I. INTRODUCTION

Thermo-acoustic instabilities are acoustic oscillations that arise when a flame is coupled to an acoustic mode of a combustion chamber. They are highly debated in the combustion community because of the dramatic consequences they can have on industrial burners [START_REF] Timothy | Combustion Instabilities in Gas Turbine Engines: Operational Experience, Fundamental Mechanisms, and Modeling[END_REF], rocket engines [START_REF] Dj | Liquid Propellant Rocket Instability[END_REF], or ramjet engines [START_REF] Yu | Low-frequency pressure oscillations in a model ramjet combustor[END_REF] to name a few. In order to control these instabilities, it is of prime importance to understand the diverse phenomena at play [START_REF] Ducruix | Combustion Dynamics and Instabilities: Elementary Coupling and Driving Mechanisms[END_REF][START_REF] Schadow | Combustion instability related to vortex shedding in dump combustors and their passive control[END_REF][START_REF] Mcmanus | A review of active control of combustion instabilities[END_REF] which participate to the necessary condition summed up in the Rayleigh [7] criterion: pressure and heat release variations have to be in phase for the acoustic instability to develop. For propagating flames, a canonical experiment was designed in vertical tubes [START_REF] Searby | A parametric acoustic instability in premixed flames[END_REF] and highlights the namely first and secondary thermo-acoustic instabilities, that have an influence on the flame topology in addition to the emission of acoustic waves. During the primary instability, an increasing acoustic field flattens Darrieus-Landau self-wrinkled flames by parametric restabilization. During the secondary instability, the more intense acoustic field generates new small scale wrinkles through a Faraday like mechanism: the parametric resonance. In both cases, the fundamental acoustic mode of the tube is the most likely excited (i.e. with a wavelength four times larger than the tube length). The interaction mechanism between acoustic waves and flame dynamics originally proposed by Markstein [START_REF] Markstein | Interaction of Flow Pulsations and Flame Propagation[END_REF]10] has been analytically [START_REF] Pelcé | Vibratory instability of cellular flames propagating in tubes[END_REF][START_REF] Bychkov | Analytical scalings for flame interaction with sound waves[END_REF], numerically [START_REF] Denet | Numerical Study of Premixed Flames Parametric Acoustic Instability[END_REF][START_REF] Yáñez | The acoustic-parametric instability for hydrogen-air mixtures[END_REF], and experimentally [START_REF] Baillot | Parametric Response of a Conical Flame to Acoustic Waves[END_REF][START_REF] Aldredge | Experimental evaluation of Markstein-number influence on thermoacoustic instability[END_REF][START_REF] Yanez | Flame instability of lean hydrogen-air mixtures in a smooth openended vertical channel[END_REF] studied. A main outcome is that acoustic instabilities are more likely to emerge when the Markstein number is low [START_REF] Searby | A parametric acoustic instability in premixed flames[END_REF], (e.g rich propane-air flame [START_REF] Al Sarraf | Darrieus-Landau instability and Markstein numbers of premixed flames in a Hele-Shaw cell[END_REF]). Controlling the primary instability in tubes, by using an acoustic source (loudspeaker) or an acoustic damper (sub-millimeter aperture) at the closed end of the tube, allowed to study the Darrieus-Landau intrinsic wrinkling [START_REF] Darrieus | Propagation d'un front de flamme[END_REF][START_REF] Landau | On the theory of slow combustion[END_REF] in the linear [START_REF] Clanet | First experimental study of the Darrieus-Landau instability[END_REF] and non linear [START_REF] Almarcha | Premixed flames propagating freely in tubes[END_REF] regimes.

More recently, accurate studies of the Darrieus-Landau wrinkling have been performed in an experimental device which allows to reduce the flame dynamics to a quasi-bidimensionnal one [START_REF] Al Sarraf | Darrieus-Landau instability and Markstein numbers of premixed flames in a Hele-Shaw cell[END_REF][START_REF] Almarcha | Experimental two dimensional cellular flames[END_REF][START_REF] Fernández-Galisteo | Analysis of premixed flame propagation between two closely-spaced parallel plates[END_REF][START_REF] Al Sarraf | Quantitative Analysis of Flame Instabilities in a Hele-Shaw Burner[END_REF][START_REF] Hye | Unsteady propagation of premixed methane/propane flames in a mesoscale disk burner of variable-gaps[END_REF][START_REF] Alexeev | Experimental Study on Cellular Premixed Propane Flames in a Narrow Gap between Parallel Plates[END_REF][START_REF] Almarcha | Interface dynamics, pole trajectories, and cell size statistics[END_REF][START_REF] Sharif | Premixed-gas flame propagation in Hele-Shaw cells[END_REF]. The two dimensional dynamics has been shown to be similar to that of freely propagating flames [START_REF] Al Sarraf | Darrieus-Landau instability and Markstein numbers of premixed flames in a Hele-Shaw cell[END_REF]. In particular, the front corrugations undergo a complex motion causing permanent fluctuations of the total flame area. Due to the induced fluctuating heat release, one would expect the emergence of acoustic instabilities involving the acoustic modes of the burner in the same way as it does with the tubes. Surprisingly, acoustic instabilities were reported in the experiments solely when the gap was large enough (more than 7mm), or for glass plates sufficiently thick (19mm) [START_REF] Aldredge | Experimental evaluation of Markstein-number influence on thermoacoustic instability[END_REF][START_REF] Martinez-Ruiz | Vessel confinement contributions to thermoacoustic instabilities of premixed flames[END_REF][START_REF] Veiga-López | Experimental analysis of oscillatory premixed flames in a hele-shaw cell propagating towards a closed end[END_REF]. For smaller gap or smaller thickness of the glass plates, no oscillating instability was obtained, one possible reason being that viscous losses and acoustic loss at walls are too large to allow pressure fluctuations to excite the acoustic mode of the burner [START_REF] Aldredge | Saffman-taylor Influence On Flame Propagation In Thermoacoustically Excited Flow[END_REF]. However, in most of the Hele-Shaw burners, some transient pseudo-periodic oscillations of the flame speed have been observed just after the ignition of the flame of the flame front (see Alexeev et al. [START_REF] Alexeev | Experimental Study on Cellular Premixed Propane Flames in a Narrow Gap between Parallel Plates[END_REF] Fig. 3 or Jang et al. [START_REF] Hye | Unsteady propagation of premixed methane/propane flames in a mesoscale disk burner of variable-gaps[END_REF] Fig. 3 or Al Sarraf et al. [START_REF] Al Sarraf | Darrieus-Landau instability and Markstein numbers of premixed flames in a Hele-Shaw cell[END_REF] supplementary material Fig. S2). As remarked by Alexeev et al. [START_REF] Alexeev | Experimental Study on Cellular Premixed Propane Flames in a Narrow Gap between Parallel Plates[END_REF], these oscillations are not corresponding to the fundamental acoustic mode that is usually observed in tubes, indicating that the coupling with the burner seems to be different.

The purpose of the present paper is to investigate the emergence of velocity and pressure oscillations in Hele-Shaw burners. This phenomenon is acting on the wrinkling of flames and change this way their whole dynamics. We demonstrate that the thermo-acoustic instability usually observed in tubes is limited and also that it can be overcome by another new oscillating instability that emerges from a coupling with the structural modes of the burner walls. In §II we introduce the apparatus used to tackle this problem. In §III, we report the diverse oscillating instabilities appearing in the diverse configurations of the Hele-Shaw cell. In §IV, the structural modes of the burner are studied both analytically and experimentally, and it is shown that these modes are excited during the flame propagation. In §V we show that flow oscillations can be induced inside the burner by forcing these structural modes. These oscillations act on the flame in a similar fashion than in the classical thermo-acoustic scenario. Finally, a summary of the results and a short discussion about their potential applications is given in §VI.

II. EXPERIMENTAL APPARATUS

The Hele-Shaw burner studied here is composed of two plates, one 19mm thick glass plate and another plate chosen in a set of plates made of either glass or PPMA, with thickness between 5mm and 19mm. The two plates are 500mm large and 1500 mm high, separated by a 5mm gap. By taking a different thickness or material for each plate, we expect to reduce the coupling between the plates, and keep the thicker one static to simplify the analysis. The Hele-Shaw burner is oriented vertically, closed at the bottom and on the two sides, and open at the top. A mechanical forcing method is used to study the structural modes of the burner by positioning an electrodynamic vibration exciter (SmartShaker K2004E01) on one plate (see Fig. 1). In order to measure the vibrating response of the plates, several accelerometer are positionned on the plates in order to measure their vibrating response. In addition, electret microphones are inserted in the cell sides at height 0, 30, 60, 90 and 120cm to measure pressure fluctuations. In order to initiate the flame propagation, the gas mixture is initially injected by a flow line from the bottom of the burner. The flow rate and the equivalence ratio (ϕ) of the combustible mixture are controlled with less than 1% error thanks to a PC-interface connected to Bronkhorst EL-Flow series mass-flow regulators. The flow of reactive gas is then stopped and the flame is ignited at the top and starts its downward propagation.

