Amaury Mathis

François Courvoisier
email: francois.courvoisier@femto-st.fr

Luc Froehly

Luca Furfaro

Maxime Jacquot

Pierre-Ambroise A Lacourt

John Michael Dudley

Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams

come   L'archive ouverte pluridisciplinaire

The use of high power lasers for material ablation is widely applied in a broad range of technologies in material processing [START_REF] Phipps | Laser Ablation And Its Applications[END_REF] . With appropriate selection of laser wavelength, power and pulse duration, many impressive examples of both surface and volumetric processing in a variety of materials have been developed, at scales from centimeters to nanometers. [START_REF] Pronko | [END_REF][3][4] A particular challenge in laser material processing, however, is machining structures both at the surface and within the sample that have longitudinally-varying characteristics, because this requires simultaneous control of beam steering and sample rotation and translation. This becomes especially difficult when the desired spatial features have micron or sub-micron scale. 5 In this paper, we present a solution to this problem using accelerating femtosecond laser beams to machine both curved surface profiles and curved trenches on micron scales.

For diamond and silicon samples, we machine initially square sample edges to curved circular profiles, and we also show experimental results in silicon where accelerating beams have been applied to writing curved trenches to ~ 80 µm depth within the bulk sample. In fact, although our experiments have been performed in the femtosecond regime, accelerating beams can be generated with laser beams of any temporal duration (even continuous wave). Our results therefore extend the technology of laser material processing into a regime where curved features are generated from the intrinsic properties of the laser beam itself rather than as a result of geometric sample rotation.

Accelerating beams are a class of laser beam that consist of a strongly-localized high intensity lobe that follows a propagation trajectory that displays curvature in a dimension transverse to its propagation. Since their experimental demonstration in 2007, 6 accelerating beams have been applied to particle sorting and nonlinear optics, 7,8 and they have recently attracted much attention as it has been shown that they represent a fundamentally original class of solution to Maxwell's equations for spatial beam propagation 9,10 and that they represent a stationary solution in the nonlinear regime. 11 They can be readily synthesized from a Gaussian laser beam through the application of an appropriate spatial phase function, and their properties can be interpreted usefully in terms of the properties of optical caustics. 10,[12][13][14][15][16] In our experiments, we generate accelerating beams from a 5 kHz amplified Ti:Sapphire laser emitting 100 fs pulses at a central wavelength of 800 nm, by applying appropriate spatial phase to the Gaussian laser beam using an optically-addressed spatial light modulator (SLM). We design a phase profile to generate an accelerating laser beam in the form of a primary high intensity lobe that takes the form of a curved cylinder or tube as it propagates. Our phase mask design uses the combination of two 1-dimensional masks at 90 degrees. 6 The phase masks were designed in order to produce an accelerating beam in the first order of diffraction of our SLM. An imaging system is used to place the Fourier-filtered image of the SLM phase profile at the back focal plane of a high numerical aperture (NA 0.8) microscope objective, as shown on figure 1(a). The accelerating beam with main intensity lobe transverse dimensions of ~2 µm is produced in the focal region of the microscope objective. The 3D surface of isointensity at half-maximum which highlights the primary lobe only is shown in red in figure 1(a). To aid in the visualization of the complete spatial field profile, the figure on the right plots the isosurface at 5% of the maximal intensity at a viewing angle selected to illustrate the acceleration profiles of the subsidiary lobes of the Airy functions. for arbitrary acceleration profiles even in the non-paraxial regime. 10,17,[START_REF] Gilmore | Catastrophe theory for scientists and engineers[END_REF] The high intensity accelerating tube separates two regions: on the outside of the circle, there is subsidiary structure corresponding to the Airy function minima, while on the inside there is a shadow zone of zero intensity. This structure naturally suggests application in machining convex surfaces which is the case considered here. As shown in figure 1(a), we position the sample profile in the plane of the beam acceleration so that the high intensity tube is the first area of the beam that encounters the surface to be machined. It is this part of the beam that therefore induces ablation, while the other lobes of the beam freely propagate outside the sample.

Sample positioning was performed by visualization through the focusing microscope objective on a CCD camera as in reference 19. Samples were placed on a gimbal mount to ensure planarity better than 1 mrad, and positioned using a motorized stepping motor with 100 nm resolution. In the back focal plane of the microscope objective, the beam polarization is parallel to x axis, but the polarization of the accelerating beam in the vicinity of the main intensity lobe is in the xz plane perpendicular to the beam trajectory. The pulse duration was 120 fs at the sample site.

We have applied this approach to machine curved surfaces of sample materials that were both transparent and opaque at the 800 nm pump wavelength used. For transparent materials, we selected diamond slides, and for the opaque material, we used silicon wafers. A selection of results is shown in Fig. It is easy to confirm from the SEM images that the curved profiles in both diamond and silicon are in quantitative agreement with the trajectory of the main intensity lobe. In particular, the precise shapes of the machined profiles are determined by the form of the beam profile that follows the fluence isocontour at the ablation threshold. Indeed, the bottom images in Fig. 2 show a side view of the SEM images of the samples to permit a quantitative comparison with the threshold fluence isocontour profile for each material (~1.1 J/cm 2 for diamond 21 and ~1J/cm 2 for silicon 20 ). There is clearly excellent visual agreement; more detailed analysis shows that the machined surface follows the beam contour to within ±0.5

µm. The processed surfaces exhibited roughness on the order of 200 nm (diamond) and 500 nm (silicon) with no evidence of deleterious structures such as cracks or voids.

We also investigated the possibility of using the accelerating beams directly incident upon the surface in order to machine curved trenches within a sample. Figure 3 shows the SEM images of trenches machined in a 100 µm thick silicon sample using a 120 µm radius beam. In Fig. 3(a) we show results at fixed energy of 11 µJ/pulse using a sample translation of 100 µm/s as a function of increasing (back and forth) passes along a 4 mm line. The sample was diced before imaging. It is clear that the machining process evolves downwards into the sample from the surface, and that trench depth increases with the number of passes.

Significantly, we see that the debris deposition using the accelerating beam is highly asymmetric, with debris mainly redeposited on the surface of the sample away from the position of the primary intensity lobe. A possible explanation for this could be that the ablated plume is directed perpendicular to the ablated surface itself 22 , and we anticipate that this phenomenon could be advantageous in applications such as laser dicing. In Fig. 3(b) we

show similar results but studying the variation of trench properties as a function of energy with a fixed number of passes. With this side view, we also compare the form of the machined trenches with the profile of the accelerating beam (dashed line) and we see excellent agreement.

In conclusion, we have shown in this paper how accelerating beams can be used for ultrafast laser processing of curved profiles in transparent and opaque materials, applied to both surface processing and trench machining. Although our results have been obtained using femtosecond pulses where the thermally affected zone is reduced compared to laser processing techniques using longer pulses, the essential approach introduced here is applicable more widely provided that the thermally affected zone is smaller than the curvature of the target trajectory. We anticipate a broad range of applications in different technological fields such as the processing of flat panels and precision photonic components

Figure 1 (

 1 Figure 1(b) shows experimental measurements of the beam profile along the

2 .

 2 Scanning electron micrograph (SEM) images of the results obtained for diamond are shown in Fig.2(a), where we machined a circular profile of 70 µm radius on a diamond slide of 50 µm thickness. The process used here consisted of moving the sample into the 11 µJ energy beam in 0.3 µm steps along the x-direction, and machining at each position with one (back and forth) pass along the edge at a translation speed 10 µm/s. Results for silicon are shown in Fig.2(b). The higher ablation rate for silicon 20 allowed translation using larger steps of 0.5 µm while the other parameters were kept identical.

Acknowledgements:

  The authors thank B. Guichardaz, L. Gauthier-Manuel and R. Salut for important technical assistance. We acknowledge funding from the Région of Franche-Comté and the Agence Nationale de la Recherche, contracts 2011-BS04-010-01 NANOFLAM, 2011-BS09-026-02-SMART_LASIR, ANR-09-BLAN-0065 IMFINI.

Figure 1 :

 1 Figure 1: (a) Schematic of our experimental setup showing focusing of shaped accelerating beam into our sample. (b) Experimental measurements of accelerating beam profile along direction of propagation i.e. the xz plane in a side view, and viewed from the top in the xy plane at the point of peak intensity at z = 0. The white line shows the target circular beam trajectory with radius R=120 µm.

Figure 2 :

 2 Figure 2: Machining of (a) diamond and (b) silicon using circular accelerating beams. Top row subfigures show 3D view, bottom row subfigures show images with the sample oriented for quantitative profile measurements. In (a) the profile (beam) radius is 70 µm and the sample thickness is 50 µm. In (b) the profile (beam) radius is 120 µm and the sample thickness is 100 µm. In the bottom row we compare the form of the machined surface with the expected profile using the calculated ablation threshold fluence isocontour.

Figure 3 :

 3 Figure 3: Machining of curved trenches in silicon using accelerating beams. In (a) we show the variation in trench properties for different number of passes at fixed energy of 11 µJ. In (b) we use a fixed number of 5 passes and study the variation in the trench properties with incident energy.


F Courvoisier
email: francois.courvoisier@femto-st.fr

A Mathis

L Froehly

R Giust

L Furfaro

P A Lacourt

M Jacquot

J M Dudley

Sending femtosecond pulses in circles: highly non-paraxial accelerating beams

We use caustic beam shaping on 100 fs pulses to experimentally generate non-paraxial accelerating beams along a 60 degree circular arc, moving laterally by 14 µm over a 28 µm propagation length. This is the highest degree of transverse acceleration reported to our knowledge. Using diffraction integral theory and numerical beam propagation simulations, we show that circular acceleration trajectories represent a unique class of non-paraxial diffraction-free beam profile which also preserves the femtosecond temporal structure in the vicinity of the caustic.

Transversally accelerating beams such as Airy beams exhibit curved trajectories at their point of maximum intensity. Since this intensity maximum is strongly localized over an extended distance along the trajectory, they constitute a novel class of "non-diffracting" beam and have attracted wide interest for applications in linear and nonlinear optics [1][2][3][4][5]. Studies of accelerating beams, however, have to date been confined only to the paraxial regime, for small deviation angles typically < 10 degrees.

In this letter, we apply a caustic-based approach to accelerating beam synthesis [6,7] to generate accelerating beams in the non-paraxial regime. We consider in particular a circular trajectory, accelerating 100 fs pulses at 800 nm along a 60 degree arc, moving a 1 µm primary intensity lobe laterally by 14 µm over a 28 µm propagation distance. To our knowledge, this is the highest degree of transverse acceleration reported to date. We also analyze the spatio-temporal beam properties using diffraction integral theory, and show how the primary lobe size is linked to the local radius of curvature of the trajectory. Moreover, we show that in the non-paraxial regime, it is only the circular propagation trajectory that can be considered as genuinely diffraction-free. We also numerically show that the temporal pulse duration is preserved in the vicinity of the primary lobe.

Our experiments applied caustic beam shaping to a Gaussian beam from a Ti:Sapphire laser at 800 nm and generating 100 fs pulses [8]. The beam (with near-uniform spatial phase) was incident upon an optically-addressed SLM (Hamamatsu PAL-SLM) to which an appropriate caustic-phase function was written. In contrast to Fourierbased techniques, the caustic phase function is applied directly to the incident beam. The starting point with this approach is the desired acceleration trajectory () cz along the propagation direction z. With () cz defined, the corresponding SLM phase mask () y Φ to apply in the transverse beam plane y is determined from:

2 1/ 2 / (1 ) d / dy k c c ′′ Φ = +
where k is the wave vector and the derivative c dc / dz ′ =

. The beam characteristics in z and y are related using the Legendre transform of the caustic [7]. The beam was imaged and demagnified by a factor of 278 using a lens and a microscope objective (numerical aperture 0.8) in a 4f configuration to allow us to obtain caustic trajectories of micron dimensions (suitable for applications in micromachining and nonlinear optics). The plane at which the SLM is imaged defines the point from where the micron-scale accelerating beam is generated, and all results below show characterization of the demagnified beam after this plane. Recording the experimental images used additional × 55 imaging of the beam on a CCD camera.

