Steven J

Walter V

Loys B

Genetic parameters for litter size in sheep: natural versus hormone-induced oestrus

The litter size in Suffolk and Texel-sheep was analysed using REML and Bayesian methods. Litters born after hormonal induced oestrus and after natural oestrus were treated as different traits in order to estimate the genetic correlation between the traits. Explanatory variables were the age of the ewe at lambing, period of lambing, a year*flock-effect, a permanent environmental effect associated with the ewe, and the additive genetic effect. The heritability estimates for litter size ranged from 0.06 to 0.13 using REML in bi-variate linear models. Transformation of the estimates to the underlying scale resulted in heritability estimates from 0.12 to 0.17. Posterior means of the heritability of litter size in the Bayesian approach with bivariate threshold models varied from 0.05 to 0.18. REML estimates of the genetic correlations between the two types of litter size ranged from 0.57 to 0.64 in the Suffolk and from 0.75 to 0.81 in the Texel. The posterior means of the genetic correlation (Bayesian analysis) were 0.40 and 0.44 for the Suffolk and 0.56 and 0.75 for the Texel in the sire and animal model respectively. A bivariate threshold model seems appropriate for the genetic evaluation of prolificacy in the breeds concerned.

sheep / litter size / oestrus induction / heritability / genetic correlation

INTRODUCTION

Litter size (LS) is economically the most important trait in lamb meat production [START_REF] Olesen | Application of a multipletrait animal model for genetic evaluation of maternal and lamb traits in Norwegian sheep[END_REF] but it also has an important indirect effect on the improvement of other traits. A higher litter size allows more selection pressure on other economically important traits [START_REF] Shaat | Genetic trends for lamb weights in flocks of Egyptian Rahmani and Ossimi sheep[END_REF]. Because the heritability of LS is usually low, a selection on phenotype will be quite ineffective in improving litter size. The use of estimated breeding values, using BLUP and including information from relatives will substantially accelerate genetic progress. For this reason, the focus of the breeding programme in Belgian meat sheep is on improving the number of lambs born and appropriate genetic parameters are needed for the breeding value estimation procedure. In Belgian pedigree flocks of Suffolk (S) and Texel (T), the practice of hormonal induction of oestrus, followed by natural mating or artificial insemination (AI), is relatively common. In the period under study (1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002), 10% (S) and 24% (T) of all litters were born after hormonal treatment. There is no indication that these proportions have changed since. The use of hormones (such as Pregnant Mare Serum Gonadotrophin, PMSG) in sheep is known to cause an additional variation in litter size [START_REF] Bodin | Variability of litter size in French sheep following natural or induced ovulation[END_REF][START_REF] Calus | Mogelijkheden van intensieve schapenhouderij. Possibilities of intensive sheep breeding[END_REF]. The effect of hormone administration varied with the level of prolificacy of the breed, the seasonal state of the ewe (anoestrus vs. oestrus) and the dosage. Consequently, the question was raised how litter size after natural oestrus (LSN) and litter size after induced oestrus (LSI) could be combined in a genetic evaluation of natural prolificacy. The average flock size of pedigree sheep breeders in Belgium is small, and discarding the litters after hormonal oestrus induction would lead to a significant loss of information. Moreover, the practice of AI would have been out of the picture because the litters resulting from AI, which is usually done after induced oestrus, would not have been processed in the genetic evaluation for litter size.

Genetic parameters for LSI are scarce in the literature as compared to estimates for LSN. In the Lacaune ewe lambs, the heritability of LSI was 0.05 and 0.06 in 2 data sets and the genetic correlation between the two types of litter size was 0.39 [START_REF] Bodin | Estimation of genetic parameters of the Lacaune ewe lamb litter size after fecundation on natural and induced oestrus[END_REF]. Other references on the heritability of litter size after induced oestrus and on the genetic correlation with natural oestrus in livestock were not found. With a correlation of less than unity and a difference in variance, it might not be optimal to treat "type of oestrus" as a fixed effect of LS.

Litter size in sheep, defined as the total number of lambs born per lambing is expressed in discrete numbers (1, 2, 3, 4 and 5). In many studies, LS is analysed by a linear model and variance components are obtained by REMLmethods. Arguments are that (1) non-linear models have no big advantages in goodness of fit or predictive ability as compared to linear models and (2) more computing time is required in non-linear models, which might be prohibitive for routine calculations [START_REF] Hagger | Genetic and environmental influences on size of first litter in sheep, estimated by the REML method[END_REF]. Different non-linear models for litter size have been proposed, mostly sire models [START_REF] Konstantinov | Evaluation of Dormer sires for litter size and lamb mortality using a threshold model[END_REF][START_REF] Olesen | A comparison of normal and non-normal mixed models for number of lambs born in Norwegian sheep[END_REF] but also animal models [START_REF] Matos | Genetic analysis of discrete reproductive traits in sheep using linear and nonlinear models: I. Estimation of genetic parameters[END_REF]. The results indicate that non-linear models are able to explain a larger proportion of the variation and increase the accuracy of prediction as compared to linear models. Especially for traits with low heritability and low incidence of some categories, a non-linear model becomes more advantageous compared to a linear model [START_REF] Cameron | Selection indices and prediction of genetic merit in animal breeding[END_REF][START_REF] Olesen | A comparison of normal and non-normal mixed models for number of lambs born in Norwegian sheep[END_REF]. However, the analytical representation of non-linear models becomes difficult. A Bayesian analysis of an ordered, categorical trait is feasible using Markov chain Monte Carlo-methods (MCMC). MCMC methods allow for inferences to be made about the joint or marginal distribution of fixed effects, random effects and variance components. The Gibbs sampler, as an implementation of MCMC, has demonstrated its usefulness for inference on genetic parameters for litter size in sheep [START_REF] Altaribba | Bayesian inference of variance components for litter size in Rasa Aragonesa sheep[END_REF][START_REF] Yazdi | Bayesian analysis of birth weight and litter size in Baluchi sheep using Gibbs sampling[END_REF] and on twinning and ovulation rate in cattle [START_REF] Van Tassell | Bayesian analysis of Twinning and Ovulation rates using a multiple-trait threshold model and Gibbs sampling[END_REF]. However, simulation studies [START_REF] Hoeschele | Estimation of variance components of threshold characters by marginal posterior modes and means via Gibbs sampling[END_REF][START_REF] Luo | Bayesian inference for categorical traits with an application to variance component estimation[END_REF] with threshold animal models have shown convergence problems with the Gibbs sampler. This problem is related to the extreme category problem (ECP), which occurs when all observations within a level of an effect are in the same category. It turned out that sire threshold models are more reliable than animal threshold models [START_REF] Hoeschele | Estimation of variance components of threshold characters by marginal posterior modes and means via Gibbs sampling[END_REF][START_REF] Luo | Bayesian inference for categorical traits with an application to variance component estimation[END_REF].

