Michèle Cohen

Johanna Etner

Meglena Jeleva

A Chateauneuf

R Giraud

J Y Jaffray

E Karni

P Klibanoff

Dynamic Decision Making when Risk Perception depends on Past Experience *

The aim of the paper is to propose a preferences representation model under risk where risk perception can be past experience dependent. A first step consists in considering a one period decision problem where individual preferences are no more defined only on decisions but on pairs (decision, past experience). The obtained criterion is used in the construction of a dynamic choice model under risk. The paper ends with an illustrative example concerning insurance demand. It appears that our model allows to explain modifications in the insurance demand behavior over time observed on the insurance markets for catastrophic risk and difficult to justify with standard models.

Introduction

Decision theory under risk had for a long time focused mainly on the impact of different risk and wealth perceptions on the agents' optimal decisions. In these classical studies, risk perceptions, as well as utility functions, depend only on the considered decision characteristics (pairs probabilities-outcomes) and thus can not be influenced by outside factors. However, some psychological studies [START_REF] Slovic | The Perception of Risk[END_REF] point out the fact that risk perception may be strongly influenced by the context in which the individuals are when they take their decisions. Context can take different forms: (i) it can correspond to past experience concerning the same decision problem (relevant in insurance decisions as noticed in [START_REF] Kunreuther | Mitigation disaster losses through insurance[END_REF] and Brown, Hoyt (2000)) or other events independent on the relevant decision problem as weather or general economic situation (a relation between weather conditions and stock market behaviour have been enlightened by [START_REF] Hirshleifer | Good Day Sunshine: Stock Returns and the Weather[END_REF], (ii) it can also correspond to anticipatory feelings about some future states [START_REF] Caplin | Psychological expected utility theory and anticipatory feeling[END_REF], (iii) it can be related to the decision outcomes presentation (leading then to the framing effect pointed out by [START_REF] Tversky | Rational Choice and the Framing of Decisions[END_REF].

In this paper, we focus on the context created by past experience, corresponding to a sequence of events occurring up to the moment of the decision. This past experience can concern different events : (i) past realizations of the decision-relevant events (as accidents when an insurance decision is considered) or (ii) realization of events, others than the decision relevant ones (such as temperature or health status when a stock demand decision is considered).

The influence of past experience on decisions appears in particular on insurance markets for catastrophic risk. It appears that in California, before the earthquake in 1989, 34 % of individuals consider that insurance against earthquake is useless. After the earthquake, they are only 5% and 11% of the non insured individuals subscribed an insurance contract [START_REF] Kunreuther | Mitigation disaster losses through insurance[END_REF]. These results are confirmed by an empirical study from Brown and Hoyt (2000) that reveals a strong positive correlation between the number of flood insurance contracts subscribed in a year in a given State of the US and losses due to flood in the same State the previous year. A relation between past experience and insurance demand appears also in experimental studies when individuals are well informed about the probability of loss realization and about the independence of losses in successive periods (McClelland, Schulze, Coursey 1993, [START_REF] Ganderton | Buying insurance for disaster-type risk: experimental evidence[END_REF][START_REF] Papon | L'influence de la durée d'engagement et du vécu dans les décisions d'assurance: deux études expérimentales[END_REF]). Their results are less clear-cut: if the existence of a strong correlation between past damages and insurance demand is well established, its sign is less clear. Indeed, two opposite effects can be identified corresponding to availability biais and gambler fallacy in the sense of [START_REF] Tversky | Availability: a heuristic for judging frequency and probability[END_REF]. The availability biais corresponds to an overestimation of the probability of an event that recently occurred and implies an increase in insurance demand after a natural disaster, this demand being low after a long period without a catastrophe. The opposite occurs with the gambler's fallacy effect: individuals underestimate the probability of repetition of the event that they just observed and thus buy less insurance after a loss. When events are independent over time, behaviors that we have just described cannot be explained in the standard expected utility model. Indeed, in this model, past experienced losses lead only to a wealth decrease and not to a probability assesment modification.

Relaxing the axiom of context independence of preferences under uncertainty can allow the rationalization of some decisions considered as inconsistent with respect to the existing criteria because reflecting unstable preferences as for instance the modification of insurance demand against catastrophic risk after the occurrence of a catastroph.

The aim of the paper is to propose a preferences representation model under risk where risk perception can be past experience dependent. A first step consists in considering a one period decision problem where individual preferences are no more defined only on decisions but on pairs (decision, past experience). The obtained criterion is used in the construction of a dynamic choice model under risk.

The underlying model of decision making under risk that is used here is the RDU (Rank dependent Utility), proposed by [START_REF] Kahneman | Prospect Theory : an Analysis of Decision under Risk[END_REF], Quiggin (1982) and [START_REF] Yaari | The Dual Theory of Choice under Risk[END_REF]. This model has the advantage to allow a non linear treatment of probabilities, in addition to a non linear treatment of outcomes. When one period decisions are considered, we adapt the RDU axiomatic system of [START_REF] Chateauneuf | Comonotonicity axioms and rank-dependent expected utility theory for arbitrary consequences[END_REF] to represent preferences on pairs (decision, past experience).

To better capture the long term impact of past experience on decisions, after the preferences representation on a point of time, we model intertemporal decisions. RDU model can generate intertemporal consistency. To rule out this problem, we use the recursive model of [START_REF] Kreps | Temporal resolution of uncertainty and Dynamic choice theory[END_REF]. In the latter model, risk aversion is characterized by a standard utility function and the agents' past experience is summarized by a sequence of monetary payoffs, resulting from the past decisions and the lottery realizations. In the present paper are introduced additional aspects: (i) risk perception is non linear and depends on a probability transformation function; (ii) past experience does not reduce anymore to the only payoffs, but is characterized by a sequence of events, corresponding more generally to individuals' past experience. To achieve this preferences representation, we assume that preferences at a point of time are represented by the "past experience dependent" RDU previously axiomatized and modify the dynamic consistency axiom of KP in order to apply to states and not to payments.

The paper starts with the "past experience dependent" preferences representation at a point of time. We propose an axiomatic foundation for "past experience dependent" rank dependent utility under risk. In the third section, we consider a dynamic choice problem and prove a representation theorem. Section four contains an illustrative example.

Behavior at a point of time

In this section, we consider a static problem. We propose an axiomatic representation of preferences by a rank dependent expected utility which takes into account the agent's past experience.

Notations and definitions

Decision problem is characterized by a set of risky perspectives in which an agent has to make his choice and by a set of states that characterize the agent's past experience.

Let Z denote a set of outcomes. We assume that Z is a non empty connected compact and metric space and L is the set of lotteries over Z.

S is a set of realized states, assumed nonempty, compact, connected separable topological space. An element of S ×L will be called a "past experience dependent lottery".

% is a binary relation on S × L which denotes the preference relation of a decision maker. We denote by  the strict preference and by ∼ the indifference.

Axiom 1 % is a weak order on S × L.

The preferences representation of % on S × L will be built in two steps. We start with the preference relation on s × L and its representation by a Rank Dependant Utility (RDU) model. Then, we give some additional assumptions to achieve a RDU preferences representation on S × L.

Preferences on s × L and RDU

In this section we consider the restrictions of % to s × L that we denote by % s . For a given state s, we face a standard decision problem under risk.

The RDU representation of the preference relation % s is obtained by the following axioms, proposed in [START_REF] Chateauneuf | Comonotonicity axioms and rank-dependent expected utility theory for arbitrary consequences[END_REF].

From Axiom 1, it follows directly that % s is a weak order on s × L.

Axiom 2 (Continuity) For a given s ∈ S, let P n = (s, L n ) , P = (s, L) , Q = (s, L 0 ) ∈ s × L, with P n weakly converging to P , then ∀n,

P n % s Q ⇒ P % s Q and ∀n, P n -s Q ⇒ P -s Q.
For a given s, it is possible to completely order the space Z. The definition of the first order stochastic dominance (SD1) becomes L SD1 L 0 if and only

if P L (z ∈ Z, (s, z) Â s (s, x)) ≥ P L 0 (z ∈ Z, (s, z) Â s (s, x)) ∀x ∈ Z.
The next axiom guarantees that % s preserves first order stochastic dominance.

Axiom 3 For any L, L 0 ∈ L such that L SD1 L 0 , (s, L) % s (s, L 0 ) . Axiom 4 (Comonotonic sure-thing principle) For any s ∈ S, let lotteries P = ((s, z i ) p i ) , Q = ((s, y i ) p i ) be such (s, z i 0 ) ∼ s (s, y i 0 ), then P % s Q implies P 0 % s Q 0 , for lotteries P 0 , Q 0 obtained from lotteries P and Q by merely replacing the i th 0 common pair (s, z i 0 ), by a common pair (s, x i 0 ) again in i th 0 rank both in P 0 and Q 0 .

