Guillaume Gibert

Gérard Bailly

Frédéric Elisei

Evaluation of a virtual speech cuer

This paper presents the virtual speech cuer built in the context of the ARTUS project aiming at watermarking hand and face gestures of a virtual animated agent in a broadcasted audiovisual sequence. For deaf televiewers that master cued speech, the animated agent can be then superimposed -on demand and at the reception -in the original broadcast as an alternative to subtitling. The paper presents the multimodal text-to-speech synthesis system and the first evaluation performed by deaf users.

Introduction

Listeners with hearing loss and orally educated typically rely heavily on speechreading based on lips and face visual information. However speechreading alone is not sufficient due to the lack of information on the place of tongue articulation and the mode of articulation (nasality or voicing) as well as to the similarity of the lip shapes of some speech units (so called labial sosies such as [u] vs. [y] for French). Indeed, even the best speechreaders do not identify more than 50 percent of phonemes in nonsense syllables [START_REF] Owens | Visemes observed by hearing-impaired and normal-hearing adult viewers[END_REF] or in words or sentences [START_REF] Bernstein | Speech perception without hearing[END_REF]. Cued Speech (CS) was designed to complement speechreading. Developed by [START_REF] Cornett | Cued Speech[END_REF][START_REF] Cornett | Le Cued Speech. Aides manuelles à la lecture labiale et perspectives d'aides automatiques[END_REF] and adapted to more than 50 languages [START_REF] Cornett | Cued Speech, manual complement to lipreading, for visual reception of spoken language[END_REF], this system is based on the association of speech articulation with cues formed by the hand. While uttering, the speaker uses one of his hand to point out specific positions on the face with a hand shape). Numerous studies have demonstrated the drastic increase of intelligibility provided by CS compared to speechreading alone [START_REF] Nicholls | Cued Speech and the reception of spoken language[END_REF] and the effective facilitation of language learning using CS [START_REF] Leybaert | Phonology acquired through the eyes and spelling in deaf children[END_REF][START_REF] Leybaert | The role of Cued Speech in language processing by deaf children: an overview[END_REF]. A large amount of work has been devoted to CS perception but few works have been devoted to CS synthesis. We describe here a multimodal text-to-speech system driving a virtual CS speaker and its first evaluation by deaf users.

The multimodal text-to-speech system

The multimodal text-to-speech system developed in the framework of the ARTUS project converts a series of subtitles into a stream of animation parameters for the head, face, arm and hand of a virtual cuer and an acoustic signal. The control, shape and appearance models of the virtual cuer (see Figure 1) have been determined using multiple multimodal recordings of one human cuer. The different experimental settings used to record our target cuer and capture its gestures and the complete text-to-Cued speech system are described in our JASA paper [START_REF] Gibert | Analysis and synthesis of the 3D movements of the head, face and hand of a speaker using cued speech[END_REF].

-

[b ]

[u]

[r] -Figure 1: Chronogram for the word "Bonjour" pronounced and cued by our virtual speaker.

Evaluation

A first series of experiments have been conducted to evaluate the intelligibility of this virtual cuer with skilled deaf users of the French cued speech. The first evaluation campaign was dedicated to segmental intelligibility and the second one to long-term comprehension.

Segmental intelligibility

This test was conducted to assess the contribution of the cueing gestures in comparison with lip reading alone.

Minimal pairs

The test mirrors the Modified Diagnostic Rime Test developed for French [START_REF] Peckels | Le test de diagnostic par paires minimales. Adaptation au francais du 'Diagnostic Rhyme Test' de W.D. Voiers[END_REF]: the minimal pairs do not here test acoustic phonetic features but gestural ones. A list of CVC word pairs has thus been developed that test systematically pairs of consonants in initial positions that differ almost only in hand shapes: we choose the consonants in all pairs of 8 subsets of consonants that are highly visually confusable [START_REF] Summerfield | Visual perception of phonetic gestures. Modularity and the motor theory of speech perception[END_REF]. The vocalic substrate was chosen so as to cover all potential hand positions while the final consonant was chosen so that to avoid rarely used French words or proper names, and test our ability to handle coarticulation effects. Due to the fact that minimal pairs cannot be found in all vocalic substrates, we end up with a list of 196 word pairs. Conditions Minimal pairs are presented randomly and in both order. The lipreading-only condition is tested first. The cued speech condition is then presented in order to be able to summon up cognitive resources for the most difficult task first. Stimuli In order to avoid a completely still head, head movements of the lipreading-only condition are those produced by the text-to-cued speech synthesizer divided by a factor of 10 (in fact the head accomplished on average 16.43% of the head/hand contact distance). We did not modify segmental nor suprasegmental settings that could enhance articulation. Subjects Height subjects were tested. They are all hearing impaired people who have practised French CS (FCS) since the age of 3 years. Results Mean intelligibility rate for "lipreading" condition is 52.36%. It is not different from haphazard way of response that means minimal pairs are not distinguishable. Mean intelligibility rate for "CS" condition is 94.26%. The difference in terms of intelligibility rate between these two conditions shows our virtual cuer gives significant information in terms of hand movements. In terms of cognitive efforts, the "CS" task is easier: the response time is significantly different F(1,3134)=7.5, p<0.01 and lower than for the "lipreading" one.

Comprehension

To evaluate the global comprehension of our system, we asked the same 4 subjects to watch a TV program where subtitles were replaced by the incrustation of the virtual cuer. Ten questions were asked. The results show all the information is not perceived. On average, the subjects replied correctly to 3 questions. The difficulties of the task (proper names, high speaking rate) could explain these results. We conducted further experiments using a Tobii© eye tracker. We asked 4 deaf people to watch a TV program divided in 2 parts: one part subtitled and another part with the inlay of a cuer video. The results show the subjects spend 56.36% of the time on the teletext and 80.70% on the video of the cuer with a significant difference F(1,6)=9.06, p<0.05. A control group of 16 hearing people spend 40.14% of the time reading teletext. No significant difference was found.

Conclusions

The observation and recordings of CS in action allow us to implement a complete text-to-Cued Speech synthesizer. The results of the preliminaries perceptive tests show significant linguistic information with minimal cognitive effort is transmitted by our system. This series of experiments must be continued on more subjects and other experiments must be added to quantify exactly the cognitive effort used. Discourse segmentation and part of speech emphasis by prosodic cues (not yet implemented) is expected to enlighten this effort.

Figure 2 :

 2 Figure 2: Eye gaze for one subject captured during the comprehension test using an eye tracker system: (left) teletext, (right) video superimposed.

Acknowledgements

The authors thank Martine Marthouret, Marie-Agnès Cathiard, Denis Beautemps and Virginie Attina for their help and comments on building the perceptive tests. We also want to thank the 25 subjects who took part to the evaluation.


Oxana Govokhina

Gérard Bailly
email: gerard.bailly@icp.inpg.fr

) Gaspard Breton

Paul Bagshaw
email: paul.bagshaw@francetelecom.fr

TDA: A new trainable trajectory formation system for facial animation

Keywords:

A new trainable trajectory formation system -named TDAfor facial animation is here proposed that dissociates parametric spaces and methods for movement planning and execution. Movement planning is achieved by HMM-based trajectory formation. This module essentially plans configurations of lip geometry (aperture, spreading and protrusion). Movement execution is performed by concatenation of multi-represented diphones. This module is responsible for selecting and concatenating detailed facial movements that best obey to the target kinematics of the geometry previously planned. Movement planning ensures that the essential visual characteristics of visemes are reached (lip closing for bilabials, rounding and opening for palatal fricatives, etc) and that appropriate coarticulation is planned. Movement execution grafts phonetic details and idiosyncratic articulatory strategies (dissymetries, importance of jaw movements, etc) to the planned gestural score. This planning scheme is compared to alternative planning strategies using articulatory modeling and motion capture data.

Introduction

Embodied conversational agents -virtual characters as well as anthropoid robots -should be able to compute facial movements from symbolic input in order to speak with human partners. This symbolic input minimally consists in the phonetic string with phonemic durations. It can be enriched with more phonological information, facial expressions, or paralinguistic information that has an impact on speech articulation (mental or emotional state). A trajectory formation model has thus to be build that computes articulatory parameters from such a symbolic specification of the speech task. These articulatory parameters will then drive the plant (the shape and appearance models of a talking face or the control model of the robot). Human interlocutors are very sensitive to discrepancies between the visible and audible consequences of articulation [START_REF] Dixon | The detection of audiovisual desynchrony[END_REF][START_REF] Mcgurk | Hearing lips and seeing voices[END_REF] and have strong expectations on articulatory variability [START_REF] Sereno | Acoustic analysis and perceptual data on anticipatory labial coarticulation in adults and children[END_REF][START_REF] Whalen | Coarticulation is largely planned[END_REF] resulting from the underspecification of articulatory targets and planning. The proper modeling of coarticulation in speech benefits to the intelligibility of the agent and is in fact a challenging issue for trajectory formation systems. We propose here a trajectory formation system that builds on the task dynamics model [START_REF] Saltzman | A dynamical approach to gestural patterning in speech production[END_REF] but combines two up-to-date trajectory formation systems operating on two different representation spaces of the movement for handling separately movement planning and execution.

State-of-the-art

The most popular trajectory formation system used in facial animation computes a sequence of contextual articulatory targets that are connected or weighted by temporal functions. The most simple model consists in blending prototypical phonemic targets according to phoneme-specific coarticulation functions [START_REF] Cohen | Modeling coarticulation in synthetic visual speech[END_REF]. Similarly Reveret et al [START_REF] Revéret | MOTHER: a new generation of talking heads providing a flexible articulatory control for video-realistic speech animation[END_REF] have adapted the more speech-specific Ohman's model [START_REF] Öhman | Numerical model of coarticulation[END_REF] that distinguishes between vowel-and consonant-specific coarticulation functions. Statistical N-phones models only considering one target per phoneme have also been proposed. The trajectory is then built by adding general constraints on movements such as minimal jerk [START_REF] Ezzat | Trainable videorealistic speech animation[END_REF]. For tongue control, Okadome et al [START_REF] Okadome | Articulatory movement formation by kinematic triphone model[END_REF] added dynamic parameters sampled at targets in order to cope with inter-gestural phasing. This could be considered as the first step towards more general statistical modeling of phoneme-specific gestures in context using the generation abilities of HMM [START_REF] Tamura | Text-to-audio-visual speech synthesis based on parameter generation from HMM[END_REF] The availability of large motion capture data and the possibility of storing and retrieving video segments has also popularized concatenation-based techniques [START_REF] Huang | Triphone-based unit selection for concatenative visual speech synthesis[END_REF][START_REF] Minnis | Modeling visual coarticulation in synthetic talking heads using a lip motion unit inventory with concatenative synthesis[END_REF]. Please note finally the use of inverse acoustics for lip-sync [START_REF] Yehia | Facial animation and head motion driven by speech acoustics[END_REF] that should possibly exploit the acoustic trace of coarticulation to recover the visible one. Using point-light displays, Bailly et al [START_REF] Bailly | Evaluation of movement generation systems using the point-light technique[END_REF] compared key proposals using common training and test material. More recently Govokhina et al [START_REF] Govokhina | Evaluation de systèmes de génération de mouvements faciaux[END_REF] added an HMM-based trajectory formation to the set of key proposals driving a full videorealistic talking head. In both studies HMM-and concatenation-based techniques reach a level of adequacy close to original motion. The long-term coherence of the trajectory offered by HMM-based technique seems however to outperform a pure concatenation.

