Dynamic Decision Making when Risk
Perception depends on Past Experience∗
Michèle Cohen
CES, University Paris 1
Johanna Etner
GAINS, University of Maine and CES, University Paris 1
Meglena Jeleva
GAINS, University of Maine, and CES, University Paris 1
October 2, 2006

Abstract
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.
∗

We thank A. Chateauneuf, R. Giraud, J. Y. Jaﬀray, E. Karni, P. Klibanoﬀ, J. C.
Vergnaud and the participants of FUR XII for valuable comments and useful suggestions.

11Introduction

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 (Slovic 2000) point out the fact that risk perception may bestrongly influenced by the context in which the individuals are when theytake 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 Kunreuther (1996) and Brown, Hoyt (2000)) or otherevents 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 Hirshleifer, Shumway 2003), (ii) it canalso correspond to anticipatory feelings about some future states (Caplin,
Leahy 2001), (iii) it can be related to the decision outcomes presentation(leading then to the framing eﬀect pointed out by Tversky and Kahneman1986).
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 thedecision-relevant events (as accidents when an insurance decision is considered) or (ii) realization of events, others than the decision relevant ones (suchas 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 thenon insured individuals subscribed an insurance contract (Kunreuther 1996).
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 past2

experience and insurance demand appears also in experimental studies when
individuals are well informed about the probability of loss realization andabout the independence of losses in successive periods (McClelland, Schulze,
Coursey 1993, Ganderton et al. 2000, Papon 2004). Their results are lessclear-cut: if the existence of a strong correlation between past damages andinsurance 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 Tversky and Kahneman (1973). 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 unstablepreferences 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 firststep consists in considering a one period decision problem where individual
preferences are no more defined only on decisions but on pairs (decision, pastexperience). 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 Kahneman-Tversky (1979),
Quiggin (1982) and Yaari (1987). This model has the advantage to allow a
non linear treatment of probabilities, in addition to a non linear treatmentof outcomes. When one period decisions are considered, we adapt the RDU
3

axiomatic system of Chateauneuf (1999) 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 ruleout this problem, we use the recursive model of Kreps and Porteus (1978). 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 thepresent paper are introduced additional aspects: (i) risk perception is nonlinear and depends on a probability transformation function; (ii) past experience does not reduce anymore to the only payoﬀs, but is characterized by asequence 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 inorder 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.

2

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 whichtakes into account the agent’s past experience.

2.1

Notations and definitions

Decision problem is characterized by a set of risky perspectives in which anagent has to make his choice and by a set of states that characterize theagent’s past experience.
4

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 relationof a decision maker. We denote by Â the strict preference and by ∼ theindiﬀerence.
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 bya Rank Dependant Utility (RDU) model. Then, we give some additional
assumptions to achieve a RDU preferences representation on S × L.

2.2

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 thefollowing axioms, proposed in Chateauneuf [1999].
From Axiom 1, it follows directly that %s is a weak order on s × L.
Axiom 2 (Continuity) For a given s ∈ S, let Pn = (s, Ln ) , P = (s, L) , Q =(s, L0 ) ∈ s × L, with Pn weakly converging to P , then ∀n, Pn %s Q ⇒ P %s Q
and ∀n, Pn -s Q ⇒ P -s Q.
For a given s, it is possible to completely order the space Z. The definitionof the first order stochastic dominance (SD1) becomes L SD1 L0 if and only
if PL (z ∈ Z, (s, z) Âs (s, x)) ≥ PL0 (z ∈ Z, (s, z) Âs (s, x)) ∀x ∈ Z.
The next axiom guarantees that %s preserves first order stochastic dominance.
Axiom 3 For any L, L0 ∈ L such that L SD1 L0 , (s, L) %s (s, L0 ) .

5Axiom 4 (Comonotonic sure-thing principle) For any s ∈ S, let lotteries P = ((s, z i ) pi ) , Q = ((s, y i ) pi ) be such (s, z i0 ) ∼s (s, y i0 ), then P %s Q
implies P 0 %s Q0 , for lotteries P 0 , Q0 obtained from lotteries P and Q byi0i0
merely replacing the ith
0 common pair (s, z ), by a common pair (s, x ) again
00
in ith0 rank both in P and Q .