Axiom 5 (Comonotonic mixture independance axiom) For any s ∈ S, and for any lotteries P = ((s,

z i ) p i ) and Q = ((s, y i ) q i )), For any p ∈ [0, 1] , for any a, b, c, d ∈ Z • P 1 = (1 -p) ¡ s, z min ¢ + p (s, a) ∼ s Q 1 = (1 -p) ¡ s, y min ¢ + p (s, b)
and

P 2 = (1 -p) ¡ s, z min ¢ + p (s, c) ∼ s Q 2 = (1 -p) ¡ s, y min ¢ + p (s 0 , d) imply ∀α ∈ [0, 1] , αP 1 + (1 -α) P 2 ∼ s αQ 1 + (1 -α) Q 2 • P 1 = (1 -p) (s, z max ) + p (s, a) ∼ s Q 1 = (1 -p) (s, y max ) + p (s 0 , b) and P 2 = (1 -p) (s, z max ) + p (s, c) ∼ s Q 2 = (1 -p) (s, y max ) + p (s 0 , d) imply ∀α ∈ [0, 1] , αP 1 + (1 -α) P 2 ∼ s αQ 1 + (1 -α) Q 2
Theorem 1 Let the preference relation % s on s×L satisfy axioms 1 -5, then there exist an increasing function ϕ s : [0, 1] → [0, 1], with ϕ s (0) = 0, ϕ s (1) = 1 and a utility function, v s : Z → R, which is increasing, continuous, and unique up to an affine transformation such that:

∀L, L 0 ∈ L, (s, L) % s (s, L 0 ) iff V s (L) ≥ V s (L 0 ) with V s (L) = n X i=1 Ã ϕ s ( n X j=i p j ) -ϕ s ( n X j=i+1 p j ) ! × v s (z i ) Proof. Chateauneuf [1999]
We now consider the general preferences comparing lotteries in different contexts (S × L). Let us notice that the preferences on S × L induce a preference on S × Z but in no case a preference on S alone. At this stage, the payoffs evaluation depends not only on z but also on s. The objective of this paper is to emphasize the link between risk perception and individual context. In order to isolate this feature, we assume that only perceptions depend on s. This assumption needs more discussion, mainly with respect to the state-dependant model of [START_REF] Karni | Decision Making under Uncertainty: The case of statedependant preferences[END_REF]. The main feature of Karni's model is that the evaluation of a given amount of money may strongly depend on the state in which the individual is when receiving this amount. Here, the states we consider are not of the same type: they are already realized (past) states, and not future ones. It seems then more realistic to assume that they are more likely to influence risk perception than future monetary evaluations.

The following axiom guarantees that the payoffs evaluations do not depend on past experience.

Axiom 6 For any s, s 0 ∈ S and any z ∈ Z, (s, z) ∼ (s 0 , z) .

Let us notice that axioms 3 and 6 induce the existence of a preference relation on Z independent on S. To simplify notations, we can then write z ≥ z 0 instead of (s, z) º (s, z 0 ) for all s ∈ S.

The following preferences representation theorem can then be formulated.

Theorem 2 Under Axioms 1 -6, the weak order % on S ×L is representable by a function V : S × L → R. For any s, s 0 ∈ S and any L,

L 0 ∈ L , (s, L) % (s 0 , L 0 ) ⇔ V (s, L) ≥ V (s 0 , L 0 ) where V (s, L) = n P i=1 Ã ϕ s ( n P j=i p j ) -ϕ s ( n P j=i+1 p j ) ! v(z i ).
Proof. The generalization of the preferences representation of the restrictions % s on % is allowed by the unicity of the probability transformation function ϕ s (p) and the independence of the utility function on s.

We can see that a decision maker with ϕ s (p) < p systematically underestimates the probabilities of the favorable events and then is called pessimist under risk. Moreover, we obtain the following result. Corollary 1 Let s, s 0 ∈ S. ϕ(s, p) ≥ ϕ(s 0 , p) for any p ∈ [0, 1] if and only V (s, L) ≥ V (s 0 , L) for any L ∈ L.

Proof. Let L = (z 1 , p 1 ; z 2 , p 2 ; ...; z n , p n )with z 1 ≤ ... ≤ z n Note that this axiom implies that for any z, z 0 ∈ Z such that z ≥ z 0 , (s, z) % s (s, z 0 ) .

(i) ⇒: V (s, L)-V (s 0 , L) = n P i=2 Ã ϕ s ( n P j=i p j ) -ϕ s 0 ( n P j=i p j ) ! (v(z i ) -v(z i-1 )) ≥ 0 if ϕ(s, p) ≥ ϕ(s 0 , p) for any p ∈ [0, 1];
(ii) ⇐: if there exists p 0 such that ϕ(s, p 0 ) < ϕ(s 0 , p 0 ) then for

L 0 = (z 1 , 1 -p 0 ; z 2 , p 0 ), V (s, L 0 ) < V (s 0 , L 0 ).
This result implies that if there exists a realized state that induces pessimistic risk perception, then an individual will dislike any decision in this context, with respect to a state where his risk perception is less pessimistic.

Dynamic choice

In this section, we consider a dynamic choice problem under risk where risk perception and utility of the outcomes may depend on agents' past experience. Preferences at a point of time are represented as in the previous section of the paper, by a past experience dependent RDU. It is now well known (see for instance [START_REF] Machina | Dynamic Consistency and Non-Expected Utility Models of Choice under Uncertainty[END_REF]) that preferences representations models that do not verify the independence axiom can not verify at the same time dynamic consistency, consequentialism and reduction of compound lotteries. To preserve dynamic consistency, as in Epstein, Wang (1994) and Epstein, Schneider (2003), a recursive model is adopted here. More precisely, we modify the [START_REF] Kreps | Temporal resolution of uncertainty and Dynamic choice theory[END_REF] model in order to introduce both risk perception and past experience dependence.

Some notations and definitions

We consider a discrete and finite sequence of times t = 1, ..., T. Z t is the set of possible payoffs at time t. To simplify, we assume that Z t = Z which is a compact interval of R for any t from 1 to T . A payoff realized at time t is denoted by z t . Decision maker past experience at time t is characterized by a sequence of events, relevant for the considered decision and denoted by s t . More precisely, s t = (e 0 , e 1 , ..., e t ) where e τ is the event that occurred at time τ with e τ ∈ E τ , the set of all possible events at time τ and e 0 ∈ E 0 the set of all possible past experiences. S t is then the set of possible histories up to time t verifying the recursive relation: S 0 = E 0 and S t = S t-1 × E t . We denote by M (E t ) the set of distributions on E t .

At period T , L T is the set of distributions on Z T endowed with the Prohorov metric. X T , the set of risky perspectives in which agent has to make his choice, is assumed to be the set of closed non empty subsets of L T , endowed with the Hausdorff metric.

By recurrence, we define, L t , the set of probability distributions on

C t =Z t ×X t+1 × M (E t+1
) with X t+1 the set of closed non empty subsets of L t+1 .

At each period, the nature chooses a probability distribution on E t+1 . The agent cannot influence this distribution. Given this distribution, the agent has to choose a lottery in the set L t . The assumption of compound lotteries reduction is made between distributions on wealth and events for a fixed period. However, this assumption is relaxed between two consecutive periods.

For each period t, % t denotes a binary relation on s t × L t for a given s t . We denote  t the strict preference and ∼ t the indifference.

Axiom 7 (1bis) % t is a complete order on s t × L t .

We assume that % t verifies axioms 2 -5 on s t × L t . Under these previous axioms, we can represent preferences at a point of time in the similar way to theorem 1: Lemma 1 For any s t in S t , axioms 1bis, 2-5 are necessary and sufficient for there to exist, for each t, a bounded continuous function v t :

s t × Z t × X t+1 × E t+1 -→ R and a continuous function ϕ t : s t × [0, 1] -→ [0, 1] such that for L t , L 0 t ∈ L t , (s t , L t ) % t (s t , L 0 t ) if and only if V t (s t , L t ) ≥ V t (s t , L 0 t ) with V t (s t , L t ) = n P i=1 Ã ϕ t (s t , n P j=i p j ) -ϕ t (s t , n P j=i+1 p j ) ! × v t (s t , z t , x t+1 , e t+1 ) ≡ RDv t (s t , L t ) .
The proof comes immediately from the previous section. Let us notice that utility function v t depend on past experience s t whereas in a static problem, v did not depend on state s. This comes from the fact that in a dynamic choice, future perspective depend on past experience. We will precise this point in the next section.

Temporal consistency and the representation theorem

We adapt the KP temporal consistency axiom to our context in the following manner.

Axiom 8 (Temporal consistency) We consider the degenerate distributions, ∆ e t in M (E t ) and for a given distribution δ e t+1 in ∆ e t+1 , the degenerate distributions on Z t ×X t+1 ×δ e t+1 . Then, for all t, s t ∈ S t ,

e t+1 ∈ E t+1 , z t ∈ Z t , x t+1 , x 0 t+1 ∈ X t+1 , (s t , z t , x t+1 , e t+1 ) % t ¡ s t , z t , x 0 t+1 , e t+1 ¢ iff (s t+1 , x t+1 ) % t+1 ¡ s t+1 , x 0 t+1 ¢ with s t+1 = (s t , e t+1 ) at period t + 1.
Let us consider the two following lotteries:

The temporal consistency axiom sets that if the degenerate lottery (z t , x t , e t ) is preferred to the degenerate lottery (z 0 t , x 0 t , e t ) for a decision maker with past experience s t , then if at time t, e t is realized, the decision maker will choose x t between x t and x 0 t . In the same way, if at time t + 1, he chooses x t , then he cannot at time t, strictly prefer (z 0 t , x 0 t , e t ) to (z t , x t , e t ).