HMM Synthesis

Trainable trajectory formation systems

We sketch below the basic features of these two trajectory formation systems.

Concatenation of multi-represented segments

Synthesis by concatenation consists in selecting and concatenating pre-recorded segments. Phonological features are first used to select candidate segments: besides a simple phonemic match, phonotactic constraints (phonemic context, position in the syllable or word) and/or compliance with higher-level phonological structure [START_REF] Taylor | Speech synthesis by phonological structure matching[END_REF] are often added. Then a DTW algorithm find an optimal path through this lattice of candidate units that best gathers units with minimal selection and concatenation costs. Two penalties are often considered to build the selection cost: a penalty that depends on the degree of adequacy of the considered unit with the phonological constraints and a penalty that considers the distance between parameters characterizing the selected unit with parameters computed with an external prediction model. So parameters computed by a prosodic model are often considered [START_REF] Hunt | Unit selection in a concatenative speech synthesis system using a large speech database[END_REF].

The concatenation cost depends on the match between parameters of adjacent units across the boundary. Both static and dynamic cues are often considered.

HMM-based synthesis

The principle of speech synthesis by HMM was first introduced by Donovan for acoustic speech synthesis [START_REF] Donovan | Trainable speech synthesis[END_REF]. This was extended to audiovisual speech by the HTS working group [START_REF] Tamura | Text-to-audio-visual speech synthesis based on parameter generation from HMM[END_REF]. The HMM-trajectory synthesis technique comprises training and synthesis parts.

Training. An HMM and a duration model for each state are first learned for each segment of the training set. The input data for the HMM training is a set of observation vectors. The observation vectors consist of static and dynamic parameters, i.e. the values of articulatory parameters and their derivatives.

The HMM parameter estimation is based on ML (Maximum-Likelihood) criterion [START_REF] Tokuda | Speech parameter generation algorithms for HMM-based speech synthesis[END_REF]. The ML estimation is achieved using a particular EM (Expectation Maximization) algorithm known as the Baum-Welch recursion algorithm. Usually, for each phoneme in context, a 3-state left-to-right model with single Gaussian diagonal output distributions and no skips is learned. The state durations of each HMM are modeled by single Gaussian distributions. A second training step may also be added to factor out similar output distributions among the entire set of states.

Synthesis.

The synthesis is performed as follows. The phonetic string to be synthesized is first chunked into segments and a sequence of HMM states is built by concatenating the corresponding segmental HMMs. State durations for the HMM sequence are determined so that the output probability of the state durations are maximized [START_REF] Yoshimura | Duration modeling for HMMbased speech synthesis[END_REF].

From the HMM sequence with the proper state durations assigned, a sequence of observation parameters is generated using a specific ML-based parameter generation algorithm [START_REF] Zen | An introduction of trajectory model into HMM-based speech synthesis[END_REF]. This algorithm exploits the dynamic parameters included in the observations in training as well as in synthesis: the generated trajectory reflects both the means and covariances of the output distributions of a number of frames before and after each of the frames. By this way, this algorithm may incorporate implicitly part of long-term coarticulation patterns.

Comments

Note here that HHM synthesis imposes some constraints on the distribution of observations for each state. The ML-based parameter generation algorithm requires single Gaussian diagonal output distributions [note however the use of a Gaussian mixtures in 21]. It will thus best operate on an observation space that has compact targets and characterize targets with maximally independent parameters.

The proposed trajectory formation system

TDA (Task Dynamics for Animation), the trajectory formation system we propose, combines the advantages of both HMMand concatenation-based techniques. The proposed system is motivated by articulatory phonology [START_REF] Browman | Articulatory gestures as phonological units[END_REF] and its first implementation by the task dynamics model [START_REF] Saltzman | A dynamical approach to gestural patterning in speech production[END_REF]. Articulatory phonology put forward underspecified gestures as primary Planning gestures by HMM synthesis. We have shown elsewhere [START_REF] Govokhina | A new trainable trajectory formation system for facial animation[END_REF] that trajectory formation based on contextdependent phone HMMs (context is here limited to the following viseme -3 visemes for vowels and 4 for consonants, see Figure 2) outperforms both in objective and subjective terms concatenative synthesis and phoneme or diphone HMMs, when all these systems are trained to generate directly articulatory parameters. When trained on geometric parameters, these systems generate also targets that are more discriminated (see Figure 2). The correlation between original trajectories and those generated by all systems is substantially higher when considering geometry (see Table 1 and Figure 3). This confirms previous studies that promote constrictions as the best characteristics for speech planning [START_REF] Bailly | Learning to speak. Sensori-motor control of speech movements[END_REF].

Consonantal visemes Vocalic visemes Articulation

Table 1: Mean correlations (±standard deviations) between observed and predicted trajectories using different systems and representations.

Executing gestures by concatenative synthesis. If diphone

HMMs generate smooth trajectories while preserving visually relevant phonetic contrasts, concatenative synthesis has the intrinsic properties of capturing inter-articulatory phasing and idiosyncratic articulation. Concatenative synthesis also intrinsically preserves the variability of natural speech (see Figure 2 and Figure 3).

Evaluation

We compare the adequacy of different trajectory formation systems in generating articulatory trajectories that best integrate with a natural audio sound given its phonemic transcription and phoneme durations (see Table 1). The TDA system is compared to the original articulation and four other systems that directly generate articulatory trajectories: phoneme-HMM, contextual phoneme-HMM and concatenation of diphones with or without weighting candidates by their similarity with trajectories first computed by the contextual phoneme-HMM. The performance of the concatenation system is substantially increased when considering a selection cost using target parameters computed HMM trajectory planner. This is true whenever considering geometry or articulatory planning space. The performance of the current implementation of the TDA is however deceptive: the articulatory generation often degrades the quality of the planned geometric characteristics. If the TDA compensates well for the bad planning of movement during syntactic pauses, it often degrades the timing (see Figure 3). We are

Figure 3. Comparing trajectory formation systems (blue: natural reference; red: concatenation/selection TDA; green: contextual phoneme-HMM) with a natural test stimulus (blue).From top to bottom: geometric parameters: lip aperture, width and protrusion; articulatory parameters: jaw aperture, lips rounding/spreading. Major discrepancies between TDA and contextual phoneme-HMM are enlighten

currently reconsidering the procedure that warps stored articulatory segments to planned gestures

Conclusions and perspectives

The TDA system is a trajectory formation system for generating speech-related facial movement. It combines a HMM-based trajectory formation system responsible for planning long-term coarticulation in a geometric space with a trajectory formation system that selects and concatenates segments that are best capable of realizing this gestural score. Contrary to most proposals, this system builds on motor control theory -that identifies distinct modules for planning and execution of movements -and implements a theory of control of speech movements that considers characteristics of vocal tract geometry as primary cues of speech planning. This clear dichotomy between planning and execution provides a possible route towards sets of MPEG4 compatible talking faces where the encoder specifies geometric features and the decoder is responsible for computing the speakerspecific facial deformations given these constraints. The TDA system parallels proposals made for acoustic synthesis where a prosodic model helps a concatenative speech synthesis system for selecting appropriate acoustic segments and maintains a global structural coherence of the synthetic stimuli. In the future we will examine the possibility to extend our approach to joint audiovisual synthesis and enhance the ability of our planning module to cope with longterm coarticulation patterns.

Figure 1 .

 1 Figure 1. The proposed trajectory formation system TDA.

Figure 2 .

 2 Figure 2. Projecting the target consonantal and vocalic visemes on the first discriminant plane (set using natural reference) for various systems and two different parametric representations: articulatory versus geometric. From top to bottom: phoneme HHM, diphone HMM, concatenative synthesis and natural reference.Targets are more discriminated using the geometric representation using whatever system.

  Articulation Geometry Phoneme-HMM 0.61±0.11 0.77±0.07 Contextual phoneme-HMM 0.69±0.10 0.83±0.07 Concatenation of diphones 0.61±0.15 0.78±0.07 Concatenation with HMM guide 0.63±0.15 0.81±0.06 TDA 0.59±0.16 0.81±0.06

Figure 4 :

 4 Figure 4: Motion capture data and videorealistic clone mimicking recorded articulation.


Gérard Bailly
email: gerard.bailly@icp.inpg.fr

Jan Gorisch
email: jan.gorisch@icp.inpg.fr

Generating German intonation with a trainable prosodic model

Keywords: speech synthesis, prosody, evaluation

A trainable prosodic model called SFC (Superposition of Functional Contours), proposed by Holm and Bailly, is here confronted to German intonation. Training material is the publicly available Siemens Synthesis Corpus that provides spoken utterances for high-quality speech synthesis. We describe the labeling framework and first evaluation results that compares the original prosody of test sentences of this corpus with their prosodic rendering by the proposed model and state-of-the-art systems available on-line on the web.

Introduction

The trainable prosodic model SFC (Superposition of Functional Contours) has been developed by Holm and Bailly [START_REF] Bailly | SFC: a trainable prosodic model[END_REF][START_REF] Holm | Implémentation d'un modèle morphogénétique de l'intonation[END_REF][START_REF] Holm | Learning the hidden structure of intonation: implementing various functions of prosody[END_REF]. It implements a theoretical model of intonation initially sketched by Aubergé [START_REF] Aubergé | Developing a structured lexicon for synthesis of prosody[END_REF][START_REF] Aubergé | Generation of intonation: a global approach[END_REF] that promotes an intimate link between phonetic forms and linguistic functions: metalinguistic functions acting on different discourse units (thus at different scopes) are directly implemented as global multiparametric contours. These metalinguistic functions refer to the general ability of intonation to demarcate phonological units and convey information about the propositional and interactional functions of these units within the discourse. This trainable prosodic model has been confronted to speech styles (from read speech to spoken maths) and different languages including French, Galician or more recently Chinese [START_REF] Chen | A superposed prosodic model for Chinese textto-speech synthesis[END_REF]. German is of most interest because of its rich morphology and its potentially deep recursive syntactic embedding. Analysis of German prosody notably induces Schreuder and Gilbers [START_REF] Schreuder | Recursive patterns in phonological phrases[END_REF] to question the Strict Layer Hypothesis [START_REF] Selkirk | Phonology and syntax: the relation between sound and structure[END_REF] and claim for the existence of recursive prosodic phrases. While most quantitative models of German intonation that have been so far applied to speech synthesis use a phonological representation with few levels when not limited to prosodic phrases [START_REF] Heuft | Synthesizing prosody: a prominence-based approach[END_REF][START_REF] Mixdorff | Building an integrated prosodic model of German[END_REF]. We describe here our first efforts in confronting the SFCthat may potentially capture rich embedded performance structures [START_REF] Holm | Performance structures of mathematical formulae[END_REF] -to German intonation. Our first parameterization of the SFC using limited training material is evaluated against state-of-the-art text-to-speech systems available on the web.

