Legesse Lemecha Obsu

Maria Laura

Delle Monache

Paola Goatin

Semu Mitiku Kasa

Macroscopic traffic flow optimization on roundabouts

Keywords: Road traffic on networks, Macroscopic models, Riemann problem, Optimization, Scalar conservation laws Réseaux routiers, Modèles macroscopiques, Problème de Riemann, Optimisation, Lois de conservation scalaires

The aim of this paper is to propose an optimization strategy for traffic flow on roundabouts using a macroscopic approach. The roundabout is modeled as a sequence of 2 × 2 junctions with one mainline and secondary incoming and outgoing roads. We consider two cost functionals: the total travel time and the total waiting time, which give an estimate of the time spent by drivers on the network section. These cost functionals are minimized with respect to the right of way parameter of the incoming roads. For each cost functional, the analytical expression is given for each junction.

Introduction

The first macroscopic traffic flow models are due to the seminal works of Lighthill and Whitham [START_REF] Lighthill | On kinematic waves. ii. a theory of traffic flow on long crowded roads[END_REF] and, independently, Richards [START_REF] Richards | Shock waves on the highway[END_REF]. They proposed a fluid dynamic model for traffic flow on an infinite single road, using a non-linear hyperbolic partial differential equation (PDE). This model has, then, been extended to initial boundary value problems in [START_REF] Bardos | First order quasilinear equations with boundary conditions[END_REF] and developed for concave fluxes in [START_REF] Floch | Explicit formula for scalar non-linear conservation laws with boundary condition[END_REF]. More recently, several authors suggested models for networks, see [START_REF] Coclite | Traffic flow on a road network[END_REF][START_REF] Colombo | Road network with phase transition[END_REF][START_REF] Delle Monache | A PDE-ODE model for a junction with ramp buffer[END_REF][START_REF] Garavello | The Cauchy problem at a node with buffer[END_REF][START_REF] Herty | A novel model for intersections of vehicular traffic flow[END_REF] and references therein. These models consider different type of solutions and some of them have been used for optimization of traffic flow of networks, see for example [START_REF] Chitour | Traffic circles and timing of traffic lights for cars flow[END_REF][START_REF] Cascone | Optimization of traffic on road networks[END_REF][START_REF] Cutolo | Traffic optimization at junctions to improve vehicular flows[END_REF][START_REF] Gugat | Optimal control for traffic flow networks[END_REF][START_REF] Herty | Modeling, simulation, and optimization of traffic flow networks[END_REF]. In this article, we focus on optimization problems for roundabouts. We consider the model introduced in [START_REF] Delle Monache | A PDE-ODE model for a junction with ramp buffer[END_REF] and extend it to roundabouts. Roundabouts can be modeled as a sequence of 2×2 junctions. In particular, each junction is composed by one mainline and two secondary roads connected through a node. On the mainline we apply a macroscopic approach using the Lighthill-Whitham-Richards (LWR) model, while on the incoming secondary road a buffer of infinite size and capacity is used, whose evolution is described by an ordinary differential equation (ODE) that depends on the difference between the incoming and the outgoing fluxes on the lane. The outgoing secondary road is modeled as a sink. At each junction the Riemann problem is uniquely solved using a right of way parameter for the incoming fluxes, and solutions are constructed via wave-front tracking. The aim of this paper is to derive the analytical expression of two cost functionals, the Total Travel Time (TTT) and the Total Waiting Time (TWT), and to optimize them through a suitable choice of the right of way parameter. The TTT and the TWT give an estimate of the time spent by drivers in the network sections (TTT) or in the queues at the buffers (TWT). The cost functionals are computed analytically for a simple network consisting of a single 2x2 junction. This paper is organized as follows. Section 2 describes the junction model and its extension to roundabouts. Section 3 describes in detail the construction of the Riemann Solver at junctions. Finally, in Section 4 we introduce the cost functionals and compute their analytical expression.

Mathematical Model

In this work we consider a roundabout joining three roads as illustrated in Figure 1. A roundabout can be seen as a periodic sequence of junctions and it can be represented by an oriented graph, in which roads are described by arcs and junctions by vertexes. Each road is modeled by an interval I i = [a i , b i ] ⊂ R, i = 1, 2, a i < b i with the possibility of either a i = -∞ or b i = +∞. In particular, in our case, each junction can be models as a 2×2 junction, see Figure 2. To recover the behavior of the roundabout periodic boundary conditions are introduced on the At each junction we will consider the model introduced in [START_REF] Delle Monache | A PDE-ODE model for a junction with ramp buffer[END_REF]: the evolution of the traffic flow in the mainline segments is given by the scalar hyperbolic conservation law:

J 1 J 2 J 3
f in F out F in f out J i (a) Part of the roundabout. J I 1 I 2 R 1 R 2 (b) Corresponded junction.
∂ t ρ i + ∂ x f (ρ i ) = 0, (t, x) ∈ R + × I i i = 1, 2, 3, (2.1)
where

ρ i = ρ i (t, x) ∈ [0, ρ max ]
is the mean traffic density, ρ max the maximal density allowed on the road and the flux function f : [0, ρ max ] → R + is given by following flux-density relation:

f (ρ) =    ρv f if 0 ≤ ρ ≤ ρ c , f max ρ max -ρ c (ρ max -ρ) if ρ c ≤ ρ ≤ ρ max ,
with v f the maximal speed of the traffic, ρ c = f max v f the critical density and f max = f (ρ c ) the maximal flux value. Throughout the paper, for simplicity, we will assume ρ max = 1 and v f = 1.

In Figure 3 is an example of flux function satisfying these hypotheses.

ρ ρ c ρ max f max f (ρ) v f f max ρmax-ρc

Inria

The incoming lane of the secondary roads entering the junctions are modeled with a buffer of infinite size and capacity. This choice is made to avoid backward moving shocks at the boundary, for details see [START_REF] Delle Monache | A PDE-ODE model for a junction with ramp buffer[END_REF]. In particular, the evolution of the queue length of each buffer is described by an ODE:

dl i (t) dt = F i in (t) -γ r1,i (t), t ∈ R + i = 1, 2, 3, (2.2)
where l i (t) ∈ [0, +∞[ is the queue length, F i in (t) the flux entering the lane and γ r1,i (t) the flux exiting the lane. The outgoing lane is considered as a sink that accepts all the flux coming from the mainline. No flux from the incoming lane is allowed on the outgoing stretch of same road. The Cauchy problem to solve is then:

         ∂ t ρ i + ∂ x f (ρ i ) = 0, (t, x) ∈ R + × I i , i = 1, 2, 3 dl i (t) dt = F i in (t) -γ r1,i (t), t ∈ R + , i = 1, 2, 3 ρ i (0, x) = ρ i,0 (x), on I i , i = 1, 2, 3 l i (0) = l i,0 i = 1, 2, 3, (2.3)
where ρ i,0 (x) are the initial conditions and l i,0 the initial lengths of the buffers. This will be coupled with an optimization problem at the junctions that gives the distribution of traffic among the roads. We define the demand d(F i in , l i ) of the incoming lane for the secondary roads, the demand function δ(ρ i ) on the incoming mainline segment at each junction, and the supply function σ(ρ i ) on the outgoing mainline segment at each junction as follows.

d(F i in , l i ) = γ max r1,i if l i (t) > 0, i = 1, 2, 3 min (F i in (t), γ max r1,i ) if l i (t) = 0, i = 1, 2, 3 (2.4)
δ(ρ i ) = f (ρ i ) if 0 ≤ ρ i < ρ c , i = 1, 2, 3 f max if ρ c ≤ ρ i ≤ 1, i = 1, 2, 3 (2.5
)

σ(ρ i ) = f max if 0 ≤ ρ i ≤ ρ c , i = 1, 2, 3 f (ρ i ) if ρ c < ρ i ≤ 1, (2.6)
where γ max r1,i is the maximal flow on the incoming lane R 1,i , i = 1, 2, 3. Moreover, we introduce β ∈ [0, 1] the split ratio of the outgoing lane R 2,i , and γ r2,i (t) = βf (ρ i (t, 0-)) i = 1, 2, 3 its flux.

Definition 1 (See [START_REF] Delle Monache | A PDE-ODE model for a junction with ramp buffer[END_REF].) Consider a junction J with two incoming roads I 1 =] -∞, 0[ and R 1 and two outgoing roads

I 2 =]0, +∞[ and R 2 . A triple (ρ 1 , ρ 2 , l) ∈ 2 i=1 C 0 R + ; L 1 ∩ BV(R) × W 1,∞ (R + ; R + ) is an admissible solution to (2.3) if 1. ρ i is a weak solutions on I i , i.e., ρ i : [0, +∞[×I i → [0, 1], i = 1, 2, such that R + Ii ρ i ∂ t ϕ i + f (ρ i )∂ x ϕ i dxdt = 0, i = 1, 2, (2.7)
for every ϕ i ∈ C 1 c (R + × I i ). 2. ρ i satisfies the Kružhkov entropy condition [START_REF] Kružhkov | First order quasilinear equations with several independent variables[END_REF] on (R × I i ), i.e., for every k ∈ [0, 1] and for all

ϕ i ∈ C 1 c (R + × I i ), t > 0, R + Ii (|ρ i -k|∂ t ϕ i + sgn (ρ i -k)(f (ρ i ) -f (k))∂ x ϕ i )dxdt + Ii |ρ i,0 -k|ϕ i (0, x)dx ≥ 0; i = 1, 2.
(2.8)

3. f (ρ 1 (t, 0-)) + γ r1 (t) = f (ρ 2 (t, 0+)) + γ r2 (t).
4. The flux of the outgoing mainline f (ρ 2 (t, 0+)) is maximum subject to

f (ρ 2 (t, 0+)) = min (1 -β)δ(ρ 1 (t, 0-)) + d(F in (t), l(t)), σ(ρ 2 (t, 0+)) , (2.9)
and 3.

5. l is a solution of (2.2) for a.e. t ∈ R + .

Remark 1 A parameter q is introduced to ensure uniqueness of the solution, that is, q ∈ ]0, 1[ is a right of way parameter that defines the amount of flux that enters the outgoing mainline from each incoming road. In particular, when the priority applies, qf (ρ 2 (t, 0+)) is the flux allowed from the incoming mainline into the outgoing mainline, and (1 -q)f (ρ 2 (t, 0+)) the flux from the onramp.

Riemann Problem the junction

In this section we recall briefly the costruction of the Riemann Solver at a junction as introduced in [START_REF] Delle Monache | A PDE-ODE model for a junction with ramp buffer[END_REF] and then we apply it to our particular case to recover the expressions of the cost functionals. We will focus only on one junction J. We fix constants

ρ 1,0 , ρ 2,0 ∈ [0, 1], l 0 ∈ [0, +∞[, F in ∈ ]0
, +∞[ and a priority factor q ∈ ]0, 1[. The Riemann problem at J is the Cauchy problem (2.3) where the initial conditions are given by ρ 0,i (x) ≡ ρ 0,i in I i for i = 1, 2. We define the Riemann Solver by means of a Riemann Solver RSl, which depends on the instantaneous load of the buffer l. For each l the Riemann Solver RSl is constructed in the following way.

