on a single and indivisible good. We could also mention the work of Arnott & Stiglitz (see [START_REF] Arnott | Randomization with asymmetric information[END_REF]) for a comparison between random and deterministic insurance contracts that a monopolist offers -again -to a finite number of buyers with unknown preferences. More recently, and in a more general framework, Carlier proved in [START_REF] Carlier | Problèmes de calcul des variations issus de la théorie des contrats[END_REF] an existence result of optimal stochastic contracts in a relaxed version of what we call the Rochet-Choné problem (i.e. the multidimensional screening problem). However, the author does not discuss the profitability of introducing lotteries.

It seems, according to this literature, that this question has been analyzed so far only in discrete models (a finite number of different agents), and often with zero production costs. The novelty of our paper is that we find the same kind of answers in a continuous framework à la Rochet & Choné, thanks to a calculus of variations formulation.

Before entering the mathematical details of the principal's optimization problem, we discuss some assumptions in a rather qualitative way. The agents will be supposed to be described by a type θ and the goods the monopolist may produce by a quality x. Both θ and x belong to a same Euclidean space R d (or R d + ) and the utility that an agent of type θ derives from consuming a good x is supposed to be the scalar product θ • x. Each coordinate θ i can be viewed as a marginal utility to the i th quality of the good. Besides the utility function, the other element of the model is the production cost C(x). This cost is supposed to be convex in x. We consider, as in [START_REF] Choné | Ironing, Sweeping and Multidimensional Screening[END_REF], the purely quadratic production cost, i.e. C(x) = 1 2 |x| 2 . Both these choices (linear utility and quadratic production costs) are the most classical and the most significant ones, since they already capture all the qualitative features of the solution of more general cases.

We then consider the fact that goods qualities may include a random part, and we will describe it by introducing a new variable y ∈ R + standing for instance for their variance (or for the standard deviation). More precisely, we think of a good as a random variable X where x = E[X] stands for the mean and

y = E[|X -x| 2 ] = E[X 2 ] -|E[X]| 2 is
a scalar variance. We suppose that consumer agents have still a linear utility function of the form θ • x + αy, where α < 0 is a new parameter in the description of agents' type, standing for their risk aversion (see Section 2 for a more detailed discussion). Notice that we are assuming that the utility is linearly decreasing in y, which means that y must be chosen as a parametrization of the randomness of the good such that agents are linearly averse with respect to it. If by chance in modeling a situation we find that agents are linear with respect to the standard deviation instead of the variance, we should call y the standard deviation. But this is, up to now, just a matter of naming variables.

The role played by y would be more important if looking at the production cost. Due to the random components in this extended model, we will call it C R (x, y). We have in mind at least two toy models for the dependence of C R on y.

• There are some cases where C R depends increasingly on y, say C R (x, y) = |x| 2 + λy for λ > 0, or |x| 2 + λy 2 . This is for instance the case where the monopolist sells black boxes containing an unknown collection of goods, randomly sorted according to a certain distribution (think of closed boxes containing a dozen of wine bottles, randomly selected by the seller). If the cost for each of them is quadratic, the cost for producing one of such boxes is proportional to E[X 2 ] and not to |E[X]| 2 . This means that the cost is of the

form |E[X]| 2 + (E[X 2 ] -|E[X]| 2 ) = |x| 2 + y.
• There are also cases where C R depends decreasingly on y, say C R (x, y) = |x| 2 + λy for λ < 0. This happens when the producer needs special machines for guaranteeing more precise goods qualities, and he does not really chooses to introduce variances, even if he can pay for reducing it. For the model to make sense, the monopolist must be able to manage at the same time, and at his will, the production of possibly different levels of variance, exactly as he needs to be able to produce different levels of quality. This is surely reasonable in large-scale firms, using different machines in different factories.

In this paper we will mainly focus on the case λ = 0, i.e. C R (x, y) = |x| 2 . This case can be obtained as a combination of the two previous effects, but more important reasons suggest to consider it as key case. First of all, we already pointed out that we use a parametrization of the variable y so as to guarantee that the utility functions of agents are linear. If C R really depends on y, when specifying a dependence (for instance C R (x, y) = |x| 2 + λy) we would be fixing a very specific case where both production costs and utilities are linear in the same parametrization, while there is no modeling reason to suppose that. Second, the case λ > 0 is not a new problem : it is just the Rochet-Choné Problem (see [START_REF] Choné | Ironing, Sweeping and Multidimensional Screening[END_REF]) with an extra dimension. The only point which was not investigated in [START_REF] Choné | Ironing, Sweeping and Multidimensional Screening[END_REF] is the fact that we look at consumers in R d + ×R -and goods in R d + × R + , i.e. we explicitly account for a characteristic that is negatively perceived by agents, but the formalism would be the same. Third, we will see that the main results of the paper consist in examples and conditions where the optimal strategy for the risk neutral principal is exactly to offer goods with nonzero values of y (i.e. lotteries). This makes the case λ < 0 less interesting. Indeed, it is clear that whenever there exist examples where the seller offers costless lotteries, the profit he gets from lotteries increases if λ < 0. Moreover, it is clear from the choice λ = 0 that the label "variance" does not play any special role any more, in the sense that this last characteristic may be thought of as any extra feature of the good that consumers dislike. It is easy to re-think at the whole problem in these terms : if the monopolist may produce goods with qualities (x, y), and the value of y does not affect the production cost, but negatively affects the utilities of the consumers, will he use this "undesirable" quality y? It is widely accepted that examples of this kind have been used in real life by some companies. It was for instance the case of third-class seats in some trains, which were just more unconfortable than second-class ones without saving space for the railway company. It seems a common strategy of the firms to create low-quality goods just to attract consumers that would otherwise not enter the market. For these reasons, we will from now on refer to the quality y as the undesirable quality and to α as the aversion parameter (and not specifically risk aversion). And, for the sake of generality, we will drop the random terminology (risk, lotteries, . . . ). Actually, thanks to these last considerations, the problem we study and the (partial) answers we give may switch very easily from the random interpretation to the more abstract and general one of an undefined "undesirable" quality. This is one of the main features of this very simplified model, which does not want to compete with more sophisticated ones that have been investigated concerning Principal-Agent problems involving random contracts. It is the case of the problems studied by Carlier in [START_REF] Carlier | Problèmes de calcul des variations issus de la théorie des contrats[END_REF] (where the author focuses on existence issues for more general utility functions and "randomness", i.e. not only considering mean and variance of stochastic contracts), as well as the work of Carlier, Ekeland & Touzi ( [START_REF] Carlier | Optimal derivatives design for meanvariance agents under adverse selection[END_REF]), in which, nevertheless, the principal cannot control the level of the undesirable quality y.

In section 2 we will first discuss more precisely the maximization problem for the principal, which we have not yet introduced. We will then jump to the extended problem (i.e. adding the components y and α) and present some considerations on it. Section 3 and 4 will be devoted to the description of some classes of examples where the solution of the extended problem really uses y (hence it does not coincide with the solution of the classical problem), in the scalar and the multidimensional cases, respectively. The paper will be closed by a conclusive section with comments on the result we got.

Classical and extended problems

We start by recalling the classical problem studied by Rochet and Choné, including its powerful calculus of variations formulation.