III. SELF INDUCED VIBRO-ACOUSTIC OSCILLATIONS

We analyze the downward propagation of propane-air flame with equivalence ratios in the range 0.7 to 1.5 with each plates set. This range corresponds to the range where the flame can propagate inside the Hele-Shaw cell of 5mm gap without thermal extinction due to heat loss on the walls [START_REF] Al Sarraf | Darrieus-Landau instability and Markstein numbers of premixed flames in a Hele-Shaw cell[END_REF]. Acoustic instability usually appears during propagation in tubes [START_REF] Searby | A parametric acoustic instability in premixed flames[END_REF]. At the expected frequency, the burner height L corresponds to the quarter of the acoustic wavelength. The corresponding frequency in our apparatus would be c/4 L ≈ 50Hz. With PMMA plates, such a frequency is not observed, but some low frequency oscillations appear instead. A typical pressure signal recorded during the propagation of a stoichiometric flame is reported on Fig. 3(a). The periodogram of this signal is drawn in Fig. 3(b) and indicates the emergence of oscillations around 11Hz, which is too low to be associated with pure acoustic waves. When two 19mm glass plates are used instead, some acoustic oscillations around 50Hz appear for equivalence ratios 1.4 and 1.5. In order to figure out which of the acoustic(-like) instability frequency is likely to appear, we report on table I the measured frequency for each set of plates. We observe that the frequency of the pressure oscillations is increasing when using glass plates instead of PMMA plates. The frequency is also increasing with the plate thickness, until classical acoustic instability emerges when both plates are of glass and are 19mm thick. These observations provide the evidence that a coupling between the flame propagation and the structure of the burner is at play. Table I. Frequencies of the self-induced oscillations appearing in a Hele-Shaw cell with a 19mm glass plate on one side and one with characteristics reported on the first line on the other. The range of unstable propane-air equivalence ratio is reported for each case.

In order to have a better understanding of the oscillations appearing in the cell we focus on the 5mm glass plate configuration. The burner is prepared with two accelerometers on the thin plate to measure plate vibrations in addition to a microphone at the bottom of the cell to measure the pressure fluctuations. One accelerometer is positionned at the top of the plate (x = 250mm, y = 0mm) and the other one at (x = 250mm, y = 800mm). In order to trigger self-induced oscillations, the burner is then filled with a rich (ϕ = 1.4) propane-air mixture. The signals measured

IV. PLATE VIBRATION

In order to understand the frequencies observed on table I, we plot on Fig. 4 the response of the burner deformation to a vibration excitation when the 5mm glass plate is used. As can be seen, in the range 30 → 180Hz, the oscillation amplitude of the 19mm thick plate is always at least one order of magnitude smaller than that of the thin plate. Thanks to this observation, we can consider that for this frequency range, the response of the structure can be reduced to the thinner plate one. We identify this way the resonant modes and frequencies of the thin plate, including the previous self induced 63Hz oscillations appearing when the rich propane air flame propagates freely inside. This frequency can be approximated theoretically using the Rayleigh method. The detailed calculation is presented hereafter. Following the Kirchhoff plate theory, the free vibrations of an isotropic plate are governed by the biharmonic wave equation [START_REF] Love | The Mathematical Theory of Elasticity[END_REF]:

w tt + D ρh ∇ 4 w = 0 (1)
where w(x, y, t) is the transverse displacement of the plate of material density ρ, Young modulus E and Poisson's ratio ν. h is the thickness of the plate and D = Eh 3 / 12 1ν 2 its bending stiffness. This equation cannot be solved analytically in a general case. However, providing the appropriate boundary conditions the resonant modes of the plate can be approximated using the Rayleigh method [START_REF] Leissa | Vibration of Plates[END_REF]. The plate modes for the transverse displacement w(x, y, t) = W (x, y) exp(iωt) are considered as the product of two beam modes W (x, y) = X(x)Y (y). In the present apparatus the plate is considered as simply supported on the bottom side and on the two vertical sides and free on the top side, leading to the following boundary conditions :

X (x = 0) = X (x = a) = Y (y = b) = 0 ∂ 2 X ∂x 2 x=0 = ∂ 2 X ∂x 2 x=a = ∂ 2 Y ∂y 2 y=b = 0 (2) ∂ 2 Y ∂y 2 y=0 = 0, ∂ 3 Y ∂y 3 y=0 = 0 (3)
This leads to one linear system for each direction x and y whose eigenvalues (respectively eigenvectors) correspond to the resonant wavenumbers (respectively resonant mode shapes) of the plate in the corresponding direction. The resulting vibration amplitude writes: where α 1 = mπ/a is the wavenumber of the m th resonant mode in the x direction and α 2 (n) is the wavenumber of the n th mode in the y direction. α 2 (n) is the solution of tan(α 2 b) = tanh(α 2 b) which cannot be solved analytically but is well approximated by α 2 (n) = (4n + 1) π/4b. The approximate pulsation of the resonant mode (m, n) is then given by :

W (x, y) = sinh (α 2 b) sin (α 2 b) sin (α 2 (y -b)) + sinh (α 2 (y -b)) sin (α 1 x) (4)
ω(m, n) = π 2 m a 2 + n + 1/4 b 2 D ρh (5)
The first four resonant frequencies given by eq.5 for the 5mm thickness glass plate (E = 69GP a, ρ = 2500kg

• m -3 , ν = 0.25) are f (m = 1, n = 1) = 58Hz, f (m = 1, n = 2) = 77Hz , f (m = 1, n = 3) = 107Hz, f (m = 1, n = 4) = 148Hz
which are in good agreement with the previously experimentally measured resonant frequencies (see Fig. 1(b)). To ensure that these resonant modes correspond to the measures, the shape of the latter is analyzed by measuring the local acceleration A(x, y) along the x and y directions for these four resonant frequencies. The measured relative amplitude G = A(y)/ max(A(y)) is then compared to the mode shape given by eq.4 (Fig. 5). As predicted analytically, the measured mode in the x-direction is m = 1 for these first four resonant frequencies whereas in the y-direction we get n = 1 for f = 63Hz, n = 2 for f = 86Hz, n = 3 for f = 116Hz, n = 4 for f = 156Hz (see comparison on Fig. 5). This ensures that the structural modes identified in Fig. 1(b), correspond actually to the eigenmodes of the thinner plate. The slight differences between theoretical and experimental frequencies arise from the connection of the plates to the frame. An increase in the tightening of the plates resulting in a slight increase of the experimental frequency.

A similar analysis has been performed with the 8mm PMMA plate, and a similar result was obtained, but the first excited mode that emerged was the mode (m=1, n=0) with 10Hz frequency. In the next section it is shown that these plate modes may influence the flame propagation by generating flow oscillations in the combustion chamber.

V. VIBRO-ACOUSTIC COUPLING

To observe the influence of the plate vibrations on the flame dynamics, we analyze the flame propagation when an eigenmode of the structure is excited. To this end, the shaker frequency is adjusted to one of the eigenfrequencies previously measured. The combustion chamber is filled with a mixture of propane-air at equivalence ratio ϕ = 0.8. We chose this equivalence ratio in order to be out of the range of self-excitation reported on table I. The flame dynamics is recorded using two high speed cameras (Photron fastcam mini ax-200). The camera frame rate is adjusted to . Flow speed oscillations experienced by the flame when one of the eigenfrequencies of the plate is enforced (to compare to Fig. 5 be much higher than the oscillation frequency. For each of the four eigenfrequencies identified previously, the flame front coordinates are extracted from these images by way of a python image processing (opensource library scikitimage) with subpixel accuracy (extraction of the bottom part of the front by thresholding methods). From the front coordinates, the mean flame front position at each instant is then computed as y mean (t) = 1/a a 0 y(x, t)dx. Its time derivative corresponds to the oscillating component of the flame velocity:

v a = d(y mean (t)) dt - y mean (t f ) -y mean (t 0 ) t f -t 0 (6)
As the flame front is advected by the flow, v a corresponds to the flow oscillations induced by the plate vibrations. Such flow oscillations are then reconstructed along the y axis as it is shown in Fig. 6 for the first four plate eigenfrequencies. We observe that the envelop of the oscillations is in agreement with the plate modes. The temporal analysis shows that flow oscillations are in phase with the pressure oscillations measured with electret microphones, and in quadrature with the acceleration of the plates. This is in agreement with a flow induced by the volume variations inside the Hele-Shaw gap: the flow in phase with the velocity of the plate vibrations. In addition, the absence of phase shift between the pressure fluctuations and the flow oscillations indicates that a Poiseuille flow assumption is satisfied.

We can now investigate the influence that these flow oscillations have on the flame shape [START_REF] Pelcé | Vibratory instability of cellular flames propagating in tubes[END_REF]. As shown by Searby and Rochwerger [START_REF] Searby | A parametric acoustic instability in premixed flames[END_REF], under periodic flow oscillations, each mode amplitude Φ( k, t) of the flame shape is well described by :