The aim of this study was to study the genetic parameters of, and between, natural and induced litter size (LSN and LSI) in Suffolk and Texel sheep using different models and different approaches.

MATERIALS AND METHODS

Animals

Litter size records from Suffolk and Texel in Belgium were selected from the National Association of Meat Sheep Breeders database (NAMSB). This database holds pedigree information and performance data from 20 breeds of sheep. All information in the system was supplied by the breeders and included the identification of ewes and rams, the date of mating (only for a limited number of litters) and the type of oestrus (natural or hormonally induced). At birth, lambs were individually recorded and the identification number, the date of birth, a code for mortality and the dam identification number were registered. Litter size was defined as the "total number of lambs born per ewe at lambing". Barren ewes were not recorded in the database.

Prior to analysis, data checks were made to discover incomplete or aberrant data. Records with obvious typing errors or with missing information were removed (3% of the records across breeds). The mating date was not generally available for analysis and this made the removal of records of ewes, which returned on induced oestrus impossible. Also, no details were given on the hormonal treatment at oestrus induction. For instance, active components and dosage of hormones were unknown and could not be included in the analysis.

From the complete data sets, only a subset (for each breed) was used in the analysis. Ewes were retained in the data only when their sire had at least two daughters with records and when among his daughters, the records of both types of oestrus were represented. The latter constraint could be met by subsequent records from the same ewe or by records from different ewes. Also, a minimum of three records per year*flock (YF)-effect was imposed for each trait.

This editing resulted in datasets with 1955 and 17 199 ewes descending from 249 and 1767 Suffolk and Texel rams respectively. All known ancestors of the ewes were included in the pedigree, resulting in 4123 Suffolk and 29 877 Texel animals in the analyses.

Models

Preliminary least squares analyses were conducted [START_REF]SAS/STAT User's Manual, Version 6[END_REF] for the traits separately, to find significant explanatory variables. The records were attributed to 6 classes according to the age of the ewe at lambing (AL), viz. 1 to 5 and 6 years and older. Records were also classified in two periods of lambing (PL) only in the Texel breed. Period 1 was the 30-day period covering the lambing peak, whereas period 2 contained all other litters. The PL-effect is particularly important in breeds with extended mating periods [START_REF] Hagger | Genetic and environmental influences on size of first litter in sheep, estimated by the REML method[END_REF]. PL and AL were included as fixed effects in the model. A fixed year*flock-effect (YF) was included in the preliminary least squares analysis, together with a random sire effect.

For variance components estimation with REML and for Gibbs sampling, PL and AL-effects were included as fixed whereas the YF-effect was modelled as the random effect. Handling the YF as a random effect avoided convergence problems of the Gibbs sampler in threshold models [START_REF] Hoeschele | Estimation of variance components of threshold characters by marginal posterior modes and means via Gibbs sampling[END_REF][START_REF] Luo | Bayesian inference for categorical traits with an application to variance component estimation[END_REF]. Other random factors were the additive genetic effect of the animal in animal models (AM) or the sire-effect in sire models (SM) and a permanent environmental (PE) effect of the ewe. Random effects were assumed independent from each other, but a covariance was fitted between the same effect in the two traits. The PE-effect of the ewe affects the performance in consecutive litters. The significance of the PE-effect was evaluated in the REML analysis by fitting four models to the data; model I included the animal and PE effect for both traits (LSN and LSI). In models II to IV the PE effect for one or both traits was not fitted. The models can be considered as nested models and were compared using the Akaike Information Criterion (AIC). The AIC was computed from the likelihood value by properly accounting for the number of independently adjusted parameters and the rank of the matrix of fixed effects [START_REF] Wada | Selecting statistical models with information statistics[END_REF]. REML-estimates in the linear animal models were obtained using VCE4.2.5 [START_REF] Groeneveld | VCE4 User's Guide and Reference Manual[END_REF][START_REF] Neumaier | Restricted maximum likelihood estimation of covariances in sparse linear models[END_REF] by fitting bivariate AM and treating litter size as a normally distributed trait. Heritability estimates and the variance of the YF and PE-effect were calculated as ratios to the phenotypic variance. Heritability estimates, calculated on the observed scale can be used to approximate the heritability for the underlying variable by the following formula

h 2 U = h 2 Cat *           m j=1 i 2 j p j -          m j=1 i j p j          2                    m-1 j=1 z j i j+1 -i j          2
where h 2 U is the heritability of the underlying variable and h 2 Cat the heritability of the observed variable [START_REF] Gianola | Theory and analysis of threshold characters[END_REF]. The number of categories m equals 3, namely litters of singles, twins and triples and this implies 2 thresholds. Each category ( j) has an incidence of p j and a mean value of i j , z j is the height of the standard normal distribution corresponding with the proportion p j . Also the variance proportion of the PE-effect was rescaled in the same way. Genetic correlations and PE correlations estimated in the observed scale are direct estimators of the correlations in the underlying variable [START_REF] Gianola | Theory and analysis of threshold characters[END_REF].

Inference on the genetic parameters for litter size was also made by a Bayesian approach. LSN and LSI were analysed in bi-variate threshold AM and SM, assuming an underlying variable and two thresholds for each trait. Inference on the parameters of interest was based on the marginal posterior distributions. The posterior distributions were obtained by Markov chain Monte Carlo using Gibbs sampling (GS), implemented in MTGSAM [START_REF] Van Tassell | Multiple Trait Gibbs Sampler for Animal Models: Flexible Programs for Bayesian and Likelihood-Based (Co)Variance Component Inference[END_REF]. The technique of data augmentation was used to improve convergence [START_REF] Sorensen | Gibbs sampling in quantitative genetics[END_REF]. Single chains of 1 000 000 and 500 000 samples were generated for the Suffolk and Texel respectively. Prior distributions for the variances and covariances were inverted Wishart distributions with expected values of 0.10 for the heritability, 0.05 for the PE-effect and YF-effect and 0.50 for the correlations. The residual covariance was set to zero because the observations on LSN and LSI are mutually exclusive. Prior distributions for the thresholds were uniform distributions with starting values calculated from the data [START_REF] Van Tassell | Multiple Trait Gibbs Sampler for Animal Models: Flexible Programs for Bayesian and Likelihood-Based (Co)Variance Component Inference[END_REF].