General setting Let us see more in details the very famous problem studied by Mussa & Rosen (see [START_REF] Mussa | Monopoly and product quality[END_REF]) in the one dimensional case and by Armstrong and Rochet & Choné for the case of multidimensional types (see respectively [START_REF] Armstrong | Multiproduct nonlinear pricing[END_REF][START_REF] Choné | Ironing, Sweeping and Multidimensional Screening[END_REF]). It is a nonlinear pricing model in a monopolistic situation with adverse selection. The principal (also the monopolist, the firm, the seller...) knows the global distribution f of agents types on a convex open bounded domain Ω ⊂ R d + . This domain will be supposed for simplicity to be a rectangle and the distribution of agents over it will be described through its density, i.e. a positive function on Ω with integral equal to 1, that we still denote by f . In this way we consider a continuum of buyers. Any agent with type θ ∈ Ω derives the utility u θ (x, t) := θ • xt from consuming a good of quality x ∈ Q ⊂ R d + at price t. The monopolist has a strictly convex cost C : x ∈ Q → C(x) for producing a good of quality x.

The optimization of the monopolist's strategy is performed on possible pricing of the goods he sells : the monopolist may choose tariffs t : Q → R + , taking into account that any agent θ buys a good x(θ) selected so as to maximize θ • xt(x); she also has the possibility of not buying anything if for every good x she gets θ • xt(x) < 0; globally each agent buys either zero or one good (or, equivalently, she buys exactly one good but there also exists a zero good 0 ∈ Q whose production cost is 0, which gives utility 0 to everybody, and such that the principal may not charge for it, i.e. t(0) = 0 is prescribed). In this way the profit for the firm is

Ω t(x(θ)) -C(x(θ)) f (θ)dθ.
Thus, the principal's optimization problem has another optimization problem embedded: the agent's optimization problem.

A more standard viewpoint in incentive theory, which by the way avoids the problem of possible multiple goods x maximizing the utility (net of price) of the agent, is the following : agents buy either one or zero good and, in the other hand, the firm directly chooses a family of contracts (x(θ), t(θ)) θ∈Ω (i.e. produces x(θ) and sells it at price t(θ) to the agent of type θ), and has to take into account some incentive and participative constraints.

The first one is known as the Incentive Compatibility (IC) constraint and basically ensures that agent of type θ will really choose x(θ) at price t(θ) when she looks at the product line (x(θ ′ ), t(θ ′ )) θ ′ ∈Ω . The second one is the Individual Rationality constraint (IR), modeling the fact that agents do not buy if they get a utility lower than what they would get if they bought nothing, i.e. they have a reservation utility equal to zero. We define the two constraints as:

(IC) θ ∈ argmax{θ • x(θ ′ ) -t(θ ′ ), θ ′ ∈ Ω} (IR) θ • x(θ) -t(θ) ≥ 0.
The classical problem then reads as

sup Ω t(θ) -C(x(θ)) f (θ)dθ, (x, t) satisfies (IC)

and (IR) .

Notice that it is well-known that this problem admits a unique solution.

Calculus of variations A very powerful tool to study the above problem is the following idea : introduce (as in [START_REF] Armstrong | Multiproduct nonlinear pricing[END_REF][START_REF] Choné | Ironing, Sweeping and Multidimensional Screening[END_REF]) the potential (or surplus function)

v(θ) := max{θ • x(θ ′ ) -t(θ ′ ), θ ′ ∈ Ω} and define K D = {v : Ω → R + , v is convex, and ∇v ∈ R d + }.
Then the problem may be re-stated in terms of v, in the following way :

(P D ) sup v∈K D J D (v) = Ω θ.∇v(θ) -v(θ) -C(∇v(θ)) f (θ)dθ.
It is a calculus of variations formulation subject to convexity constraints. This problem, after being introduced in [START_REF] Choné | Ironing, Sweeping and Multidimensional Screening[END_REF], has been studied by Carlier and Lachand-Robert in [START_REF] Carlier | Régularité des solutions d'un problème variationnel sous contrainte de convexité[END_REF] as far as regularity results are concerned. Surprisingly, we will really need their C 1 regularity theorem. Notice that this formulation is very useful to compute the solution, either explictely (see section 3) or numerically ( [START_REF] Choné | Ironing, Sweeping and Multidimensional Screening[END_REF][START_REF] Ekeland | An algorithm for computing solutions of variational problems with global convexity constraints[END_REF]). From now on we refer to the solution of (P D ) as the classical solution, in opposition to the solution of the extended problem that we introduce in details in the next paragraph.

The extended problem We now add a component to good qualities and to agent types. They are denoted by y and α, respectively, and we impose y ≥ 0 and α ≤ 0. In particular, α is non-positive (we will impose that it belongs to a bounded interval of R -, α ∈ A := [-κ, 0]) and it models agent's aversion to the new (one-dimensional) characteristic of the good quality.

In the sequel, we refer to the component α as the aversion parameter and to the corresponding one-dimensional characteristic of the good as the undesirable quality or, more simply, we call it y. In the present work we look at Von Neumann Morgensten (VNM) agents and the case where the y component of the good quality does not lead to any production cost.

In other words, we consider utilities of the type u θ,α (x, y, t) = θ • x + αyt, i.e. the utility that any agent of type (θ, α) ∈ Ω × A derives from buying (x, y) ∈ R d+1 + at price t. As before, the distribution h of agent types on Ω × A is given (mathematically, it is an absolutely continuous measure on Ω × A, but we will as well denote by h its density). When integrating according to this density we will equivalently write h(θ, α)dθdα or dh(θ, α). We will always tacitly assume that the θ-marginal of h will be the same density f that we consider in the classical problem (i.e. A h(θ, α)dα = f (θ) for every θ ∈ Ω).

As we mentioned before, we consider the case where the production cost (denoted in this case by C R ) does not depend on y, and we take C R (x, y) = C(x). Introducing w(θ, α)

:= max{θ • x(θ ′ ) + αy(α ′ ) -t(θ ′ ), (θ, α ′ ) ∈ Ω × A and the set K R = {w : Ω × A → R + , w is convex, and ∇w ∈ R d+1 + },
we can directly jump to the variational formulation of the problem, which is

(P R ) sup v∈K R J R (w) = Ω×A (θ, α).∇w(θ, α) -w(θ, α) -C(∂ θ w(θ, α)) dh(θ, α).
Notice that, due to the fact that the C-part of the cost (the strictly convex one, which is given by the production cost) does not depend on ∂ α w, this problem does not coincide with a multidimensional version of Mussa-Rosen's problem (i.e. with a Rochet-Choné problem; see Section 4). We will not enter in this paper the problem of existence of a maximizer for (P R ), which is a little bit trickier than that for (P D ) or for Rochet-Choné, due to the lack of bounds on ∂ α w. Anyway, a maximizer does exist, see for instance [START_REF] Carlier | Calculus of variations with convexity constraint[END_REF].

The following -straightforward -proposition states that (P R ) is really an extension of (P D ).

Proposition 2.1. If the solution w of (P R ) is such that ∂ α w = 0, then it coincides with the solution of (P D ) and sup (P R ) = sup (P D ). In the other hand, if h is a measure concentrated on {α = 0}, then (P R ) = (P D ).

In the sequel we analyse some easy features of the maximization problem (P R ), focusing on the question whether the principal really uses positive values of the new quality y. The two following results show that adverse selection will "control" the use of y, even if not preventing it completely. We will interprete later these results in terms of discrimination (or screening) and correlation. Proposition 2.2. If w is a solution of (P R ) such that ∂ α w ≥ c, for some nonnegative constant c, then c = 0.