1. Define Γ 1 = f (ρ 1 (t, 0-)), Γ 2 = f (ρ 2 (t, 0+)), Γ r1 = γ r1 (t);
2. Consider the space (Γ 1 , Γ r1 ) and the sets O 1 = [0, δ(ρ 1 )], O r1 = [0, d(F in , l)];

3. Trace the lines (1 -β)Γ 1 + Γ r1 = Γ 2 ; and Γ 1 = q 1-q Γ r1 ;

4. Consider the region

Ω = (Γ 1 , Γ r1 ) ∈ O 1 × O r1 : (1 -β)Γ 1 + Γ r1 ∈ [0, Γ 2 ] . (3.1)
Different situations can occur depending on the value of Γ 2 :

• Demand limited case: 4(a).

Γ 2 =(1 -β)δ(ρ 1 (t, 0-)) + d(F in , l). We set Q to be the point ( Γ1 , Γr1 ) such that Γ1 = δ(ρ 1 (t, 0-)), Γr1 = d(F in , l) and Γ2 = (1 -β)δ(ρ 1 (t, 0-)) + d(F in , l), as illustrated in Figure
• Supply limited case:

Γ 2 = σ(ρ 2 (t, 0+)).
We set Q to be the point of intersection of (1 We define the function τ as follows, for details see [START_REF] Garavello | Traffic Flow on Networks: Conservation Laws Model[END_REF].

-β)Γ 1 + Γ r1 = Γ 2 and Γ 1 = q 1-q Γ r1 . If Q ∈ Ω, we set ( Γ1 , Γr1 )=Q and Γ2 = Γ 2 , see Figure 4(b); if Q / ∈ Ω, we set ( Γ1 , Γr1 )=S and Γ2 = Γ 2 , where S is the point of the segment Ω ∩ (Γ 1 , Γ r1 ) : (1 -β)Γ 1 + Γ r1 = Γ 2 closest to the line Γ 1 = q 1-q Γ r1 see Figure 4(c). Inria δ(ρ 1 ) Γ 2 = (1 -β)Γ 1 + Γ r1 Γ 1 = q 1-q Γ r1 Q S (
Definition 2 Let τ : [0, 1] → [0, 1] be the map such that: • f (τ (ρ)) = f (ρ) for every ρ ∈ [0, 1]; • τ (ρ) = ρ for every ρ ∈ [0, 1] \ {ρ c }.
Theorem 1 Consider a junction J and fix a right f way parameter q ∈ ]0, 1[. For every ρ 1,0 , ρ 2,0 ∈ [0, 1] and l 0 ∈ [0, +∞[ there exists a unique amdmissible solution (ρ 1 (t, x), ρ 2 (t, x), l(t)) in the sense of Definition 1 satisfying the priority (possibly in an approximate way). Moreover, there exists a unique couple

(ρ 1 , ρ2 ) ∈ [0, 1] 2 such that ρ1 ∈ {ρ 1,0 }∪]τ (ρ 1,0 ), 1] if 0 ≤ ρ 1,0 ≤ ρ c , [ρ c , 1] if ρ c ≤ ρ 1,0 ≤ 1; f (ρ 1 ) = Γ1 , (3.2) and ρ2 ∈ [0, ρ c ] if 0 ≤ ρ 2,0 ≤ ρ c , {ρ 2,0 } ∪ [0, τ (ρ 2,0 )[ if ρ c ≤ ρ 1,0 ≤ 1; f (ρ 2 ) = Γ2 . (3.3)
For the incoming road the solution is given by the wave (ρ 1,0 , ρ1 ), while for the outgoing road the solution is given by the wave (ρ 2 , ρ 2,0 ). Furthermore, for a.e. t > 0, it holds

(ρ 1 (t, 0-), ρ 2 (t, 0+)) = RS l(t) (ρ 1 (t, 0-), ρ 2 (t, 0+)).
For the proof see [START_REF] Delle Monache | A PDE-ODE model for a junction with ramp buffer[END_REF].

Cost Functionals

In this section we define the cost functionals and derive their expressions. We introduce the Total Travel Time (T T T ) on the road network and the Total Waiting Time (T W T ) on the incoming lanes of the secondary roads, which are defined as follows:

T T T (T ) = T 0 Ii ρ(t, x)dxdt + T 0 l(t)dt + Ii ρ(T, x)dx + l(T ) (4.1)
T W T (T ) = T 0 l(t)dt + l(T ) (4.2)
for T > 0 that we will take sufficiently big to that the solution is stabilized. Our aim is to derive the explicit form of these cost functionals to study their dependence on the right of way parameter q. Since the solutions of such optimization problems cannot be analytically computed in general, we focus on the case of the junction showed in Figure 2(b) with I 1 = [-1, 0] and

I 2 = [0, 1].
We suppose that the network and the buffer are empty at t = 0 and we assume that the following boundary data are given: f in the inflow on the incoming mainline, f out the outflow on the outgoing mainline and F in the incoming flux of the secondary road. Moreover we assume

F in ≤ f max = γ max r1
and f out ≤ f max . Now, we can solve the corresponding initial-boundary value problem. The first step is to compute the demand and supply functions of the roads. We have δ(ρ

1,0 ) = 0, d(F in , l) = min(F in , γ max r1 ) = F in and σ(ρ 2,0 ) = f max . Then we can compute Γ 2 : Γ 2 = min (1 -β)δ(ρ 1,0 ) + d(F in , l), σ(ρ 2,0 ) = F in .
It is straightforward to see that the problem is demand limited, hence the optimal point is the point at maximal demands. Thus it follows Γ1 = 0, Γ2 = F in and Γr1 = F in , from which we can derive ρ1 = 0 and ρ2 = F in . Since we are demand limited we also have l(t) = 0. The solution in the x -t plane looks as in Figure 5. The wave produced by the junction problem interacts with the right boundary x = 1 at time t 1 = 1. Moreover at x = -1, the boundary condition enforces the creation of an additional wave at t = 0 with speed equal 1. This produces a density ρ1 equal at f in which reaches the junction at the same time t 1 = 1. The solution looks as in Figure 5. At t 1 = 1 we solve a new Riemann problem at the junction with initial densities

Inria f in f out F in ρ 2,0 ρ 1,0 ρ2 ρ1 t -1 1 x t 1
ρ(1, x) = ρ1 if x < 0, ρ2 if x > 0.
We assume that the splitting ratio β ∈ (0, 1) is fixed and that 0 ≤ ρ1 < ρ c and 0 ≤ ρ2 < ρ c . The demand and supply functions on the respective roads are δ(ρ

1 ) = f in , d(F in , l 0 ) = min(F in , γ max r1 ) = F in , σ(ρ 2 ) = f max .
Computing Γ 2 from these we obtain

Γ 2 = min (1 -β)δ(ρ 1 ) + d(F in , l), σ(ρ 2 )
The case (1 -β)δ(ρ 1 ) + d(F in , l) ≤ σ(ρ 2 ) is demand limited and thus vehicles enter the junction without restrictions. That is, the roundabout is not congested and hence we do not consider this situation. Since we want to minimize our functionals with respect to the right of way parameter, from now on we will only focus on the supply constrained case and hence, we make the following assumption.

First assumption on data

f max ≤ (1 -β)f in + F in (4.3)
With this assumption then it is straightforward to compute the value of Γ 2 = f max and hence,

Γ 1 = q 1 -βq f max and Γ r1 = 1 -q 1 -βq f max .
Moreover, let us assume the following conditions

A 1 ) Γ 1 = q 1 -βq f max < δ(ρ 1 ) = f in , A 2 ) Γ r1 = 1 -q 1 -βq f max < d(F in , l) = F in .
The following are solutions of the Riemann problem at the junction.

a) If both A 1 and A 2 are satisfied, then q 1 -βq f max , 1 -q 1 -βq f max , f max is the solution. b) If δ(ρ 1 ) = f in ≤ q 1 -βq f max , then f in , f max -(1 -β)f in , f max is the solution of Riemann problem. c) If d(F in , l) = F in ≤ 1 -q 1 -βq f max , then f max -F in 1 -β , F in , f max is the solution.
From condition b) at f in ≤ q 1 -βq f max solving for q we get

q ≥ q 2 = f in f max + βf in (4.4) Similarly, from condition c) at d(F in , l) = F in = 1 -q 1 -βq f max solving we have q ≤ q 1 = F in -f max βF in -f max (4.5)
To see the relation between q 1 and q 2 with respect to the feasible set we take their difference.

q 2 -q 1 = f in f max + βf in - F in -f max βF in -f max = f max [f max -(1 -β)f in -F in <0 ] (f max + βf in )(βF in -f max <0 ) . (4.6)
This holds true because of (4.3) and implies q 1 ≤ q 2 , see Figure 6.

Γ r1 d(F in , l) Γ 1 δ(ρ 1 ) Γ 2 = (1 -β)Γ 1 + Γ r1 Γ 1 = q 1 1 -q 1 Γ r1 Γ 1 = q 2 1 -q 2 Γ r1
Figure 6: Relationship between q 1 and q 2

We will analyze the different cases that can occur according to the different values of q.

4.1 Case q 1 < q < q 2

We solve now the Riemann problem at the junction at t 1 . Since we assume that we are supply limited and that q 1 < q < q 2 it is straightforward to have the following fluxes at the junction:

Inria Γ 1 = q 1 -βq f max , Γ r1 = 1 -q 1 -βq f max and Γ 2 = f max . It follows that ρ 1 = 1 - (1 -f max )q 1 -βq (4.7)
and that the wave speed λ is

λ(ρ 1 , ρ 1 ) = f in (1 -βq) -qf max (1 -βq)(f in -1) + (1 -f max )q (4.8)
The characteristic x = λ(t -1) crosses the boundary x = -1 at

t 2 = λ 1 -λ = 1 - (1 -βq)(f in -1) + (1 -ρ c )q f in (1 -βq) -qf max = = 1 -(β + f max )q -(1 -f max )q f in (1 -βq) -qf max = 1 -(1 + β)q f in (1 -βq) -qf max (4.9)
In the outgoing road ρ 2 = ρ c which produces a wave with positive speed 1 as can be seen in the 7

f in f out F in ρ 2,0 ρ 1,0 ρ2 ρ1 ρ 2 ρ 1 t -1 1 x t 1 t 2 Figure 7: Junction problem at t 1 .
The value of the corresponding flux from the buffer is Γ r1 = (1 -q)f max 1 -βq . For all β and

q ∈]0, 1[ we have (1 -q) 1 -βq < 1, so the length of buffer satisfies l = F in -Γ r1 = F in - (1 -q)f max 1 -βq > 0 Since (1 -βq)F in > (1 -q)f max or q(f max -βF in ) > f max -F in then q > f max -F in f max -βF in = q 1 .
Therefore,

l(t) = (F in - (1 -q)f max 1 -βq )(t -1) > 0. (4.10)
The length of buffer increases linearly.