Sequences of variance proportions and correlations were computed from the variances and covariances in the original chains. These sequences were further analysed with GIBANAL [START_REF] Van Kaam | Analyzing program for Markov chain Monte Carlo sequences[END_REF]. The burn-in period was determined for each parameter separately, as the number of samples before crossing the mean value of the chain for the second time. The burn-in samples were removed, leaving a "stationary phase" of the chain for further processing. Serial correlations were computed between samples and this was repeated at different lags to determine how many samples should be omitted between any two samples (thinning). The more correlated samples are, the more thinning is needed, resulting in smaller final sets. The final set was formed when the serial correlation between the samples was at the most 0.10. Location parameters of the posterior distribution (mean, mode, median and standard deviation) were computed from this subset of independent GS-samples.

RESULTS

Some details on the structure of the datasets are given in Table I. On average, 6.1 and 12.6 litters per year*flock were counted in S and T respectively. Only 2% (S) and 6% (T) of the YF-effects showed no variation in litter size because all litters were singles.

On average, sires had 7.9 (S) and 9.7 (T) daughters in the dataset with an average of 20.6 and 22.5 litters per sire. These numbers corresponded with an average lifetime production of 2.56 (S) and 2.31 (T) litters per ewe. In the Suffolk, the ewes had 1.99 litters after natural oestrus and 0.57 after induced oestrus, whereas in Texel the values were 1.59 and 0.73, respectively. From all active ewes, 27% (S) and 28% (T) had records for both types of LS. Most ewes exclusively had records on LSN (60% in S and 54% in T) and the remaining 13% and 19% of the ewes only had records on LSI.

The average litter size after hormonal treatment was +0.04 (S) and +0.10 (T) lamb/litter higher than litter size after natural oestrus (Tab. I). Also, litter size variance was significantly increased after hormonal induction by 16% in the Suffolk and 27% in the Texel. Hormonal treatment of the ewes resulted in less litters with singles and more litters of three lambs being born, compared to natural breeding circumstances. In the Texel, also more twins were born after oestrus induction.

The preliminary analysis of variance of LSN and LSI resulted in models explaining between 19 and 31% of the total variance. The univariate model for LSI in the Suffolk breed showed the lowest R 2 and a non-significant effect of the sire (Tab. II).

The comparison of models in the REML analysis, using the AIC (Tab. III), led to a different best choice in the two breeds. In the Suffolk, a model without a PE-effect for LSI was most likely. In the Texel, model I, with an additive genetic effect and a permanent environmental effect for both traits, had the lowest AIC.

The heritability estimates for litter size on the observed scale, obtained with REML, ranged from 0.06 to 0.13 depending on the breed and the model used. Somewhat higher heritability estimates were obtained for LSN when the permanent environmental effect was not included in the model. This effect was not observed in LSI. In Suffolk, the heritabilities of LSI and LSN were comparable whereas in Texel, LSN had a higher heritability than LSI. Standard errors of the estimates were of the order of 0.01 to 0.03 with higher values in the Suffolk breed and for LSI.

Rescaling of the estimates to the underlying scale inflates the values for the heritabilities to values in the range of 0.13 to 0.17. The fraction of PE variance (only modelled in the Texel) was between 0.03 and 0.05 in the observed scale and increased to 0.07 to 0.08 after rescaling. Repeatabilities of litter size are thus between 0.20 (LSI) and 0.25 (LSN) in the Texel. REML-estimates of the genetic correlation between litter size traits ranged from 0.57 to 0.64 in the Suffolk and from 0.75 to 0.81 in the Texel. Standard errors in the Suffolk were about 4-times larger than in the Texel. The unconstrained estimate for the PE correlation was 1.26 (S) and 0.72 (T). The result in S can theoretically not be called REML.

The results obtained with Gibbs sampling are presented in Tables IV andV and marginal posterior distributions are depicted in Figures 1 and2. The burnin periods varied from 5 to 928 samples, so most samples in the chain were kept for further processing. The final sample size for the different components varied between 321 and 20 000. After thinning, serial correlations between samples within each chain were at the most 0.10. The values in the final samples could therefore be considered sufficiently independent.

In the Suffolk, AM and SM yielded similar posterior distributions. Distributions were fairly symmetric with the mean, mode and median close to each other and they appeared close to normal. The posterior mean of the heritability of litter size was 0.10 and 0.08 for LSN and 0.11 and 0.10 for LSI in AM and SM respectively. The PE-effect accounted for about 0.06 to 0.09 of the variance and the YF-effect for 0.05 to 0.06. In the Texel, posterior distributions appeared close to normal. The differences were noted between AM and SM for the location parameters. In the AM, the posterior mean of the heritability of litter size was 0.18 for LSN and 0.11 for LSI. In the SM, corresponding values for the heritability were 0.07 and 0.05. The posterior means for the proportion of variance of the PE-effect were 0.06 and 0.16 for LSN and 0.05 and 0.10 for LSI in AM and SM respectively. For the YF-effects, the means of the posterior distribution ranged from 0.05 to 0.08 and there was no difference between AM and SM.

Table IV. Marginal posterior location parameters (mean, mode, median and standard deviation) and final sample characteristics (burn in, sample size (N) and serial correlation (corr)) for the heritability (h 2 ), variance fraction of the permanent environmental effect (PE 2 ), variance fraction of the year*flock effect (YF 2 ) and genetic (r g ), permanent environmental correlation (r pe ), year*flock correlation (r yf ) for litter size after natural oestrus (LSN) and litter size after induced oestrus (LSI) in the Suffolk. The posterior means of the genetic correlation between LSN and LSI, were 0.44 (AM) and 0.40 (SM) in the Suffolk and 0.72 (AM) and 0.56 (SM) in the Texel (Figs. 1 and2). The mean of the distribution of the correlation between PE-effects was 0.41 (AM) and 0.50 (SM) in the Suffolk and 0.44 (AM) Table V. Marginal posterior location parameters (mean, mode, median and standard deviation) and final sample characteristics (burn in, sample size (N) and serial correlation (corr)) for the heritability (h 2 ), variance fraction of the permanent environmental effect (PE 2 ), variance fraction of the year*flock effect (YF 2 ) and genetic (r g ), permanent environmental correlation (r pe ), year*flock correlation (r yf ) for litter size after natural oestrus (LSN) and litter size after induced oestrus (LSI) in the Texel. Posterior standard deviations on the variance fractions (i.e. heritability, proportion of the PE-effect, proportion of the YF effect) were between 0.01 and 0.03. For the correlations, the order of magnitude of the standard deviations was between 0.05 and 0.17

DISCUSSION

Data issues

Both the quality of field data and the distribution of effects over animals cannot be controlled. Imbalance in the data and/or disconnectedness may cause biased estimates. In order to minimise the risk of bias, data were edited and "sparse cells" were removed from the dataset. From Figures 3 and4, the cumulative % of sires appearing in 1, 2, 3, to n different YF-effects, can be evaluated. The percentage of sires appearing in only one YF-effect is 10.2% for LSN and 37% for LSI in the Suffolk. In the Texel breed, these percentages are respectively 12.4% and 30%. This indicates that most sires are represented in at least 2 YF-effects. In the Suffolk, one sire had observations on LSN in 44 different YF-effects. In the Texel, the highest count of YF-effects for one sire was 166. The distribution of sires over YF-effects could be interpreted as a minimal evaluation of the genetic connectedness. Also the genetic relationships between ewes and between ewes and rams will contribute to the genetic links.

