Greqam

Mohamed Belhaj
email: mbelhaj@ec-marseille.fr.

Frédéric Deroïan

Frédéric Deroian
email: eric.deroian@univmed.fr.

Risk taking under heterogenous risk sharing

Keywords: JEL Classification Numbers: D85, C72, G11 Risk taking, Heterogenous risk sharing, Strategic Interactions, Bonacich measure

We revisit the common view that risk sharing enhances risk taking in the context of heterogenous risk sharing in a small economy. Under low volumes of transfers, we express individual risk level in terms of Bonacich measure. We find that heterogeneity combined to strategic interaction imply that risk sharing enhances risk taking only in average. However, under high transfer volumes, risk sharing may reduce risk taking. We also provide conditions under which agents under or over invest with respect to the risk allocation maximizing the sum of profits.

Introduction

In many economic contexts, a redistribution of incomes in a society of risk averse agents affects risk taking. For instance, in labor markets, unemployment insurance encourages workers to seek higher productivity jobs because they are more willing to endure the possibility of unemployment; similarly, redistributive taxation enhances entrepreneurship 1 . Hence, the general message is that social insurance enhances individual risk taking. The conclusion is of sizeable importance, since it has an impact on innovation and growth. This article revisits this issue in a context of heterogenous risk sharing and strategic interactions. A typical example is that of a small economy, say a developing village, in which agents insure themselves against risk by sharing revenues 2 . First, strategic interactions can emerge from the sharing of correlated incomes. Second, informal insurance may be heterogenous across agents. For instance, if there is no formal institution to enforce a redistribution mechanism, heterogeneity in risk sharing may arise because of self-enforcing mechanisms and trust (for instance social sanctions may be heterogenous), and also the heterogeneity in information flows and moral hazard, in income correlations, in geographic costs, increasing costs to group size, etc.

The main objective of this article is to examine how risk sharing heterogeneity affects individual incentives to take risk. To proceed, we consider a society of risk-averse agents. Each agent has one unit to invest in a specific project. Projects are developed 1 See Acemoglu andShimer [1999, 2000], [START_REF] Mayshar | Should government subsidize risky private projects?[END_REF], [START_REF] Kanbur | Risk taking and taxation: an alternative perspective[END_REF], [START_REF] Zeira | Risk reducing fiscal policies and economic growth[END_REF], [START_REF] Boadway | Optimal linear income taxation in models with occupational choice[END_REF], [START_REF] Sinn | Social insurance, incentives and risk taking[END_REF], [START_REF] Garcia-Penalosa | Redistribution and entrepreneurship with Schumpeterian growth[END_REF].

2 [START_REF] Townsend | Risk and insurance in village India[END_REF] emphasizes the importance of informal insurance networks in Indian villages; similarly, [START_REF] Udry | Risk and insurance in a rural credit market: an empirical investigation in Northern Nigeria[END_REF] documents that the majority of transfers take place between neighbors and relatives in Northern Nigeria -see also [START_REF] Rosenzweig | Risk, implicit contracts and the family in rural areas of lowincome countries[END_REF], [START_REF] Murgai | Localized and incomplete mutual insurance[END_REF], [START_REF] Fafchamps | Risk Sharing Networks in Rural Philippines[END_REF], [START_REF] Dercon | risk sharing networks and insurance against illness[END_REF], [START_REF] Fafchamps | The Formation of Risk Sharing Networks[END_REF], De [START_REF] Weerdt | Social Identity and the Formation of Health Insurance Networks[END_REF]. Recent theoretical advances about risk sharing in developing economics have studied the impact of the structure of the social insurance networks on the volume of transfers, whether rules are self-enforcing [START_REF] Ambrus | Consumption risk sharing in Social Networks[END_REF], [START_REF] Bloch | Reciprocity in Groups and the Limits to Social Capital[END_REF]Bloch et al. ( , 2008)) )) or not (Bramoullé andKranton (2006, 2007), and [START_REF] Gallegati | The Asymmetric Effect of Diffusion Processes: Risk Sharing and Contagion[END_REF]). With regard to this theoretical literature, our work allows for endogenous income, and heterogenous transfers across agents.

through investment in a portfolio of two correlated technologies A and B. We assume without loss of generality that technology A is more profitable but entails more risk.

After investing, incomes are realized, and then agents proceed to transfers.

As a first key ingredient of our model, we consider mean-variance utility functions, and we assume that technologies are correlated. Hence, agents are not only exposed to idiosyncratic factors, but also to systemic factors. In this context, diversification via resources pooling enables to reduce specific risk.

The second key ingredient concerns the rule of transfers, that we assume exogenous and binding 3 . We represent the structure of transfers by a matrix Λ = [λ ij ], where λ ij represents the share of agent j's income that agent j transfers to agent i. To focus on pure risk sharing and rule out wealth effects4 , we assume that expected after-transfer revenues are equal. This assumption implies that the matrix is row-stochastic. Moreover, it is column-stochastic by budget constraint (in total the matrix is bi-stochastic).

We ask the following questions. How does the heterogeneity of risk sharing drive individual risk choice? Does more revenue sharing enhance risk taking? Do agents under-invest or over-invest with respect to what is socially optimal? Last, how does cash transfers affect the overall investments in the more risky technology?

We first point out that if the excess return of technology A relative to B is positively (resp. negatively) correlated among agents, risk levels are strategic substitutes (resp.

3 The issue of self-enforcement is beyond the scope of this article. Actually, this problem is limited by two factors in developing countries. First, agents may not exchange their whole income with neighbors (of course, a self-enforcing mechanism may have designed low-volume transfers). Second, agents generally know each other for a long time and do not often move during their lives, which enforces trust and cooperation (social sanction are strong). "For developing countries, most researchers [...] argue[...] that information and enforcement problems are likely to be small between the members of a village and this creates a favorable environment for cooperation", [START_REF] Dercon | risk sharing networks and insurance against illness[END_REF]. complements). We then focus on the case of strategic substitutability, which is more plausible (in particular this is the case if technology B is risk free).

We gradually introduce heterogeneity in income transfers. As a benchmark, we first examine the (full and partial) equal sharing rule, which stipulates that agents share equally an identical proportion of their realized incomes. In the polar case in which projects are not correlated, equal sharing enhances investments in (the higher risk/return) technology A. Due to correlations between transfers, correlated incomes affect the attractiveness of technology A under risk sharing. We therefore state a condition (condition C 0 thereafter) which guarantees that technology A stays more attractive under full equal sharing, and we assume that this condition holds for the rest of the paper. In particular, this condition guarantees technology A stays more attractive under partial equal sharing.

Then we turn to the general case. The transfer matrix can be interpreted as a network, in which the value of connection ij is the share that agent j gives to agent i. First we consider the case in which own shares are homogenous. We find that equilibrium risk levels are homogenous (but not profits) and decreasing in the value of own share. That is, two transfer matrices with same homogenous own share but different off-diagonal elements generate the same equilibrium risk profile. Hence, heterogenous risk levels only emerge from heterogenous own shares.

Second, we examine the case of heterogenous own shares. In general, an optimal risk profile exists and it is unique. We then separate the space of transfer matrices in two regions, region R in which risk sharing is weak, region R in which risk sharing is large. Our main theoretical results concern region R. In this region, risk levels can be interpreted as a Bonacich measure of a transformation of the transfer matrix 5 . Furthermore, with risk sharing, agents take more risk than in isolation. Last, we prove that increasing revenue sharing enhances risk taking, but only in average; typically certain agents may take less risk.

In region R, in which risk sharing is large, simple examples illustrate that more revenue sharing can reduce the average risk level. Moreover, certain risk level may be lower that the level of risk taken by an isolated agent. Therefore, a deep conclusion is that depending on the volume of risk sharing, heterogeneity in risk sharing may affect risk taking in opposite directions.

Then we focus on weak risk sharing. We examine the tension between individually optimal risks and the risk profile maximizing the sum of individual payoffs. We first notice that the game may generate either positive or negative externalities. This arises from a simple tradeoff: when some agent increases investment in technology A, this has an ambiguous effect on the profits of agents who will receive some transfer from her, by raising both expected means and volatility of future transfer.

We first characterize the efficient risk profile as a Bonacich measure defined over a matrix which aggregates all these externalities. Then we turn to the comparison of the average equilibrium risk level and the average efficient risk profile. Given that the sign of externalities is endogenous, agents may either under-invest or over-invest with respect to social welfare. Actually, agents under-invest (resp. over-invest) in situations in which they do not internalize that an increase of their risk level would promote (resp. reduce) significantly the return of technology A for others -because the induced increase in expected means dominates (resp. is dominated by) the raise of variance. We do relate the comparison of the average equilibrium risk level and the average efficient risk profile to conditions on the transfer matrix. We find that if each agent's own share is larger than (resp. is lower than) the sum of squares of shares she transfers to others, then agents under-invest (resp. over-invest) in technology A with respect to the efficient profile.

Finally, an extension examines the issue of cash transfer or wealth redistribution. We determine the impact of a modification of initial wealths on the total amount of investments that the society devotes to technology A. The result has important implications for economic policy. As a first illustration, we present a set of matrices for which the identity of eligible households does not matter. This case happens to be that of societies with homogenous own shares. In a second application, one agent receives a positive shock on her initial wealth, for instance as being eligible to cash transfer by an aid program. We characterize the agent to be selected in order to generate a maximal increase in the overall investments made in technology A. Typically, agents with maximal equilibrium risk (before the treatment) or with lowest own share may not be those to select.

