Eric Gloaguen
email: e.gloaguen@brgm.fr

Yannick Branquet

Alain Chauvet

Vincent Bouchot

Luc Barbanson

J L Vigneresse

Tracing the magmatic/hydrothermal transition in regional low-strain zones: The role of magma dynamics in strain localization at pluton roof, implications for intrusion-related gold deposits

Keywords: magmatic-to-hydrothermal transition, pluton dynamics, strain localization, granite roof, mechanical instabilities, reduced intrusion-related gold deposit, Variscan, NW Spain, Boborás granite

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Tracing the magmatic/hydrothermal transition in regional low-strain zones: The role of magma dynamics in strain localization at pluton roof, implications for intrusion-related gold deposits

Introduction

Mechanical instabilities at the roof of a magmatic intrusion (the term "roof" describes the apical envelope before erosion of an intrusion) appear to be a major structural prerequisite for localizing the magmatic-to-hydrothermal transition. Roof instabilities favour morphological features like cupolas, apexes and dyke swarms, which appear to be critical when mineralization is formed, as in the case of reduced intrusion-related gold deposits (e.g. [START_REF] Goldfarb | Distribution, Character, and Genesis of Gold Deposits in Metamorphic Terranes[END_REF][START_REF] Hart | Mineral deposits of Canada: A Synthesis of Major Deposit Types, District Metallogeny, the Evolution of Geological Provinces, and Exploration Methods[END_REF]. This class of gold deposit has only been recognized as a separate feature from the orogenic gold deposit class since 1999 and, as such, is in a juvenile state of understanding [START_REF] Hart | Mineral deposits of Canada: A Synthesis of Major Deposit Types, District Metallogeny, the Evolution of Geological Provinces, and Exploration Methods[END_REF]. Reduced intrusion-related gold deposits (RIRGD) include various deposit styles, such as skarns, disseminations, replacements, breccias, stockworks or, more commonly, intrusion-hosted sheeted arrays of thin quartz veins. Such veins with a lowsulphide Au-Bi-Te-W signature may develop within, beyond, or above the thermal aureole of the pluton. The intrusions have moderately low primary oxidation states, making them into reduced, ilmenite-series [START_REF] Ishihara | The granitoid series and mineralization[END_REF] granitoids. In such mineralized systems, the emplacement-to-cooling cycle of reduced felsic intrusions is assumed to be coeval and genetically related to gold deposition. RIRGD are therefore systems where the magmatic-tohydrothermal transition is demonstrated by fluid-inclusion studies [START_REF] Baker | Fluid inclusion characteristics of intrusion-related gold mineralization, Tombstone-Tungsten magmatic belt, Yukon Territory Canada[END_REF]; they are generally centred on small plutons, plutonic apexes or dyke swarms, around which a cracked thermal aureole was developed [START_REF] Baker | Fluid inclusion characteristics of intrusion-related gold mineralization, Tombstone-Tungsten magmatic belt, Yukon Territory Canada[END_REF][START_REF] Stephens | Structural and mechanical controls on intrusion-related deposits of the Tombstone Gold Belt, Yukon, Canada, with comparisons to other vein-hosted ore-deposit types[END_REF]. Such intrusions are weakly deformed as they postdate regional ductile shearing and peak metamorphism. However, despite favourable conditions, to our knowledge no structural studies (including gravimetry) exist that are dedicated to such deposits, accounting for intrusion dynamics as structural controls of such magmatic-hydrothermal mineralized systems.

The magmatic-to-hydrothermal transition is a key process in ore-forming systems related to igneous activity (see review in [START_REF] Halter | The magmatic to hydrothermal transition and its bearing on ore-forming systems[END_REF]. While this transition is well known from mineralogical and geochemical investigations, it remains difficult to assess via structural and textural analyses. This assertion is particularly valid for plutonic-and hydrothermalrelated systems, as plutons emplaced close to high-strain regional shear zones (e.g. the footwall of detachments planes in metamorphic core complexes) may preserve magmatic and submagmatic textural gauges, but on-going high solid-state strain and rapid exhumation make it difficult accurately to establish a possible genetic link between magma injection and the subsequent hydrothermal system (e.g. [START_REF] Menant | The North Cycladic Detachment System and associated mineralization, Mykonos, Greece: Insights on the evolution of the Aegean domain[END_REF]. In contrast, although magmatic and submagmatic textural gauges are subtle features due to low strain, plutons and their related ore-bearing systems emplaced far from major shear zones are good candidates for showing the magmatic-to-hydrothermal transition from a structural and textural point of view.

A weakly deformed magmatic system also provides an opportunity for evaluating the contribution of magma dynamics to the textural framework and plumbing system of a magmatic-to-hydrothermal transition.

In this paper we present a detailed structural and micro-structural analysis of the roof of the Late Variscan Boborás granite and its associated gold mineralization, the Brués deposit in northwestern Spain. The Boborás granite, dated at 318 ± 5 Ma [START_REF] Gloaguen | Apports d'une étude intégrée sur les relations entre granites et minéralisations filoniennes (Au et Sn-W) en contexte tardiorogénique (Chaîne Hercynienne, Galice centrale, Espagne)[END_REF] is part of a roughly 30 km-long NNE-SSW alignment of four plutons, displaying a strong spatial association with gold mineralization located in and around the plutons (Fig. 1). Gold-bearing quartz veins in the granite, dykes or pegmatites, exhibit the typical low-sulphide Au-Bi-Te-W signature of RIRGD and contain locally magmatic brines. The Brués deposit on the northwestern edge of the Boborás pluton is the most important deposit of the area [START_REF] Asensio Pérez | Las mineralizaciones de oro asociadas al granite de Boborás (Brués, Orense). Gold mineralizations associated to the Boborás granito (Brués, Orense)[END_REF][START_REF] Gloaguen | Relations between Au / Sn-W mineralizations and late Variscan granite: Preliminary results from the Schistose Domain of Galicia-Trás-os-Montes Zone, Spain[END_REF][START_REF] Gloaguen | Apports d'une étude intégrée sur les relations entre granites et minéralisations filoniennes (Au et Sn-W) en contexte tardiorogénique (Chaîne Hercynienne, Galice centrale, Espagne)[END_REF]Fig. 2). It is composed of a high density of near-parallel granite dykes and sills crosscutting micaschists at the roof of the pluton. Steeplydipping dykes hosting gold-bearing quartz veins, trend N060°E close to the pluton and turn gradually to N100°E farther west. Veins from this deposit result from at least four successive deformation events with fluid infill. The deposition of gold and associated minerals occurred during the latest stages of the hydrothermal system. This network of dykes, sills and associated gold-bearing quartz veins is not observed elsewhere around the pluton.

We discuss the key parameters controlling fluid flow and mineral deposition, such as interaction between regional stress and pluton dynamics, the underlying root zone and anisotropy within the cracked thermal aureole. These parameters may explain why only some areas of the intrusion are mineralized, in other words why the mineralization does not cover the entire surface of the intrusion roof.

Geological setting

The Galician Variscan segment of the Ibero-Armorican orogenic belt (Fig. 1a) is composed of three major tectonic units [START_REF] Ribeiro | Contribution à l'étude tectonique de Trás-os-Montes oriental[END_REF][START_REF] Díez Balda | Autochthonous sequence. Structure[END_REF]Ribeiro et al., 1990aRibeiro et al., , 1990b) ) as a result of a long-lived and complex geological history (e.g. [START_REF] Martínez Catalán | A rootless suture and the loss of the roots of a mountain chain: The Variscan belt of NW Iberia[END_REF]. From base to top, we find (Fig. 1b): i) para-autochtonous schist and quartzite of the Schistose Domain of Galicia Trás-os-Montes [START_REF] Ribeiro | Contribution à l'étude tectonique de Trás-os-Montes oriental[END_REF][START_REF] Marquínez García | La geología del área esquistosa de Galicia Central (Cordillera Herciniana, NW de Espana)[END_REF][START_REF] Arenas | Tectonostratigraphic units in the complexes with mafic and related rocks of the NW of the Iberia[END_REF][START_REF] Farias | Aportaciones al conocimiento de la litoestratigrafía y estructura de Galicia Central[END_REF][START_REF] Farias Arquer | El Paleozoico Inferior de la Zona de Galicia-Tràs-os-Montes (Cordillera Herciniana, NW de España)[END_REF]; and ii) an allochthonous unit including the Ordenes complex, the main nappe complex of the Galician area [START_REF] Martínez Catalán | Thrust and detachment systems in the Ordenes Complex (northwestern Spain): Implications for the Variscan-Appalachian geodynamics[END_REF][START_REF] Ábalos | Variscan and Pre-Variscan Tectonics[END_REF]. The region records three main deformation events, forming part of a long-lived Variscan history in response to subduction, continental collision and post-crustalthickening extension (e.g. [START_REF] Matte | La chaîne varisque parmi les chaînes paléozoïques péri atlantiques, modèle d'évolution et position des grands blocs continentaux au Permo-Carbonifère[END_REF][START_REF] Lagarde | Granites et collision continentale : l'exemple des granitoïdes carbonifères dans la chaîne hercynienne ouest-européenne[END_REF][START_REF] Martínez Catalán | Space and time in the tectonic evolution of the north-western Iberian Massif: Implications for the Variscan belt[END_REF], 2009). The study area (Fig. 1b) is mainly affected by ductile deformation events (D2, D3) and a D4 event, which largely overprint an earlier D1 event.

D2 is related to the emplacement of nappes expressed by a regional shallow-dipping cleavage and prograde metamorphism active between 345-320 Ma [START_REF] Dallmeyer | Diachronous Variscan tectonothermal activity in the NW Iberian Massif: Evidence from 40 Ar/ 39 Ar dating of regional fabrics[END_REF].

D3 corresponds to E-W horizontal compression [START_REF] Díaz García | La estructura y la evolución metamórfica en un corte comprendido entre Carballiño y Forcarei. Area esquistosa de Galicia Central, NW de España. The structure and metamorphic evolution in a section between Carballiño and Forcarei[END_REF][START_REF] Gloaguen | Apports d'une étude intégrée sur les relations entre granites et minéralisations filoniennes (Au et Sn-W) en contexte tardiorogénique (Chaîne Hercynienne, Galice centrale, Espagne)[END_REF][START_REF] Fernández | Kinematic evolution and structural styles of fold-and-thrust belts[END_REF] associated with north-verging detachments occurring between 330 and 315 Ma. D3 is characterized by large-scale upright folds and associated S3 axial cleavage, and by subvertical strike-slip shear zones (i.e. the Serra do Suido shear zone on Fig. 1b) [START_REF] Martínez Catalán | Variscan exhumation of a subducted Palaeozoic continental margin, The basal units of the Ordenes Complex, Galicia, NW Spain[END_REF]. This event is also coeval with the emplacement of numerous granites. Four generations of plutons (G1 to G4; [START_REF] Capdevila | Les différents types de granites hercyniens et leur distribution dans le nord ouest de l"Espagne[END_REF], well constrained by crosscutting relationships (Fig. 1b), intrude the two abovementioned tectonic units that are mainly composed of micaschist, quartzite, feldspathic schist and paragneiss intercalations [START_REF] Marquínez García | Estudio geológico del área esquistosa de Galicia Central (zona de Lalín-Forcarei-Beariz)[END_REF][START_REF] Marquínez García | La geología del área esquistosa de Galicia Central (Cordillera Herciniana, NW de Espana)[END_REF][START_REF] Farias | Aportaciones al conocimiento de la litoestratigrafía y estructura de Galicia Central[END_REF][START_REF] Pereira | Cisalhamentos hercínicos e controlo das mineralizações de Sn-W, Au e U na Zona Centro-Ibérica, em Portugal[END_REF]. G1 granodiorite is assumed to be coeval with D2 [START_REF] Castro | Palaeozoic Magmatism[END_REF] whereas G2 (two-mica) and G3 (biotitedominant) granites are syn-D3 [START_REF] Barrera Morate | Ourense/Verín[END_REF][START_REF] Farias Arquer | El Paleozoico Inferior de la Zona de Galicia-Tràs-os-Montes (Cordillera Herciniana, NW de España)[END_REF]. Finally, D4 is represented by the emplacement of large rounded G4 granodiorite plutons (e.g. the Ribadavia pluton; Figs. 1b and2) that are related to generalized late-orogenic crustal extension [START_REF] Aranguren | Granite laccolith emplacement in the Iberian arc: AMS and gravity study of the La Trojiza pluton (NW Spain)[END_REF]. The emplacement of G4 plutons is estimated around 298 ± 2 Ma, the age of the neighbouring G4 Ourense pluton (Guttiérez-Alonso et al., 2011).

