CLASSIFICATION OF ABSOLUTELY DICRITICAL FOLIATIONS OF CUSP TYPE

We give a classification of absolutely dicritical foliation of cusp type, that is, the germ of singularities of complex foliation in the complex plane topologically equivalent to the singularity given by the level of the meromorphic function y 2 +x 3 xy .

An important problem of the theory of singularities of holomorphic foliations in the complex plane is the construction of a geometric interpretation of the so-called moduli of Mattei of these foliations [START_REF] Mattei | Modules de feuilletages holomorphes singuliers[END_REF]. These moduli appear when one considers a very special kind of deformations called the unfoldings. Basically, the moduli of Mattei are precisely the moduli of germs of unfoldings of a given singular foliation. One of the major difficulty one meets looking at the mentionned geometric description is the lack of basic examples in the litterature. Actually, except when the foliation is given by the level of an holomorphic function, there exist none exemple. The purpose of the following article is not to solve the problem of Mattei even for the class of singularities we consider here but to describe this one as accuratly as possible in order to prepare the attack of the problem of moduli of Mattei.

The absolutely dicritical foliations of cusp type are good candidates to begin this study for the following reasons:

(1) their transversal structure, which usually is a very rich dynamic invariant [START_REF] Loray | Pseudo-groupe d'une singularité de feuilletage holomorphe en dimension deux[END_REF], is very poor and can be completely understood. (2) their number of Mattei moduli is 1.

(3) the topology of their leaves is more or less trivial. Some results in the article might be quite easily extended to a larger class of absolutely dicritical foliations up to some technical and confusing additions. The risk would have been to miss the very first objective of this paper, that is, to give an example.

A germ of singularity of foliation F in C 2 , 0 is said to be absolutely dicritical if there exists a sequence of blowing-up E such that E * F is regular and transverse to each irreducible component of the exceptionnal divisor E -1 (0). It is of cusp type if two successive blowing-up are sufficient. In that case the exceptionnal divisor E -1 (0) is the union of two irreducible components P 1 (C) of respective self-intersection -2 and -1. We denote them respectively D 2 and D 1 . The expression cusp type insists on the fact that the special leaf that passes trough the singular point of the divisor is analytically equivalent to the cuspidal singularity y 2 + x 3 = 0. The simplest example of an absolutely dicritical foliation is given by the levels of the rationnal function near (0, 0)

f = y 2 + x 3 xy .
We associate to F a germ σ ∈ Diff ((D 2 , p) , (D 1 , p)) as in the picture above. It is defined by the property that x ∈ D 2 and σ (x) ∈ D 1 belongs to the same local leaf. This germ is called the transversal structure of F . This is the very first invariant of such a foliation. For the rationnal function above, the transversal structure σ reduces to the identity map in the standard coordinates associated to E.

The main result of this article is the following one: for any foliation F that is absolutely dicritical of cusp type we consider it topological class Top (F ) , that is the set of all foliations topologically equivalent to F . The moduli space Top (F ) / ∼ of F is defined as the quotient of Top (F ) by the analytical equivalence relation. Now we have, Theorem 1. The class Top (F ) is equal to the set of all absolutely dicritical foliations and its moduli space Top (F ) / ∼ can be identified with the functionnal space C {z} up to the action of C * defined by ǫ • (z → f (z)) = ǫ 2 f (ǫz) .

In this theorem, the germ of convergent series f is the image of the transversal structure σ by the Schwarzian derivative

S (σ) = 3 2 σ ′′′ σ ′ -σ ′′ σ ′ 2
. A quick lecture of the theorem would suggest that the transversal structure σ is the sole invariant of the foliation, which is not exactly true as it is highlighted in theorem [START_REF] Grauert | Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen[END_REF].

We have to mention that it does exists a lot of absolutely dicritical foliations. Following a result due to F. Cano and N. Corral [START_REF] Cano | Absolutely dicritical foliations[END_REF], the process E does not contain any obstruction to the existence of absolutely dicritical foliations. In other words, for any sequence of blowing-up E, there exists an absolutely dicritical foliation whose associated process of blowing-ups is exactly E.

Topological classification.

The topological classification is trivial as stated in a proposition to come in the sense that two absolutely dicritical foliations of cusp type are topologically equivalent. To prove this fact, we describe below the model foliations from which the absolutely dicritical foliations are build.

1.1. Model foliations. Let us consider the following model foliations -F 2 is given by the gluing of two copies of C 2

C 2 = (x 1 , y 1 ) C 2 = (x 2 , y 2 )
glued by x 2 = 1 y1 and y 2 = y 2 1 x 1 whose the neighborhood of x 1 = y 2 = 0 is transversaly foliated by y 1 = cst and x 2 = cst. Topologically, this is a foliated neighborhood of a Riemann surface of genus 0 whose selfintersection is -2.

-F 1 is given by the gluing of two copies of C 2

C 2 = (x 3 , y 3 ) C 2 = (x 4 , y 4 )
glued by x 4 = 1 y3 and y 4 = y 3 x 3 whose the neighborhood of x 3 = y 4 = 0 is transversaly foliated by y 3 = cst and x 4 = cst. Topologically, this is a foliated neighborhood of a Riemann surface of genus 0 whose selfintersection is 1.

Following [START_REF] Camacho | Fibered neighborhoods of curves in surfaces[END_REF], any neighborhood of a Riemann surface A of genus 0 embedded in a manifold of dimension two with A • A = -2 (resp. -1) and foliated by a transverse codimension 1 foliation is equivalent ot F 2 (resp. F 1 ). From this, it is easy to show that any C 0 , C ∞ , C ω -isomorphism between two Riemann surfaces A 1 and A 2 as before can be extended in a neigborhood of A 1 and A 2 as a C 0 , C ∞ , C ωconjugacy of the foliations.

Topological classification. Let us first recall the following lemma:

Lemma 2. Let σ be a germ in Diff P 1 , a , i.e., a germ of automorphism of a neighborhood of a in P 1 . Then there exists h a global homeomorphism of P 1 such that h and σ coincide in a neighborhood of a.

Proof. Let S 1 be a small circle around a in a domain where σ is defined. Its image σ (S 1 ) is a topological circle. Consider S 2 a second circle such that the disc bounded by S 2 contains S 1 and σ (S 1 ) . The two coronas bounded respectively by S 1 and S 2 and σ (S 1 ) and S 2 are homeomorphic. Actually, there exists an homeomorphism h of the two coronas such that h

S2 = Id h S1 = σ.
Therefore, we can define the homeomorphism h the following way: in the disc bounded by S 1 , we set h = σ; in the corona bounded by S 1 and S 2 , h = h; everywhere else we set h = Id. Clearly, h satifies the properties in the lemma.

H * 0 (E * F 0 ) = E * F 1 H * 1 (E * F 0 ) = E * F 1 and H 0 | D2 = Id and H 1 | D1 = h. Since h = σ -1 0 •σ 1 , the automorphism H 1 •H -1
0 of E * F 0 let invariant each leaf of E * F 0 . Now, adapting the argument of the previous lemma yields the existence of H a global homeomorphism of E * F 0 defined in a neighborhood of D 1 letting invariant each leaf such that H and

H 1 • H -1 0 coincide in a neighborhood of p. Thus H -1 • H 1 •H -1
0 is equal to Id in a neighborhood of p. Therefore the collection H -1 • H 1 and H 0 glue in a global homeomorphism between E * F 0 and E * F 1 . This homeomorphism can be blown down in a neighborhood of C 2 and is a C 0 -conjugacy of the foliations F 0 and F 1 . Now, if F 0 is topologically equivalent to an absolutely dicritical foliation of cusp type, a theorem of C. Camacho and A. Lins Neto and P. Sad [START_REF] Camacho | Topological invariants and equidesingularization for holomorphic vector fields[END_REF] ensures that the process of reduction of F 0 is the one of an absolutely dicritical foliation. Since, they also shared the same dicritical components, F 0 is absolutely dicritical of cusp type.

Moduli space.

Consider a germ of biholomorphism φ written in the coordinates of the model foliations (x 3 , y 3 ) = φ (x 1 , y 1 ) , φ (0, 0) = (0, 0) . Suppose that it send the foliation defined by y 1 = cst to the one defined by y 3 = cst and that the curve x 1 = 0 is send to a curve transverse to x 3 = 0. With such a biholomorphism we can consider the foliation obtained by gluing the two models foliations F 2 and F 1 with the application φ

D 2 D 1 φ F F 2 1
Following a classical result due to Grauert, this gluing is analytically equivalent to the neighborhood of the exceptionnal divisor obtained by a standard process of two successive blowing-ups [START_REF] Grauert | Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen[END_REF]. The obtained foliation can be blown down in an absolutely dicritical foliation of cusp type at the origin of C 2 .

Remark 4. Key remark. Two foliations obtained by such an above gluing with the respective biholomorphisms φ and ψ are analytically equivalent if and only if there exists an automorphism Φ 2 of the foliation F 2 and Φ 1 of the foliation F 1 such that

φ = Φ 1 • ψ • Φ 2 .
Let us fix σ ∈ Diff (C, 0). We consider the following biholomorphisms g σ (x 1 , y 1 ) = (x 1 + σ (y 1 ) , σ (y 1 )) and Φ α (x 1 , y 1 ) = (x 1 (1 + αy 1 ) , y 1 )

The composition g σ • φ α send the foliation y 1 = cst on it-self. Thus we can denote by F σ,α the foliation obtained by the above gluing

F σ,α := F 1 F 2 / p ∼ g σ • Φ α (p) .
Now, moving the parameter α, we obtain an analytical family of absolutely dicritical foliations. Actually, the following property holds.

Theorem 5. The germ of deformation (F σ,α ) α∈(α0,C) for α in a neighborhood of α 0 in C is a germ of equisingular semi-universal unfolding of F σ,α0 in the sense of Mattei [START_REF] Mattei | Modules de feuilletages holomorphes singuliers[END_REF]. In particular, for any germ of equisingular unfolding (F t ) t∈(C p ,0) with p parameters such that F t | t=0 ∼ F σ,α0 there exists a map α : (C, α 0 ) → (C p , 0) such that for all t F t ∼ F σ,α(t) .