The article is organized as follows. Section 3 describes the model. Section 4 studies the existence and characterization of equilibrium, and some comparative statics. Section 5 examines efficiency issue, and section 6 is an extension exploring cash transfer issue. The last section concludes. All proofs are in the appendix.

The model

We consider a three-stage game in which, first, agents choose their level of risk, second, incomes are realized, and third agents share simultaneously some part of their revenues.

To fix ideas, consider a village economy, in which risk averse farmers have one unit of land to use for the plantation of potatoes and strawberries (the latter variety having higher expected means and being more risky). After planting, incomes are realized and farmers share some part of their revenues.

Formally, the society contains a finite set N = {1, 2, • • • , n} of agents. Each agent i has one unit to invest in two technologies A and B. The return Y j of a technology j ∈ {A, B} is random with expected mean µ j and variance σ 2 j . Agents are risk-averse and care about expected means and variance of the returns, i.e. the utility of agent

i investing in technology Y j is U i (Y j ) = E(Y j ) -κ 2 V ar(Y j
). We suppose that both technologies are intrinsically attractive, i.e. µ j > κ 2 σ 2 j for j = A, B. Further, technology A has greater expected return than technology B and is more risky, that is, µ A > µ B and σ 2 A > σ 2 B (one possible interpretation is that technology A is an innovation). Let σ AB represent the covariance between technologies A and B in a same project. Now we define correlations between projects. We let τ A (resp. τ B ) represent the correlation between a project that uses technology A (resp. B) and a distinct one that uses technology A (resp. B). Further, τ AB is the covariance between technologies A and B in two different projects. Economically, τ A , τ B , τ AB correspond to systemic risks, while σ AB , σ 2 A , σ 2 B incorporate both specific and systemic risk. A strategy for agent i is a scalar x i ∈ [0, 1]. It represents the share of investment that agent i devotes to technology A. The study of corner solutions is beyond the scope of this article; we obtain solutions in the interval [0, 1] for a large range of parameters.

We interpret x i as the level of risk chosen by agent i. Note that if agents had a binary choice between A and B, x i ∈ [0, 1] could be interpreted as a probability, and we would solve a Bayesian equilibrium. Choosing some level x i , agent i selects therefore

technology Y i = x i Y A + (1 -x i )Y B . We let X = (x 1 , x 2 , • • • , x n ) denote a strategy
profile of the society.

To insure against risk, agents share part of their realized incomes with others. In our setting, the realization of individual incomes is observable by all others. Furthermore, the sharing rule is exogenous and binding. Transfers are described by a transfer matrix

Λ = [λ ij ],
where λ ij ∈ [0, 1] represents the share of the realized income of agent j that agent j transfers to agent i. Parameter λ ii represents agent i's own share. By budget constraint, λ ii + j =i λ ji = 1 for all i. Furthermore, to isolate risk sharing from wealth effects, we suppose that ex ante after-sharing revenues are equal. This implies that own share plus the proportions of revenues that agent i receives from others sum up to 1, i.e. λ ii + j =i λ ij = 1 for all i. Therefore, the matrix Λ is bi-stochastic. This set of rules encompasses the possibility of ex ante asymmetric bilateral transfers 6 . Among all possibilities covered by our setting, a polar case is the egalitarian sharing of realized incomes7 .

6 In line with the existence of asymmetric transfers, the redistributive role of transfers has been emphasized, as in [START_REF] Lucas | Motivations to remit: evidence from Botswana[END_REF] or [START_REF] Azam | Migrants' remittances and the household in Africa: A review of evidence[END_REF]. Recent work on child fostering focuses on the role of extended family ties as key determinant of insurance against persistent shocks, such as how to deal with the loss of a parent or husband, like in [START_REF] Duflo | Grandmothers and granddaughters: old-age pensions and intrahousehold allocation in South Africa[END_REF] or [START_REF] Ksoll | Family Networks and Orphan Caretaking in Tanzania[END_REF].

Having defined the matrix of transfers, we relate individual profits to transfers. We consider a profile of choices X

= (x 1 , x 2 , • • • , x n ). Agent i's profit writes: π i (X) = j λ ij E(Y j ) - κ 2 j k λ ij λ ik cov(Y j , Y k ) (1)
where

E(Y j ) = µ A x j + µ B (1 -x j )
and, letting symbol 1 quote for the indicator function,

cov(Y j , Y k ) = x j x k σ 2 A × 1 {j=k} + τ A × 1 {j =k} + x j (1 -x k ) + x k (1 -x j ) σ AB × 1 {j=k} + τ AB × 1 {j =k} +(1 -x j )(1 -x k ) σ 2 B × 1 {j=k} + τ B × 1 {j =k} 3 Equilibrium
We search for the existence and characterization of Nash equilibria. Formally, a profile X * is a (pure) Nash equilibrium if it satisfies that, for all i, for all x i ∈ [0, 1],

π i (x * i , x * -i ) ≥ π i (x i , x * -i ).
The system of first order equations is linear and writes as:

λ ii x * i + F j =i λ ij • x * j = H + Kλ ii (2) with F = τ A +τ B -2τ AB σ 2 A +σ 2 B -2σ AB , H = µ A -µ B κ +τ B -τ AB σ 2 A +σ 2 B -2σ AB , and K = σ 2 B -τ B +τ AB -σ AB σ 2 A +σ 2 B -2σ AB .
Parameter F , which will be of major interest throughout the paper, can be written

cov((A-B) i ,(A-B) j ) cov((A-B) i ,(A-B) i ) ;
that is, F is equal to the ratio of the systemic risk of excess return of A over B to its total risk. If the excess return of technology A relative to B is positively (resp. negatively) correlated among agents, risk levels are strategic substitutes (resp. complements). We assume that F > 0, which is the most plausible case (in particular this is the case when technology B is risk free). Equation (2) shows that agents take into account correlations between projects. Further, as F > 0, individual actions are strategic substitutes, which will be a crucial feature of our analysis.

We introduce the n × n matrix Γ = [γ ij ], with γ ii = 0 for all i, and γ ij = λ ij λ ii for all i, j = i. The element γ ij is equal to the ratio of the share that agent j gives to agent i over agent i's own share. It is the matrix of interactions, over which the Bonacich measure will be defined8 .

Risk sharing with homogenous own shares

We begin with the program of an isolated agent; that is, for all i, j = i, λ ii = 1, λ ij = 0. Individual decisions take into account the following factors. First, technology B is attractive since it is less volatile than technology A. Second, agents are incited to invest some share in technology A for two reasons: for diversification purpose, i.e. variance reduction, even if covariance limits diversification, and because technology A has higher expected return than technology B. These factors shape the optimal level of risk of agent i, denoted x e * :

x e * = µ A -µ B κ + σ 2 B -σ AB σ 2 A + σ 2 B -2σ AB
We note that x e * = H + K, that we assume positive. And under mild conditions on the parameters of the game, we have x e * ≤ 1. In essence, Markovitz's message about diversification is effective, although limited by the covariance of the lotteries.

Let us turn to socialized agents. We consider here the egalitarian sharing of realized incomes. This corresponds to a transfer matrix Λ F ES with λ F ES ij = 1 n for all i, j (where 'F ES' quotes for 'Full Equal Sharing'). Adapting the first order equation (2) to matrix Λ F ES and solving, we obtain

x F ES = nH + K 1 + (n -1)F (3)
We denote ∆µ c = κ

(σ 2 A -σ 2 B )τ AB +(τ B -τ A )σ AB +σ 2 B τ A -σ 2 A τ B (σ 2 A -τ A )+(σ 2 B -τ B )-2(σ AB -τ AB )
. The following condition will be useful:

Condition C 0 µ A -µ B > ∆µ c
In terms of parameters H, K, F , the condition writes (1-F )H -F K > 0 or equivalently

H + K < H F .
We also note that the combination of conditions H + K > 0 and C 0 imply that H > 0. In the polar case in which projects are not correlated, equal sharing enhances investments in technology A. Due to correlations between transfers, correlated incomes affect the attractiveness of technology A9 . Condition C 0 guarantees that the equilibrium risk level under full equal sharing exceeds the equilibrium risk level of some isolated agent; i.e. x F ES > x e * . We assume that condition C 0 holds until the end of the article.

We extend the analysis to partial equal sharing. The typical case is that is which a fixed proportion, say τ 0 , of incomes is collected and equally redistributed. Then, agent i receives 1 - (n-1) n τ 0 y i + τ 0 n j =i y j . Denoting λ 0 = 1 -(n-1) n τ 0 , this sharing-rule can be represented by the transfer matrix Λ P ES (λ 0 ) such that λ P ES ii = λ 0 , for all i, and

λ P ES ij = 1-λ 0
n-1 , for all i, j = i (where 'P ES' quotes for 'Partial Equal Sharing') -with 1-λ 0 n-1 = τ 0 n . Straightforward computation gives:

x P ES (λ 0 ) = H + Kλ 0 (1 -F )λ 0 + F (4)
Noticing that λ 0 is a decreasing function of τ 0 , the equilibrium level of risk is decreasing in the value of λ 0 . In particular, it is larger than the risk taken by an isolated agent.

In a word, in the context of egalitarian redistribution, more taxation implies more risk sharing. This leads to an increase of risk taking.