Proof. Let w ∈ K R be a solution of (P R ) such that ∂ α w ≥ c. We are only concerned with perturbations of the type

w ε (θ, α) = w(θ, α) -ε(α + κ), for 0 ≤ ε ≤ c. Since ∂ θ w ε = ∂ θ w and ∂ α w ε = ∂ α w -ε,
it is clear that w ε is convex and increasing. The constant κ has been inserted in order to guarantee the other admissibility constraint w ε ≥ 0. To see that it is satisfied it is sufficient to check it in the corner of Ω × A where all the coordinates are minimal. In this point we have both w = 0 and c(α + κ) = 0, so that w ε ∈ K R . Then, we easily get

J R (w ε ) -J R (w) = εκ ≥ 0, ∀ε ∈ [0, c].
This contradicts the optimality of w except when c = 0 (so that ε = 0).

An interesting consequence of proposition 2.2 is that the monopolist does not offer products involving nonzero y when he cannot use them to discriminate (which is the case if the value of y is the same for every good he sells), i.e. to screen the agents. If one restricted the offer of the monopolist adding the constraint y = constant, then the solution would be the classical one.

Moreover, the same proposition proves that the minimal value of y among the goods that are really present in the market must be 0. The principal will not increase the value of y for every good just for the sake of consumers' unhappiness, it will not be profitable to him! There is another general case where we can prove the principal does not offer goods with undesirable qualities. Indeed, the following proposition says that he does not use the undesirable quality as soon as the distribution of types θ and the one of aversions α are independent. Proposition 2.3. If h(θ, α) = f (θ)g(α), then the classical solution is the unique minimizer of (P R ).

Proof. Assume that w is a solution of (P R ). Define v(θ) = Ω×A w(θ, α)g(α)dα. Since v does not depend on α, we get J R (v) = J D (v). On the other hand, using

(v ′ (θ)) 2 ≤ ∂ θ w(θ, α) 2 g(α)dα (Jensen's inequality), -α∂ α w ≥ 0 (since α ≤ 0 and ∂ α w ≥ 0), θv ′ (θ)f (θ)g(α)dθdα = θ ∂ θ w(θ, α)f (θ)g(α)dθdα (by linearity),
one has:

J R (v) = Ω×A - 1 2 (v ′ (θ)) 2 + θv ′ (θ) -v(θ) f (θ)g(α)dθdα ≥ Ω×A - 1 2 (∂ θ w(θ, α)) 2 +(θ, α)•∇w(θ, α)-w(θ, α) dh(θ, α) = J R (w).
Hence, the maximization may be restricted to those convex functions that only depend on θ, and, among them, the maximizer is v. Also remark that the inequality -α∂ α w ≥ 0 is strict for functions truly depending on α which proves that v is the unique solution of (P R ).

In the next sections, our study consists in describing situations where of the extended problem is not the classical one. We look successively at the scalar and the multidimensional cases.

3 When the principal should use the undesirable quality : the 1D case

In this part we provide an example where the seller offers goods with various undesirable qualities y, hence using it for screening purpose. In particular, he uses nonzero values of y.

Since Problem (P R ) is pretty much involved in the general setting, and we are convinced that it is not easy to classify general cases where the solution of the extended problem is or is not the classical one, we first build an example in the simple case d = 1. Notice that this means that the original problem (i.e. without y and α) is one-dimensional, and it actually becomes twodimensional after adding these new variables. The simplest example we can provide is the case of a quadratic production cost and a uniform distribution of type θ on segment Ω = [θ min , θ max ]. The next proposition is well-known and describes the explicit classical solution in a class of cases, which includes this one. We denote by F the primitive of f , i.e. F (θ) = θ θ min f (s)ds. Proposition 3.1. Suppose that f is a positive regular density on [θ min , θ max ], with θmax θ min f (θ)dθ = 1, that the production cost is quadratic, i.e. C(x) = 1 2 x 2 . Then, the solution v of problem (P D ) is characterized by

v(θ min ) = 0, v ′ (θ) = θ - 1 -F (θ) f (θ) , provided the inequalities θ min f (θ min ) ≥ 1 and f ′ (θ)(1 -F (θ)) ≥ -2f (θ) 2 hold on [θ min , θ max ].
In particular, for the uniform density on [START_REF] Armstrong | Multiproduct nonlinear pricing[END_REF][START_REF] Arnott | Randomization with asymmetric information[END_REF], these assumptions are satisfied and v(θ) = 1 2 θ 2 -2θ + 3 2 . Although the computation of v is standard, we provide it for the sake of completeness.

Proof. The key point for solving (P D ) is the integration by parts

θmax θ min vf = θmax θ min v ′ (1 -F ),
where we used F (θ max ) = 1 and v(θ min ) = 0 (we may assume inf v = 0 and, v being increasing, the minimum si realized at θ min ). We set u = v ′ and we have:

J D (v) = J(u) = θmax θ min 1 2 u(θ) 2 -θu(θ) f (θ) + u(θ) 1 -F (θ) dθ.
Then we easily get that the minimal value would be realized by choosing:

u(θ) = θ - 1 -F (θ) f (θ) , θ a.e.,
provided this choice is possible. This means that u(θ) must be the derivative of a function v satisfying the constraints. Since v must be increasing and convex, then u must be positive and increasing. It is sufficient to compute its derivative, to require that it is positive on [θ min , θ max ], and to impose u(θ min ) ≥ 0. This last condition gives θ min -1 f (θ min ) ≥ 0, and the condition on

u ′ gives 0 ≤ u ′ (θ) = 1 - -f (θ) 2 + (1 -F (θ))f ′ (θ) f (θ) 2 = 2 + (1 -F (θ))f ′ (θ) f (θ) 2 ,
which is known as the hazard condition. The assumptions on F exactly ensure that these conditions are satisfied. For the particular case of the uniform density on [START_REF] Armstrong | Multiproduct nonlinear pricing[END_REF][START_REF] Arnott | Randomization with asymmetric information[END_REF], just replace F (θ) = θ -1 and f (θ) = 1 and deduce from this equation that v ′ (θ) = 2(θ -1). We then obtain the expression of v above and finally check that v ∈ K D .

Note that it is a simple case without bunch (or equivalently perfect screening i.e. the monopolist offers different contracts to different types).

Let us now look at the extended problem (P R ) for the quadratic production cost. Our goal is to find some cases where the solution of (P R ) is not the classical solution v of (P D ). The strategy starts by introducing, for all ε > 0, the perturbations

w ε (θ, α) = v(θ + εl(α)), ∀(θ, α) ∈ Ω × A,
where l(α) = (α + a) + = max(α + a, 0) and a ∈ (-κ, 0) is given. If necessary, one can extend v outside Ω by affine extensions before defining w ε . Notice that w 0 = v and that, due to the convexity and monotonicity properties of l, w ε is convex and increasing as well. Proposition 3.2. Let v be the classical solution of (P D ) for a certain density of types f . Assume that v is C 2 and that there exists B ⊂ Ω such that

B f (θ)dθ > 0 and ∀θ ∈ B, v ′′ (θ) (θ -v ′ (θ)) -v ′ (θ) > 0.
Then there exists a joint distribution h on Ω×A, with θ-marginal equal to f , such that J R (w ε ) > J R (v) for small ε > 0. In particular sup (P R ) < sup (P D ) and the solution of

(P R ) is not w(θ, α) = v(θ). An explicit example is obtained if one takes Ω = [1, 2], κ = 1, a ∈ (0, 1), f = 1 and h(θ, α) = 1 a 1 (θ,α)∈R 1 + 1 1-a 1 (θ,α)∈R 2 ,
where R 1 and R 2 are the two regions of Fig. 1.