Modelling issues

In the preliminary least squares analysis, about 20 to 30% of the variation in litter size could be attributed to different explanatory variables. The lowest coefficient of variation was obtained in the Suffolk and for the LSI trait. This set was the most limited in terms of the number of records and this might explain the poorer modelling. There is probably scope for improvement of the model by including other explanatory variables. The preceding litter size (at birth or at weaning), the weaning to mating interval or a variable describing the body condition of the ewe could possibly explain non-genetic variation. However, this information is currently not recorded by the NAMSB and an extension of the current data collection system in the flock book will be necessary. This extension is only advisable if the extra information can reduce the residual variance in the model which would result in a higher heritability and more genetic improvement.

The increase of the mean and the variance of litter size due to hormonal treatment is also noted in most of the French sheep breeds [START_REF] Bodin | Variability of litter size in French sheep following natural or induced ovulation[END_REF] and in Rasa Aragonesa sheep [START_REF] Altaribba | Bayesian inference of variance components for litter size in Rasa Aragonesa sheep[END_REF]. An adverse effect of hormonal treatment on LS has been found with a reduction of average LS only in prolific breeds, such as the Romanov, Bleu du Maine, the Flemish breed and in Flemish × Texelcrosses [START_REF] Bodin | Variability of litter size in French sheep following natural or induced ovulation[END_REF][START_REF] Calus | Mogelijkheden van intensieve schapenhouderij. Possibilities of intensive sheep breeding[END_REF]. However, the effects of oestrus induction on LS found in our data might be slightly biased due to the following reasons: firstly, the ewes were not randomly assigned to oestrus induction. In Suffolk and Texel, the ewes treated with hormones before AI or planned matings were probably "preselected" on general appearance and this could have been a disturbing effect. Secondly, the ewes returning on induced oestrus cannot be distinguished (in our data) from ewes mated after normal, i.e. natural oestrus. The ewes returning on induced oestrus were included as "natural" but an impact of the hormonal treatment on the next fertilising cycle cannot be excluded. We think, however, that the disturbing effects are limited and that the data were sufficiently good for genetic parameter estimation.

The separation of the random PE-effect from the residual term is most critical for LSI because on average, only 1.45 (S) and 1.57 (T) records per ewe for LSI were available (i.e. 1124 records by 774 ewes in S and 12 550 records by 7972 ewes in T). For LSN, 2.28 (S) and 1.95 (T) litters per ewe were available (3885 records by 1701 ewes in S and 27 265 records by 14 009 ewes in T) which makes the separation of the PE-effect from the residual term feasible. With REML-techniques, almost no variance was attributed to the PE-effect, especially for LSI in S. Although S and T do not differ so much in the number of records per trait, per ewe, differences in the amount of genetic covariance between animals and in pedigree depth may also have affected the separation of variance in different components. In S, less daughters per sire were present than in T (on average 7.9 versus 9.7 ewes/sire) and also pedigree depth was lower in S as compared to T (3.5 versus 6 generations of known ancestors on average). The dataset properties mentioned above might explain why the "best" REML-model was different in S and T.

In the initial Markov chains, burn-in periods were of variable length but considerably shorter than in other studies where up to 5000 and 10 000 samples were discarded [START_REF] Altaribba | Bayesian inference of variance components for litter size in Rasa Aragonesa sheep[END_REF][START_REF] Yazdi | Bayesian analysis of birth weight and litter size in Baluchi sheep using Gibbs sampling[END_REF]. A flexible choice of the burn-in period is indicated to save samples. Initial serial correlations between samples were high, indicating that samples were strongly autocorrelated. The usage of data augmentation is known to increase auto-correlation in the chain [START_REF] Sorensen | Gibbs sampling in quantitative genetics[END_REF] so that more wastage of samples occurs. In general, final sample sizes were smaller in the runs involving animal models compared to sire models. Another difference was seen for posterior distributions involving the PE-effect. In the sire models, the smallest samples were obtained for chains involving one or both PE-effects. This might indicate a higher autocorrelation of the Gibbs samples for the permanent environmental effects than for the genetic or the year*flock effect. In the animal models, chains involving genetic effects were more autocorrelated and resulted in smaller final samples for the heritabilities and genetic correlations.

Simulation studies with threshold sire and animal models [START_REF] Luo | Bayesian inference for categorical traits with an application to variance component estimation[END_REF] indicated convergence problems in the animal models and this was related to the extreme category problem. Convergence problems were not encountered in our analyses, possibly as the result of treating the YF-effect as random [START_REF] Hoeschele | Estimation of variance components of threshold characters by marginal posterior modes and means via Gibbs sampling[END_REF][START_REF] Luo | Bayesian inference for categorical traits with an application to variance component estimation[END_REF]. Also the proportion of YF-effects with a zero variance was restricted in our data (2% in S and 6% in T) so that the impact of extreme categories was limited.

Genetic parameters

A mean value of 0.07 for h 2 (LSN) in sheep can be computed from REMLestimates published before 1995 [START_REF] Fogarty | Genetic parameters for live weight, fat and muscle measurements, wool production and reproduction in sheep: a review[END_REF]. Recent REML-estimates for litter size in sheep ranged from 0.05 to 0.26 [START_REF] Bromley | Genetic correlations for litter weight weaned with growth, prolificacy, and wool traits in Columbia, Polypay, Rambouillet and Targhee sheep[END_REF][START_REF] Fadili | Estimation of additive and non-additive genetic parameters for reproduction, growth and survival traits in crosses between the Moroccan D'man and Timadhite sheep breed[END_REF][START_REF] Hagger | Genetic and environmental influences on size of first litter in sheep, estimated by the REML method[END_REF][START_REF] Lee | Parameter estimates for number of lambs born at different ages and for 18-month body weight of Rambouillet sheep[END_REF][START_REF] Matika | Genetic parameters in Sabi sheep[END_REF][START_REF] Matos | Genetic analysis of discrete reproductive traits in sheep using linear and nonlinear models: I. Estimation of genetic parameters[END_REF]. Our REML heritability estimates for LSN are in agreement with these studies. Heritability estimates for LSI in Lacaune ewe lambs were 0.05 and 0.06 in two datasets with 1328 and 2201 records, respectively. Only the last estimate was significantly different from zero [START_REF] Bodin | Estimation of genetic parameters of the Lacaune ewe lamb litter size after fecundation on natural and induced oestrus[END_REF] and this value corresponds well with our REML-estimates of the heritability of LSI in S and T.