We extend the analysis further to transfer matrices with heterogenous off-diagonal elements. We thus define the family of transfer matrices with homogenous own shares L HOS (λ 0 ) = {Λ/λ ii = λ 0 , ∀i}. A natural issue consists in ordering such matrices in terms of their associated equilibrium risk profile. In fact, for any matrix Λ ∈ L HOS (λ 0 ), the equilibrium risk level is homogenous and only related to the own share of the matrix Λ:

x

HOS (λ 0 ) = H + Kλ 0 (1 -F )λ 0 + F (5)
(this stems directly from equation ( 2), exploiting that Λ is row-stochastic) The following proposition summarizes the results of the whole subsection:

Proposition 1 In societies with homogenous own shares, the equilibrium level of risk is homogenous and decreasing in the value of own share. Furthermore, the equilibrium level of risk exceeds that of isolated agents.

Proposition 1 confirms that for societies with homogenous own shares, more revenue sharing enhances risk taking. We observe that, in societies with homogenous own shares, risk level is only related to own share. However, profits depend on the whole distribution of transfers. Let

λ -i = (λ i1 , λ i2 , • • • , λ ii-1 , λ ii+1 , • • • , λ in ).
We say that "the distribution λ -i is more diversified than the profile λ -i whenever j =i λ 2 ij > j =i λ 2 ij ". Then, it is easily shown that, when own shares are homogenous, more diversification in the distribution λ -i is always beneficial to agent i's equilibrium profit (proof are available upon request). This means that agents are better off in the society with transfer matrix Λ P ES (λ 0 ) than in any other society with the same homogenous own share.

Risk sharing with heterogenous own shares

Until now, we considered risk sharing with homogenous own shares. We found that risk levels are only affected by the value of own share, and we confirmed that more risk sharing enhances risk taking, even in the presence of strategic interactions. We will now focus on heterogenous risk sharing, and we will discuss the view that risk sharing enhances risk taking.

We separate the space of all bi-stochastic matrices in two complementary regions parameterized by F . We let R be the set of bi-stochastic matrices such that λ ii ∈ F 1+F , 1 for all i, and R be the set of bi-stochastic matrices such that for some i, λ ii ≤ F 1+F . We note that no own share is below one half in region R.

We consider a n × n matrix M with nonnegative elements m ij , any column-vector a = (a 1 , a 2 , • • • , a n ) with nonnegative real components and we let J denote the column vector of ones. We define B a (M ; α) as the solution of I -αM B a = a, α ∈] -1, 1[ (we write B(M ; α) if a = J). When both α > 0 and the greatest modulus of eigenvalues of M is smaller than 1 α , B a (M ; α) can be interpreted as a vector of weighted (by a) Bonacich centrality measure defined over the network with link m ij representing the intensity of the connection from agent i to agent j. Indeed,

B a (M ; α) = ∞ k=0 α k M k a (6)
is a well-defined quantity, and B a i (M ; α) measures the sum of the values of paths from agent i to others through the network, where paths of length k toward agent j are weighted by a j α k (where the value of a path represents the product of link intensities). This measure (actually, a slightly modified version) was introduced in [START_REF] Bonacich | Power and centrality: a family of measures[END_REF]. Our model containing strategic substitutes, we will consider the case α ∈] -1, 0[. In such a case, the quantity given in equation ( 6) is also well-defined, but the contribution of the network to the measure is ambiguous. Odd (resp. even) paths contribute negatively (positively) to the measure.

Remark 1 Theorem 1, pp. 1408, in [START_REF] Ballester | Who'swho in networks. Wanted: the key player[END_REF] provides a formulation in terms of Bonacich centrality measure (i.e. a formulation using a positive decay parameter) over another interaction matrix, under some restrictions of our game. We choose not to use that formulation for two reasons. First, the specification is non linear, making also difficult to interpret it. Second, our solutions are valid under less restrictive environment than those required to obtain a positive decay parameter. We will then speak about 'Bonacich measure', without reference to the notion of centrality 10 10 When the matrix of cross effects is binary, a centrality measure can be defined over its comple-(we keep the reference to Bonacich, as it conveys the notion of network, and further the measure is technically of same nature).

We define the vector D such that D i = 1 λ ii . Inverting the linear system presented in equation ( 2), we obtain the following theorem:

Theorem 1 A unique equilibrium X * = (I + F Γ) -1 (HD + KJ) exists 'almost everywhere' in the space of admissible values of F . Furthermore, if Λ ∈ R, a unique solution X * always exists and can be written

x * i = H F - (1 -F )H -F K F B i (Γ; -F ) (7)
for all i ∈ N , where B i (Γ; -F ) is given by equation ( 6) and belongs to ]0, 1[. We note that if own shares exceed one half, Bonacich measures are well defined 11 .

Equilibrium risk levels are lower than unity if

H F ≤ 1 (i.e. µ A -µ B < κ(τ A -τ AB )
), what we assume for simplicity.

Remark 2 In region R, inverting the system presented in equation ( 2), one expects a characterization of equilibrium risk levels as a weighted Bonacich measure (since the constant H + Kλ ii contains an idiosyncratic component). Theorem 1 expresses the solution as a simple (non weighted) Bonacich measure. This is specific to the fact that the constant is an affine function of λ ii and that the transfer matrix is row-stochastic (the proof is in lemma 1).

A lesson of theorem 1 is that under weak risk sharing, interactions are sufficiently low to guarantee that x * i ∈]H + K, H F [. This means that in any society with transfer matrix in R, every agent takes more risk at equilibrium than in isolation.

mentary matrix (an off-diagonal element of the complementary matrix is 1 if and only if the associate element is 0 in the initial matrix) -see [START_REF] Corbo | The Importance of Network Topology in Local Contribution Games[END_REF], [START_REF] Bramoullé | Strategic Substitutes and Networks[END_REF], [START_REF] Bloch | Pricing on networks[END_REF]. 11 Angelucci and De Giorgi show that "for every 100 pesos transferred by Progresa to the eligible households, the consumption of ineligible households increases by approximately 11 pesos". Since the proportion of treated household is greater than one half, this suggests that own shares are high proportions of revenues.

We proceed to some comparative analysis of transfer matrices. Two issues emerge when we try to replicate proposition 1 to societies with heterogenous own shares. First, how to define more revenue sharing? We generalize the idea of 'more revenue sharing' as follows. Starting from any society, revenue sharing increases when all own shares are diminished and no other share is diminished, in a way that preserves bi-stochasticity.

Formally:

Definition [more revenue sharing]

Consider two transfer matrices Λ, Λ = Λ+Θ such that for all i, θ ii =j =i θ ij =j =i θ ji . There is more revenue sharing in matrix Λ than in Λ if for all i, θ ii ≤ 0 and for all i, j = i, θ ij ≥ 0.

Second, for societies with homogenous own shares, individual and average risk taking coincide. In opposite, risk levels are differentiated for societies with heterogenous own shares. A natural question that arises is: does more revenue sharing always increase individual risk taking, or eventually in average? The following theorem compares the risk taking behaviors of two such societies 12 in the region of weak risk sharing:

Theorem 2 Consider two transfer matrices Λ, Λ ∈ R. If there is more revenue shar-

ing in Λ than in Λ, then i x * i ≥ i x * i .
Theorem 2 confirms that more revenue sharing promotes risk taking in average in region with weak risk sharing. Actually, more revenue sharing does not necessarily lead to an increase of all individual risk levels. To illustrate, let Λ = Λ + Θ be such that there is more revenue sharing in Λ than in Λ and such that θ 1j = 0 for all j. Then, combining first order equations applied to agent 1 in both matrices, one obtains easily that there exists some agent, say i 0 , such that x * i 0 < x * i 0 . Moreover, we note that, by continuity, the result holds if there is strictly more revenue sharing in Λ than in Λ (we just have to select a sufficiently small perturbation of the matrix Θ that yields θ 11 < 0 12 Comparing transfer matrices in which diagonal elements vary in opposite directions is difficult.

The point can be tackled by perturbing locally a society with homogenous own shares. We find that any small modification, that induces an decrease of diagonal elements in average, leads to more risk taking in average. Results are available upon request.

while preserving both signs of all other terms and bi-stochasticity of the new transfer matrix).

In the region with large risk sharing R, more revenue sharing may decrease risk taking in average. Simple examples illustrate the problem in region R. Suppose H = .5, K = 0, F = .9 and consider the following matrices: Λ = Remark 3 When one of the two matrices Λ or Λ is homogenous, the theorem still holds as soon as the other matrix lies in region R. Considering any transfer matrix

Λ ∈ R, we denote λ = min i λ ii , λ = max i λ ii , x = x * (λ) and x = x * ( λ). Then, x ≤ 1 n i x * i ≤ x. A direct implication is that there exists a value λ 0 ∈ [min i λ ii , max i λ ii ] such that x * (λ 0 ) = 1 n i x * i .
Furthermore, there exists at least one agent i 0 (resp. j 0 ) with x * i 0 > x (resp. x * j 0 < x).

To sum up, fixing any transfer matrix in the region of weak risk sharing R, we provide two broad sets of conditions under which the average level of risk taking is enhanced by a (inequality preserving) modification of the structure of risk sharing.

First, this occurs if the modification induces a decrease of all diagonal elements without decreasing any non diagonal element, and such that the resulting matrix stays in region R; second, this occurs if the modification induces a decrease of all diagonal elements until some homogenous value not larger than the lowest own share of the former matrix, irrespective of the structure of the modification out of the diagonal, and irrespective of the locus of the resulting matrix (it can be in both regions).

The region of large risk sharing R is more uncertain. Simple examples suggest that the variance of risk levels is higher than in the region of weak risk sharing, and more risk sharing can reduce risk taking; even the average risk level can be lower than the optimal risk of an isolated agent.

4 Over-versus under-investment w.r.t. social welfare

We consider the risk profiles that maximize the sum of profits in the society, W = i π i .