Since J R (v) = J D (v) = sup (P D ) and using proposition 2.1 we immediately deduce the next corollary. The previous result states that the principal takes advantage of the costless and undesirable quality to discriminate. Indeed, screening considerations and the possibility to ease the incentive constraints outperform the desire to provide every agent with fully desirable goods (y = 0). Before proving proposition 3.2, we would like to comment a bit such particular case. The coupling between the classical type θ and the aversion characteristic α is concentrated on the two regions R 1 (low types have low aversion parameters) and R 2 (high types have higher aversion parameters) represented in Fig. 1. From the proof we will infer that in the present example the principal derives profit from offering undesirable goods to those agents having both low types θ and low aversion α (i.e. in R 1 ). Hence, in the present case, a correlation aspect appears: the example we provide has the same structure than the one obtained in [START_REF] Maskin | Monopoly with incomplete information[END_REF] and [START_REF] Bolton | Contract Theory[END_REF] in a discrete framework, where optimal contracts are random when the correlation between -α and θ is positive. We now prove proposition 3.2.

Proof. First we compute the derivatives of w ε :

∂ θ w ε = v ′ and ∂ α w ε = εl ′ v ′ ,
where l is differentiable (i.e. almost everywhere). We already noticed the admissibility of w ε , which follows from the convexity and monotonicity of l and v. Now, define j(ε) := J R (w ε ) and compute the first variation

j ′ (0)= Ω×A v ′′ (θ) (θ -v ′ (θ)) -v ′ (θ) l(α) + αv ′ (θ)l ′ (α) dh(θ, α). (3.1)
Assume that there exists a set B satisfying the same assumptions as in the proposition. Then we can choose a density h of the following form :

h = 1 ã f (θ)1 [-ã,0] + 1 κ-a f (θ)1 [-κ,-a] (α)
, for some small value ã ∈ (0, a]. In this case it is clear that the integral in (3.1) will be immediately restricted to the set B × (-ã, 0], where the first term is positive. Since the goal is to get j ′ (0) > 0, we only need to choose ã small enough. In this case, the term involving αv ′ (θ)l ′ (α) will be negligible (since the density h tends to concentrate around the set {α = 0}), and

lim ã→0 v ′′ (θ) (θ -v ′ (θ)) -v ′ (θ) l(α)dh = l(0) (v ′′ (θ) (θ -v ′ (θ)) f (θ)dθ > 0.
Thus, we finally get j ′ (0) > 0.

To prove the second statement of the proposition, we simply compute the variation replacing v (who is of course C 2 in the uniform case) and h by their expression

j ′ (0) = Ω×A (3 -2θ)l(α) + α(θ -1)l ′ (α) dh(θ, α) = a 2 16 > 0.
Notice that in this particular case, B = [1, 3 2 ] and ã = a. Remark 3.4. We emphasize that, in the case where the production cost in the extended problem has the form C R (x, y) = C(x) + λy for some λ > 0 one can anyway build an explicit example from the uniform density f . More precisely, for all λ > 0, there exists M = M λ > 0 and a density h on [START_REF] Armstrong | Multiproduct nonlinear pricing[END_REF][START_REF] Arnott | Randomization with asymmetric information[END_REF] × [-M, 0] providing a non-classical solution. This density may be chosen of the usual form, taking 0

< a < M, with λ < 1 2 a, h(θ, α) = 1 a 1 (θ,α)∈B×[-a,0] + 1 M -a 1 (θ,α)∈Ω\B×[-M,-a] .
Notice anyway that for a fixed set of aversion parameter A this is possible for some λ only. Remark 3.5. Note that the assumption that v should be C 2 is not really needed to get such a result. Indeed we could suppose that v is only C 1 and apply a similar technique as the one we use for proving proposition 4.1 in the next section.

The multidimensional case

In this section we look at the multidimensional case, which has been mainly studied in [START_REF] Armstrong | Multiproduct nonlinear pricing[END_REF][START_REF] Choné | Ironing, Sweeping and Multidimensional Screening[END_REF]. Since less information on the solution in this case is known than in the 1D one, we will try to use the minimal regularity and the minimal structure results on the optimal v that have been investigated. Proposition 4.1. Suppose that f is a density on a convex domain Ω ⊂ R d such that the optimal function v in the Rochet-Choné problem (P D ) is a C 1 convex function, with Ω 0 := {v = 0} having non-empty interior. Then there exists a density h on Ω × A such that the solution of (P R ) is not w(θ, α) = v(θ).

For the sake of this proposition, we need two things: C 1 regularity of v and the fact that there exists a flat zone {v = 0}. General C 1 regularity results are proven in [START_REF] Carlier | Régularité des solutions d'un problème variationnel sous contrainte de convexité[END_REF] for the solutions of a wide class of variational problems under convexity constraints, including the one of Rochet & Choné. As far as the structure of v is concerned, [START_REF] Choné | Ironing, Sweeping and Multidimensional Screening[END_REF] proves that if d = 2, v has the shape described in Fig. 2 when f is an exponential density, and check numerically the same behavior for the uniform density. For numerical computations showing this kind of behavior, we also refer to [START_REF] Ekeland | An algorithm for computing solutions of variational problems with global convexity constraints[END_REF]. Anyway the crucial feature of the classical solution we need is that the region Ω 0 := {v = 0} has non-empty interior. This statement means that the seller excludes a set of consumers. Actually, it has been proven by Armstrong in [START_REF] Armstrong | Multiproduct nonlinear pricing[END_REF] that for any dimension d ≥ 2, the set Ω 0 has positive measure as soon as Ω is strictly convex; yet, the same proof easily generalizes to other classes domains, for instance to any straight rectangle (i.e. a rectangle whose edges are parallel to the coordinate axes). We thus easily derive the following corollary.

Corollary 4.2. For every regular density f of types on a domain Ω which is either strictly convex or a straight rectangle in R d , for d ≥ 2, there exists a density h on Ω×A such that the optimal contract contains undesirable goods.

We will concentrate the proof of proposition 4.1 in the case of a straight rectangle in 2D, which is the one that has been most investigated so far, but the reader may easily notice that the procedure may be extended to higher dimensions and to different shapes of Ω, since we only perform local variations.

Proof. The strategy for looking at the multidimensional case is similar to the 1D one. We suppose that the solution is a function v depending on θ ). The function w ε is convex and increasing with respect to each variable, provided that v is convex and increasing. We notice that ∇ θ w ε (θ, α) = ∇v(θ+εel(α)), ∂ α w ε = ε∇v(θ+εel(α))•el ′ (α), for a.e. (θ, α).

θ 1 θ 2 Ω 0 v = 0 Ω B v = v(θ 1 +θ 2 ) Ω 1 v strictly convex
Hence we can write J R (w ε ) and we have

J R (w ε ) = θ•∇v(z θ,α ) + εαl ′ (α)∇v(z θ,α )•e -v(z θ,α ) - |∇v| 2 2 (z θ,α ) dh(θ, α) = (-k(z θ,α ) -ε(l(α) -αl ′ (α))∇v(z θ,α ) • e) dh(θ, α)
where z θ,α denotes θ + εel(α) and k(z

) := 1 2 |∇v(z)| 2 -z • ∇v(z) + v(z).