Moreover at time t 1 the interaction between the wave in the outgoing road and the boundary at x = 1 can generate an additional wave if F in > f out . When this is the case, in fact, there is a wave with negative speed which can interact with other waves between [0, 1]. Since our purpose is to study the interactions among waves we make the following:

Second assumption on data

F in > f out . (4.11)
When (4.11) is satified there is a wave with negative speed created at (t 1 , 1). This new wave creates a density

ρ 3 = 1 - f out (1 -f max ) f max , (4.12)
and has a wave speed λ(ρ 2 , ρ 3 )

λ(ρ 2 , ρ 3 ) = F in -f out ρ2 -ρ 3 = F in -f out F in -1 + f out (1-f max ) f max = f max (F in -f out ) (F in -1)f max + (1 -f max )f out . (4.13)
This wave with equation x = λ(t -1) + 1 intersect the characteristic line x = t -1 at a point P as in Figure 8. Solving the system we get the coordinates of the point P = (t P , x P ):

f in f out F in ρ 2,0 ρ 1,0 ρ2 ρ1 ρ 2 ρ 1 ρ 3 t -1 1 x t 1 P t 2
t P = t 3 = λ -2 λ -1 = f max (F in -f out ) -2((F in -1)f max + (1 -f max )f out ) f max (F in -f out ) -((F in -1)f max + (1 -f max )f out ) = = f max f out + 2f max -2f out -f max f out f max -f out (4.14) x P = 1 1 -λ = (F in -1)f max + (1 -f max )f out (F in -1)f max + (1 -f max )f out -f max (F in -f out ) = = f max f out + f max -f max F in -f out f max -f out . ( 4

.15)

Inria Once we have determined the coordinates of P, we can solve the classical Riemann Problem at P:

∂ t ρ + ∂ x f (ρ) = 0 ρ(t, x) = ρ c if x < x p , ρ 3 if x > x p , (4.16)
where x p is given by (4.15). Note that ρ c < ρ 3 , hence, from the Rankine-Hugoniot jump condition we can compute the characteristic speed of the wave emanating from the point P

λ(ρ c , ρ 3 ) = f (ρ 3 ) -f (ρ c ) ρ 3 -ρ c = f max f max -1 . (4.17)
The solution of the classical Riemann Problem at P is

ρ(t, x) = ρ c if x < λ(t -t p ) + x p , ρ 3 if x > λ(t -t p ) + x p , (4.18)
where λ and t p are respectively as in (4.17) and (4.14). The characteristic line with speed λ starting from P interacts with the junction at x = 0. This line intercepts the t-axis at

t 4 = t p - 1 λ x p = t p + 1 -f max f max x p = (f max ) 2 -f max F in + f max -f out f max (f max -f out ) . ( 4

.19)

The situation at t = t 4 looks as in Figure 9 f in f out We can then, solve at t 4 another Riemann problem at the junction:

F in ρ 2,0 ρ 1,0 ρ2 ρ1 ρ 2 ρ 1 ρ 3 t -1 1 x t 1 t 4 P t 2
ρ(t 4 , x) = ρ 1 if x < 0, ρ 3 if x > 0. (4.20)
To do so, we compute:

d(F in , l) = γ max r1 , (4.21) δ(ρ 1 ) = f max , (4.22) σ(ρ 3 ) = f out . (4.23)
In order to limit the complexity of the computation we fix

γ max r1 = f max . It follows Γ 2 = min((1 -β)δ(ρ 1 ) + d(F in , l), σ(ρ 3 )) = min((1 -β)f max + f max , f out ) = f out , Γ1 = q 1 -βq f out and Γr1 = 1 -q 1 -βq f out
Since by assumption it holds f out < f max the soution of the Riemann problem at the junction lies inside the feasible region as shown in Figure 10. The solution of Riemann Problem at junction is given by

Γ r d(F in , l) = f max Γ 1 δ(ρ 1 ) = f max Γ 2 = (1 -β)Γ 1 + Γ r Γ 1 = q 1-q Γ r Q
( Γ1 , Γr1 , Γ2 ) = q 1 -βq f out , 1 -q 1 -βq f out , f out
From this we can uniquely recover the corresponding values of the queue length and of the densities of the mainline. Since the flux exiting the secondary road is Γr1 = 1 -q 1 -βq f out then the queue length is

l(t) = l(t 4 ) + F in - 1 -q 1 -βq f out (t -t 4 ) > 0. (4.24)
Thus, the length of the buffer increases linearly.

On the outgoing mainline there is no new wave created since Γ2 = f out = f (ρ 3 ). And on the incoming mainline we have

Γ1 = q 1 -βq f out ⇒ q 1 -βq f out = 1 -ρ1 1 -ρ c f max which gives ρ1 = 1 - (1 -f max )qf out (1 -βq)f max . (4.25)
On the incoming main road of a roundabout ρ c ≤ ρ 1 ≤ 1 so ρ c ≤ ρ1 ≤ 1, and the pair (ρ 1 , ρ1 ) produces a wave with negative speed on the incoming road of the roundabout. The speed is given by λ(ρ

1 , ρ1 ) = f max f max -1
. This characteristic line crosses the left boundary x = -1 at a time t = t 5 given by

t 5 = t 4 - 1 λ = t p + 1 -f max f max (x p + 1) = f max f in -f max f out -2f max + 2f out f max (f out -f max ) . ( 4

.26)

Inria Moreover, since q 1 -βq f out < δ(ρ 1 ) = f max , there is no wave produced by the interaction with the boundary x = -1 at time t 5 . The solution at t 5 looks like as in Figure 11 f in f out Depending on the priority parameter q, the waves emanating from the junction at time t 1 and t 4 could collide within the region -1 < x < 0. The wave starting from (t 1 , 0) could be slower than the one created at time t 4 and so they could collide. Solving the system generated by the characteric lines we can find the value of q for which this happens. We find that for q = q, where q is given by

F in ρ 2,0 ρ 1,0 ρ2 ρ1 ρ 2 ρ 1 ρ 3 ρ1 t -1 1 x t 1 t 4 t 5 P t 2
q = (1 -β) f in f max F in + 2f in f max + 2f in f out -f max (βf in f max + f max -f out + f in f out (2β -1)) (f max ) 2 (F in -βf in + β -1 -f out ) + f max f out (1 -β + βf in )
(4.27) the waves collide. We can distinguish two additional cases q 1 < q < q and q < q < q 2 4.1.1 The case q 1 < q < q If q 1 < q < q, the waves do not intersect in the region -1 < x < 0 and there are no new waves created. Hence, the study of the problem is concluded.

4.1.2

The case q ≤ q < q 2

As described above, the pair (ρ 1 , ρ1 ) produces a wave with negative speed λ = f max f max -1 on the incoming main road of the roundabout. This wave interacts with the wave exiting from (t 1 , 0) at point Q as in Figure 12 f in f out

F in ρ 2,0 ρ 1,0 ρ2 ρ1 ρ 2 ρ 1 ρ 3 ρ1 t -1 1 x t 1 t 4 P Q Figure 12: Solution in the case q ≤ q < q 2
The coordinates of the point Q are then

t Q = f max t 4 ((1 -βq)(f in -1) + (1 -f max )q) + (1 -f max )(f in (1 -βq) -qf max ) f max ((1 -βq)(f in -1) + (1 -f max )q) + (1 -f max )(f in (1 -β) -qf max ) (4.28)
and

x Q = f max f max -1 (t Q -t 4 ) (4.29)
Once we have determined the coordinates of point Q, we can solve classical Riemann Problem at Q.

∂ t ρ + ∂ x f (ρ) = 0 (4.30) ρ(t, x) = ρ1 if x < x Q , ρ1 if x > x Q , (4.31)
where Q is given by (4.29). By Rankine-Hugoniot jump condition we can derive

λ(ρ 1 , ρ1 ) = f (ρ 1 ) -f (ρ 1 ) ρ1 -ρ1 = f in -Γ1 ρ1 -ρ1 = f in - q 1 -βq f out f in -1 -(1-f max )qf out (1-βq)f max = = f in (1 -βq) -qf out f max (1 -βq)f max (f in -1) + (1 -f max )qf out (4.32)
where λ is the characteristic speed of the wave emanating from the intersection point Q. The solution of the classical Riemann Problem at Q is

ρ(t, x) = ρ1 if x < λ(t -t Q ) + x Q , ρ1 if x > λ(t -t Q ) + x Q , (4.33)
where λ and t Q are respectively as in (4.32) and (4.28). The wave with speed λ starting from Q cross the left boundary x = -1 at

t 5 = t Q - 1 + x Q λ (4.34)

Inria

The complete solution can be seen in Figure 13 f in f out

F in ρ 2,0 ρ 1,0 ρ2 ρ1 ρ 2 ρ 1 ρ 3 ρ1 t -1 1 x t 1 t 4 t 5 P Q Figure 13: Solution at t Q
This concludes the analysis of this case.

Case q ≥ q 2

In this subsection, we assume that assumption (A1) is not satisfied and give the analysis for the Riemann problem at junction for t ≥ t 1 . The Riemann problem to solve is

∂ t ρ + ∂ x f (ρ) = 0, ρ(t, x) = ρ1 if x < 0, ρ2 if x > 0. (4.35)
The splitting ratio β ∈ (0, 1) is fixed and 0 ≤ ρ1 < ρ c and 0 ≤ ρ2 < ρ c as in the previous section.

The demand and supply functions on the corresponded roads are given by δ(ρ

1 ) = f (ρ 1 ) = f in , d(F in , l 0 ) = min(F in , γ max r1 ) = F in and σ(ρ 2 ) = f max .
From this follows that ρ 1 = ρ1 and no wave is created in the incoming mainline. However, on the outgoing link ρ 2 = ρ c which generates a wave with positive speed. The buffer increases since l

(t) = (F in + (1 -β)f in -f max )(t -1) > 0.
The wave with positive speed 1 generated at (t 1 , 0) interacts with the wave generated from right boundary at point P = (t P , x P ) at time t = t P as described in subsection 4.1 under assumption (4.11). At the right boundary

f out = 1 -ρ 3 1 -f max f max , hence one obtain that ρ 3 = 1 - f out (1 -f max ) f max . (4.36)
At this point up to time t 4 the analysis is the same as in the previous case, so we will solve directly the Riemann problem at the junction at t 4 with the new assumption.