One particularity of the game is that the sign of externalities is endogenous. Indeed, through the transfer of intensity λ ji , an increase in x i induces higher expected return for agent j, but also higher variance13 . In consequence, whether agents over-or underinvest in the innovation with regard to social welfare is ambiguous. We will give some partial answers, yet potentially instructive. In particular, the study suggests that the structure of the transfer matrix is crucial to understand whether agents under-or over-invest in technology A.

Let Ψ denote the matrix such that ψ ij = k λ ki λ kj for all i, j. We note that ψ ij = ψ ji , and since the matrix Λ is bi-stochastic, the matrix Ψ is also bi-stochastic.

We also define the matrix Φ

= [φ ij ] such that φ ij = 0 for all i, φ ij = ψ ij
ψ ii for all i, j = i. Finally, we define the vector E such that E i = 1 ψ ii . We remark that, being bi-stochastic, the matrix Ψ associated with the transfer matrix Λ can be seen as a transfer matrix.

Hence, we obtain:

Proposition 2 A unique efficient risk profile X = (I + F Φ) -1 (HE + KJ) exists 'almost everywhere' in the space of admissible values of F . Furthermore, if Ψ ∈ R, a unique solution X always exists and can be written

xi = H F - (1 -F )H -F K F B i (Φ; -F ) (8)
for all i ∈ N , where B i (Φ; -F ) is given by equation ( 6).

When the sign of externalities is constant, it is in general easy to see if agents under invest or over invest with respect to social welfare. Here, since externalities may be either positive or negative, it is difficult to find general conditions under which the equilibrium risk profile either dominates or is dominated by the efficient profile. We will provide two sets of conditions under which we can compare the average value of efficient and equilibrium risk profiles.

Since the efficient risk profile corresponds to the transfer matrix of a modified game, the conditions under which we were able, in the preceding section, to compare equilibrium risk profiles associated to distinct transfer matrices basically hold. Hence, we obtain:

Proposition 3 Consider a matrix Λ ∈ R. The average equilibrium level of risk is lower (resp. higher) than the average efficient level of risk if those three conditions apply simultaneously: (i)

Ψ ∈ R; (ii) λ ii ≥ ψ ii (resp. λ ii ≤ ψ ii ) for all i; (iii) λ ij ≤ ψ ij (resp. λ ij ≥ ψ ij ) for all i, j = i.
Suppose now that Λ ∈ L HOS (λ 0 ), and suppose that Ψ ∈ R. If λ 0 ≥ max i ψ ii (resp.

λ 0 ≤ min i ψ ii ), the average equilibrium level of risk is lower (resp. higher) than the average efficient level of risk.

(proof omitted) Proposition 3 is the mirror of theorem 2 and remark 3 as applied to efficiency issue. The proposition indicates that if matrix Ψ is the transfer matrix of a game in which there is more (resp. less) revenue sharing than in the original game Λ, then agents under-invest (resp. over-invest). Indeed, when agents increase their risk level, this has two opposite effects on the payoff of neighbors: this increases their risk, but this also increases their expected return. Now, if agents do not exchange much with the society, the expected return effect dominates, i.e. in average, agents do not internalize that increasing their risk level would increase the overall expected return of the society. Symmetrically, if agents exchange too much with the society, the risk effect dominates, i.e. in average, agents do not internalize that decreasing their risk level would reduce the overall risk of the society.

The condition specifying whether agents under-or over-invest in technology A is only related to the transfer matrix. For instance, in the case of homogenous own shares, the condition is simple, and it compares own shares with a diversification index. Noticeably, under the reasonable condition λ 0 > 1 2 , it turns out that λ 0 > ψ ii for all i. That is, agents under-invest with respect to societal view14 .

To illustrate, we consider the following transfer matrix. We assume λ ii = λ 0 for all i, λ ij = 1-λ 0 n-1 for all i, j = i. Then, own shares are homogenous, and social links are maximally diversified, so

ψ ii = ψ 0 = λ 2 0 + (1-λ 0 ) 2 n-1
for all i. An immediate observation is that the efficient profile of payoffs is homogenous, and thus Pareto-dominates the equilibrium profile. More, we note that λ 0 > ψ 0 iff λ 0 > 1 n . Applying the theorem, we deduce the following results. First, if λ 0 > 1 n , and for ψ 0 > F 1+F , which is probably the most realistic case, the equilibrium level of risk is lower than the average efficient level of risk. Second, if λ 0 = 1 n , the equilibrium risk profile coincides with the strong efficient one, which is a good news once we have remark 4 in mind.

Remark 4 Suppose that we let agents fix by themselves the values of the transfers without cost. Simple optimization induces that there is a unique equilibrium15 . At this equilibrium, it can be shown that all agents diversify their social links at maximum, i.e.

λ * ij = 1 n for all i, j. Further, it will be seen thereafter that this sharing-rule uniquely satisfies that equilibrium risk coincides with that maximizing social welfare.

Extension: cash transfers

Formal institutions can help developing villages by transferring Cash to households.

For Progresa program, [START_REF] Angelucci | Indirect effects of an aid program: how do cash transfers affect ineligibles consumption?[END_REF] document that part of this aid is transferred by informal arrangements through the network and therefore also affects the consumption of the non-treated 16 . Beyond consumption, and in direct filiation with the main hypotheses of this article, we address the issue of the impact of Cash transfer or wealth redistribution on the overall investments in technology A.

Suppose that an institution, interested in the promotion of technology A, uses wealth redistribution or cash transfer as a policy tool. For simplicity, we will assume that initial wealths are equal to 1 for all i 17 . A shock in wealths a = (a 1 , a 2 , • • • , a n ) generates the new vector of wealths Ω = (ω 1 , ω2 , • • • , ωn ) such that ωi = 1 + a i for all

i. Let z i = 1 • x i (resp. zi = ωi • x i ) denote the amount of wealth that agent i invests in technology A and Z = (z 1 , z 2 , • • • , z n ) (resp. Z = (z 1 , z2 , • • • , zn )) the associated
profile. We let Z * (resp. Z * ) denote the equilibrium associated with wealth profile Ω (resp. Ω).

The following proposition relates the variation of the part of wealths invested in technology A to the transpose of matrix Γ: Proposition 4 Suppose that Λ ∈ R. The variation in the sum of equilibrium amounts of wealths invested in technology A writes:

i z * i -z * i = i a i H λ ii + K B i (Γ T ; -F ) (9)
16 "Started in 1997 and still ongoing, Progresas aim is improving poor households education, health, and nutrition through sizeable cash transfers. In our sample of rural villages, more than half of the households are treated. The targeted villages are small and agriculture is the main, and often sole economic activity. The exposure to natural disasters, the absence of formal credit and insurance institutions, and extensive within-village kinship relationships create incentives to engage in informal risk sharing activities. If this is the case, treated households will share part of their higher income with members of their social network through gifts or loans. Therefore, the entire village will benefit from the program."

17 Extending the analysis to heterogenous initial wealths, results are qualitatively unchanged (results are available upon request).

The institution aiming at promoting technology A should therefore take into account the structure of the matrix Γ, and target certain households rather than others. As a first illustration, we present a case in which the identity of eligible households is not an issue. Suppose that own shares are homogenous, of value λ 0 :

Example 1: suppose that λ 0 ∈ F 1+F , 1 . Then variation in the sum of equilibrium amounts of wealths invested in technology A writes as a function of the shocks on wealths and the transfer matrix (proof in appendix):

i z * i -z * i = x * (λ 0 ) i a i
One direct implication is that the amount of cash transfer matters, not the identity of eligible agents. A second implication in that the magnitude of the impact of the wealth shock on investments in technology A is related to x * (λ 0 ), the equilibrium level of risk of a society with homogenous own shares of value λ 0 ; precisely, x * (λ 0 ) is the proportion of cash transfer that will be allocated to technology A.

Example 2: this example shows that, in general, risk sharing heterogeneity makes the task of targeting eligible households adequately not obvious. Suppose that Cash transfer is given to a unique agent i 0 ; i.e., a i 0 = 1 > 0 and a j = 0 for all j = i 0 . The induced variation in the sum of wealths invested in the risky technology writes:

H λ i 0 i 0 + K B i 0 (Γ T ; -F ) (10)
Hence, if one agent is given cash transfer, an institution that would aim at maximizing the investment in the (more) risky technology should give the transfer to the agent who maximizes expression (10). That agent needs not coincide with that investing the maximal amount at X * .

Conclusion

This paper analyzes a model of risk choice in the presence of heterogenous risk sharing and strategic interactions. To focus on pure risk sharing, our model considers agents with homogenous initial wealth, homogenous expected 'after transfer wealth', and same risk aversion.

