Philippe Biane

Matrix valued Brownian motion and a paper by Polya

Introduction

This paper has two parts which are largely independent. In the first one I recall some known facts on matrix valued Brownian motion, which are not so easily found in this form in the literature. I will study three types of matrices, namely Hermitian matrices, complex invertible matrices, and unitary matrices, and try to give a precise description of the motion of eigenvalues (or singular values) in each case. In the second part, I give a new look at an old paper of G. Pólya [START_REF] Pólya | Bemerkung über die Integraldarstellung der Riemannschen ζ-Funktion[END_REF], where he introduces a function close to Riemann's ξ function, and shows that it satisfies Riemann's hypothesis. As put by Marc Kac in his comments on Pólya's paper [11], "Although this beautiful paper takes you within a hair's breadth of Riemann's hypothesis it does not seem to have inspired much further work and reference to it in the mathematical litterature are rather scant". My aim is to point out that the function considered by Pólya is related in a more subtle way to Riemann's ξ function than it looks at first sight. Furthermore the nature of this relation is probabilistic, since these functions have a natural interpretation involving Mellin transforms of first passage times for diffusions. By studying infinite divisibility properties of the distributions of these first passage times, we will see that they are generalized gamma convolutions, whose mixing measures are related to the considerations in the first part of this note.

Matrix Brownian motions

We will study three types of matrix spaces, and in each of these spaces consider a natural Brownian motion, and show that the motion of eigenvalues (or singular values) of this Brownian motion has a simple geometric description, using Doob's transform. The following results admit analogues in more general complex symmetric spaces, but for the sake of simplicity, discussion will be restricted to type A symmetric spaces. Actually the interesting case for us in the second part will be the simplest one, of rank one, but I think that this almost trivial case is better understood by putting it in the more general context. Some references for results in this section are [START_REF] Babillot | A probabilistic approach to heat diffusion on symmetric spaces[END_REF], [START_REF] Biane | La fonction zêta de Riemann et les probabilités[END_REF], [START_REF] Biane | Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions[END_REF], [START_REF] Dyson | A Brownian-motion model for the eigenvalues of a random matrix[END_REF], [START_REF] Guivarc'h | Compactifications of symmetric spaces[END_REF], [START_REF] Helgason | Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions[END_REF], [START_REF] Jones | Weyl chambers, symmetric spaces and number variance saturation[END_REF], [START_REF] Malliavin | Factorisations et lois limites de la diffusion horizontale audessus d'un espace riemannien symétrique[END_REF], [START_REF] Taylor | The Iwasawa decomposition and the limiting behaviour of Brownian motion on a symmetric space of noncompact type[END_REF], [START_REF] Taylor | Brownian motion on a symmetric space of noncompact type: asymptotic behaviour in polar coordinates[END_REF]. Let M (t) be a Brownian motion with values in this space, which is simply a Gaussian process with covariance E[T r(AM (t))T r(BM (s))] = T r(AB)s ∧ t for A, B traceless Hermitian matrices.

Let λ 1 (t) ≥ λ 2 (t) ≥ . . . ≥ λ n (t) be the eigenvalues of M (t); they perform a stochastic process with values in the Weyl chamber

C = {(x 1 , . . . , x n ) ∈ R n | x 1 ≥ x 2 ≥ . . . ≥ x n } ∩ H n where H n = (x 1 , . . . , x n ) ∈ R n n i=1 x i = 0 .
Let p 0 t be the transition probability semi-group of Brownian motion killed at the boundary of the cone C. This cone is a fundamental domain for the action of the symmetric group S n , which acts by permutation of coordinates on R n . Using the reflexion principle, one shows easily that

p 0 t (x, y) = σ∈Sn ǫ(σ)p t (x, σ(y)) x, y ∈ C where p t (x, y) = (2πt) -(n-1)/2 e -|x-y| 2 /2t and ǫ(σ) is the signature of σ. Let h be the function h(x) = i>j (x i -x j ).
Proposition 2.1. The function h is the unique (up to a positive multiplicative constant) positive harmonic function for the semigroup p 0 t , on the cone C, which vanishes on the boundary.

The harmonic function h corresponds to the unique point at infinity in the Martin compactification of C. Consider now the Doob's transform of p 0 t , which is the semigroup given by

q t (x, y) = h(y) h(x) p 0 t (x, y).
It is a diffusion semigroup on C with infinitesimal generator 1 2 ∆ + ∇ log h, ∇ • .

Proposition 2.2. The eigenvalue process of a traceless Hermitian Brownian motion is a Markov diffusion process in the cone C, with semigroup q t . We can summarize the last proposition by saying that the eigenvalue process is a Brownian motion in C, conditioned (in Doob's sense) to exit the cone at infinity.

The group SL n (C)

. This is the group of complex invertible matrices of size n × n, with determinant 1. Its Lie algebra is the space sl n (C) of complex traceless matrices. Consider the Hermitian form A, B = T r(AB * ) on sl n (C) which is invariant by left and right action of the unitary subgroup SU (n). This Hermitian form determines a unique Brownian motion with values in sl n (C).

The Brownian motion g t , on SL n (C), is the stochastic exponential of this Brownian motion, solution to the Stratonovich stochastic differential equation

dg t = g t dw t
where w t is a Brownian motion in sl n (C).

There are two remarkable decompositions of SL n (C), the Iwasawa and Cartan decompositions. The first one is SL n (C) = N AK where K is the compact group SU (n), A is the group of diagonal matrices with positive coefficients, and determinant one, and N is the nilpotent group of upper triangular matrices with 1's on the diagonal. Each matrix of SL n (C) has a unique decomposition as a product g = nak of elements of the three subgroups N, A, K. This can be easily inferred from the Gram-Schmidt orthogonalization process. If g t is a Brownian motion in SL n (C), one can consider its components n t , a t , k t . In particular, denoting by (e w1(t) , . . . , e wn(t) ) the diagonal components of a t the following holds (cf [START_REF] Taylor | The Iwasawa decomposition and the limiting behaviour of Brownian motion on a symmetric space of noncompact type[END_REF]).

Proposition 2.3. The process w 1 (t), . . . , w n (t) is a Brownian motion with a drift ρ = (-n + 1, -n + 3, . . . , n -1) in the subspace H n .