d 2 Φ d t2 + 2c d Φ d t + ω 2 0 + a 1 cos(ω a t) Φ = 0 (7)
Where ωa stands for the non dimensional pulsation of flow velocity fluctuations and c, ω 0 and a 1 are coefficients depending on the non-dimensional wavenumber of the perturbation and on the properties of the combustion reaction. Expression 7 is a damped parametric oscillator equation of eigen pulsation ω 0 and damping coefficient c. As a consequence, a flame front exposed to periodic flow velocity oscillations is prone to exhibit both parametric restabilization(leading to a flattening of the wrinkles) and parametric resonance (leading to small scale wrinkling) [8, [START_REF] Aldredge | Experimental evaluation of Markstein-number influence on thermoacoustic instability[END_REF][START_REF] Yanez | Flame instability of lean hydrogen-air mixtures in a smooth openended vertical channel[END_REF][START_REF] Searby | Acoustic Instability in Premixed Flames[END_REF]. The effect of the variable amplitude of the forcing v a (Fig. 6) along the cell on the flame front stability is analyzed by looking at the flame appearance during its propagation (see Fig. 7). For all studied forcing frequencies the flame starts its propagation as a flat flame and experiences at some locations the Darrieus-Landau wrinkling, exhibiting the cellular pattern dynamics described in [START_REF] Almarcha | Interface dynamics, pole trajectories, and cell size statistics[END_REF]. But during its propagation, the front undergoes periodic restabilization/destabilization in areas whose locations depend on the forcing frequency. Comparing Fig. 7 with Fig. 6, one can note that along the y axis, the restabilization zones correspond to the antinode zones for the flow velocity oscillations. Moreover, the restabilizations are faster in the center, at x = 250mm, where the plate vibration and the flow are maximum according to m = 1. The Searby and Rochwerger [8][12] theory predicts two different wrinkling regimes for flame front advected by flow velocity oscillations. For small v a and sufficient Froude number, a range of unstable wavenumbers is delimited by two cutoff wavenumbers: k g (respectively k c ) where gravity (respectively thermal diffusive) effects are damping the Darrieus-Landau wrinkling. Increasing the forcing intensity v a , the range of unstable wavenumbers is reduced. Above the threshold v a > v * I , the Darrieus-Landau wrinkling zone is completely suppressed for all wavenumbers. The theory predicts a secondary instability zone which appears when v a > v * the forcing v a decreases until v a < v * I and the flame is once again undergoing Darrieus-Landau wrinkling. This phenomenon is repeated each time the flame crosses an antinode of the modes, which explains the periodic restabilization/destabilization observed in Fig. 7. As the flame speed is correlated to the flame shape, we can observe on Fig. 8 that the flame speed is modulated during the propagation in presence of 156Hz vibration forcing. The velocity reaches some maxima when the Darrieus-Landau wrinking is at play, in the zone where parametric restabilization is not effective. At the contrary, the velocity decreases when the flame is flattened by the parametric restabilization. This confirms that the vibrations of the plates are efficient to induce flow oscillations in the Hele-Shaw combustion chamber and directly act on the flame flattening and on the flame speed. In addition, when the amplitude of the oscillations is sufficiently high, it is possible to reach parametric destabilization in a similar way as the secondary acoustic instability. When looking at f = 116Hz and f = 156Hz modes on Fig. 7, one can notice a small-scale cellular aspect of the flame, in the last antinode at the bottom of the burner. This small scale wrinkling is oscillating with a period twice the one of the flow oscillations. This is the evidence that it corresponds to the parametric destabilization which arises when v a > v * II . Indeed for f = 116Hz and f = 156Hz, v a is quite larger in the last antinode than in the others (see Fig. 6).

VI. CONCLUSION

The present study was motivated by the unexplained flame oscillations observed in recent studies on flame dynamics in narrow channels. More precisely, we demonstrated that these oscillations may be caused by structural modes of the burner. These modes have been studied both analytically and experimentally and the obtained results are in agreement. It has been shown that the plate vibration generates flow speed oscillations in the burner, which in turn act on the flame dynamics in a similar fashion to the oscillations induced by thermo-acoustic instability. The important difference here is that the frequency and the mode shape of the oscillations are not ruled by the burner internal geometry like for acoustic modes, but by the structure deformation modes which depends on the material and the whole geometrical characteristics of the burner (thickness, size and assembly). When focusing on the topology of the flame interface, both parametric restabilization and parametric resonance are possible to be forced, as predicted by the Searby and Rochwerger [START_REF] Searby | A parametric acoustic instability in premixed flames[END_REF] theory. It has been shown that the plate modes may be excited by the flame propagation itself and are certainly responsible for the flame speed oscillations with unexpected frequencies observed in recent experiments [START_REF] Al Sarraf | Darrieus-Landau instability and Markstein numbers of premixed flames in a Hele-Shaw cell[END_REF][START_REF] Hye | Unsteady propagation of premixed methane/propane flames in a mesoscale disk burner of variable-gaps[END_REF][START_REF] Alexeev | Experimental Study on Cellular Premixed Propane Flames in a Narrow Gap between Parallel Plates[END_REF].

These observations open new ways to study the response of flames wrinkling to flow oscillations. Indeed, the forcing method used in this study is both efficient and easy to set-up, and it allows to study the flame response to time dependent stretch [START_REF] Joulin | On The Response of Premixed Flames to Time-Dependent Stretch and Curvature[END_REF][START_REF] Clavin | High-frequency response of premixed flames to weak stretch and curvature: A variable-density analysis[END_REF] on a large range of frequencies (expandable by changing the bending stiffness of the plate). Moreover, the distance between the two plates can be easily modified giving opportunity to study the influence of Saffman-Taylor effects on the restabilization threshold v * I and on the parametric destabilization threshold v * II [START_REF] Aldredge | Saffman-taylor Influence On Flame Propagation In Thermoacoustically Excited Flow[END_REF].

Figure 1 .

 1 Figure 1. Experimental apparatus: Hele-Shaw burner consisting of two plates (one is made of glass with 19mm thickness ) separated by a 5mm gap. A shaker can force the plates. Accelerometer probes can be placed on the plates and electret microphones are inserted in the cell side.

Figure 2 .Figure 3 .

 23 Figure 2. Left: Evolution of the pressure fluctuations during propane-air flame (equivalence ratio 1.2) propagation when the second plate is a PMMA plate of 8mm of thickness. Right: associated periodogram exhibiting a 11Hz frequency.

Figure 4 .

 4 Figure 4. Vibrating accelerations measured by the two probes. The first four resonant frequencies of the thin plate are highlighted by red arrows.

Figure 5 .

 5 Figure 5. Comparison of theoretical plate mode shapes and frequencies (left) with the experimentally measured ones along the vertical with accelerometers (right).

Figure 6

 6 Figure 6. Flow speed oscillations experienced by the flame when one of the eigenfrequencies of the plate is enforced (to compare to Fig.5

Figure 7 .

 7 Figure 7. Flame propagation when the plate is forced for the first four resonant frequencies. The front wrinkling undergoes periodic restabilization/destabilization during its propagation. The restabilization (respectively destabilization) zone corresponds to the antinodes (respectively nodes) of the enforced plate modes.

  II and is called parametric instability. The two threshold v * I and v * II depend on the physico-chemical parameters associated with the flame. If for some flames v * II > v * I and in a range of forcing v * I < v a < v * II the intrinsically unstable flame undergoes a parametric restabilization. It is exactly what we observe in our experiments, in the antinodes the threshold v * I is reached and the flame remains flat (stable). Then, when the flame propagates further downwards,

Figure 8 .

 8 Figure 8. Flame propagation speed when propagating in the cell with forcing at 156Hz at play. We can observe that the velocity increases in the zones where Darrieus-Landau wrinking is at play and decreases where the wrinkles are flattened by the parametric restabilization.

We thank 'Agence Nationale de la Recherche' for funding of the ANR 'PDF' ANR-14-CE05-0006, the Excellence Initiative of Aix-Marseille University -A*MIDEX, and Labex MEC, for funding.


Romain Espinosa

Bruno Deffains

Christian Thöni
email: christian.thoeni@unil.ch.

Debiasing preferences over redistribution: An experiment †

Keywords: Redistribution, Self-serving bias, Debiasing, Experiment

We study the manipulation of preferences over redistribution. Previous work showed that preferences over redistribution are malleable by the experience of success or failure in a preceding real-effort task. We manipulate the information subjects receive about the importance of chance relative to effort in determining success. We investigate the effect of this manipulation on (i) subjects' redistribution choices affecting third parties, and (ii) preferences for redistributive taxation. Our results show that informing the subjects about the relative importance of chance after the real-effort task does not mitigate the self-serving bias in redistribution choices. Only providing full information before the real-effort task prevents the emergence of the self-serving bias.

Introduction

Free market economies tend to result in unequal allocations of resources. While all modern societies put forward some form of intervention to mitigate the gap between rich and poor, redistributive measures are subject to intense political debates. Behind the Rawlsian veil of ignorance, all citizens might unanimously agree on a redistributive system. Real world debates are complicated by the fact that individuals know their position in the socio-economic hierarchy, which gives rise to opposing views about redistribution. [START_REF] Alesina | Inoculation and Counterexplanation: Debiasing Techniques in the Perseverance of Social Theories[END_REF] show that preferences over redistribution depend on personal characteristics such as gender, race and socioeconomic status. Given that usually the poor outnumber the rich by far, it is a puzzle why democratic systems do not result in more extensive redistribution. [START_REF] Bonica | Why hasn't democracy slowed rising inequality[END_REF] explore this question with respect to the U.S. and note that one of the explanations might be increased polarization between the Democrats and the Republicans, which makes it more difficult to find a compromise. 1 While increasing inequality might be an important cause of the polarization on preferences for redistribution, [START_REF] Deffains | Political self-serving bias and redistribution[END_REF] recently argued that an additional force is at play: Building on psychological research on self-serving biases, they show experimentally that the experience of success or failure shifts individuals' views on redistribution, even in the absence of any selfish motives. In their experiment, subjects are performing a real-effort task for which they are randomly assigned to either a success or a failure condition. After this, subjects are asked to redistribute wealth from a rich to a poor subject, a task that leaves their own payoff unaffected. The design deliberately introduces ambiguity as to whether success or failure is due to internal factors (insufficient effort or ability), or due to external factors (the nature of the task). The results show that subjects who had been successful in the real-effort task redistribute less than their less successful counterparts do. In addition, a questionnaire reveals that successful subjects tend to attribute their success to internal factors, while unsuccessful subjects attribute their failure to external factors. This 'political self-serving bias' contributes to the polarization of the views on redistribution: Not only do those who are born with a golden spoon in their mouths oppose redistributive measures because they are net payers, but also because they mistakenly think they earned their position in the hierarchy.

The polarization of views on redistribution documented by [START_REF] Deffains | Political self-serving bias and redistribution[END_REF] is presumably due to the high level of ambiguity with respect to the relative importance of internal and external factors. To what extent the degree of ambiguity in the experiment approximates the real world is obviously difficult to assess. While the importance of intergenerational transmission of economic status is part of an academic debate (e.g. [START_REF] Bowles | The inheritance of inequality[END_REF], it seems not unlikely that an overestimation of the internal factors among relatively successful individuals is also pertinent outside the laboratory. More information about the relative importance of external and internal factors might thus reduce the polarization and facilitate reaching a consensus in the political process.