Lemma 1 Consider three parameters

δ ∈]0, 1[, α ∈ R, β ∈ R, such that α 1-δ δ = β. Consider also a n × n row-stochastic matrix A = [a ij ] such that a ii > δ 1+δ for all i. Define E = [e ij ]
, with e ii = 0 for all i and e ij = a ij a ii for all i, j = i. The system of equations such that, for all i,

a ii x i + δ j =i a ij x j = α + βa ii (11)
admits a unique solution

x i = α δ -α 1 -δ δ -β B i (E; -δ) (12) with B i (E; -δ) ∈]0, 1[.
Proof of lemma 1. We consider the following transformation:

v i = 1 α 1-δ δ -β α δ -x i (13)
Equation ( 11) becomes:

a ii α F -α 1 -δ δ -β v i + δ j =i a ij α F -α 1 -δ δ -β v j = α + βa ii (14)
Dividing all terms by a ii , and taking account of j =i a ij = 1 -a ii , one obtains: 15) or in matrix form (I + δE)V = J. Since a ii > δ 1+δ for all i, the matrix I + δE is strictly diagonal-dominant. The preliminary result 1 applies with M = I + δE.

v i + δ j =i e ij v j = 1 (
Inverting the system, the solution writes as a Bonacich measure

v i = B i (E; -δ) with B(E; -δ) = ∞ k=0 (-δ) k E k J. Rearranging, B(E; -δ) = I(I -δE)J + (δE) 2 (I -δE)J + (δE) 4 (I -δE)J + • • • (16)
Factorizing, one obtains:

B(E; -δ) = ∞ k=0 δ 2k E 2k • (I -δE)J (17)
Notice that δ 2k [E 2k ] ij > 0 for all k, i, j. Further, if a ii > δ 1+δ for all i, the vector (I -δE)J > 0. Hence, B(E; -δ) > 0. More, note that a solution of (I + δE)Z = J also writes Z = J -δEZ. That is, if Z > 0, clearly Z < J.

Proof of theorem 1. The matrix with diagonal element λ ii and non diagonal elements F λ ij represents the system of first order conditions. We note that the system is invertible 'almost everywhere' in terms of the parameter F ; indeed, the determinant of the system is a polynomial expression of degree not higher than n in parameter F . Hence, fixing the matrix of transfers, there exists at most n values of parameter F ∈ [0, 1] for which the system is not invertible. If this matrix is diagonal-dominant (which is the condition on own shares given in the theorem), the system is invertible. This guarantees that the solution is unique, and more, Bonacich measures are well defined.

Since the transfer matrix is row-stochastic, the equation 2 becomes:

λ ii x * i + F j =i λ ij • x * j = H + Kλ ii (18)
Hence, we can apply lemma 1 with δ = F , α = H, β = K, A = Λ. Note that in this case α 1-δ δ > β by condition C 0 .

Proof of theorem 2. We use the following lemma:

Lemma 2 (adapted from Farkas's lemma) Let Q be an n × n matrix. If the equation Q T x = J admits a positive solution, then for all y ∈ R n such that Qy ≥ 0, we have i y i ≥ 0.

We let matrix Λ F denote the matrix with diagonal element λ ii and non-diagonal elements F λ ij . We will see that the conditions of the lemma apply if we fix Q = Λ F and y = x * -x * :

First, we prove that there exists a positive solution to (Λ F ) T x = J:

The condition writes:

λ ii x i + F j =i λ ji x j = 1 (19)
Since the matrix Λ is bi-stochastic, the matrix Λ T is row-stochastic. We can therefore apply lemma 1 with δ = F , α = 1, β = 0, A = Λ T , and we conclude that there is a positive solution to the system (Λ F ) T x = J.

Second, we see that Λ F (x * -x * ) ≥ 0:

Indeed, we observe that [Λ F x] * i = H + K(λ ii + θ ii ), and Λ F x * = Λ F x * -Θx * . Then,

[Λ F x * ] i = H + K(λ ii + θ ii ) -[Θ F x * ] i . Hence, [Λ F (x * -x * )] i = Kθ ii -(θ ii x * i + F j =i θ ij x * j ) (20)
Since the matrix Λ ∈ R, theorem 1 implies that H + K < x * i < H F for all i. Recalling that all θ ij > 0, x * i + F j =i

θ ij θ ii x * j > H + K + H j =i θ ij θ ii . Since j =i θ ij θ ii = -1, x * i + F j =i θ ij
θ ii x * j > K and we are done.

Third, we apply lemma 2 and conclude that i

(x -x * i ) ≥ 0 (21)
which proves the theorem.

Proof of proposition 2. Simple computation entails:

1 κ ∂W ∂x i = µ A -µ B κ + (σ 2 B -τ AB ) j λ 2 ji -(τ A + τ B -2τ AB ) j λ ji k =i λ jk x k +(τ B -τ AB ) j λ ji k =i λ jk -(σ 2 A + σ 2 B -2τ AB )x i j λ 2 ji (22)
That is, ∂W ∂x i = 0 if and only if

ψ ii xi + F j =i ψ ij xj = H + Kψ ii (23)
Since the matrix Λ is bi-stochastic, j =i ψ ij = 1 -ψ ii ; that is, the matrix Ψ is rowstochastic. We can therefore apply lemma 1 with δ = F , α = H, β = K, A = Ψ (and thus E = Φ). In particular, ψ ii > F 1+F for all i implies that the matrix I + F Φ is strictly diagonal-dominant.

Proof of proposition 4. From the first order condition given by equation (2), we obtain:

   λ ii z * i + F j =i λ ij z * j = H + Kλ ii λ ii z * i + F j =i λ ij z * j = (1 + a i ) H + Kλ ii (24)
Thus,

z * i -z * i + F j =i γ ij (z * j -z * j ) = a i H λ ii + K (25)
Defining vector e such that e i = a i λ ii , we find in matrix form:

(I + F Γ)( Z * -Z * ) = He + Ka (26)
Denoting T = (I + F Γ) -1 , and summing all terms of the vector, we find:

i z * i -z * i = H i j T ij a j λ jj + K i j T ij a j (27)
That is:

i z * i -z * i = H j a j λ jj i T ij + K j a j i T ij (28)
Remind that B i (Γ; -F ) = j T ij . Now, since Λ ∈ R, we can develop matrix T in its series:

B i (Γ T ; -F ) = j (I + F Γ T ) -1 ij = j ∞ k=0 F k (Γ T ij ) k = j ∞ k=0 F k ((Γ ij ) k ) T = j ∞ k=0 F k (Γ ij ) k T = j (I + F Γ) -1 ij T = j (I + F Γ) -1 ji = i T ij
That is,

B i (Γ T ; -F ) = i T ij (29)
and we are done.

Proof of the result in example 1. Equation (25) writes:

λ ii (z * i -z * i ) + F j =i λ ij (z * j -z * j ) = a i H + Kλ ii (30)
summing over all agents and taking account of the fact that the matrix of transfers is row-stochastic, one obtains:

F i z * i -z * i + (1 -F ) i λ ii z * i -z * i = H i a i + K i λ ii a i
Taking account that λ ii = λ 0 for all i, the corollary follows.

.

  is more revenue sharing in Λ than in Λ in the sense of the above definition. We find x * 1 Hence, theorem 2 does not necessarily hold in region R. Furthermore, those two cases are such that certain agents can take less risk than in isolation.More, where we allowing values of x i / ∈ [0, 1], what can be interpreted as short-selling, the average level of risk itself may be lower than the risk taken in isolation. To illustrate, suppose H = .1, K = 0, F = .9023 and consider Λ =

Of course, wealth inequality can generate different risk levels; first risk aversion is in general wealth dependent, and second poor agents may not be able to access all technologies. We study another mechanism producing differentiated risk levels.

This measure has been introduced in[START_REF] Bonacich | Power and centrality: a family of measures[END_REF].[START_REF] Ballester | Who'swho in networks. Wanted: the key player[END_REF] renewed the idea in the field of economics. Two recent papers link Bonacich centrality and optimal decisions: see[START_REF] Ghiglino | Keeping up with the neighbours: social interaction in a market economy[END_REF] for a model of a pure exchange economy with a positional good; see[START_REF] Corbo | The Importance of Network Topology in Local Contribution Games[END_REF] and[START_REF] Bramoullé | Strategic Substitutes and Networks[END_REF] for models of local public goods,[START_REF] Bloch | Pricing on networks[END_REF] for a model of oligopoly with local externalities among consumers. In these latter works, as well as in ours, actions are strategic substitutes.

This is for instance the case inBramoullé and Kranton (2006, 2007). Interestingly, this particular rule happens to be the solution of optimal decisions of costless link formation in our model (see remark

thereafter), and more, this solution is efficient.

The matrix of interaction is the matrix over which the Bonacich measure is defined. This matrix is a transformation of the initial matrix of bilateral cross effects. In our game, the matrix of cross effects is Λ and the matrix of interaction is Γ. Technically, the transformation of our game echoes that was introduced in[START_REF] Ballester | Who'swho in networks. Wanted: the key player[END_REF], remark 2 pp. 1409.

Suppose for instance that τ A is high and τ B low. Then, B may become more attractive and the equilibrium levels of risk of socialized agents are lower than in isolation.

Another originality of the model is that, due to covariances, the sign of the externality that agent i's risk level generates to agent j is related to the risk levels of all agents.

This result provides a possible explanation of the lack of investment in risky innovations in developing villages[START_REF] Valente | Network models of the diffusion of innovations[END_REF]).

 15 The proof is available upon request.

We first point out that solutions generally exist and are unique. In particular, when risk sharing is not too large, optimal risk levels are linearly related to the Bonacich measures of a slight transformation of the transfer matrix.

When risk sharing is not too large, the common view that more risk sharing entails more risk taking is preserved, although only in average. Further, simple statistics over the composition of transfers indicate whether agents over-or under-invest with regard to the efficient allocation. Last, the analysis has important policy implications. For instance, we characterize how to select adequately eligible households for cash transfer or which wealth redistributions is opportune, in order to increase investments in the more profitable/risky technology.

When risk sharing is large, i.e. when risk sharing is voluminous, simple examples illustrate that the combination of strategic interactions and heterogeneity can destroy all the results: the average risk level may be reduced by an increase of transfers, and more, it may be even lower than the risk chosen by an isolated agent.

It would be interesting to test the theoretical predictions of this simple model. Moreover, some lines of research should deserve attention: relating the formation of risk sharing rules to wealth and correlated risks, to the nature of risks (health vs incomes shocks), to the interplay between risk sharing and other activities aimed at insuring against volatility (savings, extra income earnings).