Simplifying assumptions on h

Since our goal is just to provide sufficient conditions to show an example where v is not optimal (and w ε gives a better result than v), we can make some simplifying assumptions: we suppose that h(θ, α) = I K (θ)I S (α) for those pairs (θ, α) such that l(α) = 0, where K ⊂ Ω is a non-negligible given subset of Ω and S ⊂ [-a/2, 0] is an interval where l is positive. This is not so restrictive in terms of the density f , provided f is bounded from below by a positive constant on K: it is actually always possible to express f as the θ-marginal of a density h of this kind, if we adjust the rest of the density with the behavior of h on {l = 0}. This reduces the integral above to

- S dα K (k(z θ,α ) + ε(l(α) -αl ′ (α))∇v(z θ,α ) • e) dθ.
Simplifying assumptions on K Now suppose that K is actually a rectangle with two sides which are parallel to e and two orthogonal sides denoted by ∂K + and ∂K -(see Fig. 3). If we denote by d 1 the one-dimensional integration on ∂K + and ∂K -, it is not difficult to check that

d dε ε=0 K k(θ + εel(α))dθ = d dε ε=0 K+εel(α) k(θ)dθ = l(α) ∂K + k(θ) d 1 θ - ∂K - k(θ) d 1 θ
provided that k is continuous (which is the case if v ∈ C 1 ). Moreover, we have

K ∇v • e dθ = ∂K + v d 1 θ - ∂K - v d 1 θ.
Now, suppose that v = ∇v = 0 on ∂K -. This simplifies the computation and yields to

d dε ε=0 J R (w ε ) = S l(α)dα ∂K + k d 1 θ + S (l(α) -αl ′ (α)) dα ∂K + v d 1 θ . Since v is nonnegative and l(α) -αl ′ (α) = (α + a) + -α = a ≤ 2l(α) on S,
we can estimate the last term through

S (l(α) -αl ′ (α)) dα ∂K + v d 1 θ ≤ 2 S l(α)dα ∂K + v d 1 θ , and get d dε ε=0 J R (w ε ) ≥ - S l(α)dα ∂K + (k + 2v) d 1 θ . e θ 1 θ 2 Ω 0 = {v = 0} {v > 0} θ θ W V K Figure 3: Structure of the 2D example
Choice of a point θ It is now sufficient to prove that one can choose K so that k+2v = 1 2 |∇v(θ)| 2θ • ∇v(θ) + 3v(θ) is negative on ∂K + in order to have a positive derivative of J R (w ε ).

Here the key point is to suppose that v has a flat part Ω 0 with non-empty interior. Thanks to the convexity of v, this set is convex and we suppose that it includes a neighborhood of 0. Let θ be a point on the boundary of Ω 0 which is in the interior of Ω. Then, thanks to the monotonicity properties of v, the vector e is an outwards vector for the set Ω 0 , in the sense that v(θ + se) > 0 for s > 0. The vector 1 2 ∇v(θ)θ will have negative coordinates, and the same is true in a neighborhood V ⊂ Ω of θ. If -c is a constant bounding from above the components of 1 2 ∇v(θ)θ for θ ∈ V , we can write

1 2 ∇v(θ) -θ • ∇v(θ) ≤ -ce • ∇v(θ),
and the term e • ∇v(θ) is the derivative of v in the direction e. Choice of ∂K + so that k + 2v < 0 Now, let us consider the values of k + 2v on a line segment stemming from θ in the direction e, i.e. we consider (k + 2v)(θ + se) for s > 0 such that θ + se ∈ V . Let us suppose that this quantity stays always positive : this means that we have the inequality

-c d dt v(θ+se) + 2v(θ+se) ≥ 1 2 ∇v(θ + se) -(θ+se) •∇v(θ+se)+2v(θ+se) ≥ 0 , which implies that d dt v(θ + se) ≤ 2 c v(θ + se).
By Gronwall Lemma, this inequality implies that v(θ + se) ≤ e 2 c s v(θ) = 0, which is in contradiction with the fact that θ is on the boundary of Ω 0 , i.e. that v(θ + se) > 0.

This implies that there exists a point θ of the form θ + se in the neigborhood V where k + 2v is negative. The same will be true for all the points in a neighborhood W of θ. In this way, if one chooses a rectangle K such that one of the two sides orthogonal to e is completely included in Ω 0 and the other is included in W , the derivative of J R (w ε ) is positive.

Notice that in this proof, we use again profitable (from the seller viewpoint) contracts where goods with positive y are sold to rather low type agents (i.e. the lowest type agents among those who enter the market). Actually, as one can see from Fig. 3, the agents who are concerned are those who are close to the region Ω 0 , i.e. to the consumers who stay out of the market.

Summary and conclusions

In this paper, we have been concerned with providing partial progress on the possible profitability of random (or undesirable) contracts, for a monopolist facing risk averse (or averse) buyers with multidimensional types. In comparison with the existing literature which is focused on the random interpretation, we stressed the fact that, in this simplified model, randomness is just an instance of undesirable qualities. This is why we feel free to switch from one terminology to the other.

We first defined an extended version of the multidimensional screening problem mainly studied by Rochet & Choné. We discussed several interpretations of the addition of an undesirable quality and we highlighted situations where the seller either can or cannot improve his profit by offering undesirable goods (or lotteries).

The optimal policy is not random in the case where agents type θ and aversion α are independent. The correlation between the distributions of agents type and aversion seems to be at the heart of the answer. We built profitable examples involving the undesirable quality (risk, say) when the correlation is of the following form: low-type consumers are less risk averse. This can be seen both in the 1D and in the multidimensional examples we provided, where we exploited the presence of poorly risk averse agents close to the "out-of-the-market" zone Ω 0 . What we get is consistent with the results presented in [START_REF] Bolton | Contract Theory[END_REF][START_REF] Maskin | Monopoly with incomplete information[END_REF]. Yet, the novelty of this paper is to analyze these same questions with a continuum of agents, in a Rochet & Choné framework.

The non profitability conclusion is also true when the monopolist is added a constraint forcing him to offer constant risk, i.e. removing the possibility to screen the agents (or discriminate). We saw that, if the principal really uses random contracts (and he only does use them for pure screening purposes), then the randomness he proposes (i.e. the undesirable characteristic y) will take different values, including 0. Anyway, by providing examples in both the scalar and the multidimensional cases, we emphasized the role of the undesirable quality to screen the agents, and we have seen that the screening considerations sometimes outperform aversion.

1 Figure 1 :

 11 Figure 1: Structure of the 1D example

Figure 2 : 1 √ 2 , 1 √ 2

 21212 Figure 2: The shape of the classical solution for the exponential distribution

Acknowledgements

The authors wish to thank G. Carlier for suggesting the topic of this article as well as for very helpful discussions and I. Ekeland for his active interest in the problem and interesting suggestions. The second author acknowledges the support of the ANR project "Evolution and variational methods applied to economics and finance".


Matthias ; Langhorst
email: m.langhorst@zeiss.de

Zeiss Carl

Gmbh Microimaging

Joerg ; Schaffer

Bernhard ; Goetze

Microimaging

Matthias F Langhorst

Matthias F Langhorst

Carl Zeiss

Structure brings clarity

Keywords: Optical sectioning, structured illumination, aperture correlation, confocal microscopy, fluorescence microscopy Optical sectioning, structured illumination, aperture correlation, confocal microscopy, fluorescence microscopy, superresolution microscopy

Biological samples are three dimensional and therefore optical sectioning is mandatory for microscopic images to precisely show the localization or function of structures within biological samples. Today, researchers can choose from a variety of methods to obtain optical sections. This article focuses on structured illumination microscopy, which is a group of techniques utilizing a combination of optics and mathematics to obtain optical sections: A structure is imaged onto the sample by optical means and the additional information thereby encoded in the image is used to calculate an optical section from several acquired images.

Different methods of structured illumination microscopy (mainly grid projection and aperture correlation) are discussed from a practical point of view, concentrating on advantages, limitations and future prospects of these techniques and their use in cell biology. Structured illumination can also be used to obtain superresolution information if structures of higher frequency are projected onto the sample. This promising approach to superresolution microscopy is also briefly discussed from a user's perspective.