The SFC prosodic model

In this section we sketch briefly the main features of this model. For more details please refer to Bailly et al [START_REF] Bailly | SFC: a trainable prosodic model[END_REF]. The SFC trainable prosodic model directly encodes metalinguistic functions by phonetic events -i.e. overlapping multiparametric contours -without any intermediate surface representation. Input. These metalinguistic functions refer to the general ability of prosody to segment, structure, emphasize or encode semantic or pragmatic cues associated with speech units. As emphasized by Aubergé [START_REF] Aubergé | A Gestalt morphology of prosody directed by functions: the example of the step by step model developed at ICP[END_REF], other linguistic agents (morphosyntax, semantics, etc) collaborate with prosody to encode this information. These metalinguistic functions apply to units of variable sizes (discourse, sentence, clause, group, word, syllable, phoneme). The set of these metalinguistic functions [see intonation and its uses in 13] is quite open and most of the parameterization of the SFC resides in the identification of the metalinguistic functions used in the training material and the speech units they apply to. Prosodic contours. The SFC postulates that these metalinguistic functions are encoded via multiparametric contours. The extend of each contour equals to the scope of the functions, i.e. the contours are coextensive to the speech units carrying the functions. Since the same metalinguistic function (e.g. segmentation) may apply to speech units of various sizes -and potentially embedded -the elementary multiparametric contours associated with each unit and each function (one unit may carry several functions, such as a word being emphasized and having a particular role in the syntactic structure that merits a specific spotlight) overlap. The parallel encoding of these several metalinguistic functions by overlapping contours is simply done by superposing and adding these elementary contributions by parameter-specific operators (see illustration in Figure 1): for f0, addition in the log-domain; for duration, addition of z-scores of rhythmic units, etc. Mapping functions to contours. Considering prosodic contours as the superposition of elementary contours is a many-to-one ill-posed problem that requires regularization schemes. The Fujisaki model [START_REF] Fujisaki | A generative model for the prosody of connected speech in Japanese[END_REF], for example, imposes constraints on the shape of these elementary contours (exponential responses of second-order filters to impulses and square waves). The SFC model does not impose such lowlevel constraints, but relies only on the consistency between different instantiations of the same discourse function on different units of different sizes within the corpus. Contour generators The instantiation of a given function on a unit -the calculation of one elementary multiparametric contour -is performed by so-called contour generators. A contour generator generates thus a family of contours, a set of prosodic clichés [START_REF] Fónagy | Clichés mélodiques[END_REF] that develop on units of different size but encode the same metalinguistic function. General-purpose contour generators have been developed in order to be able to generate a coherent family of contours indexed only by their scopes. These contour generators are implemented as simple feed-forward neural networks and an original analysis-bysynthesis method has been developed to train these networks that best predict the target prosodic stylization given the metalinguistic labels [START_REF] Bailly | SFC: a trainable prosodic model[END_REF].

Training the model

The prosodic database

We used speech data from the speaker ai of the SI1000P database commercially available from ELRA. The SI1000P recordings were done to provide material for high quality concatenate speech synthesis. It contains 1000 newspaper sentences read by two German professional broadcasting announcers in studio quality together with the laryngographic signal and the glottal pulse stream. Parts of the corpus were labeled and segmented phonemically (SAM-PA) and prosodically (borders+ accents). For this experiment we labeled carefully 80 utterances of the corpus. 70 sentences were used as training material. The 10 remaining sentences were used for evaluating the trained prosodic model (see § 3).

Annotating utterances with metalinguistic functions

The first metalinguistic function is sentence modality (we have shown elsewhere [START_REF] Morlec | Generating prosodic attitudes in French: data, model and evaluation[END_REF] that prosodic attitudes in general are characterized/encoded by prosodic clichés). Its scope is the whole sentence. In our corpus all sentences are declarative and markers thus cue beginning and ending of each sentence. For instance, the sentence shown in Figure 1 is first annotated with the mark DC (for declaration) as below: [Schmücker verwahrte sich jedoch gegen den Vorwurf die Regierung besitze kein energiepolitisches Koncept.] DC In our work, we always consider metalinguistic functions responsible for giving cues to the syntactic structure of sentences in the discourse. We thus annotate dependency relations between chunks [ see also [START_REF] Bachenko | A computational grammar of discourse-neutral prosodic phrasing in English[END_REF][START_REF] Bailly | Integration of rhythmic and syntactic constraints in a model of generation of French prosody[END_REF][START_REF] Pynte | Prosodic breaks and attachment decisions in sentence parsing[END_REF], for dependency structure analysis and prosodic correlates of attachment/branching of syntactic constituents]. We consider four kinds of dependency relations that may link constituents (words, groups, phrases, clauses): left dependency (DG, dépendence à gauche) linking the head of a sub-tree (the "governor" or "mother") with its immediately linearly preceding dependent unit ("sister"), right dependency (DD, dépendence à droite) linking the governor with its immediately following dependent unit, interdependency (IT) linking two adjacent units headed by the same governor and independency (ID) when none of the preceding simple relations can be identified. The syntactic parse we use is thus very simplified and can be accomplished using a chink and chunk technique [20, see also 21, for the use for syntax/prosody mapping]. For instance, the sentence shown in Figure 1

is further parsed as below: [[Schmücker] DG [[[verwahrte sich] DD [jedoch]] IT [gegen den Vorwurf]]] ID [[die Regierung] DG [[besitze] DD [kein [energiepolitisches] DG [Konzept]]]]

We then added markers for morphological decomposition. In the example, the word "energiepolitisches" is then parsed as:

[[energie] AM [politisches]]

German has lexical stress. So markers are also added to signal lexical stress position within each morpheme. In the example, the words "Schmücker" and "Regierung" are further marked as below:

[[Schmü] AC [cker]] and [[Regie] AC [rung]]

The last metalinguistic function is emphasis. Some demonstrative (e.g. "dieser"), negations (e.g. here "kein") or newly introduced proper nouns (e.g. "Schmücker") receive emphatic stress. In the example, the word "kein" receives the marker EM (for emphasis):

[kein] EM The Figure 1 displays the elementary melodic and rhythmic contours generated by the 8 contour generators (responsible for generating multiparametric contours encoding DC, DD, DG, IT, ID, AM, AC and EM) on the different scopes. It also displays the results of the supersposition and addition of these elementary contours in comparison with the original training material.

Prosodic stylization

We analyze and generatemultiparametric prosodic contours, i.e. we model themelody and rhythmic organization of the utterance. These contours capture the prosodic characteristics of the syllables of each utterance. Each syllable is characterized by a melodic movement [stylized by three F0 values on the vocalic nucleus as initially proposed by 22] and a lengthening factor (that will stretch or compress all phonemic segments of that syllable using z-scoring, see [START_REF] Barbosa | Generation of pauses within the z-score model[END_REF]).

Mapping metalinguistic functions to elementary multiparametric contours

The SFC (i.e. the eight contour generators) is then trained using phonetic and associated annotations from 70 utterances. As stated above, the iterative training consists in adjusting the contour generators so that their combined outputs best predicts observed multiparametric contours. Prosodic stylization of 10 test utterances (see Appendix) is then predicted using the trained SFC. Prediction performance on training and test material is given in Table 1. Such numbers are difficult to find in the literature… They could be compared to the correlation coefficients published by Mixdorff & Jokish [START_REF] Mixdorff | Implementing and evaluating an integrated approach to modeling German prosody[END_REF]: 0.55 for F0 parameters and 0.82 for durations.

Table 1: RMS errors (correlation coefficients) pour training and test material. F0 mean is set to 80 Hz for that speaker.

Parameter

Subjective evaluation

In order to situate this preliminary prosodic model with reference to available implementations, we collected prosodic characteristics of outputs of 6 state-of-the-art text-to-speech synthesizers (see Table 2): these systems will be anonymously named Alien1..6 in the following. The 10 test sentences were submitted to each online text-to-speech server and the synthetic audio files collected. Automatic f0 detection and phonemic alignment was then corrected by hand. The same prosodic stylization as used by SFC was finally performed to gather 6 alternative prosodic contours for each test sentence. Note that we compensate summarily for differences in voice registers of the different systems: we compute the mean f0 for each alien system and scale it to the mean f0 of our target speaker ai using a simple rule of three.

Evaluation procedure

We compared the synthetic prosody computed by our 7 different prosodic systems with the natural prosody using TD-PSOLA resynthesis of the natural test signals (procedure similar to [START_REF] Boula De Mareüil | A joint intelligibility evaluation of French text-to-speech systems: the EvaSy SUS/ACR campaign[END_REF]). Note that the stimuli driven natural prosody is synthetic: prosody for all systems is characterized by segmental durations and three F0 values on the vocalic nucleus. We do add automatically residual micromelody as proposed by Monaghan [START_REF] Monaghan | Extracting microprosodic information from diphones --a simple way to model segmental effects on prosody for synthetic speech[END_REF] and implemented as an option in the SFC [START_REF] Bailly | SFC: a trainable prosodic model[END_REF] The evaluation paradigm combines advantages of Mean Opinion Score (MOS) ratings and preference tests. We ask our Subjects to position our synthetic stimuli -identified as colored icons -in a geometric plane whose abscissa is the MOS scale. As they can listen to stimuli as many times as they want, they can compare stimuli by pairs, position stimuli already ordered in some part of the plane and further refine their judgment (this procedure has been used by Pfitzinger [START_REF] Pfitzinger | Local speech rate as a combination of syllable and phone rate[END_REF] for studying perceived speech rate of short segments). When they are satisfied with their ranking, the stimuli are once again played in decreasing MOS order before confirmation. Each subject ranks thus the ten sets of 8 stimuli arbitrarily thrown on 10 successive test planes.