APPENDIX

for all i, j and n j=1 m ij = 1 for all i.

Preliminary result 1 If a n×n matrix M is strictly diagonal-dominant, the equation M Z = J admits a unique solution.


Anke M Brock
email: anke.brock@enac.fr

Céline Classe
email: celine.classe@sii.fr

Laura Duphil
email: laura.duphil@univ-tlse3.fr

Florent Fabas
email: ffabas@sii.fr

Liam Guigo
email: liam.guigo@alumni.enac.fr

Thomas Holstein
email: thomas.holstein@sii.fr

Christophe Lounis
email: christophe.lounis@sii.fr

Ilyes Reguig
email: ilyes.reguig@univ-tlse3.fr

Caroline Senaux
email: caroline.senaux@alumni.enac.fr

Making ATIS accessible for pilots who are deaf or hard of hearing

INTRODUCTION

Deaf and Hard of Hearing (DHH) pilots can easily fly in an uncontrolled space, where radio use is not required to communicate with air traffic control (ATCO). However, DHH pilots generally cannot fly without any assistance in controlled airspace, where radio use is required. An important service for pilots in general aviation is ATIS (Automatic Terminal Information Service), a vocal message containing essential information, such as weather data, active runways, available approaches and any other information needed by pilots. Pilots usually listen to ATIS before contacting control, which reduces controller workload and decreases frequency occupancy. However, since this is an audio-based service it is currently not accessible to DHH pilots. D-ATIS (datalink) allows transmitting written information, but it is currently only used by large airports. As a result, alternative communication methods between DHH pilots and air traffic controllers have been developed. Current communication methods that DHH pilots use are light gun signals, a tool used by ATCO to communicate with aircraft during communications malfunctions. These lights emit different colored beams and can be flashed or steady with different meanings for aircraft in flight or on the ground. A second method depends on a hearing copilot (radio copilot) onboard to communicate with ATCO [START_REF] Major | Able flight at purdue university: Case studies of flight training strategies to accommodate student pilots with disabilities[END_REF] or listen to ATIS, who then transmits the audio information to the flying pilot by writing on a whiteboard. These methods are not always possible in the controlled airspace of a large, crowded airport and thus present the primary barrier for DHH pilots in pursing an Airline Transport Pilot job or a non-commercial activity [START_REF] Tinio | Perceiving the Communication Methods between Deaf Pilots and Air Traffic Control[END_REF]. The FANS4all association (Future Air Navigation System for All, https://fans4all.org/) aims at making it possible for DHH pilots to fly in controlled airspaces. One challenge is the accessibility of ATIS for DHH pilots. In this paper, we focus on the work conducted to make ATIS more accessible on the side of the user interface (i.e. the information presented to the DHH pilot).

RELATED WORK Technology to support communication of DHH people (outside aeronautics)

DHH people are largely excluded from communication with hearing peers. In the area of Assistive Technology, many researchers have investigated how to support the communication of DHH users through technology. Deaf people largely rely on the use of sign language. One way to support this through technology is using sign language datasets [START_REF] Bragg | The fate landscape of sign language ai datasets: An interdisciplinary perspective[END_REF]). An alternative is the generation of sign language avatars [START_REF] Brock | Learning three-dimensional skeleton data from sign language video[END_REF]. Moreover, interactive systems allow presenting environmental sounds to DHH users through visual or tactile modalities often on mobile devices [START_REF] Jain | Homesound: An iterative field deployment of an in-home sound awareness system for deaf or hard of hearing users[END_REF]. For our work we conclude that the tactile sens can easily be used for alarms, but it is more convenient to provide detailed information (such as names, or numbers) using the visual modality as text or symbols. Moreover, mobile and wearable solutions have proven to be interesting for DHH users.

Making aeronautics more inclusive

In aeronautics, some studies have investigated the possibility of using technology to allow sensory impaired people to fly [START_REF] Valéry | Hear me Flying! Does Visual Impairment Improve Auditory Display Usability during a Simulated Flight?[END_REF]. Blind pilots currently use a sonification system, the soundflyer, which sonifies two dimensions of the aircraft attitude, i.e. pitch and bank angles. The soundflyer sonification consists in modulating the features (i.e. frequency, rhythm, inter-aural balance) of a sinusoidal pure tone which is continuously displayed to the pilot via his headphones [START_REF] Valéry | Can an aircraft be piloted via sonification with an acceptable attentional cost? a comparison of blind and sighted pilots[END_REF]. In contrast to these technological advances for blind pilots, no system is used for DHH pilots to the best of our knowledge. A promising avenue is the implementation of multimodal cockpit assistants [START_REF] Lounis | Flight eye tracking assistant (feta): Proof of concept[END_REF] or mobile assistants [START_REF] Simon | Application mobile d'aide à la navigation aérienne : une perspective de conception participative[END_REF] to support pilots in difficult situations.

DESIGN PROCESS

To make ATIS accessible to DHH pilots, several aspects need to be addressed. In the current project, we focus on the user interface, i.e. how to display the information that is currently presented using audition with a different sensory modality. To design a system that meets the users' needs, in our case DHH pilots of General Aviation, we applied a user-centered design approach involving users at every stage of the design process to ensure satisfying their needs. This approach is in line with prior projects on mobile assistants for General Aviation pilots [START_REF] Simon | Application mobile d'aide à la navigation aérienne : une perspective de conception participative[END_REF]. Due to restrictions during the COVID crisis, we had to adapt the process to comply with the local instructions and some steps had to be conducted online. Since there are few DHH pilots in France (and none in Toulouse) we extended our user population to hearing pilots. 16 pilots (including 3 who were DHH) participated in our project. Our design process consisted in four different phases: exploration, ideation, elaboration and evaluation.

Exploration Phase

The goal of the exploration phase is to better understand the context and user needs. As a first step, a workshop with DHH pilots and designers allowed to better understand the current situation and problems of DHH pilots. Following this workshop, the objective of the project was defined as designing and developing an accessible ATIS system that 1) delivers the complete information currently delivered via radio / phone, 2) has a comparable or shorter acquisition time as ATIS via radio / phone, 3) has a comparable error rate as ATIS via radio / phone, 4) is usable during all flight phases, 5) notifies pilots about changes (update of ATIS) and 6) allows to select a new airport with a comparable or better time than ATIS via radio / phone. There was no constraint regarding the device to be used. A second step consisted in an observation flight with two hearing pilots. Since the audio quality of ATIS was bad, we concluded that a written version of ATIS might even be of interest for hearing pilots. Third, we conducted interviews with 14 pilots (including one DHH pilot). This allowed us to prepare 17 scenarios which involved the use of ATIS and possible challenges, such as accessing the ATIS before the flight, accessing ATIS of an airport that was not initially on the flight plan during the flight, equipment malfunctioning, forgetting to consult ATIS before contacting ATCO, forgetting the frequency, etc.

Ideation Phase

The goal of the ideation phase is the creation of ideas based on the observations from the exploration phase. We conducted two brainstormings. First, an on-site brainstorming with 10 students from ENAC who were either studying to become pilots or aeronautical engineers, and second an on-line brainstorming with one DHH pilot and 6 pilot students. In both brainstorming sessions we asked the following two questions: 1) How should ATIS information be visualized ? and 2) How should ATIS be selected ? At the end of the on-site brainstorming session, we asked participants to sketch ideas for an accessible ATIS system. Ideas for different types of devices were proposed, such as printing paper strips or using a mobile device (see Figure 1 for examples).

Elaboration Phase

During the elaboration phase, we followed an iterative design process, allowing us to converge from initial paper-based prototypes to a functional prototype on an Android tablet. Initially, we explored designs for four different device types: mobile phone, tablet, smartwatch and an embedded cockpit system. Videos of the prototypes were presented to ten users online and asynchronously. We analyzed and prioritized the collected feedback. Based on this feedback we made the following design decisions. First, we decided to focus on the design of an application for a tablet. Indeed, it presents the advantage over an embedded system that no modifications to the cockpit are required. Moreover, the display size is more adapted for visualizing ATIS information than the smaller size of phone or smartwatch. Finally, pilots are already used to carry on tablets while flying [START_REF] Simon | Application mobile d'aide à la navigation aérienne : une perspective de conception participative[END_REF]. Second, we learned that a radar view should be included. Finally, we noticed that a few symbols needed to be improved. Figure 2 shows the different iterations of the mobile app interface. Prototype 1 was presented on-site to four pilots in a TB20 simulator and to two pilots in a wooden cockpit simulator. Following the feedback from the users, modifications were made in Prototype 2 such as including automatic update of ATIS and displaying the version number through letters A,B,C,D etc.

Prototype 2 (Adobe XD) was presented to six users online in an experience map. This allowed to choose interaction techniques such as the pie menu for the radar view, a virtual keyboard with enlarged keys and gesture recognition for the input of OACI code as well as visual notifications. Prototype 3 (Adobe XD) was presented online and asynchronous as videos to four pilots (including one DHH pilot). Feedback was collected through the use of a questionnaire. It allowed us to identify modifications that were considered in the final prototype for instance regarding choice of colors and symbols. Prototype 4 was implemented in React Native on Samsung Galaxy Tabs S6 Lite with Android OS. It made use of the GPS position of the tablet and touch recognition. ATIS update was handled using HTTP communication. The interface was slightly redefined based on the user feedback in previous steps, but also due to constraints due to moving from and Adobe XD prototype to an Android implementation.