The other decomposition is the Cartan decomposition SL n (C) = KA + K, where A + is the part of A consisting of matrices with positive nonincreasing coefficients along the diagonal. In order to get the Cartan decomposition of a matrix g ∈ SL n (C), take its polar decomposition g = ru with r positive Hermitian, and u unitary, then diagonalize r which yields g = vav ′ with v and v ′ unitary and a diagonal, with positive real coefficients which can be put in nonincreasing order along the diagonal. These coefficients are the singular values of the matrix g. This decomposition is not unique since the diagonal subgroup of SU (n) commutes with A, but the singular values are uniquely defined. Call (e a1(t) , . . . , e an(t) ) the singular values of the Brownian motion g t , with a 1 ≥ a 2 ≥ . . . ≥ a n . They form a process with values in the cone C. Let us mention that this stochastic process can also be interpreted as the radial part of a Brownian motion with values in the symmetric space SL n (C)/SU (n). We will now give for the motion of singular values a similar description as the one of eigenvalues of the Hermitian Brownian motion. For this, consider a Brownian motion in H n , with drift ρ, killed at the exit of the cone C. This process has a semigroup given by p 0,ρ t (x, y) = e ρ,y-x -t ρ,ρ /2 p 0 t (x, y) . Proposition 2.4. The function

h ρ (y) = i>j (1 -e 2(yj-yi) )
is a positive harmonic function for the semigroup p 0,ρ t , in the cone C, and vanishes at the boundary of the cone.

It is not true that this function is the unique positive harmonic function on the cone; indeed the Martin boundary at infinity is now much larger and contains a point for each direction inside the cone, see [START_REF] Guivarc'h | Compactifications of symmetric spaces[END_REF]. The Doob-transformed semigroup

q ρ t (x, y) = h ρ (y) h ρ (x) p 0,ρ t (x, y) is a Markov diffusion semigroup in the cone C, with infinitesimal generator 1 2 ∆ + ρ, • + ∇ log h ρ , ∇• .
Note that it can also be expressed as

1 2 ∆ + ∇ log hρ , ∇• with hρ (y) = i>j sinh(y j -y i )
(see [START_REF] Jones | Weyl chambers, symmetric spaces and number variance saturation[END_REF]).

Proposition 2.5. The logarithms of the singular values of a Brownian motion in SL n (C) perform a diffusion process in the cone C, with semigroup q ρ t . As in the preceding case, we can summarize by saying that the process of singular values is a Brownian motion with drift ρ in the cone C, conditioned (in Doob's sense) to exit the cone at infinity, in the direction ρ. Let e iθ1 , . . . , e iθn be the eigenvalues of a matrix in SU (n), which can be chosen so that i θ i = 0, and

θ 1 ≥ θ 2 ≥ . . . ≥ θ n , θ 1 -θ n ≤ 2π. These conditions determine a simplex ∆ n in H n ,
which is a fundamental domain for the action of the affine Weyl group on H n . Recall that the affine Weyl group W is the semidirect product of the symmetric group S n , which acts by permutation of coordinates in H n , and of the group of translations by elements of the lattice (2πZ) n ∩ H n .

One can use the reflexion principle again to compute the semigroup of Brownian moton in this simplex killed at the boundary. One gets an alternating sum over the elements of W , p 0 t (θ, ξ) = w∈ W ǫ(w)p t (θ, w(ξ)). The infinitesimal generator is 1/2 × the Laplacian in the simplex, with Dirichlet boundary conditions. It is well known that this operator has a compact resolvent, and its eigenvalue with smallest module is simple, with an eigenfunction which can be chosen positive. Consider the function

h u (θ) = j>k (e iθj -e iθ k ).
Proposition 2.6. The function h u is positive inside the simplex ∆ n , it vanishes on the boundary, and it is the eigenfunction corresponding to the Dirichlet eigenvalue with smallest module on ∆ n . This eigenvalue is λ = (n -n 3 )/6. The Doob-transformed semigroup

q u t (x, y) = h u (y) h u (x) e -λt p 0 t (x, y) is a Markov diffusion semigroup in ∆ n , with infinitesimal generator 1 2 ∆ + ∇ log h u , ∇• -λ.
Proposition 2.7. The process of eigenvalues of a unitary Brownian motion is a diffusion with values in ∆ n with probability transition semigroup q u t . Again a good summary of this situation is that the motion of eigenvalues is that of a Brownian motion in the simplex ∆ n conditioned to stay forever in this simplex.

2.4. The case of rank 1. In the next section we will need the simplest case, that of 2 × 2 matrices. Consider first the case of Hermitian matrices. The process of eigenvalues is essentially a Bessel process of dimension 3, with infinitesimal generator 1 2 For these last two examples, we shall write a spectral decomposition of the generator L i , i = 1, 2, of the form

d 2 dx 2 + 1 x d dx on ]0,
(2.1) f (x) = Φ i λ (x) Φ i λ (y)f (y) dm i (y) dν i (λ) i = 1, 2 for every f ∈ L 2 (m i )
, where m i is measure for which L i is selfadjoint in L 2 (m i ), and the functions Φ i λ are solutions to L i Φ i λ + λΦ i λ = 0 and ν i is a spectral measure for L i .

For

L 1 = 1 2 d 2 dx 2 on [0, π] with Dirichlet boundary conditions, m 1 (dx) is Lebesgue measure on [0, π], and L 1 is selfadjoint on L 2 ([0, π)]. Furthermore Φ 1 λ (x) = sin( √ 2λx) and (2.2) ν 1 (dλ) = 1 π ∞ n=1 δ n 2 /2 (dλ). For L 2 = 1 2 d 2 dx 2 + d dx on [0, +∞[, the measure m 2 (dx) = e 2x dx, and Φ 2 λ (x) = e -x sin( √ 2λ -1x) λ > 1/2.
The spectral measure is Note that the spectral decompositions, and in particular the measures ν i , depend on the normalisation of the fonctions Φ λ . We have made a natural choice, but it does not coincide with the usual normalisation of Weyl-Titchmarsh-Kodaira theory, see [START_REF] Coddington | Theory of ordinary differential equations[END_REF].

(2.3) ν 2 (dλ) = 1 π √ 2λ -1 dλ λ > 1/2 on [
3. MacDonald's function and Riemann's ξ function 3.1. Pólya's paper. In his paper [START_REF] Pólya | Bemerkung über die Integraldarstellung der Riemannschen ζ-Funktion[END_REF], Pólya starts from Riemann's ξ function

ξ(s) = s(s -1)π -s/2 Γ(s/2)ζ(s)
where ζ is Riemann's zeta function. Then ξ is an entire function whose zeros are exactly the nontrivial zeros of ζ.

Putting s = 1/2 + iz yields

ξ(z) = 2 ∞ 0 Φ(u) cos(zu)du with (3.1) Φ(u) = 2πe 5u/2 ∞ n=1 (2πe 2u n 2 -3)n 2 e -πn 2 e 2u
and the function Φ is even, as follows from the functional equation for Jacobi θ function; furthermore Φ(u) ∼ 4π 2 e 9u/2-πe 2u u → +∞ so that Φ(u) ∼ 4π 2 (e 9u/2 + e -9u/2 )e -π(e 2u +e -2u ) u → ±∞.

This lead Pólya to define a "falsified" ξ function

ξ * (z) = 8π 2 ∞ 0
(e 9u/2 + e -9u/2 )e -π(e 2u +e -2u ) cos(zu)du.