Most biological samples are three dimensional, therefore, microscopy has to precisely relay not only the localization of a given structure in the lateral dimensions (x, y), but also in depth (z). Fluorescence images from a conventional widefield microscope show not only the plane of focus, but structures below and above the focal plane also add to the image. If those structures are close to the focal plane, they can be perceived as defocused, or, if they are far away from the focal plane, they add homogenous intensity to the background, thereby decreasing the signal-to-background-ratio (SBR). Therefore, widefield microscopy can only yield clearly defined results when used on thin samples (sample thickness close to the focal depth of the objective used). With increasing sample thickness, out-of-focus information will render the resulting image impossible to interpret. Thicker samples have to be imaged using a microscopic technique offering optical sectioning, i.e. a way to reject out-of-focus light and maintain only in-focus information in the final image. Three dimensional rendering and precise localization in 3D can only be applied to optically sectioned images. Furthermore, advanced imaging modalities like spectral detection or fluorescence lifetime imaging also require optical sectioning for unambiguous interpretation.

Today, a variety of methods are available for optical sectioning microscopy [START_REF] Conchello | Optical sectioning microscopy[END_REF]. They can be grouped in three large categories: a) techniques based purely on optical means to prevent outof-focus light from being detected, b) techniques which rely purely on computational means to subsequently remove out-of-focus light from three dimensional widefield data sets, and c) techniques using optical means to encode additional information in the image which can then be used to greatly facilitate computation of an optical section from the raw data.

The most commonly used optical sectioning microscope is the point scanning confocal microscope. In these instruments an aperture ("pinhole") in a conjugate image plane is used to prevent out-of-focus light from reaching the detector. In confocal microscopes, the detector "sees" only photons from the focal plane, therefore, point-scanning confocal microscopes are ideally suited to generate high contrast optical sections even in the presence of a lot of out-offocus intensity, i.e. with thick and highly fluorescent samples [START_REF] Wright | Introduction to confocal microscopy[END_REF]. However, point-scanning is inherently slow and can only be sped up to a certain point until the pixel dwell time becomes so short that no useful image can be generated any more. Therefore, to speed up confocal image acquisition several approaches to parallel scanning of multiple points are now available like spinning disk or line scanning confocals [START_REF] Graf | Live cell spinning disk microscopy[END_REF][START_REF] Wolleschensky | High-speed confocal fluorescence imaging with a novel line scanning microscope[END_REF]. However, parallel scanning sacrifices a bit of sectioning quality due to out-of-focus light reaching the detector through e.g. neighbouring pinholes (pinhole crosstalk) for the gain in speed. Confocal microscopes normally use lasers for excitation, therefore, the choice of dyes is limited by the available laser lines. Tuneable ("white light") laser sources promise to also overcome this limitation. Deconvolution (DCV), on the other hand, imposes the least requirements on the hardware for image acquisition. Any motorized fluorescence microscope with a digital camera for image capture can be used. The optical section is calculated offline using algorithms based on either the properties of the imaging system (magnification, NA, medium etc.) to calculate a theoretical point-spread function (PSF) or on an empirical point spread function measured under conditions close to the actual experiment [START_REF] Wallace | A workingperson's guide to deconvolution in light microscopy[END_REF]. This method captures not only the infocus signal coming from the sample, but also all the out-of-focus information, therefore, DCV offers the best possible light efficiency not sacrificing any signal. Therefore, it is ideally suited for imaging of very sensitive, living samples. However, the quality of the final computed optical section depends on the quality of the PSF used (for review see [START_REF] Swedlow | Quantitative fluorescence microscopy and image deconvolution[END_REF]). For thicker samples, local variations within the sample (e.g. refractive index variations in different parts of a tissue) cannot be measured or modeled and, therefore, DCV will presumably introduce artifacts when used on thicker inhomogeneous samples. DCV always requires acquisition of a three dimensional data set (z-stack) at high sampling rates. Furthermore, computation time for large datasets can still be considerable. single sections and, therefore, acquisition of a z-stack is not mandatory. Furthermore, SIM computation is very robust and, therefore, less likely to induce structural artifacts in the final section. Furthermore, computation of the sectioned image is very fast, making online display of the sectioned data possible.

Structured Illumination for optical sectioning:

A variety of approaches to structured illumination have been published. They all rely on the same principle of operation: A known pattern is positioned / generated in a conjugate plane to the focal plane and projected onto the sample (Figure 1). In focus, the excitation intensity is modulated along the pattern and the emitted fluorescence will reproduce the pattern. Out-offocus, the amplitude of the projected pattern is decreasing according to the modulation transfer function [START_REF] Neil | Method of obtaining optical sectioning by using structured light in a conventional microscope[END_REF], thus, the image of the pattern will be blurred and the modulation of excitation intensity over the field of view is clearly reduced. Therefore, fluorescence emission from structures outside the focal plane is much less modulated. This allows discrimination of in-focus information from out-of-focus information by a simple comparison of several images with the pattern at different positions. Fluorescence from structures which lie outside the focal plane will not vary in intensity between the different images, while fluorescence from infocus structures will display a clear variation in intensity depending on the position of the excitation pattern. By comparing the intensities between the different images on a pixel by pixel basis a sectioned image can be quickly calculated (Figure 2). Different patterns can be used in structuring the excitation light. Laser speckles will provide a quasi random pattern [START_REF] Ventalon | Quasi-confocal fluorescence sectioning with dynamic speckle illumination[END_REF], laser interference will produce a very well defined pattern [START_REF] Neil | Wide-field optically sectioning fluorescence microscopy with laser illumination[END_REF] or a multipoint mask can be used [START_REF] Benedetti | Method for the acquisition of images by confocal microscopy[END_REF]. Depending on the density of the pattern used, the number of images needed for reconstruction changes. Sparse patterns provide a high modulation contrast, but a higher number of images is needed to unambiguously reconstruct the sectioned image. The simplest pattern to use is a Ronchi ruling, i.e. a grid with a 1:1 mark/space ratio [START_REF] Neil | Method of obtaining optical sectioning by using structured light in a conventional microscope[END_REF]. With such a regular 1:1 grid, a minimum of three images per plane is necessary to calculate an optical section. Use of this approach in fluorescence microscopy was pioneered by Wilson and coworkors [START_REF] Neil | Method of obtaining optical sectioning by using structured light in a conventional microscope[END_REF] and by now commercial instruments are available on the market (ApoTome from Carl Zeiss, OptiGrid from Qioptiq).

For non-coherent illumination, one can approximate the axial response (FWHM) of such a grid projection SIM system by the following equation [START_REF] Neil | Method of obtaining optical sectioning by using structured light in a conventional microscope[END_REF]: The optical section thickness achieved by structured illumination is therefore determined by the grid frequency and the NA and magnification of the objective used. For optical sectioning, relatively low grid frequencies can be used. Similar to the optimal pinhole diameter in confocal microscopes, the optimal grid frequency for a given objective is a compromise between SNR of the final image and sectioning capability. For different objectives, different grid frequencies will be needed to achieve optimum sectioning. On the other hand, some samples may demand thicker sections to achieve a better SBR in the final images. So even with a given objective, a choice of grids may be useful to optimize imaging parameters according to the sample.

( )       - ⋅ × ⋅ = -
For the calculation of the optical section, different algorithms can be used (reviewed by [START_REF] Heintzmann | Structured illumination methods[END_REF]).