Pilot Study

We evaluated the prototype in a preliminary study with four pilots, including one DHH pilot. All pilots tested the application in a Flight Simulator with Microsoft Flight Simulator 2020 and then replied to a SUS questionnaire [START_REF] Brooke | SUS: A "quick and dirty" usability scale[END_REF]. The three hearing pilots and an additional fourth pilot participated in a comparison between audio-based ATIS and our mobile app. We also collected qualitative feedback. DHH and hearing participants appreciated the application and considered it useful. The SUS scores were above 75 except for the first participant who encountered many technical problems during his test. According to [START_REF] Bangor | An Empirical Evaluation of the System Usability Scale[END_REF], SUS values above 75 correspond to a good, respectively excellent usability. The comparison of audio-based ATIS and our application showed that the mobile app was quicker to use (mean gain of time: 13s). However, SUS and time scores would need to be verified with a larger number of participants. Qualitative feedback allowed us to identify aspects to further improve the application in the future. For instance, some symbols need to be improved to facilitate comprehension. Moreover, users requested to add direct interaction, such as touching and moving elements or zooming with pinch gestures. This is in line with interaction that is already used in mainstream tablet applications. However, direct manipulation might be difficult during turbulences and this would need to be studied further. Pilots also appreciated that the mobile app included features beyond ATIS (such as wind information and a radar view). It inspired them to aim even larger for a more holistic application that includes features beyond ATIS, similar as for instance FlyMate [START_REF] Simon | Application mobile d'aide à la navigation aérienne : une perspective de conception participative[END_REF].

DISCUSSION

This project is a first step towards a more accessible ATIS developed through a user-centered design process with DHH and hearing pilots. However, the current project only addresses the user interface and some important challenges such as automatic speech recognition from radio-based ATIS still need to be addressed. Since, there are currently few DHH pilots in France (and none in Toulouse) this work has partially been conducted with sighted pilots. Beyond involving them as substitute users, we believe that an ATIS mobile app might be interesting for them as well since it allows including supplementary features and overcoming low audio-quality of ATIS.

CONCLUSION AND FUTURE WORK

In this paper we present the user-centered design, implementation and preliminary evaluation of an accessible tablet-based ATIS application. This project is part of a bigger project, involving Fans4All association, SII and researchers from Universities Rennes, Tarbes and ENAC, and we will continue the work on making piloting more accessible to DHH pilots. In the future, it would be interesting to conduct a controlled user-study with additional participants in a flight simulator or an aircraft. Moreover, it would be necessary to work on the certification and regulations to make this usable in the cockpit. We hope that this work will inspire other researchers and designers to work towards making aeronautics more accessible for people with impairments.

Figure 1 :

 1 Figure 1: Examples of sketches from the brainstorming session from left to right: 1) strip-based interface, 2) sketch of the terrain, 3) mobile app interface.

Figure 2 :

 2 Figure 2: Iterations of the mobile app interface. Prototypes 2 to 4 are composed by two views: left: flight preparation, right: ATIS consultation.

ACKNOWLEDGMENTS

First, we would like to thank all DHH and hearing pilots who participated in the project. We are grateful for the support by Fans4All association and SII. This project was also conducted during the M2IHM in Toulouse, and we would like to thank Sylvain Pauchet and Nicolas Saporito for their helpful feedback.


Antoine N André
email: antoine.andre@u-picardie.fr

Guillaume Caron
email: guillaume.caron@cnrs.fr

Photometric Visual Gyroscope for Full-View Spherical Camera

This article presents a visual gyroscope based on the spherical representation of dual-fisheye cameras. By using the fully available view of the scene thanks to a dual-fisheye camera and projecting the photometric information on a sphere, a highly precise orientation estimation along with a great convergence domain can be achieved. This method is validated with the help of images taken from the PanoraMIS dataset to evaluate quantitatively the performances.

Introduction

A Visual Gyroscope (VG) estimates the 3D orientation of a camera from a captured image with respect to landmarks in structured environments [START_REF] Li | Quasi-globally optimal and near/true real-time vanishing point estimation in manhattan world[END_REF] or to a reference image [START_REF] Hadj-Abdelkader | Closed form solution for rotation estimation using photometric spherical moments[END_REF]. This classical estimation problem is the heart of many computer vision applications, for example upright correction of spherical panoramas [START_REF] Li | Quasi-globally optimal and near/true real-time vanishing point estimation in manhattan world[END_REF], stabilization of 360 degrees (deg) videos [START_REF] Kopf | 360°video stabilization[END_REF] and robot motion estimation [START_REF] Caron | Spherical Visual Gyroscope for Autonomous Robots using the Mixture of Photometric Potentials[END_REF].

VG algorithms consider two kinds of information, direct or indirect. An Indirect VG (IVG) leverages image features, either handcrafted as patches around image points [START_REF] Hadj-Abdelkader | Closed form solution for rotation estimation using photometric spherical moments[END_REF][START_REF] Kopf | 360°video stabilization[END_REF] or learnt to estimate the optical flow [START_REF] Kim | Self-supervised optical flow derotation network for rotation estimation of a spherical camera[END_REF]. Instead, a Direct VG (DVG) almost considers directly pixel brightness of the whole image as input of a 3D rotation optimization method [START_REF] Makadia | Rotation recovery from spherical images without correspondences[END_REF]. DVGs almost consider directly pixel brightness because the latter usually undergo a transform to another space before estimating the 3D rotation, e.g. a spherical Fourier transform to solve for the rotation with phase correlation [START_REF] Makadia | Rotation recovery from spherical images without correspondences[END_REF] or a Mixture of Photometric Potentials * G. Caron is also with Université de Picardie Jules Verne, MIS, France. This work is carried out as part of the Interreg VA FCE ADAPT project "Assistive Devices for empowering disAbled People through robotic Technologies" (adapt-project.com). The Interreg FCE Programme is a European Territorial Cooperation programme that aims to fund high quality cooperation projects in the Channel border region between France and England. The Programme is funded by the European Regional Development Fund. (MPP) transform to optimize for the rotation minimizing the Sum-of-Squared-Differences between two MPPs [START_REF] Caron | Spherical Visual Gyroscope for Autonomous Robots using the Mixture of Photometric Potentials[END_REF]. Despite the absence of features, DVGs could recently reach very large estimation domains such as up to 360 deg in static environment [START_REF] Caron | Spherical Visual Gyroscope for Autonomous Robots using the Mixture of Photometric Potentials[END_REF]. But the rotation estimation error is still of the order of 1 deg (for real-time or close to real-time setups) whereas Neural Networks could recently reach 0.3 deg [START_REF] Kim | Self-supervised optical flow derotation network for rotation estimation of a spherical camera[END_REF], though within a much tighter estimation domain of [-5; 5] deg around each axis. This error level is still problematic as it gets to be accumulated over time within 360 deg video stabilization [START_REF] Kopf | 360°video stabilization[END_REF] or visual odometry [START_REF] Pathak | Distortion-resistant spherical visual odometry for UAV-based bridge inspection[END_REF].

We believe the rather low accuracy of state-of-the-art real-time DVGs is due to the transformation of the pixelic information. Hence, this article proposes a new DVG to improve the rotation estimation accuracy by directly considering pixel brightness of captured images lifted on a sphere, named PVG (Photometric Visual Gyroscope). PVG implements a new hybrid multi-resolution/sub-sampling scheme in two levels allowing to decrease significantly the best known estimation errors [START_REF] Kim | Self-supervised optical flow derotation network for rotation estimation of a spherical camera[END_REF] together with a much larger convergence domain, as assessed by the evaluation made on many images available in the PanoraMIS dataset [START_REF] Benseddik | Panoramis: An ultra-wide field of view image dataset for vision-based robot-motion estimation[END_REF].

The rest of the paper presents the PVG algorithm (Sec. 2) and the evaluation of its linearity and accuracy results compared to the state-of-the-art (Sec. 3) before conclusion.

Spherical Photometric Visual Gyroscope

This section presents first the method to project dualfisheye images on a sphere (Cartesian representation). Then, the PVG's optimization algorithm is detailed.

Spherical representation of dual-fisheye images

Dual-fisheye cameras can easily be modeled using a unified central projection model involving two main steps that are (i) the projection on a unit sphere and (ii) a perspective projection on the image plane [START_REF] Geyer | A unifying theory for central panoramic systems and practical implications[END_REF]. The process used to calibrate and combine the two dual-fisheye cameras have already been described in [START_REF] Benseddik | Panoramis: An ultra-wide field of view image dataset for vision-based robot-motion estimation[END_REF]. Once the calibration of the two cameras have been done, the two parts of the dual-fisheye images are then projected on a single sphere.

In order to do this, an isocahedron (20 equilateral triangles and 12 vertices) is firstly considered, before being subdivided n times. The unit norm points

c X Si = [X Si , Y Si , Z Si ] ⊤ ∈ R 3 , i ∈ [1, P ]
resulting from this subdivision are uniformly spaced on the whole sphere surface [START_REF] Li | A full-view spherical image format[END_REF] (partially presented in Fig. 2).

Alignment method based on pixelic information

Consider I S ( c X S ) ∈ [[0, 255]], the spherical image brightness 1 in the spherical image I S at the orientation r ∈ R 3 (i omitted for compactness). The input of the PVG is the stacking of all spherical brightness as vector I S (r) ∈ [[0, 255]] P ×1 :

I S (r) = [. . . , I S ( c X S ), . . .] ⊤ . (1)
Vector r represents the camera 3D rotation with axis w = [w X , w Y , w Z ] ⊤ ∈ R 3 : ||w|| = 1 and angle θ ∈ R.