Axiom 5 (Comonotonic mixture independance axiom) For any s ∈S, and for any lotteries P = ((s, z i ) pi ) and Q = ((s, y i ) qi )),For any p ∈ [0, 1] , for any a, b, c, d ∈ Z¡¡¢¢• P1 = (1 − p) s, z min + p (s, a) ∼s Q1 = (1 − p) s, y min + p (s, b)¡¡¢¢
and P2 = (1 − p) s, z min + p (s, c) ∼s Q2 = (1 − p) s, y min + p (s0 , d)
imply ∀α ∈ [0, 1] , αP1 + (1 − α) P2 ∼s αQ1 + (1 − α) Q2• P1 = (1 − p) (s, z max ) + p (s, a) ∼s Q1 = (1 − p) (s, y max ) + p (s0 , b)
and P2 = (1 − p) (s, z max ) + p (s, c) ∼s Q2 = (1 − p) (s, y max ) + p (s0 , d)
imply ∀α ∈ [0, 1] , αP1 + (1 − α) P2 ∼s αQ1 + (1 − α) Q2
Theorem 1 Let the preference relation %s on s×L satisfy axioms 1 - 5, thenthere exist an increasing function ϕs : [0, 1] → [0, 1], with ϕs (0) = 0, ϕs (1) =1 and a utility function, vs : Z → R, which is increasing, continuous, andunique up to an aﬃne transformation such that:∀L, L0 ∈ L, (s, L) %s (s, L0 ) iﬀ Vs (L) ≥ Vs (L0 ) withVs (L) =

n
X
i=1

Ã

!
n
n
X
X
ϕs (
pj ) − ϕs (pj ) × vs (z i )
j=i

j=i+1

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.
6

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, mainlywith respect to the state-dependant model of Karni (1985). 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 thanfuture monetary evaluations.
The following axiom guarantees that the payoﬀs evaluations do not depend on past experience.
Axiom 6 For any s, s0 ∈ S and any z ∈ Z, (s, z) ∼ (s0 , 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 writez ≥ 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, s0 ∈ S and any L, L0 ∈ L ,(s, L) % (s0 , L0 ) ⇔ V (s, L) ≥ V (s0 , L0 )
Ã
!
n
n
n
P
P
P
ϕs ( pj ) − ϕs (pj ) v(z i ).
where V (s, L) =
i=1

j=i

j=i+1

Proof. The generalization of the preferences representation of the restrictions %s on % is allowed by the unicity of the probability transformation
function ϕs (p) and the independence of the utility function on s.
We can see that a decision maker with ϕs (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, s0 ∈ S. ϕ(s, p) ≥ ϕ(s0 , p) for any p ∈ [0, 1] if and only
V (s, L) ≥ V (s0 , L) for any L ∈ L.
7

Proof. Let L = (z1 , p1 ; z2 , p2 ; ...; zn , pn )with z1 ≤ ... ≤ zn Note that thisaxiom implies that for any z, z 0 ∈ÃZ such that z ≥ z 0 , (s,!z) %s (s, z 0 ) .
n
n
n
P
P
P
ϕs ( pj ) − ϕs0 ( pj ) (v(z i ) − v(z i−1 )) ≥(i) ⇒: V (s, L)−V (s0 , L) =
i=2

j=i

j=i

0 if ϕ(s, p) ≥ ϕ(s0 , p) for any p ∈ [0, 1];(ii) ⇐: if there exists p0 such that ϕ(s, p0 ) < ϕ(s0 , p0 ) then for L0 =(z1 , 1 − p0 ; z2 , p0 ), V (s, L0 ) < V (s0 , L0 ).

This result implies that if there exists a realized state that induces pessimistic risk perception, then an individual will dislike any decision in thiscontext, with respect to a state where his risk perception is less pessimistic.

3

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 sectionof the paper, by a past experience dependent RDU. It is now well known (seefor instance Machina 1989) 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
Kreps and Porteus (1978) model in order to introduce both risk perception
and past experience dependence.