Lemma 2 Axioms 1bis, 2-5 and 8 are necessary and sufficient for there to exist functions v t as in previous lemma and, for fixed {v t }, unique functions

u t : {(s t , z t , γ) ∈ S t × Z t × R : γ = RDv t+1 (s t+1 , L t+1 )} → R
which are strictly increasing in their third argument and which satisfy v t (s t , z t , x t+1 , e t+1 ) = u t (s t , z t , L t+1 maxRDv t+1 (s t+1 , L t+1 )) for all s t ∈ S t , e t+1 ∈ E t+1 , z t ∈ Z t , x t+1 ∈ X t+1 .

Proof.

Axioms 1bis-5 and 8 are hold, lemma 3 fix V t (s t , L t ) . Then,

V t+1 ((s t , e t+1 ) , L t+1 ) = V t+1 ¡ (s t , e t+1 ) , L 0 t+1 ¢ =⇒ V t (s t , L t+1 ) = V t ¡ s t , L 0 t+1 ¢
for a given e t+1 (axiom TC). Consequently, u t is strictly increasing in its third argument.

(ii) If V t and u t are given with u t is strictly increasing in its third argument, V t verifies lemma 3. Then axioms 1bis-6 hold. u t is increasing in its third argument then V t (s t , L) ≥ V t (s t , L 0 ) ⇐⇒ u t (s t , z t , V t+1 (s t+1 , L)) ≥ u t (s t , z t , V t+1 (s t+1 , L 0 )) ⇐⇒ V t+1 ((s t , e t+1 ) , L) ≥ V t+1 ((s t , e t+1 ) , L 0 ). Then axiom 8 holds.

As we can see, in the dynamic problem, utility functions v t depend on past experience s t . Indeed, at time T , utility function does not depend on past experience. But, at time T -1, the certainty equivalent of lottery, given by u T -1 , depends on past experience s T -1. Consequently, v T -1 directly depends on past experience. Recursively, at each period, certainty equivalent depends on past experience and then utility function too.

Theorem 3 Axioms 1bis, 2-5 and 8 are necessary and sufficient for there to exist a continuous function v : S T × Z T → R and, for t = 0, ..., T -1, continuous functions u t : S t × Z t × R → R, strictly increasing in their third argument, so that, v T (s T , z T ) = v (z T ) and, recursively v t (s t , z t , x t+1 , e t+1 ) = L t+1 maxu t (s t , z t , RDv t+1 (s t+1 , L t+1 )),

then, for all

s t ∈ S t , L t , L 0 t ∈ L t , (s t , L t ) % t (s t , L 0 t ) iff RDv t (s t , L t ) ≥ RDv t (s t , L 0 t ) with RDv t (s t , L t ) = n P i=1 Ã ϕ t (s t , n P j=i p j ) -ϕ t (s t , n P j=i+1 p j ) ! × v t (s t , l t ).
Proof. We adapt the demonstration of the theorem in Kreps and Porteus.

Notice that the representation needs v, the functions u t and the functions ϕ t to implicitly define functions v t . As in Kreps and Porteus, it introduces the concept of timing of resolution of uncertainty. This representation can explain some dynamic behaviors not explain by the standard Expected Utility model. We propose in the next section an illustration.

An insurance demand illustration

In this section, we study the implications of the previous model for multiperiod demand decisions on the insurance market. It appears that introducing a relation between realized damages and risk perception gives an explanation for the observed insurance demand against catastrophic risk.

We study the optimal insurance demand strategy of an individual for 3 periods of time (years). The individual faces a risk of loss of amount L at each period. There exist a perfectly competitive insurance market proposing insurance contracts at a fair premium corresponding to the estimated expected loss. Insurance contracts are subscribed for one period. Consequently, the individual has to choose an amount of coverage at each period. We assume that for one period the estimated probability of incurring a loss is p and that losses in successive periods are independent: P(loss at period t/ loss in period t -1) = p.

We can represent the decision tree as following: At each period, the insurance contract C t is characterized by a pair (indemnity I t , premium Π t ). Only proportional coverage is considered:

I t = α t L with α t ∈ [0, 1].
The premium for a coverage rate α t is Π t = α t pL. At period t = 0, the agent receives a certain wealth, z 0 , and he is in the state s 0 . Past experience is resumed by the sequences of events {damage, no damage}. We denote by e t the event "damage at period t" and by e 0 t the event "no damage at period t". At each period, individual has to choose α t .

At a point of time, the probability transformation function is assumed to be the following:

ϕ t (p, s t ) = p t P τ =0
e τ with e 0 = 1 2 , e t = 1 2 and e 0 t = 0 for t = 1, 2. In this case, the individual is optimistic at period 0 and modifies his risk perception with respect to damages, occurring or not: the occurrence of a damage modifies his risk perception and he becomes less optimistic, if no damages occurs, his risk perception does not change.

To better isolate the risk perception arguments, we assume that preferences at the final period are represented by a linear utility function under certainty.

Moreover, for simplicity, we suppose that the individual is neutral toward the time of resolution of uncertainty (in the sense of Kreps and Porteus), so that u t (s t , z t , RDv t+1 (s t+1 , L t+1 )) = RDv t+1 (s t+1 , L t+1 ) .

The dynamic choice problem solves in several steps. Note that, due to the linear utility assumption, only corner solutions will prevail.

(i) For each terminal node, we have to compute v (s 3 , z 3 ) = z 3 , with z 3 the wealth at the final period.

(ii) For each final decision node, we have to evaluate the individual utility and maximize it.

For example, at node L 3 , for a coverage rate α 3 , the utility writes:

V 3 (s 3 , L 3 ) = RDv (s 3 , L 3 ) = v ¡ s 3 , z 3 ¢ + £ v (s 3 , z 3 ) -v ¡ s 3 , z 3 ¢¤ × ϕ 3 (1 -p, s 3 ) with z 3 = z 2 -Π 3 -L + I 3 = z 2 -L + α 3 L (1 -p), z 3 = z 2 -Π 3 = z 2 -α 3 Lp, and ϕ 3 (1 -p, s 3 ) = (1 -p) e 0 +e 1 +e 2 = (1 -p) 3 2 .
At node L 0 2 , we have to pay attention to the value of v 2 (s 0 2 , z 0 2 ) = u 2 (s 0 2 , z 0 2 , RDv 3 ) and v 2 (s 0 2 , z 0 2 ) = u 2 (s 0 2 , z 0 2 , RDv 3 ) since, in the RDU framework, we have to rank utility.

In our example, v 2 (s 0 2 , z 0 2 ) < v 2 (s 0 2 , z 0 2 ) . Thus, we obtain that

V 2 (s 0 2 , L 0 2 ) = RDv 2 (s 0 2 , L 0 2 ) = v 2 ³ s 0 2 , z 0 2 ´+ h v 2 (s 0 2 , z 0 2 ) -v 2 ³ s 0 2 , z 0 2 ´i × ϕ 2 (1 -p, s 0 2 ) with v 2 ³ s 0 2 , z 0 2 ´= z 0 2 -pL = z 1 -L (1 + p)+α 0 2 L (1 -p) , v 2 (s 0 2 , z 0 2 ) = z 0 2 -L ³ 1 -(1 -p) 1/2 ´= z 1 -L ³ 1 -(1 -p) 1/2 ´-α 0 2 pL and ϕ 2 (1 -p, s 0 2 ) = (1 -p) e 0 +e 0 1 = (1 -p) 1/2 . Then, V 2 (s 0 2 , L 0 2 ) = z 1 + Lp h (1 -p) 1/2 -2 i + α 0 2 L (1 -p) 1/2 h (1 -p) 1/2 -1 i
As utility is a decreasing function of the coverage rate, the optimal coverage is null and the value of utility becomes

V 2 (s 0 2 , L 0 * 2 ) = z 1 + Lp h (1 -p) 1/2 -2 i (2)
(v) At period 1, we have two nodes which values are:

u 1 (s 1 , z 1 , RDv 2 ) = z 1 -2pL with z 1 = z 0 -Π 1 -L + I 1 and u 1 (s 1 , z 1 , RDv 2 ) = z 1 + Lp h (1 -p) 1/2 -2 i with z 1 = z 0 -Π 1 .
Then, at node L 1 , for a coverage rate α 1 , the utility writes:

V 1 (s 1 , L 1 ) = RDv 1 (s 1 , L 1 ) = v 1 ¡ s 1 , z 1 ¢ + £ v 1 (s 1 , z 1 ) -v 1 ¡ s 1 , z 1 ¢¤ × ϕ 1 (1 -p, s 1 ) with v 1 ¡ s 1 , z 1 ¢ = z 1 -2pL = z 0 -Π 1 -L+I 1 -2pL = α 1 L (1 -p)+z 0 - L (1 + 2p) , v 1 (s 1 , z 1 ) = z 1 +Lp h (1 -p) 1/2 -2 i = z 0 -α 1 pL+Lp h (1 -p) 1/2 -2 i and ϕ 1 (1 -p, s 1 ) = (1 -p) e 0 = (1 -p) 1/2 . Then, V 1 (s 1 , L 1 ) = α 1 L (1 -p) 1/2 h (1 -p) 1/2 -1 i + z 0 -L (1 + 2p) + L h p (1 -p) 1/2 + 1 i (1 -p) 1/2
As utility is a decreasing function of the coverage rate, the optimal coverage is zero and the value of utility becomes

V 1 (s 1 , L * 1 ) = z 0 -L (1 + 2p) + L h p (1 -p) 1/2 + 1 i (1 -p) 1/2 (3)
To summarize, the results are the following:

• α 1 = 0;

• α 2 = 0 if no loss at period 1;

α 2 ∈ [0, 1] if loss at period 1;

• α 3 = 0 if no loss at periods 1 and 2; α 3 = 1 if loss at periods 1 and 2;

α 3 ∈ [0, 1] else.
In this illustration, the individual chooses not to buy insurance contract in the first period. In the second period, he choose not to be covered only if he had not damage. Finally, in the third period, if he had never had an accident, he chooses not buying insurance contract, if he had two consecutive accidents, he decides to buy a full coverage and in the intermediate cases, he is indifferent.

This example underlines the fact that what is important for the decision maker is not only the event occurring in the period directly preceding the moment of the decision, but the all sequence of events, that is all the past experience.

Let us now compare the predictions of our model with those of some standard modelizations.

• the particular case when ϕ t (p, s t ) = p : this corresponds to the standard version of a recursive expected utility model.

The results are then the following: α i ∈ [0, 1] for any i = 1, 2, 3. The individual is indifferent between different amounts of coverage and this, at any period and for any experienced damage.

• we consider here an alternative non consequentialist way to model dynamic choices: the resolute choice model proposed by [START_REF] Mcclennen | Rationality and Dynamic Choice[END_REF].

In this model, all the strategies are evaluated at the root node and compared according to the root node preferences. We consider 2 cases: the case when one-shot preferences are EU and the case when one-shot preferences are RDU. The results are then the following:

(i) for EU preferences, α i ∈ [0, 1] for any i = 1, 2, 3. The individual is indifferent between different amounts of coverage and this, at any period and for any experienced damage.

(ii) for RDU preferences, α i = 0 or 1 for any i.

-for ϕ (p) < p, complete coverage at any period and for any experienced loss is preferred to a strategy consisting in buying insurance only after experiencing a loss; -for ϕ (p) > p, no coverage at any period and for any experienced loss is preferred to a strategy consisting in buying insurance only after experiencing a loss.

Concluding remarks

The insurance demand example shows that our model allows to explain the modifications in the insurance demand behaviour over time observed for catastrophic risk and given in the introduction. It well appears that past experience have a cumulative effect on decisions: an individual can maintain constant its insurance demand after one occurrence of the loss and modify it only after two, or more consecutive loss events. In this example, we assumed that at any period observing a loss renders the individual more pessimistic. This explains a behaviour in accordance with the availability bias. The gambler's fallacy attitude is explained if the individual becomes more and more optimistic after experiencing losses.

The comparison with other modelizations shows that neither the recursive model alone, nor the RDU model alone can justify all the observed pattern of behaviour.

The insurance example corresponds to the particular case when past experience (context) is composed by the decision-relevant events. Considering different events, that do not directly influence the outcomes (as weather condition in investment decisions) will make even easier to underline the new insights of the present model because of the complete absence of these events in the preferences representations of the standard models.

Figure 1 :

 1 Figure 1:

Figure 2 :

 2 Figure 2:

Then,

As utility is an increasing function of the coverage rate, the optimal coverage is the full coverage and the value of utility becomes

In the same way, we obtain that

(iii) At period 2, we have four nodes which values are:

(iv) We repeat step (ii). Then, at node L 2 , for a coverage rate α 2 , the utility writes:


D Munteanu

S Moindjie

J L Autran

A water tank muon spectrometer for the characterization of low energy atmospheric muons

Keywords: Atmospheric muons, low energy muons, cosmic rays, muon flux, muon spectrometer, sea level muon intensity, differential spectrum, integral spectrum

In this work, a water tank muon spectrometer was designed, assembled and operated to measure the energy distribution of low energy atmospheric muon flux induced by cosmic-rays at sea level in the energy range 100-500 MeV. The principle of this experiment is to use water as muon moderator inserted between two coincidence detectors to select the cutoff energy below which muons can no longer be detected. The differential and integral muon spectra are then derived from successive measurements by varying the liquid height within the water tank.

The instrument was entirely characterized and modeled in terms of detector efficiency, cutoff energy and counting rate. Experimental data are reported for the energy distribution of muon flux at sea level (43°N of latitude) and finally compared with literature survey.

Introduction

Muons are the most abundant energetic charged particles at sea level where they arrive with an average flux of about ≈1 muon per square centimeter and per minute and with an average energy of about 4 GeV [START_REF] Cecchini | Atmospheric muons: Experimental aspects[END_REF]. Their characterization provides important information on the physics of primary cosmic rays and on the production mechanisms of atmospheric cascades. An abundant literature exists for muons at ground level and underground but the overwhelming majority of works rather concerns high-energy physics with energy ranges above a few GeV and beyond [START_REF] Grieder | Cosmic Rays at Earth[END_REF]. Literature surveys on atmospheric muons [START_REF] Grieder | Cosmic Rays at Earth[END_REF][START_REF] Allkofer | A survey on the recent measurements of the absolute vertical cosmic-ray muon flux at sea level[END_REF][START_REF] Rastin | An accurate measurement of the sea-level muon spectrum within the range 4 to 3000 GeV/c[END_REF][START_REF] Kremer | Measurements of Ground-Level Muons at Two Geomagnetic Locations[END_REF][START_REF] Naumov | Atmospheric muons and neutrinos[END_REF][START_REF] Gaisser | Atmospheric muons and neutrinos[END_REF] show that there is a clear lack of experimental data typically below a few hundred MeV. In another field of interest, i.e. in radiation effects on modern electronics, the interest for low energy muons is rapidly growing due to the potential role that they can play in failure mechanisms of nanoscale digital integrated circuits [START_REF] Autran | Soft Errors: From Particles to Circuits[END_REF][START_REF] Munteanu | Modeling and Simulation of Single-Event Effects in Digital Devices and ICs[END_REF]. Even if the research effort has so far focused mainly on the errors induced by atmospheric neutrons in electronics [START_REF] Autran | Soft-errors induced by terrestrial neutrons and natural alpha-particle emitters in advanced memory circuits at ground level[END_REF][START_REF] Leray | Effects of atmospheric neutrons on devices, at sea level and in avionics embedded systems[END_REF][START_REF] Autran | Real-time soft-error testing of 40 nm SRAMs[END_REF][START_REF] Semikh | The plateau de bure neutron monitor: Design, operation and Monte-Carlo simulation[END_REF][START_REF] Autran | Real-time soft-error rate measurements: A review[END_REF], the effects of muons can no longer be neglected. Indeed, recent investigations demonstrated the importance of these particles in the production of soft errors in the most advanced CMOS technologies [START_REF] Sierawski | Muon-induced single event upsets in deep-submicron technology[END_REF][START_REF] Sierawski | Effects of scaling on muon-induced soft errors[END_REF][START_REF] Serre | Effects of Low Energy Muons on Electronics: Physical Insights and Geant4 Simulation[END_REF][START_REF] Roche | Technology downscaling worsening radiation effects in bulk: SOI to the rescue[END_REF]. The probability of memory upsets was shown to increase near the energy region corresponding to muon stopping in the circuit semiconductor material, indicative of direct ionization effects [START_REF] Sierawski | Effects of scaling on muon-induced soft errors[END_REF]. The characterization of low energy muon distributions at ground level is thus a challenging issue for the prediction of radiation effects in microelectronics; this corresponds to the initial motivation of the present work.

In this context, we recently developed a muon telescope to measure the cosmic-ray induced atmospheric muon flux at sea level (43°N of latitude) in terms of vertical muon intensity and zenithal angle dependence [START_REF] Autran | Characterization of Atmospheric Muons at Sea Level Using a Cosmic Ray Telescope[END_REF]. However, the energy distribution of the muon flux cannot be measured with the instrument developed in [START_REF] Autran | Characterization of Atmospheric Muons at Sea Level Using a Cosmic Ray Telescope[END_REF]. In the present work, we thus developed a new instrument, called "water tank muon spectrometer" to precisely measure the energy distribution of atmospheric muons in the range of hundreds of MeV. The principle of this experiment is to use water as muon moderator sandwiched between two detectors in coincidence to select the cutoff energy below which the muons will no longer be detected. It is then possible to determine the energy distribution of muon flux by varying the water height in the water reservoir. This paper details the main characteristics of this instrument including its complete characterization and modeling and reports our first measurements of the muon differential and integral flux in the range 100-500 MeV conducted at sea level in Marseille (43° N of latitude).