The brightness vector I * S is built from the desired image I * S , as I S (r) is built with (1) from I S . PVG is designed to minimize the Sum of Squared Differences (SSD):

C (p) = 1 2 ||I S (r) -I * S || 2 , (2)
within a Levenberg-Marquardt optimization method computing iteratively the rotation increment δr ∈ R 3 from the spherical image Jacobian L I S ∈ R P ×3 :

δr (it) = -(H + µ diag (H)) -1 L I S ⊤ (I S (r) -I * S ), (3)
with H = L I S ⊤ L I S , at each iteration it until convergence. In (3), µ ∈ R * + is the damping parameter. L I S links the variations of spherical image brightness to the 3 camera pose DoF, thanks to the Optical Flow Constraint Equation (OFCE), valid in Lambertian scenes [START_REF] Horn | Determining optical flow[END_REF]. L I S is the stacking of interaction matrices L I S ( c X S ) evaluated for all c X S , each composing spherical image gradients 1 Any other quantization could be considered.

x 0 x x N c X Si X SN

Image plane

Sampled sphere Figure 2. Spherical representation of the dual-fisheye camera: partial view of the subdivided isocahedron, a pixel projected on the sphere (blue), its neighbors along the x direction (red), backprojection of the neighbors to the image plane (pink).

∇I Sc X S ∈ [[-255, 255]] 3 to the geometric interaction ma- trix Lc X S = [ c X S ] × (skew-symmetric matrix of c X S [3]): L I S ( c X S ) = -∇I ⊤ Sc X S Lc X S . (4)

Spherical image gradient computation

In order to approximate as closely as possible the gradient directly on the sphere, the three derivatives of ∇I Sc X S (i.e. ∂I S ∂X S , ∂I S ∂Y S and ∂I S ∂Z S ) are computed using finite differences. The sampling used for this finite difference gradient computation is defined from the image plane to sphere projection pr -1 ξ () (see [START_REF] Caron | Photometric visual servoing for omnidirectional cameras[END_REF] for the detailed projection function and intrinsic parameters) of the size of one pixel at the principal point of coordinates u 0 and v 0 , as follow:

∆ X S = ∆ Y S = ∆ Z S = pr -1 ξ   u 0 + 1 v 0 1   -   0 0 1   (5)
Then, for each spherical pixel, three neighborhoods are defined, one for each axis. Each of these N -neighbors (with N = 6 typically) are calculated accordingly to the following equation that presents the case of the neighbors computation along the sphere abscissa axis:

X SN =    (XS+k∆X S , Y S , Z S ) T ∥(X S +k∆ X S , Y S , Z S ) T ∥ -N 2 ≤ k ≤ N 2 , k ̸ = 0    (6)
The same procedure is then applied similarly for the neighbors along the two others directions Y SN and Z SN . With this neighbors computation, a fine cartesian spherical gradient evaluation of the images can be done, completing the PVG optimization expression presented previously.

Hybrid visual alignment method

As PVG uses pixels lifted to the equivalent sphere, the image resolution should match the isocahedron subdivision level. For n subdivisions levels, P = 1 2 (20 • 4 n ) + 2 points form the sphere. Input images usually don't match the number of spherical points by a large factor (see Tab. 1). However, by considering the full image resolution, better accuracy can be achieved, thanks to an oversampling of image gradients, whereas the convergence domain is smaller (as assessed by experiments, Sec. 3). Inversely, re-sampling the image accordingly to the (lower) number of spherical pixels (see Tab. 1, 2nd row) leads to a larger convergence domain, though for a poorer accuracy.

To combine both the adequate spherical sampling and the high accuracy granted by the super sampling, PVG implements a hybrid method: first, PVG runs with re-sampled imaged, then it uses the original images (i.e. oversampled).

Performance of the measure

The method has been implemented leveraging the dualfisheye camera model and tools of the libPeR base C++ library 2 to study PVG's performance. In order to do so, the publicly available dataset PanoraMIS 3 has been used [START_REF] Benseddik | Panoramis: An ultra-wide field of view image dataset for vision-based robot-motion estimation[END_REF]. presented experiments use the third sequence (Sequence 3) that contains images of multiple 360 deg pure rotation around a single axis with a rotation increment of 2.5 deg between successive images. This acquisition is repeated five times at different locations (named CP i for i ∈ [0, 4]), thus giving a set of 720 images (144 images per full rotation). PVG's results are also compared with the prior visual gyroscope leveraging the MPP of spherical images (Sec. 1). In all the experiments n = 5 subdivision levels for the sphere, and the damping parameter (3) is set as µ = 0.01.

Accuracy

The first experiment evaluates PVG's accuracy (closeness of agreement between a test result and the accepted reference value). For this, the orientation between each successive pair of images I k and I k+1 is estimated. All successive rotation ground truths are 2.5 deg plotted as the black line in Figure 3 that shows a high accuracy with an average estimation error of 0.0981 deg (std: 0.0442 deg), when considering the full image. Tab. 2 shows the contribution in accuracy of PVG over the MPP-based gyroscope [START_REF] Caron | Spherical Visual Gyroscope for Autonomous Robots using the Mixture of Photometric Potentials[END_REF].

However, when considering images with the same number of pixels as the number of points sampling the sphere, 2 https://github.com/PerceptionRobotique/libPeR_base 3 https://mis.u-picardie.fr/ ˜panoramis the accuracy decreases to reach an average error of 0.79 deg. This result confirms the interest in considering full resolution images to achieve the best accuracy.

Linearity and convergence domain

To avoid being in a particular case when studying the convergence domain of the method, the experiments are using the whole set of images, giving a more accurate experimental evaluation of the convergence width. As the dataset contains full rotations around a single axis at five different locations, the experiment is split in five sub-studies, one for each considered location of the camera. For each location, the image at mid-rotation (180 deg) is set the reference image and, then, PVG aligns its spherical representation to each image in the [-180, 180] deg range.

To highlight the interest of the hybrid alignment (Sec. 2.4), the experiment is split in two parts, in order to study the convergence domains when either, first, considering re-sampled images or, second, full images. Fig. 4 presents the aggregation of the five sub-experiments to provide a global study of the convergence domain width. On the other hand, Fig. 5 presents the results of linearity when full resolution images are considered As a summary, if Fig. 4 highlights a loss in accuracy (with an error level of 0.79 deg) compared to using the full images (especially when considering small angles that are not detected), it is nothing but the counterpart of a ±45 deg convergence domain, much larger than the ±12.5 deg convergence domain when using full images (Fig. 5).

This convergence domain can be summarized with the representation of the cost function [START_REF] Caron | Photometric visual servoing for omnidirectional cameras[END_REF]. Fig. 6a shows it together with the domain where the gradient descent-like method of 3 is results in an accurate orientation estimation. These results show the main interest in considering the hybrid PVG to benefit a greater convergence domain and a high accuracy. Though the 90 deg convergence domain width of the hybrid PVG is 4 times narrower than the domain of the MPP-based gyroscope, the hybrid PVG increases 37 times the estimated rotation accuracy.

to translation

the characteristics of PVG known in the of pure rotation, a final experiment studies the robustness to translation. Fig. 6b presents the results of hybrid PVG to align the reference image of the first location CP 0 with images at any location within the demonstrated convergence domain. PVG's accuracy decreases as the distance to CP 0 increases, with a mean error of 2.57 deg overall. This error increase, compared to considering images of pure rotations only, shows the sensitivity of PVG to translation, however still more accurate than the MPP-based gyroscope in the presence of pure rotation (Tab. 2).

Conclusion

This article presented a Photometric Visual Gyroscope based on dual-fisheye images. By projecting the captured images and their photometric information on a sphere, the method presents a high level of accuracy of up to 0.0981 deg in a ±45 deg convergence domain. To achieve these results, this article presented a method to account for the image resolution, before its projection on the sphere, enhancing its convergence domain. Future works target the validation of the method by extending the gyroscope to the servoing of a robot arm embedding a dual-fisheye camera.

Figure 1 .

 1 Figure 1. (a) Dual-fisheye image taken from the PanoraMIS dataset and used as full-view spherical data for the PVG. (b) Dualfisheye camera used for the data acquisition.

Figure 3 .

 3 Figure 3. Accuracy test carried over the whole images data-set, with a ground truth of 2.5 deg between successive images.

(a) Resulting mean convergence and linearityFigure 4 .

 4 Figure 4. Convergence and linearity study in case of images resampled before estimating their orientations once aggregated to provide a global conclusion.

Figure 5 .

 5 Figure 5. Linearity test carried with full resolution images.

  Figure 6. (a) Cost function evaluation between a reference image and images in the [-90, 90] deg range: convergence domain with (1) re-sampled images and with (2) full resolution images. (b) Mean and std orientation estimation error, function of 5 locations.

Table 1 .

 1 Re-sampling table of images to match the number of spherical pixels depending on the icosahedron subdivision level.

	Level	of	Nb. of fea-	Scale factor	Resulting	New Nb. of
	subdiv. n		tures P	s	size	pixels
	Full image	-	1	1280 × 720 921600
	5		10242	9.48	135 × 76	10260

Table 2 .

 2

	Used method	PVG	MPP
	Accuracy (deg)	0.0981	3.15
	Convergence domain (deg)	±45	±180
	Computing time (ms)	883.49 57160.00

PVG (hybrid) versus MPP-based gyroscope

[START_REF] Caron | Spherical Visual Gyroscope for Autonomous Robots using the Mixture of Photometric Potentials[END_REF]

, both applied to the pure rotation images of the PanoraMIS dataset.