A very simple approach was proposed by Neil et al. [START_REF] Neil | Method of obtaining optical sectioning by using structured light in a conventional microscope[END_REF]. With three images of an equal phase shift of 2/3π, an optical section can be calculated on a pixel-per-pixel basis using the following equation:

( ) ( ) ( ) [ ] 2 1 2 3 2 2 3 1 2 2 1 I I I I I I I OS - + - + - =
This simple calculation will only result in an artifact-free image if the three images were taken with exactly the assumed phase shift. Any inaccuracy in grid movement will result in residual grid lines visible in the sectioned image. A more general approach to calculation of the optical section was proposed by Schaefer et al. [START_REF] Schaefer | Structured illumination microscopy: Artefact analysis and reduction utilizing a parameter optimization approach[END_REF] and later also by Tkaczyk et al [START_REF] Tkaczyk | High resolution, molecular-specific, reflectance imaging in optically dense tissue phantoms with structured illumination[END_REF]. Their approach has the advantage of working with arbitrary phase shifts. Furthermore, this general approach allows the use of more than three images. In this case, a least squares approach can be used to produce results with greater accuracy and improved SNR [START_REF] Schaefer | Structured illumination microscopy: Artefact analysis and reduction utilizing a parameter optimization approach[END_REF]. Schaefer et al. also considered several potential artifacts. Fluctuations in light source intensity will lead to intensity differences between the three images which are unrelated to the position of the grid. Similarly, bleaching will result in intensity losses which have to be taken into account during calculation. Continuous bleaching of the whole field of view will result in a gradual loss in intensity over time, which may be corrected by normalization of the raw data images by their mean intensity as suggested by Cole et al [START_REF] Cole | Time-domain whole-field fluorescence lifetime imaging with optical sectioning[END_REF]. Heterogenous bleaching will occur when different structures / dyes display different decay rates. As mentioned above, inexact grating shifts may result in residual grid lines in the sectioned image. Finally, all commercial instruments use rectangular grating patterns instead of the ideal sinusoidal ones, which also is a reason for artifacts in the final image. Schaefer and colleagues modeled all these sources of potential artifacts and included a parameter-optimization approach to achieve efficient artifact removal [START_REF] Schaefer | Structured illumination microscopy: Artefact analysis and reduction utilizing a parameter optimization approach[END_REF] which was incorporated into the Zeiss ApoTome software.

Today, many experiments in cell biology demand quantification of the final results. As the resulting optical section in SIM is based on calculations, the quantifiability of SIM images has to be scrutinized. Most suggested algorithms for calculating optical sections from SIM data include non-linear steps at some point. In many cases the influence of these non-linear steps is negligible, and the result will be linear with respect to the emitted light intensity [START_REF] Tkaczyk | High resolution, molecular-specific, reflectance imaging in optically dense tissue phantoms with structured illumination[END_REF][START_REF] Barlow | Quantization of widefield fluorescence images using structured illumination and image analysis software[END_REF]. However, when the signal is low, the non-linear steps will lead to severe deviations from a linear relationship between the emitted fluorescence and the intensity in the sectioned image.

Therefore, quantification of SIM images with low SNR (e.g. in situation with very strong outof-focus background) should be treated with caution [START_REF] Heintzmann | Structured illumination methods[END_REF].

The theoretical sectioning capabilities of a SIM system are slightly worse than a true confocal instrument [START_REF] Karadaglic | Image formation in structured illumination wide-field fluorescence microscopy[END_REF]. The practical performance of any SIM system is, however, limited by the signal-to-background ratio. Sectioning in a SIM system depends on sufficient grid contrast.

Strong of out-of-focus fluorescence which overwhelms a weak in-focus signal can lead to insufficient grid contrast, making reliable calculations of an optical section impossible. As a rule of thumb, the grid lines have to be clearly visible in the eyepieces for SIM sectioning to work. In thick, highly scattering samples, the grid contrast can be completely lost -then a true point scanning confocal might be the only instrument able to capture optical sections. This is closely linked to another problem of SIM -limited dynamic range of the final image. The raw data images contain both in-focus and out-of-focus information. During calculation of the optical section, the out-of-focus information is removed, and only the in-focus intensity remains in the final sectioned image. How much dynamic range this final sectioned image still offers depends again on the signal to background ratio and also on the dynamic range of the camera used. Only cameras with a high dynamic range will allow capturing sufficient in-focus information even in the presence of strong out-of-focus intensity. This can be improved by high dynamic range (HDR) imaging. In this technique several images with increasing exposure time are combined to calculate one HDR image with increased dynamic range including both very bright and very dim structures. HDR imaging can be combined with SIM acquisitions to increase the dynamic range of the final optical section (Figure 3).

The necessity to acquire at least three images per optical section imposes a clear limit on the time resolution of any SIM system. If living structures are observed, one must assure that no significant movement will occur in between the acquisition of the three images -otherwise artifacts are inevitable.

Some recent publications demonstrate ways to improve structured illumination systems

beyond what is possible at the moment. The use of digital micromirrors or spatial light modulators promise to improve the flexibility of SIM setups [START_REF] Fukano | Whole-field fluorescence microscope with digital micromirror device: Imaging of biological samples[END_REF]. Adaptive optics may be used to improve the performance on thick samples [START_REF] Debarre | Adaptive optics for structured illumination microscopy[END_REF]. Area detectors with "smart pixels" in combination with illumination structures continuously moving at a defined frequency may allow on-chip calculation of the sectioned image [START_REF] Mitic | Optical sectioning in wide-field microscopy obtained by dynamic structured light illumination and detection based on a smart pixel detector array[END_REF], which may allow the study of highly dynamic processes and/or may also offer a higher dynamic range in the final sectioned image, once such detectors become available at sensitivities sufficient for use in microscopy.

In summary, structured illumination microscopy provides an easy-to-use and most of all easyto-add tool for optical sectioning. The necessary modification to a standard widefield fluorescence microscope is very simple, as only a slider with the grid has to be added to the field stop position in the reflected light beam path. SIM systems work with normal white light illumination; therefore, the full range of dyes can be used. The low excitation intensities necessary for widefield illumination also lead to low bleaching rates, making SIM a suitable technique for sensitive samples. Acquisition of optical sections is relatively quick, as only three images per plane have to be acquired and computation time is negligible. A big advantage of SIM systems is that they capture both a complete widefield image and all information for an optical sectioned image simultaneously. This allows direct comparison of

Structured illumination for dynamic events:

To push structured illumination into the domain of live cell imaging is the promise of another technique, called aperture correlation. Again, this technique was pioneered by T. Wilson and coworkers [START_REF] Juskaitis | Efficient real-time confocal microscopy with white light sources[END_REF][START_REF] Wilson | Confocal microscopy by aperture correlation[END_REF]. They used a spinning disk approach and combined that with the high light efficiency of structured illumination [START_REF] Neil | A light efficient optically sectioning microscope[END_REF]. In the original approach, the sample was illuminated through a spinning disk with a random pinhole pattern allowing up to 50% of the incident light to pass through to the sample. As in other spinning disc systems, emitted light from the sample is directed back to and detected through the same disk which is located in a conjugate image plane. As the openings on the disk are placed very close together in this approach, the image detected through the disk is not purely confocal. A significant amount of out-of-focus light can pass through neighboring pinholes. Therefore, the image detected through the disk consists of the in-focus information overlaid by a pronounced out-of-focus contribution. To calculate an optical section, this widefield contribution has to be subtracted.