Results

We launched the evaluation campaign on April 10th. By the time of the submission, 24 German listeners participated in the MOS test. 9 listeners have experience with synthetic speech and prosody while 15 have none. The preliminary results are shown in Figure 3. Four groups emerge from this first evaluation: the SFC has a statistically significant different mean from the natural prosody and alien systems 1, 5 and 6 (Anova analysis using anova1 and multcompare with hsd option Matlab® procedures). Experience with synthesis/prosody has no significant influence on the results. For instance, the proposed system generates an acceptable prosody that lies in the heading set of state-of-the-art systems we were able to input. These results should be interpreted with caution. On one hand, we compare prosody computed by raw state-of-the-art TTS systems with one computed by a prosodic system which receives enriched text. Although one can imagine that most of theses systems compute some accentual, morphological and syntactic information, the linguistic front-end involved in the computation of this information is not prone to errors and delivers impoverished data compared to the hand-labeled data delivered to the SFC. On the other hand, these preliminary modeling and evaluation results aim at confronting the SFC to a new language and helping us identifying theoretical or modeling flaws.

Figure 2. Java interface for subjective evaluation. Subjects should position stimuli (the natural utterance + synthetic renderings by 7 different prosodic models, including the one proposed here) on a MOS scale. Subjects can listen to each stimulus as many times as required (simple-click) for taking their decision. Positioning is done by drag and drop. Figure 3. Comparative results of the MOS test. The proposed model lies in the heading set of state-of-the-art systems.

Conclusions

We described a first confrontation of a trainable prosodic model, the SFC (recently proposed by Holm and Bailly) to German intonation. The rich morphological productivity of the German language and its highly embedded syntax fit quite well our theoretical framework. The first results are objectively and subjectively encouraging but the generated prosody is still far from natural prosody -as the ones generated by the state-of-the-art systems we collected. Of course the evaluation procedure we used allows and perhaps favors such discrimination. Potential improvements of the generated prosodic contours are: large differences occur because SFC only produces one prototypical cliché per metalinguistic function. A preliminary analysis of gross deviations between predicted and natural prosodic parameters suggests that several patterns can concurrently be used by speakers to encode the same function. Although this can be handled within the SFC by adding more metalinguistic functions, we want first to understand if this choice is arbitrary or contextual. The strict independence between the elementary contours should also be questioned. This is certainly especially true for the anchoring of contours

. Names and web sites of the systems used in the evaluation experiment (output gathered on 10 March 2006).

encoding metalinguistic functions on larger units than the word. This anchoring should probably include lexical stress positions. Despite its crude assumptions and potential refinements, the SFC is flexible enough to automatically capture essential prosodic regularities of the language it observes given general assumptions on metalinguistic functions of intonation.

Figure 1 .

 1 Figure 1. Comparing original and predicted prosodic contours. Left: f0. Right: syllable lengthening. For each caption: top: superposition of predicted (yellow) and original (green) contours; bottom: elementary contours predicted for each discourse function used to encode the linguistic structure of the utterance; the prediction is obtained by superposing and adding these elementary contours. Horizontal axis represents the syllable count

Proser http://www.atip.de/german/technologie/tts/proseronline.htm AT&T http://public.research.att.com/~ttsweb/tts/demo.php Festival CSLU http://cslu.cse.ogi.edu/tts/demos/ Cepstral http://www.cepstral.com/demos/ DFKI Mary http://mary.dfki.de/online-demos/online-speech-synthesis/speech_synthesis Loquendo http://actor.loquendo.com/actordemo/default.asp?language=en Table 2

Acknowledgements

We warmly thank Harald Hoege at Siemens and Valerie Mapelli at ELRA for helping us getting rapidly this precious database.

Appendix


G Gibert
email: gibert@icp.inpg.fr

G Bailly
email: bailly@icp.inpg.fr

F Elisei
email: elisei@icp.inpg.fr

Evaluating a virtual speech cuer

Keywords: Cued speech, evaluation, audiovisual speech synthesis

This paper presents the virtual speech cuer built in the context of the ARTUS project aiming at watermarking hand and face gestures of a virtual animated agent in a broadcasted audiovisual sequence. For deaf televiewers that master cued speech, the animated agent can be then superimposed -on demand and at the reception -on the original broadcast as an alternative to subtitling. The paper presents the multimodal text-to-speech synthesis system and the first evaluation performed by deaf users.

Introduction

Listeners with hearing loss and orally educated typically rely heavily on speechreading based on lips and face visual information. However speechreading alone is not sufficient due to the lack of information on the place of tongue articulation and the mode of articulation (nasality or voicing) as well as to the similarity of the lip shapes of some speech units (so called labial sosies as [u] vs. [y]). Indeed, even the best speechreaders do not identify more than 50 percent of phonemes in nonsense syllables [START_REF] Owens | Visemes observed by hearing-impaired and normal-hearing adult viewers[END_REF] or in words or sentences [START_REF] Bernstein | Speech perception without hearing[END_REF]. Cued Speech (CS) was designed to complement speechreading. Developed by Cornett [START_REF] Cornett | Cued Speech[END_REF][START_REF] Cornett | Le Cued Speech, in Aides manuelles à la lecture labiale et perspectives d'aides automatiques[END_REF] and adapted to more than 50 languages [START_REF] Cornett | Cued Speech, manual complement to lipreading, for visual reception of spoken language. Principles, practice and prospects for automation[END_REF], this system is based on the association of speech articulation with cues formed by the hand. While uttering, the speaker uses one of his hand to point out specific positions on the face (indicating a subset of vowels) with a hand shape (indicating a subset of consonants). The French CS (FCS) system is described in Figure 2 and Figure 3. Note that the basic CS coding unit is the CV sequence. Isolated vowels and consonants (resp. not preceded by a consonant or not followed by a vowel) are respectively coded with a default hand position or hand shape (indicated by stars in Figure 2 and Figure 3). Numerous studies have demonstrated the drastic increase of intelligibility provided by CS compared to speechreading alone [START_REF] Nicholls | Cued Speech and the reception of spoken language[END_REF][START_REF] Uchanski | Automatic speech recognition to aid the hearing impaired: Prospects for the automatic generation of cued speech[END_REF] and the effective facilitation of language learning using FCS [START_REF] Leybaert | Phonology acquired through the eyes and spelling in deaf children[END_REF][START_REF] Leybaert | The role of Cued Speech in language processing by deaf children: an overview[END_REF].

A large amount of work has been devoted to CS perception but few works have been devoted to CS synthesis [see rule-based systems described in 1, 10]. We describe here a multimodal text-to-speech system driving a virtual FCS speaker and its first evaluation by deaf users.

The multimodal text-to-speech system

The multimodal text-to-speech system developed in the framework of the ARTUS project [START_REF] Bailly | ARTUS : synthèse et tatouage audiovisuel des mouvements d'un personnage animé virtuel pour l'accessibilité d'émissions télévisuelles aux téléspectateurs sourds comprenant la Langue Française Parlée Complétée[END_REF] converts a series of subtitles into an acoustic signal and a stream of animation parameters for the head, face, arm and hand of a virtual cuer. The control, shape and appearance models of the virtual cuer have been determined using multiple multimodal recordings of one human speaker using cued speech.

Figure 1: Superimposition of the ARTUS virtual speaker in a

broadcast produced by ARTE.

Data collection and modelling

The different experimental settings used to record our target cuer and capture its gestures are described in a previous paper [START_REF] Gibert | Analysis and synthesis of the 3D movements of the head, face and hand of a speaker using cued speech[END_REF]. These settings include (a) intensive motion capture with good time resolution (120Hz) and high accuracy (0.1mm) of 113 infrared (IR) reflecting markers glued on the face and hand of the subject (see Figure 4), when cueing 238 sentences; (b) video capture of 247 coloured beads glued on the face of the subject (see Figure 5), when cueing simple syllables; and (c) scans of her head, moulds of her hand and her teeth. Using this data, accurate shape and appearance models of the head and face of the subjects have been developed [START_REF] Elisei | Capturing data and realistic 3D models for cued speech analysis and audiovisual synthesis[END_REF]: errors are close to the millimetre for both models. Both shape and appearance models of the face and hand are driven by quasi-articulatory parameters emerging from statistical analysis of the geometric degrees-of-freedom of the observed shapes.

Multimodal text-to-speech synthesis

The COMPOST text-to-speech synthesis system [START_REF] Bailly | COMPOST: a server for multilingual text-to-speech system[END_REF] has been parameterized and specific modules have been added to deal with FCS generation. These add-ons include the following items: Linguistic processing and mark-up. The subtitles have little punctuation. A specific module considers the beginning of each text fragment as a potential sentence beginning and discards unlikely hypotheses. Synchronization marks equal to the time codes of retained start fragments are then inserted. The rhythmic model will then adapt inter-sentences pause durations to fulfil when possible these meeting points. Prosody. Even though cuers are able to minimize the impact of the adding of the gesture modality on speech rate, the coding of isolated consonants (in the onset of a consonantal cluster or in the coda) using side hand position and the inertia of the arm impose a slower speech rate and an hyperarticulated pronunciation of complex syllables. Intonation is also affected. The trainable prosodic model SFC [START_REF] Bailly | SFC: a trainable prosodic model[END_REF] was thus trained using data from experiment (a). For the three broadcasts tested so far, only four sentences were not pronounced in the time interval devoted to their display, with an average delay of 120ms. Note that rules governing the positioning of time stamps of subtitles usually consider the number of letters to display and not the time to read them. Multimodal concatenative synthesis. Synthetic gestures and sound are produced by selecting, smoothing and concatenating multi-represented multimodal segments. Because of the specific coordination between face and hand gestures during Cued Speech production, the system proceeds in two steps: two types of segments are considered and synchronized by phasing gestural and acoustic landmarks according to specific rules [START_REF] Gibert | Analysis and synthesis of the 3D movements of the head, face and hand of a speaker using cued speech[END_REF]: "polysounds" that capture the signal and facial gestures between two stable acoustic targets (sounds such as glides are thus enclosed in larger units) and "dikeys" that encompass the arm, hand and head gestures between two successive hand position targets. Note that head movements of our cuer contribute significantly to the hand/face constrictions: if the hand carries out most of the path towards the hand position on the face, the head itself accomplishes on average 16.43% of that distance. Such an enhanced contribution of posture to discourse structure has also been reported for native signers [START_REF] Brentari | A prosodic model of sign language phonology[END_REF]. Multi-represented segments are selected by a classical dynamic programming using specific selection and concatenation costs. The concatenation costs take into account the relative contribution of each quasi-articulatory parameter to the variance of the facial and hand shapes. Figure 7: The virtual cuer: from motion capture to videorealistic animation.

Video-realistic animation

A video-realistic model of the hand of our cuer has been developed using casts and skinning procedures (see Figure 6). Generic models of the lips, skull, teeth and eyes have been adapted to the subject's morphology. The resulting highdefinition shape model is properly textured using blending of multiple cylindrical textures controlled by the quasiarticulatory parameters. The resulting videorealistic speech cuer is presented in Figure 7. It is controlled by the hand and face models developed using motion capture data (cf. § 2.1) and thus combines a high and accurate surface resolution (more 10000 vertices for articulated hand and face meshes) with nice time resolution (motion capture data at 120Hz).