Figure 1 shows results obtained for a target nonparaxial trajectory corresponding to a circle of radius 35 µm. Figure 1(a) (top) compares the target caustic (dashed white line) with results from numerical propagation of a Gaussian beam to which an ideal caustic phase profile is applied. These numerical results confirm that we can indeed achieve non-paraxial acceleration with our setup. The results in Fig. 1(a) (bottom) provide experimental confirmation. We see very good agreement between experiment and the numerical results, both in terms of the transverse localization of the beam intensity along the target caustic trajectory, as well as the variation of the local intensity with propagation along the caustic (we discuss this more below). Residual scattering and fringes seen inside the caustic in experiment arise from unfiltered zero-order diffraction, but this does not significantly modify the field localization along the caustic itself. Note that in our experiments, the plane of polarization was perpendicular to the acceleration trajectory [the yz plane in Fig 1(b)] and so vectorial effects are expected to have negligible influence on propagation. The experimental acceleration profile is maintained over an arc > 60° measured over the range z ~ 5-40 µm where the main lobe size does not vary by more than 10%. The lateral displacement from the SLM to the trajectory minimum is 14 µm over a propagation distance of 28 µm. Although we show results here for a circle, other cases of non-paraxial caustics (e.g non paraxial polynomials) have also been generated and display similar characteristics.

To interpret these results further, Fig. 1(b) shows details of the caustic geometry beyond the SLM image plane, defining a normal direction to the caustic (u) along which we can extract and plot the local intensity profile at any point M. Figure 1(c) shows different intensity profiles along the caustic at two points P and Q, comparing numerical results (line) and experimental measurements (circles), with good agreement.

The different values of maximum intensity at P and Q and the intensity variation along the caustic arise from the use of finite numerical aperture Gaussian beam illumination. Figure 1(b) shows how this can be readily understood in terms of simple geometrical arguments. Specifically, the tangent rays from the SLM image plane corresponding to points P (low intensity) and Q (high intensity) arise respectively from points in the wings and near the center of the incident Gaussian beam. It is thus readily seen that it is the intensity difference between the two points on the initial beam that feeds in directly to the intensity profile along the accelerating profile.

The above qualitative description can be quantified using analysis based on the non-paraxial Sommerfeld diffraction integral. Using this approach, we derive the intensity profile at any point in the vicinity of the caustic as follows:

1/3 2/3 2 2 0 2 2 2 ( ) ( ) Ai () 21 M M c kz k Iu Iy u k Rz c π   ′′     = -     ′ +    (1)
Here, = (, ()) is the point at which the intensity () is calculated, and the normal coordinate u is as defined in Fig. 1(b). Ai is the Airy function, 0 ( ) is the intensity of the incident beam corresponding to the ray from the SLM image plane tangent to M, and

2 3/2 ( ) (1 ) / Rz c c ′ ′′ = +
is a local radius of curvature geometrically defined for every point on the trajectory.

Primes indicate differentiation with respect to z.

The non-paraxial intensity given by Eq. ( 1) is compared to the experimental results (no free parameters) at points P and Q in Fig. 1(c). The agreement at Q is very good in the vicinity of the primary lobe, but we see deviations for the subsidiary lobes, and also for point P. In fact, this is expected since Eq. ( 1) is derived assuming a slowly varying incident beam profile and the caustic at P and the side lobes are generated from points on the beam away from the slowly varying intensity maximum. Better agreement at all points on the caustic would be obtained with a larger input beam and SLM. We also note that the result in terms of an Airy function is obtained using a third-order phase expansion which is valid for caustic trajectories with non-vanishing second derivatives c′′ . For such cases, our analysis extends into the non-paraxial regime the universal association of the Airy function with arbitrary acceleration profiles [7]. Our analysis can be generalized to other classes of trajectory using higher order phase expansions (using e.g. Pearcy integrals), but the results are qualitatively similar to the Airy function in yielding beam properties in terms of a primary central lobe with lower amplitude oscillations [START_REF] Nye | Natural focusing and fine structure of light[END_REF].

We also use Eq. ( 1) to consider more generally the nature of "non-diffracting" beams in the non-paraxial regime. To this end, we derive the transverse size (FWHM) of the primary intensity lobe in the direction normal to the caustic as:

() = Δ[() 2 2
⁄ ] 1/3 with Δ = 1.630 the half maximum of the intensity |Ai| 2

10

. From this result, we see that the condition for a "nondiffracting" trajectory [i.e. a z-invariant ()] is only achieved when the local radius of curvature () is constant. This corresponds to requiring that the chosen trajectory () is in fact a circle, and a consequence of this is that all trajectories that are not circular, including the well-known parabolic beam trajectory used to initially generate accelerating beams, are in fact not ideally diffraction-free outside the paraxial regime. Significantly, we note that analysis based on Maxwell equations has recently presented a first-principles demonstration of this result [ ].

This non-diffracting nature of the circular caustic is confirmed in Fig. 2(a) which plots the evolution with propagation distance of the primary lobe size (), comparing experiment (circles) with the analytic prediction for () (dashed line), as well as results extracted from numerical beam propagation (solid line). The caustic maintains constant width 1.1 ± 0.1 µm over a significant distance of ~20 µm. For comparison, a Gaussian beam of the same size (as beam waist) would have a Rayleigh range of only 3.4 µm. Also note that the lobe of a parabolic beam with minimum size 1.1 µm would actually increase to 1.4 µm over the range shown, where a circular trajectory maintains constant beam size.

The shaded region in the figure is used to show the onset of the regime where numerical beam propagation indicates that the effects of finite numerical aperture of the incident beam induces deviation from diffraction-free behavior. Additional results are shown in Fig. 2(b) where the result in Eq. ( 1) for the variation of the intensity maximum along the caustic IM (u = 0) is compared with experimental and numerical results. We see good agreement in the diffraction-free regime, further providing confirmation of our analysis and interpretation. The high acceleration in the non-paraxial regime raises a natural question of the effect on the spatiotemporal structure of the beam in the femtosecond regime [START_REF] Kaganovsky | [END_REF]. We have numerically investigated this using the non-paraxial plane wave spectrum model of propagation, neglecting SLM material dispersion (valid for pulses longer than ~10 fs). At four different points on the caustic, Fig. 2(c) shows snapshots of the structure of a 15 fs pulse at 800 nm as it propagates. A link to a movie of this propagation is in the figure caption. Figure 3 shows the spatio-temporal intensity plotted in a reference frame of co-moving time and spatial distance from the caustic parallel to the y axis. This view clearly shows how the propagation along the caustic is associated with locallyvarying pulse front tilt and spatio-temporal structure, but significantly, in the non-diffracting regime at z = 15 µm, 25 µm and 35 µm, the temporal structure of the primary lobe in the vicinity of the caustic intensity maximum is invariant. This is seen in the figure where we plot the profile on the target caustic (at u = 0) as the black curve.

On the other hand, outside the non-diffracting regime for z = 55 µm, we see significant pulse temporal broadening.

The pulse widths at these points are given in the figure. The major result of this paper has been to experimentally generate non-paraxial non-diffracting beams along a circular trajectory. Our results have been interpreted with numerical and analytical studies, with the latter providing insight into the nature of diffractionfree propagation in the non-paraxial regime. Within the diffraction free regime, the temporal structure at the caustic maximum is preserved, and we expect our results to impact on uses of highly accelerating beams in nonlinear optics.

Fig. 1 (

 1 Fig. 1 (a) Numerical (top) and experimental (bottom) intensity distributions of a circular accelerating beam. The corresponding target caustic is shown as the white dashed line. (b) Propagation geometry of the caustic beam. (c) Comparison of the numerical (line) and experimental (circles) intensity profile normal to the caustic at points P and Q as indicated. In (c) we also show the analytic result from Eq. 1 (dashed line).

Fig. 2 (

 2 Fig. 2 (a) Evolution of the primary lobe size with propagation. The dashed corresponds to the analytical model of Eq. (1), the solid line to the numerical propagation and the circles to experimental results. (b) Evolution of the intensity along the caustic with the propagation distance. (c) Snapshots of a 15 fs pulse at various points along its propagation. The corresponding propagation times are 40, 80, 140 and 230 fs. [Media 1]

Fig. 3 .

 3 Fig. 3. Intensity distribution of a 15 fs pulse at propagation distances as shown as a function of co-moving time and distance from the caustic.

We thank the Université de Franche Comté, the Région Franche-Comté and the Agence Nationale de la Recherche contract 2011-BS04-010-01 NANOFLAM.


Mostapha Diss
email: diss@gate.cnrs.fr

Patrizia Pérez-Asurmendi
email: patrizia.perez@eco.uva.es

Consistent collective decisions under majorities based on differences

Keywords: Majorities based on difference of votes, Majorities based on difference in support, Probability, Transitivity, Triple-acyclicity

The main criticism to the aggregation of individual preferences under majority rules refers to the possibility of reaching inconsistent collective decisions from the election process. In these cases, the collective preference includes cycles and even could prevent the election of any alternative as the collective choice. The likelihood of consistent outcomes under two classes of majority rules constitutes the aim of this paper. Specifically, we focus on majority rules that require certain consensus in individual preferences to declare an alternative as the winner. In the case of majorities based on difference of votes, such requirement asks to the winner alternative to obtain a difference in votes with respect to the loser alternative taken into account that individuals are endowed with weak preference orderings. Same requirement is asked to the restriction of these rules to individual linear preferences, whereas in the case of majorities based on difference in support, the requirement has to do with the difference in the sum of the intensities for the alternatives in contest.

Introduction

Since Condorcet (1785) introduced the Voting Paradox, it is well known that the aggregation of transitive individual preferences under simple majority rule could lead to inconsistent collective preferences. Recalling the classical example, consider a three-alternative election with alternatives x 1 , x 2 , x 3 and three individuals endowed with the following rankings x 1 x 2 x 3 , x 2 x 3 x 1 and x 3 x 1 x 2 , where, for instance, x 1 x 2 x 3 means that x 1 is preferred to x 2 , x 2 is preferred to x 3 and x 1 is preferred to x 3 . For each pair of alternatives, each individual casts a vote for her/his preferred alternative following just assumed orderings. Adding up these votes, alternatives x 1 , x 2 and x 3 defeat x 2 , x 3 and x 1 respectively, by two votes to one. In that voting situation, there is a cycle on the ordering induced by the strict collective preference. In such a case, that preference fails on transitivity and on triple-acyclicity given the requirements of such conditions. To illustrate, assume that alternative x 1 defeats x 2 and x 2 defeats x 3 ; x 1 defeats x 3 whenever the strict collective preference is transitive whereas x 3 does not defeat x 1 whenever the strict collective preference is triple-acyclic.

Consider now the following voting process' outcome: x 1 defeats x 2 and it is indifferent to x 3 , and x 2 is also indifferent to x 3 . In this case, the weak collective preference fails on consistency. Notice that the strict preference associated with that weak preference behaves right but the indifference relation associated with the weak preference fails on transitivity.