The Boborás granite

General features (Figs. 1 and 2)

The Boborás granite is a small north-south elongate elliptical intrusion (7 x 2.5 km) emplaced in the para-autochthonous unit consisting of monotonous micaschist including scarce dark quartzite and greywacke with numerous intercalations of quartz-segregation veins. A minimum emplacement depth of 9.3 km was estimated by means of fluid-inclusion studies on gold-bearing quartz veins [START_REF] Gloaguen | Apports d'une étude intégrée sur les relations entre granites et minéralisations filoniennes (Au et Sn-W) en contexte tardiorogénique (Chaîne Hercynienne, Galice centrale, Espagne)[END_REF]. It is a peraluminous S-type monzogranite containing no magnetite (average magnetic susceptibility of 53x10 -6 S.I). The rock is homogeneous and has an equigranular texture where biotite is dominant over muscovite. The pluton does not seem to be deformed, and as a first approximation appears to be "strain free".

Around the pluton, contact metamorphism is marked by andalusite commonly replaced by quartz and muscovite. At its southern edge, the Boborás granite is intruded by the post-D3, G4 Ribadavia pluton (Figs. 1b and2).

Gravity and 3D geometry

A survey of 472 gravity measurements covered the Boborás granite area with a Lacoste & Romberg (G847) gravimeter (precision of ± 0.05 mGal). Elevations were determined using elevation benchmarks and a baro-altimeter with a precision of ± 0.5 m equivalent to ~0.1 mGal. The gravity data, referred to the Vigo international gravity station (n°105921 of the Bureau Gravimétrique International), were subjected to the classical corrections with a reduction density of 2.7. The kriging-interpolated contour map of the complete Bouguer anomaly (Fig. 3a) shows perturbations caused by the G2 and G3 plutons on a broad E-W negative gradient. The highest negative values to the east correspond to the large Carballiño G2 pluton. The regional anomaly was estimated by graphic smoothing of the Bouguer anomaly assuming broadly parallel regional isovalues (Fig. 3b), yielding a gently eastwarddipping plane with an estimated mean gradient of 0.78 mGal/km. The residual Bouguer anomaly was then calculated and its isovalues were contoured. Due to the large anomaly related to the Carballiño G2 granite, the 0 mGal contours around the small G3 plutons could not be obtained. In order to minimize this effect, a local regional anomaly was recalculated giving a closer approximation of the residual Bouger anomaly for the Boborás granite (Fig. 3c). The resulting data were used to perform a three-dimensional inversion [START_REF] Cordell | Iterative three dimensional solution of gravity anomaly using a digital computer[END_REF][START_REF] Vigneresse | Use and misuse of geophysical data to determine the shape at depth of granitic intrusions[END_REF] based on a realistic density contrast of -0.03 between the granite and the hosting micaschist.

In the resulting depth-contour map (estimated error of ~15 %) and N-S section across the Boborás pluton (Fig. 3d), the pluton appears as a laccolith with a present-day mean thickness of 1-1.5 km, and with two probable feeder zones located close to the northern and southern edges of the pluton. Between these feeders, the central zone of the pluton displays a swelling floor bordered by two thin zones shown by arrows on Fig. 3d. The eroded upper part of the pluton remains unknown, but, the residual-anomaly area being greater than the outcropping surface, it is likely that a significant part of the pluton remains below the present-day surface.

In the Brués gold deposit area this assumption is strengthened by the very shallow dip (about 15° NW) of the granite-schist contact. The numerous occurrences of G3 pegmatite pods and dykes, generally concentrated at the top of intrusions, also attest to a nearby pluton roof (see [START_REF] London | Pegmatites[END_REF]. Therefore, very little granite has been removed by erosion, and we can conclude that the present-day outcropping surface is close to the initial roof of the granite body.

Structures around the pluton

The micaschist is dominated by a nearly constant, N-S trending S2 cleavage with a mean dip of about 40° to the west (Fig. 2). Locally, S2 was intensively folded during D3, resulting in a near-horizontal N-S-trending L3 crenulation related to upright folds (Fig. 2). S3 cleavage trajectories are crosscut by the northern end of the pluton, clearly suggesting a discordant pluton (Fig. 2). Around the pluton, D3 is shown by: i) Boudin-like G2 aplite-pegmatite bodies deformed during roughly N-S stretching (Fig. 4a); ii) A sparse N-S, near-horizontal L3 lineation that developed in the thermal aureole where cordierite with rotated inclusion trails (Fig. 4b), and truncated andalusite-sillimanite porphyroblasts are observed (Fig. 4c), both indicate a top-to-the-north ductile shearing. Neither ductile detachments, nor high-strain mylonitic zones have been reported or recognized in the Boborás granite area (Fig. 2).

Dykes, sills and veins systems

Numerous sills, dykes and veins are exposed in the host micaschist around the Boborás granite. Structural, mineralogical and textural observations allow distinguishing four main types: G2 pegmatite and aplite; G3 granite dykes and sills; G3 pegmatite and aplite; and hydrothermal quartz veins.

(i) The most important network of intrusive sills and dykes is represented by pegmatite and aplite, varying in size from several metres to over a kilometre, and found all along the western border of the G2 Carballiño granite (only the most important bodies are shown on Fig. 2).

Since they appear to be genetically associated to the G2 magmatism and are crosscut by G3 granite (Figs. 5b and6a), they are labelled as G2 aplite-pegmatite and not further described.

(ii) Conversely, G3 sills and dykes are well-developed only northwest of the Boborás pluton (Figs. 2 and5a). Two groups of G3 granite injections into micaschist are defined: 1) G3 granite sills , and 2) high-angle north-dipping G3 granite dykes (Figs. 6b andc). The G3 granite sills are connected with numerous G3 high-angle granite dykes (Fig. 6c) with straight or irregular walls (Figs. 5b andd); some of them display a typical staircase geometry (Fig. 6b). The G3 dykes are connected to the Boborás pluton (Fig. 5b).

(iii) Several G3 pegmatites and/or aplites are observed in the Boborás granite, generally close to the roof. They are well represented mostly at the pluton border (e.g. the Xurenzas, Salon-Sobreido road stations on Fig. 2), and appear either as 10-cm-thick isolated vugs filled with coarse-grained pegmatite, or as 1 m-thick dykes of layered aplite-pegmatite.

The G3 granitic dykes are commonly bordered by centimetre to metre-scale quartz veins (Fig. 7a), either inside the dyke or along the dyke wallrock (Fig. 5c andd). At the Brués gold deposit (Figs. 2 and5), the quartz veins can be sorted into extensional and shearedextensional veins, hereafter called "shear veins". A few specific veins, filled with quartz and feldspar, correspond to hybrid shear-extensional fractures (Figs. 7e-f). Extensional veins are near-vertical (Fig. 2), commonly with an "en échelon" geometry consistent with a northverging normal motion (Figs. 5c and7c-d). The quartz in these veins is undeformed and milky to translucent in colour. The shear veins, however, have a laminated texture of alternating ~0.5 cm-thick ribbons of quartz and sulphide (Fig. 7b). At the Brués gold deposit, the shear veins exclusively occur along and inside the G3 dykes; they have steep northerly dips and their overall attitude and shear criteria demonstrate a north-verging normal motion (Figs. 7a andb).

At the pluton scale (Fig. 2), the extensional veins are preferentially concentrated along the north edge. Here, the magmatic G3 dykes and quartz veins gradually rotate from N060°E close to the pluton to N090°E farther west (Figs. 2 and5a) and systematically dip to the north.

The near-vertical extensional veins follow the same trend. Around the pluton, another type of shear vein is observed, corresponding to N120°E-trending left-lateral veins conjugated with N60°E-trending right-lateral veins (e.g. Fondo de Vila station on Fig. 2). These trends were also observed for the G3 pegmatites and aplites (Fig. 2), but the kinematics are often difficult to assess in such rocks. Right dihedral (Fig. 2) analysis of all shear veins measured around the Boborás pluton indicates that the vein pattern is compatible with a single stress tensor, implying horizontal E-W compression associated with horizontal N-S extension (Fig. 2).

Microfabrics in the Boborás pluton, dykes and sills

Structural observations on the Boborás G3 pluton, dykes and sills show little evidence of flow criteria. In the granite, a discreet sub-horizontal mineral lineation marked by N-S aligned elongate biotite is weakly developed at its roof. Under the microscope, submagmatic deformation features are characterized by numerous fractured plagioclases infilled by quartz and muscovite (Fig. 6e). The quartz inside the fractures is in continuity with that in the matrix, occasionally forming a single grain. Horizontal and north-south directed feldspar crystals are truncated, attesting to a local N-S extension (Fig. 6e). Numerous myrmekites are also observed at plagioclase/K-feldspar boundaries. In all these rocks, quartz forms large grains with little plastic deformation such as chessboard patterns, particularly abundant in the vicinity of the pluton roof (Fig. 6g). No evidence of overprinting deformation is observed, except in a few sectors where low-to medium-temperature dynamic recrystallization is seen.

Incipient solid-state deformation in granite and dykes is attested by wavy extinction, twins and kink bands affecting biotite, muscovite and plagioclase (Fig. 6f), subgrains in quartz and some well-developed fractures affecting all granite-forming minerals and infilled with muscovite (Fig. 6h).

Microfabrics within hydrothermal veins

Vein infill succession

The three types of quartz veins (extensional, hybrid and shear veins) are composed of quartz, K-feldspar, apatite, muscovite-phengite and sulphides [START_REF] Gloaguen | Apports d'une étude intégrée sur les relations entre granites et minéralisations filoniennes (Au et Sn-W) en contexte tardiorogénique (Chaîne Hercynienne, Galice centrale, Espagne)[END_REF]. Veins are built-up with four generations of quartz, each one corresponding to a deformation stage. The first generation, Q1 (~70% of the vein infill) is commonly coeval with K-feldsparmuscovite ± apatite (Figs. 7e andf) and locally scheelite (Asencio Pérez et al., 2000). Q2 quartz can be elongate or euhedral -in overgrowth on Q1-and corresponds to the reopening of Q1 veins. Q2 quartz deposition was followed by the growth of masses of arsenopyrite crystals along the Q1 and Q2 grain boundaries. Subsequent deformation increments created spaces along quartzarsenopyrite contacts or fractures inside the arsenopyrite crystals and are filled by Q3 quartz (Fig. 8a), often associated with pyrite and muscovite. This Q3 quartz locally is the matrix of brecciated borders of some shear veins. Finally, the last increment of deformation resulted in fractures and geodic cavities in arsenopyrite to be filled by undeformed and clear Q4 quartz grains (Fig. 8a), locally including electrum, Bi-Te sulphides and phengite.

Internal textures

Within the extensional veins, quartz is essentially of the Q1 type and occurs as large euhedral grains normal to vein walls. Abundant overgrowths are also observed. Some veins are geodic with centimetre-sized automorphic Q1 quartz crystals; they can contain magmatic brines as halite-and carbonates-bearing inclusions with a salinity of 35-40 wt. % NaCl equiv., which homogenize in the liquid phase at > 336° C [START_REF] Gloaguen | Apports d'une étude intégrée sur les relations entre granites et minéralisations filoniennes (Au et Sn-W) en contexte tardiorogénique (Chaîne Hercynienne, Galice centrale, Espagne)[END_REF]. Some extensional veins show single-phase sulphide domains composed of arsenopyrite (Fig. 7d).

A few hybrid veins, intermediate between tension and shear (Figs. 7e andf), display large euhedral quartz and K-feldspar grains, with a marked elongation oblique to the veins walls.