Before proving the above result, let us recall that an unfolding of a given foliation F is a germ F of codimension 1 foliation in C 2+p , 0 transversal to the fiber of the projection

C 2+p , 0 → (C p , 0) , π : (x, t) → t such that F| π -1 (0) ∼ F .
The equisingularity property is a quite technical property to state. However, it means basically that the topology of the process of desingularization of the family of foliation F| t=α does not depend on α. For the details, we refer to [START_REF] Mattei | Modules de feuilletages holomorphes singuliers[END_REF].

Proof.

Step 1 -Let us prove that the deformation (F σ,α ) α∈(C,α0) of F σ,α0 is induced by an unfolding. We can make the following thick gluing

F := F 1 × (C, α 0 ) F 2 × (C, α 0 ) / (x 1 , y 1 , α) → (g σ • Φ α0 ) • Φ -1 α0 • Φ α (x 1 , y 1 ) , α .
where F i × (C, α 0 ) stands for the product foliation: its leaves are the product of a leaf of F i and of an open neighborhood of α 0 in C. The codimension 1-foliation F comes clearly with a fibration defined by the quotient of the map π : (p, α) → α whose fibers are transverse to the foliation. Thus, the above gluing is an unfolding. Now, the restriction

F| π -1 (α0) = F 1 F 2 / (g σ • Φ α0 ) is equal to F σ,α0
. Finally, it is equisingular by construction. Thus, it satisfies all the properties of an equisingular unfolding in the sense of Mattei.

Step 2 -Let us consider the sheaf Θ whose base is the exceptionnal divisor E -1 (0) = D = D 2 ∪ D 1 of tangent vector fields to the foliation E * F σ,α0 and to the divisor E -1 (0). The cohomological group H 1 (D, Θ) represents the finite dimensionnal C-space of infinitesimal unfoldings. Following [START_REF] Mattei | Modules de feuilletages holomorphes singuliers[END_REF], there exists a Kodaira-Spencer map like that associate to any unfolding with parameter in (C p , 0), its Kodaira Spencer derivative which is a linear map from C P to H 1 (D, Θ). The unfolding is semi-universal as in the theorem above if and only if its Kodaira Spencer derivative is a linear isomorphism. We consider the covering of the exceptionnal divisor E -1 (0) by two open sets U 1 and U 2 where U 1 and U 2 are respectively tubular neighborhood of D 1 and D 2 . It is known that this covering is acyclic with respect to the sheaf Θ, i.e, H 1 (D i , Θ) = 0. Therefore, following [START_REF] Godement | Théorie des faisceaux[END_REF] to compute the cohomological group H 1 (D, Θ) we can use this covering, that is to say, the following isomorphism

(2.1) H 1 (D, Θ) ≃ H 0 (U 1 ∩ U 2 , Θ) H 0 (U 1 , Θ) ⊕ H 0 (U 2 , Θ)
.

In view of the glued construction of

F σ,α0 , a 0-cocycle X 12 in H 0 (U 1 ∩ U 2 , Θ) is trivial in H 1 (D, Θ) if and only if the cohomological equation (2.2) X 12 = X 1 -(g σ • Φ α0 ) * X 2 admits a solution where X 1 ∈ H 0 (U 1 , Θ) and X 2 ∈ H 0 (U 2 , Θ). Now, it is known [10][3] that the dimension of the C spaceH 1 (D, Θ) is 1.
Thus, to prove the result, it is enough to show that the image of the deformation (F σ,α ) α∈(α0,C) by the foliated Kodaira-Spencer map is not trivial in H 1 (D, Θ). The foliation F σ,α is obtained from F σ,α0 by gluing with the automorphism

Φ -1 α0 • Φ α (x 1 , y 1 ) = x 1 1 + αy 1 1 + α 0 y 1 , y 1 .
Thus, its image by the Kodaira-Spencer map is the cocycle

∂ ∂α Φ -1 α0 • Φ α α=α0 = x 1 y 1 1 + α 0 y 1 ∂ ∂x 1
Hence, the unfolding is semi-universal if and only if the equation

(2.3) x 1 y 1 ∂ ∂x 1 + • • • = X 2 -(g σ • Φ α0 ) * X 1
has no solution. This equation can be more precisely written in the following way

x 1 y 1 ∂ ∂y 1 + • • • = A 2 (x 1 , y 1 ) x 1 ∂ ∂x 1 -(g σ • Φ α0 ) * A 1 (x 3 , y 3 ) x 3 ∂ ∂x 3
where A 1 and A 2 are functions defined respectively in U 1 and U 2 . Let us write the Taylor expansion of

A 2 = ij a 2 ij x i 1 y j 1 . In the coordinates (x 2 , y 2 ) the function A 2 is written A 2 = ij a 2 ij x 2i-j 2 y i 2 .
Therefore, if a 2 ij = 0 then 2i -j ≥ 0 and the monomial term y 1 cannot appear in the Taylor expansion of A 2 . In the same way, the Taylor expansion of

A 1 = ij a 1 ij x i 3 y j 3 , satisfies a 1 ij = 0 ⇒ i ≥ j.
Since X 1 vanishes along the exceptionnal divisor whose trace in U 1 is the diagonal x 3 = y 3 , we have A 1 = (x 3 -y 3 ) Ã1 . Thus, in the coordinates (x 1 , y 1 ), X 1 is written

X 1 = Ã1 (x 1 (1 + α 0 y 1 ) + σ (y 1 ) , σ (y 1 )) (x 1 (1 + α 0 y 1 ) + σ (y 1 )) x 1 ∂ ∂x 1 .
If Ã1 (0, 0) = 0 then the term y 1 x 1 ∂ ∂y1 of the Taylor expansion of the cocycle (2.3) cannot come from X 1 . However, if Ã1 (0, 0) = 0 then A 1 cannot be global. Therefore, the equation (2.3) cannot be solved, which proves the result.

We observe that F σ,α is an unfolding over the whole C. Actually in the course of the above proof, we obtain a more precise result Corollary 6. More generally, for any germ of function A (x, y) with A (0, 0) = 0, the C-space H 1 (D, Θ) for the foliation

F 1 F 2 / (x 1 , y 1 ) → (x 1 A (x 1 , y 1 ) + σ (y 1 ) , σ (y 1 )) .
is generated by the cocycle the image of x 1 y 1 ∂ ∂y1 through the isomorphism (2.1). In particular, any deformation of the form

ǫ → (F 1 F 2 ) ǫ/ (x 1 , y 1 ) → (x 1 A ǫ (x 1 , y 1 ) + σ (y 1 ) , σ (y 1 ))
where ∂Aǫ ∂y1 (0, 0) does not depend on ǫ is locally analytically trivial.

As an easy consequence of the corollary, we obtain a theorem of normalization of the construction of absolutely dicritical foliations of cusp type.

Theorem 7. Any absolutely dicritical foliation of cusp type is equivalent to some F σ,α .

Proof. Let us consider F an absolutely dicritical foliation of cusp type and let E be its associated reduction. Since along each component of the exceptionnal divisor the foliation is purely radial, there exists two automorphisms Φ 1 and Φ 2 that conjugates F respectively to the models F 1 and F 2 in the neighborhood of respectively D 1 and D 2 . The cocycle of gluing is thus written

Φ 1 • Φ -1 2 .
Applying if necessary a global automorphism of Φ 1 that let invariant each leaf, we can suppose that Φ 1 • Φ -1 2 send the exceptionnal divisor x 1 = 0 on the line x 3 = y 3 . Since the cocycle conjugates the foliations F 2 and F 1 , it can be written

(x 1 , y 1 ) → (x 1 A (x 1 , y 1 ) + σ (y 1 ) , σ (y 1 )) .
for some σ ∈ Diff (C, 0) and some A ∈ C {x 1 , y 1 } with A (0, 0) = 0. Applying if necessary an automorphism of F 2 defined by (ǫx 3 , ǫy 3 ) for some ǫ = 0, we can suppose that A (0, 0) = 1. Now we can write the cocycle

(x 1 , y 1 ) → x 1 1 + αy 1 + Ã (x 1 , y 1 ) + σ (y 1 ) , σ (y 1 ) .
where no term of the form ay 1 appears in Ã. According to the corollary, the deformation parametrized by ǫ and defined by the gluing cocycle

(x 1 , y 1 ) → x 1 1 + αy 1 + ǫ Ã (x 1 , y 1 ) + σ (y 1 ) , σ (y 1 )
is locally analytically trivial. Thus the foliation obtained setting ǫ = 1 and ǫ = 0 are analytically equivalent and setting ǫ = 0 yields a cocycle of the desired form.

The couple (α, σ) is unique up to conjugacies fixing any point of the exceptionnal divisor. However, once we authorize any kind of conjugacies, this couple is not unique anymore. But the ambiguity can be described. Proposition 8. Two normal forms F σ,α and F γ,α ′ are conjugated if and only if there are two homographies h 0 and h 1 such that

(2.4)    σ = h 1 • γ • h 0 2 5 α -3 2 σ ′′ (0) σ ′ (0) = 2 5 α ′ -3 2 γ ′′ (0) γ ′ (0) h ′ 0 (0) - h ′′ 0 (0) h ′ 0 (0)
Proof.