The idea that the Voting Paradox 'certainly could not be very likely to ever be observed in realistic situation' stated by [START_REF] Gehrlein | Condorcet's Paradox and anonymous preference profiles[END_REF], promotes the probabilistic study of the occurrence of that paradox and of their consequences under different aggregation rules. In several studies, it is assumed an a priori probability model to estimate the likelihood of different voting situations, derived the conditions under which the paradox or the effects of that appear and reached probabilities through combinatoric calculus. In this context, stand out the studies about simple majority rule [START_REF] Gehrlein | Condorcet's Paradox and anonymous preference profiles[END_REF][START_REF] Fishburn | The Paradox of Voting: Effects of individual indifference and intransitivity[END_REF][START_REF] Gehrlein | Condorcet's Paradox[END_REF], supermajority rules [START_REF] Balasko | The probability of Condorcet cycles and super majority rules[END_REF][START_REF] Tovey | Probabilities of preferences and cycles with super majority rules[END_REF] or scoring rules [START_REF] Gehrlein | Robustness of positional scoring over subsets of alternatives[END_REF][START_REF] Gehrlein | Constant scoring rules for choosing one among many alternatives[END_REF][START_REF] Gehrlein | Condorcet's Paradox[END_REF][START_REF] Cervone | Which scoring rule maximizes Condorcet efficiency under IAC?[END_REF], among others. In other ones, these probabilities are calculated following the Montecarlo simulation methodology. Specifically, the study of the cyclical and intransitive collective decisions under the simple majority rule are carried out in [START_REF] Campbell | A measure of the importance of cyclical majorities[END_REF], [START_REF] Klahr | A computer simulation of the Paradox of Voting[END_REF], [START_REF] Demeyer | The probability of a cyclical majority[END_REF] and [START_REF] Jones | Condorcet winners and the Paradox of Voting: Probability calculations for weak preference orders[END_REF].

This paper is devoted to analyze and compare the probabilities of consistent collective decisions over three alternatives for two different classes of majorities rules: majorities based on difference of votes [START_REF] García-Lapresta | Majority decisions based on difference of votes[END_REF][START_REF] Llamazares | The forgotten decision rules: Majority rules based on difference of votes[END_REF][START_REF] Houy | Some further characterizations for the forgotten voting rules[END_REF]) and majorities based on difference in support (García-Lapresta and Llamazares 2010). Given two alternatives, these majorities based on differences focus on requiring to an alternative, to be declared the winner, to reach a number of votes or a support that exceeds the number of votes or the support for the other alternative in a quantity fixed before the voting process. Therefore, the difference between these two classes of majorities restricts to the types of individual preferences considered. In the first case, individual preferences are understood as crisp preferences, i.e. given a pair of alternatives individuals declare if they prefer an alternative to another one or if they are indifferent between them. Here, we distinguish between the case where individuals are endowed with weak preferences and the case where individuals are endowed with linear orderings. In the second case, individual preferences are understood as reciprocal preference relations, i.e. given two alternatives, individuals declare the degree with which they prefer an alternative to another one, in other words their intensities of preference, by means of numerical values in the unit interval.

Coming back to the consistent collective decisions analyzed here, we specifically calculate the probabilities of transitive and triple-acyclic strict collective preferences and the corresponding ones of transitive weak collective preferences for the three specifications of majorities based on differences stated before, as the proportion of collective decisions that fulfill each of such consistency conditions over the total number of possible collective decisions.

To calculate the probabilities of consistent outcomes under majorities based on difference of votes taken into account weak and linear individual preferences respectively, we consider that each individual preference ordering is equiprobable by embracing the Impartial Anonymous Culture (IAC) condition [START_REF] Gehrlein | Condorcet's Paradox and anonymous preference profiles[END_REF] to describe the likelihood of the possible individual orderings. On the one hand, assuming weak or linear individual orderings jointly with the IAC condition allow to know the total number of possible collective preferences (again, [START_REF] Gehrlein | Condorcet's Paradox and anonymous preference profiles[END_REF]). On the other hand, the number of consistent profiles is calculated by means of Ehrhart polynomials, a method recently introduced in the social choice literature by [START_REF] Wilson | Probability calculations under the IAC hypothesis[END_REF] and [START_REF] Lepelley | On Ehrhart polynomials and probability calculations in voting theory[END_REF] in order to estimate the probabilities of some voting paradoxes under the IAC condition.

In the case of the calculations of the probabilities of consistent outcomes under majorities based on difference in support, we propose to apply the Montecarlo simulation methodology to estimate such probabilities inspired by the studies in [START_REF] Campbell | A measure of the importance of cyclical majorities[END_REF], [START_REF] Klahr | A computer simulation of the Paradox of Voting[END_REF], [START_REF] Demeyer | The probability of a cyclical majority[END_REF] and [START_REF] Jones | Condorcet winners and the Paradox of Voting: Probability calculations for weak preference orders[END_REF].

Specifically, we generate the individual reciprocal preference relations for the case of three alternatives. Each individual intensity of preference is understood as a continuous random variable in the unit interval consistently built with a specific transitivity condition over the individual's reciprocal preference relations. Then, we fix the required difference in support and aggregate these individual preferences with the corresponding majority based on difference in support. We derive the resultant collective ordering of alternatives and evaluate its consistency. Finally, we iterate that procedure to estimate desired probabilities as the number of consistent orderings over the total number of simulated collective orderings.

The methodology proposed here allows us to hypothesize about a relationship between the type of individual preferences assumed under each rule and the likelihood of inconsistent collective decisions. Moreover, we set forth our results for majorities based on difference of votes with previous ones on simple majority [START_REF] Gehrlein | Condorcet Paradox and the Condorcet efficiency of voting rules[END_REF] and [START_REF] Lepelley | Condorcet's Paradox for weak preference orderings[END_REF]). We also are able to analyze the impact of the types of transitivity conditions for individual reciprocal preferences on the probability of consistent collective decisions under majorities based on difference in support. In addition, we compare our results on probabilities with some theoretical ones about the consistency of the collective preferences under majorities based on difference in support (Llamazares et al 2013;Llamazares and Pérez-Asurmendi 2013).

The paper is organized as follows. Section 2 describes the theoretical framework followed in this paper and introduces majorities based on difference of votes and majorities based on difference in support. Sections 3 and 4 provide the results about the probability of consistent collective decisions under majorities based on difference of votes with linear preferences and with weak preferences, respectively. Section 5 is devoted to the simulated probabilities in the case of majorities based on difference in support. Section 6 discusses the results and concludes.

Preliminaries

Consider a set of alternatives X = {x 1 , x 2 , x 3 } in an election with m individuals. Let S be a binary relation on X, i.e. a subset of the cartesian product X × X. In what follows, x i Sx j stands for (x i , x j ) ∈ S, i.e. when x i is in the relation S with x j . S -1 is the inverse relation of S defined by x i S -1 x j ⇔ x j Sx j and S c is the complement relation of S defined by x i S c x j ⇔ ¬(x i Sx j ). Given two binary relations S and T , the intersection of S and T is also a binary relation defined by

x i (S ∩ T )x j ⇔ (x i Sx j ∧ x i T x j ). A binary relation S on X is 1. reflexive if ∀x ∈ X, xSx, 2. symmetric if ∀x i , x j ∈ X, x i Sx j ⇒ x j Sx i , 3. asymmetric if ∀x i , x j ∈ X, x i Sx j ⇒ ¬(x j Sx i ), 4. antisymmetric if ∀x i , x j ∈ X, (x i Sx j ∧ x j Sx i ) ⇒ x i = x j , 5. complete if ∀x i , x j ∈ X, x i Sx j ∨ x j Sx i , 6. transitive if ∀x i , x j , x l ∈ X, (x i Sx j ∧ x j Sx l ) ⇒ x i Sx l , 7. triple-acyclic if ∀x i , x j , x l ∈ X, (x i Sx j ∧ x j Sx l ) ⇒ ¬(x l Sx i ).
A weak preference R is a complete binary relation on the set of alternatives X. The strict preference P associated with R is the asymmetric binary relation on X defined by P = (R -1 ) c and the corresponding indifference relation I is the reflexive and symmetric binary relation on X defined by I = R ∩ R -1 . P(X) is the set of strict preferences. A weak ordering is a transitive weak preference whereas a linear ordering is also antisymmetric.

From definitions above it is well know that any weak ordering implies a transitive strict preference relation and a transitive indifference relation. Moreover, any transitive strict preference is also a triple-acyclic preference relation. Notice that the converse is not true.

Given that the social decision between two alternatives is given by either a strict preference relation or an indifference relation, and that three alternatives are in contest, we consider the 27 cases in Table 1 as possible social outcomes.

Table 1 Possible social outcomes in a three alternative election.

1. x 1 P x 2 x 2 P x 3 x 1 P x 3 14.

x 1 P x 2 x 2 Ix 3 x 1 Ix 3 2. x 1 P x 3 x 3 P x 2 x 1 P x 2 15. x 1 Ix 2 x 2 Ix 3 x 1 P x 3 3. x 2 P x 1 x 1 P x 3 x 2 P x 3 16. x 1 Ix 2 x 2 Ix 3 x 3 P x 1 4. x 2 P x 3 x 3 P x 1 x 2 P x 1 17. x 1 Ix 3 x 3 P x 2 x 1 Ix 2 5. x 3 P x 1 x 1 P x 2 x 3 P x 2 18. x 2 P x 1 x 1 Ix 3 x 2 Ix 3 6. x 3 P x 2 x 2 P x 1 x 3 P x 1 19. x 1 Ix 2 x 2 P x 3 x 1 Ix 3 7. x 1 P x 2 x 2 Ix 3 x 1 P x 3 20. x 1 P x 2 x 2 P x 3 x 1 Ix 3 8. x 2 P x 1 x 1 Ix 3 x 2 P x 3 21. x 3 P x 1 x 1 P x 2 x 2 Ix 3 9. x 3 P x 1 x 1 Ix 2 x 3 P x 2 22. x 2 P x 3 x 3 P x 1 x 1 Ix 2 10. x 1 Ix 2 x 2 P x 3 x 1 P x 3 23. x 3 P x 2 x 1 P x 3 x 1 Ix 2 11. x 2 Ix 3 x 3 P x 1 x 2 P x 1 24. x 2 P x 1 x 1 P x 3 x 2 Ix 3 12. x 1 Ix 3 x 3 P x 2 x 1 P x 2 25. x 3 P x 2 x 2 P x 1 x 1 Ix 3 13. x 1 Ix 2 x 2 Ix 3 x 1 Ix 3 26. x 1 P x 2 x 2 P x 3 x 3 P x 1 27. x 2 P x 1 x 1 P x 3 x 3 P x 2
Our interest focuses on the frequency of consistent social outcomes given the 27 possible outcomes above. We distinguish among three cases of consistent outcomes; the case of weak orderings corresponding to the first thirteen outcomes, the case of transitive strict preferences corresponding to the first nineteen and the case of triple-acyclic strict preferences corresponding to the first twenty-fifth outcomes.

Individual preferences

We consider that individuals compare the alternatives on X by pairs and declare their preferences by means of values r p ij ∈ [0, 1], where r p ij > 0.5 means that alternative x i is somewhat preferred to x j by the individual p, whereas r p ij < 0.5 signifies that alternative x j is somewhat preferred to x i by the individual p. At that point, we distinguish between the general case in which preferences are represented by reciprocal preferences and the particular case of that, referred to as crisp preferences.