The Riemann problem to solve is then

ρ(t 4 , x) = ρ1 if x < 0, ρ 3 if x > 0,
coupled at the junction with the following demands and supply functions

d(F in , l) = γ max r1 = f max , (4.37)
δ(ρ 1 ) = f in , (4.38)
σ(ρ 3 ) = f out . (4.39)
We can now compute

Γ 2 = min (1 -β)δ(ρ 1 ) + d(F in , l), σ(ρ 3 ) = min (1 -β)f in + f max , f out = f out and Γ1 = q 1 -βq f out .
According to the value of f in two cases can occur at this point. If f in < f out the solution of the Riemann Problem at the junction is given by ( Γ1 ,

Γr1 , Γ2 ) = (f in , f out -(1 -β)f in , f out ). From this it is straightforward to see that ρ1 = ρ 1 since f in = ρ1 for v f = 1.
No new waves are created. If f in > f out the solution of the Riemann problem at the junction becomes

( Γ1 , Γr1 , Γ2 ) = q 1 -βq f out , 1 -q 1 -βq f out , f out .
From this we can uniquely recover the corresponding values of the densities:

ρ1 = 1 - (1 -f max )qf out (1 -βq)f in (4.40)
such that ρ c ≤ ρ1 ≤ 1. The pair (ρ 1 , ρ1 ) produces a wave with negative speed on the incoming main road of the roundabout, that is

λ(ρ 1 , ρ1 ) = f in -f out ρ1 -ρ1 = (f in -f out )(1 -βq)f in (1 -βq)((f in ) 2 -1) + (1 -rc)f out . (4.41)
This characteristic line with initial point (t 4 , 0) and speed λ crosses the left boundary x = -1 when t = t 5 . That is,

t 5 = t 4 - 1 λ = t 4 - (1 -βq)((f in ) 2 -1) + (1 -f max )f out (f in -f out )(1 -βq)f in . (4.42)
Since q 1 -βq f out < f in , there is no interaction with the boundary x = -1 at time t 5 . On the outgoing mainline of the roundabout Γ2 = f out = f (ρ 3 ). Hence no new wave is created on the outgoing link. Hence, the solution looks as in Figure 14.

Inria f in f out F in ρ 2,0 ρ 1,0 ρ2 ρ1 ρ 2 ρ 3 ρ1 t -1 1 x t 1 t 4 P
Figure 14: Solution in the case q ≥ q 2

The flux that leaves the buffer towards the junction is

Γr1 = 1 -q 1 -βq f out . Substituting this in the ODE we have l = F in - 1 -q 1 -βq f out > 0 since 1 -q 1 -βq f out < f out < F in . Integrating it gives l(t) = l(t 4 ) + F in - 1 -q 1 -βq f out (t -t 4 ) > 0. (4.43)
Thus, the queue length increase linearly. This complete the analysis for this case.

4.3

The case q ≤ q 1

In this subsection, we assume that assumption (A2) is not satisfied and give the analysis for the Riemann problem at junction for t ≥ t 1 , as in the subsection 4.2. In order to do so, we compute the demand and supply functions at

t = t 1 d(F in , l) = F in , (4.44)
δ(ρ 1 ) = f in , (4.45)

σ(ρ 2 ) = f max , (4.46)
coupled with the following Riemann problem at the junction

ρ(t 1 , x) = ρ1 if x < 0, ρ2 if x > 0.
Since we suppose that only assumption A 1 holds true then Γ1 , Γr1 , Γ2 =

f max -F in 1 -β , F in , f max .
From the value of Γr1 we can solve the ODE (2.2) and find l(t) = 0. From Γ1 we can find the new value of the density

ρ 1 = (1 -β)f max -(1 -ρ c )(f max -F in ) (1 -β)f max = f max (f max -β -F in ) + F in (1 -β)f max (4.47)
and since 0 ≤ ρ1 ≤ ρ c then ρ c ≤ ρ 1 ≤ 1. Moreover, the wave created has a negative speed λ:

λ(ρ 1 , ρ 1 ) = f in -Γ1 ρ1 -ρ 1 = ((1 -β)f in -(f max -F in ))f max (1 -β)f max (f in -1) + (1 -ρ c )(f max -F in ) (4.48)
The characteristic line with speed λ crosses the boundary x = -1 at time:

t 2 = 1 - 1 λ = (1 -β)f max -((1 -f max ) + f max )(f max -F in ) ((1 -β)f in -(f max -F in ))f max = F in -βf max (f in -βf in -f max + F in )f max (4.49)
In the outgoing link of the mainline we obtain ρ 2 = ρ c which produces a wave with positive speed equal to 1 in x ∈ [0, 1]. This wave interacts with a wave coming from the boundary since it is still possible that F in > f out . In particular, this happens when q < f max -f out f max -βf out . We procede now as in the subsection 4.1 and we obtain the coordinates of the point of intersection P , which are given by (4.15) and (4.14). The problem is the same as in subsection 4.2 up to time t 4 . At this point we solve again the Riemann problem at the junction

ρ(t 4 , x) = ρ 1 if x < 0, ρ 3 if x > 0.
coupled with the following values of demands and supply functions: From this it follows Γ 2 = min((1-β)δ(ρ 1 )+d(F in , l), σ(ρ 3 )) = min((1-β)f max +F in , f out ) = f out , Γ1 = q 1 -βq f out and Γr1 = 1 -q 1 -βq f out . From this we can uniquely recover the corresponding value of the densities. In the incoming mainline we have

d(F in , l) = F in , ( 4
ρ1 = 1 - (1 -f max )qf out (1 -βq)f max such that ρ c ≤ ρ1 ≤ 1.
This produces a wave of negative speed λ = f max f max -1 on the incoming road that intersects the boundary x = -1 at

t 5 = t p + 1 -f max f max (x P + 1).
Since q 1 -βq f out < f in = f max , there is no interaction with the boundary x = -1 at time t 5 .

On the outgoing link there is no wave generated since Γ2 = f out = f (ρ 3 ). Hence, the solution looks as in Figure 15.

Inria f in f out F in ρ 2,0 ρ 1,0 ρ2 ρ1 ρ 2 ρ 3 ρ1 t -1 1 x t 1 t 4 P
Figure 15: Solution in the case q ≤ q 1

Last thing to do is to compute the queue length at t 4 . To do so, we insert the value of Γr1 inside the ODE (2.2) and we solve it obtaining

l(t) = F in - 1 -q 1 -βq f out (t -t 4 ) > 0. (4.53)
This concludes our analysis for this case. We are now ready to give the explicit version of the cost functionals for each different case.

Total Waiting Times and Total Travel Times

• q 1 ≤ q ≤ q

We compute the T W T as follows

T W T (T, q) = t4 t1 F in - (1 -q)f max 1 -βq )(t -1) dt + T t4 l(t 4 ) + F in - 1 -q 1 -βq f out (t -t 4 ) dt + l(t 4 ) + F in - 1 -q 1 -βq f out (T -t 4 ). ( 4

.54)

Concerning T T T (T ), it is given by a constant plus a term depending on the priority, that we denote by T T T (T, q):

T T T (T, q) = A1 ρ1 (t, x)dtdx + A2 ρ 1 (t, x)dtdx + A3 ρ1 (t, x)dtdx + t4 t1 F in - (1 -q)f max 1 -βq )(t -1) dt + T t4 l(t 4 ) + F in - 1 -q 1 -βq f out (t -t 4 ) dt + l(t 4 ) + F in - 1 -q 1 -βq f out (T -t 4 ) + x2 x1 (ρ 1 + ρ 3 ) dx (4.55)
where the areas are defined by

A 1 = 1 2 (t 2 -1) = 1 2 1 -(1 + β)q f in (1 -βq) -qf max -1 A 2 = 1 2 (t 5 + t 4 -t 2 -1)
A 3 = 1 2 (t 5 -t 4 ) and T = t 5 , as in Figure 16.

f in f out F in ρ 2,0 ρ 1,0 ρ2 ρ1 ρ 2 ρ 1 ρ 3 ρ1 t -1 1 x t 1 t 4 P t 3 A 1 A 2 A 3 t 5
Figure 16: Area of integration in the case q 1 ≤ q ≤ q Inria • q ≤ q ≤ q 2

The T W T is as in (4.54), since it does not depend on the wave interactions but only on the queue length. The T T T (T ) is given by the constant term plus T T T (T, q) = A1+A2+A5 ρ1 (t, x)dtdx + A3 ρ 1 (t, x)dtdx + A4+A6+A7 ρ1 (t, x)dtdx

+ t4 t1 F in - (1 -q)f max 1 -βq )(
f in f out F in ρ 2,0 ρ 1,0 ρ2 ρ1 ρ 2 ρ 1 ρ 3 ρ1 t -1 1 x t 1 t 4 t 5 P Q A 1 A 2 A 5 A 6 A 7 A 4 A 3 t Q Figure 17
: Area of integration in the case q ≤ q ≤ q 2

The areas are defined by

A 1 = 1 2 (t Q -1), A 2 = 1 2 (t Q -1)(x Q + 1) A 3 = 1 2 (x Q -x Q t 4 ) RR n°8291 A 4 = 1 2 (t Q -t 4 )(-x Q ) A 5 = 1 2 (t 5 -t Q )(x Q + 1) A 6 = 1 2 (t 5 -t Q )(x Q + 1)
A 7 = (t 5 -t Q )(-x Q ) as in Figure 17.

• q ≥ q 2

The T W T is given by

T W T (T, q) = t4 t1 F in + (1 -β)f in -f max )(t -1) dt + T t4 (F in + (1 -β)f in -f max )(t 4 -1) + F in - 1 -q 1 -βq f out (t -t 4 ) dt + (F in + (1 -β)f in -f max )(t 4 -1) + F in - 1 -q 1 -βq f out (T -t 4 ). ( 4

.57)

The T T T (T ), as usual, is instead calculated by the constant term plus T T T (T, q) = A1 ρ1 (t, x)dtdx + A2 ρ1 (t, x)dtdx

+ t4 t1 F in + (1 -β)f in -f max )(t -1) dt + x2 x1 (ρ 1 + ρ 3 ) dx + T t4
(F in + (1 -β)f in -f max )(t 4 -1) + F in -1 -q 1 -βq f out (t -t 4 ) dt + (F in + (1 -β)f in -f max )(t 4 -1) + F in -1 -q 1 -βq f out (T -t 4 ) (4.58)

and the area for this case are A 1 = 1 2 (t 5 -t 4 ) and A 2 = 1 2 (t 5 -t 4 ) as shown in the Figure 18.

• q ≤ q 1

The T W T is computed as

T W T (T, q) = T t4 F in - 1 -q 1 -βq f out (t -t 4 )dt + F in - 1 -q 1 -βq f out (T -t 4 ) (4.59)
while the T T T (T, q) is given by

T T T (T, q) = A1 ρ 1 (t, x)dtdx + A2 ρ1 (t, x)dtdx + T t4 F in - 1 -q 1 -βq f out (t -t 4 )dt + F in - 1 -q 1 -βq f out (T -t 4 ) + x2 x1
(ρ 1 + ρ 3 ) dx (4.60)

where A 1 = 1 2 (t 5 -t 4 ) = A 2 as shown in Figure 19.