In Suffolk, rescaled REML-estimates are higher than the posterior mean obtained with GS whereas in Texel there is good agreement between the rescaled REML-estimates and posterior means or modes obtained with GS.

Posterior heritabilities obtained with the Gibbs sampler in AM for Suffolk were close to the posterior mean of 0.08 reported in Rasa Aragonesa sheep [START_REF] Altaribba | Bayesian inference of variance components for litter size in Rasa Aragonesa sheep[END_REF]. The corresponding Texel results were above this value. GS-results in SM in this study were lower than the posterior heritabilities for litter size (0.26) in Sabi sheep [START_REF] Matika | Genetic parameters in Sabi sheep[END_REF] and in Baluchi sheep evaluated at three ages (0.34 to 0.43) [START_REF] Yazdi | Bayesian analysis of birth weight and litter size in Baluchi sheep using Gibbs sampling[END_REF]. In the latter study, no PE-effect was fitted which is different from our model.

Variance proportions of the PE-effect of litter size are usually small and in the range of 0.01 to 0.03 [START_REF] Bromley | Genetic correlations for litter weight weaned with growth, prolificacy, and wool traits in Columbia, Polypay, Rambouillet and Targhee sheep[END_REF][START_REF] Matos | Genetic analysis of discrete reproductive traits in sheep using linear and nonlinear models: I. Estimation of genetic parameters[END_REF]. Also in our REML analyses, very little variance was attributed to the permanent environmental effect whereas with GS, the distribution of the PE-proportions had a mean between 0.05 and 0.16. Also, more variance was attributed to PE-effects in SM as compared to AM and this was seen in both breeds.

The genetic correlation estimates between LSN and LSI obtained with REML (0.57 to 0.81), were above the value of 0.39 reported in Lacaune ewe lambs using a sire model [START_REF] Bodin | Estimation of genetic parameters of the Lacaune ewe lamb litter size after fecundation on natural and induced oestrus[END_REF]. Genetic correlations obtained with SM and GS corresponded well with the results obtained by Bodin [START_REF] Bodin | Estimation of genetic parameters of the Lacaune ewe lamb litter size after fecundation on natural and induced oestrus[END_REF].

The differences in genetic correlation estimates found in this study were partly due to the use of an animal versus a sire model. But, breed differences might also exist. In prolific breeds, hormonal induction is known to depress average LS while increasing its variance and this might lead to differences in genetic correlation estimates between breeds. The moderate to high genetic correlation, but different from unity, implies that the two types of litter size should be considered as different traits. Possibly, different sets of genes are in play. Another reason for the genetic correlation being different from one might be that insufficient information is provided by the data.

The REML PE-correlation estimate has a large standard error. From the GS results, it is clear that the PE-correlation was difficult to assess accurately.

Final sample size of the PE-effect was smaller than for the YF or genetic effect, indicating a slower mixing of the chain. Slower mixing of the chain is caused by less information coming from the data. In the case of the covariance between PE-effects, information has to come from subsequent observations of the two traits in the same animal. Only 26% (S) and 28% (T) of the ewes have records for both types of oestrus, which means that the within-animal covariance is difficult to estimate and larger standard errors on the covariances and correlations can be expected.

IMPLICATIONS

The magnitude of the heritabilities (0.08 to 0.18 for LSN and 0.05 to 0.11 for LSI) indicate that genetic improvement of litter size is achievable. Usage of a threshold model will have an advantage over the use of a linear model because the heritabilities in the threshold models were higher than in the linear model. Accounting for the categorical nature of litter size is thus recommended.

A moderately high and positive genetic correlation between natural and induced litter size was found in both breeds and values ranged from 0.40 to 0.81. Selection for higher (natural) litter size can also be based on observations of LSI, but progress will be slower. Although natural oestrus is more "informative" about litter size than induced oestrus, the latter technique is needed to apply AI or planned matings. These reproductive techniques allow the spread of the best genes in the population and provide genetic links in the data. Not discarding the litters after oestrus induction, but including them as a correlated trait, makes the best use of the information. This can be achieved by implementing a bivariate threshold model for litter size in the Belgian Suffolk and Texel.

  .

Figure 1 .

 1 Figure 1. Marginal posterior distribution of the genetic correlation between natural and induced litter size in Suffolk sheep, animal model (a) and sire model (b).

Figure 2 .

 2 Figure 2. Marginal posterior distribution of the genetic correlation between natural and induced litter size in Texel sheep animal model (a) and sire model (b).

Figure 3 .

 3 Figure 3. Cumulative percentage of sires for the number of year*flocks per sire (Suffolk).

Figure 4 .

 4 Figure 4. Cumulative percentage of sires for the number of year*flocks per sire (Texel).

Table I .

 I Description of datasets used in the analysis of litter size in the Suffolk and Texel.

		Suffolk	Texel
	N • of records	5009	39815
	n • of records after natural oestrus	3885	27265
	n • of records after induced oestrus	1124	12550
	Total n • of animals in pedigree of animal model	4123	29877
	N • of sires of ewes	249	1767
	N • of ewes	1955	17203
	with record(s) only after natural oestrus	1181	9231
	with record(s) only after induced oestrus	254	3194
	with records after natural + induced oestrus	520	4778
	Year*flock effects	817	3170
	with zero variance for litter size	16	195
	Average number of lambs born per ewe lambing
	natural oestrus	1.65	1.48
	induced oestrus	1.69	1.58
	Variance of lambs born per ewe lambing
	natural oestrus	0.32	0.30
	induced oestrus	0.37	0.38
	Distribution of LSN (%)
	Single	39.5	54.1
	Twin	55.9	43.5
	Triplet	4.6	2.4
	Distribution of LSI (%)
	Single	38.7	48.3
	Twin	53.7	44.9
	Triplet	7.7	6.8

Table II .

 II Least squares analysis of litter size after natural oestrus (LSN) and litter size after induced oestrus (LSI) in Suffolk and Texel sheep. Number of records, number of levels in each effect and coefficient of determination of the model (R 2 ). Probability values of the F-test are given in brackets.

		Number of Age of ewe Year*Flock records (AL) (YF)	Period of lambing (PL)	Sire	R 2
	Texel
	LSN	27265	6	2588	2	1767 0.28
			(0.001)	(0.001)	(0.001)	(0.001)
	LSI	12550	6	1276	2	1767 0.31
			(0.04)	(0.0001)	(0.0001) (0.0001)
	Suffolk
	LSN	3885	6	700	-	252 0.30
			(0.009)	(0.004)		(0.08)
	LSI	1124	6	162	-	252 0.19
			(0.10)	(0.001)		(0.30)

Table III .