1. Crisp preferences: the values of r p ij are restricted to the set of discrete values {0, 0.5, 1}. If r p ij = 1, individual p prefers alternative x i to alternative x j , whereas if r p ij = 0, individual p prefers x j to x i . If r p ij = 0.5, individual p is indifferent between both alternatives. Condition r p ij + r p ji = 1 guarantees that the preference of individual p is a weak preference. Moreover, the conditions

(r p ij = 1 ∧ r p jl = 1) ⇒ r p il = 1, (r p ij = 0.5 ∧ r p jl = 0.5) ⇒ r p il = 0.5,
assure that the preference of individual p is a weak ordering. Individual linear orderings could also be represented in this framework by including the following condition: r p ij ∈ {0, 1} ∀i = j. Thus, individuals could only be indifferent between an alternative and itself. 2. Reciprocal preferences: the values of r p ij belong to the unit interval [0, 1] with the following interpretation: r p ij > 0.5 indicates that the individual p prefers the alternative x i to the alternative x j , the more the nearer is the value of r p ij to 1 that represents the maximum degree of preference for x i over x j ; conversely, r p ij < 0.5, means that individual p prefers alternative x j to x i , the more the nearer is the value of r p ij to 0 that represents the maximum degree of preference for x j over x i ; finally, r p ij = 0.5 stands for the indifference between x i and x j for individual p. The reciprocity of these preferences is described by the condition r p ij + r p ji = 1. To avoid the possibility of having incoherent individual preferences, we need to assume some kind of rationality condition. But, in this framework, several concepts could be taken to ensure such rationality requirement (see, among others, [START_REF] Zadeh | Similarity relations and fuzzy orderings[END_REF][START_REF] Dubois | Fuzzy Sets and Systems: Theory and Applications[END_REF][START_REF] Dasgupta | Transitivity and fuzzy preferences[END_REF][START_REF] García-Lapresta | Individual-valued preferences and their aggregation: Consistency analysis in a real case[END_REF]. Here, we consider the following transitivity conditions for reciprocal preference relations.

Definition 1 We say that individual p is (a) 0.5-transitive if ∀ i, j, l ∈ {1, 2, 3} (r p ij > 0.5 ∧ r p jl > 0.5) ⇒ r p il > 0.5, (b) min-transitive if ∀ i, j, l ∈ {1, 2, 3} (r p ij > 0.5 ∧ r p jl > 0.5) ⇒ r p il ≥ min{r p ij , r p jl }, (c) am-transitive if ∀ i, j, l ∈ {1, 2, 3} (r p ij > 0.5 ∧ r p jl > 0.5) ⇒ r p il ≥ r p ij + r p jl /2, (d) max-transitive if ∀ i, j, l ∈ {1, 2, 3} (r p ij > 0.5 ∧ r p jl > 0.5) ⇒ r p il ≥ max{r p ij ,
Individual preferences are collected in a vector where each vector-element represents the preferences of an individual. Assuming m individuals 1 and taking into account the above distinction among linear, weak and reciprocal preferences, a profile of linear orderings is a vector (R 1 , . . . , R m ) ∈ L(X) m , being L(X) the set of all linear orderings; a profile of weak orderings is a vector (R 1 , . . . , R m ) ∈ W(X) m , being W(X) the set of all weak orderings; and a profile of reciprocal preferences is a vector (R 1 , . . . , R m ) ∈ R(X) m , being R(X) the set of all reciprocal preference relations.

Majorities based on differences

Given the nature of the three types of individual preferences examined in Subsection 2.1, we also consider three different specifications of majorities based on differences relying on the types of individual preferences that we take into account in each case. In what follows, we refer to these majorities as majorities based on difference of votes with linear orderings, majorities based on difference of votes with weak orderings and majorities based on difference in support for reciprocal preference relations.

The concept of majorities based on difference of votes was introduced in García-Lapresta and [START_REF] García-Lapresta | Majority decisions based on difference of votes[END_REF] and was later axiomatically characterized in [START_REF] Llamazares | The forgotten decision rules: Majority rules based on difference of votes[END_REF], and subsequently in [START_REF] Houy | Some further characterizations for the forgotten voting rules[END_REF]. These rules involve crisp preferences, i.e. given a pair of alternatives, individuals could declare their preference for one of them or their indifference between both alternatives.

Under these majorities, an alternative, say x i , is declared the winner if the number of individuals who prefer that alternative, to the other one, say x j , exceeds the number of individuals who prefer x j to x i in a difference of votes, fixed before the election process. Assuming m individuals, that difference could take any integer value in {0, . . . , m -1}. These majorities are located between simple majority rule where the difference of votes is zero and unanimity where the difference of votes is the total number of individuals m minus one. Moreover, if the indifference state is ruled out from individual preferences, these majorities are equivalent to supermajority rules.

Taking into account the former majorities based on difference of votes, we introduce in Definition 2 the majorities based on difference of votes with linear orderings and in Definition 3 the majorities based on difference of votes with weak orderings. In what follows, the symbol # stands for the cardinality of a set.

Definition 2 (Majorities based on difference of votes with linear orderings or M L k majorities) Given k ∈ {0, 1, . . . , m-1}, the majority based on difference of votes with linear orderings or M L k majority is the mapping

M L k : L(X) m -→ P(X) defined by M L k (R 1 , . . . , R m ) = P L k ,
where

x i P L k x j ⇔ #{p | r p ij = 1} > #{p | r p ji = 1} + k .
The indifference relation associated with P L k is as follows:

x i I L k x j ⇔ #{p | r p ij = 1} -#{p | r p ji = 1} ≤ k .
Example 1 Let R I and R II be the following individual linear preference orderings over the alternatives on X = {x 1 , x 2 , x 3 }.

R I = Ñ 0.5 1 1 0 0.5 1 0 0 0.5 é , R II = Ñ 0.5 0 1 1 0.5 1 0 0 0.5 é . Consider the profile (R 1 , R 2 , R 3 , R 4 , R 5 ) where R p = ® R I if p = 1, 2, 3, R II if p = 4, 5.
Assuming a required difference of votes k equal to 1 and applying the corresponding M L 1 majority we have

|#{p | r p 12 = 1} -#{p | r p 21 = 1}| = |3 -2| ≤ 1 ⇒ x 1 I L 1 x 2 , #{p | r p 23 = 1} = 5 > #{p | r p 32 = 1} + 1 = 0 + 1 ⇒ x 2 P L 1 x 3 , #{p | r p 13 = 1} = 5 > #{p | r p 31 = 1} + 1 = 0 + 1 ⇒ x 1 P L 1 x 3 .
Definition 3 (Majorities based on difference of votes with weak orderings or M k majorities) Given k ∈ {0, . . . , m -1}, the majority based on difference of votes with weak orderings or M k majority is the mapping M k : W(X) m -→ P(X) defined by M k (R 1 , . . . , R m ) = P k , where

x i P k x j ⇔ #{p | r p ij = 1} > #{p | r p ji = 1} + k .
The indifference relation associated with P k is as follows:

x i I k x j ⇔ #{p | r p ij = 1} -#{p | r p ji = 1} ≤ k .
Example 2 Let R I and R II be the following individual weak preference orderings over the alternatives on X = {x 1 , x 2 , x 3 }.

R I = Ñ 0.5 1 0.5 0 0.5 0 0.5 1 0.5 é , R II = Ñ 0.5 0.5 1 0.5 0.5 1 0 0 0.5 é . Consider the profile (R 1 , R 2 , R 3 , R 4 , R 5 ) where R p = ® R I if p = 1, 2, 3, R II if p = 4, 5.
Assuming a required difference of votes k equal to 2 and applying the corresponding M 2 majority we have

#{p | r p 12 = 1} = 3 > #{p | r p 21 = 1} + 2 = 0 + 2 ⇒ x 1 P 2 x 2 , |#{p | r p 23 = 1} -#{p | r p 32 = 1}| = |2 -3| ≤ 2 ⇒ x 2 I 2 x 3 , |#{p | r p 13 = 1} -#{p | r p 31 = 1}| = |2 -0| ≤ 2 ⇒ x 1 I 2 x 3 .
In García-Lapresta and Llamazares ( 2010), majorities based on differences of votes were extended to the framework of reciprocal preferences allowing individuals to declare their degrees of preferences over pairs of alternatives. Majorities based on difference in support allow us to aggregate each profile of reciprocal preferences into a strict collective preference P k over the set of alternatives. Under these rules, the winner alternative is required to reach a support that exceeds the support for the other alternative in a quantity, fixed before the voting process. Formal definition for these majorities is as follows.

Definition 4 (Majorities based on difference in support or M k majorities (García-Lapresta and Llamazares 2010)) Given k ∈ [0, m), the majority based on difference in support or M k majority is the mapping M k : R(X) m -→ P(X) defined by M k (R 1 , . . . , R m ) = P k , where

x i P k x j ⇔ m p=1 r p ij > m p=1 r p ji + k. (2)
The indifference relation associated with P k is defined by:

x i I k x j ⇔ m p=1 r p ij - m p=1 r p ji ≤ k. (3)
Notice that when considering crisp preferences, the expression in (2) goes for defining the majorities based on difference of votes with linear and weak preference orderings.

Example 3 Let R I and R II be the following reciprocal preference relations over the alternatives on X = {x 1 , x 2 , x 3 }.

R I = Ñ 0.5 1 0.9 0 0.5 0.6 0.1 0.4 0.5 é , R II = Ñ 0.5 0.8 1 0.2 0.5 0.7 0 0.3 0.5 é . Consider the profile (R 1 , R 2 , R 3 , R 4 , R 5 ) where R p = ® R I if p = 1, 2, 3, R II if p = 4, 5.
Assuming a required difference in support k equal to 1.75 and applying the corresponding M 1.75 majority we have

5 p=1 r p 12 = 4.6 > 5 p=1 r p 21 + 1.75 = 0.4 + 1.75 ⇒ x 1 P 1.75 x 2 , 5 p=1 r p 23 - 5 p=1 r p 32 = |3.2 -1.8| ≤ 1.75 ⇒ x 2 I 1.75 x 3 , 5 p=1 r p 13 = 4.7 > 5 p=1 r p 31 + 1.75 = 0.3 + 1.75 ⇒ x 1 P 1.75 x 3 .
3 Probability of consistent collective decisions under majorities based on difference of votes with linear orderings

In this section the results about the probabilities of consistent collective decisions under M L k majorities are introduced under IAC assumption. Given that voters are endowed with complete linear preference orderings, there are six possible preference orders that they might have,

x 1 x 2 x 3 (m 1 ) x 1 x 3 x 2 (m 2 ) x 2 x 1 x 3 (m 3 ) x 2 x 3 x 1 (m 4 ) x 3 x 1 x 2 (m 5 ) x 3 x 2 x 1 (m 6 ) (4)
where m i is the number of voters with the associated linear preference ordering.

In this framework, a voting situation is a vector m

= (m 1 , m 2 , m 3 , m 4 , m 5 , m 6 ) such that 6 i=1 m i = m.
As the IAC condition is assumed, all possible voting situations m are equally liked to be observed. [START_REF] Gehrlein | Condorcet's Paradox and anonymous preference profiles[END_REF] showed that for m agents and 3 alternatives, the total number of voting situations m is given by the expression:

ψ(m) = (m + 1)(m + 2)(m + 3)(m + 4)(m + 5) 120 .
(5)

Probabilities of triple-acyclic strict preferences under majorities based on difference of votes with linear orderings

To calculate the probability of triple-acyclic strict preferences under M L k majorities we focus on the cases from 1 to 25 in Table 1. Specifically, we first calculate the probability of cyclic strict preferences, i.e. the probability of having preferences like the ones described in the cases 26 and 27 (see again Table 1). Thereafter, we obtain the probability of triple-acyclic cases as 1 minus the probability of cyclic strict preferences.