This study builds on the method of [START_REF] Deffains | Political self-serving bias and redistribution[END_REF] and investigates to which extent the political self-serving bias is stable to adding information about the relative importance of external factors. Like [START_REF] Deffains | Political self-serving bias and redistribution[END_REF] we elicit behavior in (i) a disinterested dictator game, where participants redistribute money among two other participants, and (ii) a game in which subjects choose redistributive taxation affecting their own future payoffs. For both dependent variables we confirm the earlier findings and show that-when there is high ambiguity in terms of the relative importance of external and internal factors-a political self-serving bias arises. We then investigate the stability of the bias. To do so we provide information about the relative importance of the external factors. In particular, we inform the participants that their success or failure in the real-effort task was almost fully determined by external factors. We provide this information either before or after the realization of success or failure. We find that the self-serving bias survives if the information is provided after the realization of success or failure. If, however, we provide the information that the deck is stacked before the realization of success or failure, then little, if any, self-serving bias in redistributive preferences remains. The decrease in the self-serving bias seems to result from different behavioral responses of over-and underachievers. When redistributive choices do not affect their personal interest, it is mainly the overachievers who close the gap by increasing redistribution. When redistributive choices affect their own payoff, then it is primarily the underachievers who reduce their preferred tax rate.

The rest of the paper is organized as follows. In Section 2, we discuss the related literature, Section 3 describes the experiment and the hypotheses. In Section 4 and 5 we discuss the results and Section 6 concludes.

Theoretical and experimental background

Preferences for redistribution have received great attention in the experimental literature. A first line or research investigated the role of self-interest [START_REF] Hoffman | Entitlements, rights, and fairness: An experimental examination of subjects' concepts of distributive justice[END_REF][START_REF] Bjerk | In front of and behind the veil of ignorance: An analysis of motivations for redistribution[END_REF][START_REF] Jiménez-Jiménez | The effect of initial inequality on meritocracy: A voting experiment on tax redistribution[END_REF][START_REF] Gerber | The role of ignorance in the emergence of redistribution[END_REF], social preferences [START_REF] Tyran | A little fairness may induce a lot of redistribution in democracy[END_REF][START_REF] Ackert | Social preferences and tax policy design: Some experimental evidence[END_REF][START_REF] Balafoutas | Equality, equity and incentives: An experiment[END_REF], or social identity [START_REF] Klor | Social identity and preferences over redistribution[END_REF] for the determination of redistributive systems. [START_REF] Durante | Preferences for redistribution and perception of fairness: An experimental study[END_REF] provide estimates for the relative importance of selfish and fairness motives. [START_REF] Kittel | Knowledge, power, and self-interest[END_REF] build on their framework and add the role of power as a determinant for redistributive preferences. [START_REF] Eisenkopf | Unequal opportunities and distributive justice[END_REF] study the effect of unequal access to education on the demand for redistribution.

In a second strand of studies, economists have sought to determine how individuals are willing to redistribute under uncertainty about their position in the income distribution [START_REF] Schildberg-Hörisch | Is the veil of ignorance only a concept about risk? An experiment[END_REF]. In a survey study, [START_REF] Karadja | Richer (and holier) than thou? The effect of relative income improvements on demand for redistribution[END_REF] informed respondents about their relative position in the income distribution and show that respondents who are richer than they initially thought demand less redistribution. A series of studies have investigated how preferences for redistribution are affected by economic experience [START_REF] Frohlich | Choosing justice in experimental democracies with production[END_REF][START_REF] Cabrales | It is Hobbes, not Rousseau: An experiment on voting and redistribution[END_REF][START_REF] Cappelen | The pluralism of fairness ideals: An experimental approach[END_REF][START_REF] Großer | Redistribution and market efficiency: An experimental study[END_REF].

One of the main discussions developed in this literature concerns the relative importance of luck and effort in the determination of individual outcomes [START_REF] Kataria | Risk, entitlement and fairness bias: Explaining preferences for redistribution in multi-person setting[END_REF][START_REF] Cappelen | Fairness and the proportionality principle[END_REF]).2 [START_REF] Konow | Fair Shares: Accountability and Cognitive Dissonance in Allocation Decisions[END_REF] shows that individuals act according to the accountability principle, i.e., they use redistribution systems to compensate (bad) luck and to reward efforts. Evidence suggests, however, that individuals tend to deviate from the accountability principle when they have interests at stake: [START_REF] Babcock | Biased judgments of fairness in bargaining[END_REF] and Babcock and Loewenstein (1997) show that people tend to exhibit a self-serving bias, i.e., their view about fairness is distorted to their advantage (see also [START_REF] Rodriguez-Lara | Self-interest and fairness: Self-serving choices of justice principles[END_REF][START_REF] Ubeda | The consistency of fairness rules: An experimental study[END_REF][START_REF] Gino | Motivated bayesians: Feeling moral while acting egoistically[END_REF]. [START_REF] Deffains | Political self-serving bias and redistribution[END_REF] demonstrate the existence of a political self-serving bias: when individuals succeed (fail) in a task, they are more (less) likely to attribute the outcome to their effort rather than luck. Consequently, these distorted attributions lead to diverging redistributive preferences.

Our experimental design is also related to a strand of literature studying situations in which subjects' decisions do not affect their own payoff, i.e., they act in the role of an impartial spectator [START_REF] Croson | Social preferences and moral biases[END_REF][START_REF] Konow | Is fairness in the eye of the beholder? An impartial spectator analysis of justice[END_REF][START_REF] Tinghög | Are individuals luck egalitarians?an experiment on the influence of brute and option luck on social preferences[END_REF][START_REF] Konow | Equity versus equality: Spectators, stakeholders and groups[END_REF][START_REF] Luhan | Money or morality: Fairness ideals in unstructured bargaining[END_REF][START_REF] Cassar | A matter of perspective: How failure shapes distributive preferences[END_REF].

The demonstration of the existence of self-serving biases for redistribution raises the question of how to mitigate the gap between biased individuals. [START_REF] Larrick | Debiasing. Blackwell handbook of judgment and decision making[END_REF] distinguishes two main techniques for debiasing individuals in general: using one bias to offset another (i.e., rebiasing), or changing the decision environment. The effectiveness of these techniques has been shown to vary with the type of bias under consideration. [START_REF] Arkes | Costs and benefits of judgment errors: Implications for debiasing[END_REF] discriminates between three types of judgment errors: strategy-based judgment errors (subjects take suboptimal decisions because elaborating more subtle strategies is too costly), association-based judgment errors (misperception of the probabilities of the states of the world), and psychophysically based errors (like e.g. sunk cost effects or framing effects). Strategy-based judgment errors are usually corrected by incentivizing individuals to develop more elaborated strategies (e.g., increasing the rewards associated to correct reactions). Psychophysically based errors can be compensated by shifting the reference point considered by individuals in their S-shaped utility function.

Remedies to mitigate association-based judgment errors like the self-serving bias mainly consist in three types of strategies: rebiasing, forcing participants to consider alternative reasoning, and emphasizing elements necessary to form correct beliefs. First, rebiasing is a relatively unpopular and risky strategy. Rebiasing requires indeed a good assessment of the pre-existing bias and a good calibration of the counter-bias. Should one of these two elements fail, rebiasing can either overshoot and generate an inverted bias, or fall short and fail at resorbing the bias. For instance, [START_REF] Jolls | Debiasing through law[END_REF] propose to use availability heuristics to correct for optimistic beliefs of consumers: by telling them anecdotes, one can increase their perceived probability of some states of the world. Second, forcing participants to consider alternative reasoning has been shown to mitigate biases [START_REF] Slovic | On the psychology of experimental surprises[END_REF][START_REF] Koriat | Memory reasons for confidence[END_REF]Anderson, 1982[START_REF] Anderson | Abstract and Concrete Data in the Perseverance of Social Theories: When Weak Data Lead to Unshakeable Beliefs[END_REF][START_REF] Lord | Considering the opposite: A corrective strategy for social judgment[END_REF]Babcock et al., 1997). A third approach proposes to make the information necessary for correct inference more salient (e.g., [START_REF] Nisbett | The use of statictical heuristics in everyday inductive reasoning[END_REF]: the objective is to em-phasize useful information in order to increase the likelihood that the information is taken into account when individuals form their beliefs. In our experiment, we focus on this third debiasing strategy and provide subjects with specific information about the relative importance of internal and external factors.

Experimental Design

Our experimental protocol starts with inducing the experience of success or failure in a realeffort task. We define success (failure) as performing above (below) the median performance in the real-effort task. In our design, success and failure is almost exclusively determined by a random draw, assigning subjects to either a hard or an easy task. We conduct three treatments, varying in the degree to which subjects are informed about the relative importance of luck and effort in determining success and failure. In the Baseline treatment, we replicate [START_REF] Deffains | Political self-serving bias and redistribution[END_REF] and inform subjects only about the existence of the hard and easy task and the random assignment procedure. We do not inform the subjects whether they were assigned to the hard or easy task. In the other treatments, we inform the subjects about the task they were assigned to and the relative importance of the random task assignment in determining success and failure. We conducted two treatments, differing with respect to the timing of the information. In treatment PostTask, subjects learn after the realization of success and failure that the random task assignment was almost fully determining the outcome. In treatment PreTask, subjects receive this information prior to the real-effort task. Subsequently we elicit the main dependent measure-redistributive preferences-in two games, the Disinterested Dictator Game and the Redistributive Taxation Game. In the following, we describe the elements of the experimental protocol in more detail.