The feldspars are commonly fractured parallel to the wall rock, suggesting a crack-seal growth mechanism [START_REF] Ramsay | The crack-seal mechanism of rock deformation[END_REF]. The central part of the vein is devoid of feldspar grains and presents thin fractures filled with white micas, and sulphide or oxide grains.

By contrast, quartz and arsenopyrite grains in shear veins are laminated and form irregular ribbons (Fig. 7b). Quartz grains define a N-S trending lineation that may be observed on the shear vein surfaces. The first quartz generation (Q1) forms elongate comb-grains whose longaxes are parallel or at a low angle to the vein walls. The quartz grains are highly deformed with features resulting from intense dynamic recrystallization (Fig. 8). Extinction is strongly wavy and high-angle conjugate shear bands are common in the Q1 grains (Figs. 8b-c and 9ab). Small strain-free grains formed along the quartz grain boundaries and, inside the grains, along the conjugate shear bands (Figs. 8c, 8g and9). Shearing may induce local rotation of Q1 quartz crystals or aggregates (Fig. 9a). Several vugs in deformed Q1 quartz are filled by euhedral Q2 quartz (Figs. 8j,8k), attesting to the presence of micro pull-apart structures affecting Q1. However, Q2 mainly corresponds to very elongate grains parallel or oblique to the vein walls, between Q1 combs (Figs. 8c, 8i, 8g and 9b). Q2 quartz grains have far fewer fluid and solid inclusions than Q1 and are unaffected by dynamic recrystallization, having weak wavy extinction, no sub-grains and no conjugate shear bands. Locally, the elongate Q2 grains have parallel fluid-inclusion bands at an angle of 85 to 70° to the long grain axes (Figs. 8d andf). Some Q2 fibres have a very sharp boundary with neighbouring Q1 quartz (Fig. 8b), implying a different mode of crystallization.

Quartz [c]-axis fabrics from shear veins

Universal-stage measurements were made on quartz shear veins for both Q1 and elongate Q2 grains (Fig. 9), carefully avoiding several measurements on a single deformed comb grain.

Measurements of long-axes of the Q1 and Q2 grain confirmed that they are parallel to the shear vein border (Fig. 9d ande), this parallelism being better defined for Q2 grains. The average aspect ratio (length/thickness) is 8 for Q2 grains and only 4 for Q1 quartz. Most [c]axes of the Q1 grains (Fig. 9d) lie close to the rim of the diagram, indicating a high-angle to the vein border. Such a distribution is generally observed in the case of comb quartz.

Interestingly, many of the Q1 [c]-axes near the edge of the diagram are parallel to Q1 grain long-axes, indicating a Q1 grains rotation induced by normal shearing (Fig. 9a). Nevertheless, suggest two different modes of nucleation that will be discussed in 6.3 below.

Interpretation

Granite emplacement during N-S stretching: associated magmatic to submagmatic strain gauges

According to our gravity data, the Boborás pluton is a small and thin elliptical laccolite. From these data, two main root zones are identified at the north and south ends of the pluton, suggesting that it was formed by the coalescence of intrusions ascending from these feeders (e.g. D" [START_REF] D"eramo | Coalescence of lateral spreading magma ascending through dykes: a mechanism to form a granite canopy (El Hongo pluton, Sierras Pampeanas, Argentina)[END_REF]. The northern root zone is also attested by the high density of granite dykes and sills anywhere observed around the pluton. This may also have been the case for the southern root, but late crosscutting by the G4-pluton removed these features (Figs. 1 and2). This may imply a N-S-directed magma flow of melts from each root, further suggested by the faint sub-horizontal mineral lineation marked by biotites at the pluton roof.

The D3 E-W compression event led to N-S horizontal stretching marked by a N-S L3 stretching lineation (boudinaged G2 pegmatite and truncated andalusite and sillimanite in the thermal aureole, Figs. 4a andc), associated with top-to-the-north shearing (syntectonic cordierite porphyroblasts, Fig. 4b). At the Brués deposit, trends of G3 granite dykes are normal to the N-S stretching direction (Fig. 2), some having a staircase geometry (Fig. 6b).

Such geometries suggest that the traps capturing the injected magma were coeval with a north-verging motion of the micaschist hanging-wall. This suggests that, at the end of D3, the top-to-the-north shearing locally evolved to a north-verging normal motion in specific areas like the edges of N-S-elongated plutons. This may partially explain the lack of S3-cleavage-trajectory deflection at the north and south ends of the G3 plutons. This north-verging normal motion persisted during magma crystallization as demonstrated by the truncated feldspars (Fig. 6e) considered to be markers of submagmatic deformation and melt relocation during the late stage of granite crystallization (e. g. [START_REF] Hibbard | Deformation of incompletely crystallized magma systems: granitic gneisses and their tectonic implications[END_REF][START_REF] Bouchez | Submagmatic microfractures in granites[END_REF]. The numerous myrmekites observed are considered to have been formed by Caand Na-replacement of K-feldspar during low-strain deformation in a sub-magmatic state (e.g. [START_REF] Simpson | Evidence for deformation-induced K-feldspar replacement by myrmekite[END_REF][START_REF] Vernon | A Practical Guide to Rock Microstructures[END_REF].

High-temperature low-strain gauges

Deformation from the magmatic to solid-state condition is recorded by chessboard patterns (Fig. 6g) that are particularly abundant near the granite roof. Quartz chessboard patterns result from prism plus basal subgrain boundaries, revealing the activity of both a and prism cslip. The latter is restricted to the β-quartz stability field of high-temperature quartz and is thus dependant on P-T conditions [START_REF] Mainprice | Dominant c slip in naturally deformed quartz: implications for dramatic plastic softening at high temperature[END_REF][START_REF] Kruhl | Prism-and basal-plane parallel subgrain boundaries in quartz: a microstructural geothermobarometer[END_REF][START_REF] Kruhl | Prism-and basal-plane parallel subgrain boundaries in quartz: a microstructural geothermobarometer: Reply[END_REF]. Plotting the minimum pressure obtained from fluid-inclusion studies (2.5 kbar; [START_REF] Gloaguen | Apports d'une étude intégrée sur les relations entre granites et minéralisations filoniennes (Au et Sn-W) en contexte tardiorogénique (Chaîne Hercynienne, Galice centrale, Espagne)[END_REF] in the P-T diagram given by [START_REF] Kruhl | Prism-and basal-plane parallel subgrain boundaries in quartz: a microstructural geothermobarometer[END_REF], a minimum temperature of 630° C is needed for the formation of chessboard patterns in quartz in the Boborás granite within the β-quartz stability field. This provides strong evidence for subsequent deformation at subsolidus temperatures shortly after the beginning of crystallization [START_REF] Kruhl | Prism-and basal-plane parallel subgrain boundaries in quartz: a microstructural geothermobarometer[END_REF][START_REF] Stipp | The eastern Tonale fault zone: a "natural laboratory" for crystal plastic deformation of quartz over a temperature range from 250 to 700°C[END_REF]. Quartz grains with their irregular shapes, lobate boundaries, "pinning" features and large embayments are consistent with grain-boundary migration that calls for incipient dynamic recrystallization at high temperature and low strain rate [START_REF] Jessell | Grain-boundary migration microstructures in a naturally deformed quartzite[END_REF][START_REF] Passchier | Microtectonics[END_REF].

Fracturing during the magmatic-to-hydrothermal transition

During the final stages of granite roof crystallization, extensional and shear quartz veins begin to form close to granite dykes and the granite roof. The first Q1 quartz crystallized as comb quartz mainly normal to the vein walls but also parallel to them as shown by the [c]-axis diagram (Fig. 9). As we observed (Figs. 8 and9), quartz with such orientation is a candidate for being affected by conjugate shear bands under subsequent shearing [START_REF] Van Daalen | Orientation analysis of localized shear deformation in quartz fibres at the brittle-ductile transition[END_REF]. The Q1 quartz is strongly deformed by normal shear and locally affected by grain rotation, now appearing as ribbons parallel to the vein borders with conjugate shear bands. Q2 quartz does not contain such bands and occurs within Q1 micro pull-apart structures as thin ribbons parallel to the vein border (Fig. 8). The quartz grain shapes, [c]-axis diagrams and crack-seal bands marked by fluid-inclusion bands are consistent with stripped bedding-Q2 veins formed in pull-apart cracks within Q1 quartz during normal shearing, which may correspond to bedding subparallel striping and opening of the Q1 shear veins (B-type of [START_REF] Koehn | Shear sense indicators in striped bedding-veins[END_REF][START_REF] Bons | A review of the formation of tectonic veins and their microstructures[END_REF], compatible with the north-verging normal faulting. This is coherent with local geodic (comb) Q2 quartz observed within deformed Q1 quartz (Figs. 8a,8j and 8k). Because striped Q2 bedding veins form within Q1 quartz, they lack the typical inclusion trails and inclusion bands of host rock slices generally observed in such microstructures [START_REF] Koehn | Shear sense indicators in striped bedding-veins[END_REF]. The differences between Q1 and Q2 aspect ratios, deformation and [c]-axis orientations are compatible with a Q2 crystallization postdating more deformed Q1, with Q2 crystallizing in cracks within Q1 during shearing. The growth of Q2 quartz may thus be syn-tectonic and polyphase veins infills also demonstrate a succession of deformation increments during the life of the hydrothermal system.

Consequently, we assume that the Boborás granite and its associated Brués gold deposit accurately record the magmatic-to-hydrothermal transition during a persistent and single lowstrain (i.e. lack of mylonites and detachment planes) regional deformation. Hereafter, we will discuss the role of pluton dynamics (e.g. feeder zones, dykes, etc.) in the deformation localization process.

Discussion

Role of pluton dynamics in incremental strain localization

Combined with a low regional deformation intensity, we propose that the zone feeding the north edge of the pluton controlled the magmatic-hydrothermal system (Fig. 10). Successive magma injections through this feeder likely triggered mechanical instabilities along the northern contact of the pluton. Inflexions of the dyke trends, seen on Fig. 5a, suggest variations in stress orientation throughout the granite roof. The probable cause of such variations was interference between regional and magmatic stress fields, again underlining the role of pluton dynamics. The magmatic to solid-state microstructures observed in the granite roof record numerous deformation patterns that typically characterize successive mechanical instabilities during decreasing temperature (e.g. [START_REF] Pawley | The development of contrasting structures during the cooling and crystallisation of a syn-kinematic pluton[END_REF]. The deformation continued after granite roof crystallization and became more localized, marked by quartz veins recording the magmatic-to-hydrothermal transition (e.g. [START_REF] Mustard | The magmatic-hydrothermal transition at Timbarra: Implications for the genesis of intrusion-related gold deposits[END_REF].

Implication for reduced intrusion-related gold deposits (RIRGD)

The polyphase veins also exhibit a succession of deformation increments and fluids infills, with the deposition of gold and associated minerals occurring at the end of the hydrothermal system life cycle. Since several stages of deformation and infill are required for creating an efficient trap for gold, such as fractured sulphide-bearing quartz veins acting as a electrochemical trap (e.g. [START_REF] Möller | Electrochemical accumulation of visible gold on pyrite and arsenopyrite surfaces[END_REF], a succession of mechanical instabilities throughout the life of a hydrothermal system is required for efficient gold deposition.