The main result of [START_REF] Pólya | Bemerkung über die Integraldarstellung der Riemannschen ζ-Funktion[END_REF] is Théorème 3.1. The function ξ * is entire, its zeros are real and simple. Let N (r), (resp. N * (r)) denote the number of zeros of ξ(z) (resp. ξ * (z)) with real part in the interval [0, r], then N (r) -N * (r) = O(log r).

Recall that the same assertion about the zeros of the function ξ (without the statement about simplicity, beware also that s = 1/2 + iz) is Riemann's hypothesis. Recall also the well known estimate

N (r) = r 2π log(r/2π) - r 2π + O(1).
Pólya's results rely on the intermediate study of the function

G(z, a) = ∞ -∞
e -a(e u +e -u )+zu du from which ξ * is obtained by

ξ * (z) = 2π 2 (G(iz/2 -9/4, π) + G(iz/2 + 9/4, π))
Pólya shows that G(z, a) has only purely imaginary zeros, (as a function of z) and the number of these zeros with imaginary part in [0, r] grows as r π log r a -r π + O(1). The results on ξ * are then deduced through a nice lemma which played a role in the history of statistical mechanics (the Lee-Yang theorem on Ising model), as revealed by M. Kac [11]. We shall now concentrate on G(z, a). In particular, for a = π, the function ξ(z) = G(iz/2, π) is another approximation of ξ which has many interesting structural properties.

MacDonald functions.

The function denoted G(z, a) by Pólya is actually a Bessel function. Indeed, MacDonald's function, also called modified Bessel function (see e.g. [START_REF] Andrews | Special functions. Encyclopedia of Mathematics and its Applications[END_REF]), given by

K µ (x) = ∞ 0 t µ-1 e -x 2 (t+t -1 ) dt x > 0, µ ∈ C. satisfies K z (2x) = G(z, x).
The function G(z, a) is therefore essentially a MacDonald function, as noted by Pólya. MacDonald function is an even function of µ and satisfies

(3.2) 2µ x K µ (x) = K µ+1 (x) -K µ-1 (x) (3.3) -2 d dx K µ (x) = K µ+1 (x) + K µ-1 (x)
The first of these equations is used by Pólya in a very clever way to prove that the zeros (in z) of G(z, x) are purely imaginary.

3.3. Spectral interpretation of the zeros. From (3.2), (

( µ x -d dx )K µ = K µ+1 (-µ x -d dx )K µ = K µ-1 from which one gets K µ = (-µ+1 x -d dx )( µ x -d dx )K µ = ( d 2 dx 2 + 1 x d dx -µ 2 x 2 )K µ . 3.3)
This differential equation will give us a spectral interpretation of the zeros of G(z, x). Change variable by ψ µ (x) = K µ (e x ) to get

(3.4) (- d 2 dx 2 + e 2x )ψ µ = -µ 2 ψ µ
Since K µ vanishes exponentially at infinity, the spectral theory of Sturm-Liouville operators on the half-line (see e.g. [START_REF] Coddington | Theory of ordinary differential equations[END_REF], [START_REF] Levitan | Introduction to spectral theory: selfadjoint ordinary differential operators[END_REF]) implies that the squares of the zeros of µ → ψ µ (y) are the eigenvalues of d 2 dx 2 -e 2x on the interval [y, +∞[ with the Dirichlet boundary condition at y, the functions ψ µ being the eigenfunctions. Since this operator is selfadjoint and negative the zeros are purely imaginary, and are simple.

This spectral interpretation of the zeros of MacDonald function is well known [START_REF] Titchmarsh | Eigenfunction Expansions Associated with Second-Order Differential Equations[END_REF], I do not know why Pólya does not mention it.

3.4. H = xp. Equation (3.4) can be put into Dirac's form, indeed the equations

d dx + 1 2 + e x f = γg -d dx + 1 2 + e x g = γf imply - d 2 dx 2 + e 2x f = (γ 2 - 1 4 )f.
Using the change of variables u = e x , we get

u d du + 1 2 + u f = γg -u d du + 1 2 + u g = γf.
Remark that this Dirac system yields a perturbation of the Hamiltonian H = xp considered by Berry et Keating [START_REF] Berry | The Riemann zeros and eigenvalue asymptotics[END_REF], in relation with Riemann's zeta function.

3.5. Asymptotics of the zeros. General results on Sturm-Liouville operators allow one to recover the asymptotic behaviour of the spectrum, thanks to a semiclassical analysis, see e.g. [START_REF] Levitan | Introduction to spectral theory: selfadjoint ordinary differential operators[END_REF]. One can get a more precise result using the integral representation of K iµ . Pólya gives the asymptotic estimate This estimate can be obtained by the stationary phase method, writing

K x+iy (2a) = 1 √ 2πy e -π
K z (2a) = ∞ -∞
e zt-2a cosh(t) dt.

Making a contour deformation we get

K z (2a) = -A -∞ e zt-2a cosh(t) dt + i π/2 0 e z(-A+it)-2a cosh(-A+it) dt + A -A e z(t+i π 2 )-2ai sinh(t) dt -i π/2 0 e z(A-it+i π 2 )-2a cosh(A-it+i π 2 ) dt + ∞ A e zt-2a
cosh(t) dt and Pólya's estimate can be obtained by standard methods, which give also estimates for the derivatives of MacDonald's function. Finally the zeros of y → K iy (2a) behave like the solutions to

y log y a -y - π 4 = (n + 1 2 )π n integer
The number of zeros with imaginary part in [0, T ] is thus T π log T a -T π + O(1).

Probabilistic interpretations

We will now give interpretations of the functions ξ and ξ using first passage times of diffusions.

4.1. Brownian motion with a drift. The first passage time at x > 0 of Brownian motion started at 0 follows a 1/2 stable distribution i.e.,

P (T x ∈ dt) = x e -x 2 2t √ 2πt 3 dt with Laplace transform E[e -λ 2 Tx/2 ] = e -λx .
Adding a drift a > 0 to the Brownian motion gives a first passage distribution

P a (T x ∈ dt) = x e -x 2 2t √ 2πt 3 e ax-a 2 t 2 dt
with Laplace transform

E a [e -λ 2 Tx/2 ] = e -x √ λ 2 +a 2 +ax .
This is a generalized inverse Gaussian distribution. In particular, its Mellin transform is

E a [T s x ] = (x/a) s K -1/2+s (ax) K -1/2 (ax) = (x/a) s π/ax e ax K -1/2+s (ax)
which gives a probabilistic interpretation of MacDonald's function (as a function of s) as a Mellin transform of a probability distribution.

Three dimensional Bessel process.

There exists a similar interpretation of the ξ function, which is discussed in details in [START_REF] Biane | La fonction zêta de Riemann et les probabilités[END_REF], [START_REF] Biane | Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions[END_REF], for example, Consider the first passage time at a > 0 of a three dimensional Bessel process (i.e., the norm of a three dimensional Brownian motion) starting from 0. The Laplace transform of this hitting time is

E[e -λ 2
2 Sa ] = λa sinh λa .