Real-Effort Task

In the real-effort task, subjects were presented with sequences of binary digits, with four to thirteen digits each. Subjects had to count and indicate the number of ones in each sequence. There were five consecutive screens with a time limit of 30 seconds per screen. Correct answers were rewarded. In each session, half of the subjects were randomly assigned to the 'hard task', the other half to the 'easy task'. In the hard task, screens were made of 25 lines of binary digits. In the easy task, the last six lines were not displayed, and counted as correctly solved. The theoretical maximum number of tokens was identical in both tasks, but the tasks were designed such that it was very unlikely that a subject with the hard task would earn more tokens than a subject with the easy task. After completion of the five screens, we used the total number of tokens earned in the real-effort task to perform a median split of the subjects within a session. Subjects who earned more tokens than the median were told that they performed above median. The wording on the screen was: Congratulations! You performed better than most of the participants! For the subjects with below median performance the message read: What a pity! You performed worse than most of the participants! In the following, we label above median performers as overachievers, and below median performers as underachievers. Participants were told that the maximum possible earnings were the same in both tasks. While the two tasks clearly varied in difficulty, even the easy task was designed such that none of the participants managed to solve it perfectly, given the time limit. In addition, participants could not observe other participants' tasks. Consequently, in absence of information about the assigned tasks, participants were unable to deduce which task they were assigned to. Information about the task assignment was our treatment variation, which we will describe next.

Treatments

We conducted three treatments differing in the amount and the timing of information about the degree to which subjects' relative performance was determined by internal or external factors:

• Baseline: This treatment replicates the procedures by [START_REF] Deffains | Political self-serving bias and redistribution[END_REF], where no additional information was provided. Note that in all our treatments participants are aware of the fact that (i) there is a hard and an easy task, and (ii) there is random assignment to task with equal probabilities.

• PostTask : In this treatment, participants receive additional information after the realeffort task and the realization of under-/overachiever. First, we informed subjects about the task they were assigned to (hard/easy). Second, we explained that in our previous sessions all participants with the hard task ended up as underachievers and vice versa. We based this information on the perfect separation observed in previous experimental research using the same real-effort task [START_REF] Deffains | Political self-serving bias and redistribution[END_REF]. Finally, we told participants that the game was designed such that it was "virtually impossible" to end up as overachiever (resp. underachiever) when given the hard (resp. easy) task.3

• PreTask : In this treatment we gave participants the same information as in PostTask, but participants received the information prior to the real-effort task. They were told that the game was designed such as to produce full separation. After the real-effort task, when participants were informed whether they performed better or worse than the median, we also told them the nature of the task they received.

Disinterested Dictator Game

For the Disinterested Dictator Game (DDG) we selected two participants (the 'targets') randomly among all participants of the session. The remaining participants (the 'disinterested dictators') had then the possibility to redistribute tokens from the wealthier to the poorer target. To avoid comparison, disinterested dictators were not informed about the absolute level of the two targets' payoffs. Instead, they could reallocate the difference between the wealthier and the poorer subject's payoffs from the former to the latter. In particular, they had to indicate a percentage of redistribution. Figure 1 shows an excerpt of the experimental interface. When subjects entered the stage, they were shown the two bars on the left indicating that the two targets' payoffs differ, but not allowing any inference about the absolute payoffs. Subjects select a redistribution level on a 12-point equidistant scale reaching from 0% (no redistribution) to 100% (transfer all the money at disposal from the wealthier target to the poorer target). 4 Upon choosing a redistribution percentage, the two bars on the right appeared as a visual confirmation of the selected action (Figure 1 shows the case of 45.5% redistribution). Subjects could revise their decision and get an updated bar chart until they were satisfied with the result. All participants were told that the decision of one disinterested dictator in the session would be randomly selected and implemented. Participants were also explicitly told that redistribution would concern the two targets only, and that all other subjects would not be affected by any redistribution mechanism in this task. Prior to the decision, disinterested dictators were reminded that targets may have faced different tasks. After all disinterested dictators made their choice, one redistribution proposal was randomly selected and implemented. Payoff information from the first game was withheld until the end of the experiment.5

Redistributive Taxation Game

After the DDG, participants were given a new set of instructions for the Redistributive Taxation Game (RTG). In this game, subjects decided upon redistributive taxation for income to be earned in a real-effort task later in the experiment. Participants were told that they were going to perform a series of real-effort tasks different from the one before.6 They were told that they could earn up to 80 tokens for the whole series of tasks, and that their final income would be subject to redistribution within a group of four participants. Before engaging in the real-effort task, participants in a group collectively decided on the redistributive rule for their group.

The redistribution rule was a simple redistributive tax scheme among the four participants in the group. Subjects could opt for a tax rate on their future real-effort task earnings from 0% to 100% by increments of 10 percentage points. All taxed revenue was redistributed in equal parts to the four participants. Consequently, with a tax of zero all subjects keep their earnings, while a tax of 100% means that all subjects in the group of four earn the same. Before taking a decision, participants went through extensive control questions with hypothetical situations to ensure that redistribution rules were well understood. (See Figures D1 to D4 in Appendix D for the screenshots.)

The decision about the redistributive tax was chosen by a random dictator mechanism. All subjects indicated their preferred tax rate, and in each group the decision of one randomly drawn subject was implemented.7

Hypotheses

Previous findings show that the ambiguity in the Baseline protocol can induce a self-serving bias, which affects the demand and the supply of redistribution [START_REF] Deffains | Political self-serving bias and redistribution[END_REF]. Our goal is to replicate this finding and to investigate whether adding information about the relative importance of luck in determining success and failure can serve as a debiasing device, such that differences in the redistribution decisions between the two groups disappear.

First, we expect our protocol to generate a self-serving bias among participants when causal attributions are unclear. We expect over-and underachievers to have different views about the extent to which internal and external factors affect the outcome when they lack additional information. Previous research showed that, when participants have diverging views about causal attributions, they also diverge in their redistribution preferences. The accountability principle discussed by [START_REF] Konow | Fair Shares: Accountability and Cognitive Dissonance in Allocation Decisions[END_REF] implies that participants tend to reward effort and to compensate for bad luck. While [START_REF] Konow | Fair Shares: Accountability and Cognitive Dissonance in Allocation Decisions[END_REF] provides evidence at the micro-level, the results by [START_REF] Alesina | Fairness and Redistribution[END_REF] suggest that similar mechanisms are at play at the macro-level.

In our experiment, the self-serving bias predicts that succeeding (failing) in the task will increase one's propensity to believe that success is driven by effort (luck). Konow's accountability principle postulates that in redistribution choices react to the degree to which an observed distribution is caused by luck or effort. Combining the accountability principle with the self-serving bias, we predict that overachievers (underachievers) are less (more) supportive of redistribution, because they perceive the outcome to the real-effort task to result mainly from effort (luck). This prediction was confirmed in [START_REF] Deffains | Political self-serving bias and redistribution[END_REF] and we use our Baseline condition to test whether their results replicate. Consequently, in this treatment we expect overachievers to redistribute less in the DDG and to opt for lower tax rates in the RTG.

The self-serving bias emerges in Baseline because causal attributions are unclear. In the two main treatments, we provide information about the mechanism leading to success or failure.

Informing the participants about the mechanisms that determine the outcome should reduce the differences in the causal attributions across groups. Following Konow's accountability principle, groups with similar beliefs on the role of luck and effort should then display similar redistribution choices. In both PostTask and PreTask , causal attributions are extensively explained to the participants when they learn that the success and failure in the real-effort task is almost purely randomly determined. This leaves little room for diverging views on causal attributions. If participants process this information, overachievers and underachievers should have similar beliefs and should therefore choose similar levels of redistribution, both in the DDG and the RTG.

Previous works in social psychology (e.g., Guenther and Alicke 2008) suggest, however, that informing participants about causal attributions after they have formed their beliefs may not have the desired effect. The concept of belief perseverance states that people often tend to cling to their convictions when presented with contradictory information. In a seminal work, [START_REF] Ross | Perseverance in self-perception and social perception: biased attributional processes in the debriefing paradigm[END_REF] showed that participants who received positive feedback about their performance in a task were more likely to keep giving credit to the task than those who received negative feedback, even when they were told that the feedback was fabricated. According to the authors, perseverance takes place because individuals spontaneously form beliefs to explain the original feedback, and later they are reluctant to revise their beliefs in light of new information. In our PostTask treatment participants experience success or failure and, similar to the design of [START_REF] Ross | Perseverance in self-perception and social perception: biased attributional processes in the debriefing paradigm[END_REF], receive information about the mechanism ex-post. If belief perseverance is at play, we should expect similar results as in Baseline, i.e., the political self-serving bias, once could publicly announce their desired level of redistribution, and bargaining would stop in case of unanimous agreement on a tax rate. Both the random dictator and the result of the bargaining were potentially relevant for the outcome of the RTG. With 50% probability a group's final tax rate was determined by the random dictator rule; with the remaining probability it was determined by the bargaining. In the following we will focus on the private decision, as this is not confounded with strategic concerns.

established, might be immune to information about the causal mechanism.

Finally, in the treatment PreTask we inform subjects before entering the real-effort stage. This is arguably the strongest debiasing intervention, and we hypothesize that this treatment prevents the emergence of the self-serving bias, and, consequently produces similar levels of redistribution in both the DDG and the RTG.

Results

We ran seventeen sessions with 24 participants. In each session, two subjects were selected as targets for the DDG game and did not take a decision in this game. For the analysis, we dropped these observations from the sample, which leaves us with a total of 374 subjects. All sessions were run at the University of Strasbourg (January and June 2017). The sessions lasted about 45 min, and participants earned on average 13.7 euro. Seven sessions were devoted to the Baseline treatment, seven to the PostTask treatment, and three to the PreTask treatment. We start presenting the results with the real-effort task, followed by the Disinterested Dictator Game and the Redistribution System Game.

Real-effort task

The experiment starts with a real-effort task, in which subjects are asked to count the number of ones in binary sequences. This provides us with a measure for individual performance. Figure 2 shows the average performance of subjects in the two tasks. Subjects assigned to the easy task earned substantially more tokens than those in the hard task (16.5 vs. 10.3). The standard error spikes in Figure 2 indicate that the difference is highly significant (p < 0.001, Wilcoxon ranksum test). In the treatments Baseline and PostTask subjects are in the identical situation when commencing the real-effort task, while in PreTask subjects were informed that the task was designed to achieve separation. This information might have affected their motivation to exert effort. We use OLS regressions to test whether the performance in the real-effort task reacts to the treatment variations. We find very similar results in all treatments (see Table A2 in the Appendix).