 III REML estimates for the heritability (h 2 ), fraction of the permanent environmental effect (PE 2 ) and genetic (r g ) and environmental correlation (r pe ) for litter size after natural oestrus (LSN) and litter size after induced oestrus (LSI) in the Suffolk and Texel. For the different linear models the Akaike Information Criterion (AIC) is given, standard errors are given between brackets.

	Breed / parameter		Model
	Suffolk	I	II	III	IV	Rescaled*
	LSN
	h 2	0.06 (0.02) 0.06 (0.01) 0.07 (0.02) 0.07 (0.01)	0.12
	PE 2	0.02 (0.02) 0.01 (0.00)	-	-	0.03
	LSI
	h 2	0.07 (0.03) 0.07 (0.03) 0.07 (0.03) 0.07 (0.03)	0.16
	PE 2	0.01 (0.02)	-	0.00 (0.02)	-	-
	(LSN,LSI)
	r g	0.57 (0.31) 0.64 (0.28) 0.59 (0.27) 0.59 (0.26)	-
	r pe	1.26 (0.00)	-	-	-	-
	AIC	7758	7755	7756	7755
	Texel
	LSN
	h 2	0.09 (0.01) 0.10(0.01) 0.13 (0.01) 0.13 (0.01)	0.17
	PE 2	0.05 (0.01) 0.05 (0.01)	-	-	0.08
	LSI
	h 2	0.06 (0.01) 0.08 (0.01) 0.07 (0.01) 0.08 (0.01)	0.13
	PE 2	0.03 (0.01)	-	0.03 (0.01)	-	0.07
	(LSN,LSI)
	r g	0.75 (0.10) 0.80 (0.07) 0.81 (0.08) 0.75 (0.06)	-
	r pe	0.72 (0.23)	-	-	-	-
	AIC	57792	57797	57810	57811	-

* Rescaled parameters are presented for model I in Texel and model II in Suffolk.

ACKNOWLEDGEMENTS

The Federal Ministry of Agriculture, Medium and Small Enterprises, DG6 -Research and Development, provided funding for conducting this study. The National Association of Meat Sheep Breeders is acknowledged for supplying the data.


Guillaume Hollard
email: guillaume.hollard@univ-paris1.fr

P De Donder

B Grofman

M Le Breton

A Kirman

J F Laslier

S Martin

H Moulin

J Roemer

K Van Der Straeten

An alternative approach to valence advantage in spatial competition. *

Keywords: Spatial voting, Valence advantage, J.E.L. Classification: D72, D71

come

An alternative approach of valence advantage in spatial competition

1 Introduction.

The Downsian model constitutes a benchmark for the analysis of electoral competition. It predicts a close race between the two candidates and minimal differentiation in chosen platforms at equilibrium (Downs, 1957). At equilibrium, both parties or candidates choose the median platform. However, the empirical analysis of elections shows that differentiation exists1 . To explain this fact, a first line of research introduces uncertainty in the position of the median voter (i.e. candidates have imperfect information on the voters' positions) as well as partisan preferences (i.e. candidates care not only about winning but also about defending a particular program) [START_REF] Ashworth | Valence Competition and Platform Divergence[END_REF] . Adding both uncertainty and partisan preferences for the candidates allows us to predict differentiation at equilibrium. An alternative line of research examines the impact of removing the hypothesis of symmetric candidates. Candidates can be differently evaluated, even if they adopt the same policy positions. Voters then evaluate the candidates not only on the "policy issues", but also according to characteristics that are unanimously positively or negatively evaluated. For example, all agents prefer candidates who are not corrupt , have more charisma, greater intelligence, superior character, or better handshaking skills. Stokes (1963) was the first to underline the importance of those non spatial aspects. He called them "valence issues" [START_REF] Ansolabehere | Valence politics and equilibrium in spatial election models[END_REF] . It is rather intuitive that the candidate with a valence disadvantage will lose the election if he chooses a platform identical to his opponent's. He thus has a clear incentive to choose a A common feature to these models is to assume that the utility of a voter increases by a fixed amount when he considers the position of the favored candidate i.e. valence advantage results in adding a constant amount of utility. Such an additive valence-advantage means that the utility is separable, i.e. that valence-issues have no link with the policy issues. This hypothesis is clearly quite restrictive, since we can validly think that candidates' qualities can interact with the evaluation by the agents of their policy positions. The degree of corruption of a candidate matters more if he proposes a high level of public spending. This justifies a multiplicative form for the valence-advantage. This paper investigates the consequences of moving from an additive form to a multiplicative form. The main result proves that the less favored candidate is supported by voters whose preferred platform is in an unbounded set in the additive case, but a bounded set in the multiplicative one. It is important to note that our result holds for any dimension of the policy space (while it is often assumed that the policy space is of dimension 1), and any distance (while much of the literature is restricted to the Euclidean distance).

A direct and striking consequence of this result for dimension 1 is that the electorates of both sides on the equilibrium spectrum support the most favored candidate. We get an "ends against the middle" split of the population. This is in sharp contrast with the additive case which predicts a split into two intervals. This result holds for the higher dimensions. For example in dimension 2, voters supporting the less favored candidates are located within a ball. This offers a new explanation for situations in which a candidate, or alternative, is supported by extreme voters from both the left and the right.

As in the additive case, the favored candidate is guaranteed to win the election if his valence advantage is above a threshold. No equilibrium in pure strategy exists if the valence advantage is below that threshold.

The paper is organized as follows : Section 2 presents the model and main assumptions and Section 3 presents the main results. We conclude in Section 4.

The model

A voter i is identified with his preferred platform, or bliss point, a i in the policy space R n . A candidate j chooses a policy platform x j ∈ R n . In the basic spatial model of electoral competition, a voter i votes for candidate j whose announced platform x j is the closest to a i . If candidate j is elected, the utility function of voter i is:

u(x j , a i ) = -x j -a i . (1)
When valence advantages are introduced, each candidate has an intrinsic quality independent of the chosen platform. An additional parameter θ j > 0 thus accounts for candidate j valence. The higher θ j , the more appreciated candidate j.

When the valence advantage is modeled by adding a constant, the utility is written as the following:

u ad (θ j , x j , a i ) = -x j -a i + θ j (2)
As explained, we introduce a multiplicative valence advantage. The utility function is now written:

u(θ j , x j , a i ) = - 1 θ j x j -a i (3)
Note that x → x is any norm on R n . Most of our results don't depend on the choice of the norm. We define k = θ 1 θ 2 and assume, without loss of generality, that 0 < θ 2 < θ 1 , i.e. k > 1. Then candidate 1 gets a clear advantage.