Let S ′ a be an independent copy of S a , and let W a = S a + S ′ a ; then the density of the distribution of W a is obtained by inverting the Laplace transform. One gets

P (W a ∈ dx) = ∞ n=1 (π 4 n 4 x/a 4 -3π 2 n 2 /a 2 )e -π 2 n 2 x/2a
P (γ ω,c ∈ dt) = c -ω Γ(ω) t ω-1 e -t/c dt = Γ ω,c (dt)
where ω and c are > 0 parameters. The Laplace transform is

E[e -λγω,c ] = (1 + λ/c) -ω .
The gamma distributions form a convolution semigroup with respect to the parameter ω, i.e., Γ ω1,c * Γ ω2,c = Γ ω1+ω2,c . The Lévy exponent of the gamma semigroup is

ψ c (λ) = log(1 + λ/c) = ∞ 0 (1 -e -λt )
e -ct t dt so that this is the semigroup of a subordinator with Lévy measure e -ct /t dt.

The generalized gamma convolutions are the distributions of linear combinations, with positive coefficients, of independent gamma variables, and their weak limits.

One can also characterize the generalized gamma convolutions as the infinitely divisible distributions with a Lévy exponent of the form

ψ(λ) = ∞ 0 ψ c (λ)dν(c)
for some positive measure ν which integrates 1/c at ∞. This measure is called the Thorin measure of the generalized gamma distribution. The variables T x and W a of the preceding paragraph are generalized gamma convolutions. Indeed it is easy to check, using the computations of section 4.3, that W a has a generalized gamma convolution as distribution, with Thorin measure e -ct dc π c -a 2 /2 .

Final remarks.

We can now make a connection between the preceding considerations and those of the first part of the paper. Indeed, the Thorin measures associated with the variables T x and W a can be expressed as spectral measures associated with the generators of Brownian motion on matrix spaces. The hitting times of Brownian motion with drift are related with the radial part of Brownian motion in the symmetric space SL 2 (C)/SU (2), whereas the hitting times of the Bessel three process are related with the Brownian motion on the unitary group SU (2). The precise relations are contained in formulas (2.2), (2.3), (4.1), (4.2). Thus the Riemann ξ function, which is the Mellin transform of a hitting time of the Bessel three process, as in section 4.2, and the Polya ξ function from section 3.1, which appears as Mellin transform of hitting time of Brownian motion with drift, are related in this non obvious way.

2. 1 .

 1 Hermitian matrices. Consider the space of n × n Hermitian matrices, with zero trace, endowed with the quadratic form A, B = T r(AB).

2. 3 .

 3 Unitary matrices. The Brownian motion with values in SU (n) is obtained by taking the stochastic exponential of a Brownian motion in the Lie algebra of traceless anti-Hermitian matrices, endowed with the Hermitian form A, B = -T r(AB).

2 d 2 2 d 2

 2222 +∞[, obtained from Brownian motion killed at zero, of infinitesimal generator 1 dx 2 with Dirichlet boundary condition at 0, by a Doob transform with the positive harmonic function h(x) = x In the case of the group SL 2 (C), or the symmetric space SL 2 (C)/SU (2), which is the hyperbolic space of dimension 3, the radial process has infinitesimal generator condition at 0, by a Doob's transform with the function 1 -e -2x . Finally the last case is Brownian motion in SU (2), where the eigenvalue process takes values in [0, π] and has an infinitesimal generator 1 dx 2 + cot x d dx obtained by a Doob transform at the bottom of the spectrum from 1 2 d 2 dx 2 on [0, π] with Dirichlet boundary conditions at 0 and π, by the function sin(x).

e

  -iΦ + O(e -π 2 y y |x|-3/2 ) in the strip |x| ≤ 1 uniformly as y → ∞, where Φ = y log y a -y -π 4 .

δ n 2 4

 24 /a 2 (dc) ; whereas T x is distributed as a generalized gamma convolution with Thorin measure (

  Infinite divisibility. The distributions of T x and W a are infinitely di--π 2 n 2 t/a 2 dt therefore the variable W a has the distribution of a subordinator, with Lévy measure 2 -π 2 n 2 t/a 2 dt, taken at time 1. Observe however that the process (W a ) a≥0 is not a subordinator. 4.4. Generalized gamma convolution. The gamma distributions are

	4.3. visible. Indeed	log E a [exp(-λ 2 2 T x )] = -x √ = x ∞ λ 2 + a 2 + ax 0 (e -λ 2 2 t -1) e -a 2 2 √ 2πt 3 dt t
	which shows that T x is a subordinator with Lévy measure
		e -a 2 2 t √ 2πt 3	dt.
	Similarly
	log E[exp(-λ 2 2 W a )] = 2 log(λa/ sinh(λa)) = 2 ∞ 0 (e -λ 2 2 t -1) ∞ n=1 e ∞ e
		n=1
			2 dx
	from which one can compute the Mellin transform
		E[W s a ] = 2(2a 2 /π) s ξ(2s).
	The function 2ξ thus has a probabilistic interpretation, as Mellin transform of
	π 2 W 1 .


Oguz Özgür Karadeniz
email: oguzozgur.karadeniz@student.kuleuven.be

Stefan Schiffner
email: stefan.schiffner@uni.lu

Get to Know Your Geek: Towards a Sociological Understanding of Incentives Developing Privacy-Friendly Free and Open Source Software

In this paper we sketch a road map towards a sociological understanding of software developers' motivations to implement privacy features. Although there are a number of studies concerning incentives for developers, these accounts make little contribution to a comprehensive sociological understanding, as they are either based on a simplistic view of FOSS development in terms of altruism vs. utilitarianism, or are focused on individual psychological factors, leaving room for research that takes into account the complex social context of FOSS development. To address this gap, we propose a mixed methods approach, incorporating the strengths of qualitative and quantitative techniques for a comprehensive understanding of FOSS development as a social field. We then sketch how we envision developing a game theoretic approach based on the gathered data to analyze the situation in the field with respect to privacy features and propose relevant changes in policy and best practices.

Introduction

Motivation. The GDPR [START_REF]679 of the European Parliament and of the Council of 27 april 2016 on the protection of natural persons with regard to the processing of personal data and on the free movement of such data, and repealing directive 95/46/EC[END_REF], and the European Commission's Ethics Guidelines for Trustworthy AI [START_REF]High Level Expert Group on AI: Ethics Guidelines for Trustworthy AI[END_REF] indicate the importance of data privacy in contemporary society. With GDPR, developers are now legally obliged to implement Privacy by Design (PbD), or in the lingua of the legal text, data protection by design, in their development process. However, privacy breaches continue to occur, despite policy initiatives and guidelines that legally incentivize or nudge developers towards privacy-friendly features. In fact, as pointed out by ENISA's threat landscape [START_REF] Enisa | ENISA threat landscape report 2018 -15 top cyberthreats and trends[END_REF], threats continue to rise as the landscape becomes more and more complex.