3.1

Some notations and definitions

We consider a discrete and finite sequence of times t = 1, ..., T. Zt is the setof possible payoﬀs at time t. To simplify, we assume that Zt = Z which is
a compact interval of R for any t from 1 to T . A payoﬀ realized at time t
is denoted by zt . Decision maker past experience at time t is characterized
by a sequence of events, relevant for the considered decision and denoted byst . More precisely, st = (e0 , e1 , ..., et ) where eτ is the event that occurred attime τ with eτ ∈ Eτ , the set of all possible events at time τ and e0 ∈ E0 the8set of all possible past experiences. St is then the set of possible histories up
to time t verifying the recursive relation: S0 = E0 and St = St−1 × Et . We
denote by M (Et ) the set of distributions on Et .
At period T , LT is the set of distributions on ZT endowed with the Prohorov metric. XT , 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 LT , endowedwith the Hausdorﬀ metric.
By recurrence, we define, Lt , the set of probability distributions on
Ct =Zt ×Xt+1 × M (Et+1 ) with Xt+1 the set of closed non empty subsetsof Lt+1 .
At each period, the nature chooses a probability distribution on Et+1 .
The agent cannot influence this distribution. Given this distribution, theagent has to choose a lottery in the set Lt . 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 st × Lt for a given st .
We denote Ât the strict preference and ∼t the indiﬀerence.
Axiom 7 (1bis) %t is a complete order on st × Lt .
We assume that %t verifies axioms 2 - 5 on st × Lt .
Under these previous axioms, we can represent preferences at a point oftime in the similar way to theorem 1:
Lemma 1 For any st in St , axioms 1bis, 2-5 are necessary and suﬃcientfor there to exist, for each t, a bounded continuous function vt : st × Zt ×
Xt+1 × Et+1 −→ R and a continuous function ϕt : st × [0, 1] −→ [0, 1] suchthat for Lt , L0t ∈ Lt , (st , Lt ) %t (st , L0t ) if and only if Vt (st , Lt ) ≥ Vt (st , L0t )
with
Ã
!
n
n
n
P
P
P
ϕt (st , pj ) − ϕt (st ,Vt (st , Lt ) =pj ) × vt (st , zt , xt+1 , et+1 ) ≡
i=1

j=i

j=i+1

RDvt (st , Lt ) .

The proof comes immediately from the previous section. Let us noticethat utility function vt depend on past experience st whereas in a static
9

Figure 1:
problem, v did not depend on state s. This comes from the fact that ina dynamic choice, future perspective depend on past experience. We will
precise this point in the next section.

3.2

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,∆et in M (Et ) and for a given distribution δ et+1 in ∆et+1 , the degenerate
distributions on Zt ×Xt+1 ×δ et+1 . Then, for all t, st ∈ St , et+1 ∈ Et+1 , zt ∈ Zt ,xt+1 , x0t+1 ∈ Xt+1 ,¡¢¡¢(st , zt , xt+1 , et+1 ) %t st , zt , x0t+1 , et+1 iﬀ (st+1 , xt+1 ) %t+1 st+1 , x0t+1
with st+1 = (st , et+1 ) at period t + 1.
Let us consider the two following lotteries:
The temporal consistency axiom sets that if the degenerate lottery (zt , xt , et )is preferred to the degenerate lottery (zt0 , x0t , et ) for a decision maker with past10

experience st , then if at time t, et is realized, the decision maker will choose
xt between xt and x0t . In the same way, if at time t + 1, he chooses xt , thenhe cannot at time t, strictly prefer (zt0 , x0t , et ) to (zt , xt , et ).
Lemma 2 Axioms 1bis, 2-5 and 8 are necessary and suﬃcient for there toexist functions vt as in previous lemma and, for fixed {vt }, unique functions
ut : {(st , zt , γ) ∈ St × Zt × R : γ = RDvt+1 (st+1 , Lt+1 )} → R
which are strictly increasing in their third argument and which satisfy
vt (st , zt , xt+1 , et+1 ) = ut (st , zt , Lt+1 maxRDvt+1 (st+1 , Lt+1 )) for all st ∈St , et+1 ∈ Et+1 , zt ∈ Zt , xt+1 ∈ Xt+1 .
Proof.
Axioms 1bis-5 and 8 are hold, lemma 3 fix Vt (st , Lt ) . Then,¡¢¡¢
Vt+1 ((st , et+1 ) , Lt+1 ) = Vt+1 (st , et+1 ) , L0t+1 =⇒ Vt (st , Lt+1 ) = Vt st , L0t+1
for a given et+1 (axiom TC). Consequently, ut is strictly increasing in its
third argument.
(ii) If Vt and ut are given with ut is strictly increasing in its third argument, Vt verifies lemma 3. Then axioms 1bis-6 hold. ut is increasing in
its third argument then Vt (st , L) ≥ Vt (st , L0 ) ⇐⇒ ut (st , zt , Vt+1 (st+1 , L)) ≥ut (st , zt , Vt+1 (st+1 , L0 ))⇐⇒ Vt+1 ((st , et+1 ) , L) ≥ Vt+1 ((st , et+1 ) , L0 ). Then axiom 8 holds.
As we can see, in the dynamic problem, utility functions vt depend on pastexperience st . Indeed, at time T , utility function does not depend on pastexperience. But, at time T − 1, the certainty equivalent of lottery, given byuT −1 , depends on past experience sT −1. Consequently, vT −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 : ST × ZT → R and, for t = 0, ..., T − 1,continuous functions ut : St × Zt × R → R, strictly increasing in their thirdargument, so that, vT (sT , zT ) = v (zT ) and, recursively
vt (st , zt , xt+1 , et+1 ) = Lt+1 maxut (st , zt , RDvt+1 (st+1 , Lt+1 )),11then, for all st ∈ St , Lt , L0t ∈ Lt , (st , Lt ) %t (st , L0t ) iﬀ RDvt (st , Lt ) ≥RDvt (st , L0t )
Ã
!
n
n
n
P
P
P
ϕt (st , pj ) − ϕt (st ,pj ) × vt (st , lt ).
with RDvt (st , Lt ) =
i=1

j=i

j=i+1

Proof. We adapt the demonstration of the theorem in Kreps and Porteus.