The water tank muon spectrometer

Experimental setup

The experimental setup is presented in Figure 1. The muon spectrometer was developed using a dedicated metallic tower structure capable to support heavy load (two metric tons of water in storage tanks) between the two detectors installed at the top and the bottom of the structure. The two detectors are square plastic scintillators (surface of 50x50 cm 2 , thickness of 5 cm, housed in 1.5 mm of aluminum plate). The plastic scintillator is made of polyvinyltoluene (PVT) and fabricated by Eljen Technology (EJ-200) [20]. This plastic is highly sensitive to charged particles: its typical stopping power for 1 GeV muons is 2.132 MeV/cm [21]. Each scintillator has a photomultiplier tube (PM) directly integrated in the assembly; the tube is placed in a housing positioned in the bulk material and optically coupled using a high refractive index optical coupling medium at the bottom of the housing (polished surface with an optical quality). The PMs are Hamamatsu model R6427 with 28 mm tube diameter, 25 mm round photocathode and 10-stage photomultiplier. The assembly (PM+scintillator) is mounted in an aluminum housing (thickness 1.5 mm) that ensures a permanent light sealing. This solution also offers a simplification of detector design by eliminating support hardware to maintain contact between the scintillator material and the PM. between the front and back detectors separated by a distance of 2.815 m. PM signals as well as time-of-flights converted in voltage pulses using a time-to-amplitude converter (TAC) are digitized using multi-channel analyzers (MCAs based on 16k ADCs). For piloting the experiment, we have developed dedicated Visual Basic software that allows the acquisition of the muon count rate as function of the water level in the water tank. The two polyethylene water tanks have a thickness of about 3 mm except the bottom of the tanks, which has a thickness of 9 mm. The second tank is installed on a stainless steel plate about 2 mm thick.

The variation of the water level was carried out manually by filling or emptying the tanks by a step of 100 L, which corresponds to a water height of 9 cm. An additional layer of lead with a thickness of 50 mm was optionally added just below the top detector in order to verify that low energy gamma rays and electrons are satisfactorily rejected in the detector response.

Spectrometer calibration

In order to optimize the spectrometer operation and to be sure that the spectrometer detects and counts only atmospheric muons, a careful calibration of the instrument has been performed, very similarly to that carried out in [START_REF] Autran | Characterization of Atmospheric Muons at Sea Level Using a Cosmic Ray Telescope[END_REF] for the cosmic ray telescope. The calibration procedure consists in the following three steps:

(i) the detection threshold of the pulses coming out of each PM has been fine-tuned (around 130 mV), which ensures that the low-energy peak (due to the PM noise and the contribution of ambient gamma radiation) is well rejected, the counting rate thus only corresponding to the contribution of the sole charged atmospheric muons (see Fig. 3 in [START_REF] Autran | Characterization of Atmospheric Muons at Sea Level Using a Cosmic Ray Telescope[END_REF] for details) ;

(ii) the nominal value of the supplying voltage (around 1500 V) for each PM has been carefully selected in order to operate in the middle of the plateau of the PM counting rate -voltage characteristics ;

(iii) a calibration of the electronic acquisition chain has been also conducted to determine the MCA channel number versus time proportionality. The MCA was calibrated to a resolution of approximately 160 bins/ns.

Finally, since a detector may sometimes miss (i.e. do not detect) a muon passing through, the efficiency, η, of each detector has been carefully determined using the calibration procedure described in [START_REF] Autran | Characterization of Atmospheric Muons at Sea Level Using a Cosmic Ray Telescope[END_REF]. We found η F = 99% and η B = 99% for the front and the back detector, respectively.

Spectrometer modeling and simulation

Cutoff energy

To be detected by the back detector, muons must cross the instrument with a minimal energy, depending on the level of water in the tanks, called cutoff energy (E min ). Extensive simulations using TRIM (Transport of Ions in Matter) Monte Carlo simulation code [START_REF] Ziegler | The Stopping and Range of Ions in Matter[END_REF] have been performed to determine the cutoff energy of the spectrometer for each water level in the tanks. In these simulations we have taken into account the exact 1D stack of materials that the muons pass through from the metal roof of the building to the back detector, including the first floor (air and concrete slab) and the different materials of the spectrometer. Muons have been emulated in TRIM by applying a simple "mass scaling" to protons, as suggested in [START_REF] Tang | SEMM-2: A new generation of single-event-effect modeling tools[END_REF].

We estimate that signal pulses produced by a particle depositing around 2 MeV in the scintillator material of the detector can pass the constant fraction discriminator. Figure 3 and4 illustrate TRIM simulations used to determine the cutoff energy for a cumulative water level in the tanks of 150 cm. We consider an incident muon arriving in perpendicular incidence on the building roof with an incident energy of 374 MeV. The muon pass through the successive layers from the metallic roof to the back detector and its energy decreases until going to zero in the back detector where it definitely stops and is detected by the coincidence detection procedure. The variation of the muon energy and the energy lost in each layer are shown in Fig. 3 and4, respectively; the different layers crossed by the muon are also indicated in these figures. Our simulations show that muons with lower incident energies are stopped in the spectrometer layers before reaching the back detector (i.e. they will not be detected by the spectrometer), while muons with higher incident energies are systematically detected by the back detector. This means that the cutoff energy of the spectrometer for a water level of 150 cm is E min = 374 MeV. This value should be slightly higher for oblique tracks. The same procedure was used to determine the cutoff energy of the spectrometer for each level of water in the water tanks. Figure 5 shows the calculated cutoff energy of the spectrometer as a function of the cumulative height of the water level in the two water tanks.

Modeling of the spectrometer counting rate

The spectrometer counting-rate has been numerically estimated from a model directly derived from the work of Sullivan [START_REF] Sullivan | Geometrical factor and directional response of single and multi-element particle telescopes[END_REF] and Thomas and Willis [START_REF] Thomas | Analytical derivation of the geometric factor of a particle detector having circular or rectangular geometry[END_REF]. Figure 6 introduces the notations used in the following. For an anisotropic distribution of incident muons I (m -2 s -1 sr -1 ) following a power cosine law with the zenithal angle θ:

I = 𝐼 ! 𝑐𝑜𝑠 ! (𝜃) (1)
, the counting rate CR (s -1 ) of the spectrometer fixedly pointing in the vertical direction can be easily expressed as [START_REF] Thomas | Analytical derivation of the geometric factor of a particle detector having circular or rectangular geometry[END_REF]: where I 0 is the vertical muon flux (expressed in m -2 s -1 sr -1 ) integrated over the energy range of the instrument, η F and η B are the front and back detector efficiencies, respectively, and Ζ is the distance between the two square detectors of surface 4L 2 (see Figure 6).

CR = 𝐼 ! 𝜂 ! 𝜂 ! ! !!! ! ! ! ! ! !! ! ! ! ! ! !! ! ! !!! 𝑑𝑥 ! ! !! ! !! ! !! ! !! 𝑑𝑥 ! 𝑑𝑦 ! 𝑑𝑦 ! (2)
Assuming that the zenith angle distribution of muons follows Eq. ( 1) with a fixed value of exponent n in the energy range of interest of the instrument, Eq. ( 2) directly gives the value of I 0 for each experimental counting rate CR corresponding to a given water height in the tank, i.e. to a given cutoff energy E min .

x y

M 1 (x 1 ,y 1 ,Z) q z z O M 2 (x 2 ,y 2 ,0) Muon flux intensity [m -2 s -1 sr -1 ] I(q) = I 0 cos n (q) FRONT detector BACK detector dS 1 dS 2 2L 2L d Front detector efficiency h F Back detector efficiency h B Spectrometer counting rate CR [s -1 ] cos(q) = Z/d
For n = 2, Eq. ( 2) can be expressed analytically and the counting rate of the instrument is then given by:

𝐶𝑅 !!! ! = 𝐼 ! 𝜂 ! 𝜂 ! ×2𝐿 ! ! !!! ! ! ! !!! ! ×𝑎𝑟𝑐𝑡𝑎𝑛 !! ! ! !!! ! -𝑍×𝑎𝑟𝑐𝑡𝑎𝑛 !! ! (3)
where the superscript "a" is for "analytical" and the subscript "n=2" indicates that this expression is restricted to this particular value of parameter n. The experimental data are obtained using an additional setup based on a plastic shelf described in [START_REF] Autran | Characterization of Atmospheric Muons at Sea Level Using a Cosmic Ray Telescope[END_REF]. The curve labeled "Model" is obtained from Eq. ( 3) with I 0 = 100 µ m -2 .s -1 .sr -1 and n = 2.0, as experimentally obtained in [START_REF] Autran | Characterization of Atmospheric Muons at Sea Level Using a Cosmic Ray Telescope[END_REF].