Moreover, as pointed in the Ethics Guidelines for Trustworthy AI [START_REF]High Level Expert Group on AI: Ethics Guidelines for Trustworthy AI[END_REF], legal compliance, e.g. with the GDPR, does not guarantee that a system meets any ethical standards. In order to meet privacy standards, software developers need to move beyond checklist compliance towards a continuous critical approach concerning the product/service and its social impact. Hence software developers need to continually prioritize and reassess the implementation of secure features, rather than a superficial compliance to law.

In this paper, we seek to pave the way for a better sociological understanding of software developers' motivations when taking implementation decisions for privacy-friendly and secure software. In order to reduce direct monetary and other external motives, we focus on free and open source software development (FOSS). In the first part of the paper, we present a literature review indicating the lack and importance of a sociological understanding of this issue. Following this, we offer a methodological approach based on mixed methods, which is not only appropriate for gaining a better understanding of FOSS developers' motives, but also for bringing about positive social transformation.

Background. FOSS development represents an immense contribution to technology and, through this, to society. Open source software has often contributed to more secure and privacy-friendly software either as its main goal, such as the development of PGP3 for email encryption and Tor4 for anonymity protection, or by setting high standards, such as the Linux kernel 5 , and the Firefox web browser 6 . Some open source software has been developed and maintained for decades, and has thus become a practical alternative for commercial products. At other times the contribution has been more indirect, as in cases where the success of certain superior aspects of open source products led to improvements in the same direction in closed source competitors, such as end-to-end encryption for messenger apps.

Many open source licensing models allow commercial applications to make use of them. This is arguably a double-edged sword, since it adds a potential layer of vulnerability to every service that uses them. As code is open to the public in FOSS projects, it is easier to find vulnerabilities, but due to their dependence on voluntary contributions, maintenance is often more difficult, potentially exposing applications and services that use them. On the other hand, FOSS developers also have the potential to facilitate the development of privacy-friendly and secure software, such as programs to identify vulnerabilities and assess risks [START_REF] Kilani | Will the GDPR affect open source projects?[END_REF], or by setting high standards as mentioned above. Finally, open source software usually comes with 'no warranty' despite popular belief in their superiority in this respect, which can be erroneous at times [START_REF] Coleman | Coding Freedom: The Ethics and Aesthetics of Hacking[END_REF][START_REF] Bezroukov | A second look at the cathedral and the bazaar[END_REF]. This puts FOSS in a very critical role in terms of privacy and security, since, although it represents a very important contribution and potential improvement, its perceived security can be misleading given the reluctance among developers [START_REF] Bednar | Engineering privacy by design: Are engineers ready to live up to the challenge[END_REF][START_REF] Spiekermann | Inside the organization: Why privacy and security engineering is a challenge for engineers[END_REF] to implement PbD.

Our contribution. In this paper, we argue that in order to bring about social transformation with regards to the production of privacy-friendly and secure software and services, a more formal model of the motivations of FOSS developers is necessary. We detail the steps towards a model that allows for a better sociological understanding of the issue, which can be used to render advice for policy changes. Firstly, we point to the areas where legal incentives fell short of preventing breaches or the development of needlessly privacy invasive services. Secondly, we present the literature on developer motivations to highlight the need for a sociological understanding of the motivations of FOSS developers. We will then offer our own methodological approach for a sociological inquiry into this problem based on mixed methods, combining the strengths of different quantitative and qualitative techniques for a more comprehensive view of the cultural context being investigated [START_REF] Creswell | Research design: Qualitative, quantitative, and mixed methods approaches[END_REF]. Following this, we describe how game theory can be used to construct a model that draws from the findings of a sociological research into this field. Finally, we offer a thought experiment that illustrates our approach.

Related Work

FOSS Development and Developer Motivation. In his seminal work "The Cathedral and the Bazaar" [START_REF] Raymond | The cathedral and the bazaar[END_REF], Eric Raymond compares the FOSS development scene to a crowded bazaar where individuals with multiple underlying agendas interact. According to him, the gift culture that characterizes the motivations of FOSS developers is ultimately self-interested rather than altruistic. Despite competition and the self-interest of the participants, FOSS development culture results in a harmonious system and better and more secure software than commercial development [START_REF] Lin | The future of sociology of FLOSS[END_REF][START_REF] Raymond | The cathedral and the bazaar[END_REF]. Raymond's work can be considered as a step towards an understanding of FOSS as a social field with its own relatively autonomous dynamics, as it identifies motivations and goals in FOSS development other than altruism, such as recognition and gaining rank within the community. However, as Lin argues, Raymond's account is a long way from capturing the diversity of the field that it acknowledges or addressing how individual efforts can work harmoniously to develop sophisticated software within this complexity [START_REF] Lin | The future of sociology of FLOSS[END_REF].

Raymond's work had a tremendous influence on academic and popular discourses on FOSS. The work led to the popular belief that open source is in many ways superior to commercial, epitomized by the phrase "Given enough eyeballs, all bugs are shallow", commonly attributed to Linus Torvalds [START_REF] Bezroukov | A second look at the cathedral and the bazaar[END_REF]. Coining the term "vulgar Raymondism", Nikolai Bzrukov severely criticizes this idealization for ignoring many problematic aspects of FOSS development ranging from developer burnout, hyper-inflated egos among developers and disagreements within the community [START_REF] Bezroukov | A second look at the cathedral and the bazaar[END_REF][START_REF] Bezroukov | Open source software development as a special type of academic research: Critique of vulgar Raymondism[END_REF].

With the influence of Raymond's work, many researchers in the field tried to situate motivations and incentives in the field in terms of self-interest and altruism. For example, Lerner and Tirole [START_REF] Lerner | Some simple economics of open source[END_REF][START_REF] Lerner | The open source movement: Key research questions[END_REF] formulate motivations underlying contribution to open source in terms of career advancement and reputation. According to them, contribution to open source is connected with an expectation of delayed return in career and economic terms due to gain in reputation and an associated ego boost. Bonaccorsi and Cristina also mention a combination of altruistic or self-interested motives, but they also find that these motivations differ between firms and individuals. According to their survey, individual program-mers have a tendency to have social and altruistic motives, such as contribution to the movement and belief in non-proprietary code [START_REF] Bonaccorsi | Altruistic individuals, selfish firms? the structure of motivation in open source software[END_REF]. Others such as Hars and Ou, and Roberts et al. differentiate between intrinsic and extrinsic motivations, as they associate intrinsic motivations with societal contribution and extrinsic motivations with economic and reputational gains [START_REF] Roberts | Understanding the motivations, participation, and performance of open source software developers: A longitudinal study of the Apache projects[END_REF][START_REF] Ou | Working for free? motivations for participating in open-source projects[END_REF][START_REF] Benbya | Understanding developers' motives in open source projects: A multi-theoretical framework[END_REF].