Figure 8: Mean intelligibility rate for each subject for "lipreading" and "Cued Speech" conditions.Whereas intelligibility rate is not different from haphazard way (red dot line) for the "lipreading" condition, it is up to 94% for the "Cued Speech" condition.

Figure 9: Response time for each subject for "lipreading" and "Cued Speech" conditions. Even there is variability between subjects, the "lipreading" response time is doubled compared to the "Cued Speech" response time.

Evaluation

A first series of experiments are been conducted to evaluate the intelligibility of this virtual cuer with skilled deaf users of the French cued speech. This first evaluation campaign is dedicated to segmental intelligibility and the second one to comprehension of content.

Segmental intelligibility

This test was conducted to assess the contribution of the cueing gestures in comparison with lip reading alone. Minimal pairs. The test mirrors the Modified Diagnostic Rime Test developed for French by Peckels and Rossi [START_REF] Peckels | Le test de diagnostic par paires minimales. Adaptation au francais du 'Diagnostic Rhyme Test' de W.D. Voiers[END_REF]: the minimal pairs do not here test acoustic phonetic features but gestural ones. A list of valid French CVC word pairs has thus been developed that test systematically pairs of consonants in initial positions that differ almost only in hand shapes (ex: [bal] vs. [mal], [siR] vs. [tiR]): we choose the consonants in all pairs of 8 subsets of consonants (see Figure 2) that are highly visually confusable [START_REF] Summerfield | Visual perception of phonetic gestures, in Modularity and the motor theory of speech perception[END_REF]. The vocalic substrate was chosen so as to cover all potential hand positions while the final consonant was chosen so that to avoid rarely used French words or proper names, and test our ability to handle coarticulation effects. Due to the fact that minimal pairs cannot be found in valid French CVC words, we end up with a list of 196 word pairs. Conditions. Minimal pairs are presented randomly and in both order. The lipreading-only condition is tested first. The cued speech condition is then presented in order to be able to summon up cognitive resources for the most difficult task first (see the lower reaction times for CS stimuli in Figure 9).

Stimuli.

In order to avoid a completely still head, head movements of the lipreading-only condition are those produced by the text-to-cued speech synthesizer divided by a factor of 10. No attempt is made to modify segmental or suprasegmental settings that enhance articulation. Although some subjects have cochlear implants that enhance their speech decoding capacity, sound generation was turned off. Subjects. Eight subjects were tested. They are all hearing impaired people who have practised French Cued Speech since 3 years old. Results. Mean intelligibility rate for "lipreading" condition is 52.36% (see Figure 8). It is not different from haphazard way of response that means minimal pairs are not distinguishable. Mean intelligibility rate for "Cued Speech" condition is 94.26%. The difference in terms of intelligibility rate between these two conditions shows our virtual cuer gives significant information in terms of hand movements. In terms of cognitive efforts, the "Cued Speech" task is easier: the response time is significantly different (one factor repeated measure ANOVA (F(1,3134)=7.5, p<0.01)) and lower than for the "lipreading" one (see Figure 9).

Comprehension

To evaluate the global comprehension of our system, we asked the same subjects to watch a TV program where subtitles were replaced by the superimposition of the virtual cuer (see Figure 1). Ten questions were asked. The results show all the information is not perceived. On average, the subjects correctly replied to 3 questions. The difficulties of the task (proper names, high speaking flow …) could explain these results.

We conducted further experiments using a Tobii® eye tracker system in order to understand previous results. We asked 4 other deaf people to watch a video consisted of 2 parts: on the first part, the video of a real cuer and on the second part the video of our virtual cuer. There is no significant difference on the time spent on the mouth area for both modalities F(1,6)=0.22, p=0.65. Then, we asked the same 4 deaf people to watch the TV program that was half subtitled and half commented by the virtual cuer (see Figure 10). The results show deaf people spend 56.36% of the time on the teletext and 80.70% on the overlay area for the cuer with a significant difference F(1,6)=9.06, p<0.05. A control group consisting of 16 normal-hearing subjects watches the same audiovisual sequence that was entirely subtitled. They spend 40.14% of the time reading teletext. There is no significant difference with the deaf subjects.

Conclusions

The recordings and analysis of the performance of a real FCS speaker allow us to implement a complete text-to-cued speech synthesizer. The results of the preliminaries perceptive tests show that significant linguistic information with minimal cognitive effort is transmitted by our system. This series of experiments must be continued on more subjects. More evaluation and modelling work is required to quantify and reduce the cognitive effort devoted to CS decoding.

Figure 2 :

 2 Figure 2: Coding subsets of consonants with hand shapes

Figure 3 :

 3 Figure 3: Coding subsets of vowels with hand positions.

Figure 4 :

 4 Figure 4: Motion capture using a VICON® system with 12 cameras, 50 beads glued on the hand and 63 on the face.

Figure 5 :

 5 Figure 5: Video recordings of the face with 247 beads.

Figure 6 :

 6 Figure 6: Left: meshing the cast of the hand. Right: controlling the skinned model with the shape model built from motion capture.

Figure 10 :

 10 Figure 10: Eye gaze captured during the comprehension test using an eye tracker system for the subtitled (left) and video superimposed (right) broadcast audiovisual sequence.

Acknowledgements

The authors thank Martine Marthouret, Marie-Agnès Cathiard, Denis Beautemps and Virginie Attina for their help and comments on building the perceptive tests. We also want to thank the 28 subjects who took part to the evaluation.


Olivier Cailloux

Beatrice Napolitano
email: beatrice.napolitano@dauphine.fr

M Remzi Sanver

Compromising as an equal loss principle

Keywords: Axiomatic Analysis, Ex-Post Compromise, Bargaining, Egalitarianism, Arbitration

A social choice rule aggregates the preferences of a group of individuals over a set of alternatives into a collective choice. The literature admits several social choice rules whose recommendations are supposed to reflect a compromise among individuals. We observe that all these compromise rules can be better described as procedural compromises, i.e., they impose over individuals a willingness to compromise but they do not ensure an outcome where everyone has effectively compromised. We revisit the concept of a compromise in a collective choice environment with at least three individuals having strict preferences over a finite set of alternatives. Referring to a large class of spread measures, we view the concept of compromise from an equal loss perspective, favoring an outcome where every voter concedes as equally as possible. As such, being a compromise may fail Pareto efficiency, which we ensure by asking voters to concede as equally as possible among the Pareto efficient alternatives. We show that Condorcet consistent rules, scoring rules (except antiplurality) and Brams-Kilgour compromises (except fallback bargaining) all fail to ascertain an outcome which is a compromise. A slight restriction on acceptable spread measures suffices to extend the negative result to antiplurality and fallback bargaining. This failure also prevails for social choice problems with two individuals: all well-known two-person social choice rules of the literature, namely, fallback bargaining, Pareto and veto rules, short listing and veto rank, fail to pick ex-post compromises. We conclude that there is a need to propose and study rules that satisfy this equal loss, or outcome oriented, notion of a compromise.

Introduction

In a typical social choice problem, several individuals express their preferences over a set of alternatives and one shall be picked as the collective outcome. Although the literature admits several Social Choice Rules (SCRs) with different properties, there is a common understanding that collective choices must reflect "compromises". One of the first to explicitly refer to a SCR as a compromise is Sertel (1986) introducing the majoritarian compromise. This SCR, further analysed by [START_REF] Sertel | The majoritarian compromise is majoritarianoptimal and subgame-perfect implementable[END_REF], is a rediscovery of a method suggested by James W. Bucklin at the beginning of the 20 th century [START_REF] Erdélyi | Control complexity in bucklin and fallback voting: A theoretical analysis[END_REF]. Starting from everyone's ideal alternative, it falls back to the voters' second, third and more generally k-th best, until one of the alternatives considered appears among the first k best for a majority. [START_REF] Brams | Fallback bargaining[END_REF] generalise this concept and introduce a class of SCRs called q-approval fall-back bargaining, where q is the required quantity of support that can vary from a single voter up to unanimity. Different choices of q lead to different SCRs. Considering n voters, the choice of q = 1 corresponds to the plurality rule, q = ⌈ n 2 ⌉ coincides with the majoritarian compromise and q = n represents a bargaining procedure called fall-back bargaining, which has been further analysed by [START_REF] Kibris | Bargaining over a finite set of alternatives[END_REF] and [START_REF] Congar | A characterization of the maximin rule in the context of voting[END_REF]. We will refer to these rules as Brams and Kilgour (BK) compromises with threshold q.

As Özkal-Sanver and Sanver ( 2004) discuss, the concept of compromising is mostly understood as the trade off between the number of voters supporting an alternative (i.e., the quantity of support) and the distance of that alternative from the supporters' ideal alternative (i.e., the quality of support). This trade off, which is explicit for q-approval fall-back bargaining, is also the basis for several other SCRs such as the median voting rule proposed by [START_REF] Bassett | Robust voting[END_REF] and further analysed by [START_REF] Gehrlein | On some limitations of the median voting rule[END_REF] or the Condorcet practical method described by [START_REF] Nurmi | Voting paradoxes and how to deal with them[END_REF]. [START_REF] Merlin | Compromise rules revisited[END_REF] identify and analyse a large class of compromise rules that explore this trade off.

One can argue that a collective choice per se implies a compromise. After all, except extreme cases such as dictatorships, a SCRs operates on the principle that all voters could fall back from their ideal position. Whether all voters effectively do fall back and whether this fall is "equal among voters" is the subject of our analysis. In what follows, we will present examples where they do not, which could be considered counter to the spirit of compromising.1

Consider the following example. At P , all BK compromises, except when q = n (i.e. fall-back bargaining), will ignore the single voter and will pick a as the collective outcome.

As a matter of fact, almost every SCR will ignore this "marginal minority" and choose a in this situation. While this choice is defensible on the grounds of qualified majoritarianism, it is questionable whether a can be qualified as a compromise. Observe that b receives unanimous support when each voter falls back one step from his ideal point. The question becomes more compelling when a remains the collective choice even if the ignored group is much larger. When q ∈ 1, n 2 +1 , all BK compromises pick a, and, again, it does not appear as a compromise as 51 voters reach their best alternative while the remaining 49 voters have to be contented with their worst one.

Observe that all these SCRs impose to voters a willingness to compromise, but do not effectively ensure a compromise as the collective choice. In fact, the term "compromise" in this literature refers to procedural compromises that differ from outcome oriented compromises, a conceptual distinction that seems to be overlooked in the literature. This can also be viewed as a distinction between ex-ante and ex-post compromises, where the profile is the source of uncertainty.