Inria f in f out F in ρ 2,0 ρ 1,0 ρ2 ρ1 ρ 2 ρ 3 ρ1 t -1 1 x t 1 t 4 P A 1 A 2 t 5
Figure 18: Area of integration in the case q ≥ q 2

f in f out F in ρ 2,0 ρ 1,0 ρ2 ρ1 ρ 2 ρ 1 ρ 3 ρ1 t -1 1 x t 1 t 4 P t 3 A 1 A 2 t 5
Figure 19: Area of integration in the case q ≤ q 1 RR n°8291

Figure 1 :

 1 Figure 1: Roundabout considered in the article.

Figure 2 :

 2 Figure 2: Detail of the network modeled in the article

Figure 3 :

 3 Figure 3: Flux function considered.

Figure 4 :

 4 Figure 4: Solutions of the Riemann Solver at the junction.

Figure 5 :

 5 Figure 5: Solution of the initial-boundary value problem for t ∈ [0, t 1 ].

Figure 8 :

 8 Figure 8: Solution at t 1 .

Figure 9 :

 9 Figure 9: Solution at time t = t 4

Figure 10 :

 10 Figure 10: Solution of the Riemann problem at the junction at t 4 .

Figure 11 :

 11 Figure 11: Solution of the problem at t 5

  .50) δ(ρ 1 ) = f max , (4.51) σ(ρ 3 ) = f out . (4.52)

  t -1) dt +

						x2
						(ρ 1 + ρ 3 ) dx
						x1
	+	T t4	l(t 4 ) + F in -	1 -q 1 -βq	f out (t -t 4 ) dt
	+ l(t 4 ) + F in -	1 -q 1 -βq	f out (T -t 4 ).	(4.56)

RR n°8291

Γ r1 d(F in , l) Γ 1 δ(ρ 1 ) Γ 2 = (1 -β)Γ 1 + Γ r1 Q (a) Demand limited case Γ r1 d(F in , l) Γ 1 δ(ρ 1 ) Γ 2 = (1 -β)Γ 1 + Γ r1 Γ 1 = q 1-q Γ r1 Q (b) Supply limited case: intersection inside Ω Γ r1 d(F in , l) Γ 1 δ(ρ 1 ) Γ 2 = (1 -β)Γ 1 + Γ r1 Γ 1 = q 1-q Γ r1 Q S Γ r1 d(F in , l) Γ 1

Obsu & Delle Monache & Goatin & Kasa

route des Lucioles -BP 93 06902 Sophia Antipolis Cedex Publisher Inria Domaine de Voluceau -Rocquencourt BP 105 -78153 Le Chesnay Cedex inria.fr ISSN 0249-6399

Acknowledgments

This research was supported by the ERC Starting Grant 2010 under the project "TRAffic Management by Macroscopic Models" and by "International Science Program, Sweden" (ISP) and "African Mathematics Millennium Science Initiative" (AMMSI).

Inria


Subjective risk and disappointment (Last version*)

March 2013 Thierry Chauveau** ABSTRACT: If an investor does care for utilities -and not for monetary outcomesstochastic dominances should be expressed in terms of utility units ("utils"). If so, any "rational" investor may be characterized by an elementary utility function -called canonical utility function-which is such that the partial weak order induced by stochastic dominance over utils is as "close" to the weak order of preferences as possible. As a consequence, the random utilities of the available prospects do not violate the second-order stochastic dominance property. Substituting utils for monetary units leads to substitute "subjective" risk for "objective" risk à la [START_REF] Rothschild | Increasing Risk; I A De…nition[END_REF]. A weakened independence axiom may then be set over comparable prospects, i.e. those which exhibit the same canonical expected utility. This leads to a fully choice-based theory of disappointment. The functional is lottery-dependent [START_REF] Becker | Lottery Dependent Utility[END_REF]. When constant marginal utility is assumed, it is but the opposite to a convex measure of risk [START_REF] Föllmer | Convex measures of risk and trading constraints[END_REF]. It may be viewed as a theoretical justi…cation for choosing this measure of risk. JEL classi…cation: D81. Key-words: disappointment, risk-aversion, subjective risk, risk premium, expected utility. RESUM E : Si un agent économique rationnel est moins sensible au niveau de son revenu qu'à l'utilité de celui-ci, e¤ectuer des tests de dominance stochastique n'a de sens que si ces tests portent sur les utilités de revenus aléatoires. On peut laors montrer que cet agent est caractérisé par une fonction d'utilité élémentaire telle que les utilités des revenus aléatoires ne violent jamais la dominance stochastique de second ordre. Le préodre induit par la dominance stochastique est alors aussi proche que possible du préordre des préférences et l'on peut généraliser la notion de risque à la Rothschild et Stiglitz en raisonnant en terme de risque "subjectif" où les valeurs sont exprimées en utilités. On peut en…n poser un axiome d'indépendance a¤aibli qui n'est valide que pour les revenus aléatoires "comparables" c'est-à-dire ceux qui ont la même espérance d'utilité. Cette nouvelle axiomatisation du comportement d'un individu en univers incertain aboutit à une théorie de la déception où la fonctionnelle représentant les préférences est "loterie-dépendante" [START_REF] Becker | Lottery Dependent Utility[END_REF]. Si l'utilité marginale de l'investisseur est constante, la nouvelle fonctionnelle n'est que l'opposé d'une mesure de risque convexe (Föllmer and Shied 2002) et elle peut constituer la justi…cation de celle-ci. JEL classi…cation: D81. Mots-clés : déception, aversion pour le risque, risque subjectif, prime de risque, utilité espérée.

Introduction

According to many theories of decision making under risk, preferences may be represented by a functional which depends on the utility of a random outcome. This means that an investor is sensitive but to the utility of his wealth. Equivalently, one can say that, to take a …nancial decision, he only needs to know the values of the utilities of the outcomes he may get. A well-known example is the case when expected utility theory (henceforth EU theory) is valid: the value of a random prospect is a probability weighted average of the utilities of the possible outcomes. When marginal utility is variable, an investor is risk-averse (prone) if his elementary utility function is concave (convex). In the particular case of constant marginal utility, he is risk-neutral. Finally, in any case, the investor takes into account but an average of the results of a gamble to which he is sensitive. In other words, whatever his attitude towards risk (risk-aversion, risk-proneness or neutrality), he behaves in the same way. This looks as a paradox. A second paradox is that checking for the rationality of the behaviour of an investor is often undertaken through implementing a second-order stochastic dominance test where monetary values are taken into account. This is clearly inconsistent if the investor cares for utilities rather than for monetary values.

By contrast, if an investor behaves according to a theory such as that of Loomes and Sugden (henceforth L&S) (1986), he is considered as risk-neutral as long as he averages utilities. He is risk-averse/prone if and only if (henceforth i¤) his welfare includes expected elation/disappointment in addition to the expected utility of his wealth. Elation/disappointment depends on the gap between the utility of the actual outcome and the expected utility of the prospect. Moreover, it becomes easy to de…ne stochastic dominance with respect to "utils" -i.e. units of welfare-instead of monetary units using the frame of L&S (1986). Risk may then be de…ned in accordance with the new de…nition of second-order stochastic dominance.

Unfortunately, the theory of L&S (1986) lacks an axiomatic basis, and so do most of the other disappointment models developed since that time. As a consequence, it seems of interest to develop a fully choice-based theory of disappointment which will allow for a satisfactory distinction between risk-averse, risk-neutral and risk-prone investors.

The main idea of this paper is that the utility function of an investor may be inferred from a comparison between the weak order which represents his preferences and the partial preorder induced by second-order stochatic dominance over utils. The "closer" to the former is the latter, the more likely the utility function is to represent preferences. Of course, to make sense, the previous statement requires that the closeness between the two weak orders will have …rst been de…ned -which will be done in Section 2-.

Next, a partition of the set of the random prospects will be made. Each subset then includes prospects exhibiting the same expected utility. The independence axiom will be set over each subset of prospects and a standard representation theorem will be used to de…ne, for any subset, a lottery-dependent functional. Finally, preferences will be represented by a functional which encom-2 Documents de Travail du Centre d'Economie de la Sorbonne -2012.63 (Version révisée) passes, as particular cases, that of EU theory and that of Loomes and Sugden. Unlike most of its predecessors, this lottery-dependent model preserves a fully choice-based approach and is testable. Moreover, the corresponding decision theory is endowed with two interesting properties: (i ) when constant marginal utility is assumed, the certainty equivalent of a prospect is but the opposite to a convex measure of risk à la [START_REF] Föllmer | Convex measures of risk and trading constraints[END_REF] and (ii) it provides a mathematical expression of the value of a risk premium as a function of risk aversion and of a quantity of risk. However, to get the last property, one must leave aside the usual de…nition of risk -which is a scalar-and split any risk premium into elementary risk premia, each of which may be identi…ed to the product of a quantity of risk by risk aversion.

The rest of this article is organized as follows: …rst, stochastic dominance is revisited and the de…nitions of subjective risk and of rational investors are clari…ed. Next, the axiomatization of a general theory of disappointment is developed. Section 4 concludes.

Revisiting stochastic dominance 2.1 Preliminary de…nitions

From now on, we consider a set of random prospects, labelled W, whose outcomes are monetary and belong to a bounded interval of R, say [a; b]. An element of W will be labelled e w and its cumulative distribution function F e w (:). If a random prospect e w has a discrete support {w 1 ; w 2 ; :::; w K }, it will also be denominated [w 1 ; w 2 ; :::; w K ; p 1 ; p 2 ; :::; p K ] where p k = Pr ( e w = w k ). A probability mixture of e w 1 and e w 2 , will be denoted e w Simultaneously, partial weak orders may be de…ned, independently of preferences: they include …rst and second-order stochastic dominance (henceforth FSD and SSD). FSD (SSD) will be denoted -1 ,(-2 ) with 1 ( 2 ) for strict dominance. A partial weak order induced by FSD (SSD) is consistent with the total weak order of preferences if we have the following implication: e w 1 -1 e w 2 ) e w 1 e w 2 ( e w 1 -2 e w 2 ) e w 1 e w 2 ). It is often argued that a "good" theory of decision making under risk must be endowed with two properties: (P roperty a) For any investor, the weak order induced by …rst-order stochastic dominance is consistent with the weak order of preferences. (P roperty b) For any risk-averse investor, the weak order of preferences is consistent with the weak order induced by second-order stochastic dominance. As already said, this way of reasoning is somewhat paradoxical since, on the one hand, an investor is assumed to be sensitive to the utility of an outcome but, on the other hand, the consistency of his behaviour is checked with a test of stochastic dominance over monetary outcomes. Hence it seems preferable to substitute utils for monetary values i.e. to de…ne subjective stochastic dominances as indicated below:

De…nition 1 (First-order and second-order subjective stochastic dominance (henceforth FSSD and SSSD)). ). Of course, looking at levels of outcomes may be equivalent to looking at utilities. This happens to be the case when …rst-order dominance is taken into account. Indeed, FSD is a property which is conservative through the change of random variable: e ! = u ( e w). , then, the n.e.u. function u (:) is not consistent with the weak order of preferences . If an investor cares for utilities, his n.e.u. function will be consistent with his weak order of preferences and it becomes of interest to consider a "subjective" notion of risk whose de…nition will be consistent with SSSD -i.e. with SSD over utilities-. Actually, the de…nition given by Rothchild and Stiglitz (henceforth R&S) (1970) for "objective" risks may be transposed in the following way:

De…nition 3 (subjective risk). Let u (:) be a n.e.u. function. The random prospect e w 1 is subjectively less risky than the random prospect e w 2 if the two following conditions are met:

(a) e w 1 dominates e w 2 by SSSD (i.e. e w 2u 2 e w 1 ) and

(b) R b a u 0 (x) (F e w1 (x) F e w2 (x
))dx = 0. Note that SSSD has been substituted for SSD in condition (a) whereas the integral condition (b) coincides with that of R&S (1970) if marginal utility is constant. As a consequence, if e w 1 is subjectively less risky than e w 2 , then its expected utility is equal to that of e w 2 . Indeed, we have the following equality:

R b a u 0 (x) (F e w1 (x) F e w2 (x))dx = (E [u ( e w 1 )] E [u ( e w 2 )]) (1)
We now give another example: consider the preorder of preferences which is represented by the following functional:

U ( e w) def = R b a (u (x) + E(u (x) E [u ( e w)]
)) dF e w (x) (2). where the following conditions are met: (a)

E (0) = 0, (b) 0 < E 0 (x) < 1, (c) E 00 (x) < 0 and (d) sup E 0 (z)
1. 2 The functional (2) has been introduced by L&S (1986) and it will be called, from now on, a LS-functional. As shown in the next proposition, the elementary utility function of a LS-functional is consistent.