While the research Raymond's work inspired sought to shed more light on motivations to participate in open source, research focusing on motivations in implementations of PbD features is more sparse. More recent work by Spiekermann [START_REF] Spiekermann | Inside the organization: Why privacy and security engineering is a challenge for engineers[END_REF] and Bednar et al. [START_REF] Bednar | Engineering privacy by design: Are engineers ready to live up to the challenge[END_REF] further indicate the need for research and intervention in the field by documenting the reluctance to implement PbD. Using both qualitative and quantitative data, their findings indicate organizational and individual components to the low motivation for ethical system development and implementing privacy features [START_REF] Bednar | Engineering privacy by design: Are engineers ready to live up to the challenge[END_REF][START_REF] Spiekermann | Inside the organization: Why privacy and security engineering is a challenge for engineers[END_REF].

FOSS and Non-Monetary Incentives. Yuwei Lin points out that the reductionist instrumentalist approach to developer motivations is inadequate, since FOSS development is not characterized by a universal activity: the actors in FOSS development assume different identities (e.g. leader, follower, developer, user, hacker) and competitive and cooperative strategies that change between different social circles [START_REF] Lin | The future of sociology of FLOSS[END_REF]. Moreover, as Gabriella Coleman notes, individual motivations and ethical commitments do not remain constant, but change over the course of projects, adding another level of complexity to the mosaic of ethical and motivational parameters [START_REF] Coleman | Coding Freedom: The Ethics and Aesthetics of Hacking[END_REF]. In addition, there are other actants in the field such as corporations, NGOs, legal administrative bodies and platforms such as GitHub 7 and SourceForge8 which further complicate matters.

Following from these criticisms, it can be argued that an account of developer motivations needs to move beyond the assumption that monetary and legal incentives will work universally as sources of motivation, and take into account the diverse social contexts of FOSS developers and sources of motivation specific to them. In this regard, following Pierre Bourdieu [START_REF] Bourdieu | The forms of capital[END_REF], recognizing the non-monetary forms of capital in the field of FOSS development can be valuable in understanding motivations. For example, recognition and reputation, which are forms of symbolic capital, have been referred as an alternative to monetary capital by Roberts and other researchers following him [START_REF] Roberts | Understanding the motivations, participation, and performance of open source software developers: A longitudinal study of the Apache projects[END_REF][START_REF] Ou | Working for free? motivations for participating in open-source projects[END_REF][START_REF] Benbya | Understanding developers' motives in open source projects: A multi-theoretical framework[END_REF]. This type of symbolic capital is not necessarily antagonistic to commercial interests in FOSS development 9 , since the reputation and experience gained in open source projects are reasonably aligned with developoers' career progression goals. However, this brings the pitfall of adopting an overly reductionist approach which assumes that every decision in the field can be explained in terms of career goals and ignoring the rest of the cultural aspects of FOSS. As, e.g., Gabriella Coleman argues in Coding Freedom [START_REF] Coleman | Coding Freedom: The Ethics and Aesthetics of Hacking[END_REF], participation in FOSS development and hacking has important ethical and political aspects. Similarly, Linus Torvalds's autobiography 'Just for Fun' [START_REF] Diamond | Just for Fun: The Story of an Accidental Revolutionary[END_REF] can be seen as a narrative on how 'fun' can be considered a source of motivation in FOSS. Bednar et al. [START_REF] Bednar | Engineering privacy by design: Are engineers ready to live up to the challenge[END_REF] find that developers describe implementation of privacy features using terms such as 'inconvenient', 'not pleasing', 'challenging', 'nightmare', or 'interesting', 'exciting' and 'satisfying' respectively. As these findings show, the forms of capital and sources of motivation in software development involve diverse elements, including symbolic capital (reputation, recognition), politics, ethics, excitement, convenience and fun.

Research Agenda

As stated in the previous sections, the field of FOSS development has many types of non-monetary rewards, incentives, values and goals related to developers' motivation. Due to this diversity and dependence on social context, an inquiry into motivations in this field needs to address these diverse and field-specific parameters. The research we propose hopes to bring about a social change towards the adoption of PbD principles in open source developer communities, based on a theoretical framework combining game theory and field theory. Our work examines FOSS development as a relatively autonomous field, with its specific rules, goals, values, and forms of capital, in order to understand the motivations of FOSS developers in implementing software features, and to effectively model objective functions based on them to bring about change towards a more favorable equilibrium.

Model Assumptions

Game theory assumes an individual that develops rational strategies consisting of actions oriented towards a goal [START_REF] Sensat | Game theory and rational decision[END_REF]. In sociology, the notion of rational individual is used not as an accurate and realistic representation, but as a simplification that allows for accounting for complex mechanisms [START_REF] Adams | The unknown James Coleman[END_REF]. Despite this acknowledgment, this assumption is criticized for being too narrow, and rarely coinciding with social reality, ultimately defeating its purpose [START_REF] Adams | The unknown James Coleman[END_REF].

Goldthorpe (1998) [START_REF] Goldthorpe | Rational action theory for sociology[END_REF] (as cited in Glaesser 2013 [START_REF] Glaesser | Using rational action theory and Bourdieu's habitus theory together to account for educational decision-making in England and Germany[END_REF]), who favors rational action theory over the concept of habitus for large-scale data, acknowledges that actors do not act purely on rational terms in individual cases. He puts forth a notion of subjective rationality based on the idea that [...] actors may hold beliefs, and in turn pursue courses of action, for which they have 'good reasons' in the circumstances in which they find themselves, even though they may fall short of the standard of rationality that utility theory would presuppose. [START_REF] Glaesser | Using rational action theory and Bourdieu's habitus theory together to account for educational decision-making in England and Germany[END_REF] This view contextualizes actors' courses of action in terms of their social circumstances, which also includes the information available to them. As actors do not necessarily have the resources or disposition to access the 'best information imaginable', they usually act on the available information [START_REF] Glaesser | Using rational action theory and Bourdieu's habitus theory together to account for educational decision-making in England and Germany[END_REF]. They may be aware or unaware of their lack of information or the costs of accessing additional information and thus act on incomplete information [START_REF] Glaesser | Using rational action theory and Bourdieu's habitus theory together to account for educational decision-making in England and Germany[END_REF].

The assumption of rational choice is similarly criticized within game theory since rational action is sometimes dependent on extra-game circumstances including 'history of past interactions, existing customs and practices, or contingently salient features of the particular instantiation of the game' [START_REF] Sensat | Game theory and rational decision[END_REF]. This echoes the criticisms from sociology, as it implies that a simplified and universalized model of an individual acting optimally on the best information ignores the diversity of individual and social contexts.