Once candidates have announced their platforms x 1 and x 2 , the electorate splits

in three parts Ω 1 (x 1 , x 2 , k), Ω 2 (x 1 , x 2 , k) and I(x 1 , x 2 , k)
. Ω i is the set of voters who support candidate i and I is the set of voters who are indifferent between the two candidates. Formally, we have:

Ω 1 (x 1 , x 2 , k) = a ∈ R n ; 1 θ 1 x 1 -a < 1 θ 2 x 2 -a (4) = {a ∈ R n ; x 1 -a < k x 2 -a } Ω 2 (x 1 , x 2 , k) = a ∈ R n ; 1 θ 2 x 2 -a < 1 θ 1 x 1 -a (5)
= {a ∈ R n ; k x 2 -a < x 1 -a } I(x 1 , x 2 , k) = {a ∈ R n ; k x 2 -a = x 1 -a }
Voters, i.e. bliss points, are distributed on R n , according to a density distribution function f . The function f is assumed to be bounded on R n , and we denote by X the support of f , i.e. f (x) = 0 for x / ∈ X. The proportion of voters who prefer candidate j to his opponent is then given by the function S j (x 1 , x 2 , k), where:

S j (x 1 , x 2 , k) = Ω j f (a)da. ( 6
)
Note that the set

I = I(x 1 , x 2 , k) is of negligible importance, i.e. I f (a)da = 0.
In the additive case, we replace u(θ j , x j , a i ) by u ad (θ j , x j , a i ) = -x j -a i + θ j .

We set b = θ 1 -θ 2 > 0. We define the sets:

Ω ad 1 (x 1 , x 2 , b) = {a ∈ R n ; x 1 -a < b + x 2 -a } and Ω ad 2 (x 1 , x 2 , b) = {a ∈ R n ; x 1 -a > b + x 2 -a } Then, S ad j (x 1 , x 2 , k) = Ω ad j f (a)da.
Since the candidates are only office-motivated, we can then assume that the utility of candidate j is formally given by:

v j (x 1 , x 2 ) =          0 if S j (x 1 , x 2 , k) < 1/2 1 2 if S j (x 1 , x 2 , k) = 1/2 1 if S j (x 1 , x 2 , k) > 1/2 (7)
The game of asymmetric electoral competition is played sequentially and its main elements are summarized by the couple (k, f ). The chronology of the game is the following: first the two candidates announce their platforms x 1 and x 2 ; secondly the electors vote for their preferred candidate. In the next section, we solve the game sequentially.

Main results

We first explore the electorate's split both in the additive and multiplicative cases, when the candidates have already announced their platforms. The first step of the game of asymmetric electoral competition, i.e. the existence of equilibria, is then analyzed in the second subsection.

The electorate's split in the additive and multiplicative cases

The multiplicative and the additive cases mainly differ in the split of the electorate between those who support candidate 1, the advantaged candidate, and those who support candidate 2. Technically, we explore the properties of the sets Ω 1 and Ω 2 . Our main proposition proves that a substantial difference exists for any norm and any dimension of the policy space (proposition 1). This result is particularly striking if we restrict to the Euclidean norm (proposition 2). We also provide an explicit comparison in the particular, but popular, case where the policy space is of dimension 1 (corollary 1).

Proposition 1 For every norm on R n , we have:

(i) The set Ω 2 (x 1 , x 2 , k) is a bounded open set in R n . It is non-empty iff x 1 = x 2 (ii) The set Ω ad 2 (x 1 , x 2 , b) is unbounded if b < x 2 -x 1 , and empty if b ≥ x 2 -x 1 . Proof. (i) If a ∈ Ω 2 (x 1 , x 2 , k), then k x 2 -a < x 1 -a = x 1 -x 2 + x 2 -a ≤ x 1 -x 2 + x 2 -a thus (k -1) x 2 -a < x 1 -x 2 i.e. If a ∈ Ω 2 (x 1 , x 2 , k), then x 2 -a < 1 k -1 x 1 -x 2 (8)
Thus Ω 2 (x 1 , x 2 , k) is a bounded set. It is clearly an open set, according to (5).

(ii) If x 1 = x 2 , then Ω 2 (x 1 , x 2 , b
) is clearly empty. We assume now that x 1 = x 2 .

We set u = x 2 -x 1 x 2 -x 1 . -For a = x 2 + λu, with λ > 0, we have x 2 -a = λ since u = 1. Moreover:

x 1 -a = x 2 + λu -x 1 = (x 2 -x 1 ) 1 + λ x 2 -x 1 = x 2 -x 1 1 + λ x 2 -x 1 = x 2 -x 1 + λ x 1 -a -x 2 -a -b = x 2 -x 1 +λ-λ-b = x 2 -x 1 -b > 0 if b < x 2 -x 1 Thus x 1 -a > x 2 -a + b if a = x 2 + λu, for every λ > 0, when b < x 2 -x 1 . It shows that Ω ad 2 (x 1 , x 2 , b) is unbounded if b < x 2 -x 1 . -When b ≥ x 2 -x 1 , we have x 1 -a = x 1 -x 2 + x 2 -a ≤ x 1 -x 2 + x 2 -a ≤ b + x 2 -a thus Ω ad 2 (x 1 , x 2 , b) is empty.
Voters supporting the disadvantaged party have their bliss points included in a bounded set in the multiplicative case. Our next proposition shows how the set Ω 2 (x 1 , x 2 , k) can be constructed, in the case of an Euclidean norm.

Proposition 2 With the Euclidean norm, the set

Ω 2 (x 1 , x 2 , k) is a hyperball in R n centered at c(x 1 , x 2 , k) = k 2 k 2 -1 x 2 -1 k 2 -1 x 1 of radius r(x 1 , x 2 , k) = k k 2 -1 x 1 -x 2 . Let ∆ denote the line (x 1 , x 2 ). The set Ω 2 (x 1 , x 2 , k
) is a hyperball tangent to the cone of vertex x 1 , axis ∆, and angle 2α such that sin α = 1 k .

Proof See Appendix.

Figure 1:

Ω 2 (x 1 , x 2 , k)
The Downsian model as the limit of the asymmetric game It is natural to ask how the model behaves when parties tend toward symmetry, i.e. when k → 1 + .

We have Ω 2 = {a ∈ X; k x 2 -a < x 1 -a } -→ {a ∈ X; x 2 -a < x 1 -a }, as k → 1 + , i.e. the set Ω 2 tends to the half-space delimited by the median hypersurface4 between x 1 and x 2 as in the symmetrical model.

Corollary 1 In a one dimensional setting,

Ω 2 (x 1 , x 2 , k) =] kx 2 +x 1 k+1 ; kx 2 -x 1 k-1 [ if x 1 < x
We can observe that Ω 1 = R \ Ω2 is a non convex set. It is a situation of "ends against the middle". Candidate 1 gets support of voters located at both ends of the electoral spectrum (see Figure 2).