Step 1 -In view of our gluing construction and following the key remark (4), the existence of a conjugacy implies that there exist two automorphisms of respectively F 2 and F 1 written

Φ 2 = (x 1 A 2 (x 1 , y 1 ) , h 0 (y 1 )) and Φ 1 = (x 3 A 1 (x 3 , y 3 ) , h 1 (y 3 )) such that (x 1 (1 + αy 1 ) + σ (y 1 ) , σ (y 1 )) = Φ 1 • x 1 1 + α ′ y 1 + γ (y 1 ) , γ (y 1 ) • Φ 2 .
First, we obviously get the following relation σ = h 1 • γ • h 0 . Moreover, if we look at the first component of the above relation we get

x 1 (1 + αy 1 ) + σ (y 1 ) = x 1 A 2 (x 1 , y 1 ) 1 + α ′ h 0 + γ • h 0 × A 1 x 1 A 2 (x 1 , y 1 ) 1 + α ′ h 0 + γ • h 0 , γ • h 0
If we compute the derivative ∂ ∂x1 of the above relation and then set x 1 = 0, we get

1 + αy 1 = A 2 (0, y 1 ) 1 + α ′ h 0 × γ • h 0 ∂A 1 ∂x 1 (γ • h 0 , γ • h 0 ) + A 1 (γ • h 0 , γ • h 0 ) (2.5)
(1) Now, since Φ 1 preserve the curve y = x, we obtain

A 1 (x, x) = h 1 (x) x Thus, A 1 (0, 0) = h ′ 1 (0) . Setting y 1 = 0 in the relation above, we get 1 = A 2 (0, 0) A 1 (0, 0) . Therefore, A 2 (0, 0) = 1 h ′ 1 (0) . Now, let us write the Taylor expansion of A 1 A 1 (x 3 , y 3 ) = h ′ 1 (0) + rx 3 + sy 3 + • • • . Since, A 1 (x, x) = h1(x)
x , we have r

+ s = h ′′ 1 (0)
2 . Now, the biholomorphism (x 3 A 1 (x 3 , y 3 ) , h 1 (y 3 )) is global: therefore, it can be push down and extended at the origin of C 2 as a local automorphism written

(x, y) → xA 1 x, y x , h 1 y x xA 1 x, y x .
The second component of this expression is written

y αx + βy h ′ 1 (0) x + rx 2 + sy + • • • where α = 1 h ′ 1 (0) and β = - h ′′ 1 (0) 2h ′ 1 (0) 2 .
It is extendable at (0, 0) if and only if the expression in parenthesis can be holomorphically divided by αx + βy. Looking at the first jet of these expressions leads to

β α s h ′ 1 (0) = 0 =⇒ s = βh ′ 1 (0) α = - h ′′ 1 (0) 2
Finally, we have r = h ′′ 1 (0) . (2) In the same way, let us write the Taylor expansion of

A 2 (x 1 , y 1 ) = 1 h ′ 1 (0) + uy 1 + vy 2 1 + • • • . The second component of the expression of Φ 2 in the coordinates (x 2 , y 2 ) is y 2 x 2 2 h 2 0 1 x2 A 2 y 2 x 2 2 , 1 x2
which is equal to

y 2 (α ′ x 2 + β ′ ) 2 αx 2 2 + ux 2 + v + y 2 (• • • ) where α ′ = 1 h ′ 0 (0) and β ′ = - h ′′ 0 (0) 2h ′ 0 (0) 2 . Since it is extendable at x 1 = -β ′ α ′ , there exists a constant Γ such that α ′ x 2 + β ′ 2 = Γ αx 2 2 + ux 2 + v
Hence, we have the equality u = 2

αβ ′ α ′ = - h ′′ 0 (0) h ′ 0 (0)h ′ 1 (0) . Now, we can identified the coefficient of the equation (2.5) It is α = A 2 (0, 0) γ ′ (0) h ′ 0 (0) ∂A 1 ∂x 1 (0, 0) + h ′′ 1 (0) 2 γ ′ (0) h ′ 0 (0) + α ′ h ′ 0 (0) h ′ 1 (0) +uh ′ 1 (0) = 3 2 γ ′ (0) h ′ 0 (0) h ′′ 1 (0) h ′ 1 (0) - h ′′ 0 (0) h ′ 0 (0) + α ′ h ′ 0 (0) .
Using the relation σ = h 1 • γ • h 0 the above equality can be formulated as in the theorem.

Step 2 -We suppose that the conclusion of the statement is satisfied. Let us suppose that

h 1 (z) = z α + βz h 0 (z) = z a + bz Then we set A 2 (x 1 , y 1 ) = α + 2 αb a y 1 + αb a 2 y 2 1 A 1 (x 3 , y 3 ) = α + βy 3 (α + βx 3 ) 2 .
In view of the computations done in the first step, the two automorphisms Φ 1 and Φ 2 associated to A 1 and A 2 can be extended on U 1 and U 2 , tubular beighborhood of D 1 and D 2 . Moreover, we obtain the following relation

(x 1 (1 + αy 1 + ∆ (x 1 , y 1 )) + σ (y 1 ) , σ (y 1 )) = Φ 1 • x 1 1 + α ′ y 1 + γ (y 1 ) , γ (y 1 ) • Φ 2
where ∆ does not contain any monomial term in y 1 . Now, using the proposition (6), we see that the deformation defined by ǫ → (x 1 (1 + αy 1 + ǫ∆ (x 1 , y 1 )) + σ (y 1 ) , σ (y 1 ))

is analytically trivial, which ensures the theorem.

Theorem 9. The moduli space of absolutely dicritical foliations of cusp type can be identified with the functionnal space C {z} up to the action of C * defined by

ǫ • (z → σ (z)) = ǫ 2 σ (ǫz) .
Proof. We can consider the following family parametrized by Diff (C, 0)

σ ∈ Diff (C, 0) → F 3 2 σ ′′ (0) σ ′ (0) ,σ
.

It is a complete family for absolutely dicritical foliations of cusp type: in any class of absolutely dicritical foliation of cusp type there is one that is analytically equivalent to one of the form

F 3 2 σ ′′ (0) σ ′ (0) ,σ
. Indeed, considering the foliation

F α ′ ,γ , we can choose h 0 such that 2 5 α ′ -3 2 γ ′′ (0) γ ′ (0) h ′ 0 (0) - h ′′ 0 (0) h ′ 0 (0) = 0. Therefore, setting σ = γ • h 0 ensures that F α ′ ,γ and F 3 2 σ ′′ (0) σ ′ (0) ,σ are analytically equivalent. Moreover, if F 3 2 σ ′′ 0 (0) σ ′ 0 (0) ,σ0
and

F 3 2 σ ′′ 1 (0) σ ′ 1 (0)
,σ1

are analytically equivalent then there exists ǫ ∈ C * and an homographie h 1 such that

(2.6) σ 0 (z) = h 1 • σ 1 • (ǫz) .
Indeed, the second homographie h 0 that appears in the proposition (8) has to be linear for the relations (2.4) ensures that h ′′ 0 (0) = 0. Thus, h 0 is written z → ǫz for some ǫ. To simplify the relation (2.6), we use the Schwartzian derivative which is a surjective operator defined by S :

   Diff (C, 0) → C {z} y → 3 2 y ′′′ y ′ -y ′′ y ′ 2
and satisfying the following property: the relation (2.6) is equivalent to S (σ 0 ) (z) = ǫ 2 S (σ 1 ) (ǫz). Therefore, the moduli space of absolutely dicritical foliation of cusp type is identified via the Schwartzian derivative to the quotient of C {z} up to the action of

C * ǫ • (z → σ (z)) = ǫ 2 σ (ǫz) .
As mentionned in the introduction, this theorem does not state that the transversal structure σ is the sole analytical invariant of an absolutely dicritical foliation of cusp type. Indeed, the action of the group of conjugacies act transversaly to the transverse structures σ and to the moduli of Mattei α. The family

F 3 2 σ ′′ (0) σ ′ (0)
,σ is a complete tranversal set for this action the group of conjugacies.

Orbits of the action of

α ∈ C F 3 2 σ ′′ (0) σ ′ (0) ,σ σ ∈Diff(C, 0)
As a consequence of the above description of the moduli space of absolutely dicritical foliations, we should be able to prove the existence of a non algebrizable absolutely dicritical foliation using technics developped in [START_REF] Genzmer | Existence of non-algebraic singularities of differential equation[END_REF].

3. Formal normal forms for 1-Forms.

It is known [START_REF] Cano | Absolutely dicritical foliations[END_REF] that the valuation of a 1-form ω with an isolated singularity defining an absolutely dicritical foliation of cusp type is 3. Up to some linear change of coordinates, we can suppose that the singular point of the foliation after one blowing-up has (0, 0) for coordinates in the standard coordinates associated to the blowing-up. Moreover, since the foliation is generically transversal to the exceptionnal divisor of the blowing-up of 0 ∈ C 2 , 0 , the homogeneous part of degree 3 of ω is tangent to the radial form ω R = xdy -ydx. Thus there exists an homogeneous polynomial function of degree 2 P 2 such that

ω = P 2 ω R + i≥4 (A i (x, y) dx + B i (x, y) dy) .
After one blowing-up, the singular locus is given by the solutions of P 2 (1, y) = 0 and P 2 (x, 1) = 0 in each chart. Thus P 2 is simply written ay 2 for some constant a = 0. After on blowing-up (x, t) → (x, tx), the linear part near (0, 0) of the pull-back form is written

A 4 (1, 0) + t ∂A 4 ∂t (1, 0) + tB 4 (1, 0) dx + xB 4 (1, 0) dt + xA 5 (1, 0) dx.
The absolutely dicritical property ensures that this linear part is non trivial and tangent to the radial vector field tdx + xdt. Hence, the following relations hold

A 4 (1, 0) = A 5 (1, 0) = 0 and ∂A 4 ∂t (1, 0) + 2B 4 (1, 0) = 0
Finally, the form ω is written

ω = y 2 ω R + -2αx 3 + yQ 2 (x, y) ydx + αx 4 + yQ 3 (x, y) dy + (A 5 (x, y) dx + B 5 (x, y) dy) + • • •
where α = 0.