In their original approach, Juskaitis et al. suggested to capture a second, pure widefield image through a blank sector of the disc. A scaled subtraction of this widefield image from the through-the-disk image would yield an optical sectioned image [START_REF] Juskaitis | Efficient real-time confocal microscopy with white light sources[END_REF][START_REF] Wilson | Confocal microscopy by aperture correlation[END_REF][START_REF] Neil | A light efficient optically sectioning microscope[END_REF]. This approach required tight synchronization of the camera acquisition with the disk rotation to match the disk sectors. Furthermore, two images had to be acquired, which might still lead to certain motion artifacts. These disadvantages were overcome in a second generation design [23], which allows simultaneous capture of both images. To this end, the spinning disk is positioned in the beampath at a slight angle and the side of the disk facing the sample is reflective. Thus, light rejected at the disk can be injected into a second imaging beampath. This can be captured by a second camera or -using additional optics -it can be imaged sideby-side with the through-the-disk image on one camera [23] (Figure 4). A similar approach to simultaneous capture of the conjugate and non-conjugate image was also proposed for the programmable array microscope [START_REF] Heintzmann | A dual path programmable array microscope (PAM): Simultaneous acquisition of conjugate and non-conjugate images[END_REF]. Simultaneous capture of both the conjugate (in-focus + widefield) and non-conjugate (mainly out-of-focus) information allows reliable and artifactfree calculation of optical sections even of highly dynamic events (Figure 4).

The optical sectioning properties of such a system are comparable to other SIM approaches [START_REF] Wilson | Confocal microscopy by aperture correlation[END_REF]. As the in-focus information is again imaged against the background of the out-of-focus information, the practical limit is again set by the SBR. In thick, highly fluorescent samples, both images will become more and more identical and the dynamic range of the final sectioned image will decrease until no sectioned image can be reliably calculated any more.

However, the technique maintains the high light efficiency of other structured illumination approaches, so that normal white light sources can be used. The instrumentation is more complicated than the one necessary for grid projection, but still much simpler than that of laser-based Nipkow-disk instruments, thus representing a cost-effective entry point to optical sectioning live cell imaging. The technique has been developed into a commercial product by the inventors and Aurox Ltd., which will be distributed by e.g. Carl Zeiss.

Structured Illumination for super-resolution imaging:

The recent years have seen the advent of several techniques able to overcome the classical resolution limit in fluorescence microscopy (Reviewed in [START_REF] Hell | Far-field optical nanoscopy[END_REF] and [START_REF] Henriques | What hides behind the raleigh limit?[END_REF]). Of all technical approaches to overcome this barrier in fluorescence microscopy, a special variant of SIM proves to be a very versatile approach to super-resolution microscopy [START_REF] Heintzmann | Breaking the resolution limit in light microscopy[END_REF]. In high resolution SIM (HR-SIM), a grid pattern with a grid frequency close to the theoretical maximum (i.e. the cut-off frequency of the optical transfer function of the system) is projected onto the sample. This finely structured light will interact with fine structures in the sample (created by the distribution of fluorophors in the sample) to create an interference pattern reminiscent of a moiré pattern in the acquired image -a pattern with greater periodicity than both original patterns. This emerging moiré pattern formed by the unknown sample pattern and the known grid pattern can be used to extract information beyond the normal resolution limit in fluorescence microscopy (Figure 5). By acquiring images at different known grid positions, structures below the resolution limit can be first visualized as different moiré versions of their former selves and, finally, the original structure can be reassigned once the known grid pattern is removed [START_REF] Gustafsson | Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy[END_REF][START_REF] Gustafsson | Extended resolution fluorescence microscopy[END_REF]. While in optical sectioning SIM most calculations can be done in real space, the calculations of a HR-SIM image are done in frequency (Fourier) space. Details of the calculation of a HR-SIM image are beyond the scope of this review as recent excellent articles cover these aspects in depth [START_REF] Heintzmann | Structured illumination methods[END_REF][START_REF] Gustafsson | Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination[END_REF][START_REF] Carlton | Three-dimensional structured illumination microscopy and its application to chromosome structure[END_REF].

For HR-SIM, a grid with a high grid frequency has to be projected onto the sample. To be able to use grids with the highest frequencies possible, phase grids in combination with laser illumination are used in HR-SIM, as the transmission grids used in optical sectioning SIM would not yield sufficient modulation contrast at high frequencies like this. Direct interference of coherent laser beams is another approach to create patterns of suitable frequency. In contrast to optical sectioning SIM, this pattern has to be moved not only laterally but must also be rotated. Care has to be taken, that the whole system including the sample is exceptionally stable during acquisition of the HR-SIM dataset.

The pattern projected onto the sample can interact with structures finer than the diffraction limit; the pattern projection itself is, however, limited by the diffraction limit. Therefore, the maximal increase in resolution achieved by HR-SIM is twofold in all dimensions. Although this gain in resolution is moderate, it is enough to enter dimensions where important biological questions wait for answers. For example, recent publications have shown that the resolution increase by HR-SIM is enough to gain important new information on chromosome structure [START_REF] Carlton | Three-dimensional structured illumination microscopy and its application to chromosome structure[END_REF][START_REF] Schermelleh | Subdiffraction multicolor imaging of the nuclear periphery with 3D structured illumination microscopy[END_REF] and HR-SIM will probably provide new information on the organization of other subcellular components in future.

The big advantage of HR-SIM is again its ease-of-use. In contrast to other high resolution microscopy systems, the use of common dyes is straightforward in HR-SIM. Fixation and In all cases, SIM systems are distinguished by their simple and robust instrumentation and their ease-of-use. SIM will continue to provide exciting new possibilities for cell biology over the next years.

  Structured illumination microscopy (SIM) used for optical sectioning can be considered as an intermediate between deconvolution and confocal microscopy. It does not require the sophisticated instrumentation of confocal microscopes, but gives reliable results even with thicker samples. It uses additional information encoded in the image by structuring the excitation light to compute an optical section from several raw data images. SIM works on

  λ wavelength in nm, n refractive index of the immersion medium, α aperture angle of the objective (NA = n sinα) in the sample plane With M magnification of the objective, ν grid frequency, b instrument-dependent magnification factor of the reflected light beam path, NA numerical aperture of the objective

  with the sectioned image and also subsequent deconvolution of the SIM data.

  similar to normal fluorescence microscopy, only special care has to be taken that the sample is fixed thoroughly enough to preserve all structures at the new resolution level. Dye-labeling must be stable enough to survive the acquisition of multiple images required for a high resolution reconstruction without too much bleaching. But if those few precautions are observed, multilabel and multicolor experiments are straightforward to carry out using the full range of normal dyes, fluorescent proteins and antibody conjugates already available in all labs. This makes HR-SIM the most versatile and robust of all new high resolution microscopy techniques.Conclusions:Over the last years, structured illumination in the form of grid projection systems has developed into an established technique in fluorescence imaging. It is now recognized as an attractive alternative to classical confocal microscopes for optical sectioning. But structured illumination is not limited to optical sectioning -with the new techniques presented the strengths of the SIM approach now extend to live cell imaging and to high resolution imaging.

Figure 1 :

 1 Figure legends:

Figure 2 :

 2 Figure 2: Calculation of an optical section using SIM. Three raw images with the grid pattern

Figure 3 :

 3 Figure 3: The dynamic range of the final optical section can be increased by HDR imaging.

Figure 4 :

 4 Figure 4: Calculation of an optical section using the latest design of the aperture correlation

Figure 5 :Figure 1 :Figure 2 :Figure 3 :Figure 4 :Figure 5 :

 512345 Figure 5: Principle of HR-SIM: Overlay of an unknown pattern (A) with a known high

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Acknowledgements:

We thank Megan MacNeil and Ingo Kleppe for valuable help during preparation of the figures.

Conflict of interest:

All authors are with Carl Zeiss MicroImaging GmbH, a manufacturer of microscope systems also offering structured illumination systems of the kind discussed here. However, the use of trade names is for product identification purposes only and does not imply endorsement.