To define an ex-post compromise, we adapt a concept of equal losses that considers allocation of continuous utilities. This principle is used for bargaining [START_REF] Chun | The equal-loss principle for bargaining problems[END_REF][START_REF] Chun | The lexicographic equal-loss solution[END_REF] and bankruptcy problems [START_REF] Herrero | The three musketeers: four classical solutions to bankruptcy problems[END_REF]. We introduce two definitions of compromise. In both of them, we pick a spread measure that determines how equally a given vector of numbers is distributed and we propose to make a collective choice where voters give up from their ideal points "as equally as possible". The difference between the two is that one, called egalitarian compromise, insists on equality at the expense of Pareto efficiency while the other, called Paretian compromise, is constrained to pick among the Pareto efficient alternatives. We show that Pareto efficient SCRs cannot ensure egalitarian compromises under any spread measure. We prove that several well-known SCRs such as Condorcet extensions, scoring rules, q-approval fall-back bargaining, all fail to be Paretian compromises under any spread measure. We provide examples for which being a Paretian compromise would necessitate to pick an alternative that is, although Pareto optimal, ranked very low by all voters. Such alternative would never be picked by any of the above-mentioned SCRs.

We conclude that the equal-loss principle appears adequate for collective choice problems with at least three individuals, when egalitarianism, in the sense of conceding equally, is a major concern. Imagine a situation where the head of a laboratory needs to decide which project to fund and she asks for the preferences of the laboratory members. The workplace harmony is extremely important and, in order to avoid conflicts, the winning project must be equally supported by all members.

Consider now a situation with only two voters. As the vast literature on the ultimatum game [START_REF] Güth | More than thirty years of ultimatum bargaining experiments: Motives, variations, and a survey of the recent literature[END_REF] suggests, mutual consent is hard to obtain when one individual sees injustice at the levels of mutual losses. The equal-loss principle seems to be crucial in this new scenario.

Collective choice models with two individuals can be interpreted as bargaining or arbitration problems. While the bargaining interpretation necessitates an explicitly defined disagreement outcome [START_REF] Kibris | Bargaining over a finite set of alternatives[END_REF], the arbitration interpretation [START_REF] Sprumont | Intermediate preferences and rawlsian arbitration rules[END_REF] remains within the classical collective choice environment with no explicit disagreement outcome. In this paper, we consider the latter interpretation.

Arbitration rules are thoroughly discussed by [START_REF] Barberà | Compromising on compromise rules[END_REF]. As prominent examples, we have fallback bargaining proposed by [START_REF] Brams | Fallback bargaining[END_REF], the veto-rank and short listing procedures analysed by de [START_REF] De Clippel | On the selection of arbitrators[END_REF] and the Pareto-and-veto rules analysed by [START_REF] Laslier | A solution to the twoperson implementation problem[END_REF]. These models consider discrete alternatives which are not contained by the classical [START_REF] Nash | The bargaining problem[END_REF] bargaining environment with convex utilities. We make the same assumption. However, as [START_REF] Mariotti | Nash bargaining theory when the number of alternatives can be finite[END_REF] and [START_REF] Nagahisa | An axiomatization of the kalai-smorodinsky solution when the feasible sets can be finite[END_REF] illustrate, the two worlds can be interconnected, as we do for the equalloss principle of [START_REF] Chun | The equal-loss principle for bargaining problems[END_REF] and [START_REF] Chun | The lexicographic equal-loss solution[END_REF]. The arbitration environment presents an instance where the equal-loss principle could matter and it is rather surprising to discover that most interesting SCRs used as arbitration solutions fail to be Paretian compromises.

The rest of the paper is organised as follows. Section 2 presents the basic notions and notation. Section 3 introduces egalitarian compromises and Paretian compromises, two concepts that turn out to be logically incompatible. Section 4 shows that with at least three individuals, many SCRs fail to pick a compromise. Section 5 considers the two-individual case, showing that most SCRs of the literature fail to pick compromises. Section 6 makes some concluding remarks.

Basic notions and notation

Consider a finite set N of individuals with #N = n ≥ 2 and a finite set A of alternatives with #A = m ≥ 3. We write L(A) for the set of linear orders over A. A generic element ≻ i of L(A) stands for a preference of i ∈ N . This implies that, given any x ̸ = y ∈ A, precisely one of x ≻ i y and y ≻ i x holds while x ≻ i x holds for no x ∈ A. Moreover, x ≻ i y and y

≻ i z implies x ≻ i z ∀x, y, z ∈ A. A profile P : N → L(A) associates with each individual i ∈ N a preference P (i) = ≻ i . A Social Choice Rule (SCR) is a mapping f : L(A) N → 2 A \ {∅}. We write r ≻i (x) = #{y ∈ A | y ≻ i x} + 1 for the rank of x ∈ A at ≻ i ∈ L(A).
We denote by λ ≻i (x) = r ≻i (x) -1 the loss in terms of ranks for i ∈ N with preference ≻ i , when x is elected instead of the best alternative for i. The mapping λ P : A → 0, m -1 N assigns to each x ∈ A the loss vector λ P (x) = (λ ≻i (x)) i∈N induced by the election of x. The double brackets denote intervals in the integers.

We are interested in measuring the spread of loss vectors. To this end, we define a spread measure σ : 0, m -1 N → R + as a function that associates a spread value to every possible loss vector. We write Σ for the set of spread measures σ that satisfy, for every l ∈ 0, m -1 N , σ(l) = 0 ⇔ l i = l j ∀i, j ∈ N . Thus, the spread of l gets its lowest value 0 in case of perfect equality and only in this case. Note that this condition incorporates the minimal requirement to identify a spread measure and leads to the largest set of spread measures one could define. As discussed in Section 4.4, this flexibility has the advantage of making our results more general.

Given any distinct x, y ∈ A, we say that x Pareto dominates y at P ∈ L(A) N (or equivalently y is Pareto dominated by x at P ) iff x ≻ i y, ∀i ∈ N . We denote by PO

(P ) = {x ∈ A | ∀y ∈ A \ {x}, ∃i ∈ N | x ≻ i y} the set of Pareto optimal alternatives at P . A SCR f is Paretian iff f (P ) ⊆ PO(P ) ∀P ∈ L(A) N .

Egalitarian versus Paretian compromises

Egalitarian compromises

We let arg min X (σ • λ P ) = {x ∈ X | ∀y ∈ X : σ(λ P (x)) ≤ σ(λ P (y))} denote the minimal elements of X ⊆ A according to (σ • λ P ). In other words, arg min X (σ • λ P ) denotes the alternatives in X whose loss vectors are the most equally distributed according to the spread measure σ.

In what follows, we define some classes of SCRs that we are interested in analysing.

Definition 3.1 A SCR f is an Egalitarian Compromise (EC) iff ∃σ ∈ Σ | ∀P ∈ L(A) N we have f (P ) ⊆ arg min A (σ • λ P ). Definition 3.2 A SCR f is Egalitarian Compromise Compatible (ECC) iff ∃σ ∈ Σ | ∀P ∈ L(A) N we have f (P ) ∩ arg min A (σ • λ P ) ̸ = ∅.
Under a SCR that is EC (resp., ECC), all (resp., some) winners are among the alternatives with most equally distributed losses. Clearly, EC is a subclass of ECC. Perhaps less obviously, being ECC (or EC) is incompatible with being Paretian. This will be deduced from the following proposition, which will also be useful to prove other theorems. Proposition 3.1 For n ≥ 2, m ≥ 3, there exists a profile P ∈ L(A) N and an alternative a m such that ∀i ∈ N : r ≻i (a m ) = m, and such that ∀σ ∈ Σ : arg min A (σ

• λ P ) = {a m }; hence, arg min A (σ • λ P ) ∩ PO(P ) = ∅.
Proof Consider the following profile P :

1 a 1 a 2 . . . a m-1 a m n -1 a π (1) a π (2) . . . a π (m-1) a m ,
where π is the following permutation over 1, m -1 :

π(i) = i + 1 if i ∈ 1, m -2 1 if i = m -1 .
In P , a m is the only alternative such that r ≻i (a m ) is independent of i; hence, σ(λ P (a)) > 0, ∀a ∈ A \ {a m }, ∀σ ∈ Σ. Thus, the set arg min A (σ • λ P ) consists of the sole element a m , and, because a m is Pareto dominated, arg min A (σ

• λ P ) ∩ PO(P ) = ∅.
Our main result for section 3.1 follows easily.

Theorem 3.1 For n ≥ 2, m ≥ 3, no Paretian SCR is ECC. Proof Proving this amounts to show that ∀σ ∈ Σ, ∃P ∈ L(A) N | PO(P ) ∩ arg min A (σ • λ P ) = ∅.
Suffices to use proposition 3.1, which asserts that there exists a profile P such that ∀σ ∈ Σ : arg min A (σ • λ P ) ∩ PO(P ) = ∅.

Paretian compromises

Having seen the tension for a SCR being Paretian and ECC, we investigate the consequences of inverting the order of priorities by insisting that at least some of the winning alternatives are Pareto optimal, and considering the most equally distributed loss vectors among those.

We consider two classes of SCRs. Observe that arg min PO(P ) (σ•λ P ) denotes the set of Pareto optimal alternatives whose loss vectors are the most equally distributed according to the spread measure σ.

Definition 3.3 A SCR f is a Paretian Compromise PC iff ∃σ ∈ Σ | ∀P ∈ L(A) N we have f (P ) ⊆ arg min PO(P ) (σ • λ P ). Definition 3.4 A SCR f is Paretian Compromise Compatible PCC iff ∃σ ∈ Σ | ∀P ∈ L(A) N we have f (P ) ∩ arg min PO(P ) (σ • λ P ) ̸ = ∅.
Again, it is clear that PC is a subclass of PCC. It will also probably come with no surprise that for a SCR, being PC is incompatible with being ECC, as being PC requires to be Paretian, which permits to use theorem 3.1. On the other hand, it is less immediate that being EC is incompatible with being PCC, because being PCC does not require to be Paretian. This is however true.

Theorem 3.2 For n ≥ 2, m ≥ 3, no SCR is both EC and PCC.

Proof Considering the profile P of proposition 3.1, with a m the alternative mentioned there, any EC f and any σ ∈ Σ, suffices to prove that f (P ) ∩ arg min PO(P ) (σ

• λ P ) = ∅.
First, from proposition 3.1 we have that {a m } ∩ PO(P ) = ∅, hence {a m }∩ arg min PO(P ) (σ

• λ P ) = ∅.
Second, because f is EC, for some σ, f (P ) ⊆ arg min A (σ • λ P ). Using proposition 3.1 again, we see that arg min A (σ

• λ P ) = {a m }, hence f (P ) = {a m }.
That f (P ) ∩ arg min PO(P ) (σ • λ P ) = ∅ follows from these two facts.

It is interesting to note that the incompatibility is not complete, however.

Remark 3.1 For n ≥ 2, m ≥ 3, there exist SCRs that are both ECC and PCC, such as the SCR that selects the whole set of alternatives at every profile. However, this SCR fails to be Paretian, as is any SCR that is ECC.

Which SCRs are compromises?