The structural controls of RIRGD are considered "intermediate" between the strong variability of vein orientations in porphyries and the strong regularity of lode orientation in orogenic gold deposits [START_REF] Stephens | Structural and mechanical controls on intrusion-related deposits of the Tombstone Gold Belt, Yukon, Canada, with comparisons to other vein-hosted ore-deposit types[END_REF]. These authors demonstrated the dominant control of regional stress fields without the involvement of major faults and large-scale shear zones, as in the case for orogenic gold deposits. Our results show that both pluton dynamics and favourably located magma-feeding zones have to be looked for in the exploration of such mineralized systems. In other words, the stages of pluton construction have to be carefully examined as they might control and located on-going mineralization and ore volume. [START_REF] Eldursi | Numerical investigation of transient hydrothermal processes around intrusions: Heat-transfer and fluid-circulation controlled mineralization patterns[END_REF] numerically modelled hydrothermal processes during pluton emplacement for the Boborás granite, according to which, favourable areas for mineralization are located along the upper edges and roof of the pluton. They also showed that highly permeable fractured thermal aureoles (10 -15 m²) may restrict the mineralization in the upper corner of the Boborás pluton roof, and that mineral deposition can develop before and during the hottest phase of pluton emplacement. Our results agree with this hydrothermal modelling, particularly in explaining why only some parts of intrusion are mineralized. The mineralizing fluids were discharged and focused around high-permeability zones that are controlled by pluton dynamics, thus resulting in mechanical instabilities and strain localization.

Conclusions

Field and microstructural analysis of the Boborás granite, emplaced at a minimum depth of 9.3 km in a D3 E-W-directed shortening and N-S extension event, demonstrate that this thin laccolith (1-1.5 km) was emplaced at the end of D3. The regional D3 stress combined with magma inflow from the underlying root zone located a succession of mechanical instabilities in the northern part of the pluton roof. The Boborás pluton and its Brués gold deposit record a complete succession of magmatic and hydrothermal structures and textures, spanning the magmatic-to-hydrothermal transition during cooling of the pluton. Based on these data, we established a coherent model where gold mineralization is coeval with pluton emplacement, magma dynamics and crystallization. Our results strongly argue for studying pluton dynamics and growth (e.g. locating feeder zones) when evaluating the structural controls of an intrusion-related gold deposit. For a long time, the interference between regional and pluton (volume-related) stress fields has been recognized as a major factor in pluton emplacement (Brun and Pons, 1981). In the same way, we suggest that such interference is critical for most ore deposits (Au, Sn, W) related to granite intrusions. Such "mineralizing" interference is favourably expressed in the case of regional low-strain zone. Conversely, such interference should play a minor role in ore formation when mineralizing granitic magma is emplaced in localized high-strain shear zones.

Figure captions

a

  second maximum is close to the center of the diagram and shows Q1 [c]-axes normal to the direction of shear. The [c]-axis diagram of Q2 (Fig. 9e) is different and shows several [c]-axes close to the centre of the diagram. The four maxima of Q2 [c]-axes are at high angles with the grain long-axes, and mainly toward the local extension direction (north -X) with an angle of about 60° with the shear vein dip. These [c]-axis diagram differences between the two quartz generations

Fig. 1 :

 1 Fig. 1: The study area in the western part of the Variscan belt before opening of the Bay of Biscay. (A) Location in Western Europe; abbreviations: AM, Armorican Massif, MC, Massif Central, CZ, Cantabrian Zone, WALZ, West-Asturian-Leonese Zone, GTOMZ, Galicia-Trásos-Montes Zone, CIZ, Centro Iberian Zone, OMZ, Ossa-Morena Zone, SPZ, South Portuguese Zone; (B) Geological sketch map of the Schistose Domain in Galicia Trás-os-Montes Zone (modified from Barrera Morate et al, 1989, and from González Cuadra et al., 2006 and Sizaret et al., 2009 for the Beariz granite).

Fig. 2 :

 2 Fig. 2: Geological map of the Boborás granite area (modified from the 1:50,000-scale Ourense sheet 187 and the 1:200,000-scale Oursense/Verin sheet 17/27 of the Geological map of Spain, and field work by Gloaguen, 2006). Orientation diagrams (lower hemisphere, equal area) of G3 Boborás granite dykes, quartz veins and G3 aplite-pegmatite dykes around the Boborás pluton are indicated.

Fig. 4 :

 4 Fig. 4: Photographs of outcrops and microstructures indicating deformation kinematics in host-micaschists during D3. (a) Boudin-like layered aplite-pegmatite related to G2 granite; the near-vertical S3 crenulation cleavage turns around the magmatic body (Hammer for scale). (b)

Fig. 5 :

 5 Fig. 5: (a) Geological map of the northern edge of the Boborás granite and its associated granitic dyke network. (b) Interpretative cross-section within the incised Viñao river valley of the Boborás granite roof at Brués gold deposit; cross-section drawn from field and underground mapping. (c) Detailed view of the roof geometry of the Bobóras granite showing the relationship between "en échelon" extensional veins and shear veins. (d) Cross section within micaschist on top of granite in vertical road cuts along the N541 road, which show intrusive bodies and late deformation affecting magmatic bodies.

Fig. 6 :

 6 Fig. 6: Photographs showing morphology and deformation within intrusive bodies. (a) Early

Fig. 7 :

 7 Fig. 7: (a) Re-used granite dyke intruding micaschist with a north-normal motion. A N088°E trending quartz-arsenopyrite shear vein dipping 60°W is emplaced within a normal fault along the dyke. (b) Detailed view of the previous photograph. Foliation in shear vein is well-marked

Fig. 8 :

 8 Fig. 8: Texture and filling of a north-normal quartz-arsenopyrite-bearing shear vein

Fig. 9 :

 9 Fig. 9: Microstructures, [c]-axis study of Q1 and Q2 quartz generation from the north normal

Fig. 10 :

 10 Fig. 10: Integrative conceptual model of the Boborás G3 granite emplacement and the associated Brués hydrothermal gold system (not to scale). (a) The first stage is the

G2 pegmatite sill cross-cut by a G3 Boborás granite sill. Concrete corresponds to the upper part of the ditch of the national highway. (b) Granite dyke with staircase border consistent with north-verging shearing during granite emplacement. (c) Boborás granite dyke and sill intruding host micaschist. The shape of the connection between dyke and sill is compatible with a north-normal motion during granitic melt injection. The main schistosity is S2, and S3 is marked by a N-S horizontal lineation crenulation, not visible at this scale. (d) Stretched granite sill. The beginning of N-S boudinage (neck) is marked by a quartz extensional vein. (e) Breaking plagioclase filled by magmatic quartz showing a north-normal motion during granite dykes emplacement. (f) Plagioclases with kink bands. (g) Chessboard texture within large quartz grain from a granite dyke. (h) Solid-state fractures filled by muscovite (arrows) at the granite roof.

Acknowledgements

This study had the scientific and financial support from the CNRS and BRGM through the National Research Group on Ore Deposits (GDR n°2458, TRANSMET). We particularly thank J. Pons for constructive discussions. Our work also benefited from fruitful discussions with all co-workers of the research group involved in the GDR TRANSMET project.

Constructive reviews by J.L. Bouchez, M. Tubía, an anonymous reviewer and S. Paterson (on a former version) helped in greatly improving the manuscript.

emplacement of the G3 Bobóras granite, along a rheological discontinuity represented by the contact between the Carballiño G2 granite and micaschist. This emplacement and the location of the underlying root zone may have created a mechanical instability at the pluton"s roof.

Together with the regional stress, this favoured injection of the earliest granite dyke controlled by the on-going north-directed shearing. The bulk shear strain (double arrows) was very low, as attested by the lack of mylonites and detachment planes. (b) The deformation/emplacement history then continued with on-going granite and dyke emplacement, earlier emplaced granites and dykes undergoing sub-magmatic deformation.

During this stage, granite sills and dykes (also affected by sub-magmatic deformation) were emplaced. (c) Vein and dyke system formation at the granite roof was achieved with the emplacement of syntectonic quartz veins, mainly close to the granite dykes and still controlled by the north-verging shearing and the mechanical instability. At this stage, many granite-dyke wall rocks were reactivated to form hydrothermal shear veins. The injection process and pluton growth increment may have continued at depth in the root zone area.


Sébastien Courtin

Boniface Mbih
email: boniface.mbih@unicaen.fr

Issofa Moyouwou

Susceptibilty to coalitional strategic sponsoring The case of parliamentary agendas

Keywords:

It usually happens that the alternatives to be voted on in committees are chosen or sponsored by some particularly active committee members. For example, in parliaments, some representatives and some government members are known to be especially active in introducing bills on which the whole committee will later vote. It appears that parliamentary agendas -namely amendment and successive elimination voting rules -are vulnerable to strategic behavior by groups of individuals introducing motions which are not their most preferred alternatives. Our aim in this paper is to evaluate how frequently this type of behavior is susceptible to arise.

Introduction

Most of the literature on strategic voting studies the manipulation of individual preferences given the issue, that is the set of feasible alternatives. The main result on this topic is the Gibbard-Satterthwaite theorem (Gibbard 1973 and[START_REF] Satterthwaite | Strategy-proofness and Arrow's conditions: existence and correspondence theorems for voting procedures and social welfare functions[END_REF]. It states that every nondictatorial voting procedure selecting a unique outcome is vulnerable to strategic manipulation of preferences, in the sense that one can …nd some con…guration of preferences at which some individual has an incentive to misrepresent her preferences.

Besides, some authors are concerned with a related but di¤erent line of investigation, which takes account of a variable issue; in other words the set of alternatives open to the vote is not given a priori. This line must be distinguished from the literature on the strategic e¤ects of agenda manipulation, which takes into consideration the strategic ordering of alternatives (see [START_REF] Banks | Sophisticated Voting Outcomes and Agenda Control[END_REF][START_REF] Miller | Committees, Agendas, and Voting[END_REF].

One aspect of the variable issue analysis is strategic candidacy. More precisely, in an electoral context, strategic candidacy concerns the opportunity for a potential candidate who cannot win the election given the preferences of the voters, to opt in or to opt out, in order to secure another candidate she ranks higher in her preferences than the one who would win the election.

Recently, [START_REF] Osborne | A model of political competition with citizencandidates[END_REF] and [START_REF] Besley | An economic model of representative democracy[END_REF] independently develop, in the context of representative democracy, the same basic model of electoral competition in which the political process is modeled as a three-stage game; they further assume that running as a candidate in the election is costly. In stage 1, each citizen decides whether or not to become a candidate; in the second stage, all citizens vote over the set of declared candidates; and in stage 3, the winner can implement her favorite policy and receive bene…t from her position. Their main result states that there exist situations in which candidates, with no chance of winning, enter the election in order to a¤ect the identity of the winner, even though such entry is both optional and costly.

In this same context, another important work is due to Dutta, Jackson andLe Breton (2000, 2001); they particularly focus on the incentive of candidates to strategically a¤ect the outcome of a voting procedure. They give speci…c results for voting procedures based on sequential pairwise elimination of alternatives, but they also provide more general results. They show that the outcome of every nondictatorial voting rule satisfying a unanimity condition is susceptible to be a¤ected by strategic candidacy, not only when the set of voters and candidates are distinct, but also when they overlap.

Another aspect of the variable issue approach, early introduced by [START_REF] Majumdar | Choice and revealed preference[END_REF], and later extensively discussed by [START_REF] Dutta | On strategic manipulation of issues in group decision[END_REF], is the problem of strategic sponsoring of alternatives. In a famous example (see Section 2), Majumdar considers seven voters (the choosers) two of whom are sponsors (two political opponents or two especially energetic members of a committee who usually move all resolutions), and he assumes that all voters always vote according to their sincere preferences. This process of sponsoring takes place in two steps. First, sponsors choose the issue, and second the alternatives in the chosen issue are then submitted to the vote. The example shows that under simple majority rule, it can be advantageous for sponsors to select alternatives non sincerely, i.e. alternatives that do not come highest in their preference orderings. [START_REF] Dutta | On strategic manipulation of issues in group decision[END_REF] provide a generalization of the result of Majumdar. They prove that under a large class of voting procedures some voting situations will give individuals opportunities of strategic sponsoring.

One can easily imagine actual situations where the process of sponsoring takes place. The intuition behind the mechanisms we describe mainly refer to parliamentary agendas: in parliaments, some representatives and some government members introduce bills on which the whole committee will later vote. But many other examples can be found in various …elds. This is the case for the pre-selection of candidates applying for a position in a …rm or some other organization (e.g. lecturers in universities); it usually happens that a limited set of reporters chooses among candidates, and those who pass this step then constitute the set of alternatives submitted to the vote of all committee members. Another example can be found in the area of sports; for the election of the "Ballon d'or" of the journal "France Football", some football -or soccer -players are …rst selected by a limited set of French journalists and then there is a …nal vote by a larger international set of journalists. A similar but more sophisticated process can be found for the selection of the "NBA All-Star Games" players.