Experimental results: sea level measurements

The muon spectrometer has been assembled indoors, on the ground floor of the experimental hall of a steel-frame industrial building with a single floor (concrete slab, thickness 10 cm) and a metal roof. The building is located on the Aix-Marseille University campus of Saint-Jérôme (Marseille, France, +43.337732°N, +5.412108°E, altitude 118 m). To check the validity of the spectrometer counting-rate modeling (Eq. 3), we performed preliminary measurements by varying the distance Z between the two detectors, using a plastic resin shelf with seven regularly spaced shelves. In this case, the two detectors were temporarily removed from the spectrometer setup and aligned vertically on two shelves of this complementary characterization setup. Figure 7 shows the results of the model characterization: the experimental data are in very good agreement in a distance range between 40 cm and 3 m with the results of simulation using the analytical model (Eq. ( 3)) presented above, taking values I 0 = 100 µ m -2 .s -1 .sr -1 and n = 2.0, as experimentally obtained in [START_REF] Autran | Characterization of Atmospheric Muons at Sea Level Using a Cosmic Ray Telescope[END_REF]. After this preliminary step, the muon spectrometer was continuously operated for around nine months for the acquisition of experimental data. Figure 8 shows the variations of the instrument counting rate as a function of the energy cutoff E min determined by the height of MeV, the corresponding muon momentum ranges from 0.176 to 0.596 GeV/c). Considering the first derivative of this I 0 (E min ) relationship (using the best linear fit on three consecutive points delimiting a small energy interval), we deduced the muon differential flux values reported in Figure 9. Experimental data measured in other published works [START_REF] Allkofer | A survey on the recent measurements of the absolute vertical cosmic-ray muon flux at sea level[END_REF][START_REF] Rastin | An accurate measurement of the sea-level muon spectrum within the range 4 to 3000 GeV/c[END_REF][START_REF] Kremer | Measurements of Ground-Level Muons at Two Geomagnetic Locations[END_REF] are also shown for comparison. Our values ideally complete the muon spectrum in an energy domain where only a few and sparse data were previously reported, especially below 200 MeV. These data are also consistent with previously published data and fits, in particular with the parabola-fitting model on a log-log scale given by [START_REF] Kempa | Low energy muons in the cosmic radiation[END_REF]:

log I θ = a 𝑙𝑛 ! 𝑝 + 𝑏 𝑙𝑛 𝑝 + 𝑐 ( 4
)
where p is the muon momentum in GeV/c and a, b, and c are three fitting parameters.

The analytical fit shown in Figure 9 has been obtained with Eq. ( 4) considering the values of a, b, and c reported in [START_REF] Kempa | Low energy muons in the cosmic radiation[END_REF] for a vertical zenith angle: a = -0.1292, b = -0.266 and c = -2.600. We observe that our data satisfactory agree with this analytical fit in the full energy domain covered by the instrument.

Conclusion

In this work, we developed an original setup, called "water tank muon spectrometer", to

Figure 1 .

 1 Figure 1. (a) Front view of the water tank muon spectrometer composed from two square

Figure 2 .

 2 Figure 2. Schematics of the electronics acquisition chain for measuring the time-of-flight of

Figure 3 .

 3 Figure 3. TRIM simulation showing the muon energy variation through the water tank 151

Figure 4 .

 4 Figure 4. TRIM simulation showing the muon energy loss through the water tank

Figure 5 .

 5 Figure 5. Muon cutoff energy of the spectrometer as a function of the cumulative height of

  Muon cutoff energy (MeV) Cummulative height of the water level in the tanks (cm) Without lead layer With 5 cm-thick lead layer

Figure 6 .

 6 Figure 6. Schematics of the front and back detectors, and definition of axes, angles and

Figure 7 .

 7 Figure 7. Counting rate as a function of the distance between the front and back detectors.

  counts/min) Distance between front and back detectors (m) Experimental data Analytical model -Eq.(3)

Figure 8 .

 8 Figure 8. Experimental counting rates (with and without the 50 mm-tick lead layer) as a

  in the water tanks. Two series of data have been acquired, with and without the 50 mm-thick lead layer. Each point corresponds to a value averaged over one week (168 hours). These results show a quasi-linear decreasing relationship between the counting rate and the instrument cutoff energy. They also show that the presence of the additional lead layer in the instrument material stack does not alter this linear dependency and only shift by ~50MeV the energy domain explored by the instrument.

Figure 9 .

 9 Figure 9. Muon spectrum at sea level deduced from data of Figure 8 and measured in other

  measure the energy distribution of low energy atmospheric muon in the energy range 100 to 500 MeV (momentum range 0.176 to 0.596 GeV/c) not yet covered by other ground-level instruments. The instrument use water as muon moderator and a controllable water height up to 2m to vary the muon cutoff energy. The differential flux for vertical muons at sea level (43°N of latitude) has been deduced from experimental measurements combined with the numerical modeling of the instrument. Differential flux values obtained between 100 and 500 MeV are consistent with previously published data and parabola fitting model available in literature. A short-term development will automate the filling and emptying of the water tanks, in order to make the measurements fully controllable by computer. With such an automated instrument, middle-term work will consist in a continuous muon flux monitoring to evidence possible flux variations as a function of various environment parameters.

Acknowledgments

This work has been conjointly supported by France's General Directorates DGA and DGE, under convention #132906128 (EVEREST project). The authors would like to thank T. Saad Saoud, A. Bchini, F. Derivaux and J. Beranger for their technical assistance during the development of the spectrometer.


Ulrik F Friis

An Goosens

Ana M Giménez-Arnau

Carola Lidén

Elena Giménez-Arnau

Ian R White

Jose H Alfonso

Wolfgang Uter

Jeanne D Johansen

Ulrik Fischer
email: ulrik.fischer.friis@regionh.dk

Self-testing for contact allergy to hair dyes -a 5-year follow-up multicentre study

Keywords: self-testing, self-test, hair dyes, allergy alert test, skin alert test

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

Introduction

Hair dyes contain different substances, some with allergenic properties [START_REF] Boonchai | Contact sensitizers in commercial hair dye products sold in Thailand[END_REF][START_REF] Hamann | p-Phenylenediamine and other allergens in hair dye products in the United States: a consumer exposure study[END_REF][START_REF] Yazar | Potent skin sensitizers in oxidative hair dye products on the Swedish market[END_REF][START_REF] Thorén | Contact allergens in 'natural' hair dyes[END_REF], like p-phenylenediamine (PPD) (CAS no. 106-50-3), toluene-2,5-diamine (CAS no. 95-70-5), p-aminophenol (CAS no. 123-30-8), resorcinol (CAS no. 108-46-3) and in hair bleaching products the persulfates (ammonium-, potassium-and sodium persulfate (CAS no. 7727-54-0; 7727-21-1 and 7775-27-1)) [START_REF] Kirchlechner | Survey of sensitizing components of oxidative hair dyes (retail and professional products) in Germany[END_REF][START_REF] Schwensen | Occupational contact dermatitis in hairdressers: an analysis of patch test data from the Danish Contact Dermatitis Group, 2002-2011[END_REF][START_REF] Søsted | Contact allergy to common ingredients in hair dyes[END_REF][START_REF] Yazar | p-Phenylenediamine and other hair dye sensitizers in Spain[END_REF].

In 2006 the European Commission's Scientific Committee on Consumer Products (SCCP) published a memorandum on substances in hair dyes and their skin sensitising properties, in which 27 substances met the EU criteria for classification as a skin sensitizer (R43/H317) [START_REF]Memorandum on Hair Dye Substances and their Skin Sensitising Properties[END_REF]. In 2013 they updated this memorandum, which now contains a list of 114 hair dye chemicals, for which 13 substances were classified as extreme, 23 as strong, and 20 as moderate sensitizers, based on OECD test guideline methods. This meant that 56 substances used in hair dyes met the EU criteria as a skin sensitizer (R43/H317) [START_REF]Memorandum on hair dye Chemical Sensitisation[END_REF]. The SCCP stated that substances fulfilling classification as R43/H317 may not be safe for the consumer, especially the ones that are categorized as extreme and strong sensitizers. This means that there is a risk for costumers of becoming sensitized, which is in accordance with the clinical experience that hair dyes sometimes cause severe allergic symptoms in consumers/customers (6;7) In 2011, a multicentre study was conducted in order to describe how hair dye manufacturers instructed consumers to do a self-test (also called 'allergy alert test' by industry) prior to dyeing their hair to identify individuals who are likely to react upon subsequent hair dyeing [START_REF] Thyssen | Self-testing for contact sensitization to hair dyes--scientific considerations and clinical concerns of an industry-led screening programme[END_REF]. A number of concerns were raised concerning the variability in instructions between products and producers and generally questioning the safety and validity of this tool. The EU Commission held a workshop on 'Skin Sensitivity Test' in April 2011 to help manufacturers address these issues. The aim of this study was to perform a 5 year follow-up to see if manufacturers still recommend a self-test and if so, if the procedures have been standardized.