Proposition 10. The 1-form ω is formally equivalent to a 1-form written

y 2 ω R + αx 3 (xdy -2ydx) + ax 3 ydy + n≥5 x n-1 ((a n x + b n y) dx + (c n x + d n y) dy)
where a 5 = 0. Moreover, this formal normal form is unique up to change of coordinates tangent to Id.

Proof. The action of a change of coordinates φ n : (x, y) → (x, y) + (P n , Q n ) where P n and Q n are homogeneous polynomial functions of degree n does not modify the jet of order n + 1 of ω. Moreover, the action on the homogeneous part of degree n + 2 is written

J n+2 φ * n ω = J n+2 ω +y 2 x ∂Q n ∂x -y ∂P n ∂x + Q n dx + x ∂Q n ∂y -x ∂P n ∂x + P n dy
We are going to verify that the linear morphism defined by

L : (P n , Q n ) → x ∂Q n ∂x -y ∂P n ∂x + Q n , x ∂Q n ∂y -x ∂P n ∂x + P n
from the set of couples of homogeneous polynomial functions of degree n to itself is a one to one correspondance. To do so, let us compute the kernel of this morphism and let us write

P n = n i=0 p i x i y n-i and Q n = n i=0 q i x i y n-i . The coefficients of the components of L (P n , Q n ) on the monomial term x i y n-i are q i (i -1) -p i+1 (i + 1) i = 0..n -1 q n (n -1) i = n and -p i (n -i -1) + q i-1 (n -i + 1) i = 1..n p 0 (n -1) i = 0.
If (P n , Q n ) is in the kernel then q n = 0 and p 0 = 0. Moreover, applying the above relation with i = 1 and i = n -1 yields p 2 = 0 and q n-2 = 0. Now for i = 1..n -1 but i = n -2, a combination of the relations above ensures that

0 = q i (i -1) -q i (i + 1) n -i n -i -2 = q i n -i -2 (2 -2n)
Thus q i = 0 for i = 0..n -1. Therefore (P n , Q n ) = 0 and L is an isomorphism. Thus, we can choose φ n such that

J n+2 φ * n ω = x n-1 ((a n x + b n y) dx + (c n x + d n y) dy) . Clearly the composition φ 2 •φ 3 •• • • is formally convergent, which proves the propo- sition.

Absolutely dicritical foliation admitting a first integral.

In this section, we study absolutely dicritical foliations that admit a meromorphic first integral. Such an existence can be completely red on the transverse structure.

Theorem 11. Let F be an absolutely dicritical foliation of cusp type with σ as transverse structure. Then F admits a first integral if and only if there exists two non constant rationnal functions R 1 and R 2 such that

R 1 • σ = R 2 .
Notice that the existence of R 1 and R 2 does not depend on the equivalence class of σ modulo homographies.

Proof. Suppose that F admits a meromorphic first integral f . After blowing-up, the function f is a non constant rationnal function in restriction to each component of the divisor. Since for any point p, p and σ (p) belongs to the same leaf, we have

f | D1 (p) = f | D2 (σ (p)) .
Now, suppose there exist two rationnal function as in the lemma. According to some previous result, there exists α and γ such that the foliation F is analytically equivalent to F α,γ . The application σ and γ are linked by a relation of the form

h 0 • σ • h 1 = γ
where h 0 and h 1 are homographies. Thus, setting

R1 = R 1 • h -1 0 and R2 = R 2 • h 1 yields R1 • γ = R2
where R1 and R2 are still rationnal. Now, let us go back to the construction of F α,γ . We glue the models F 1 and F 2 around (x 1 , y 1 ) = 0 and (x 3 , y 3 ) = 0 by

(x 1 , y 1 ) → (x 3 = x 1 (1 + αy 1 ) + γ (y 1 ) , y 3 = γ (y 1 ))
Consider for F 1 the first integral F 1 (x 1 , y 1 ) = R2 (y 1 ) and for F 2 the first integral F 2 (x 3 , y 3 ) = R1 (y 3 ). Then these two meromorphic first integrals can be glued in a global meromorphic first integral since

F 2 (x 3 , y 3 ) = F 2 (x 1 (1 + αy 1 ) + γ (y 1 ) , γ (y 1 )) = R1 (γ (y 1 )) = R2 (y 1 ) = F 1 (x 1 , y 1 ) .
Thus the absolutely dicritical foliation admits a meromorphic first integral.

In view of this result, it is easy to produce a lot of examples of absolutely dicritical foliation admitting no meromorphic first integral setting for instance

σ (z) = e z -1.
Notice that the existence of the first integral depends only on the transversal structure σ and not on the value of the moduli of Mattei α. This is consistent with the fact that along an equireducible unfolding the existence of a meromorphic first integral for one foliation in the deformation ensures the existence of such a first integral for any foliation in the deformation. Finally, since the topologically classification of absolutely dicritical foliations is trivial, the above result produce a lot of examples of couples of conjugated foliations such that only one of them admits a meromorphic first integral. Hereafter we treated a special case, that is when the transversal structure σ is an homography.

Proposition 12. Let F be an absolutely dicritical foliation of cusp type with an homographic transversal structure. Then, up to some analytical change of coordinates, F admits one of the following rationnal first integrals:

(1) f = y 2 +x 3 xy . (2) f = y 2 +x 3 xy + x
Proof. Let us consider the following germ of family of meromorphic functions with (x, y, z) ∈ C 3 , (0, 0, 0) defined by

f z = y 2 + x 3 + zx 2 y xy = a b .
For any z, the foliation associated to f z is absolutely dicritical of cusp type. Let us prove that this family is an equireducible unfolding. We consider the integrable 1-form Ω = adb -bda. It is written

2x 3 y + zx 2 y 2 -y 3 dx + xy 2 -x 4 dy + x 3 y 2 dz.
It defines an unfolding of the foliation given by f 0 with one parameter. Its singular locus is the z-axes and it is transversal to the fibers of the fibration (x, y, z) → z.

Once we blow-up the z-axe, in the chart E : (x, t, z) = (x, tx, z), the 1-form Ω is written

Ω = t (1 -zt) dx + t 2 -x dt + t 2 xdz.
Therefore, the singular locus of the pull-back foliation is still the z-axe in the coordinates (x, t, z) and in a neighborhood of x = 0 the foliation Ω is transverse to the fibration z = cst. If we blow-up again the z-axe we find

(1 -zx) dt + (1 -zt) dx + txdz
which is smooth. Since the curve x = t = 0 is invariant and since the foliation is still transverse to the fibration z = cts, the unfolding is equisingular. Now, this unfolding is analytically trivial if and only if the monomial term x 3 y 2 belongs to the ideal generated by 2x 3 y + zx 2 y 2 -y 3 and xy 2 -x 4 [START_REF] Cerveau | Formes intégrables holomorphes singulières[END_REF]. Setting z = 0 this would imply that x 3 y 2 ∈ 2x 3 y -y 3 , xy 2 -x 4 which is impossible. Thus, this unfolding is not analytically trivial and since the moduli space of unfolding of absolutely dicritical foliations is of dimension 1, it is also semi-universal. Now, let us consider a foliation F as in the proposition. Up to some linear change of coordinate, we can suppose that after the reduction process its singular point and its transversal structure are the same as the function x 2 +y 3 xy that is to say (0, 0) and Id in the standard coordinates associated to the reduction process. Let us denote by F 0 the foliation given by x 2 +y 3 xy . We are going to construct an unfolding from F 0 to F . As always since the beginning of this article, we denote by D 1 and D 2 the two exceptionnal component of the divisor. In the neighborhood of each of them, both foliation are purely radial. Thus there exists two conjugacy Φ 1 and Φ 2 defined in the neighborhood of respectively D 1 and D 2 such that

Φ * 1 F 0 = F Φ * 2 F 0 = F Φ 1 | D1∪D2 = Id Φ 2 | D1∪D2 = Id .
Since, F 0 and F have the same transversal structures, the cocycle

Φ 1 • Φ -1 2
is a germ automorphism of F 0 near the singular point of the divisor that lets fix the points of the divisor and that let globally fix each leaf. It is easy to see that one can construct an isotopy from Φ 1 • Φ -1 2 to Id in the group of germs of automorphisms of F 0 near the singular point of the divisor that let fix each point of the divisor and that let globally fix each leaf. Let us denote by Φ t this isotopy satisfying Φ 0 = Id and

Φ 1 = Φ 1 • Φ -1
2 . The unfolding defined by the following glued construction

((F 0 , D 1 ) × U ) ((F 0 , D 2 ) × U ) / (x, t) ∼ (Φ t (x) , t) .
where U is an open neighborhood of {|t| ≤ 1} links F 0 and F . The meromophic first integral f 0 of F 0 can be extended in a meromorphic first integral F of the whole unfolding [START_REF] Cerveau | Formes intégrables holomorphes singulières[END_REF]. Thus F | t=1 is a meromorphic first integral of F . By equisingularity F | t=0 and F | t=1 must have exactly the same number of irreducible components in their zeros and in their poles, which is the same number of irreducible components in the zeros and in the poles of F . They also must have the same topology since an unfolding is topologically trivial. Thus the foliation F admits a meromorphic first integral whose zero is exactly the leaf passing through the singular point of the exceptionnal divisor and whose poles are the union of two smooth curves attaching respectively to D 1 and D 2 . Thererfore up to some change of coordinates, we can suppose that F has a meromorphic first integral of the form

f = y 2 + x 3 + ∆ (x, y) a x b y c
where the Taylor expansion of ∆ (x, y) admits monomial term x i y j with 2i + 3j > 6. The absolutely dicritical property ensures that a = b = c. Therefore, we can suppose that a = b = c = 1. Let us denote by Λ λ (x, y) the homothetie Λ λ (x, y) = λ 2 x, λ 3 y . Composing by Λ λ at the right of f yields

f • Λ λ λ = y 2 + x 3 + ∆ λ (x,

y) xy

For any λ = 0, the foliation given by f and by f •Λ λ λ are analytically conjugated. But the deformation given by λ → f •Λ λ λ is an equisingular unfolding of f 0 since ∆ λ goes to 0 when λ → 0. Using the semi-universality of the family introduced at the beginning of the proof, for λ small enough, there exists some α such that the following conjugacies holds [START_REF] Camacho | Topological invariants and equidesingularization for holomorphic vector fields[END_REF]. If α = 0, applying some well-chosen homothetie, we can suppose α = 1. And f is of type (2).

f ∼ f • Λ λ λ ∼ f α . Now if α = 0 then f is of type (
Remark 13. In the last part of this article, we will prove that actually the two functions (1) and (2) of the previous result define two foliations analytically equivalent.