Proposition 2. Let u (:) be the n.e.u. function of a model where preferences are represented by a LS-functional: then u (:) is consistent with the weak order of preferences represented by the functional.

Proof. It is given in the Appendix.

Properties of consistent utility functions

Recall that the weak order induced by FSSD (SSSD) is partial. Hence the following de…nition will make sense.

De…nition 4 (comparable prospects). Two prospects are comparable with respect to u (:) -or, in short, comparable-, i¤ either e w 1 dominates e w 2 by SSSD or if e w 2 dominates e w 1 by SSSD. The subset of prospects which are comparable with respect to u (:) will be denominated

W u 2 . Let W u+ 2 (W u
2 ) consist in the subset of pairs of prospects ( e w 1 ; e w 2 ) over which the two weak orders,u 2 and , coincide (disagree). Clearly, the two subsets constitute a partition of W u 2 and the n.e.u. function u (:) is all the more a good candidate for representing the preferences of an investor that W u+ 2 is larger and W u 2 more tiny. It is easy to increase the size of W u+ 2 through making u (:) more and more concave. Unfortunately, such a process leads to an increase of the size of the subset of comparable prospects and, consequently, the size of W u 2 may also increase...This dilemma is detailed in the below proposition and corollaries.

Proposition 3. Let u (:) and v (:) be two n.e.u. functions such that u (:) is more concave than v (:). Then, we have the following implication: e w 1v 2 e w 2 ) e w 1u 2 e w 2 and the following inclusions:

W v+ 2 W u+ 2 ; W v 2 W u 2
Proof. It is given in the Appendix. Actually, we want to describe the behaviour of a rational investor, i.e. we want to rule out violations of SSSD. In other words, we are looking for a n.e.u. function which will be consistent. Actually, many n.e.u. functions are consistent. Hence, we must choose among them the "best" one. To do this, one may de…ne the closeness of two weak orders over W as follows: the weak orderu 2 is closer to the weak order than the weak orderv 2 i¤ either Documents de Travail du Centre d'Economie de la Sorbonne -2012.63 (Version révisée)

W u 2 W v 2 or W u 2 = W v 2 and W v+ 2 W u+ 2 .
In the case when two consistent n.e.u. functions, u (:) and v (:), are considered, we have

W u 2 = W v 2 = ? and u (:) is "better" than v (:) -i.e. -u 2 is closer to than -v 2 -i¤ W v+ 2 W u+ 2 .
Of course, one cannot rule out, on a priori grounds, that there may exist two consistent n.e.u. functions u (:) and v (:

) such that neither W u+ 2 W v+ 2 nor W v+ 2 W u+ 2 .
However, if we focus on functions which are either convex or concave, it can be shown that there exists a n.e.u. function, u(:), which is such that the weak orderu 2 is, among the weak orders induced by SSSD, the closest to the weak order of preferences. Indeed, let U C U denominate the subset of n.e.u. functions which are either concave or convex and U C (U C ) the subset of elements of U C which are consistent (inconsistent). As shown in the next proposition, there exists a consistent n.e.u. function which is the "best" among the elements of U C .

Proposition 4.

(a) If no violation of SSD occurs, then, the lower envelope of the functions belonging to U C is consistent and it is the most concave among the consistent and concave n.e.u. functions.

(b) If violations of SSD occur, then, the higher envelope of the functions belonging to U C is consistent and it is the less convex among the consistent and convex n.e.u. functions.

(c) In both cases, the envelope induces a weak orderu 2 which is the closest to the weak order of preferences among the partial orders induced by concave or convex n.e.u. functions.

Proof. It is given in the Appendix. Indeed, two cases may occur: either some violations of SSD exist or not. In the latter case, it can be shown that U C contains convex and concave consistent n.e.u. functions whereas, in the former one, it contains only convex functions. As a consequence, we may now set the following de…nition:

De…nition 5 (rationalizing functions). If violations of SSD occur (do not occur), the higher (lower) envelope of the consistent n. e .u. functions which are convex (concave), is called a rationalizing function, or, equivalently, one can say that it rationalizes the weak order of preferences.

Examples where a standard utility function rationalizes the preferences of an investor have been already provided. This is a strong incentive for developping a theory of decision making under risk in which each investor will be endowed with a rationalizing utility function.

A simple theory of decision making under risk

A fully choice-based theory of decision making under risk is now presented. To allevy the exposition, we focus on the case when no violations of SSD occur. As a consequence, if preferences are rationalized by a n.e.u. function, then the function is concave.

The axiomatics

The …rst step consists in assuming that preferences obey the two …rst axioms of EU theory.

Axiom e w. They also imply that there exists a continuous utility functional, U(:), mapping W on to an interval of R which represents the investor's preferences. It is de…ned up to a strictly continuous and increasing transformation. From now on, the set which includes the functionals such as U(:) will be denominated V.

To get stronger results, one (or more) additional axiom(s) must be set. In EU theory, a third axiom, namely the independence axiom is set. It is valid over the whole set W. However, to account for anomalies, one must weaken the axiom. This will now be done through setting it only for prospects which are comparable. If there exists a n.e.u. function u (:) which rationalizes the preferences of the investor, then comparable prospects will exhibit the same expected utility. Hence, we set the following two axioms:

Axiom 3 (rationality). Any investor is rational, i.e. investors'preferences are rationalized by a n.e.u. function u(:), which will be called, from now on, the canonical utility function.

Axiom 4 (weak independence axiom). The independence property is met over each subset of comparable prospects.

A well-known consequence of Axiom 4 is the following proposition: Proposition 5. Under Axioms 1 to 4, the weak order of preferences may be represented over W by the lottery-dependent functional: Proof. See, for instance, [START_REF] Fishburn | Utility Theory for Decision Making[END_REF].

U ( e w) def = R b a (x)dF e w (
From now on, we set the following normalization conditions: ). As a consequence, (:) is, from now on, unambiguously de…ned. Note that U ( e w a;b ) = u and that U u 1 ( ) = . Clearly, any random prospect e w 2 W is such that 0 U ( e w) 1 or, equivalently such that e w a;b e w u 1 ( ) (4). We now turn to some other important consequences of the above set of axioms.

(u 1 ( )) = and (
For any e w 2 W , there exists a unique real number e w 2 [0; 1] such that e w L ( e w ), or, equivalently, such that c( e w) = c e w . Its existence is a consequence of Axiom 2. Moreover, e w is well unique since (a) L ( 0 ) strictly dominates L ( ") by SSD i¤ 0 > " and that (b), by assumption, SSD is never violated. Finally, the lottery-dependent functional U (:) expresses as:

U ( e w) = e w + (1 e w )u , and we may substitute L E[u( e w)] ( e w ) for e w when ranking prospects. Now, from (4) we get that the certainty equivalent of a random prospect e w 2 W -or, equivalently, that of L ( e w )-, lies between c and u 1 ( ). As a consequence, its utility u(c( e w)) -which is also u(c(L ( e w )))-is a convex combination of u and . Hence we have: ))u = ( u ) ( e w ) + u The above functional clearly represents preferences over the whole set W and its restriction over the subset of the degenerate lotteries coincides with the canonical utility function. Two di¤erent functionals cannot coincide on the subset of degenerate lotteries since they are de…ned up to a continuous and strictly increasing transformation. Hence, U (:) is the unique preference representing functional satisfying the following condition: U ( (z)) = u(z) for z 2 [a; b]. The above discussion may be summed up in the following proposition. Proposition 6. Under Axioms 1 to 4, there exists a unique family of functions f (:)g 2[0;1] such that (i) each element of the family is a continuous and strictly increasing function mapping [0; 1] on to itself (ii) preferences of a rational investor over W may be represented by the lottery-dependent functional: Proof. See the above discussion. The lottery-dependent functional in [START_REF] Loomes | Disappointment and Dynamic Consistency in Choice under Uncertainty[END_REF], is far too general to be implemented. Hence we can try to particularize it.

u (c( e w)) = + (
U ( e w) = (E [u( e w)] u E[u( e w)] ) E[u( e w)] R b a E[u( e w)] (x)dF e w (x) u E[u( e w)] E [u( e w)] u E[u( e w)] ) ! +u E[u( e w)] (5

Regular preferences

A case of interest is when is the identity function ( ) = . If this property is shared by all the (:)'s, we shall say that preferences are regular. Clearly, the condition ( ) = is equivalent to the following one: Proof. It is a direct consequence of the de…nition of regular preferences and of Proposition 6.

Finally, note that preferences are regular if they are represented by a LSfunctional and/or when marginal utility is constant.

Properties of disappointment models with regular preferences

Many examples of models where preferences satisfy Axioms 1 to 4 can be given. We here focus on models which are endowed with regular preferences. Before we review their properties we consider z (x) as a function of the two variables, namely z and x, and, consequently, we set f (z; x) def = z (x).

Constant marginal utility

We …rst examine the case when marginal utility is constant (u (x) = x), i.e. when the functional reads:

U( e w) = c ( e w) = R b a f (E[ w]; x)dF w(x) (7).
From the properties of z (x) we get that f (z; x) is is strictly increasing and concave with respect to x and meets the following condition: f (0; 0) = 0. It is of interest to particularize f (z; x) to get a more operational speci…cation. This can be done through assessing additional conditions to preferences.

First, consider the risk premium ( w)

def = E[ w] c ( e w)
of an arbitrary prospect e w 2 W. One may assume that risk premia are translation-invariant, i.e., ( w + x) = ( w) or, equivalently, c ( e w + y) = c ( e w) + y. Under reasonable mathematical assumptions, one may show that a necessary and su¢ cient condition for (:) to exhibit the invariance property is that @f =@x + @f =@z = 1. This condition may be restated as f (z; x) = x + E (x z) where E (:) is strictly increasing and concave and meets the requirement: E (0) = 0. Finally, the functional is that of a disappointment model where elation/disappointment is an increasing and convex function of the excess of the actual outcome over its expected value. It is a particular case of the model developed by L&S (1986). It may also be viewed as the opposite to a convex measure of risk (in the sense of Föllmer and Schied ( 2002)), since one may set:

r( w) def = c ( e w) = (E[ w] + R b a E(x E[ w])dF w(x)) = U( e w) (8).
The interest of the above result is that it allows for grounding a convex measure of risk on a theory of the behaviour of economic agents towards risk. The risk controller is then assumed to behave according to Axioms 1 to 4, to exhibit constant marginal utility and to have preferences endowed with the translation invariance property.