Methods

During March 2016, a total of 40 permanent hair dye products from 21 different manufactures were bought in retail stores in eight countries (Belgium, Denmark, France, Germany, Norway, Spain, Sweden and the United Kingdom). In 39/40 (97.5%) of products from 21 manufactures (Bysalerm, Casual Spa, Clairol, Eugène, Eugene Perma Paris, FarmaVita, Frank Provost, Garnier, Hardford AB Sweden, "Jihanna Laboratorium Kosmetyczne, Joanna Sp. Poland", Llongueras Colomer, L'Oreal, L'Oreal professional, Naturigin Organic Beauty, Phergal, Schwarzkopf, Syoss, TAHE, Tigi Int. Ltd, Vitanove & Pascal Coste (Marque répère Eleclerc) and Wella), the consumer was instructed to do a self-test prior to hair dyeing.

Thirty five of the products were commercial available and five was for professional use. The products for professional use were all bought in Belgium. We investigated instructions in the product leaflets for the self-test procedure for "amount of hair dye and suggested area size", "place of test", "re-application", "rinsing", "resting time" and "definition of positive allergic reaction".

Results

There was much variation in the instructions, (Table 1). Three instructions stated that the colouring agent should be oxidized [supposedly by mixing colouring agent and developer], five suggested that a mixture of colouring agent and developer should be applied, and the rest suggested to apply only the colouring agent or dyeing product (cream). The amount of hair dye to be applied on the skin ranged from "sufficient", "small amount", "a touch of", "a pinch of", "some" to "a thin layer". The suggested site for making the selftest varied from behind the ear, the earlobe, the elbow flexure, the upper arm, to no advice at all. The suggested size of the skin area for application ranged from "not specified", to 1-2 cm 2 , size of a EURO-cent coin (2.1 cm 2 ), a 1-Danish-crown coin (3.2cm 2 ), a 1-Norwegian-crown coin (3.5 cm 2 ) and a 10-Swedishcrown coin (3.3 cm 2 ). For 10 (35.7%) of the products up to 3 applications were recommended with reapplications "after drying". No applicator was suggested for four products, a piece of cotton wool for one, and a cotton bud for the others. No rinsing was recommended for 13 (32.5%) products, for one product rinsing was to be performed after 30 minutes, for 13 products after 45 minutes, and for 12 products after 48 hours. The time before rinsing differed between the different products but also within the same manufacturer, see supplementary data. The time before reading was 48 h in all cases; for two products readings were also recommended after 45 min and after 24 h, respectively. There was no consistent description on how a positive reaction should look, but the terms redness, itchy and swelling were mentioned in most of the product leaflet instructions.

Discussion

The results of this study show that no progress has been made the last 5 years, see The size of the test area was sometimes compared with the size of coins (Table 1). [START_REF] Thyssen | Self-testing for contact sensitization to hair dyes--scientific considerations and clinical concerns of an industry-led screening programme[END_REF]. The instructions still vary considerably, more than in the previous investigation and even between different hair dye products from the same manufacturer. The sample size in 2011 was relatively small by 17 different products from 13 different manufacturers, bought in 6 countries. This follow-up study is much larger, including 40 different hair dye products from 21 different manufacturers, bought in 8 different countries, and is therefore considered to be more representative. The leaflets were analysed concerning the same issues related to performing the self-test prior to hair-dyeing as 5 years ago, namely, (a) the amount of hair dye applied, (b) the number of applications, (c) when to rinse, (d) the resting time before reading and (e) the definition of a positive reaction. In spite that several products were of the same brand and from the same manufacturer still major differences were found in the instructions, see

No instruction

Cosmetics Europe has made an on-line introduction on "how to do the 48 hour Allergy Alert Test?" [START_REF]How to do the 48 hour Allergy Alert Test?[END_REF].

Here, they briefly inform that the manufactures may have slightly different ways to perform the test, and that it is based on their experience with the particular method, but also because of the variation of product formulations, packaging and applications [START_REF]How to do the 48 hour Allergy Alert Test?[END_REF]. However, this cannot sensibly explain the major variations in instructions found in this study. Thyssen et al. concluded in 2013 that the self-test has some limitations: (i) it is not a screening test but a test; (ii) it has not been validated according to basic criteria defined by scientists; (iii) it has been evaluated in PPD allergic individuals, which is an inappropriate population for the evaluation of diagnostic performance, as the self-test is supposed to be used by persons who are not diagnosed with PPD allergy yet; (iv) skin reactions have been read by dermatologists and not by the targeted group (consumers and hairdressers); (v) hair dyes contain strong and extreme sensitizers that are left on the skin in high concentrations, potentially resulting in active sensitization; and (vi) recommendations and instructions on how to perform the hair dye self-test vary greatly, even among products from the same company, again suggesting that the basis for safe use of the test has not been determined [START_REF] Thyssen | Self-testing for contact sensitization to hair dyes--scientific considerations and clinical concerns of an industry-led screening programme[END_REF].

In April 2011 a workshop was held by the EU Commission to help manufacturers develop a uniform protocol for pre-testing and perform risk assessment. This was reported to the subgroup on skin allergies on a meeting in the group November 2012, a subgroup under the EU Commission expert working group on Cosmetic Products [START_REF] Anonymous | Minutes from the meeting of the Sub-Group on Skin Allergens[END_REF]. According to the minutes from the meeting the validation study should be launched by the industry in the first quarter of 2013 by testing selected PPD-positive hair dye allergic subject. This testing phase should last until 2015 when testing controls would be carried out. The analysis of the obtained data would be concluded by a study report in 2016. It is surprising that the study is carried out in PPD-allergic individuals again, as this is exactly a major criticism raised before: this is not the correct target group and therefore it cannot be predicted how the test will perform in consumers, who are not yet diagnosed with PPD allergy [START_REF] Thyssen | Self-testing for contact sensitization to hair dyes--scientific considerations and clinical concerns of an industry-led screening programme[END_REF].

According to the minutes from the meeting of the Sub-Group on Skin Allergens 20 November 2012 [START_REF] Anonymous | Minutes from the meeting of the Sub-Group on Skin Allergens[END_REF] some Member States experts were concerned about this approach because the safety of the self-test had not been established yet. They expressed their concerns that industry continues to advise consumers to perform the self-test even though the test has not been validated. In 2012, the status of the test was that it was being investigated and until the result of that was available, the test should not be recommended to consumers. It was also suggested to have an opinion from the SCCS on the proof of concept study [START_REF] Anonymous | Minutes from the meeting of the Sub-Group on Skin Allergens[END_REF].

Overall, the Member States experts were of the opinion that the proposed time plan was too long and that the alignment of the vast majority of the test parameters could have been done by industry much earlier.

Industry agreed that such an alignment was feasible before the completion of the proof of concept study.

Industry was to propose a harmonised approach for the parameters of the self-test by February 2013 [START_REF] Anonymous | Minutes from the meeting of the Sub-Group on Skin Allergens[END_REF].

According to the minutes from the meeting in 2012, some Member States had the same concerns as outlined above in the paper by Thyssen et al (11;14). However, the current study shows that no changes have been made, no assessment of risk to the consumer by performing the test seems to have been done in spite being announced since 2012, and advice of doing the self-test is still given to every consumer buying a hair dye product.

Conclusion

In conclusion, the concerns regarding safety and validity previously raised still remain. It is of major concern that no assessment of the risk to the consumer of performing the test has been made, and no standardization has been performed.

Figure 1 .

 1 Figure 1. Comparison between hair dye product instructions on how to perform the self-test before hair dyeing in 2011 and 2016. Figures to the left are based on data in Table 2 (n=20) in Thyssen et al, 2012[START_REF] Thyssen | Self-testing for contact sensitization to hair dyes--scientific considerations and clinical concerns of an industry-led screening programme[END_REF]. Figures to the right are based on data in the current study (n=31). The size of the test area was sometimes compared with the size of coins (Table1).

Table 1

 1

				3%
	15%		13%	None
		45%			2 times
	25%		29%	55%	2-3 times
	15%				3 times
				6%	Small area
				10%	1 cm2
	35%	45%	19%		Approx. 2 cm2
				42%	1-2 cm2
	20%		10%	13%	Approx. 3 cm2

and the supplementary data. Cosmetics Europe -The Personal Care Association (the former European Cosmetics Toiletry and Perfumery Association -COLIPA) advises hair dye manufactures to recommend to the users of their products the self-test as a precaution against any allergic reaction to permanent hair dyes

[START_REF]Frequently asked questions -3. What is the Allergy Alert Test?[END_REF]

.

Table 1 . Overview of the allergy self-test instructions found inside the 40 different hair dye products bought in the eight different countries 1 . Purchase: March 2016. n=40.

 1 A crown coin could either be a Danish ( ~3.2cm Comparison between hair dye product instructions on how to perform the self-test before hair dyeing in 2011 and 2016. Figures to the left are based on data in Table 2 (n=20) in Thyssen et al, 2012[START_REF] Thyssen | Self-testing for contact sensitization to hair dyes--scientific considerations and clinical concerns of an industry-led screening programme[END_REF]. Figures to the right are based on data in the current study (n=31). The size of the test area was sometimes compared with the size of coins (Table1).