Also note that sponsoring of alternatives must be distinguished from the rule of k names where "given a set of alternatives a committee chooses k members from this set by voting and make a list with their names, then a single individual from outside the committee selects one of the list names for the o¢ce" (see Barbera and Coelho 2004).

In this paper, we examine strategic sponsoring under two families of parliamentary agendas widely studied in the literature (see [START_REF] Rasch | Parliamentary Voting Rules[END_REF][START_REF] Dutta | Strategic candidacy and voting procedures[END_REF]Mbih, Moyouwou and Picot 2008 among others). We now describe them.

Under these two rules, alternatives are ranked according to a predetermined order, say a 1 a 2 a 3 in the three-alternative case. These rules are commonly used in Parliaments for votes on ordinary motions.

On the one hand, the Anglo-American system is based on a "two by two" procedure, namely the amendment procedure: the …rst ballot is taken between a 1 and a 2 in a pairwise majority contest, and the winner is taken against a 3 ;

On the other hand, the successive elimination rule is used in most countries of Western Europe. Alternatives are considered "one by one": the …rst vote is on alternatives a 1 ; if a 1 wins a majority, the procedure terminates; if a 1 is beaten, there is a second vote on a 2 , and so on, until there is a winner or a unique alternative left. Note that this amounts to …rst having a majority contest between a 1 and fa 2 ; a 3 g, and then between a 2 and a 3 if necessary.

Parliamentary voting procedures are vulnerable to strategic sponsoring of alternatives (see Section 2). However, it remains to evaluate how frequently this behavior is susceptible to occur. As pointed out by Pattanaik (1978, p. 187) "This is important. For, if the likelihood of such strategic voting is negligible, then one need not be unduly worried about the existence of the possibility as such."

In this contribution, for each of the procedures described above, we are interested in the quantitative signi…cance of the possibilities of strategic sponsoring. Our evaluation relies on an analytical method, as opposed to computer simulations. We compute the exact frequency of strategic sponsoring opportunities under the impartial anonymous culture (IAC), a hypothesis introduced in social choice theory by [START_REF] Kuga | Voter antagonism and the paradox of voting[END_REF] and later developed by [START_REF] Gehrlein | The probability of the paradox of voting: a computable solution[END_REF]. IAC is based on the assumption that all anonymous pro…les (see Section 2) have the same probability of occurrence. In other words, our purpose is to compute the following ratio: number of anonymous pro…les vulnerable to coalitional strategic sponsoring total number of all possible anonymous pro…les

The remainder of the paper is organized as follows: Section 2 is a presentation of the general framework, with de…nitions, assumptions and some examples; Section 3 gives a characterization of coalitional strategic sponsoring situations; next, Section 4 provides our main results about strategic sponsoring; and …nally Section 5 concludes the paper.

Notations and de…nitions

Consider a …nite set N of n voters (choosers), with n 3, and a …nite set S of sponsors, with 3. A will denote the set of alternatives, and 2 A the set of all possible nonempty subsets of A. Further, B will denote the issue, that is the set of sponsored alternatives, B 2 2 A .

The process of sponsoring, which leads to …nal outcomes, takes place in two distinct stages. First, sponsors choose, within the set A, the alternatives that will be submitted to the vote at stage two. Each sponsor independently chooses one alternative within this set, and one same alternative can be chosen by more than one sponsor. The issue B is determined as the outcome of the choice of alternatives by sponsors. At stage two the voters express their individual preference orderings over B. And given a voting procedure and the preference orderings expressed by individuals over the issue, the …nal outcome is determined.

Notice that in real life not all individuals may be potential sponsors (for example, in parliaments, bills are only introduced by some active representatives or government members). We shall thus admit that n. Moreover we assume that the sets of voters and sponsors are disjoint, i.e. S \ N = ?. It was also possible to consider the case where S N . This distinction can be illustrated by the usual ways laws are introduced in Parliaments. For example in France (and many other countries), bills are proposed by ministers -who do not participate to the vote -while private bills are proposed by representatives -who participate to the vote. In our study, in order to avoid unnecessary complexities, we only consider the case S \ N = ?, since with a …xed number of sponsors and a large electorate, both cases lead to very similar results.

Let L be the set of all possible linear orderings on A, that is the set of all complete, antisymmetric and transitive binary relations on A. R i denotes voter i 0 s preference relation and R i 2 L. A pro…le of voters' preferences is an n-tuple R N = (R 1 ; R 2 ; :::; R n ) of individual preference relations, one for each individual voter, and L N is the set of all such pro…les. In the same way for each sponsor j, let R j be j 0 s preference relation; then R S is a pro…le of sponsors' preferences, and L S is the set of all such pro…les.

We now present the strategic sponsoring behavior more formally. We …rst need some general de…nitions.

De…nition 1 A social choice function (SCF) is a mapping f from L N to A.
Including a sponsoring process leads to a notion which is more general than an SCF. For each pro…le R S of sponsors preferences over the set A of alternatives, one issue B 2 2 A is selected by the choice of a single alternative by each sponsor. And for this issue, preferences of the voters determine the …nal outcome.

De…nition 2 A social choice function with sponsoring (SCFS) is a mapping

g from 2 A L S L N to A, such that for all B; R S ; R N 2 2 A L S L N g B; R S ; R N 2 B:
Let B be the set of sponsored alternatives when sponsors choose sincerely, that is when each of them chooses her most preferred alternative. B is called a sincere issue.

De…nition 3 Let S be the set of sponsors and B the sincere issue. Given an SCFS g, a pro…le R S ; R N is unstable via coalitional strategic sponsoring if there exist some nonempty subset S 0 of S and some other issue B 0 , such that (i) B \ B 0 6 = ?;

(ii) g B; R S ; R N 6 = g B 0 ; R S ; R N ; (iii) and g B 0 ; R S ; R N R j g B; R S ; R N for all j 2 S 0 .

In words, strategic sponsoring occurs if there exists some group of sponsors S 0 (possibly a single sponsor), who have an incentive to sponsor another alternative in order to ensure that the new outcome is better from their viewpoint.

De…nition 4 A social choice function with sponsoring g is vulnerable to coalitional strategic sponsoring if there exists at least one pro…le B; R S ; R N unstable via coalitional strategic sponsoring.

In the sequel, where there is no ambiguity we shall simply write unstable via strategic sponsoring.

Note that a society choosing an alternative from a …nite set using an SCFS can be viewed as a game in normal form where (i) the set of players is the whole set of agents, sponsors and voters, although we suppose here that only sponsors can behave strategically; (ii) the set of strategies open to agents is the set of all linear orders on A, and (iii) the payo¤ function is the SCFS. With this interpretation of our context, an unstable situation via coalitional strategic sponsoring appears as a pro…le that is not a strong equilibrium point of the given game.

To illustrate the strategic sponsoring process, let us now present the (famous) example of [START_REF] Majumdar | Choice and revealed preference[END_REF].

Example 1 Suppose there are two sponsors 1 and 2 among seven voters; further there are four alternatives, a 1 ; a 2 ; a 3 and a 4 , and the SCFS is the plurality rule. Note that, in contrast to our study Majumdar assumes that the sponsors participate to the vote. Individual preferences are as follows:

1 2 3 4 5 6 7 a 4 a 3 a 1 a 3 a 3 a 2 a 4 a 1 a 2 a 2 a 4 a 1 a 1 a 2 a 2 a 1 a 3 a 2 a 4 a 3 a 1 a 3 a 4 a 4 a 1 a 2 a 4 a 3
If we suppose that the sponsors choose sincerely, then the sincere issue is B = fa 3 ; a 4 g. Then the voters express their individual preference orderings over B, and a 3 is chosen. But if sponsor 1 decides to sponsor a 1 instead of a 4 , then the new issue is B 0 = fa 1 ; a 3 g, and a 1 is chosen. And since sponsor 1 prefers a 1 to a 3 , it appears that the pro…le above is unstable via strategic sponsoring.

Up to now, all de…nitions and the example above have been given in terms of pro…les, but only for convenience. In the remainder of this work, we will be interested only in anonymous pro…les, in the sense that we do not distinguish between two pro…les that di¤er only by the identity of the individuals who express the same preference relation. In other words, the outcome of a given con…guration of individual preferences only depends on the number of individuals with some type of preference relation, and not on which individuals have that type of preferences. Thus, we assume anonymity over the set of voters and over the set of sponsors.

From now on, we shall focus on the three-alternative case. Given a set A = fa 1 ; a 2 ; a 3 g, there are exactly six linear orderings on A, labeled below:

R 1 : a 1 a 2 a 3 , R 2 : a 1 a 3 a 2 , R 3 : a 2 a 1 a 3 , R 4 : a 2 a 3 a 1 , R 5 : a 3 a 1 a 2 , R 6 : a 3 a 2 a 1 .
Then, a voting situation n is an anonymous voting pro…le obtained from a pro…le R N by rewriting it as n = (n 1 ; n 2 ; n 3 ; n 4 ; n 5 ; n 6 ), where for each k = 1; :::; 6; n k is the number of voters in N with preference relation R k . A situation is thus a 6 tuple of natural integers such that 6 k=1 n k = n. The set of all voting situations will be denoted N. Given a voting situation n 2 N and two alternatives a j and a k , n(a j ; a k ; n) is the number of voters who prefer a j to a k given situation n.

Likewise, a sponsoring situation s = (s 1 ; s 2 ; s 3 ; s 4 ; s 5 ; s 6 ) is obtained from a pro…le R S where s k is the number of individuals in S with preference relation R k and 6 k=1 s k = . The set of all sponsoring situations will be denoted S. And (a j ; a k ; s) is the number of sponsors who prefer a j to a k given situation s.

Note that every pro…le R S ; R N can be rewritten as a situation (s; n); (s; n) is then said to be associated with R S ; R N . The de…nition below follows straightforwardly from De…nitions 3 and 4.

De…nition 5 A situation is unstable (via coalitional strategic sponsoring) if it is associated with an unstable pro…le.

De…nition 6 A social choice function with sponsoring g, is vulnerable to coalitional strategic sponsoring if there exists some unstable situation under g.

In our study we consider two families of parliamentary agendas, namely the amendment and successive elimination procedures, with possibly qual-i…ed majority. An -majority contest, …rst introduced in the social choice literature by [START_REF] Slutsky | Equilibrium under majority voting[END_REF], is a rule under which, given some 2 ]0; 1[ and a voting situation n, an alternative a h is socially preferred to (at least as good as) a k if the number of individuals who prefer a h to a k is at least 1 times the number of individuals who prefer a k to a h . Moreover, ties are broken in favor of the alternatives with the greatest index, which can be written as follows:

a h beats a k , h < k ) n(a k ; a h ; n) < n(n) k < h ) n(a h ; a k ; n) n(n)
Notice that simple majority corresponds to = 1 2 . Also note that small values of give a signi…cant advantage to alternatives with greater indices. Reciprocally, large values of give an advantage to alternatives with smaller indices.

Under the -amendment procedure (denoted AP ) one assigns to a voting situation n the alternative AP (B; s; n) de…ned as follows:

(a) if B contains a single alternative x, then AP (B; s; n) = x; (b) if B is a pair of two alternatives x and y, then AP (B; s; n) is the winner of the -majority contest between x and y; (c) if B = fa 1 ; a 2 ; a 3 g then AP (B; s; n) is selected from the -majority contest between a 3 and the winner of the -majority contest between a 1 and a 2 . Then, only one of the four scenarios below holds: a 1 beats a 2 at the …rst ballot and a 1 beats a 3 at the second ballot a 1 beats a 2 at the …rst ballot and a 3 beats a 1 at the second ballot a 2 beats a 1 at the …rst ballot and a 2 beats a 3 at the second ballot a 2 beats a 1 at the …rst ballot and a 3 beats a 2 at the second ballot Under the -successive elimination procedure (denoted SE ), one assigns to a voting situation n the alternative SE (B; s; n) de…ned as follows:

(i) if B contains at most two alternatives then SE (B; s; n) and AP (B; s; n) coincide; (ii) for B = fa 1 ; a 2 ; a 3 g, voters have to compare A 1 = fa 1 g and A 2 = fa 2 ; a 3 g at the …rst ballot.