Going deeper on the strict preference described in the case 26, we notice that for such preference to exist, the numbers of voters associated with the linear orderings described in (4) have to fulfil the following conditions:

m 1 + m 2 -m 3 -m 4 + m 5 -m 6 > k , m 1 -m 2 + m 3 + m 4 -m 5 -m 6 > k and -m 1 -m 2 -m 3 + m 4 + m 5 + m 6 > k .
In other words, the strict preference in the case 26 requires a voting situation m = (m 1 , m 2 , m 3 , m 4 , m 5 , m 6 ) that fulfils the conditions given by the system of inequalities below.

(x 1 P L k x 2 , x 2 P L k x 3 and x 3 P L k x 1 ) ⇒                    m 1 + m 2 -m 3 -m 4 + m 5 -m 6 > k , m 1 -m 2 + m 3 + m 4 -m 5 -m 6 > k , -m 1 -m 2 -m 3 + m 4 + m 5 + m 6 > k , m i ≥ 0 for i ∈ {1, . . . , 6}, m -1 ≥ k ≥ 0, m 1 + m 2 + m 3 + m 4 + m 5 + m 6 = m.
Therefore, to calculate the probability of cyclic strict preferences, we need to solve the system of linear inequalities derived from conditions that the numbers of voters associated with the linear orderings in (4) have to hold for strict preferences like the ones in the cases 26 and 27 to exist.

We compute the number of voting situations that fulfil these conditions by means of the Parameterized Barvinok's algorithm [START_REF] Verdoolaege | Analytical computation of Ehrhart polynomials: Enabling more compiler analysis and optimizations[END_REF])2 . Such algorithm allows to quantify the number of integer solutions for systems of inequalities with parameters. The connection of such algorithm to Social Choice Theory was recently pointed out by [START_REF] Wilson | Probability calculations under the IAC hypothesis[END_REF] and [START_REF] Lepelley | On Ehrhart polynomials and probability calculations in voting theory[END_REF].

Given the two parameters m and k , the number of voting situations m for our system is given by bivariate quasi polynomials in m and k with 2periodic coefficients meaning that such coefficients depend on the parity of the parameters m and k . Following the notation introduced in [START_REF] Lepelley | On Ehrhart polynomials and probability calculations in voting theory[END_REF], we represent these coefficients by a list of 2 rational numbers enclosed in square brackets. To illustrate, assume the bracketed list [[a, b]

m , [b, a] m ] k .
In the case of even k , the relevant list corresponds to [a, b] m . The coefficient will be either a when m is even or b when m is odd. Accordingly, in the case of odd k , the relevant list is [b, a] m and therefore, the coefficient will be either b when m is even or a when m is odd. Thus, the coefficient will be a when m and k have the same the parity and b otherwise.

Notice that in the case of the cyclical strict preferences depicted in the cases 26 and 27 (Table 1), the number of solutions in the system of inequalities derived from the strict preference in the case 26 is the same as in the system derived from the strict preference in the case 27 given the symmetry of such cases. The program indicates that the corresponding quasi polynomial for each of these cases is as follows:

-

81
ò m k m + ï 0, 27 128 ò m , ï 27
128 , 0 ò m k k 2 + - 1 256 m 4 + ï 0, - 3 64 ò m , ï - 3 64 , 0 ò m k m 3 + ï 3 64 , - 21 128 ò m , ï - 21 128 , 3 64
ò m k m 2 + ï 0, - 9 64 ò m , ï - 9 64 , 0 ò m k m+ ï - 1 20 , 71 1280
ò m , ï 71 1280 , - 1 20 ò m k k + 1 3840 m 5 + ï 0, 1 256 ò m , ï 1 256 , 0 ò m k m 4 + ï - 1 192 , 7 384 ò m , ï 7 384 , - 1 192 ò m k m 3 + ï 0, 3 128 ò m , ï 3 128 , 0 ò m k m 2 + ï 1 60 , - 71 3840 ò m , ï - 71 3840 , 1 60
ò m k m+ ï 0, - 7 256 ò m , ï - 7 256 , 0 ò m k .
In addition, the program points out that this relation holds only if k ≤ (m -3)/3. Otherwise, the number of voting situations is zero.

We simplify3 the quasi polynomial above by considering different values of m and k . Thus, it can be deduced that the number of voting situations corresponding to each of the strict preferences represented by the cases 26 and 27 in Table 1 is given by F 1 (m, k ) if both m and k are odd (or even) and by F 2 (m, k ) if one of the parameters (m or k ) is odd and the other one is even such that:

F 1 (m, k ) = 1 3840 Å (m -3 k -4) (m -3 k ) (m -3 k + 4) (m -3 k -2) (m -3 k + 2) ã . F 2 (m, k ) = 1 3840 Å (m -3 k + 3) (m -3 k + 7) (m -3 k + 1) (m -3 k + 5) (m -3 k -1) ã .
As a consequence of the above number of voting situations and taking into account the symmetry of the strict preferences of the cases 26 and 27 in Table 1 and the total number of voting situations ψ(m) in ( 5), we introduce the probabilities of having triple-acyclic strict preferences under M L k majorities in the following result.

Proposition 1 Consider a three candidate election with m voters under M L k majority rules where each individual vote consists of a linear preference ordering on the candidates. Assuming that all voting situations are equally likely (IAC), if k ≤ (m -3)/3, the probability of triple-acyclic strict preference is as follows:

-If both m and k are odd (or even):

1 - 2F 1 (m, k ) ψ(m) .
-If one of the parameters (m or k ) is odd and the other one is even:

1 - 2F 2 (m, k ) ψ(m) .
Computed values of this probability are listed in Table 2. The values of the difference of votes k correspond to the ones that provide a probability of triple-acyclic strict preferences equal to 1. As it is previously mentioned, these probabilities indicate that the number of solutions in the systems of inequalities corresponding to the cyclic strict preferences (cases 26 and 27 in Table 1) are equal to zero when k > (m -3)/3. Some interesting facts could be emphasized from the probabilities in Table 2. First, M L 0 majority provide a probability of triple-acyclic strict preferences equal to 1 for the case of m = 4 whereas a difference of votes equal to 1 is necessary in the case of m = 3 and m = 5 to achieve such probability. We conjecture that this odd result attends to the fact that the likelihood of having ties is greater in the case of an even number of individuals than in the case of an odd number of individuals when these voters are endowed with linear orderings.

Second, the weight of the difference of votes necessary to achieve a probability of triple-acyclic strict preferences over the total number of votes equal to 1 increases with the number of individuals from m = 10 to m = 100,000. In the case of m = 10 required difference signifies a 20% of the value of m, a 32% in the case of m = 100, a 33.2% in the case of m = 1,000 and a 33.332% in the case of m = 100,000.

Third, the probabilities do not reflect small changes in the magnitude of the thresholds. See for instance, the probabilities attached to the cases m = 10, m = 100 and m = 1,000 for the values of the difference k equal to 0 and 1 and for the case of m = 100,000 in the case of k equal 0, 1 and 2.

Finally, triple-acyclic strict preferences under M L k majorities can be guaranteed with a probability of 1 for not too demanding differences of votes.

Probabilities of transitive strict preferences under majorities based on difference of votes with linear orderings

To study the probability of transitive strict preferences under M L k majorities, we follow the same methodology as the one applied in Subsection 3.1 adding the cases 20, 21, 22, 23, 24 and 25 in Table 1 to the outcomes 26 and 27 analyzed in the case of triple-acyclic strict preferences. Once we calculate the probability of non transitive strict preferences collected in the cases from 20 to 27, we determine the probability of transitive strict preferences as 1 minus the previous probability.

Ordinary preferences collected in cases from 20 to 25 are similar and hence, the number of integer solutions given by Barvinok's algorithm is the same in each of the six systems of inequalities representing these strict preferences.

For these cases, two validity domains can be distinguished. On the one hand, if k ≤ (m -3)/3, the number of voting situations is given by G 1 (m, k ) if both m and k are odd (or even) and by G 2 (m, k ) if one of the parameters (m or k ) is odd and the other one is even such that:

G 1 (m, k ) = 1 1920 (k + 1) Ä 121 k 4 -116 k 3 -200 mk 3 + 180 mk 2 - 164 k 2 + 130 m 2 k 2 + 144 k -40 m 3 k -100 m 2 k + 120 mk - 20 m 2 -80 m + 5 m 4 + 20 m 3 . G 2 (m, k ) = 1 1920 k Ä 121 k 4 -200 mk 3 -600 k 3 + 130 m 2 k 2 + 780 mk 2 + 910 k 2 -40 m 3 k -360 m 2 k -840 mk -360 k + 5 m 4 + 60 m 3 + 210 m 2 + 180 m -71 .
On the other hand, if (m -2)/3 ≤ k ≤ m -2, the number of voting situations is given by G 3 (m, k ) if both m and k are odd (or even) and by G 4 (m, k ) if one of the parameters (m or k ) is odd and the other one is even such that:

G 3 (m, k ) = 1 3840 Å (m -k -2) (m -k + 4) (m -k ) (m -k + 6) (m -k + 2) ã . G 4 (m, k ) = 1 3840 Å (m -k + 7) (m -k + 3) (m -k -1) (m -k + 5) (m -k + 1) ã .
Bearing in mind above numbers of voting situations, the results in Proposition 1 and the total number of voting situations ψ(m) in (5), we derive the probability of transitive strict preferences under M L k majorities as follows.

Proposition 2 Consider a three candidate election with m voters under M L k majority rule where each individual vote consists of a linear preference ordering on the candidates. Assuming that all voting situations are equally likely (IAC), the probability of transitive strict preferences is as follows:

1. If k ≤ (m -3)/3 -If both m and k are odd (or even):

1 - 2F 1 (m, k ) + 6G 1 (m, k ) ψ(m) .
-If one of the parameters (m or k ) is odd and the other is even:

1 - 2F 2 (m, k ) + 6G 2 (m, k ) ψ(m) . 2. If (m -2)/3 ≤ k ≤ m -2
-If both m and k are odd (or even):

1 - 6G 3 (m, k ) ψ(m) .
-If one of the parameters (m or k ) is odd and the other is even:

1 - 6G 4 (m, k ) ψ(m) .
Computed values of this probability are listed in Table 3. Going deeper on them, the weight of the required difference of votes k to guarantee a probability of transitive strict preferences equal to 1 with respect to the total number of individuals increases as the number of individuals does. To illustrate, in the case of m = 3 the required k = 1 represents around a 33.33% of the value of m whereas in the case of m = 100,000 the required k represents a 99.998% of the value of m. In fact, the required differences are very large for all the considered cases with the exception of m = 3. Even for m = 4, the difference signifies a 50% of the value of m. For not too demanding differences, the probabilities increase in the cases of m = 4, m = 5 and m = 10; this is not the case for m = 100, m = 1,000 and m = 100,000 where asking reasonable differences of votes decreases the probability of transitive strict preferences. Moreover, as in the case of the probabilities stated in Table 2, small variations in the magnitude of the differences of votes do not change, at least in a significant way, the probabilities. To illustrate, look at the probabilities of m = 100,000 with differences k equal to 0, 1, 2 and 3.