The complex social circumstances of FOSS development culture warrants taking these views into account. On one hand, the field of FOSS development is diverse, with various developer agents and other actants having a large spectrum of backgrounds and motivations. Secondly, as previous research on the issue attests, goals, norms and practices in the field are varied and contextual [START_REF] Lin | The future of sociology of FLOSS[END_REF]. To factor this complexity, brought about by the diversity of individual cognitive structures and social contexts in FOSS development, we make an assumption based on Pierre Bourdieu's concepts of field and habitus [START_REF] Bourdieu | The Logic of Practice[END_REF][START_REF] Bourdieu | Distinction: A Social Critique of the Judgement of Taste[END_REF]. Our assumption is that actors in the field are limited in their capacity of making rational and optimal choices by their habitus and specific rules and goals of the subfield.

Habitus. Bourdieu defines habitus as "systems of durable, transposable dispositions, [...] principles which generate and organize practices and representations" [START_REF] Bourdieu | The Logic of Practice[END_REF]. Habitus is a system of cognitive and motivating structures associated with past individual and class experience. Although Habitus can result in mastery in actions towards goals without conscious motivation, it also includes strategic thinking involving "an estimation of chances presupposing transformation of the past effect into an expected objective" in the light of present "objective potentialities" [START_REF] Bourdieu | The Logic of Practice[END_REF].

As Bourdieu's concept of habitus does not exclude rational choice but contextualizes it within past experience and present conditions, it has the potential to make up for the shortcomings of a purely rational model of individual in studying goal-oriented choices and motivation. One example is a study on education choices in Germany and England by Glasser and Cooper (2013), who find that young individuals make their educational choices rationally and oriented towards goals, yet these choices are also informed by their class and the culture of their educational institute, confirming habitus as a factor that affects the decision space and actions available to individuals [START_REF] Glaesser | Using rational action theory and Bourdieu's habitus theory together to account for educational decision-making in England and Germany[END_REF].

To summarize, despite the tendency to represent habitus and rational choice as polar opposites or mutually exclusive concepts, habitus is inclusive of rational choice, although based on limited information or class/individual experience. Consequently, we take both habitus and rational choice as important components of the cultures that we chose to study, informing and conditioning each other.

Specific rules and goals of the subfield. According to Pierre Bourdieu, the social world is composed of semi-autonomous systems called fields, which are not com-pletely independent but rather have their own laws and forms of capital [START_REF] Bourdieu | Distinction: A Social Critique of the Judgement of Taste[END_REF]. It follows that a field, such as literature, can have its own, non-monetary forms of capital such as recognition, prestige and autonomy [START_REF] Bourdieu | The Rules of Art: Genesis and Structure of the Literary Field[END_REF]. Fields have their own specific logic by which individuals can take positions within them or forms of capital such as monetary capital that can be converted into other forms and vice-versa [START_REF] Bourdieu | The Rules of Art: Genesis and Structure of the Literary Field[END_REF][START_REF] Bourdieu | Distinction: A Social Critique of the Judgement of Taste[END_REF].

The goals, forms of capital, the logic of their accumulation and conversion, and the rules of a field do not necessarily coincide with other fields and society as a whole. This difference affects the capacity of the agents to take rational actions in it, as well as complicating our means of interpreting these actions in purely rational psychological terms. Therefore, our second assumption is that the specific conditions, rules and goals of the field leading to courses of action do not necessarily coincide with rational choice.

Thus, our ability to analyze developer motivations in FOSS development as actions oriented towards goals, and formulate them as objective functions require taking into account the specific cognitive structures of these agents and the rules specific to the field. This context-specific information cannot be derived from macro-sociological accounts or overly-general assumptions about individual cognition. Our research addresses this issue by integrating a qualitative field study aimed towards understanding the habituses of the developers in this field, and the rules specific to the field. This insight into the contextualized motivations enabls us to use field-specific variables in our objective functions, rather than generalized notions of utilitarian or altruistic choices.

Data Collection

The research we propose uses mixed methods, i.e., combining qualitative and quantitative techniques. Due to their suitability for research projects that aspire to practical application and social transformation, mixed methods have been associated in social sciences with pragmatism [START_REF] Creswell | Research design: Qualitative, quantitative, and mixed methods approaches[END_REF][START_REF] Shannon-Baker | Making paradigms meaningful in mixed methods research[END_REF][START_REF] David L Morgan | Paradigms lost and pragmatism regained: Methodological implications of combining qualitative and quantitative methods[END_REF]. The qualitative phase of the research is an exploration of FOSS development communities as a social field. A field consists of a structured set of positions; these positions differ from spatial positions in that they include alignments based on interpersonal relations and common goals [START_REF] Bourdieu | Sociology in Question[END_REF][START_REF] Levi | The Explanation of Social Action[END_REF]. This first phase is intended to gather insight into the values, goals and forms of capital specific to the field of FOSS development, as well as the agents that take positions within it. This will be achieved through a combination of in-depth and focus-group interviews and participant observation in developer meet-up events such as conferences. While interviews will be useful in understanding individual and group motivations and shared goals, observations in developer conferences will supplement this knowledge to shed more light no interactions. The results of the first phase will facilitate selection of relevant parameters to be measured and thus will shape the quantitative inquiries that follow.

The second part of the research consists of developing questionnaires guided by feature vectors built according to the findings of the previous stage. At this stage, the individual data collected from developers will be used to examine the field using local network regression [START_REF] Mcmahan | Local network regressions[END_REF][START_REF] Levi | Some provisional techniques for quantifying the degree of field effect in social data[END_REF], and to visualize it with dimensionality reduction techniques. After mapping the field as clusters based on individual features, we will be able to determine different subfields with their respective motivating goals, capital forms and habituses. This would enable us to distill objective functions to model a two-player game (developers and users) in order to change the equilibria of the subfields to a state favorable to PbD. Our aim at this stage is to be able to address the diversity of the subfields, avoiding the pitfalls of previous research and legal incentives, while keeping the model as simple and elegant as possible. After the dissemination of tools based on our game model, the final quantitative stage of the research is an assessment of the impact of our intervention, and proposal of strategies that encourage PbD based on our findings.

Sample. Our proposed study focuses on European FOSS mobile game developers. Mobile software is significant for the issue of privacy and security as it is a platform that is very susceptible to exploitation. As mobile games are easily installed, and are used almost ubiquitously by people of all ages and backgrounds, they represent an important risk with respect to privacy and security. For our field study, we strive for maximum variation in our sample of FOSS developers in terms of age, career stage, educational background and project size. This variation will facilitate exploration of a wide variety of subfields that include developers from diverse social backgrounds.

Engineer the Game: Mechanism Design

The modeling of the FOSS field is intended to describe and understand the current situation. A mere understanding, as detailed as it might be, does not bring about change. However, the aim of our research is to produce actionable advice for policy makers. To do this, we use an approach based on mechanism design. Here we will introduce the basic notions of mechanism design and present a thought experiment how to analyse the resulting games.