It is possible to construct some others examples of absolutely dicritical foliations of cusp type with a rationnal first integral: to do so, consider a foliation of degree 1 on P 2 . These are well-known [START_REF] Cerveau | Géométrie classique de certains feuilletages de degré deux[END_REF]: they have three singular points counted with multiplicities and admit an integrating factor. For instance, the foliation given in homogeneous coordinates by the multivalued functions

[x : y : z] → x α y β z α+β or [x : y : z] → Q z 2
where Q is a non-degenerate quadratic form is of degree 1. When α and β are rationnal numbers, the foliation admits a rationnal first integral. Now consider two generic lines L 1 and L 2 . Each of them is tangent to one leaf of the foliation. We can suppose that the tangency point is different from the intersection point of L 1 and L 2 . Now, blow-up twice the tangency point on L 1 and thrice the tangency point on L 2 . The final configuration is the following

L 1 2 L -1 -2 -1 -2 -2 -1 - 2
Thus, the divisor L 1 ∪ L 2 can be contracted toward a smooth algebraic manifold. The obtained singularity is naturally absolutely dicritical of cusp type and admits a rationnal first integral. For instance, if we consider the foliation given in affine coordinates by xy = cst and L 1 : x + y = 1 and L 2 : x -y = 1, the transverse structure is equivalent to σ (t) = t + 1 and thus the foliation is equivalent to the functions of proposition (12). However, considering the foliation given by x + y 2 = cst yields the transverse structure t

→ 1- √ 1+12t+4t 2 2
which is not an homography.

5. Moduli of Mattei.

5.1.

The parameter space of the unfoldings. As already explain, the deformation α → F α,σ is an unfolding with a set of paramater equal to C. It is a natural problem to ask if two parameters define two foliations analytically equivalent. In order to do so, we introduced the following definition:

Definition 14. Let σ be an element of Diff (C, 0). An homography h with h (0) = 0 is called an homographic symetry of f if and only if there exists an homography h 1 such that (5.1)

h 1 • σ • h = σ.
We denote by H (f ) the group of homographic symetries of f .

The following result is probably known but we do not find any reference in the litterature.

Lemma 15. If H (f ) is infinite then f is an homography and H (f ) is the whole set of homographies fixing the origin.

Proof. The relation (5.1) is equivalent to the functionnal equation

f • h (z) = 1 (h ′ ) 2 f (z)
where f = S (σ) is the Schwartzian derivative of σ. Let us write h (z) = z a+bz and f (z) = n≥1 f n z n .

(1) Suppose that h ′ (0) is not a root of unity. Then applying the above relation at z = 0 leads to f (0) = 0. Now, we have

a 2 n≥1 f n z n (a + bz) n = (a + bz) 4 n≥1 f n z n .
An easy induction on n show that for any n f n = 0, thus f = 0 and σ is an homography.

(2) Suppose now h

′ (0) = 1 then n≥0 f n z n (1 + bz) n = (1 + bz) 4 n≥0 f n z n
Suppose that b = 0. If for any n ≤ N -1 we have f n = 0, let us have a look at the terms in x N +1 in the above equality. It is

-N bf N + f N +1 = 4bf N + f N +1
Thus f N = 0. Which, proves by induction that f still is equal to zero.

(3) If H (f ) is infinite, suppose it admits two elements h and g that did not commute, then [h, g] is tangent to Id but is not the Id . Thus using the above computation, f = 0. (4) Finally, if h ′ (0) is a root of unity, it is easly seen that h •(n) = Id where n is the smallest integer such that h ′ (0) n = 1. Thus, suppose that the group H (f ) is abelian and any element of finite order. We have an embedding

H (f ) -→ Aff (C)
since, the sole element tangent to Id is the identity itself. Therefore, H (f ) can be seen as abelian subgroup of Aff (C). Hence, the group has a fix point and can be seen as a subgroup of the linear transformations of C. Now let us write the relation on the Schwartzian seen at ∞ f (1/ (1/h (1/z))) = 1

h ′ 1 z 2 f 1 z .
Setting, u (z) = 1 z 4 f 1 z yields u (az + b) = 1 a 2 u (z) . Since, u = α z 4 + • • • we can consider the double primitive function U = ˜u with U (∞) = 0. This is a univalued holomorphic function defined near ∞. Finally, the function U satisfies the following functionnal relation

U (az + b) = U (z) .
But in view of the dynamics of Lin (C), it is clear that if H (f ) is infinite then U = cst and thus u = 0.

In the course of the proof of the above result, we obtain the following result Corollary 16. Let M be the quotient of C by the relation α ∼ α ′ if and only if F α,σ ∼ F α ′ ,σ then there is only two possibilities

(1) M = {0} when σ is an homography -F α,σ is then analytically to y 2 +x 3 xy .

(2) M = C / H where H is a finite subgroup of Aff (C).

Genrerically, H is reduced to {Id}.

As an obvious consequence, the functions obtained in proposition (12) define two foliations analytically equivalent. 5.2. Toward a geometric description of the moduli of Mattei. It remains to give a geometric interpretation of the parameter α. A promising approach is the following. Near the singular point of the divisor, the leaf is conformally equivalent to a disc minus two points which are the intersections between the leaf and the exceptionnal divisor. If we consider in the leaf a path linking this two points, we obtain after taking the image of this path by E, an asymptotic cycle γ as defined in [START_REF] Teyssier | Analytical classification of singular saddle-node vector fields[END_REF]which is not topologically trivial. γ Therefore, considering the family of these cycles parametrized by a transversal parameter to the foliation yields a vanishing asymptotic cycle. We claim that the moduli of Mattei should be associated to the length of this vanishing asymptotic cycle: more precisely, it should be computed by the integral of some form along this vanishing cycle. Actually, it easy to prove the following: let ω be a 1-form defining an absolutely dicritical foliation of cusp type and let η be any germ of 1 form. Then η is relatively exact with respect to ω, i.e., there exist two germs of holomorphic functions f and g such that η = df + gω if and only if the integral of η along any asymptotic cycle γ vanish. Thus, we think that in a sense that has to be worked out, the moduli of Mattei should be computed by the integral of some generator of the relative cohomology group of ω along the asymptotic vanishing cycle.

  Let us consider F 0 and F 1 two absolutely dicritical foliations of cusp type. Applying if necessary a linear change of coordinates to F 0 for instance, we can suppose that both foliations are reduced by exactly the same sequence of two blowingups E. Let us write E -1 (0) = D 2 ∪ D 1 and D 2 ∩ D 1 = {p} . Let us consider σ 0 and σ 1 in Diff ((D 2 , p) , (D 1 , p)) the transversal structures of F 0 and F 1 . According to the previous lemma, there exist h an homeomorphism of D 2 such that h = σ -1 0 • σ 1 in a neighborhood of p in D 2 . Since, along D 2 or D 1 the foliations are transverse, there exist two homeomorphisms H 0 and H 1 defined respectively in a neighborhood of D 2 and D 1 such that

Proposition 3. Two absolutely dicritical foliations of cusp type are topologically equivalent. The class Top (F ) is equal to the set of all absolutely dicritical foliations.

Proof.


Radu Vranceanu
email: vranceanu@essec.fr

Angela Sutan
email: angela.sutan@escdijon.eu

Delphine Dubart
email: dubart@essec.fr

TRUST AND FINANCIAL TRADES: LESSONS FROM AN INVESTMENT GAME WHERE RECIPROCATORS CAN HIDE BEHIND PROBABILITIES

Keywords: JEL Classification Index : C90, D81, G00 Trust, Financial transactions, Experimental economics, Investment game, External randomness

This paper shows that if a very small, exogenously given probability of terminating the exchange is introduced in an elementary investment game, more reciprocators will choose the defection strategy. Everything happens as if they "hide behind probabilities" in order to break the trust relationship. Investors do not alter their behavior in a significant way, at least not for a very small external risk. Financial assets all come with a predetermined and contractual probability that by the time when the buyer has to receive the reward for his investment, "bad luck" might have brought the asset value down to zero. In the light of the experimental findings, such trades would not provide a favorable environment for building trust.

Introduction

From Aristote to present times, philosophers and political thinkers keep on claiming that there is something rotten in the realm of financial trades. Robert Solomon, a highly respected moral philosopher, pointed out that there is huge difference between the quest for profit of the traditional producer of goods, and the "abstract greed" of financial intermediaries (Solomon,1992). In the aftermath of the Great Recession of 2007-2009, many journalists, political leaders and even a few economists argued that the main culprit for this dramatic outcome was the "unleashed greed" of main players in the financial market. For instance, Laurence Summers, in his role of the US President economic advisor, argues that during the crisis "an abundance of greed and an absence of fear led some to make investments not based on the real value of assets, but on the faith that there would be another who would pay more for those assets" [START_REF] Summers | Responding to an Historic Economic Crisis: The Obama Program, Remarks at the Brookings Institution[END_REF]. Joseph Stiglitz (2010, p. 19) also notices that "Wall Streets' high rewards and single-minded focus on making money might attract more than its fair share of the ethically challenged, but the universality of the problem suggests that there are fundamental flows in the system". If such criticism is nowadays shared by a majority of people, there are not many convincing explanations of why trades in financial assets should be more prone to this kind of egoistic behavior than trades in any other goods.