Probabilities of transitive weak preferences under majorities based on difference of votes with linear orderings

To derive the probability of transitive weak preferences under M L k majorities, we need to consider, in addition with the cases analyzed in Proposition 2, the cases from 14 to 19 in Table 1. With that, we calculate the probability of non transitive weak preferences and therefore, the probability of transitive weak preferences is determined as 1 minus the probability of non transitive weak preferences.

By symmetry arguments, the weak preferences represented in cases from 14 to 19 are similar and therefore the number of integer solutions of the six systems of inequalities corresponding to such cases is the same. Using again the Barvinok's algorithm, two validity domains can be considered. If k ≤ (m -2)/3, the number of voting situations inside system is given by H 1 (m, k ) if both m and k are odd (or even), and by H 2 (m, k ) if one of the parameters (m or k ) is odd and the other one is even such that:

H 1 (m, k ) = - 1 240 Å (k + 1) Ä 17 k 4 -30 mk 3 -22 k 3 + 20 m 2 k 2 - 28 k 2 + 30 mk 2 + 50 mk + 48 k -5 m 3 k -5 m 2 k - 5 m 3 -30 m 2 -40 m ã . H 2 (m, k ) = - 1 240 k Ä 17 k 4 -30 mk 3 -90 k 3 + 20 m 2 k 2 + 120 mk 2 + 140 k 2 -5 m 3 k -45 m 2 k -100 mk - 30 k -37 -5 m 2 -30 m .
For the second validity domain, if (m -1)/3 ≤ k ≤ m -1, this number is given by H 3 (m, k ) if both m and k are odd (or even) and by H 4 (m, k ) if one of the parameters (m or k ) is odd and the other one is even such that:

H 3 (m, k ) = 1 3840 Å (m -k + 2) (m -k ) (m -k + 4) Ä 29 k 2 + 12 mk + 94 k + 72 -m 2 + 6 m ä ã . H 4 (m, k ) = 1 3840 Å (m -k + 1) (m -k + 5) (m -k + 3) Ä 29 k 2 + 12 mk + 36 k -6 m + 7 -m 2 ä ã .
Taking into consideration the intersections between the different validity domains and using the results in Propositions 1 and 2, the probability of transitive weak preferences is as follows.

Proposition 3 Consider a three candidate election with m voters under M L k majority rule where each individual vote consists of a linear preference ordering on the candidates. Assuming that all voting situations are equally likely (IAC), the probability of transitive weak preferences is as follows:

1. If k ≤ (m -3)/3
-If both m and k are odd (or even):

1 - 2F 1 (m, k ) + 6G 1 (m, k ) + 6H 1 (m, k ) ψ(m) .
-If one of the parameters (m or k ) is odd and the other one is even:

1 - 2F 2 (m, k ) + 6G 2 (m, k ) + 6H 2 (m, k ) ψ(m) . 2. If (m -1)/3 ≤ k ≤ m -2
-If both m and k are odd (or even):

1 - 6G 3 (m, k ) + 6H 3 (m, k ) ψ(m) .
-If one of the parameters (m or k ) is odd and the other one is even:

1 - 6G 4 (m, k ) + 6H 4 (m, k ) ψ(m) .
3. If k = (m -2)/3 -Either both m and k are odd or both are even:

1 - 6H 1 (m, k ) + 6G 3 (m, k ) ψ(m) .
4. If k = m -1 -One of the parameters (m or k ) is odd and the other one is even:

1 - 6H 4 (m, k ) ψ(m) .
Analyzing the probabilities of transitive weak preferences displayed in Table 4, M L 0 majority provides the highest values for the probability of having transitive weak preferences for almost all the considered values of m. In fact, any difference of votes can be asked to guarantee a probability value of 1. Only in the cases of m = 1,000 and m = 100,000 the probability arrives to the value of 1.0000, i.e. the probability approximates to the value of 1 without reaching it4 . Even so, in both cases the required difference of votes is extremely large. Specifically, it represents a 99.9% of the value of m in the case of m = 1,000 and a 99.999% in the case of m = 100,000.

Finally, as in the previous cases stated in Tables 2 and3, the probabilities do not significantly change with small variations of the magnitude of the difference in votes. On this, see for instance the cases of k equal 2 and 3 for m equal 4, 10, 100, 1,000 and 100,000. In this section the results about the probabilities of consistent collective decisions under M k majorities are introduced under the IAC assumption. Given that voters could be indifferent between the alternatives, we have to take into account the six linear preference orderings in (4), the six possible orderings that collect the partial indifference and the one that represents the complete indifference among three alternatives. Therefore,

x 1 x 2 x 3 (m 1 ) x 1 x 3 x 2 (m 2 ) x 2 x 1 x 3 (m 3 ) x 2 x 3 x 1 (m 4 ) x 3 x 1 x 2 (m 5 ) x 3 x 2 x 1 (m 6 ) {x 1 x 2 }x 3 (m 7 ) {x 1 x 3 }x 2 (m 8 ) {x 2 x 3 }x 1 (m 9 ) x 1 {x 2 x 3 } (m 10 ) x 2 {x 1 x 3 } (m 11 ) x 3 {x 1 x 2 } (m 12 ) {x 1 x 2 x 3 } (m 13 ) (6)
where m i represents the number of voters with the associated preference ordering and {x i x j } stands for the indifference between the alternatives x i and x j .

As the IAC condition is assumed, all possible voting situations m are equally liked to be observed. For m individuals and 3 alternatives, if the indifference between alternatives is allowed, the total number of voting situations m is given by the expression:

Ψ (m) = (m + 1)(m + 2) • • • (m + 12) 12! . ( 7
)
Using the same approach applied in Section 3, the probability of consistent outcomes is calculated by means of the computation of the probability of inconsistent outcomes. As there, such probabilities are given by Ehrhart polynomials that provide the number of integer points inside the systems of inequalities that characterize each of the analyzed inconsistent outcomes.

In the framework of the preferences represented in ( 6), the complexity of the conditions makes impossible the derivation of a general mathematical representation as the one provided in Section 3. This is because for each considered consistency condition, the number of validity domains and the length of the polynomials are greater than in the cases of Section 3.

Fortunately, when the number of individuals m and the threshold k are fixed, the probabilities can be calculated for the given number of voting situations m.

Probabilities of triple-acyclic strict preferences under majorities based on difference of votes with weak orderings

In Table 5, the probabilities of triple-acyclic strict preferences under M k majorities are displayed.

The following facts can be pointed our from these results. The probabilities of having triple-acyclic strict preferences when no difference of votes is required, i.e. when M 0 majority is applied, reach very high values. Specifically, they are located between 0.9571 and 0.9989. For m = 3, m = 4 and m = 5, the needed difference of votes to achieve a probability value of 1 equals 1. Therefore, it represents a one third of the value of m in the case of m = 3, a 25% in the case of m = 4 and a 20% in the case of m = 5. For the remaining considered values, the weight of the required differences represent around one third of the value of m which means that we can guarantee with a probability of 1 the triple-acyclicity of strict preferences under M k majorities for reasonable values of the difference of votes.

Probabilities of transitive strict preferences under majorities based on difference of votes with weak orderings

In Table 6, the probabilities of transitive strict preferences under M k majorities are presented.

It is remarkable that to reach a probability of transitive strict preferences equal to 1, the weight of the required difference of votes k with respect to the total number of individuals increases as the number of individuals does. For instance, in the case of m = 4, it represents a 50% of the value of m whereas in the case of m = 1,000 it does a 99.8% of the value of m. Moreover, the required differences are too demanding for all the cases with the exception of the case of m = 3 where it represents one third of the value of m.

Probabilities of transitive weak preferences under majorities based on difference of votes with weak orderings

Probabilities of transitive weak preferences under M k majorities defined for weak orderings are displayed in Table 7. The probability value of 1 is almost achieved for the values of m equal to 100, 1,000 and 100,000. In these cases, the required differences in votes k are so high that signify a 98% of the value of m in the case of m = 100, a 99.8% in the case of m = 1,000 and a 99.998% in the case of m = 100,000.

Probabilities of consistent collective decisions under majorities based on difference in support

In this section, we provide the probabilities of reaching consistent collective decisions under M k majorities for three alternatives.

As long as the intensities of preference between each pair of alternatives can take any value in the continuous interval [0, 1], the IAC model can not be applied to provide an a priori probability to each possible voting situation. Therefore the probabilistic analysis carried out in Sections 3 and 4 turns impossible to study the case of M k majorities.

Consequently, we perform a simulation with the software Matlab to estimate these probabilities as the proportion of the number of consistent outcomes in the simulation over the total number of simulated outcomes. We generate for each of these values 100,000 outcomes to guarantee our results with a confidence level of 99% and a sampling error of less than a 0.0041%5 .

Below, we describe the methodology applied in the simulations to estimate the probability for the considered three types of consistent collective decisions under M k majorities, i.e. transitive weak preferences, transitive and tripleacyclic strict preferences. We follow that scheme taking into account each type of individual transitive reciprocal relations, i.e. 0.5-transitive, min-transitive, am-transitive and max-transitive reciprocal preference relations. Notice that the matrix in (1) representing a reciprocal preference relation is determined by the vector composed of the intensities r 12 , r 23 and r 13 .

1. We randomly generate m vectors representing the transitive reciprocal preference relations of the m individuals. Such vectors are built bearing in mind one of the considered transitivity conditions for reciprocal preference relations. 2. We compute the sum of the individuals' intensities of preference over each pair of alternatives through a vector S = (S 12 , S 23 , S 13 ) where S ij = m p=1 r p ij . 3. Having in mind the conditions in equations ( 2) and (3) and the value of k, the collective decision is evaluated over each pair of alternatives in the vector S.

P rob(|P -p| ≤ ) ≥ 0.99.

Taken into account that the sample proportion p is distributed as N Ä P, P (1 -P )/n ä , the sampling error is as follows:

= z α/2 P (1 -P )/n.
In our case, n = 100,000 and the corresponding percentile of the normal distribution for a confidence level of 99% is z α/2 = 2.57. Thus, ≤ 0.00407.

4. The collective decision in S is classified following the cases of possible collective outcomes displayed in Table 1. If it is one of the cases 26 or 27, the strict preference P k is not triple-acyclic. If it is one of the cases from 19 to 27, the strict preference P k is not transitive. Finally, if it is one of the cases from 14 to 27, the weak preference R k is not transitive. 5. This four steps are iterated 100,000 times to obtain the number of inconsistent collective decisions. Specifically, the number of simulated outcomes in which the weak preference R k is not transitive, in which the strict preference P k is not transitive and in which P k is not triple-acyclic, respectively. 6. The number of each considered type of consistent social outcomes is computed as the total number of simulated outcomes, i.e. 100,000, minus the number of inconsistent ones computed in the previous step. 7. Finally, each of the desired probabilities, i.e. the probability of transitive R k and the probability of transitive and triple-acyclic P k , is calculated as the number of consistent outcomes over the total number of simulated outcomes.

In the following, the simulated probabilities of consistent collective decisions under M k majorities are listed in tables.

Probability of transitive weak preferences under majorities based on difference in support

Tables 8, 9, 10 and 11 provide the probabilities of transitive weak preferences when reciprocal preference relations fulfil 0.5-transitivity, min-transitivity, am-transitivity and max-transitivity, respectively. When the required difference in support equals zero, the harder the transitivity condition over the reciprocal preference relations is, the higher the probabilities of having transitive weak preferences are. To illustrate, notice that the probabilities vary in between 0.8705 and 0.8835 considering 0.5-transitive reciprocal preference relations, between 0.9455 and 0.9563 taking into account In the case of m = 100,000 again, the stronger the required transitivity condition over the reciprocal preference relations is, the higher the probabilities of having transitive weak preferences are. See for instance, the probability values attached to the difference in support k = 95.32. In the case of 0.5transitive reciprocal preference relations, the probability reaches a value of 0.5913, whereas it does a value of 0.7189 in the case of min-transitive reciprocal preference relations. In the case of am-transitive reciprocal preference relations the probability achieves a value of 0.7731 while it does a value of 0.7744 in the case of max-transitive reciprocal preference relations.