If A 1 is selected then SE (B; s; n) = a 1 , otherwise SE (B; s; n)
is the winner of the -majority contest between a 2 and a 3 at the second ballot. Then, only one of the three scenarios below holds:

a 1 beats fa 2 ; a 3 g at the …rst ballot fa 2 ; a 3 g beats a 1 at the …rst ballot and a 2 beats a 3 at the second ballot fa 2 ; a 3 g beats a 1 at the …rst ballot and a 3 beats a 2 at the second ballot Further, for the latter class of procedures, every individual must compare A 1 = fa 1 g with the set A 2 = fa 2 ; a 3 g. Thus additional information on individual behavior has to be provided, explaining from R i which subset, out of A 1 = fa 1 g and A 2 = fa 2 ; a 3 g, is most preferred by voter i. We then assume that there are two types of voters in society N , with two distinct attitudes, namely maximin behavior or maximax behavior.

Then, with the successive elimination rules, we shall consider two di¤erent families of procedures:

(1) the voters are all supposed to have maximin behavior, then we denote this type of successive elimination rules, SEm ;

(2) and the voters are all supposed to have maximax behavior, then we denote this type of successive elimination rules, SEM .

The reader can easily check that these two families of rules are vulnerable to strategic sponsoring, as illustrated by the following example.

Example 2 Consider three sponsors who do not participate to the vote, …ve voters, and alternatives a 1 , a 2 and a 3 . AP 1 2 is the voting rule. Assume individual preferences are as follows:

Sponsors R k R 1 R 3 R 4 a 1 a 2 a 2 a 3 a 1 a 3 a 2 a 3 a 1 s k 1 1 1 Voters R k R 1 R 3 R 5 R 6 a 1 a 2 a 3 a 3 a 2 a 1 a 1 a 2 a 3 a 3 a 2 a 1 s k 1 1 2 1
A sincere behavior by the sponsors leads to B = fa 1; a 2 g and AP 1 2 ( B; s; n) = a 1 . However if sponsor with preference R 3 chooses a 3 rather than a 2 , then the new issue is B 0 = fa 1; a 2 ; a 3 g and AP 1 2 (B 0 ; s; n) = a 3 . We observe that a 3 is preferred to a 1 by that sponsor.

Our main purpose in this contribution to social choice theory is to compare the three families of rules described above through their vulnerability to strategic sponsoring, for any possible situation. In order to do that, we compute the ratio number of situations (s; n) unstable via coalitional strategic sponsoring total number of all possible situations (s; n)

Note that such frequency evaluations are based on the hypothesis that all possible situations (s; n) have the same probability of occurrence. Thus, as said above, we assume anonymity over the set of voters and over the set of sponsors.

In the next two sections, we characterize situations at which for each voting system considered in this paper coalitional strategic sponsoring is susceptible to occur. And these characterizations are then used to obtain formulae giving frequencies measuring the vulnerability of those systems to strategic sponsoring.

Characterizations of unstable situations

As explained in De…nition 6, a social choice function with sponsoring g, is vulnerable to coalitional strategic sponsoring if there exists some unstable situation under g. This section characterizes the sets of all situations (s; n) unstable via strategic sponsoring. More precisely, to illustrate the reasoning, we focus on the particular case where a 1 is the initial winner. For a general characterization of vulnerable situation for amendment and successive elimination procedures with maximin and maximax see the associated working paper [START_REF] Courtin | Sponsoring under parliamentary voting with anonymous voters[END_REF]. Likewise, not all proofs are given below, because they are very similar; we only provide details for the proof concerning the amendment procedure. For all the other proofs, the reader can refer to the associated working paper.

Preliminary observations on strategic sponsoring

Given a set of three distinct alternatives fa h ; a j ; a k g, let us consider a social choice function with sponsors. We assume that sponsors can choose only one alternative. Depending on sponsors preferences, several sincere issues B are conceivable. Moreover, according to a particular sincere issue, strategic behavior by the sponsors can occur in several ways.

First suppose that B consists of a unique alternative a h , that is all sponsors have a h as the same most preferred alternative. Then by the de…nition of a social choice function with sponsors, a h is selected by the voters. Since a h is trivially the best outcome for each sponsor, therefore there is clearly no way for strategic sponsoring. Now suppose that the sincere issue is B = fa h ; a j g and without loss of generality, assume that a h is elected. Then there are three types of possible collective strategic sponsoring: the new issue is fa h ; a k g and a k is …nally elected (Type 1); the new issue is fa h ; a j ; a k g and a j wins (Type 2); and the new issue is fa h ; a j ; a k g and a k wins (Type 3). Note that strategic behavior in case Type 3 is undertaken in favor of alternative a k , which is added to the sincere issue B, while in case Type 2, strategic sponsoring is in favor of a j , which was already sponsored in B. Also observe that, for strategic sponsoring to occur, the case where the new issue is fa h g is not possible since a h is already the winner. In the same way, from B = fa h ; a j g, situations where the new issue is fa j g or fa k g or fa j ; a k g are not rational, since this would require a change by a h 's sponsors, which is not possible since a h is their most preferred alternative.

Finally, suppose that B = fa h ; a j ; a k g and that a h is elected. Then there is a unique way for strategic sponsoring to occur, Type 4, that is the new issue is fa h ; a j g and a j is elected (or fa h ; a k g is the new issue and a k wins). For the same reason as above, situations where the new issue is fa h g, fa j g, fa k g or fa j ; a k g are not feasible.

In the next subsection and for each social choice function with sponsoring under study, strategic attitudes Type 1, Type 2, Type 3 and Type 4 will be useful in the proofs of characterizations results.

Unstable situations via strategic sponsoring

The statement below describes how a situation (s; n) must be in order for it to be subject to possible strategic sponsoring. For simplicity, we only consider the case where a 1 is elected.

Proposition 1 Suppose AP is the voting rule and a 1 is elected given some situation (s; n). Then (s; n) is unstable via coalitional strategic sponsoring if and only if

(1.a) or (1.d) 8 > > > > < > > > > : n1+n2+n5>n n (1.1) n4+n5+n6 n (1.2) s1+s2 1 (1.3) s4 1 (1.4) s1+s2+s4= (1.5) 8 > > > > < > > > > : n1+n2+n3>n n (1.18) n3+n4+n6 n (1.19) s1+s2 1 (1.20) s6 1 (1.21) s1+s2+s6= (1.22) or (1.b) or (1.e) 8 > > > > > > < > > > > > > : n1+n2+n5>n n (1.6) n4+n5+n6 n (1.7) s1+s2 1 (1.8) s3+s4 2 (1.9) s4 1 (1.10) s1+s2+s3+s4= (1.11) 8 > > > > > > > > < > > > > > > > > : n1+n2+n3>n n (1.23) n3+n4+n6 n (1.24) n1+n3+n4>n n (1.25) s1+s2 1 (1.26) s5+s6 2 (1.27) s6 1 (1.28) s1+s2+s5+s6= (1.29) or (1.c) 8 > > > > > > < > > > > > > : n1+n2+n3>n n (1.12) n3+n4+n6 n (1.13) n2+n5+n6 n (1.14) s1+s2 1 (1.15) s5+s6 2 (1.16) s1+s2+s5+s6=
(1.17)

Proof Necessity. Let (s; n) be an unstable situation and suppose that a 1 is elected. From the previous analysis, the sincere issue is fa 1 ; a 2 g, fa 1 ; a 3 g or fa 1 ; a 2 ; a 3 g. First suppose that the sincere issue B = fa 1 ; a 2 g. Only strategic sponsoring Type 1, Type 2 or Type 3 are conceivable.

Type 1: the new issue is fa 1 ; a 3 g and strategic sponsoring initiated by the sponsors of a 2 is in favor of a 3 (1.a). Therefore there is at least one sponsor with preference a 1 a 2 a 3 or a 1 a 3 a 2 (1.3), at least one sponsor with preference a 2 a 3 a 1 (1.4) and no sponsor with preference a 2 a 1 a 3 (s 3 = 0 in (1.5) (since such a sponsor has no incentive to favor a 3 ). Also observe that since B = fa 1 ; a 2 g, no sponsor has preference a 3 a 1 a 2 nor a 3 a 2 a 1 (s 5 = s 6 = 0 in 1.5). Given the sincere issue B, a 1 beats a 2 (1.1) and given the new issue fa 1 ; a 3 g, a 3 beats a 1 (1.2).

Type 2: the new issue is fa 1 ; a 2 ; a 3 g and strategic sponsoring is initiated by the sponsors of a 2 in favor of a 3 (1.b). Then there is at least one sponsor with preference a 1 a 2 a 3 or a 1 a 3 a 2 (1.8), at least two sponsors for a 2 (1.9) with at least one sponsor with preference a 2 a 3 a 1 (1.10). And since B = fa 1 ; a 2 g, no sponsor has preference a 3 a 1 a 2 or a 3 a 2 a 1 (s 5 = s 6 = 0 in 1.11). According to the sincere issue B, a 1 beats a 2 (1.6) and given the new issue fa 1 ; a 2 ; a 3 g, a 1 beats a 2 (1.6) and a 3 beats a 1 (1.7), (note that at the …rst ballot a 1 beats necessarily a 2 ).

Type 3: the new issue is fa 1 ; a 2 ; a 3 g and strategic sponsoring must be in favor of a 2 . However a 1 beats again a 2 at the …rst ballot. Therefore strategic sponsoring Type 3 does not occur under the amendment procedure. Note that this is a consequence of the predetermined order a 1 a 2 a 3 of pairwise majority contests. Now assume that the sincere issue B = fa 1 ; a 3 g. Only strategic sponsoring Type 1, Type 2 or Type 3 are conceivable. For Type 1 (1.d) and type 2

(1.e), the proof is similar to the one above (just replace a 2 with a 3 and a 3 with a 2 ).

Type 3: the new issue is fa 1 ; a 2 ; a 3 g and strategic sponsoring initiated by the sponsors of a 3 is in favor of a 3 (1.c). Then there is at least one sponsor with preference a 1 a 2 a 3 or a 1 a 3 a 2 (1.15), at least two sponsors for a 3 (1.16) and no sponsor with preference a 2 a 1 a 3 nor a 2 a 3 a 1 (s 3 = s 4 = 0 in (1.17). Given the sincere issue B, a 1 beats a 3 (1.12) and given the new issue fa 1 ; a 2 ; a 3 g, a 2 must beat a 1 at the …rst ballot (1.13); if not, a 1 will defeat a 3 at the second ballot. And a 3 beats a 2 at the second ballot (1.14).

Finally, assume B = fa 1 ; a 2 ; a 3 g. Then only strategic sponsoring Type 4 is conceivable.

Type 4: the new issues must be fa 1 ; a 2 g or fa 1 ; a 3 g, but a 1 beats both a 2 and a 3 . Therefore there is no strategic sponsoring Type 4 under the amendment procedure. We then consider successive elimination with maximin behavior.

Proposition 2 Suppose SEm is the voting rule and a 1 is elected given some situation (s; n). Then (s; n) is unstable via coalitional strategic sponsoring if and only if

(2.a) or (2.b) 8 > > > > < > > > > : n1+n2+n5>n n n4+n5+n6 n s1+s2 1 s4 1 s1+s2+s4= 8 > > > > < > > > > : n1+n2+n3>n n n3+n4+n6 n s1+s2 1 s6 1 s1+s2+s6= or (2.c) or (2.d) 8 > > > > > > < > > > > > > : n1+n2+n3+n5>n n n3+n4+n6 n s1+s2 1 s3+s4 1 s6 1 s1+s2+s3+s4+s6= 8 > > > > > > < > > > > > > : n1+n2+n3+n5>n n n4+n5+n6 n s1+s2 1 s5+s6 1 s4 1 s1+s2+s4+s5+s6=
And …nally, successive elimination with maximax behavior.