In this paper we bring empirical evidence in favor of the assumption according to which, by contrast with many other markets, financial transactions do not provide a favorable environment for building trust. As pointed out by many scholars, trust is one important lubricant of a social system where resource allocation is based on voluntary exchange (e.g., [START_REF] Arrow | The Limits of Organization[END_REF][START_REF] Mckean | Economics of trust, altruism and corporate responsibility[END_REF][START_REF] Noreen | The economics of ethics: A new perspective on agency theory[END_REF]. Dasgupta (2000, p.50) emphasizes that: "Trust is of much importance precisely because its presence or absence can have a strong bearing on what we choose to do and in many cases what we can do. The clause concerning the inability to monitor others' actions in my definition of trust is crucial. If I can monitor what others have done before I choose my own action, the word 'trust' loses its potency". When, for one reason or another, the trust relationship is broken, transaction costs increase dramatically and markets might not perform in a smooth way their role of allocating resources.

There is no reason to believe that all markets (a) require and (b) generate the same amount of trust. On many markets the buyer knows what he is buying at the moment when he pays for the goods, thus trust is of little utility. In markets for "experience goods" [START_REF] Nelson | Information and consumer behavior[END_REF], the buyer will learn the quality of the product only after using it. Some transactions, such as the booming e-commerce, are sequential by nature: first the buyer pays the price, then he will get the product; there is always a risk that the quality of the good may fall below the product specifications. In these markets trust appears as a convenient way to economize on various transaction costs related to seller's would-be misbehavior [START_REF] Gefen | E-commerce: the role of familiarity and trust[END_REF][START_REF] Mcknight | Developing and validating trust measures for e-commerce: An integrative typology[END_REF].

Trust is much needed on financial markets. Indeed, the buyer of a financial asset owes a promise to get more resources in the future and, as is the case for all promises, there is an element of uncertainty about the ability of the seller to keep his word. In particular, both seller's misbehavior and exogenous (natural) events can bring the future value of the asset down to zero. It is important to notice that, compared to other markets, a predetermined probability that the asset might worth nothing is contractual in financial trades: by construction, the value of a financial asset at a future time depends on the state of the nature, a random event. The price the buyer pays for the asset is the mathematical expectation of the discounted utility connected to all possible outcomes, including the outcome that the asset will worth zero.

We argue that the trust relationship between participants to exchange is deeply undermined when such "external randomness" is part of the transaction. In particular, we posit that those playing the role of the asset seller (or trustee) will be more tempted to break the trust relationship if they can "hide behind probabilities", i.e., invoke "bad luck" when they have actually implemented their selfish defection strategy. They might do so because, ex-post, those acting in the buyer's role (or trustor) can no longer distinguish whether the adverse effects are due to the trustee's decision to break the trust relationship or to the external event. In general the trust relationship builds on a mutual assessment of what "good behavior" is. We expect trustees to care about the opinion of the trustors, even if they do not know each other. Since there is no enforcement mechanism at work, this tie is fragile. If a good pretext 1 offers to them, some trustees would compromise with moral norms and behave in the most selfish way. 2In order to test for this theory, this paper develops an original variant of the traditional investment game [START_REF] Berg | Trust, reciprocity and social history[END_REF]. The latter has extensively been used to study the trust relationship. In the standard setup, an investor (player A) decides whether to transfer or not an amount of money to a reciprocator (player B). Before reaching the reciprocator, the transferred amount is augmented by the experiment administrator. Then the reciprocator decides whether or not to return some of the sum to the investor. Analyzing the problem as a one-shot, non-cooperative, sequential game, the dominant strategy for the reciprocator is to take all the money, thus the dominant strategy for the investor is not to transfer (this is the subgame perfect Nash equilibrium). However, in standard experiments, all the three possible outcomes are systematically observed: the one where the investor sends no money to the reciprocator, the partial trust outcome where the investor sends money and the reciprocator does not return a dime, and the mutual trust outcome where the investor sends money to the reciprocator and the later returns sometimes less, sometimes more than the amount initially invested. The mutual trust situation maps well the equilibrium of the cooperative game, wherein both players are better-off as compared to the non-cooperative equilibrium.

In the framework of the investment game, trust can therefore be interpreted as a (moral) value-driven way of building ties between participants to an exchange. Trust sets a natural limit on inefficient selfishness and brings about the Pareto-dominant cooperative outcome. People who reciprocate (at a cost for themselves) might do so because they want to reward a risky move of the person who placed his trust in them. Why investors send money in the first place is still a question open to debate. In general it is assumed that investors bet on the willingness of the trustees to reciprocate, and implement a strategy that maximize their expected return; their decision might also be driven by altruism, charity, or fairness considerations (see [START_REF] Camerer | Behavioral Game Theory: Experiments in Strategic Interaction[END_REF]. In Western 1 Pretext : a purpose or motive alleged or an appearance assumed in order to cloak the real intention or state of affairs (Merriam-Webster Dictionary, 2010). societies, trust in others is, in general, considered as a virtue; an investor who chooses the trust strategy might thus infer some utility only from being perceived by the others as a trustful person.

We slightly modify the original set-up by introducing an element of randomness that disconnects the outcome observed by the investor from the action of the reciprocator. Like in the standard set-up, the investor decides whether or not to send a (fixed) amount to the reciprocator. 3 If he transfers the money, a third party called Nature can decide whether to terminate the exchange and (the invested amount is then lost) or to continue it. Nature's move is random, the probability to terminate the relationship is common knowledge. If the exchange continues, the administrator multiplies the invested amount by a predetermined factor (bigger than one) and gives it to the reciprocator. The latter decide whether to keep all the money, or send back to the investor a predetermined sum (larger than the amount initially sent).

Thus, the only difference between this game and the traditional one consists in introducing an external risk of terminating the relationship between the investor and the reciprocator. Given that we want to make sure that changes in players' behaviors are induced by an alteration of the trust relationship and not by a reevaluation of the expected payoff of the investor, the external probability of termination must be very small. 4In the presence of this objective risk, when the investor sees no money forthcoming, he cannot say without ambiguity whether this is due to "bad luck" or to the reciprocator's "greed". Since the investor's assessment of the reciprocator's behavior is "blurred", the latter's feelings of shame or guilt might fade away.

Our experimental results corroborate this main assumption. The frequency of reciprocators who play the selfish strategy increases dramatically once that we introduce the objective risk. To the contrary, for a small risk, investors do not seem to anticipate the change in reciprocators' behavior; the frequency of investors who play the trust strategy does not decline in a significant way. This asymmetric response is removed for a larger objective risk.

Consequences at the macro level are straightforward: markets characterized by substantial external randomness -such as the financial market -would provide a more favorable environment for the proliferation of extremely selfish behavior, as compared to markets where sellers cannot "hide behind the probabilities" in order to deceive buyers.

These results are much in line with the transaction costs literature that shows that increasing anonymity increases the probability of opportunity and deceitful behavior [START_REF] Tazdaït | L'Analyse Economique de la Confiance[END_REF][START_REF] Milgrom | Economics, Organization and Management[END_REF]. While this literature emphasizes the role of reputation -agents extract information from observed behavior to update their beliefs about their trading partner, and cannot do so if transactions are anonymous -our paper focuses on the role of trust, that diffuse but fundamental moral tie between humans, that exists irrespective of monitoring and experience.

The paper is organized as following. The next section introduces the analytical context and the experimental design. Section 3 presents the main results. The conclusion and some policy implication are presented in the last section.

2 Context and design of the experience

Related work

There is a huge literature on trust in experimental games, such as surveyed for instance by [START_REF] Harvey | The trust paradox: a survey of economic inquiries into the nature of trust and trustworthiness[END_REF] and [START_REF] Tazdaït | L'Analyse Economique de la Confiance[END_REF].5 Some of these papers have also analyzed the interaction between trust and external risk factors. For instance, Houser, Schunk and Winter (2010) study how the personal interaction in an investment game is modified when the reaction of the reciprocator is subject to induced randomness. Firstly they measure the risk aversion of the players. Then, they made subjects play four times a trust game, where the investor is a human, while the reciprocator is either a human without known history of moves, or a human with move history, or a random computer, or a computer replicating human history of moves. Results put forward that risk attitude is an important explanatory variable in games against the computer (objective risk), but not in the trust game (between humans). [START_REF] Camerer | Tests of a sequential equilibrium reputation model[END_REF] analyze the trust behavior in a finite horizon repeated game, with uncertainty about the "type" of the trustee, who can be more or less trustworthy. Investors upgrade their beliefs by observing the moves of trustees. When playing the game several times, players tend to discover the sequential equilibrium, although some biases subside (people tend to cooperate more often than expected). [START_REF] Casari | The strategy method lowers measured trustworthy behavior[END_REF] compare the outcome of a trust game played in the "strategy method" where the reciprocator must decide whether he will send back money to the investor before the latter makes his choice, with the classical setup where the reciprocator moves after he learns the investor's choice (the "game method"). They show that the amount send back is larger in the game method as compared to the strategy method.

A significant change in interpersonal interactions induced by external randomness has already been observed by [START_REF] Dana | Exploiting moral wiggle room: experiments demonstrating an illusory preference for fairness[END_REF] in the framework of the dictator game. Under the "known" rule, the dictator can choose between the bundle (6 for him, 1 for the receiver) or the bundle (5 for him, 5 for the receiver). Subjects play the game several times and reach the standard outcome: many people are "generous", i.e., they prefer (5, 5) to (6,1). Under the "unrevealed" rule, bundles can be either (6,1) and (5,5) or, a new combination, (6,5) and (5,1). At the outset of the game, the dictator sees only his own gain from choosing a bundle (6 or 5); he does not see the gain of the receiver. In order to find out the latter, he must push on a button. What the experiments have shown is that in general players do not press the button, and choose more often the bundle that provides them the largest payoff. Not only they "hide behind risk" in order to get the largest payoff, but they do not want to have this risk removed.