Based on the performance measure we classify our subjects into overachievers (above median performance), and underachievers. While the two tasks produced widely different average outcomes, they were not fully separating. Six subjects in the hard task managed to become overachievers, and, by construction, six subjects in the easy task became underachievers. These switchers account for 3% of our sample, and, while this number seems too low to influence our result, we will account for the switchers in our analysis in various ways. 8 Our protocol induced the same level of information for both underachievers and overachievers regardless of their original task. It follows that participants were not able to deduce without further information whether they were assigned to the hard or easy task, such that presumably only the labelling as 'above the median' or 'below the median' affected their attributions. Consequently, the self-serving bias (SSB) can occur irrespective of the original task a participant was assigned to. The fact that we observe switchers raises the question whether our design was deceptive, because we told subjects that would be virtually impossible to switch. At the time we designed the experiment we based the wording of the instructions on observations from four sessions of our previous work [START_REF] Deffains | Political self-serving bias and redistribution[END_REF], where we used the same real effort task and observed perfect separation. We decided to keep the procedures constant between sessions despite the fact that we observed switchers in the experiments for the present study. In addition, the instructions read virtually impossible, which technically still concedes the possibility of switching.

Redistribution DDG

Disinterested Dictator Game

On average, subjects transfer 34.9% of the disposable tokens from the wealthier to the poorer target and most subjects redistribute at least some of the income. Figure 3 shows histograms of the redistribution decisions (by treatment and for under-and overachievers). In all cases, the most frequently chosen level is the option that almost equalizes the profits, but still leaves slightly more to the target who is initially wealthier (5/11). However, the histograms suggest that the tendency towards this option is stronger among underachievers than among overachievers in the treatments Baseline and PostTask, but not in PreTask.

Figure 4 shows the average redistribution percentage by over-and underachievers in the three treatments. In Baseline, we replicate the results of [START_REF] Deffains | Political self-serving bias and redistribution[END_REF] and observe that underachievers redistribute more than overachievers. On average, underachievers redistribute 35.9% compared to 29.0% for overachievers (p = 0.039, Wilcoxon rank-sum test).

Result 1. In absence of information, underachievers redistribute significantly more from the rich to the poor target than overachievers do.

Second, we investigate whether the gap between overachievers and underachievers disappears when we provide information about the relative importance of external and internal factors (henceforth the luck/effort ratio). Figure 4 provides mixed evidence. On one hand, we observe a similar difference in redistribution choices when the information is provided after subjects work on the real-effort task (treatment PostTask ). In this case, underachievers propose to redistribute 38.7% on average, while overachievers prefer to redistribute only 31.4% (p = 0.012, Wilcoxon rank-sum test). On the other hand, the difference between overachievers and underachievers shrinks and becomes nonsignificant (p = 0.984, Wilcoxon rank-sum test) in the treatment Pre-Task . This suggests that information prior to the real-effort task weakens the emergence of the self-serving bias.

Result 2. Underachievers and overachievers display similar redistribution preferences when information about the luck/effort ratio is provided prior to performing the real-effort task. When the information is provided ex post, overachievers redistribute significantly less from the rich to the poor target than underachievers do. 1. We estimate the marginal effect of being an overachiever in each treatment on the redistribution decision in the first game for the 374 disinterested dictators. Given that participants chose from a set of discrete alternatives, we use ordered probit estimations. We report the estimated coefficients in the table together with the marginal effects associated with the overachiever status. Given the complexity of interpreting interaction coefficients in non-linear models [START_REF] Ai | Interaction terms in logit and probit models[END_REF][START_REF] Norton | Computing interaction effects and standard errors in logit and probit models[END_REF], we report the marginal effects in percentage points in the table and detail our procedure to retrieve them in Appendix B. Specification (1) extends the results of the non-parametric tests presented above to the non-linear model. Specification (2) confirms the robustness of the results when controlling for age, gender, political orientation, and standardized performance 9 . Gender and age are insignificant. Political orientation, measured on an eleven-point scale ranging from 0 (left) to 10 (right) is highly significant with the expected sign: right-leaning subjects opt for less redistribution. Participants who performed better to the task are significantly less likely to ask for redistribution. The remaining models control for the influence of the few subjects who managed to become overachievers despite being assigned to the hard task or vice versa. In Specification (3), we include a dummy variable identifying the switchers, and in Specification (4) we estimate the models without the switchers. For both models, we find almost identical results as in Specification (2). Finally, in Specification ( 5) we instrument the overachiever status by the nature of the assigned task to estimate the marginal impacts accounting for the potential selection effect of the switchers. 10 Again we find that our results remain unchanged, giving us confidence that the observed effects are not driven by the switchers. These results are robust to separate regressions (see table A3) and to linear estimates 11 .

Redistributive Taxation Game

Our second dependent variable measures the demand for redistribution affecting a subject's own future earnings. Figure 5 displays the average tax rates per treatment (for histograms of the tax rates see Figure A1 in the Appendix). First, we replicate the findings of [START_REF] Deffains | Political self-serving bias and redistribution[END_REF] and find that, in the absence of information, underachievers ask for significantly more redistribution than overachievers do (42.7% vs. 37.6%; p = 0.048, ranksum test).

Result 3. In absence of information, underachievers opt for higher redistributive tax rates than overachievers.

Second, similarly to the DDG we observe that the difference of redistribution choices between the two groups of players remains significant when information about the luck/effort ratio is provided ex-post (p < 0.01, ranksum test), but vanishes when the information comes prior to performing the real-effort task (p = 0.521, ranksum test). Regression analyses 12 show that the difference between overachievers and underachievers is significant at 1% in the PostTask treatment, and nonsignificant in the PreTask treatment (Columns 3 and 4 of Table 2). Again, we find that the information about the luck/effort ratio prevents the emergence of the self-serving bias only when the information is provided before the real-effort task. The difference between over-and underachievers is always negative, and (at least weakly) significant in Baseline and significant in PostTask . In PreTask the point estimate is positive and far from significance. These results hold for all our specifications controlling for the influence of the switchers. Last, 9 We compute scores of standardized performance by centering and standardizing the performance scores at the session and task level.

10 To fully capture the treatment effect, we rely on an instrumental-variable approach such as suggested by [START_REF] Angrist | Identification of causal effects using instrumental variables[END_REF]. To do so, we instrument the actual treatment (overachiever or underachiever group) by the intention to treat (easy or hard task assignment). The instrumented variable overachiever is no longer correlated with the error term (due to the composition effect), because the task assignment was randomly allocated.

11 Results available upon request.

12 All regressions exclude targets from the disinterested dictator game. Redistribution in the RTG the results are robust to separate regressions (see Table A4) and partially robust to linear specifications13 .

Result 4. Overachievers and underachievers display similar redistribution choices in the RTG when information about the luck/effort ratio is provided prior to performing the real-effort task.

Overachievers ask for significantly lower levels of redistribution than underachievers do when the information is provided after the real-effort task.

Postdictions

Our hypotheses developed in Section 3.5 predicted a smaller or no bias in response to the additional information, but did not specify whether this should be due to changes in the behavior of overachievers or underachievers (or both). In the following, we present additional findings on the behavior of over-and underachievers. All findings that follow were not part of the hypotheses and are therefore postdictions. We start by investigating how the bias was mitigated in PreTask relative to the other two treatments. As mentioned in the literature review, previous experimental works on redistribution concluded that individuals tend to reward effort and to compensate bad luck. In our setting, the clarification of the causal attribution is made by revealing to participants that luck was the predominant determinant of success and failure. This information should drive participants to ask for more redistribution as they would be more likely to think that the actual distribution of payoff results from luck.

The results observed for the DDG are in line with this argument. Figure 4 suggests that the reduction in the difference between overachievers and underachievers in the DDG is driven by the increase of the redistribution by overachievers. This intuition is supported by Wilcoxon rank-sum tests that detect no statistical difference in the redistribution levels chosen by underachievers across treatments (p > 0.10), but conclude that overachievers redistribute significantly more in the PreTask condition than in Baseline (p = 0.004), or in PostTask (p = 0.012). These results are also robust to multivariate analysis.

However, for the RTG we observe opposite effects. It appears from Figure 5 that overachievers tend to select similar redistribution levels across treatments, while underachievers opt for lower redistribution levels in PreTask relative to the other two treatments. Wilcoxon rank-sum tests only reject the null hypothesis when comparing underachievers under PostTask and Pre-Task (p = 0.079). Ordered probit estimations confirm this difference and show further that underachievers ask for significantly less taxation in PreTask than in Baseline.14

Conclusion

We explored the role in information about "social mobility" on the political self-serving bias in redistributive preferences. On one hand, we found that high ambiguity in terms of the relative importance of external and internal factors is not a necessary condition for the political selfserving bias to arise. Participants exhibit such a bias (taking responsibility for success but blaming others for failure) even if we disclose that success and failure was almost exclusively due to chance, provided that the information comes after the realization of success or failure. Even with full transparency, the experience of success causes subjects to lower their willingness to redistribute, both in the disinterested dictator game and in the redistributive taxation game.

On the other hand, our experimental results suggest that the political self-serving bias does not arise when information about the importance of external versus internal factors is provided before the realization of failure or success. If subjects know the game is rigged before they engage in the real-effort task, then underachievers and overachievers display similar redistributive preferences in the disinterested dictator game, and the desired tax rates of underachievers and overachievers converge in the redistributive taxation game. The reduction in the self-serving bias seem to result from different behavioral patterns. The decrease in the disinterested dictator game appears to result from an increase in redistribution by overachievers. On the contrary, in the redistributive taxation game, underachievers seem to decrease their level of taxation. The difference in behavioral responses to the information provided before the realization of success or failure might be explained by the selfish interests at stake. Participants choose redistribution for others in the disinterested dictator game while they choose for their future self in the redistributive taxation game.