Notice that the representation needs v, the functions ut and the functions
ϕt to implicitly define functions vt . 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.

4

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 ateach period. There exist a perfectly competitive insurance market proposing insurance contracts at a fair premium corresponding to the estimatedexpected 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 periodt/ loss in period t − 1) = p.
We can represent the decision tree as following:At each period, the insurance contract Ct is characterized by a pair (indemnity It , premium Πt ). Only proportional coverage is considered: It = αt L
with αt ∈ [0, 1]. The premium for a coverage rate αt is Πt = αt pL. At periodt = 0, the agent receives a certain wealth, z0 , and he is in the state s0 . Past
experience is resumed by the sequences of events {damage, no damage}. We
12

Figure 2:

13

denote by et the event ”damage at period t” and by e0t the event ”no damageat period t”. At each period, individual has to choose αt .
At a point of time, the probability transformation function is assumed tobe the following:ϕt (p, st ) = p

t
P

eτ

τ =0

with e0 = 12 , et = 12 and e0t = 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 adamage modifies his risk perception and he becomes less optimistic, if nodamages 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 undercertainty.
Moreover, for simplicity, we suppose that the individual is neutral toward
the time of resolution of uncertainty (in the sense of Kreps and Porteus), sothat ut (st , zt , RDvt+1 (st+1 , Lt+1 )) = RDvt+1 (st+1 , Lt+1 ) .
The dynamic choice problem solves in several steps. Note that, due tothe linear utility assumption, only corner solutions will prevail.
(i) For each terminal node, we have to compute v (s3 , z3 ) = z3 ,with z3 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 L3 , for a coverage rate α3 , the utility writes:¡¢
V3 (s3 , L3 ) = RDv (s3 , L3 ) = v s3 , z3 +¡¢¤£v (s3 , z 3 ) − v s3 , z3 × ϕ3 (1 − p, s3 )with z3 = z2 − Π3 − L + I3 = z2 − L + α3 L (1 − p), z 3 = z2 − Π3 =3z2 − α3 Lp, and ϕ3 (1 − p, s3 ) = (1 − p)e0 +e1 +e2 = (1 − p) 2 .
14

Then,3

V3 (s3 , L3 ) = z2 − L + α3 L (1 − p) + L [1 − α3 ] × (1 − p) 2hi
13
= α3 L (1 − p) 1 − (1 − p) 2 + z2 − L + L (1 − p) 2
As utility is an increasing function of the coverage rate, the optimal
coverage is the full coverage and the value of utility becomesV3 (s3 , L∗3 ) = z2 − pL

(1)0
In the same way, we obtain that V3 (s03 , L0∗3 ) = z2 − pL for any α3 ∈
00
000
000∗
[0,³1], V3 (s003 , L00∗
3 )´ = z2 − pL for any α3 ∈ [0, 1] and V3 (s3 , L3 ) = z2 −L 1 − (1 − p)1/2 for α000∗
3 = 0.

(iii) At period 2, we have four nodes which values are:u2 (s2 , z 2 , RDv3 ) = z 2 − pL with z 2 = z1 − Π2 − L + I2 ,u2 (s2 , z 2 , RDv3 ) = z 2 − pL with z 2 = z1 − Π2 ,00
u2 (s2 , z 02 , RDv3 ) = z 02 − pL³ with z 02 = z1 −´ Π2 − L + I2 andu2 (s2 , z 02 , RDv3 ) = z 02 − L 1 − (1 − p)1/2 with z 02 = z1 − Π02 .