Several scholars have analyzed the contract game between a principal and an agent where the outcome depends on both the agent's effort and Nature's random draw. In presence of such objective uncertainty, if a project fails due to low effort, the agent can claim that he exerted the high effort, but he had bad luck. [START_REF] Keser | Principals' principles when agents' actions are hidden[END_REF][START_REF] Fischbacher | z-Tree: Zurich toolbox for ready-made economic experiments[END_REF] defined this situation as a "hidden action". They show that the standard theoretical hypothesis according to which the agent would accept the contract provided that his expected utility is slightly above zero does not hold in an experimental setting; indeed, in this context, agents require a "fair share" of the surplus in order to participate. [START_REF] Charness | Promises and partnership[END_REF] analyze a similar contracting game. One main result from their experiment is that pre-contractual communication limits the agent's recourse to "hidden actions". They argue that human sentiments such as shame and guilty could play an important role in containing opportunistic behavior.

Other scholars have investigated the specificity of the trust relationship in financial markets.

In a paper that combines experiments and observation of actual behavior, [START_REF] Karlan | Using experimental economics to measure social capital and predict financial decisions[END_REF] shows that persons identified as "trustworthy" in experimental games are less likely to default on loans in real life. [START_REF] Guiso | Trusting the stock market[END_REF] provide a theoretical and empirical analysis of trust as pertaining to the decision to buy shares. They notice that: "in determining whether to invest in a stock an individual has to assess not only what is the "true" distribution of returns, but also what is the possibility that the company is just a scam, that the manager steals all the proceeds, or that the broker absconds with the money instead of investing it" (Guido et al., 2008(Guido et al., : p. 2563)). Trust is then modelled as the subjective probability that investors assign to emergence of these "extreme bad events" that drive to zero the firm value, captured by the catchall situation "the firm cheats". Using Dutch and Italian micro data, they put forward that less trusting individuals are less likely to buy stock, and if they buy, they buy less. At difference with these papers that analyze the impact of trust on financial transactions (volume, diversification, investment strategies), our paper focuses on the reverse causality: how (financial) risk influences the quality of the trust relationship.

Structure of the game

The experience was cast as a one-shot sequential game between an investor (player A) and a reciprocator (player B). The two players do not know each other; the rule of the game and structure of payoffs is common knowledge. The experiment builds on the "game method", i.e. the reciprocator is called to make his choice after investor's move. The original contribution of the paper as compared to a standard investment game is the introduction of Nature, a virtual player (the computer) that can terminate the exchange between the investor and the reciprocator with a known probability p.

The sequence of decisions is the following:

• At the outset of the experience, player A gets 2 euros. He can decide to keep them or to transfer 1 euro to B. If he keeps them, the game is over.

• If he transfers the one euro, an external agent, called Nature, decides with a probability p ≥ 0 whether to terminate the exchange between player A and player B, or to continue it.

Both players know this probability. If the interaction is terminated, the investor losses his investment and the reciprocator gets nothing. Nature's action is hidden: nor the investor, neither the reciprocator know what Nature did: they only see the payoff at the end of the game.

• If the interaction is continued, the administrator multiplies the one euro amount by five and gives them to B.

• Player B has the choice between the defection strategy where he takes the whole amount, or the refund strategy, where he can keep 3 euros for himself and return two euros to player A. The game ends.

Figure 1 presents the decision tree. The upper dotted curve indicates that an investor (Player A) who at the end of the game sees that his payoff is 1 euros, cannot infer from this information whether the reciprocator has played the defection strategy or Nature has terminated the relationship. This "blurred information" might attenuate reciprocators' feelings of shame or guilt for not rewarding a person who put their trust in them.

The lower dotted curve indicates that the reciprocator (Player B) who gets nothing will not be able to infer whether this implies that the investor did not play the trust strategy or whether Nature has terminated the exchange.

Optimal and actual strategies; behavioral implications

• The reciprocator (player B)

For p = 0, the former game is identical to the elementary investment game [START_REF] Berg | Trust, reciprocity and social history[END_REF].

Under the elementary definition of rationality, the dominant strategy for the reciprocator is to take the money and leave. The traditional result in experimental economics according to which a majority of subjects choose the return strategy has been interpreted as trustworthiness.

Since the reciprocator plays after Nature, his attitude towards risk should not bear on his preferred strategy. Thus any observed change in the reciprocator's behavior when a positive probability of terminating the exchange is introduced can be directly related to the way how the investor interprets defection. When moving from p = 0 to p > 0, the investor who sees no money forthcoming can no longer identify the reason of this outcome: either the reciprocator has betrayed him, or bad luck occurred. The reciprocator might take advantage of this "wiggle room" in order to implement the defection strategy. 6We can thus introduce a more specific hypothesis to be tested:

H.1. The frequency of reciprocators playing the selfish strategy must be higher for p > 0 than for p = 0, even if p is very small.

• The investor (player A)

Taking the same elementary definition of rationality, since the optimal strategy of the reciprocator is defection, the optimal rational strategy of the first player is not to invest. Yet in experimental settings a majority of investors send the money, which has been interpreted as trust in the other. Following [START_REF] Gambetta | Can we trust?[END_REF] and [START_REF] Guiso | Trusting the stock market[END_REF], we can define trust as the positive subjective probability µ assigned by an investor to the event that player will return the investment and a reward (in our game, the reward of investing 1 euro is 2 euros). Furthermore, since people differ in their ability to trust the others, investors' own µ should follow a statistical distribution of p.d.f. f () and c.d.f. F (); the support of these distributions is [0, 1]. It is reasonable to assume that f () is single-peaked. The shape of the distribution itself depends on the settings of the problem, and mainly on p.

Since the investor must decide whether to transfer the money before the realization of the objective risk, a large objective probability of Nature terminating the exchange (p) might prompt him to play the no-trust strategy merely because for a given µ, the expected gain declines. Thus, before moving to our empirical analysis, we must analyze the relationship between the objective probability and the trust decision.

For so doing, we assume that the investor will chose the strategy that brings him the largest (expected) monetary payoff. Adding a constant monetary payoff to acknowledge that the act of giving directly increases the satisfaction of the sender, for altruistic, charity or fairness motives, would not modify the basic reasoning. Changes in the frequency of investors that follow the notrust strategy depend in a significant way on whether changes in p prompt investors to revise their beliefs about the reciprocator's behavior.

To keep this theoretical analysis simple, we follow [START_REF] Holt | Risk incentives and effects in lottery choices[END_REF] and assume that for small payoffs players are risk neutral. The expected gain of the investor who plays his "trust" strategy (i.e., invest the 1 euro) is:

E[g A |trust] = (1 -p)[(1 -µ) × 1 + µ × 3] + p × 1 = 1 + 2µ(1 -p). (1)
Player A prefers to transfer the euro to player B if the expected gain from the trust strategy is larger than the payoff from the no-trust strategy.

E[g A |trust] > [g A |self ish] ⇔ 1 + 2µ(1 -p) > 2 ⇔ 2µ(1 -p) > 1.
(2)

Condition (2) allows to determine, for a given p, which is the critical probability μ(p) below which no investor will engage in the trust strategy:

μ(p) = 1 2(1 -p) , (3)
with dμ dp = 1 2(1-p) 2 > 0. For p = 0, we have μ(0) = 0.5 In this context, investors that play the trust strategy in the elementary game (p = 0) are characterized by µ > 0.5. Notice that for p = 0.5,

we have µ = 1 : in this game, if the external risk p is bigger than 0.5, nobody will play the trust strategy.

For a given p, the frequency of investors that follow the no-trust strategy is Pr[µ < μ(p)|p] = F (μ(p); p). If investors can internalize the possible change in reciprocators' behavior if p moves from 0 to a positive value, then more investors will become mistrustful. Formally, the distribution

F (µ; p > 0) dominates the distribution F (µ; p = 0), that is F (µ; p > 0) > F (µ; p = 0) ∀µ ∈ [0, 1].
Let us denote the partial derivatives

F µ (µ, p) = ∂F (µ,p) ∂µ = f (µ, p) > 0 and F p () = ∂F () ∂p ≥ 0.
Thus the total variation in the frequency F is:

dF (µ, p) = • f (µ, p) 2(1 -p) 2 + F p (µ, p) ¸dp. ( 4
)
As already mentioned, the main issue under scrutiny is a change in p from zero to a small positive value. For an initial p = 0, we can write:

[dF (µ, p)] p=0 = [0.5f (0.5; 0) + F p (0.5, 0)] dp.

(5)

If investors no not revise their beliefs about the reciprocator's behavior when p changes, the partial derivative F p (0.5, 0) = 0, then Eq. ( 5) becomes [dF (µ, p)] p=0 = [0.5f (0.5; 0)] dp. With f (0.5; 0) < 1, it turns out that [dF (µ, p)] p=0 < 0.5dp. If changes in p do not prompt investors to revise their beliefs about the reciprocator's behavior, then small variations dp > 0 in the neiborghood of p = 0 should entail even smaller changes in the frequency of investors that follow the selfish strategy. For instance, if p is raised from 0 to 1%, it turns out that [dF (μ)] p=0 < 0.005;

for dp = 0.1, we have [dF (μ)] p=0 < 0.05.

We can introduce another testable assumption:

H.2. If moving from p = 0 to a small positive p does not induce a substantial change in investors' behavior, this implies that the latter do not anticipate subsequent changes in the reciprocator's behavior.

If changes in p prompt investors to revise their beliefs about the reciprocator's behavior, small variations dp > 0 in the neiborghood of p = 0 can entail large changes in the frequency of investors that follow the selfish strategy. Indeed, if F p (0.5, 0) = α, with α > 0, then [dF (µ, p)] p=0 > αdp.

The testable assumption is thus:

H.3. If moving from p = 0 to small positive p induces a substantial change in investors'

behavior, this implies that the latter anticipates subsequent changes in the reciprocator's behavior.

In this elementary analysis we have assumed that the investor chooses the strategy that brings him the largest (expected) monetary payoff, given his beliefs about the reciprocator's response.