Inspired by the literature on association-based judgment errors (e.g., [START_REF] Arkes | Costs and benefits of judgment errors: Implications for debiasing[END_REF], we investigated whether objective information about social mobility can help bridging the gap between the two positions. Our results are somewhat pessimistic: The bias seems to be robust against ex-post information. This puts into question whether the dissemination of empirical results on the transmission of wealth and social mobility (e.g., [START_REF] Bowles | The inheritance of inequality[END_REF] can alter views on redistribution. On the other hand, our results suggest that providing the information ex-ante eradicates the systematic differences between under-and overachievers. The crucial question for a policy application is the meaning of ex-ante. A substantial determinant of one's position in a society depends on factors already present at birth. Consequently, if ex-ante is taken literally, an intervention like the one we apply in our treatment PreTask is not feasible. A less strict interpretation of ex-ante, on the other hand, lends itself to interventions in the educational system, where there is at least a partial veil of ignorance with respect to future success and failure.

While we focus on identifying the causal chain from unequal outcomes on redistributive preferences, it is likely that in reality the converse is also the case, namely that self-serving biases affect the distribution of outcomes, as discussed in [START_REF] Alesina | The evolution of ideology, fairness and redistribution[END_REF]. It would be interesting for further research to explore the dynamics of the self-serving bias in repeated interaction. Furthermore, while we have demonstrated that ex-post information does not seem to debias redistributive preferences, we cannot conclude that the bias is immune to any kind of information. For example, participants could be informed about the very existence of the political self-serving bias before making redistributive choices. Similarly, one could consider institutional settings where individuals must consider the viewpoint of the other members of their society: overachievers could change their views if they decide after contemplating the situation of the underachievers, and vice versa.

To conclude, our results replicate earlier findings on the political self-serving bias and explore two potentially debiasing interventions. We find that the bias is surprisingly robust against expost information. Our ex-ante information treatment, on the other hand, reveal no systematic differences between over-and underachievers. The observation that the bias survives our ex-post intervention is difficult to rationalize based on theories of social preferences or the accountability principle. This suggests that our setup produces biased behavior, best explained by assuming individuals are subject to belief perseverance. The second result is more positive for theories of rational choice and suggests that, in order to produce biased beliefs, we need some degree of ambiguity in the experimental setup. The mere experience of success and failure alone does not bias redistribution choices between under-and overachievers.

A Appendix: Tables & Figures

= Φ(c 1 -Xβ) If 1 < j < J :P r[y = j|Z, under, baseline] = Φ(c j -Xβ) -Φ(c j-1 -Xβ) If j = J :P r[y = J|Z, under, baseline] = 1 -Φ(c j-1 -Xβ)
For an overachiever, the associated probabilities write:

If j = 1 :P r[y = 1|Z, over, baseline] = Φ(c 1 -Xβ -β 3 ) If 1 < j < J :P r[y = j|Z, over, baseline] = Φ(c j -Xβ -β 3 ) -Φ(c j-1 -Xβ -β 3 ) If j = J :P r[y = J|Z, over, baseline] = 1 -Φ(c j-1 -Xβ -β 3 )
The change in probabilities for overachievers for each choice in the baseline treatment is given by P r[y = j|Z, over, baseline] -P r[y = j|Z, under, baseline]. For each specific outcome, it writes:

If j = 1 :M E j (baseline) = Φ(c 1 -Xβ -β 3 ) -Φ(c 1 -Xβ) If 1 < j < J :M E j (baseline) = Φ(c j -Xβ -β 3 ) -Φ(c j-1 -Xβ -β 3 ) -Φ(c j -Xβ) + Φ(c j-1 -Xβ) If j =< J :M E j (baseline) = 1 -Φ(c j-1 -Xβ -β 3 ) -1 + Φ(c j-1 -Xβ)
We thus obtain all changes in probability associated with each choice for the baseline treatment: ∀j, M E j (baseline).

The marginal effect of being an overachiever on the redistribution level is given by the sum of proposed redistribution outcomes times the increased probability of these redistribution outcomes, which writes:

M E(baseline) = j=J j=1 j × M E j (baseline) (4)
PreTask . The computation of the marginal effects for the Pre-task treatment is similar up to a constant term β 2 which captures an overall treatment variation. We have:

If j = 1 :M E j (pretask) = Φ(c 1 -Xβ -β 2 -β 4 ) -Φ(c 1 -Xβ -β 2 ) If 1 < j < J :M E j (pretask) = Φ(c j -Xβ -β 2 -β 4 ) -Φ(c j-1 -Xβ -β 2 -β 4 ) -Φ(c j -Xβ -β 2 ) + Φ(c j-1 -Xβ -β 2 ) If j =< J :M E j (pretask) = 1 -Φ(c j-1 -Xβ -β 2 -β 4 ) -1 + Φ(c j-1 -Xβ -β 2 )
PostTask . Similarly, the post-task marginal effects are given by:

If j = 1 :M E j (posttask) = Φ(c 1 -Xβ -β 1 -β 5 ) -Φ(c 1 -Xβ -β 1 )
If 1 < j < J :M E j (posttask) = Φ(c j -Xβ -β 1 -β 5 ) -Φ(c j-1 -Xβ -β 3 -β 5 ) -Φ(c j -Xβ -β 1 ) + Φ(c j-1 -Xβ -β 1 )

If j =< J :M E j (posttask) = 1 -Φ(c j-1 -Xβ -β 1 -β 5 ) -1 + Φ(c j-1 -Xβ -β 1 )

C Instructions

Instruc(ons Part One

Hello everyone! You are about to par1cipate in an experiment, and we thank you for your par1cipa1on. This experiment will be in two parts. These instruc1ons are for the first part of the game. The instruc1ons about the second part will be distributed at the end of the first part. If the instruc1ons are not clear, or if a ques1on persists, please raise your hand and wait for an instructor to come to you. The present experiment consists of purely individual decision moments as well as group interac1ons with other individuals present in the room. At certain stages of the game, your decision may affect the gains of other par1cipants, and vice versa. Thus, it is strictly forbidden to communicate for the dura1on of the experiment. In case of viola1on of this rule, we will be obliged to exclude you from the room.

Throughout this experiment, you will obtain ECU (Experimental Currency Unit). At the end of the experiment, the total number of ECUs you have accumulated throughout the game will be converted into EURO, and you will then be returned. The conversion rate is 5 ECU for 1 Euro.

The first part of this experiment is broken down into several stages. At first, you will be randomly assigned to a task. This task will consist in coun1ng the number of 1s present in a sequence of 0s and 1s. This task will be 1med, and your gains will depend on the number of correct answers you have provided (1 ECU per correct answer). The task that will be assigned to you can be either easy (50% chance) or difficult (50% chance). Each task type contains the same number of ECUs to win.

Example: 011010. This series of digits contains 3 ones, so the correct answer is: 3.

[The following paragraph is for PRETASK treatment only.] Note that at the end of this first task, the computer will tell you if you are among the 50% of the most successful par1cipants, or the 50% of the least successful par1cipants. In reality, the experiment is made in such a way that it is almost impossible for a par(cipant who has had the difficult task to do be<er than a par(cipant who has had the easy task. Indeed, all par1cipants in previous sessions who had the hard task were consistently less successful than most par1cipants in their session. In other words, none of the par1cipants who had the difficult task was more successful than a par1cipant who had the easy task.

Once all par1cipants have completed the task assigned to them, you will need to answer a few ques1ons. Then, two par1cipants (target par1cipants) will be drawn at random. For these two par1cipants, the first part of the experiment will stop. Of these two selected par1cipants, the par1cipant with the most ECU will be called Par1cipant A, while the par1cipant with the least ECU will be called Par1cipant B.

The other par1cipants (judges) will not know the number of ECUs of these two selected players. They will, however, be able to redistribute the ECUs that player A has in addi1on to player B to player B. Each judge par1cipant may choose not to redistribute, redistribute par1ally, or redistribute completely.

Once all the judging par1cipants have made a redistribu1on choice, a solu1on will be drawn randomly and will be put in place for the two target candidates only.

D Appendix: Screenshots

Figure 1 :

 1 Figure 1: Screenshot of the Disinterested Dictator Game game.

Figure 2 :

 2 Figure 2: Performance in the hard and easy real-effort tasks.

Figure 3 :

 3 Figure 3: Histograms of the redistribution decisions in the DDG.

Figure 4 :

 4 Figure 4: Redistribution in the Disinterested Dictator Game.

Figure D1 :

 D1 Figure D1: Screenshot of the Control Questions for the Redistributive Taxation Game [1/4].

Figure D2 :

 D2 Figure D2: Screenshot of the Control Questions for the Redistributive Taxation Game [2/4].

Figure D3 :

 D3 Figure D3: Screenshot of the Control Questions for the Redistributive Taxation Game [3/4].

Figure D4 :

 D4 Figure D4: Screenshot of the Control Questions for the Redistributive Taxation Game [4/4].

Figure D5 :

 D5 Figure D5: Screenshot of the Private Choice in the Redistributive Taxation Game.

Figure D6 :

 D6 Figure D6: Screenshot of the First Round of Bargaining in the Redistributive Taxation Game.

Figure D7 :

 D7 Figure D7: Screenshot of the Second Round of Bargaining in the Redistributive Taxation Game.

Table 1 :

 1 Regressions of the demand for redistribution in the DDG.Results 1 and 2 are supported by multivariate analyses presented in Table

	Variable

Table 2 :

 2 Regressions of the demand for redistribution in the RTG Tax rate in the Redistributive Taxation Game.

	Variable

Table A1 :

 A1 Descriptive Statistics per type of participant and per treatment. Histograms of the redistribution decisions in the RTG.