Moreover, from Equation (8) we also get a decomposition of the risk premium ( w)

def = E[ w] c ( e w)
into elementary premia, which can be viewed as the contributions of the variance, the skewness, the kurtosis ... of a random prospect to the total risk premium which is demanded by an investor. If E(:) is smooth enough, one may write:

( e w) = P +1 n=2 E [( e w E [ e w]) n ] E (n) (E [ e w]
) =n! (9). The total risk premium is then an in…nite sum of elementary premia, each of which is proportional to the product of two terms: the nth order centered moment of the random variable e w, i.e. E [( e w E [ e w]) n ], and the nth order derivative of E (:) taken at point z = E [ e w]. Any even moment is but a quantity of a "symmetric" risk and its coe¢ cient must be negative if the investor is risk averse, whatever the considered de…nition of risk. An odd moment may be viewed as a quantity of an "asymmetric" risk and its coe¢ cient must be positive if the investor is risk averse. Finally, Equation (9) may be viewed as a theoretical grounding of the multimoment approach of the Capital Asset Pricing Model. Now recall that EU theory is often violated by experiments and that no general agreement has yet been found about the explaining power of its challengers, i.e. Non-EU theories. Hence it is interesting to point out that, because of its ‡exibility, the functional (9) is compatible with many of the anomalies of …nancial theory.

Variable marginal utility

We now turn to the general case of variable marginal utility. The functional U( e w) will now read:

U( e w) = u (c ( e w)) = R b a f (E[u ( w)]; x)dF w(x).
Here again f (z; x) is strictly increasing and concave with respect to x and meets the following condition: f (0; 0) = 0. Since investors care but about "utils", the risk premium of an arbitrary prospect e w 2 W is now de…ned as ( w)

def = E[u ( w)] u (c ( e w)
) and one may again assume that risk premia are translationinvariant when they are expressed in utils, i. Under reasonable mathematical assumptions, the functional may be identi-…ed to a LS-functional which expresses as:

U ( e w) def = R b a E[u( w)] (x)dF e w (x) = E[u ( w)]+ R b a E (u (x) E[u ( w)])
)dF e w (x). Elation/disappointment is an increasing and convex function of the excess of the actual outcome over its expected utility. Clearly, the above results are of interest i¤ u (:) can be elicited. This question is now going to be addressed.

The elicitation property.

We focus, in this subsection on LS-functionals, because, as it has been proved in Chauveau and Nalpas (2010), they are endowed with the elicitation property. As a preliminary, we set a new de…nition:

De…nition 6 (strong indi¤erence). Two prospects e w 1 and e w 2 are strongly indi¤erent i¤ (a) they are indi¤erent and (b) they meet the betweenness property. 3 The binary relation " e w 1 and e w 2 are strongly indi¤erent" will be labelled " e w 1 e w 2 ". The binary relation is obviously an equivalence relation over W. Clearly, strong indi¤erence implies indi¤erence in the usual sense which will be called, from now on, weak indi¤erence. The properties of LS-functionals may then be summed up in the following propositions where e w a;x p ( e w y;b 1 q ) denotes the binary lottery [a; x; 1 p; p] ([y; b; q; 1 q]).

Proposition 7 (strong indi¤erence). Consequently, the expected utility of the initial lottery -i.e. u (w)-satis…es the following equality: u (w) = E u( e w a;xn pn ) + P n i=1 (1 p i ) (10). Alternatively, one may consider a sequence of binary lotteries, f e w yn;b qn g n2N , that are de…ned as indicated below: y 0 = w; q 0 = and e w a;yn+1 qn+1 e w yn;b 1 qn and the elements of the sequence are endowed with the following property: E u( e w a;yn+1 qn+1 ) E u( e w a;yn qn ) = 1 q n+1 , or, equivalently, u (w) = E u( e w a;yn qn )

P n i=1 (1 q i ) (11). From now on, the sequences f e w a;xn pn g n2N and f e w yn;b 1 qn g n2N , will be called the canonical sequences generated by (w; ). As shown below, they respectively converge, in LS-models, towards (a) or (b). The result holds whatever the value of .

Proposition 9 (elicitation property). Let f e w a;xn pn g n2N and f e w yn;b qn g n2N be the canonical sequences binary lotteries generated by (w; ) 2 ]a; b[ ]0; 1[. Assume that preferences are represented by a LS-functional and that investors are disappointment averse. Then, fx n g n2N (fy n g n2N ) is a decreasing (increasing) sequence of real numbers converging towards a (b). The sequence f1 p n g n2N (f1 q n g n2N ) is increasing and converges towards `( 1`) where `does not depend on and is a strictly increasing function of w, mapping [a; b] on to [0; 1].

Proof. It is given in the Appendix. Finally, in LS-models, we have the following equalities: l = lim n!1

( P n i=1 (1 p i ))= = lim n!1
(1 P n i=0 (1 q i ))= and from now on we shall set l = u (w). Finally, note that Axioms 1 to 4 are, at least in principle, experimentally testable since their checking comes down to making choices between binary loteries. The number of experiments obviously depends on the desired accuracy of the elementary utility function u(:). The last remark is that EU theory is clearly a degenerate case of the above disappointment theory.

Concluding remarks

In this paper, a fully choice-based theory of disappointment has been developed which can be viewed as an axiomatic foundation of models à la L&S (1986). The above theory of disappointment may be generalized in three ways: the …rst consists in relaxing the assumption of a concave or convex n.e.u. function. One can also weaken the assumption of functions which are continuously derivable and allow for discontinuities on a …nite set of values of their argument. Last, the assumption made at the beginning of Section 3, according to which no violations of SSD occur, is convenient but not essential. and the di¤erence between the two functionals expresses as:

U ( e w 1 ) U ( e w 2 ) = Z b a (u (x) + E (u (x) 1 ))dF e w1 (x) Z b a (u (x) + E (u (x) 2 ))dF e w2 (x) = ( 1 2 ) + Z b a E (u (x) 1 ) dF e w1 (x) Z b a E (u (x) 2 ) dF e w2 (x)
and, …nally:

U ( e w 1 ) U ( e w 2 ) = T 1 + T 2
where:

T 1 = ( 1 2 ) + Z b a (E (u (x) 1 ) E (u (x) 2 )) dF e w2 (x)
and:

T 2 = Z b a E (u (x) 1 ) (dF e w1 (x) dF e w2 (x))
Straightworward calculations give:

T 1 = ( 1 2 ) Z b a ( 1 2 ) E 0 (u (x) 1 + 1 ( 1 2
)) dF e w2 (x) with 1 2 [0; 1] and: As a preliminary, recall that u (:) is more concave than v (:) if and only if u v 1 (:) is concave i.e. if there exists g (:) mapping [0; 1] on to itself and such that: u (x) = g v (x) with g 0 (:) > 0 and g 00 (:) < 0

T 2 = [E (u (x) 1 ) (F e w1 (x) F e w2 (x))] b a Z b a E 0 (u (x) 1 ) u 0 (x) (F e w1 (x) F e w2 (x))dx = E ( 1 ) (F e w1 (a) F e w2 (a)) Z b a E 0 (u (x) 1 ) u 0 (x) (F e w1 (x) F e w2 (x))dx = E ( 1 ) (F e w1 (a) F e w2 (a)) E 0 (1 1 ) Z b a u 0 (t) (F e w1 (t) F e w2 (t)) dt + Z b a E 00 (u (x) 1 ) u 0 (x) Z x a u 0 (t) (F e w1 (
The proof is grounded on the following calculations: A. We …rst assume that SSD is not violated. As a consequence, there exists at least one concave function which is consistent. It is the n.e.u. a¢ ne function de…ned by f (x) = (x a) = (b a). Hence, U C 6 = ?.

Z z a u 0 (x) (F e w1 (x) F e w2 (x))dt = Z z a g 0 (v (x)) v 0 (x) (F e w1 (x) F e w2 (x))dx = g 0 (v (x)) Z x a v 0 (t) (F e w1 (t) F e w2 (t))dt z a Z z a g 00 (v (x)) v 0 (x) Z x a v 0 (t) (F e w1 (t) F e w2 (t))dt dx = g 0 (v (z)) Z z a v 0 (t) (F e w1 (t) F e w2 (t))dt Z z a g 00 (v (x)) v 0 (x) Z x a v 0 (t) (F e w1 (
The subset U C may then include one element or more. In the …rst case U C = ff (:)g and Proposition 4 is clearly valid. We now leave aside this trivial case and assume that U C includes at least two elements, i.e. at least one strictly concave n.e.u. function.

Consider two n.e.u. functions, u (:) and v (:). Then, from Proposition 3, we get the following results:

(a) if u (:) is consistent and is more concave than v (:), then v (:) is also consistent.

(b) if u (:) and v (:) are concave and consistent, then M in (u; v) is consistent.

(c) Similarly, If v (:) is inconsistent and less concave than u (:) then u (:) is also inconsistent.

(d) if u (:) and v (:) are concave and if M in (u; v) is inconsistent, then u (:) and v (:) are inconsistent.

(e) if u (:) is more concave than v (:) then v (:) 2 hypo (u), where hypo (u) is the hypograph of u.

Let u (:) denote the lower envelope of the functions belonging to U C . We want to prove that W u 2 = ?. The proof is three-step. The …rst step consists in de…ning a consistent concave n.e.u. function u (:) which is close to u (:). First, recall that the hypograph of u (:) is de…ned as hypo (u) = T u2UC hypo(u) and, consequently, any concave n.e.u. function whose hypograph is strictly included in hypo (u) 4 is consistent. Now let u (:) be de…ned by the following equality: ( u 00 (x))

Moreover, u (:) will be strictly increasing if u 0 (x) is strictly positive. A su¢ cient condition for this is that:

< (b a) inf x2[a;b] (u 0 (x))
and, …nally, u (:) is concave and strictly increasing if the real number satis…es the below inequality: ju 0 (z) u 0 (z)j

Finally, an upper bound of is given by the following inequality: As a consequence, we get

=4

The last step consists in showing that if u (:) were not consistent, then we would get a contradiction. Indeed if u (:) were not consistent there would exist two prospects e w 1 and e w 2 such that e w 1 e w 2 and, simultaneously, there would exist z 2 [a; b], such that R z a u 0 (x) (F e w1 (x) F e w2 (x))dx > 0. In other words, there would exist a strictly positive real number such that R z a u 0 (x) (F e w1 (x) F e w2 (x))dx > 0

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  Documents de Travail du Centre d'Economie de la Sorbonne -2012.63 (Version révisée)

  x); where = E [u ( e w)] and where (:) is a continuous and increasing function mapping [a; b] on to [ (a); (b)] which is de…ned up to an a¢ ne and positive transformation.