Adding an exogenous gain to account for utility derived from altruism would not change this

outcome. Yet those playing the investor's role could also derive some utility from being perceived as trustful by the others. Should this factor play a role too, one would expect more investors to play the no-trust strategy when moving from p = 0 to p > 0 because some investors could take advantage of the now blurred relationship (the reciprocator who sees no money forthcoming no longer knows whether the investor did not send the money or bad luck occurred).

Experimental design

The experiment included a total of 90 students from two French Business Schools (ESC Dijon and ESSEC) and were conducted in the Experimental Laboratories of the two institutions in April and May 2010. All the subjects were recruited from the student population of the schools, who answered to a call for paid decision experiments. The experimental design was presented via computer interface and all interactions were computerized. The programme was written in z-Tree [START_REF] Fischbacher | z-Tree: Zurich toolbox for ready-made economic experiments[END_REF]. Instructions were common knowledge. Experiments last for less than 30 minutes.

Half of the students played the role of the investor, called player A in the experiment, half or the students played the role of the reciprocator, called player B. Participants were randomly and anonymously paired as strangers. Each student was asked to play the game three times, for p = 0, p = 0.01 and p = 0.10; the order of the three treatments was random, and pairs were recreated at each round. Nature's random move (continue/terminate relationship) was computerized, and the draw respected the displayed probability. Payoffs are exactly those indicated in the Decision Tree (Figure 1) and were paid in cash at the end of the experiment.

Table 1 indicates the location, time, number of subjects and nubmer of pairs for each session.

In sessions 1 to 3 students had a fixed role (either Player A or Player B); in session 4 we allowed them to play both roles.

Results and discussion

Tables 2 and3 summarize the main results. The first table presents investors' decisions, the second table presents reciprocators' choices, for each of the three treatements: no external risk ( p = 0), small external risk (p = 0.01) and a larger external risk (p = 0.10). In addition, Figures 2 and3 show the frequencies with which the two players have followed their trust strategies (transfer money for the investor, and pay back for the reciprocator). It should be emphasized that these frequencies apply to groups of different sizes. • Results of the standard experiment: p = 0 R0. These results are quite standard. Of the 56 pairs, players A sent money to players B for 34 times (i.e., 61% of the players adopted the trust strategy). Reciprocators paid back 18 times (53%) and defected 16 times (47%).

• Results of the test with a small probability of termination: p = 0.01.

R1. When such a small objective default probability was introduced, 31 reciprocators got the 5 euros (an insignificant drop with respect to the former situation); only 10 of them chose to pay R2. However, despite the change in the reciprocator behavior, the investor did not alter in a substantial way their behavior, as if they were not been able to anticipate the change in reciprocators' behavior. Indeed, the number of transfers from player A to player B weekly declined, from 34 in the p = 0 treatment to 31 transfers in the p = 0.01 treatment. There is no statistical difference in investors' behavior: the McNemar Chi-squared test p-value=0.66.

• Results of the test with a large probability of termination: p = 0.10.

It is important to notice that for p = 0.10, not only did investors play less often the "no trust strategy" (only 25 investors out of 56 continue to transfer), but also in our experiment Nature terminated the relationship in 6 cases. Thus the number of actual transfers to the reciprocator is much smaller in the p = 0.10 treatment than in both the p = 0 and the p = 0.01 treatments.

R3. In this treatment too, 58% of the reciprocators who get the funds played the defection strategy (i.e. 11/19), a higher frequency than in the p = 0 treatment where the defection strategy was played by 47% of the reciprocators (incremental rise in the defection rate: 11 percentage points).

R4. A substantially lower number of investors played the trust strategy (send money), from 61% (34/56) in the p = 0 treatment to 45% (25/56) in the p = 0.10% treatment, i.e. a reduction by 16 percentage points. The McNemar Chi-squared test for a substantial difference is significant at 6%.

• The Synthesis a) R1 and R3 corroborate H1: When an objective risk is introduced in a standard investment game, reciprocators hide behind probabilities and more of them implement the deception strategy.

as already mentioned, this surge in opportunistic behavior in presence of external randomness has already been revealed by Dana et al. (2000) in a different experimental set-up (dictator game).

b) The rise in the reciprocators' defection rates is higher in the very small probability treatment (p = 0.01) than in the larger probability treatment (p = 0.10). While a straightforward explanation is not easy to fit, this idiosyncratic behavior might be explained if, in the higher risk situation, reciprocators do express some form of gratitude toward investors that have taken more risk to send them money (the reciprocation effect is stronger). c) R2 corroborates H2. For a small external probability, despite a strong adjustment in reciprocators' behavior, investors do not alter in a significant way their strategies, as if they do not anticipate reciprocators' change in behavior. In Casari and Casson (2009) experiment with the trust game, investors also appeared limited in their ability to internalize changes in reciprocators' behavior.

d) According to R3, investors would correct their "error" for a larger probability of terminating the relationship. Risk aversion would induce a similar response; yet, for small gains it is highly probable that players are risk neutral. That investors who care about being perceived as trustful do also "hide behind probabilities" and chose the no-trust strategy, might provide another explanation.

Conclusion

This paper provides new evidence on the effect of external randomness on the trust relationship between two parties to a trade. In particular, we have shown that if a very small and exogenously given probability of terminating the exchange (1% to be more specific) is introduced in a elementary investment game, the frequency of reciprocators who choose the defection strategy raises from 47% to 68%. Reciprocators seem to hide behind the external risk in order to carry out their most selfish strategy. In the experiment, investors do not seem able to fully internalize this change in behavior, at least not for this very small external risk. The fact that subjects were young students, with a natural tendency to trust their colleagues, might have induced a bias. It would be interesting to run the experiment with more experienced players.

For sure, situations captured by such a simple experimental framework cannot claim to provide a comprehensive explanation for the extremely individualistic behavior observed in financial markets. In particular, financial transactions are both impersonal and not necessarily repeated.

Then, as transaction costs theory points out, incentives for the players to build reputation are weak. Yet this argument does not explain why, during the last crisis, many banks that had sold an image of champions of long-term firm-customer relationships, have betrayed their clients and took on their books excessively high amounts of risk (Diamond and Rajan, 2009; Besancenot and Vranceanu, 2011). 8 Our alternative explanation for this selfish attitude emphasizes the relationship between risk and trust. Indeed, an objective probability of default is a built-in feature of all financial transactions. Thus, when the value of an asset collapses, no one can tell whether this happened because the asset was poorly managed or because of mere bad luck. According to results of our experiment, the trust relationship is deeply undermined because the seller of the asset can "hide behind probabilities". He will resort more often to the selfish non-cooperative strategy as compared to sellers' behavior in markets where the objective risk of non-delivery is not contractual. With some caution, such systematic recourse to the defection strategy might be interpreted as "greed".

After the 2007-2009 crisis, many observers blamed the opacity of banks and complex financial assets for having provided a favorable playing ground for the most opportunistic and ruthless traders. Transaction cost literature has pointed out long time ago that increasing transaction anonymity increases the probability of selfish and deceitful behavior. Hence, increasing transparency of financial institutions has become probably the most consensual policy recommendation [START_REF] Acharya | The Financial Crisis of 2007-2009[END_REF][START_REF] Stiglitz | Freefall. Free Markets and the Sinking of the Global Economy[END_REF]. Our paper does not run against this recommendation, yet it calls for cautious optimism: since even a small external risk of value destruction suffices to harm the trust relationship, increasing transparency might have only second order effects. Only by fully removing the external risk that one could create a trust friendly environment, but this task is difficult, since risk is an intrinsic component of all financial trades.

  Figure 1: Decision Tree

Figure 2 :

 2 Figure2: Investors playing the trust strategy; reciprocators playing the refund strategy, for p = 0 and p = 0.01, in percent of total possible moves.

Figure 3 :

 3 Figure3: Investors playing the trust strategy; reciprocators playing the refund strategy, for p = 0 and p = 0.10, in percent of total possible moves.

Table 3 .

 3 Strategies played by reciprocators (players B)

	Treatment Number of pairs Number of transfers to B Percentage of transfers
	p = 0	56	34	61%
	p = 0.01	56	31	55%
	p = 0.10	56	25	45%
			Table 2. Strategies played by investors (players A)

Depending on the motive to invest, some trustors might also take advantage of this wiggle room and chose the no-trust strategy.

A fixed amount variant allows to focus on the external risk as the single control.

The reciprocator plays after the Nature's move, so his payoff is certain.

See also[START_REF] Johnson | Trust games: A meta-analysis[END_REF] for a meta-ananlysis of 143 replications of the trust game.

This possibility has been documented by[START_REF] Dana | Exploiting moral wiggle room: experiments demonstrating an illusory preference for fairness[END_REF] in the framework of a dictator game.[START_REF] Charness | Promises and partnership[END_REF] argued that the presence of an objective risk (no matter how small) provides the agent in a pricipal-agent contract game with a good pretext to pursue a hidden action, probably because his sentiments of shame or guilt are watered down to some extent.

Two pre-test sessions were performed on March, 5 2010 in Dijon with 30 students. Three variants of the game were tested: this unframed fixed payment variant, an unframed, variable payment variant (amounts transferred were allowed to vary) and a framed variant, where player B was represented as a company needing an investment.

The Goldman Sachs scandal produced by the bank's participation in packaging and selling a pool of dubious mortgage loans on request by a hedge fund manager who bet against this pool of loans is illustrative for these ethical dilemmas. For years Goldman has claimed to be concerned with the long-term interest of its clients and partners (see "Greedy until proven guilty", The Economist, April 24th, 2010).

Acknowledgement 1 The authors would like to thank an anonymous referee, Tim Cason, Aurélie Dariel, Bernard Ruffieux and participants to the First International Conference of the French Association of Experimental Economics, Grenoble, 23-24 September 2010 for their remarks that helped them to improve this paper.