					Baseline				PostTask						PreTask
				Over		Under		Over			Under			Over	Under
	Redistribution DDG		35.89			29.04		38.66			31.39			41.64	38.438
				(21)		(20.27)		(22.8)			(16.93) (24.59) (16.214)
	Redistribution RTG		42.74			37.62		45.48			34.88			36.67	38.889
			(23.61)		(30.2)		(24.51) (20.09)		(22.8)	(23.997)
	age			19.71			20.06		20.96			20.87			22.19	22.167
				(1.33)		(1.85)		(3.22)			(2.33)			(3.5)	(3.094)
	polit			4.9				4.46		4.35			4.12			4.11	3.278
				(2.25)			(2.2)		(2.22)			(2.07)			(1.6)	(2.25)
	female			51%			48%		63%			55%			47%	50%
	Means and standard deviations (in parentheses)
	.1 .2 .3 .4 Figure A1: 0 Baseline		Underachiever			Overachiever
	PostTask	.1 .2 .3 .4
		0
	PreTask	.1 .2 .3 .4
		0
		0%	10%	20%	30%	40%	50%	60%	70%	80%	90%	100%	0%	10%	20%	30%	40%	50%	60%	70%	80%	90%	100%
								Redistribution RTG

Table A2 :

 A2 Regressions of the absolute performance.

	Variable	Performance
		(1)	(2)
	PostTask	-0.421	-0.441
		(0.348)	(0.354)
	PreTask	0.662	0.510
		(0.449)	(0.463)
	Overachiever × Baseline	6.202*** 6.169***
		(0.348)	(0.348)
	Overachiever × PostTask	6.226*** 6.198***
		(0.348)	(0.348)
	Overachiever × PreTask	6.278*** 6.291***
		(0.531)	(0.532)
	Age		0.0548
			(0.0450)
	Political orientation		0.000483
			(0.0522)
	Female		-0.407*
			(0.225)
	Constant	10.35*** 9.479***
		(0.246)	(0.975)
	H 0 : Over. × Baseline=Over. × PostTask (p-value)	0.9614	0.9538
	H 0 : Over. × Baseline=Over. × PreTask (p-value)	0.9055	0.8477
	H 0 : Over. × PreTask =Over. × PostTask (p-value)	0.9353	0.8831
	Observations	408	408
	R2	0.663	0.667
	OLS, Standard errors in parentheses
	*** p<0.01, ** p<0.05, * p<0.1

Table A3 :

 A3 Separate regressions of the demand for redistribution in the DDG.

	Variable		Redistribution DDG
		(1)	(2)	(3)	(4)	(5)
	Regression for Baseline
	ME Overachiever	-6.723** -6.426** -6.540**	-6.587**	-7.742**
		(3.265)	(3.179)	(3.204)	(3.226)	(3.404)
	Regression for PostTask
	ME Overachiever	-7.878** -7.274** -7.764**	-7.820**	-7.942**
		(3.157)	(3.004)	(3.107)	(3.097)	(3.246)
	Regression for PreTask
	ME Overachiever	-0.732	-0.987	-1.766	-1.692	-3.071
		(4.621)	(4.232)	(4.649)	(4.853)	(5.219)
	Controls	No	Yes	Yes	Yes	Yes
	Switchers	.	.	Dummy	Excluded	IV
	Standard errors in parentheses
	Ordered probit estimates, and marginal effects (ME) in percentage points.
	*** p<0.01, ** p<0.05, * p<0.1

Table A4 :

 A4 Separate regressions of the demand for redistribution in the RTG.

	If j = 1 :P r[y = 1|Z, under, baseline]
	Variable		Redistribution RTG
		(1)	(2)	(3)	(4)	(5)
	Regression for Baseline
	ME Overachiever	-7.479*	-8.176*	-7.786*	-7.755*	-8.473*
		(4.332)	(4.239)	(4.285)	(4.258)	(4.387)
	Regression for PostTask
	ME Overachiever	-10.520*** -9.576*** -9.225*** -9.132*** -8.909**
		(3.523)	(3.476)	(3.512)	(3.504)	( 3.649)
	Regression for PreTask
	ME Overachiever	4.306	4.643	4.714	4.597	3.869
		(5.740)	(5.803)	(5.855)	(5.927)	(6.603)
	Controls	No	Yes	Yes	Yes	Yes
	Switchers	.	.	Dummy	Excluded	IV
	Standard errors in parentheses
	Ordered probit estimates, and marginal effects (ME) in percentage points.
	*** p<0.01, ** p<0.05, * p<0.1
	Targets of the DDG excluded.

A related strand of literature studies similar effects in bargaining[START_REF] Gächter | Moral property rights in bargaining with infeasible claims[END_REF][START_REF] Karagözoğlu | Performance information, production uncertainty, and subjective entitlements in bargaining[END_REF], dictator games[START_REF] Erkal | Relative earnings and giving in a real-effort experiment[END_REF][START_REF] Spiekermann | Objective and subjective compliance: A norm-based explanation of 'moral wiggle room[END_REF], money burning[START_REF] Fehr | Is increasing inequality harmful? Experimental evidence[END_REF], and effort provision[START_REF] Kovárík | Distributive preferences and effort provision: What determines what? Working Paper[END_REF].

In few sessions of the PostTask treatment, participants were not given this last piece of information, namely that the game was designed such as randomness almost completely determined success. We found no statistical difference neither for the demand of redistribution in the DDG (p = .806, Wilcoxon rank-sum test), nor in the RTG (p = .652), and we pool these observations for the remainder of the paper.

The associated redistribution choices were: 0%, 9.1%, 18.2%, 27.3%, 36.4%, 45.5%,

54.5%, 63.6%, 72.7%, 81.8%, 90.9%, 100%. We deliberately excluded the 50-50 split (as e.g. in[START_REF] Bellemare | Measuring Inequity Aversion in a Heterogenous Population Using Exprimental Decisions and Subjective Probabilities[END_REF].

We slightly depart from the DDG introduced in[START_REF] Deffains | Political self-serving bias and redistribution[END_REF] in three ways. First, we do not display the difference of tokens between the two targets, but let dictators choose on a percentage of redistribution. This ensures that all dictators have the same strategy space. Second, we make the consequences of the dictator's decision highly salient by means of the graphical representation. Third, participants do not learn their payoff at the end of the DGG, which minimizes spillovers on the subsequent game.

The second series of tasks consisted in memorizing pictures and short video clips. After displaying each picture or video clip for about one minute, subjects had to answer questions regarding their contents. For this second real-effort task there was no hard/easy manipulation and no median split of the participants according to performance. The sole purpose of the second real-effort task was to generate variance in the earnings among subjects in a group to render the choice over redistribution in the RTG meaningful.

After the random dictator mechanism participants were asked to bargain within their group of four to reach a consensus on the redistribution rule. The bargaining process consisted of five stages, in each of which participants

The coefficient associated with overachievers for Baseline loses in significance. Results available upon request.

All regression results are available on request.

B Marginal Effects

In Section 4, we are interested in estimating the marginal effect of being an overachiever in each treatment. To do so, we run ordered probit estimations where the equation for the latent utility writes:

where X is the matrix of control variables including a constant term.

In a linear model, the marginal effects of being an overachiever would be equal to β 3 for the Baseline treatment, β 4 for the PreTask treatment, and β 5 for the PostTask treatment. As [START_REF] Ai | Interaction terms in logit and probit models[END_REF]; [START_REF] Norton | Computing interaction effects and standard errors in logit and probit models[END_REF] note, the marginal effect of an interaction of variables in non-linear models is more complex to retrieve. We derive below how we computed the marginal effects on redistribution for our ordered probit estimations.

Let y * be the latent utility associated with the redistribution decision, where the individual can make a choice between J redistribution options, y ∈ {1, ..., J}. We have the following decision rule:

where y * is defined as in equation 1. Alternatively, we write y * = Zδ by stacking together the independent variables.

Assuming normally distributed random terms, the probability associated with each outcome is given by:

Baseline. First, we derive the marginal effect of being an overachiever in the baseline treatment. The probability of making the choice j as an underachiever is:

Instruc(ons Part Two

The second part of the experiment is about to begin. In this part, you will have to complete several tasks. These tasks are substan=ally different from those presented to you in the first part of this experiment.

At the beginning of the game, you will be put in a group of 4 par=cipants. The tasks to be performed will be individual, but all members present at the experiment will face the same tasks. From these tasks, you will get an individual income. This individual income may be taxed in part to be redistributed equally to all other members of the group (including you). All individuals in the group will be taxed with the same percentage. The tax rate can be: 0%, 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, 100%.

Example: The The redistribu=on rule can be determined in two different ways. On the one hand, you will be asked to indicate your preferred level of redistribu=on. Your answer will be strictly private and will not be communicated to other par=cipants. On the other hand, you will have to choose a level of redistribu=on unanimously with the other members of your group.

When making a collec=ve decision, you have five rounds to unanimously agree on a decision rule with the other three players in your group. Each turn, you will indicate the amount of taxa=on you wish to have. In the next round, you will learn the choice of each of the other par=cipants in your group. The decision is made if all par=cipants have indicated the same tax rate. In the event of a failure to make an unanimous decision, the par=cipants will not play the second game and will receive a lump sum, which will be indicated to you at the beginning of the collec=ve decisionmaking.

If your group has managed to agree on a common decision rule, you will play the second game. In this game you can win up to 80 ECU. The redistribu=on rule chosen will have a 50% chance of being the rule decided unanimously by the group. In the remaining 50% of cases, one of the private solu=ons will be randomly drawn to become the group's decision rule.

Individual income for the task

Level of taxation Gains from redistribution