) where e w 2 W

 2 , u(:) is the canonical utility function, u E[u( e w)] def = u(c( e w a;b E[u( e w)] ), and E[u( e w)] (:) is de…ned according to Proposition 5 and to (3). (iii) the restriction of U ( e w) over the subset of degenerate lotteries coincides with the canonical utility function or, equivalently, U ( (z)) = u(z), for any z 2 [a; b].

8

  Documents de Travail du Centre d'Economie de la Sorbonne -2012.63 (Version révisée) e w a;b L +(1 )u ( ). Indeed, we then have:U L +(1 )u ( ) = ( u ) ) + u = ( u ) + u = U(e w a;b ). Hence, preferences are regular if, given two arbitrary real numbers 2 [0; 1] and 2 [0; 1], the random prospect e w a;b is indi¤erent to L +(1 )u ( ). In other words, the utility of the certainty equivalent of any random prospect e w 2 W is a convex combination of and u whose weights coincide with those of the compound lottery L ( e w ) which is indi¤erent to e w. Hence, we get the following corollaries: Corollary 1. Under Axioms 1 to 4, preferences are regular if and only if, for any couple of real numbers ( ; ) 2 [0; 1] [0; 1], the random prospect e w a;b is indi¤erent to L +(1 )u(c( e w a;b )) ( ). Proof. See the above discussion. Corollary 2. Under Axioms 1 to 4, regular preferences may be represented over W by the lottery-dependent functional: u( e w)] (x)dF e w (x); (6) where E[u( e w)] (:) is de…ned according to Proposition 5 and the normalization conditions (3).

  e. : u (c ( e w) + y) = u (c ( e w))+u (y). 10 Documents de Travail du Centre d'Economie de la Sorbonne -2012.63 (Version révisée)

e w 1 e w 2 , c ( e w 1 ) = c ( e w 2 )13

 212 If preferences are represented by a LS-functional, two prospects e w 1 and e w 2 are strongly indi¤erent i¤ they exhibit the same certainty equivalent and the same expected utility, what formally reads: and E [u ( e w 1 )] = E [u ( e w 2 )] Proposition 8 (strong equivalents). If preferences are represented by a LS-functional, there exists exactly one binary lottery of the e w a;x p type (of the e w y;b 1 q type) which is strongly indi¤erent to e w. Lottery e w a;x p ( e w y;b 1 q ) will be called the left (right) strong equivalent of e w. The degenerate lottery (z) (the binary lottery e w a;b u(z) ) is a maximal (minimal) element in W u(z) , i.e. e w a;b u(z) z (z). Proofs. The proofs are given in the Appendix. An important property of LS-models is the elicitation property. Let w 2 [a; b] ( 2 [0; 1]) be an arbitrary level of wealth (probability). Consider the sequence of binary lotteries labelled f e w a;xn pn g n2N that meets the below requirements: x 0 = w; p 0 = and e w pn+1 is the right strong equivalent of e w a;xn pn . Clearly, fx n g n2N is a strictly decreasing sequence. The di¤erence between the expected utilities of two consecutive binary lotteries, e w a;xn pn and e w a;xn+1 pn+1 , is equal to the second weight (1 p n+1 ) of the right strong equivalent e w Recall that two prospects share the betweenness property i¤ for any 2 [0; 1] du Centre d'Economie de la Sorbonne -2012.63 (Version révisée)

2

 2 Clearly, y (x) 0 for x 2 [a; b], y 0 (x) 0 for x 2 [a; a + (b a) =2], y 0 (x) 0 for x 2 [a + (b a) =2; ] b, y (a) = y (b) = 0, a + (b a) =2 = Arg max [y (x)] and max [y (x)] = =4. A su¢ cient condition for u (:) to be concave is that:

( 1 )

 1 u 00 (x)) ; (b a) inf x2[a;b] (u 0 (x)) (Since u (a) = u (a) = 0 and u (b) = u (b) = 1, u (:) is normalized and since > 0, the hypograph of u (:) strictly includes that of u (:) and u (:) is consistent. Finally, the function u (:) is a concave n.e.u. function if (1) is met.

4

  i.e. there exists at least one value of x 2 [a; b] such that u (x) < u (x). 16 Documents de Travail du Centre d'Economie de la Sorbonne -2012.63 (Version révisée)The second step consists in looking for an upper bound for the following di¤erence:= R z a u 0 (x) (F e w1 (x) F e w2 (x))dx R z a u 0 (x) (F e w1 (x) F e w2 (x))dxIntegrating by parts, we get:= R z a (u 0 (x) u 0 (x))(F e w1 (x) F e w2 (x))dx = (u (z) u (z)) (F e w1 (z) F e w2 (z)) + R z a (u 0 (x) u 0 (x)) (dF e w1 (x) dF e w2 (x))and, consequently:j(u (z) u (z)) (F e w1 (z) F e w2 (z))j+ R z a (u 0 (x) u 0 (x)) (dF e w1 (x) dF e w2 (x))(2)The …rst term is bounded indicated as below:j(u (z) u (z)) (F e w1 (z) F e w2 (z))j j(u (z) u (z))j sup z2[a;b] ju 0 (z) u 0 (z)jWe now show that he second term may be bounded as indicated belowR z a (u 0 (x) u 0 (x)) (dF e w1 (x) dF e w2(x)) 2 sup z2[a;b] ju 0 (z) u 0 (z)j Indeed, we have R z a (u 0 (x) u 0 (x)) (dF e w1 (x) dF e w2 (x)) R z a (u 0 (x) u 0 (x)) dF e w1 (x) + R z a (u 0 (x) u 0 (x)) dF e w2 (x) and, for i = 1; 2:: R z a (u 0 (x) u 0 (x)) dF e wi (x) sup z2[a;b] ju 0 (z) u 0 (z)j R z a dF e wi (x) sup z2[a;b]

  ju 0 (z) u 0 (z)j Now, recall that sup z2[a;b] ju 0 (z) u 0 (z)j = sup

  The set of n.e.u. functions will be denoted U.

	1 (1	) e w 2 , where belongs
	to [0; 1]. The degenerate lottery whose outcome is w with certainty is (w). (strict preference) and Preferences over prospects will be denoted -, with (indi¤erence). The certainty equivalent of the prospect e w), w 2 W is labelled c ( e i.e. e w (c ( e w)). A normalized elementary utility function (henceforth n.e.u.
	function) is a continuously derivable and strictly increasing function mapping
	[a; b] on to [0; 1].

  (c) the last term is positive because E 00 (u (x)

	t) F e w2 (t))dt dx Clearly, the following inequality: [1 sup E 0 (z)] 0 implies the following equality: sign (T 1) = sign ( 1 2 ). Hence, T 1 is positive since 1 2 0. The term T 2 is also positive, since it is the sum of three positive terms: (a) E ( 1 ) (F e w1 (a) F e w2 (a)) is positive because E ( 1 ) is negative and so is (F e w1 (a) F e w2 (a)) (from SSSD) (b) the second term is positive because E 0 (1 1 ) is positive, and the integral R b positive and R x a u 0 (t) (F e w1 (t) F e w2 (t)) dt is negative (from SSSD). Finally, U ( e w 1 ) U ( e w 2 ) 0 a u 1 ) is negative, u 0 (x) is Proof of Proposition 3.

0 (t) (F e w1 (t) F e w2 (t)) dt is negative (from SSSD). 14 Documents de Travail du Centre d'Economie de la Sorbonne -2012.63 (Version révisée)

Documents de Travail du Centre d'Economie de la Sorbonne -2012.63 (Version révisée)

The proof of this statement is trivial. 4 Documents de Travail du Centre d'Economie de la Sorbonne -2012.63 (Version révisée)

The reasons for these conditions are: (a) no elation/disappointment is experimented if the actual outcome coincides with its expected value; (b and c) elation/disappointment is an increasing and concave function of the di¤erence between the utility of the actual outcome and that of its expected value; (d) disappointment must not vary too quickly.

Actually, since u (:) is consistent, we must have R z a u 0 (x) (F e w1 (x) F e w2 (x))dx < 0 and, consequently: we get: = R z a u 0 (x) (F e w1 (x) F e w2 (x))dx + R z a u 0 (x) (F e w1 (x) F e w2 (x))dx and, …nally:

Hence, if is small enough, i.e. if < 4 =3, we get a contradiction and, …nally, u (:) is well consistent.

B. We now assume that SSD is not violated, that is when W f 2 6 = ?. Then, no concave n.e.u. functions may be consistent. Indeed, if u were a consistent and concave n. e:u function, then, from Proposition 3, f (:) would be consistent, which would contradict the initial assumption. By contrast, the subset of convex n.e.u. functions is never empty since it always includes the following function: u (x) = 0 for x 2 [a; b[ and u (b) = 1. The rest of the proof is analogous to the above one.

Proof of Proposition 7. The …rst part of the proof consists in proving that, in LS-models, two in-di¤erent prospects e w 1 and e w 2 which have the same expected utility u and the same certainty equivalent c, are strongly indi¤erent. Let e w 1 and e w 2 exhibit the same expected utility u and the same certainty equivalent c. From (2) we get, for i = 1; 2:

where e w i = w 1 ; :::; w N ; p i 1 ; :::; p i N (i = 1; 2) and where u = P N n=1 p i n u (w n ). As a consequence, we have:

From (??) we get:

where c ( e w ) is the certainty equivalent of e w and, …nally: The proof of the converse is as follows. Consider two discrete prospects: where 2 [0; 1]. We must show that if e w 1 and e w 2 are strongly indi¤erent -i.e. if they have the same certainty equivalent and if they exhibit the betweenness property-, then they exhibit the same expected utility. We have, for i = 1; 2:

where:

By de…nition, we have:

Now, from (??), we get:

Substracting the above equation from the previous one and using the betweenness property yields:

or, equivalently:

and, …nally: and:

In LS-models we get:

U( e w z;

and, consequently:

U( e w z; ) = U( (z)) + ( 1) U( e w a;b ) + EXP where:

We get:

or:

and, …nally:

The above condition can be rewritten as:

where: Note that we have:

and, …nally:

) 2 E 00 (0) f (1 ) + g and the condition EXP = 0 implies u (z) = 0.

Proof of Proposition 9. If x n+1 were greater than x n , e w

and summing the members of the above equalities yields:

(1 p i ) f or n = 1; 2; :::

The above equality implies S n def = P n i=1 (1 p i ) u (w). SincefS n g n2N is an increasing sequence, it converges towards a limit u (w). As a consequence, S n S n 1 = (1 p n: ) ! 0, i.e. should be preferred to the e w xm;b 1 pm s and, consequently, that (l) should be preferred to the e w xm;b 1 pm s, that contradicts the fact that f e w xn;b 1 pn g n2N is decreasing and converges towards (l). Hence `= a and fS n g n2N converges towards = u (w). As a consequence, equality (14) is checked.