Proposition 3 Suppose SEM is the voting rule and a 1 is elected given some situation (s; n). Then (s; n) is unstable via coalitional strategic sponsoring if and only if (3.a) or (3.b) or (3.c)

8 > > > > < > > > > : n1+n2+n5>n n n4+n5+n6 n s1+s2 1 s4 1 s1+s2+s4= 8 > > > > < > > > > : n1+n2+n3>n n n3+n4+n6 n s1+s2 1 s6 1 s1+s2+s6= 8 > > > > > > > > < > > > > > > > > : n1+n2+n5>n n n3+n4+n5+n6 n n2+n5+n6 n s1+s2 1 s3+s4 2 s4 1 s1+s2+s3+s4= or (3.d) or (3.e) or(3.f ) 8 > > > > > > > > < > > > > > > > > : n1+n2+n3>n n n3+n4+n5+n6 n n1+n3+n4>n n s1+s2 1 s5+s6 2 s6 1 s1+s2+s5+s6= 8 > > > > > > < > > > > > > : n1+n2+n5>n n n3+n4+n5+n6 n n1+n3+n4>n n s1+s2 1 s3+s4 2 s1+s2+s3+s4= 8 > > > > > > < > > > > > > : n1+n2+n3>n n n3+n4+n5+n6 n n2+n5+n6 n s1+s2 1 s5+s6 2 s1+s2+s5+s6=
In the next section, these characterizations will be subsequently used to compute the likelihood of strategic sponsoring opportunities.

Strategic sponsoring occurrence

In this section and for each of the rules under study, two series of results are provided: (i) we …rst give formulae stating, for = 1 2 , the frequencies of coalitional strategic sponsoring with respect to n and and (ii) we then give general formulae stating those frequencies as a function of , n and . We also consider special cases, speci…cally when the electorate is in…nitely large or when the number of sponsors is arbitrarily …xed. All these results are derived from the characterizations of unstable situations by the use of computerised evaluations processes based on the same technique as the one in [START_REF] Gehrlein | The probability of the paradox of voting: a computable solution[END_REF], Gehrlein and Lepelley (1999) or Huang and Chua (2000).

Let us recall that the total number of situations (s; n) with three alternatives is +5 5 n+5 5

= ( +1)( +2)( +3)( +4)( +5)(n+1)(n+2)(n+3)(n+4)(n+5)
14 400

Amendment rules

Let H( ; n; ) be the likelihood that under amendment rules with -majority contests, a strategic sponsoring situation exists with n voters and sponsors. We shall …rst consider the case = 1 2 . We obtain the following results:

Proposition 4 Let AP 1 2 be the voting rule. Suppose is a …xed proportion k of n, that is = n k . Then for n 2, H( 1 2 ; n; 24k 4 n 2 144k 4 n+168k 4 290k 3 n 3 1740k 3 n 2 1810k 3 n+215k 2 n 4 + 1290k 2 n 3 +1375k 2 n 2 +74kn 5 +444kn 4 +442kn 3 +n 6 +6n 5 7n 4 )

n k ) = 8 > > > < > > > : 5kn( 24k 4 n 144k 4 290k 3 n 2 1740k 3 n 1280k 3 +215k 2 n 3 + 1290k 2 n 2 +960k 2 n+74kn 4 +444kn 3 +320kn 2 +n 5 +6n 4 ) 16(2k+n)(3k+n)(4k+n)(5k+n)(n+1)(n+5)(k+n) if n 0 mod 2 5k(
16(2k+n)(3k+n)(4k+n)(5k+n)(n+2)(n+4)(k+n)
if n 1 mod 2

For some special values of k, we have:

Corollary 1 Suppose AP 1 2 is the voting rule. Then for n 2 H( 1 2 ; n; n 5 ) = 8 > > < > > : 25n(n 5 +376n 4 +7595n 3 2400n 2 208 500n 250 000) 16(n+1)(n+5) 2 (n+10)(n+15)(n+20)(n+25) 25(n 6 +376n 5 +7588n 4 1790n 3 198 125n 2 316 250n+105 000) 16(n+2)(n+4)(n+5)(n+10)(n+15)(n+20)(n+25) if n 0 mod 2 if n 1 mod 2 H( 1 2 ; n; n 4 ) = 8 > > < > > : 5n(n 5 +302n 4 +5216n 3 +3360n 2 102 144n 118 784) 4(n+1)(n+4)(n+5)(n+8)(n+12)(n+16)(n+20)
5(n 6 +302n 5 +5209n 4 +3848n 3 95 504n 2 152 704n+43 008) 4(n+2

)(n+4) 2 (n+8)(n+12)(n+16)(n+20) if n 0 mod 2 if n 1 mod 2 H( 1 2 ; n; n 3 ) = 8 > > < > > : 15n(n 5 +228n 4 +3267n 3 +4740n 2 40 284n 46 224) 16(n+1)(n+3)(n+5)(n+6)(n+9)(n+12)(n+15)
15(n 6 +228n 5 +3260n 4 +5106n 3 36 549n 2 60 534n+13 608)

16(n+2)(n+3)(n+4)(n+6)(n+9)(n+12)(n+15) if n 0 mod 2 if n 1 mod 2 H( 1 2 ; n; n 2 ) = 8 > > < > > : 5n(n 5 +154n 4 +1748n 3 +3480n 2 10 464n 12 544) 8(n+1)(n+2)(n+4)(n+5)(n+6)(n+8)(n+10) 5(n 6 +154n 5 +1741n 4 +3724n 3 8804n 2 16 784n+2688) 8(n+2) 2 (n+4) 2 (n+6)(n+8)(n+10) if n 0 mod 2 if n 1 mod 2 H( 1 2 ; n; n) = 8 > > < > > : 5n(n 5 +80n 4 +659n 3 +1320n 2 804n 1424) 16(n+1) 2 (n+2)(n+3)(n+4)(n+5) 2 5(n 1)(n 5 +81n 4 +733n 3 +2175n 2 +1786n 168) 16(n+1)(n+2) 2 (n+3)(n+4) 2 (n+5) if n 0 mod 2 if n 1 mod 2
These results can also be stated graphically in Figure 1: Note that with any …xed proportion n k of sponsors, frequencies tend to 0 as the number of voters rises. Also notice that given a …xed number of voters, the smaller the number of sponsors as compared with the number of voters, the higher the frequencies of coalitional strategic sponsoring. For example for n = 577, the size of the french parliament, the frequencies are 0:004 for k = 5 and 0:001 for k = 2. That is for n = 577, when the number of sponsors is …ve times less than the number of voters, strategic sponsoring occurs almost three times more than when the number of sponsors is half the number of voters. Table 1 in Appendix summarizes the results above.

Let us now present the more general formulae corresponding to = 1 2 .

Proposition 5 Suppose AP 1 2 is the voting rule. Then for n 2 and 3, H( 1 2 ; n; ) = 8 > > > < > > > :

5(

n 2 4 +74n 2 3 +215n 2 2 290n 2 24n 2 +6n 4 + 444n 3 +1290n 2 1740n 144n+320 3 +960 2 1280 )

16( +1)( +2)( +3)( +4)( +5)(n+1)(n+5)
if n 0 mod 2

5(

n 2 4 +74n 2 3 +215n 2 2 290n 2 24n 2 +6n 4 +444n 3 + 1290n 2 1740n 144n 7 4 +442 3 +1375 2 1810 +168 )

16( +1)( +2)( +3)( +4)( +5)(n+2)(n+4) if n 1 mod 2
Corollary 2 In a large society,

H( 1 2 ; 1; ) = 5( 4 +74 3 +215 2 290 24) 16( +1)( +2)( +3)( +4)( +5)
For some special values of , we have:

(sponsors) 3 4 5 6 H( 1 2 ; 1; ) 0:145 0:150 0:142 0:131

These di¤erent limits show that the occurrence of strategic sponsoring, with large value of n, is maximal for 4 sponsors and then decreases as rises. Also note that H( 1 2 ; n; ) tends to 0 when tends to in…nity.

(sponsors) 3 4 5 6 H( ; 1; ) = 0:148 0:153 0:146 0:135 This means that strategic sponsoring is most likely to occur when the number of sponsors is …xed to four.

Figure 3 illustrates the following facts: (i) frequencies decrease as the number of sponsors rises; (ii) there is a strict symmetry with respect to line = 1 2 , for any number of sponsors; (iii) furthermore, the value at the local maxima remain constant whatever the number of voters and sponsors; this can easily be checked by the reader, by taking the derivative with respect to and setting it equal to 0. To summarize, we have shown that for amendment procedures, the frequency of occurrence is maximal for 4 sponsors and for equal to 5 8 or 3 8 . In the context of simple majority contests the frequencies of strategic sponsoring get smaller and smaller as the number of sponsors grows.

Successive elimination rules with maximin voters

Let K( ; n; ) be the likelihood that under successive elimination rules with -majority contests and with maximin voters, a strategic sponsoring situation exists with n voters and sponsors. As above, we …rst consider the case = 1 2 .

Proposition 7 Let SEm 1 2 be the voting rule and a …xed proportion k of n. Then for n 2, K( 1 2 ; n;

n k ) = 8 > > > > > > > < > > > > > > > :
5kn( 96k 4 n 2 804k 4 n 1368k 4 462k 3 n 3 3878k 3 n 2 7276k 3 n 1920k 3 +187k 2 n 4 + 1628k 2 n 3 +3516k 2 n 2 +1440k 2 n+78kn 5 +662kn 4 +1324kn 3 +480kn 2 +5n 6 +40n 5 +60n 4) 96k 4 n 2 456k 4 n 408k 4 462k 3 n 3 2212k 3 n 2 2526k 3 n+187k 2 n 4 + 1012k 2 n 3 +1161k 2 n 2 +78kn 5 +388kn 4 +414kn 3 +5n 6 +20n 5 +15n 4 ) 8(2k+n)(3k+n)(4k+n)(5k+n)(n+2)(n+4)(k+n) if n 1 mod 2

8(2k+n)(3k+n)(4k+n)(5k+n)(n+1)(n+3)(n+5)(k+n) if n 0 mod 2 5k(
For some special values of k, we have:

Corollary 3 Suppose SEm 1
2 is the voting rule. Then for n 2 K( 1 2 ; n; n 5 ) = 8 > > > < > > > : )

8(n+2)(n+4)(n+5)(n+10)(n+15)(n+20)(n+25) if n 0 mod 2 if n 1 mod 2 K( 1 2 ; n; n 4 ) = 8 > > > < > > > :
2(n+2)(n+4) 2 (n+8)(n+12)(n+16)(n+20) if n 0 mod 2 if n 1 mod 2 K( 1 2 ; n; n 3 ) = 8 > > < > > :
4(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+8)(n+10) 5(5n 5 +166n 4 +1207n 3 1234n 2 12 120n 3264) (4n+8)(n+4) 2 (n+6)(n+8)(n+10) if n 0 mod 2 if n 1 mod 2 K( 1 2 ; n; n) = 8 > > < > > : 5n(5n 6 +118n 5 +909n 4 +2490n 3 +22n 2 6640n 3288) 8(n+1) 2 (n+2)(n+3) 2 (n+4)(n+5) 2 5(5n 6 +98n 5 +590n 4 +964n 3 1147n 2 2982n 408) 8(n+1)(n+2) 2 (n+3)(n+4) 2 (n+5) if n 0 mod 2 if n 1 mod 2
Graphically we have in Figure 4: As for the amendment rules, Figure 4 shows that strategic sponsoring is generally less likely to occur when the number of sponsors is weak relatively to the number of voters. For the french parliament (n = 577), the frequencies are 0:016 for k = 3 and 0:027 for k = 5 (see the appendix for a table of values).