	Amount of hair dye and suggested area size	Place of test	Number of application and if mentioned when	Rinsing	Resting time	Definition of positive allergic reaction
	Apply " sufficient amount"	2cm 2 (n=1)	no suggestions (n=1)	Not specified -repeated 2-3 times (n=1),	No rinse (n=1)	48 h (n=1)	Itch, pain, burning sensation,
	(n=10)						erythema, swelling (n=1)
		1-crown coin* (n=3)	Behind the ear (n=3)	After drying -repeated 2 times (n=3),	After 48 h (n=3)	Up to 48 h (n=3)	Non-normal reaction such as itch,
							redness, or swelling at or around the
							area (n=3)
		1 Euro cent coin***	Behind the earlobe (n=1)	Not specified -repeated 2 times (n=1)	No rinse and no	Up to 48 h (n=1)	Itch, erythema. Specification: do not
		(n=1)			cover (n=1)		use once the age is <16 (n=1)
		1cm 2 (n=4)	Inner elbow (n=1)	Not specified -repeated 1 times (n=1),	No rinse (n=1)	Up to 48 h (n=1)	Burning sensation, erythema (n=1)
			no suggestions (n=1)	Not specified -repeated 2 times (n=1)	No rinse (n=1),	Up to 48 h (n=1)	Burning sensation, erythema (n=1)
			no suggestions (n=1)	Not specified -repeated 2-3 times (n=1)	No rinse (n=1),	48 h (n=1)	Itch, erythema (n=1)
			Behind the earlobe (n=1)	It's not specified -repeated 1 time (n=1),	No rinse (n=1)	Up to 48 h (n=1)	Itch, erythema. Specification: do not
							use once the age is <16 (n=1)
		1-2cm 2 (n=1)	Behind the earlobe (n=1)	Not specified -repeated 3 times (n=1)	No rinse (n=1)	Up to 48 h (n=1)	Itch, erythema (n=1)
	Sufficient amount of	1 Euro cent coin***	Behind the ear (n=3)	2 times (n=1)	No rinse (n=1)	48 h (n=1)	Itching, redness, swelling (n=1)
	colouring agent (n=4)	(n=3)		2-3 times (n=1)	No rinse (n=1)	48 h (n=1)	Itching, redness, swelling (n=1)
				Repeated (n=1)	No rinse (n=1)	48 h (n=1)	Itching, redness, swelling (n=1)
		1-2cm 2 (n=1)	Behind the ear (n=1)	2 times (n=1)	No rinse (n=1)	48 h (n=1)	Itching, redness, swelling (n=1)
	Small quantity of colouring	1cm 2 (n=1)	Inner elbow (n=1)	1 time (n=1)	After 45 min (n=1)	48 h (n=1)	Any reaction (n=1)
	agent (n=2)	1-2cm 2 (n=1)	Forearm or behind the	1 time (n=1)	After 48 h (n=1)	48 h (n=1)	Itching, redness, swelling (n=1)
			ear (n=1)
	Small quantity of unmixed	1-2cm 2 (n=1)	Behind the ear (n=1)	After drying -repeated 2 times (n=1)	After 45 h (n=1)	48 h (n=1)	Any abnormal reactions, such as
	colourant product (creme						itching, redness or swelling in or
	colourant tube) (n=1)						around the test area (n=1)
	Small amount of colour	1cm*1cm (n=1)	Inside of the elbow (n=1)	Not applicable -repeated 1 time (n=1)	After 45 min (n=1)	Up to 48 h (n=1)	Any reaction (n=1)
	cream in a thin layer (n=1)
	Small amount of colour	Not specified (n=1)	Inside of the elbow (n=1)	Not applicable -repeated 1 time (n=1)	After 45 min (n=1)	Up to 48 h (n=1)	Redness or itch (n=1)
	cream (n=1)
	Small amount of colour	Not specified (n=1)	Inside of the elbow (n=1)	Not applicable -repeated 1 time (n=1)	After 45 min (n=1)	Up to 48 h (n=1)	Any reaction (n=1)
	cream mixed with oxidizer in
	a thin layer (n=1)
	A thin layer (n=1)	1cm*1cm (n=1)	Inside of the elbow (n=1)	Not applicable -repeated 1 time (n=1)	After 45 min (n=1)	Up to 48 h (n=1)	Any reaction (n=1)
	A touch of dyeing cream	1cm 2 (n=1)	Elbow crease, upper arm	Not applicable -repeated 1 time (n=1)	After 45 min, with	Reading at 45	Tingling-, burning-, blackout feeling,
	(n=1)		(n=1)		warm water (n=1)	min and 48 h	redness, itching, eczema, bulling,
						(n=1)	swelling of eyes or face, weeping
							(n=1)
	A "pinch" amount of colour	1cm*1cm (n=1)	Inside of the elbow (n=1)	Not applicable -repeated 1 time (n=1)	After 45 min (n=1)	Up to 48 h (n=1)	Any reaction (n=1)
	cream and the activator
	(n=1)
	Not defined: some ["etwas"	1cm*1cm (n=1)	no suggestions (n=1)	Not applicable -repeated 1 time (n=1)	Rinse thoroughly	48 h (n=1)	"reactions" (n=1)
	(some)] (n=1)				after 45 min
	Apply "sufficient amount",	1-crown coin*	Behind the ear (n=2)	After drying -repeated 2 times (n=2)	Not mentioned.	48 h (n=2)	Any abnormal reactions, such as
	not mixed (n=2)	(n=2),			Area should not be		itching, redness or swelling in or
					touched, washed or		around the test area (n=2)
					covered for 48h
					(n=2)
		1-2cm 2 (n=1)	Behind the ear (n=1)	After drying -repeated 2 times (n=1)	After 48 h (n=1)	48 h (n=1)	Itching, redness or skin rash, swelling
							(n=1)
	Sufficient quantity (n=2)	1 Euro cent coin ***	Behind the ear (n=2)	After drying -repeated 2-3 times (n=2)	Not mentioned.	48 h (n=2)	Red or itchy or swollen skin (n=2)
		(n=2)			Area should not be
					touched, washed or
					covered for 48h
					(n=2)
	Small amount of colour	1cm*1cm (n=2)	In the elbow fold (n=2)	Not applicable -repeated 1 time (n=2)	After 45 min. With	48 h (n=2)	Any reaction (n=2)
	cream in a thin layer (n=2)				warm water. Avoid
					covered and
					contact with textile
					(n=2)
	Small amount of mixture	1cm 2 (n=1)	Behind the ear or elbow	Not applicable -repeated 1 time (n=1)	After 48 h (n=1)	48 h (n=1)	Redness, tingling feeling, itching,
	hair dye developer		crease (n=1)				swelling of eyes or face (n=1)
	("révélateur"-hydrogen	just mentioned	Behind the ear or elbow	Not applicable -repeated 1 time (n=1)	After 30 min (n=1)	48 h (n=1)	Redness, tingling feeling, itching
	peroxide) (n=1)	"small surface"	crease or upper arm (n=1)				(n=1)
		(n=1)
	Thin layer of mixture	1cm 2 (n=1)	Inner elbow (n=1)	1 time (n=1)	After 45 min (n=1)	48 h (n=1)	Any reaction (n=1)
	Sufficient amount of mixture	1cm 2 (n=1)	Behind the ear or elbow	1 time (n=1)	No rinse (n=1)	48 h (n=1)	Rash, stinging, itching (n=1)
	(n=1)		crease (n=1)
	Quantity unspecific of	1cm 2 (n=1)	Behind the ear or elbow	2-3 times (n=1)	No rinse (n=1)	48 h (n=1)	Any reaction (n=1)
	mixture (n=1)		crease (n=1)
	Not specified (n=1)	10-crown coin** or	Behind the ear (n=1)	Not applicable -repeated 1 time (n=1)	After 45 min (n=1)	Up to 48 h (n=1)	Swelling, itch, redness at or around
		1 cm2 (n=1)					the area (n=1)
	a small quantity of the hair	a small area of skin	behind the ear or on the	Not applicable -repeated 1 time (n=1)	After 48 hour with	24 and 48 h	Any reaction (such as swelling,
	dye as prepared for use	(n=1)	inner surface of the		soap and water	(n=1)	redness, itching, eczema, blistering
	(n=1)		forearm (n=1)		(n=1)		etc) (n=1)
	Mix a small amount of tint	1cm 2 (n=1)	Bend of your elbow (n=1)	Not applicable -repeated 1 time (n=1)	After 45 min (n=1)	Up to 48 h (n=1)	Any signs of rash, redness, swelling,
	and developer. Apply						burning, itching or similar (n=1)
	mixture to test area(n=1)
	No recommendations of an	-	-	-	-	-	-
	self-test (n=1) (n=1)

1

Belgium, Denmark, France, Germany, Norway, Spain, Sweden and the United Kingdom *

The research centre for hairdresser and beauticians is funded by an unrestricted grant from the Danish Hairdressers' and Beauticians' Union and the Danish Hairdresser Association. All the authors contributed by collecting data on hair dyes products form their home country