(iv) We repeat step (ii). Then, at node L2 , for a coverage rate α2 , theutility writes:
¡
¢ £
¡
¢¤
V2 (s2 , L2 ) = RDv2 (s2 , L2 ) = v2 s2 , z2 + v2 (s2 , z 2 ) − v2 s2 , z2 × ϕ2 (1 − p, s2 )¡¢with v2 s2 , z2 = u2 (s2 , z 2 , RDv3 ) = z 2 − pL = z1 − L (1 + p) +α2 L (1 − p) , v2 (s2 , z 2 ) = u2 (s2 , z 2 , RDv3 ) = z 2 − pL = z1 − pL − α2 pL andϕ2 (1 − p, s2 ) = (1 − p)e0 +e1 = 1 − p.
Then,
V2 (s2 , L2 ) = z1 − 2Lp
for any α2 ∈ [0, 1] .

15

At node L02 , we have to pay attention to the value of v2 (s02 , z 02 ) =u2 (s02 , z 02 , RDv3 ) and v2 (s02 , z 02 ) = u2 (s02 , z 02 , RDv3 ) since, in the RDU framework, we have to rank utility.
In our example, v2 (s02 , z 02 ) < v2 (s02 , z 02 ) . Thus, we obtain that
³
´ h
³
´i
V2 (s02 , L02 ) = RDv2 (s02 , L02 ) = v2 s02 , z20 + v2 (s02 , z 02 ) − v2 s02 , z20 × ϕ2 (1 − p, s02 )³´
with v2 s02 , z20 = z 0 2 −pL = z1 −L (1 + p)+α02 L (1 − p) , v2 (s02 , z 02 ) =´´³³z 02 − L 1 − (1 − p)1/2 = z1 − L 1 − (1 − p)1/2 − α02 pL and ϕ2 (1 − p, s02 ) =0(1 − p)e0 +e1 = (1 − p)1/2 .
Then,
h
h
i
i
V2 (s02 , L02 ) = z1 + Lp (1 − p)1/2 − 2 + α02 L (1 − p)1/2 (1 − p)1/2 − 1

As utility is a decreasing function of the coverage rate, the optimal
coverage is null and the value of utility becomeshi
1/2
V2 (s02 , L0∗
)
=
z
+
Lp
(1
−
p)
−
2
(2)
1
2
(v) At period 1, we have two nodes which values are:u1 (s1 , z 1 , RDv2 ) = z 1 − 2pLh with z 1 = z0 −i Π1 − L + I1 and
1/2
u1 (s1 , z 1 , RDv2 ) = z 1 + Lp (1 − p) − 2 with z 1 = z0 − Π1 .

Then, at node L1 , for a coverage rate α1 , the utility writes:
¡
¢ £
¡
¢¤
V1 (s1 , L1 ) = RDv1 (s1 , L1 ) = v1 s1 , z1 + v1 (s1 , z 1 ) − v1 s1 , z1 × ϕ1 (1 − p, s1 )¡¢with v1 s1 , z1 = z 1 −2pL
h = z0 −Π1 −L+I
i 1 −2pL = α1 L (1h − p)+z0 − i
1/2
L (1 + 2p) , v1 (s1 , z 1 ) = z 1 +Lp (1 − p) − 2 = z0 −α1 pL+Lp (1 − p)1/2 − 2and ϕ1 (1 − p, s1 ) = (1 − p)e0 = (1 − p)1/2 .
Then,
h
i
h
i
V1 (s1 , L1 ) = α1 L (1 − p)1/2 (1 − p)1/2 − 1 + z0 − L (1 + 2p) + L p (1 − p)1/2 + 1 (1 − p)1/2
As utility is a decreasing function of the coverage rate, the optimal
coverage is zero and the value of utility becomeshi
V1 (s1 , L∗1 ) = z0 − L (1 + 2p) + L p (1 − p)1/2 + 1 (1 − p)1/2(3)
To summarize, the results are the following:16• α1 = 0;
• α2 = 0 if no loss at period 1;
α2 ∈ [0, 1] if loss at period 1;• α3 = 0 if no loss at periods 1 and 2;
α3 = 1 if loss at periods 1 and 2;
α3 ∈ [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, heis 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 pastexperience.
Let us now compare the predictions of our model with those of somestandard modelizations.
• the particular case when ϕt (p, st ) = p : this corresponds to the standard
version of a recursive expected utility model.
The results are then the following: αi ∈ [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 McClennen, 1990.
In this model, all the strategies are evaluated at the root node and compared according to the root node preferences. We consider 2 cases: thecase when one-shot preferences are EU and the case when one-shot
preferences are RDU. The results are then the following:17

(i) for EU preferences, αi ∈ [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, αi = 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.

5

Concluding remarks

The insurance demand example shows that our model allows to explain themodifications in the insurance demand behaviour over time observed forcatastrophic risk and given in the introduction. It well appears that pastexperience 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 recursivemodel alone, nor the RDU model alone can justify all the observed patternof 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 eventsin the preferences representations of the standard models.

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