In this section we assume n ≥ 3 and leave the analysis of n = 2 to the next section.

Condorcet consistent rules

An alternative x ∈ A is a Condorcet winner at P ∈ L(A) N iff for all y ∈ A\{x},

#{i ∈ N | x ≻ i y} > #{i ∈ N | y ≻ i x}.
So each profile admits either no or a unique Condorcet winner. An SCR f is Condorcet consistent iff f (P ) = {x} at each P ∈ L(A) N that admits x as the (unique) Condorcet winner.

Theorem 4.1 Let n ≥ 3 and m ≥ 3. A Condorcet consistent SCR f is neither ECC nor PCC.
Proof Consider the following profile P , where the dots represent the sequence a 4 to a m :

n -1 a 1 a 2 a 3 . . . 1 a 3 a 2 . . . a 1 .
Consider any Condorcet consistent SCR f . Thus, f (P ) = {a 1 }. However, arg min A (σ • λ P ) = arg min PO(P ) (σ • λ P ) = {a 2 } ∀σ ∈ Σ, so there exists a profile P such that both f (P ) ∩ arg min A (σ • λ P ) and f (P ) ∩ arg min PO(P ) (σ • λ P ) are empty.

Note that Condorcet consistent rules need not be Paretian so the fact that they all fail ECC does not follow from theorem 3.1.

Scoring rules

A score vector is an m-tuple w = (w 1 , . . . , w m ) ∈ 0, 1 m with w 1 = 1, w m = 0 and w i ≥ w i+1 ∀i ∈ 1, m -1 . Given a score vector w, we write s w (x, P ) = i∈N w r≻ i (x) for the score of x ∈ A at P ∈ L(A) N . Each vector w identifies a scoring rule f w n defined as f w n (P ) = {x ∈ A : s w (x, P ) ≥ s w (y, P ) ∀y ∈ A} for every P ∈ L(A) N .

We first show that no scoring rule is ECC, for any value of n and m at least 3. Theorem 4.2 Let n ≥ 3 and m ≥ 3. No score vector w induces a scoring rule f w n that is ECC. Proof Take any score vector w. Consider the profile P of proposition 3.1. Observe that arg min A (σ • λ P ) = {a m } ∀σ ∈ Σ. However, as w 1 > w m , we have s w (a 1 , P ) > s w (a m , P ) which implies a m / ∈ f w (P ).

We call antiplurality score vector the vector w such that w i = 1, ∀i ∈ 1, m -1 , and w m = 0.

Theorem 4.3 Let m ≥ 3 and let w be the antiplurality score vector. The SCR f w n satisfies PCC for all n ≥ 3. Proof Define σ ∈ Σ as, ∀l ∈ 0, m-1 N : σ(l) = 1 iff ∃i, j ∈ N | l i ̸ = l j ; σ(l) = 0 otherwise. We show the non-emptyness of f w n (P ) ∩ arg min PO(P ) (σ • λ P ) for any profile P .

Let k = min PO(P ) {(σ • λ P )(x)} be the minimal value attained by σ • λ P over PO(P ). By construction of σ, k equals either 0 or 1.

For k = 1, take any x ∈ f w n (P ) ∩ PO(P ). This intersection is non-empty because whenever the antiplurality rule picks a Pareto dominated alternative z, it also picks all alternatives which Pareto dominate z. By definition of σ, σ(x) ≤ 1, hence, x ∈ arg min PO(P ) (σ • λ P ).

For k = 0, take any x ∈ arg min PO(P ) (σ • λ P ). As σ(λ P (x)) = 0, we have, ∀i, j ∈ N : λ P i (x) = λ P j (x), hence, ∀i, j ∈ N : r ≻i (x) = r ≻j (x). The case r ≻i (x) = m, ∀i ∈ N is ruled out by x ∈ PO(P ). Hence, r ≻i (x) ≤ m-1, ∀i ∈ N , hence, x ∈ f w n (P ). It is worth noting that the antiplurality rule f w n is not Paretian, hence fails PC for all n ≥ 3. This can be seen by picking a unanimous profile

P ∈ L(A) N with a 1 ≻ i a 2 ≻ i . . . ≻ i a m ∀i ∈ N , where arg min PO(P ) (σ•λ P ) = {a 1 } ∀σ ∈ Σ while f w n (P ) = A \ {a m }.
Theorem 4.4 Let m ≥ 3. Take any score vector w which is not the antiplurality score vector. For some n ≥ 3, the SCR f w n fails PCC.

Proof Take any m ≥ 3 and any score vector w that is not the antiplurality score vector; therefore, w m-1 < 1. Pick any n such that n ≥ m -1 and n > 1 1-wm-1 . Consider a profile P ∈ L(A) N conforming to

i = 1 a 2 . . . a m a 1 2 ≤ i ≤ m -2 a 1 . . . a m a i m -1 ≤ i ≤ n a 1 . . . a m a m-1
, where all alternatives except a m appear at least once in the last rank. Thus, for every σ ∈ Σ, we have σ(λ P (x)) > 0 ∀x ∈ A \ {a m } while σ(λ P (a m )) = 0. Moreover, a m ∈ PO(P ). Thus, arg min PO(P ) (σ • λ P ) = {a m } ∀σ ∈ Σ. On the other hand, s w (a 1 ; P ) = n -1, s w (a m ; P ) = n • w m-1 and as n > 1 1-wm-1 (or, equivalently, n -1 > nw m-1 ), we have s w (a 1 ; P ) > s w (a m ; P ), establishing a m / ∈ f w (P ), thus f w (P ) ∩ arg min PO(P ) (σ

• λ P ) = ∅ ∀σ ∈ Σ.

BK compromises

Given any k ∈ 1, m , we write n k (x, P ) = #{i ∈ N | r ≻i (x) ≤ k} for the k-support that x gets at P , that is, the number of individuals for whom the rank of alternative x ∈ A is lower than or equal to k in the profile P ∈ L(A) N . Note that n k (x, P ) ∈ 1, n is non-decreasing on k and n m (x, P ) = n. For each q ∈ 1, n , we define ρ q (x, P ) = min{k ∈ 1, m | n k (x, P ) ≥ q} as the minimal rank k at which the k-support that x gets at P is at least q. We write ρ q (P ) = min x∈A {ρ q (x, P )} for the minimal rank k at which the k-support that some alternative gets at P is at least q. A Brams and Kilgour (BK) compromise with threshold q is the SCR f q defined for each P ∈ L(A) N as f q (P ) = {x ∈ A | n ρq(P ) (x, P ) ≥ n ρq(P ) (y, P ) ∀y ∈ A}.

We first consider the BK compromise with threshold q = n, f n , which corresponds to the rule also known as fallback bargaining [START_REF] Brams | Fallback bargaining[END_REF]. Proof Define σ ∈ Σ as, ∀l ∈ 0, m -1 N : σ(l) = 1 iff ∃i, j ∈ N | l i ̸ = l j ; σ(l) = 0 otherwise. Considering any x ∈ f n (P ), let us show that x ∈ arg min PO(P ) (σ • λ P ). Because x ∈ f n (P ), x ∈ PO(P ), and therefore, suffices to show that ∀y ∈ PO(P ), σ(λ P (y)) ≥ σ(λ P (x)). Given the choice of σ, picking any y ∈ PO(P ) with y ̸ = x, suffices to show that σ(λ Proof As f n is Paretian, the proof comes straightforward from theorem 3.1. Theorem 4.7 Let n ≥ 3 and m ≥ 3. A BK compromise f q with threshold q ∈ 1, n -1 is neither ECC nor PCC.

P (y)) = 1, equivalently, that ∃i, j ∈ N | r ≻i (y) ̸ = r ≻j (y). Because x ∈ f n (P ), ρ n (P ) = ρ n (x, P ) = max N r ≻i (x). It follows from ρ n (P ) = min z∈A {ρ n (z, P )} that ρ n (y, P ) ≥ ρ n (x, P ), thus, ∃i ∈ N | r ≻i (y) ≥ ρ n (P ). Also, y ∈ PO(P ) implies that ∃j ∈ N | r ≻j (y) < r ≻j (x), thus ∃j ∈ N | r ≻j (y) < ρ n (P ). Therefore, r ≻i (y) ̸ = r ≻j (y).
Proof Consider the following profile P (also used in the proof of theorem 4.1), where the dots represent the sequence a 4 to a m :

n -1 a 1 a 2 a 3 . . . 1 a 3 a 2 . . . a 1 .
We have that f q (P ) = {a 1 }, and, because σ(λ P (a 2 )) = 0 and σ(λ P (a 1 )) > 0, neither arg min A (σ • λ P ) nor arg min PO(P ) (σ • λ P ) contain a 1 for any σ ∈ Σ.

Restrictions on sigma

The perfect equality recognition condition we adopt for spread measures, i.e., that the spread gets its lowest value 0 in case of perfect equality and only in this case, is very basic. Unless this condition is violated, Σ is the largest set of spread measures we could conceive. On the other hand, it is possible to let Σ shrink by imposing additional conditions over spread measures. Nevertheless, as the satisfaction of PC, PCC, EC, or ECC requires the existence of a spread measure, all of our negative results, namely, those expressed by Theorems 3.1, 3.2, 4.1, 4.2, 4.4, 4.6 and 4.7 prevail when Σ is restricted. In a similar vein, the positive results in Theorems 4.3 and 4.5 risk to be lost with additional conditions over spread measures. Indeed, this section shows that a mild restriction removes the positive results concerning the only two rules that we found to be compatible with any of our compromise concepts.

Definition 4.1 Given any m ≥ 4 and n ≥ max{4, m-1}, we say that a spread measure σ satisfies condition C m,n iff we have σ(m-3, m-1, m-2, . . . , m-2) < σ(m -2, m -3, . . . , 1, 0, . . . , 0).

As both vectors are n dimensional, the term m -2 repeats n -2 times in the first vector and the term 0 repeats n -m + 2 times in the second vector.

This condition imposes a very reasonable requirement on spread measures for large values of m and n. Asking for σ(1, 3, 2, 2) to be smaller than σ(2, 1, 0, 0) is demanding while asking for σ (5,7,6,6,6,6,6) to be smaller than σ(6, 5, 4, 3, 2, 1, 0) reflects a mild assumption. In any case, as we state below, several well-known spread measures of the literature (see [START_REF] Allison | Measures of inequality[END_REF] for a comprehensive account) satisfy definition 4.1. Letting l = n i=1 l i /n denote the arithmetic mean of the values of l = (l 1 , . . . , l n ), we consider the following measures:

-the mean absolute difference σ mad (l) = 1 n 2 n i=1 n j=1 |l i -l j |; -the average absolute deviation σ ad (l) = n i=1 |li-l| n ; -the standard deviation σ sd (l) = n i=1 (li-l) 2 n ; -the Gini coefficient σ G (l) = n i=1 n j=1 |li-lj | 2•n• n i=1 li .
Remark 4.1 We checked experimentally that σ mad , σ ad , σ sd and σ G all satisfy condition C m,n , for m ∈ 4, 1000 and n ∈ b, 1000 where b = max{4, m -1}.