Mechanism design originated in economic theory or, more precisely, in game theory. It takes an engineering approach to design mechanisms, i.e. incentives, toward desired objectives. By informing this approach with the qualitative and quantitative results of our field research, as sketched in Fig. 1, we acknowledge the general critique of game theory, i.e. unrealistic assumptions about the rationality of agents and over-simplification [START_REF] Adams | The unknown James Coleman[END_REF].

Games: Objective Functions and Equilibria -two examples. For our thought experiment, let's assume that we find in the analyses of our initial interviews and observations that our participants have different occupational statuses and organizational roles (e.g. junior, senior, student), and have different roles in the decision-making processes of their FOSS projects. They also have various motives for participating in FOSS (e.g. to build reputation for themselves or their organization, to have a project to show in job applications, to gain skills and knowledge, to contribute to society). They formulate the impact of their project in terms of their individual careers, contribution to the FOSS world, and society in general. Finally, let us say that we find that all of our participants think PbD features are important, but all are reluctant to implement them for differing reasons: Some state that PbD features take too much time and hinder the project, some think their skills are not sufficient for their implementation and are concerned that it will harm their reputation if they fail, and some do not want the ethical responsibility of implementing them correctly in case they fail.

Looking at these preliminary results, we develop our questionnaires. Our data can be roughly organized under habitus and forms of capital (education and skills, career, role in project or organization, reputation, money), values and goals (e.g. reputation, skills, contribution), strategies and positions (e.g. organizational and project hierarchy, career plans), and motivational factors (e.g. importance placed on PbD, time and resources or lack thereof, fear of losing reputation, wishing to contribute to society vs. fear of doing damage by developing insecure software). With the collected data and using local network regression [START_REF] Mcmahan | Local network regressions[END_REF] we will be able to cluster individuals and observe subfields formed on the basis of these features. This will help us develop a game model addressing the diversity of these clusters while remaining simple and elegant. Based on this model, we can develop tools to shift the equilibrium towards a more sensible state, followed by our assessment of impact.

c = pet c = pet p = pet 2,1 0 p = pet 0 1,2
Table 1. Payoff matrix of our example game: the privacy battle.

To illustrate the type of conclusions we envision from this process, we briefly sketch a game that reflects a very rough view on the situation that we set out in our introduction, then discuss the expected equilibria. The most basic game has two players, both of which can play two strategies: producer p, which can opt to provide a privacy-friendly product or a product without privacy protection but with equal functionality otherwise. Moreover, the consumer c might chose to buy a privacy-aware or non-privacy-aware product. In Table 1, we illustrate this game. It describes the following situations:

The producer implements a privacy-enhanced service (p = pet). In doing so, the service becomes more expensive, which might have a negative impact on the profit margin. If the consumer requires privacy (c = pet), the producer will reach some satisfaction, but at higher implementation costs, while the consumer will be fully satisfied. Hence the consumer payoffs are relatively higher. If the consumer is willing to give up privacy (c = pet ) for a lower priced product, both consumer and producer might be unsatisfied, because the consumer might not be willing to pay the higher price of the privacyenhanced product. The producer implements a service without privacy (p = pet ). By doing this, the service becomes less expensive, which might have a positive impact on the profit margin. If the consumer requires privacy (c = pet), both consumer and producer might be unsatisfied, because the consumer might not be willing to give up privacy. If the consumer is willing to give up privacy (c = pet ) for a lower-priced product, the producer will be fully satisfied but the consumer will need to surrender some of their revenue in the form of privacy.

From today's market observation, we expect a payoff matrix similar to that described above. That is, if both players opt for the more privacy-aware option c, will get their highest payoff and p some payoff lower than that. If, on the other hand, both opt for a less privacy-friendly option, p will get their highest payoff and c some lower payoff. In the remaining cases, no player will get any payoff since either the consumer is not willing to pay the higher price or is not willing to give up their privacy. Further research is needed to determine more precise figures. Note in the fundamental game theory literature, games of this form are often introduced as "battle of the sexes games".

From everyday observation, which indicates a lack of privacy-friendly services, we venture to construct the following hypothetical analysis of the game: we know that "battle of the sexes" type of games have three Nash equilibria namely two pure, where both players opt for the same privacy preference, and one mixed. While the pure strategies are utility optimal, i.e. both ensure that the sum of all revenues is maximized, both are unfair: one party will always receive the higher revenue. On the other hand, the mixed strategy is highly inefficient. From the above, we conclude that consumers settle for a lower overall revenue due to the lack of better options.

While this simple game may broadly approximate the current situation, we do not assume that such a simple game will be sufficient to describe the current motives of FOSS developers. Hence we need to collect data to come up with a somewhat more complex game.

Conclusion

In this paper, we have attempted to provide a road map for research on FOSS developer motivations, with the aim of bringing about social change towards privacy-friendly features, while taking into account and learning from the shortcomings of previous research. We argued that the problems in previous research on this topic stem from the fact that it is mostly based on assumptions about developers that do not reflect the specific conditions of the FOSS development scene. More specifically, the research we see in the literature typically assumes that the developers are hedonistic rational individuals that make self-interested choices, or assumes that they take part in FOSS development due to altruistic motives. Neither of these models successfully explains the reluctance to implement privacy features or the failure of legal incentives obliging the implementation of PbD. More recent research, such as Spiekermann (2019) [START_REF] Spiekermann | Inside the organization: Why privacy and security engineering is a challenge for engineers[END_REF] and [START_REF] Bednar | Engineering privacy by design: Are engineers ready to live up to the challenge[END_REF] [START_REF] Bednar | Engineering privacy by design: Are engineers ready to live up to the challenge[END_REF], go beyond such limited explanations and point to organizational and individual reasons underlying this reluctance, indicating the necessity for research that takes into account the specific individual and social contexts surrounding developer motivations.

To address this, we have proposed an approach combining field theory and game theory into a mixed methods research endeavour with the aim of gaining deeper insight into FOSS communities and of suggesting relevant policy changes. From field theory, we take the concept of field and habitus in order to replace the general/universal models of individual with a historically and socially contextualized understanding of FOSS developers. This approach enables us to explore the field of FOSS development and the developer motivations with respect to field specific rules, goals, forms of capital and the collective past experience of the individuals therein. The findings will be used to develop games based on social context rather than dehistoricized assumptions about developers. The insights from the field study and game model will then be used to develop guidelines, tools and policy suggestions aimed at social change towards privacy-friendly software.

Fig. 1 .

 1 Fig. 1. Our Experimental Setup

https://www.openpgp.org

https://www.torproject.org

https://www.kernel.org

https://www.mozilla.org/

https://github.com

https://sourceforge.net

Unlike, for example, avant-garde art. See[START_REF] Bourdieu | The Rules of Art: Genesis and Structure of the Literary Field[END_REF] [START_REF] Bourdieu | The Rules of Art: Genesis and Structure of the Literary Field[END_REF]