We have found some unexpected results with respect to the needed thresholds to reach probability values equal to 1 for the considered values of m from 3 to 1,000. In some of these cases, the thresholds are almost the same ones with independence of the required transitivity condition over the reciprocal preference relations (on this, see the cases of m = 3, m = 4 and m = 5). In the case of m = 100, the highest thresholds are the ones required when reciprocal preference relations are am-transitive and min-transitive, while the lowest ones are the corresponding thresholds to max-transitive and 0.5-transitive reciprocal preference relations. in the case of m = 1,000, the highest threshold needed to achieve a probability value of 1 corresponds to mintransitive reciprocal preference relations. The second highest one corresponds to 0.5-transitive reciprocal preference relations, the third one to am-transitive reciprocal preference relations and the lowest one to max-transitive reciprocal preference relations. Therefore, in these situations we can not establish a clear relationship between the strength of the transitivity condition fulfilled by the reciprocal preference relations and the size of the needed threshold to achieve a probability value of 1 of having transitive weak preferences.

Probabilities of transitive strict preferences under majorities based on difference in support

Tables 12, 13, 14 and 15 display the probabilities of transitive strict preferences P k when reciprocal preference relations fulfil 0.5-transitivity, min-transitivity, am-transitivity and max-transitivity, respectively. At a first glance, the probability of transitive strict preferences goes to value of 1 for non zero values of the threshold k for all the considered type of transitive reciprocal preference relations.

The rhythm of the convergence of these probabilities to 1 depends on the number of individuals m because in each of the individual transitivity specifications, it decreases when m increases. Notice that the strength of the transitivity condition over the reciprocal preference relations seems to play a role in that convergence. It looks that the more rational the individuals are, the smaller the thresholds different from 0 needed to induce transitive strict preferences are. The unique exception to that behavior is found when m = 100, where the threshold with attached probability value of 1 is slightly lower in the case of am-transitive reciprocal preference relations (see Table 14) than in the case of max-transitive ones (see Table 15).

To illustrate these general facts, we focus on the required thresholds for m = 3 and m = 1,000 in the four cases. In the case of 0.5-transitive reciprocal relations in Table 12, the required threshold equals 2.58 and consequently represents a 85% of the value of m. In the case of m = 1,000, the required threshold represents less than a 4.8% of the value of m. In the of min-transitive reciprocal relations in Table 13, the threshold for reaching a probability of 1 is 1.96 representing less than a 66% in the case of m = 3 whereas it symbolizes around a 3.5% in the case of m = 1,000. In the case of am-transitive reciprocal relations in Table 14, the thresholds are 1.56 and 33.10, representing a 52% and around a 3.3% of the considered numbers individuals, respectively. Finally, in the case of max-transitive reciprocal relations in Table 15, the threshold for m = 3 equals 1.51 and therefore represents around a 50% of the number of voters and in the case of m = 1,000 represents around a 2.6% of the number of voters.

Probabilities of triple-acyclic strict preferences under majorities based on difference in support

In Tables 16,17, 18 and 19, we present the results for the simulated probabilities of having triple-acyclic strict preferences when 0.5-transitive, mintransitive, am-transitive and max-transitive individual preferences are considered. Conclusions are similar to the case depicted in Subsection 5.2. First, computed probabilities are high and go to 1 for non zero values of k in all considered the cases.

Second, the weight of the needed threshold to reach a probability value of 1 relative to the number of individuals involved in the voting process increases when the value of m does. For example, consider the probabilities for the case of am-transitive reciprocal preference relations in Table 18. There, for m = 3, the probability achieves the value of 1 for a k = 0.56 that represents less than a 19% of the value of m. Instead, it represents a a 11.1% for m = 10 and less than a 1.2% for m = 1,000. Third, it seems that the harder the transitivity condition over the reciprocal preference relations is, the lower the weight of threshold required to achieve a probability of triple-acyclic strict preferences of 1 is. To illustrate, look at the weights of the thresholds of support k over the four considered cases when m = 10. In the case of 0.5-transitive reciprocal preference relations (Table 16), the needed threshold represents a 2.26% of the number of individuals. In the case of min-transitive reciprocal preference relations (Table 17), k symbolizes a 1.4% of m. In the case of am-transitive reciprocal preference relations (Table 18) the weight of the threshold is a 1.11% of the value of m. Finally, in the case of max-transitive reciprocal preference relations (Table 19), the weight of the threshold represents a 0.99% of the number of individuals. As in Subsection 5.2 , the case of m = 100 constitutes an exception to that general behavior. Again, the threshold in the case of am-transitive reciprocal preference relations (see Table 18) is softly lower than the one in the case of max-transitive reciprocal preference relations (see Table 19).

Discussion

Since now, we have computed the theoretical probabilities of consistent preferences under majorities based on difference of votes defined for both individual linear orderings and weak orderings, and the simulated probabilities of consistent preferences under majorities based on difference in support.

Notice that the results on the probabilities of triple-acyclic strict preferences under M L 0 majority in Table 2 and under M 0 majority are consistent with the corresponding ones in [START_REF] Gehrlein | Condorcet Paradox and the Condorcet efficiency of voting rules[END_REF] and in [START_REF] Lepelley | Condorcet's Paradox for weak preference orderings[END_REF].

Focusing on the results for majorities based on difference of votes with linear orderings (Section 3) and with weak orderings (Section 4) we have the following. In the cases of transitive and triple-acyclic strict preferences, the probabilities are higher considering weak than linear orderings (see Tables 2, 3, 5 and 6). In the case of transitive weak preferences, the same is true when k equals 1 and 2 and, with the exception of the case in which m = 4, also when k = 0.

Looking at the results of Subsection 5.1, notice that the probabilities under M 0 remains the same for each type of transitive reciprocal preference relation with independence of the type of consistency condition required to the collective preference. That counterintuitive result contrasts with the remaining ones in which the probability increases together with the increase of the number of social outcomes considered as consistent. That oddity dues to the following. Taking into account expression (3) the absolute value of the difference between the sum of the intensities of preference r p ij and the sum of the intensities r p ij has to be null to the indifference between alternatives to be declared. As far as Matlab generates random numbers with 15 decimal positions, the indifference between alternatives is almost impossible.

Recently some analytical studies about the consistency of majorities based on difference in support have been developed. These theoretical results rely on the needed threshold to ensure transitive and triple-acyclic strict preferences for different types of transitive reciprocal preference relations.

On the one hand, the case of transitive strict preferences is studied in Llamazares et al (2013). The results can be summarized as follows:

1. The transitivity of the strict preference can not be ensured for any threshold of support k less than m -1.

2. The transitivity of the strict preference can not be ensured for any threshold of support k less than m if the reciprocal preference relations are less demanding than am-transitive ones. 3. The strict preference is transitive for any threshold of support such that k ∈ [m -1, m) if the reciprocal preference relations are at least am-transitive ones.

On the other hand, the case of triple-acyclic strict preferences is analyzed in Llamazares and Pérez-Asurmendi (2013) with the following results:

1. The triple-acyclicity of the strict preference, in the case of 0.5-transitive reciprocal preference relations, can be guaranteed if the threshold of support k is located in 2m/3 , m where a stands for the integer part of a. 2. The triple-acyclicity of the strict preference, in the case of min-transitive and max-transitive reciprocal preference relations, can be guaranteed if the threshold of support k belongs to [m/3, m). 3. In the case of max-transitive reciprocal preference relations, it conjectures that strict preference relations are triple-acyclic if the threshold k belongs to 2m/3 /2, m .

The probabilistic results setting here complement the above theoretical ones by the following reasons. First, thresholds with associated probabilities of consistent strict preferences equal to 1 are found for all the considered types of transitive reciprocal preference relations. Second, reasonable thresholds are required to certify the consistency of the strict preference with a probability value of 1 in those cases where theoretical results asked a very high threshold to guarantee such consistency. Third, the conjecture about the needed thresholds in the case of max-transitive reciprocal preference relations seems to be true.

Specifically, in the case of transitive strict preferences with 0.5-transitive and min-transitive reciprocal preference relations, the probabilities achieve the value of 1 for the considered values of m (see Tables 12 and13, respectively) whereas as it is said before, the theoretical result asserts that no threshold guarantees the transitivity of the strict preference for such types of reciprocal preference relations.

In the cases of am-transitive and max-transitive reciprocal preference relations, the thresholds that provide a probability value of transitive strict preference relations equal to 1 are lower than the ones that guarantee the transitivity of the strict preference in the theoretical framework. To illustrate, assume m = 1,000. The theoretical result asserts that the threshold k has to belong to [999,1,000). By contrast, a probability value of 1 is achieved with a threshold of 33.10 in the case of am-transitive reciprocal preference relations and of 26.10 in the case of max-transitive reciprocal preference relations (see Tables 14 and15, respectively).

In the case of triple-acyclic strict preferences, the thresholds to reach a probability value of 1 again are much lower than the ones required in the theoretical setting. For instance, assume m = 5. The needed thresholds to certify triple-acyclic strict preferences are at least 3 in the case of 0.5-transitive reciprocal preference relations and 5/3 in the cases of min-transitive and maxreciprocal preference relations. Attending to the probabilistic analysis, the needed thresholds to achieve a probability value of 1 are 1.91 in the case of 0.5-transitive reciprocal preference relations (see Table 16), 0.90 in the case of min-transitive reciprocal preference relations (see Table 17) and 0.75 in the case of max-transitive reciprocal preference relations (see Table 19).