We write Σ Cm,n ⊆ Σ for the set of spread measures that satisfy condition C m,n .

Theorem 4.8 For all m ≥ 4, n ≥ max{4, m -1}, under Σ Cm,n , 1) f w n fails PCC when w is the antiplurality score vector; 2) the BK compromise f n fails PCC.

Proof Given any σ ∈ Σ Cm,n , let us show that there exists a profile P such that f w n ∩ arg min PO(P ) (σ

• λ P ) = ∅ and f n ∩ arg min PO(P ) (σ • λ P ) = ∅.
To that aim, consider some x, y ∈ A and some

P ∈ L(A) N with r ≻1 (x) = m -2, r ≻2 (x) = m, r ≻i (x) = m-1 ∀i ∈ N \{1, 2}, and r ≻i (y) = m-i ∀i ∈ 1, m-1 , r ≻j (y) = 1 ∀j ∈ m, n . Moreover, for each z ∈ A \ {x, y}, let r ≻i (z) = m for some i ∈ N .
Note that f w n (P ) = f n (P ) = {y}. On the other hand, λ P (x) = (m -3, m -1, m -2, . . . , m -2) and λ P (y) = (m -2, m -3, . . . , 1, 0, . . . , 0). As σ(λ P (x)) < σ(λ P (y)) (because σ ∈ Σ Cm,n ), we see that y / ∈ arg min PO(P ) (σ • λ P ).

Two-voters case

In addition to fallback bargaining (FB) [START_REF] Brams | Fallback bargaining[END_REF] (defined in section 4.3), we consider three prominent solutions of the literature. Pareto-and-Veto rules (PV) [START_REF] Moulin | The strategy of social choice[END_REF][START_REF] Abreu | Virtual implementation in nash equilibrium[END_REF][START_REF] Laslier | A solution to the twoperson implementation problem[END_REF] distribute a veto power of v 1 and v 2 alternatives to voters 1 and 2, respectively, with v 1 + v 2 = m -1. So, every voter i = 1, 2 (simultaneously) vetoes his worst v i alternatives. The SCR picks all non-vetoed and Pareto optimal alternatives.

The Veto-Rank mechanism (VR) is commonly used in the selection of arbitrators [START_REF] De Clippel | On the selection of arbitrators[END_REF]. Given a list of m (odd) alternatives (that are candidates to be arbitrators), each of the two voters (that are the two parties that must agree on an arbitrator) simultaneously vetoes his worst m-1 2 alternatives. The selected alternatives are the ones with the highest Borda score among the non-vetoed alternatives.

Again within the context of selecting arbitrators, de Clippel et al. ( 2014) propose and analyse Shortlisting (SL) where one of the two parties starts by vetoing her worst m-1 2 alternatives (m being odd), and then the second party chooses her best alternative out of the remaining ones. As the outcome of the procedure depends on the party that starts, symmetry among players is ensured by defining the solution as the union of the two outcomes where one and the other party starts.

Definition 5.1 Given any m ≥ 7, a spread measure σ ∈ Σ satisfies condition

D m iff σ(⌈ m 2 ⌉, ⌈ m 2 ⌉ -2) < σ(0, ⌈ m 2 ⌉ -1) and σ(⌈ m 2 ⌉ -2, ⌈ m 2 ⌉) < σ(⌈ m 2 ⌉ -1, 0).
For m = 7 the condition requires σ(4, 2) < σ(0, 3) and σ(2, 4) < σ(3, 0) which is reasonable in our context. When the value of m is larger, the condition appears even more convincing. As m grows, the distance between 0 and ⌈ m 2 ⌉ -1 grows, while the distance between ⌈ m 2 ⌉ and ⌈ m 2 ⌉ -2 remains constant. Requiring, for example, the spread of (15, 13) to be smaller than the spread of (0, 14) is very reasonable.

We write Σ Dm ⊆ Σ for the set of spread measures that satisfy condition D m .

Theorem 5.1 Let m ≥ 7. Under Σ Dm , FB and PV fail PCC. Furthermore, when m is odd, VR and SL also fail PCC.

Proof Take any m ≥ 7 and any σ ∈ Σ Dm . Define α = ⌈ m 2 ⌉-1 and β = ⌈ m 2 ⌉-2. It follows from σ ∈ Σ Dm , that σ(α + 1, β) < σ(0, β + 1) and σ(β, α + 1) < σ(β + 1, 0). For m odd, note that α + β + 2 = m and consider the profile P where voter i 1 has the preference

x ≻ a 1 ≻ . . . ≻ a α ≻ y ≻ b 1 ≻ . . . ≻ b β and voter i 2 has the preference b 1 ≻ . . . ≻ b β ≻ y ≻ x ≻ a 1 ≻ . . . ≻ a α .
For m even, note that α+β +3 = m, and define the profile P in the same way, except that a supplementary alternative z is added at the bottom of both rankings.

Note that σ(λ P (y)) = σ(α + 1, β) and that σ(λ P (x)) = σ(0, β + 1). Therefore, σ(λ P (y)) < σ(λ P (x)). As y is not Pareto-dominated, an SCR that uniquely picks x at P cannot be PCC. In a similar vein, at the profile P ′ which is obtained by the inversion of the preferences of i 1 and i 2 at P , an SCR that is PCC cannot pick x uniquely.

The proof will be concluded by showing that FB, PV, and (when m is odd) VR and SL all pick only x at P or at P ′ .

We readily see that FB picks only x at P (and at P ′ ) since x is the first alternative which reaches the unanimous consent. For PV, let

v i1 ≥ v i2 (thus v i1 ≥ ⌈ m-1 2 ⌉ ≥ ⌈ m-2 2 ⌉ = β + 1 and v i2 ≤ ⌊ m-1 2 ⌋ = ⌈ m-2 2 ⌉ = α)
, and consider the profile P . Observe that the first voter vetoes at least y and every b j (1 ≤ j ≤ β) while no voter vetoes x. As x Pareto-dominates every a j (1 ≤ j ≤ α), PV picks only x at P . When v i2 ≥ v i1 , a similar reasoning yields that PV picks only x at P ′ . Now let m be odd. For VR, a reasoning similar to the one applied to PV yields x as the unique choice at P : each voter vetoes her worst m-1 2 alternatives, thus i 1 vetoes y and every b j (1 ≤ j ≤ β) and i 2 vetoes every a j (1 ≤ j ≤ α). The alternative x is the only non-vetoed alternative, so it is selected as the sole winner.

Finally, SL also picks x, as it is the unique winner no matter which voter starts the veto phase. If i 1 starts, y and every b j (1 ≤ j ≤ β) get vetoed, then i 2 chooses her best alternative out of the remaining ones which is x. If i 2 starts, every a j (1 ≤ j ≤ α) get vetoed, then i 1 chooses her best alternative which is x.

Concluding remarks

We define an ex-post compromise as an outcome where individuals give up as equally as possible from their ideal points. With three or more individuals, several well known SCRs fail to pick ex-post compromises, under any reasonable meaning attributed to "giving up equally". Our findings cover Condorcet extensions and scoring rules but also BK compromises, which impose a willingness to compromise without ensuring a compromised outcome. In particular we find that: no Condorcet procedure is ECC or PCC (Theorem 4.1), no scoring rule is ECC (Theorem 4.2) or PCC except for the antiplurality rule (Theorems 4.3 and 4.4), BK compromises are neither ECC or PCC (Theorems 4.6 and 4.7) except for fallback bargaining that is PC (Theorem 4.5).

Our impossibility results are stated for the set of spread measures Σ, but they prevail for any subset of Σ. As Σ is the largest set of sensible spread measures, they are valid for any specific concept of equity one might pick. On the other hand, our possibility results on fallback bargaining and antiplurality are not propagated to subsets of Σ. In fact, as soon as a reasonably mild restriction on Σ is imposed, both rules are no longer PCC (Theorem 4.8). With two individuals and a similar restriction on Σ, all well-known two-person SCRs of the literature, namely, fallback bargaining, Pareto and veto rules, short listing and veto rank, fail to pick ex-post compromises (Theorem 5.1).

The exclusion of the equal-loss principle by almost all SCRs of the literature leads to ask whether the principle is uninteresting in a discrete social choice context. This seems to be the case for voting situations where the number of voters exceeds the number of candidates and usually every candidate is ranked last by at least one voter. In these cases, the main concern is about the support of alternatives rather than equality. On the other hand, two-person collective choice problems are typically interpreted as arbitration or bargaining situations where mutual consent is a critical element in reaching a solution. Thus, the equal-loss principle appears to be valid for two-person collective choice problems and our analysis raises the question of designing new discrete arbitration rules compatible with the equal-loss principle.

We also want to mention that different notions of compromise can be conceived. [START_REF] Börgers | Undominated strategies and coordination in normalform games[END_REF] defines as compromises all Pareto-optimal alternatives that are not the top choice of any individual. In this context a compromise does not always exist. This is also a possible approach to adopt and we thank an anonymous reviewer for this remark.

We close by noting, as one anonymous reviewer to whom we are grateful remarked, that the tension between equity and efficiency is not new in economics. In our paper we try to cast this tension in a context where it does not seem to have been considered yet. We certainly hope that this is only the beginning of a discussion that may lead to further progress in the future. In particular, viewing a compromise through the equal loss principle can be especially interesting in richer informational settings with a status-quo point, cardinal individual preferences or a continuum of alternatives.

Example 1. 1

 1 Let N be a set of n ≥ 3 voters and A a set of alternatives. L(A) represents the set of linear orders over A. Consider the following preference profile P ∈ L(A) N , 1 c b a n -1 a b c , which represents one individual who prefers c to b, b to a, hence c to a; and n-1 individuals who prefer a to b, b to c, hence a to c.

Example 1. 2

 2 Consider the following preference profile with n = 100: 49 c b a 51 a b c .

Theorem 4 . 5

 45 Let n ≥ 3 and m ≥ 3. The BK compromise f n satisfies PC.

Theorem 4 . 6

 46 Let n ≥ 3 and m ≥ 3. The BK compromise f n fails ECC.

This objection to the compromise nomenclature was raised by Jean-François Laslier during a CNRS workshop on compromising hosted by Istanbul Bilgi University in

Acknowledgements This paper is a part of the 'Polarization viewed from a social choice perspective' (POSOP) research project that is carried on under the RDI program funded by Istanbul Bilgi University. We would like to thank POSOP for the support. We also thank Jean-François Laslier who provided the inspiration and the basis for this article. Last but not least, we thank the associate editor and three anonymous reviewers for their comments and valuable suggestions.