We next come to the general formulae, with = 1 2 . Proposition 8 Let SEm 1 2 be the voting rule. Then for n 2, K( 1 2 ; n;

) = 8 > > > > > > > < > > > > > > > :
5( 5n 3 4 +78n 3 3 +187n 3 2 462n 3 96n 3 +40n 2 4 +662n 2 3 +1628n 2 2 3878n 2 804n 2 +60n 4 +1324n 3 +3516n 2 7276n 1368n+480 3 +1440 2 1920

)

8( +1)( +2)( +3)( +4)( +5)(n+1)(n+3)(n+5)( +4)( +5)(n+1)(n+3)(n+5)
if n 0 mod 2

5(

5n 2 4 +78n 2 3 +187n 2 2 462n 2 96n 2 +20n 4 +388n 3 + 1012n 2 2212n 456n+15 4 +414 3 +1161 2 2526 408)

8( +1)( +2)( +3)( +4)( +5)(n+2)(n+4)
if n 1 mod 2

Corollary 4 In a large society,

K( 1 2 ; 1; ) = 5(5 4 +78 3 +187 2 462 96) 8( +1)( +2)( +3)( +4)( +5)
For some special values of , we have: (sponsors) 3 4 5 6 K( 1 2 ; 1; ) 0:252 0:303 0:313 0:307 Again, as for the amendment rule, K( 1 2 ; n; ) tends to 0 when tends to in…nity. However, it is worth noting that for maximin the maximal occurrence when n tends to in…nity is for …ve sponsors (this number was four for the amendment rules).

As for the amendment rules, we give formulae when n tends to in…nity with some special values of and present the associated plots (Figure 5 and Figure 6). Formulae for the general case can be found in the associated working paper. the three-dimensional plot. Then under successive elimination rules with maximin behavior, strategic sponsoring occurrence is maximal for = 5 at around p 2 1 and decreases with simple majority contests as gets larger.

Successive elimination rules with maximax voters

Let J( ; n; ) be the likelihood of strategic sponsoring situations under successive elimination rules with -majority contests and with maximax voters a exists. Again, we …rst consider the = 1 2 case.

Proposition 10 Let SEM 1 2 be the voting rule. Then for n 2, J( 1 2 ; n

; k) = 8 > > > > > > > < > > > > > > > :
5kn( 72k 4 n 2 288k 4 n 96k 4 454k 3 n 3 2856k 3 n 2 5352k 3 n 3840k 3 +261k 2 n 4 + 2004k 2 n 3 +4508k 2 n 2 +2880k 2 n+94kn 5 +696kn 4 +1512kn 3 +960kn 2 +3n 6 +12n 5 +4n 4)

16(2k+n)(3k+n)(4k+n)(5k+n)(n+1)(n+3)(n+5)(k+n) if n 0 mod 2 5k( 72k 4 n 2 192k 4 n+264k 4 454k 3 n 3 1904k 3 n 2 1482k 3 n+261k 2 n 4 + 1336k 2 n 3 +1283k 2 n 2 +94kn 5 +464kn 4 +402kn 3 +3n 6 +8n 5 11n 4 ) 16(2k+n)(3k+n)(4k+n)(5k+n)(n+2)(n+4)(k+n) if n 1 mod 2
For some special values of k, we have:

Corollary 5 Suppose that SEM 1 2 . Then for n 2 J( 1 2 ; n; n 5 ) = 8 > > > < > > > :
25n( 3n 6 +482n 5 +10 009n 4 +910n 3 284 500n 2 777 000n 540 000 )

16(n+1)(n+3)(n+5) 2 (n+10)(n+15)(n+20)(n+25)
25( 3n 6 +478n 5 +8834n 4 21 340n 3 250 925n 2 305 250n+165 000 )

16(n+2)(n+4)(n+5)(n+10)(n+15)(n+20)(n+25) if n 0 mod 2 if n 1 mod 2 J( 1 2 ; n; n 4 ) = 8 > > > < > > > :
5n( 3n 6 +388n 5 +6964n 4 +9056n 3 125 248n 2 370 176n 270 336 )

4(n+1)(n+3)(n+4)(n+5)(n+8)(n+12)(n+16)(n+20)
15( n 6 +128n 5 +2007n 4 2024n 3 39 920n 2 48 000n+22 528 )

4(n+2)(n+4) 2 (n+8)(n+12)(n+16)(n+20) if n 0 mod 2 if n 1 mod 2 J( 1 2 ; n; n 3 ) = 8 > > > < > > > :
15n( 3n 6 +294n 5 +4441n 4 +10 314n 3 39 492n 2 141 912n 111 456 )

16(n+1)(n+3) 2 (n+5)(n+6)(n+9)(n+12)(n+15)
15( 3n 6 +290n 5 +3730n 4 +972n 3 45 693n 2 55 566n+21 384 )

16(n+2)(n+3)(n+4)(n+6)(n+9)(n+12)(n+15) if n 0 mod 2 if n 1 mod 2 J( 1 2 ; n; n 2 ) = 8 > > < > > :
5n(3n 6 +200n 5 +2440n 4 +7408n 3 4048n 2 35 904n 32 256) 8(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+8)(n+10) 5(3n 6 +196n 5 +1961n 4 +2516n 3 11 252n 2 14 928n+4224) 8(n+2) 2 (n+4) 2 (n+6)(n+8)(n+10)

if n 0 mod 2 if n 1 mod 2 J( 1 2 ; n; n) = 8 > > < > > : 5n(3n 6 +106n 5 +961n 4 +3062n 3 +2540n 2 2760n 3936) 16(n+1) 2 (n+2)(n+3) 2 (n+4)(n+5) 2 15(n 1)(n 5 +35n 4 +273n 3 +701n 2 +470n 88) 16(n+1)(n+2) 2 (n+3)(n+4) 2 (n+5)

if n 0 mod 2 if n 1 mod 2

We then plot the above frequencies in Figure 7: And as for the two previous procedures, the frequency of coalitional strategic sponsoring, for a given value of n, is generally greater when the number of sponsors is small as compared with the number of voters. For the french parliament(n = 577), the frequencies are 0:009 for k = 5 and 0:002 for k = 1 (see the appendix for a table of values).

We now examine the general case for simple majority.

Proposition 11 Suppose that SEM 1 2 is the voting rule. Then for n 2 and 3, J( 1 2 ; n; ) = 8 > > > > > > > < > > > > > > > : For special values of , we have: (sponsors) 3 4 5 6 J( 1 2 ; 1; ) 0:172 0:188 0:184 0:174

Once again, the likelihood of strategic sponsoring is maximal for 4 sponsors. Note that J( 1 2 ; n; ) also tends to 0 as tends to in…nity.

As for the two previous procedures general formulae are given in the associated working paper.

And for some arbitrary values of when n tends to in…nity, we obtain: Then under successive elimination rules with maximax, strategic sponsoring occurrence is maximal when = 4, at around 0:66 and with = 1 2 , the smaller the number of sponsors as compared with the number of voters, the higher the frequencies.

Concluding discussion

The goal of this paper was to compare three families of parliamentary rules through the study of their vulnerability to strategic sponsoring. In order to do that, we were concerned with computing the frequency of voting situations at which parliamentary rules with majority contests are vulnerable to coalitional strategic sponsoring of alternatives. Our results show how these frequencies change according to changes in the values of the number of voters, the number of sponsors and the quali…ed majority. Further, it appears that for all three families of rules: (i) frequencies tend to 0 as the number of sponsors rises, ceteris paribus, that is with a …xed number of voters and given any value of ; (ii) however, notice that in actual situations -as emphasized by Dutta and Pattanaik (1978, p. 169) -the number of sponsors is generally very small as compared with the number of voters, and then frequencies are higher.

It also appears that for every possible quali…ed majority , and given the number of sponsors and the number of voters, frequencies are always higher with the two versions of successive elimination than with amendment. Now, comparing successive elimination with either maximin or maximax does not lead to a clear conclusion of this kind. However, we observe that for

  Su¢ciency. It is straightforward that strategic sponsoring Type 1 occurs under constraints (1.a) or (1.d); strategic sponsoring Type 2 occurs under constraints (1.b) or (1.e); and strategic sponsoring Type 3 occurs under constraints (1.c).

Fig. 1

 1 Fig.1 Strategic sponsoring under amendment with n k sponsors

Fig. 3

 3 Fig.3 Strategic sponsoring under amendment for a large electorate

Fig. 4

 4 Fig. 4 Strategic sponsoring under maximin with n k sponsors

Fig. 6

 6 Fig. 6 Strategic sponsoring under maximin for a large electorate

Fig. 7

 7 Fig. 7 Strategic sponsoring under maximax with n k sponsors

Fig. 9

 9 Fig. 9 Strategic sponsoring under maximax for a large electorate A close look at Figure 9 con…rms that for small values of , there is a local maximum around = 0:66. But as the number of sponsors gets larger, there is a shift of this maximum towards values of that are closer to 1.

Table 1

 1 Strategic sponsoring with n k sponsors with alpha = 1=2 Am endm ent Successive with m axim in Successive with m axim ax

SébastienCourtin et al.

Sébastien Courtin et al. From the characterizations of unstable situations above, we also derive general formulae stating those frequencies as a function of , n and . Since these formulae are not easily interpretable, the reader can refer to the associated working paper. However, Figure 3 below gives an illustration of this type of relations when the number of sponsors and the number of voters are large. We then give simpler formulae below, speci…cally when the electorate is in…nitely large and the number of sponsors is arbitrarily …xed.

Proposition 6 Suppose that AP is the voting rule, changes from 3 to 6 and n tends to in…nity, then )

The expressions above can be plotted as follows in Figure 2: The four curves reveal a strict symmetry with respect to line = 1 2 , a local minimum. Moreover, the four frequency curves have the same two local maxima, at = 3 8 = 0:375 and at = 5 8 = 0:625. In other words, this means that strategic sponsoring is more likely to occur when the quota is 37:5% or 62:5%. We can also note that the frequencies rise between 3 and 4 sponsors and then decrease for 5 and 6 sponsors. At = 3 8 or 5 8 ,

Proposition 9 Suppose that SEm is the voting rule. Then

Fig. 5 Strategic sponsoring under maximin for a large electorate

Figure 5 shows that all the curves have their local maxima around = p 2 1. However when the number of sponsors rises, the maxima move towards the ordinates axis.

For example for = 4, the maximum is obtain at = 0:414, and at = 0:412 for = 6. Further, frequencies are highest for 5 sponsors, and the maximum (32:8 15 %) is reached at = 0:413. We can also analyze the evolution of the frequencies when the number of sponsors rises, by studying Proposition 12 Suppose that SEM is the voting rule. Then 115 3 355 4 +311 5 +4) 63 2( 480 +1230 2 1425 3 +725 4 121 5 +75) 63 if 0

And graphically in Figure 8 and Figure 9: The four frequency curves reveal a local maximum at around = 0:66. But contrary to the maximin case, the maxima move at the further from the ordinates axis as the number of sponsors rises. For example for = 3 and = 5, the maximum is obtain at = 0:661 and at = 0:665, respectively. Moreover the frequency is highest for = 4, and for = 0:664, strategic sponsoring frequencies grow up to 21:3 35 %.

Sébastien Courtin et al. small values of , frequencies are higher for maximin than for maximax, as illustrated with four sponsors in Figure 10 (see the appendix).

Besides, note that some other results can be derived from the characterizations. We observe that for none of the voting rules under study does the set of strategic sponsoring opportunities coincide with the set of Condorcet cycles. However, for amendment rule, strategic sponsoring Types 3 and 4 requires the presence of a Condorcet cycle, while there is no such relation for the other two rules. In the same way, it can also easily be checked that with single-peaked individual preferences (voters and sponsors), strategic sponsoring occurs under the three procedures.

Finally, it is worth noting that these procedures have been studied in the literature in the context of strategic voting (see [START_REF] Favardin | Some further results on the manipulability of social choice rules[END_REF][START_REF] Mbih | On the responsiveness of parliamentary social choice functions[END_REF] and it may be interesting to consider some slightly di¤erent context by speci…cally assuming that sponsors participate to the …nal vote, and examining the possibility of combined strategic sponsoring and strategic voting.

Appendix

In this appendix, we give a …gure comparing the three rules for large electorates and we also gives tables summarizing the results given in Section 4.