Attending to the results under the probabilistic approach, the conjecture in Llamazares and Pérez-Asurmendi (2013) seems to be true. In Table 20, we provide some other examples that support such idea. For instance, look at the case of m = 100. The probability of triple-acyclic strict preference relations equals 1 for k = (2 • 100)/3 /2 = 33.

  r p jl }. The preferences of each individual over the alternatives in X = {x 1 , x 2 , x 3 } can be represented using a 3 × 3 matrix R p =

Table 2

 2 Probability of triple-acyclic P L k .

	m →	3	4	5	10	100	1,000	100,000
	k ↓
	0	0.9643	1	0.9524	0.9860	0.9462	0.9384	0.9375
	1	1		1	0.9860	0.9462	0.9384	0.9375
	2				1	0.9605	0.9403	0.9375
	32					1	0.9628	0.9378
	332						1	0.9406
	33,332							1

Table 3

 3 Probability of transitive P L k .

	m →	3	4	5	10	100	1,000	100,000
	k ↓
	0	0.9643	0.9524	0.9524	0.9161	0.9293	0.9366	0.9375
	1	1	0.9524	0.9762	0.9161	0.9293	0.9366	0.9375
	2		1	0.9762	0.9580	0.9175	0.9348	0.9375
	3			1	0.9580	0.9175	0.9348	0.9375
	8				1	0.9088	0.9298	0.9374
	98					1	0.9101	0.9366
	998						1	0.9290
	99,998							1

Table 4

 4 Probability of transitive R L k .

	m →	3	4	5	10	100	1,000	100,000
	k ↓
	0	0.9643	0.7619	0.9524	0.8462	0.9280	0.9366	0.9375
	2	0.6786	0.7143	0.6667	0.6703	0.9062	0.9346	0.9375
	3		0.7143	0.7619	0.6703	0.9062	0.9346	0.9375
	4			0.7619	0.6503	0.8818	0.9327	0.9375
	9				0.9101	0.8292	0.9287	0.9374
	99					0.9997	0.8136	0.9366
	999						1.0000	0.9276
	99,999							1.0000
	4 Probabilities of consistent collective decisions under majorities
	based on difference of votes with weak orderings

Table 5

 5 Probability of triple-acyclic P k .

	m →	3	4	5	10	100	1,000	100,000
	k ↓
	0	0.9956	0.9989	0.9929	0.9873	0.9627	0.9577	0.9571
	1	1	1	1	0.9992	0.9720	0.9589	0.9572
	2				1.0000	0.9794	0.9601	0.9572
	3				1	0.9851	0.9613	0.9572
	33					1	0.9852	0.9576
	333						1	0.9611
	33,333							1

Table 6

 6 Probability of transitive P k .

	m →	3	4	5	10	100	1,000	100,000
	k ↓
	0	0.9692	0.9626	0.9531	0.9422	0.9519	0.9565	0.9571
	1	1	0.9901	0.9884	0.9632	0.9443	0.9554	0.9571
	2		1	0.9971	0.9851	0.9399	0.9542	0.9571
	3			1	0.9942	0.9383	0.9531	0.9571
	8				1	0.9516	0.9484	0.9570
	98					1	0.9597	0.9560
	998						1	0.9471
	99,998							1

Table 7

 7 Probability of transitive R k .

	m →	3	4	5	10	100	1,000	100,000
	k ↓
	0	0.9801	0.8538	0.8920	0.9072	0.9511	0.9565	0.9571
	2	0.7890	0.7429	0.7014	0.6980	0.9218	0.9540	0.9571
	3		0.8648	0.8167	0.6814	0.9049	0.9527	0.9571
	4			0.9176	0.7234	0.8871	0.9514	0.9571
	9				0.9919	0.7954	0.9444	0.9570
	99					1.0000	0.7782	0.9559
	999						1.0000	0.9436
	99,999							1.0000

Table 8

 8 Probabilities of transitive R k for 0.5-transitive reciprocal preference relations.

	m →	3	4	5	10	100	1,000	100,000
	k ↓
	0	0.8835	0.8781	0.8763	0.8751	0.8706	0.8705	0.8728
	2.97	1	0.9836	0.9485	0.7630	0.5927	0.7826	0.8655
	3.81		1	0.9964	0.9065	0.5481	0.7544	0.8633
	4.70			1	0.9748	0.5312	0.7248	0.8609
	7.95				1	0.6729	0.6266	0.8518
	26.40					1	0.7012	0.7954
	84.83						1	0.6156
	95.32							0.5913

Table 9

 9 Probabilities of transitive R k for min-transitive reciprocal preference relations.Table 10 Probabilities of transitive R k for am-transitive reciprocal preference relations.Table 11 Probabilities of transitive R k for max-transitive reciprocal preference relations.

	m →	3	4	5	10	100	1,000	100,000
	k ↓
	0	0.9563	0.9513	0.9505	0.9487	0.9468	0.9476	0.9455
	2.96	1	0.9831	0.9528	0.8015	0.7220	0.8814	0.9400
	3.84		1	0.9966	0.9205	0.6776	0.8576	0.9383
	4.80			1	0.9809	0.6552	0.8307	0.9366
	8.32				1	0.7559	0.7432	0.9291
	27.23					1	0.7702	0.8831
	95.32						1	0.7189
	m →	3	4	5	10	100	1,000	100,000
	k ↓
	0	0.9751	0.9710	0.9712	0.9694	0.9676	0.9670	0.9692
	2.97	1	0.9812	0.9492	0.8087	0.7738	0.9117	0.9650
	3.91		1	0.9971	0.9225	0.7295	0.8904	0.9636
	4.91			1	0.9813	0.7043	0.8683	0.9622
	7.48				1	0.7370	0.8112	0.9582
	27.90					1	0.7873	0.9180
	83.01						1	0.7967
	95.32							0.7731
	m →	3	4	5	10	100	1,000	100,000
	k ↓
	0	0.9777	0.9766	0.9751	0.9741	0.9727	0.9726	0.9744
	2.98	1	0.9842	0.9561	0.8219	0.7867	0.9189	0.9708
	3.84		1	0.9964	0.9241	0.7450	0.9003	0.9696
	4.72			1	0.9782	0.7188	0.8805	0.9681
	7.56				1	0.7495	0.8190	0.9637
	25.47					1	0.7720	0.9298
	80.00						1	0.8030
	95.32							0.7744
	min-transitive reciprocal preference relations, between 0.9670 and 0.9751 bear-
	ing in mind am-transitive reciprocal preference relations and between 0.9726
	and 0.9777 in the case of max-transitive reciprocal relations.

Table 12

 12 Probabilities of transitive P k for 0.5-transitive reciprocal preference relations.

	m →	3	4	5	10	100	1,000	100,000
	k ↓
	0	0.8835	0.8781	0.8763	0.8751	0.8706	0.8705	0.8728
	2.58	1	0.9999	0.9995	0.9916	0.8512	0.8287	0.8668
	2.97		1	1.0000	0.9970	0.8661	0.8268	0.8659
	3.24			1	0.9985	0.8759	0.8257	0.8654
	5.17				1	0.9465	0.8267	0.8611
	14.51					1	0.9295	0.8440
	47.79						1	0.8293

Table 13

 13 Probabilities of transitive P k for min-transitive reciprocal preference relations.Table 14 Probabilities of transitive P k for am-transitive reciprocal preference relations.Table 15 Probabilities of transitive P k for max-transitive reciprocal preference relations.

	m →	3	4	5	10	100	1,000	100,000
	k ↓
	0	0.9563	0.9513	0.9505	0.9487	0.9468	0.9476	0.9455
	1.96	1	0.9999	0.9997	0.9968	0.9377	0.9271	0.9421
	2.49		1		0.9993	0.9514	0.9255	0.9414
	2.61			1	0.9995	0.9533	0.9254	0.9413
	3.47				1	0.9704	0.9248	0.9398
	12.51					1	0.9793	0.9291
	35.24						1	0.9251
	47.79							0.9311
	m →	3	4	5	10	100	1,000	100,000
	k ↓
	0	0.9751	0.9710	0.9712	0.9694	0.9676	0.9670	0.9692
	1.56	1	0.9999	0.9997	0.9973	0.9605	0.9551	0.9672
	1.85		1	1.0000	0.9990	0.9655	0.9542	0.9668
	2.07			1	0.9995	0.9697	0.9538	0.9665
	3.10				1	0.9845	0.9538	0.9653
	9.17					1	0.9828	0.9600
	33.10						1	0.9570
	47.79							0.9629
	m →	3	4	5	10	100	1,000	100,000
	k ↓
	0	0.9777	0.9766	0.9751	0.9741	0.9727	0.9726	0.9744
	1.51	1	1.0000	0.9999	0.9981	0.9689	0.9619	0.9727
	1.66		1	0.9999	0.9989	0.9712	0.9614	0.9725
	1.93			1	0.9996	0.9758	0.9608	0.9723
	2.78				1	0.9870	0.9607	0.9715
	9.72					1	0.9897	0.9662
	26.10						1	0.9626
	47.79							0.9690

Table 16

 16 Probabilities of triple-acyclic P k for 0.5-transitive reciprocal preference relations.

	m →	3	4	5	10	100	1,000	100,000
	k ↓
	0	0.8835	0.8781	0.8763	0.8751	0.8706	0.8705	0.8728
	1.32	1	0.9999	0.9998	0.9967	0.9479	0.9013	0.8763
	1.50		1	1.0000	0.9983	0.9546	0.9052	0.8768
	1.91			1	0.9998	0.9681	0.9135	0.8778
	2.65				1	0.9837	0.9270	0.8794
	8.29					1	0.9834	0.8926
	26.60						1	0.9291

Table 17

 17 Probabilities of triple-acyclic P k for min-transitive reciprocal preference relations.Table18Probabilities of triple-acyclic P k for am-transitive reciprocal preference relations.

	m →	3	4	5	10	100	1,000	100,000
	k ↓
	0	0.9563	0.9513	0.9505	0.9487	0.9468	0.9476	0.9455
	0.67	1	0.9998	0.9997	0.9971	0.9812	0.9583	0.9467
	0.88		1	1.0000	0.9993	0.9817	0.9612	0.9471
	0.90			1	0.9993	0.9906	0.9614	0.9472
	1.40				1	0.9906	0.9675	0.9481
	4.56					1	0.9913	0.9540
	14.99						1	0.9687
	26.60							0.9799
	m →	3	4	5	10	100	1,000	100,000
	k ↓
	0	0.9751	0.9710	0.9712	0.9694	0.9676	0.9670	0.9692
	0.56	1	1.0000	0.9998	0.9986	0.9854	0.9743	0.9698
	0.67		1	1.0000	0.9995	0.9876	0.9758	0.9700
	0.83			1	0.9999	0.9905	0.9775	0.9702
	1.11				1	0.9941	0.9804	0.9706
	3.34					1	0.9936	0.9730
	11.68						1	0.9811
	26.60							0.9911

Table 19

 19 Probabilities of triple-acyclic P k for max-transitive reciprocal preference relations.

	m →	3	4	5	10	100	1,000	100,000
	k ↓
	0	0.9777	0.9766	0.9751	0.9741	0.9727	0.9726	0.9744
	0.67	1	1.0000	0.9999	0.9998	0.9909	0.9804	0.9753
	0.72		1	1.0000	0.9999	0.9918	0.9808	0.9753
	0.75			1	0.9999	0.9922	0.9811	0.9754
	0.99				1	0.9951	0.9834	0.9758
	3.58					1	0.9963	0.9786
	9.95						1	0.9845
	26.60							0.9943

Table 20

 20 Probabilities of triple-acyclic P k in the case of max-transitive reciprocal preference relations with k = 2m/3 /2, m .

	m	3	4	5	6	7	8	48	49	50	99	100	101
	k	1	1	1.5	2	2	2.5	16	16	16.5	33	33	33.5

The free software to calculate the integer points under the Parameterized Barvinok's algorithm can be found in http://freecode.com/projects/barvinok.

Such simplification is done with Maple software.

Notice that this value also appears in Tables

5, 7, 12, 13, 14 and 15 with the same meaning.

Assuming a proportion of consistent outcomes P on the population of a 50%, the proportion p in a random sample of size n ≥ 30 for a confidence level of 99%, diverges from the one of the population in an error of less than :

We conclude with a comment about the comparison among the probabilities of consistent preferences under the considered three specifications of the majorities based on differences. Notice that the method used to calculate these probabilities differs from the cases of the two specifications of majorities based on difference of votes to the case of majorities based on difference in support. In the first cases, the probability is calculated by counting the integer points given by systems of inequalities whereas in the second case the probability is simulated by Montecarlo techniques. Accordingly, we can not make any quantitative comparison between the results obtained in the first two cases and in the third one.

Acknowledgements The authors are grateful to Ahmad Fliti, José Luis García-Lapresta, Bonifacio Llamazares, Ana Pérez Espartero and Rachid Seghir for their valuable suggestions and comments. This work is partially supported by the Spanish Ministry of Economy and Competitiveness (Projects ECO2012-32178 and ECO2012-34202).