Analysis of the game of asymmetric spatial competition

Before analyzing the game, some further notations are required. We define the set X * 1 player 1's strategies that guarantee him or her, to win the election.

X * 1 = {x 1 ∈ X; ∀x 2 ∈ X, v 1 (x 1 , x 2 , k) = 1} (9) = x 1 ∈ X; ∀x 2 ∈ X, S 1 (x 1 , x 2 , k) > 1 2
We can define similarly X * ad 1 in the additive case.

Proposition 3 With the Euclidean norm, if X is a compact set, then X * 1 is an open, convex set.

Proof. See Appendix.

We now turn to the analysis of Nash equilibria: Proposition 4 For every norm on R n , in both multiplicative and additive cases, one of the following two assertions holds:

-The set X * 1 is not empty and for any

x 2 ∈ X, (x 1 , x 2 ) is a Nash equilibrium if and only if x 1 ∈ X * 1 . -The set X *
1 is empty and there exists no Nash equilibrium in pure strategy.

Proof. We have to show that (

x 1 , x 2 ) is a Nash equilibrium iff x 1 ∈ X * 1 . If (x 1 , x 2 ) is a Nash equilibrium, then v 1 (x 1 , x 2 ) ≥ v 1 (x 2 , x 2 ) = 1, thus v 1 (x 1 , x 2 ) = 1 Then 0 = v 2 (x 1 , x 2 ) ≥ v 2 (x 1 , x ′ 2 ) for all x ′ 2 ∈ X As v 1 = 1 -v 2 , it means that 1 = v 1 (x 1 , x 2 ) ≤ v 1 (x 1 , x ′ 2 ) for all x ′ 2 ∈ X i.e. v 1 (x 1 , x ′ 2 ) = 1 for all x ′ 2 ∈ X i.e. x 1 ∈ X * 1 . Conversely, if x 1 ∈ X * 1 and x 2 ∈ X, we have v 1 (x 1 , x 2 ) = 1 ≥ v 1 (x ′ 1 , x 2 ) for all x ′ 1 ∈ X v 2 (x 1 , x 2 ) = 0 ≥ v 2 (x 1 , x ′
2 ) for all x ′ 2 ∈ X which proves that (x 1 , x 2 ) is a Nash equilibrium.

Proposition 5 For every continuous distribution function f on the set X, there

exist positive numbers k * (f ), k * * (f ), with 1 ≤ k * (f ) ≤ k * * (f ) and k * (f ) < +∞ such that: (i) X * 1 (k) = ∅ if k < k * (f ) (ii) X * 1 (k) = ∅ and X * 1 (k) = X if k * (f ) < k < k * * (f ) (iii) X * 1 (k) = X if k > k * * (f ) . Moreover, k * * (f ) < +∞ if and only if X is bounded. Proof -Assume that 0 < k < k ′ . If a ∈ Ω 2 (x 1 , x 2 , k ′ ) then x 1 -a > k ′ x 2 -a ≥ k x 2 -a thus a ∈ Ω 2 (x 1 , x 2 , k) Hence if k < k ′ , then Ω 2 (x 1 , x 2 , k ′ ) ⊆ Ω 2 (x 1 , x 2 , k) Since X * 1 (k) = x 1 ∈ X; Ω 2 (x 1 ,x 2 ,k) f (a)da < 1 2 , ∀x 2 ∈ X then k < k ′ ⇒ X * 1 (k) ⊆ X * 1 (k ′ ) For 0 < k < 1, X * 1 (k) = ∅ since S 1 (x 1 , x 1 , k) = 0, ∀x 1 ∈ X.
1) Let us show that if X is bounded, then X * 1 (k) = X for k high enough. In R n all the norms are equivalent, thus there exist γ 1 > 0 and γ 2 > 0 such that

γ 1 x eucl ≤ x ≤ γ 2 x eucl for every x ∈ R n .
According to (8),

Ω 2 (x 1 ,x 2 ,k) f (a)da ≤ B(x 2 ; x 1 -x 2 k-1 ) f (a)da and x 2 -a < x 1 -x 2 k-1 ⇒ x 2 -a eucl < δ k-1 x 1 -x 2 eucl where δ = γ 2 γ 1 . Thus Ω 2 (x 1 ,x 2 ,k) f (a)da ≤ B(x 2 ; x 1 -x 2 k-1 ) f (a)da ≤ B eucl (x 2 ; δ k-1 x 1 -x 2 eucl ) f (a)da ≤ vol B eucl (x 2 ; δ k-1 x 1 -x 2 eucl ) . max K f ≤ const. δ k-1 x 1 -x 2 eucl n ≤ const 1 k-1 n uniformly in x 1 , x 2 (because X is bounded).
where lim k→∞

1 k-1 n = 0 thus for k high enough Ω 2 (x 1 ,x 2 ,k) f (a)da < 1 2 for all x 1 ∈ X, x 2 ∈ X, i.e. X * 1 (k) = X. 2) We show now that X * 1 (k) is not empty for k high enough, even if X is unbounded. -According to (8), we know that a ∈ Ω 2 (x 1 , x 2 , k) ⇒ x 2 -a < 1 k-1 x 1 -x 2 , i.e. Ω 2 (x 1 , x 2 , k) ⊂ B x 2 ; x 1 -x 2 k-1 . -If a ∈ Ω 2 (x 1 , x 2 , k), x 1 -x 2 ≤ x 1 -a + a -x 2 < x 1 -a + 1 k-1 x 1 -x 2 thus x 1 -a ≥ 1 -1 k-1 x 1 -x 2 = k-2 k-1 x 1 -x 2 It means that x 1 -a < k-2 k-1 x 1 -x 2 ⇒ a ∈ Ω 1 (x 1 , x 2 , k) i.e. Ω 1 (x 1 , x 2 , k) ⊃ B x 1 ; (k-2) x 1 -x 2 k-1 if k > 2.
-Let us fix x 1 ∈ X, and assume that k > 3.

There exists η > 0 such that B(x This proposition proves that an equilibrium does not always exist in our general framework but existence is guaranteed in a broader range of cases than in the symmetrical model. In that sense, our model is a real generalization of the symmetrical model.

1 ;η) f (a)da > 1 2 • If x 1 -x 2 > 2η, then (k-2) k-1 x 1 -x 2 ≥ 1 2 x 1 -x 2 > η, thus v 1 (x 1 ; x 2 ) > 1 2 , i.e. candidate 2 loses the election if x 1 -x 2 > 2η. • If x 1 -x 2 ≤ 2η, then v 2 (x 1 ; x 2 ) ≤ M.C n δ x 1 -x 2 k-1 n ≤ M.C n 2δη k-1 n < 1 2 if k is high
Proposition 6 If the distribution f admits a Condorcet winner, then k * (f ) = 1.

Proof. If there is a Condorcet winner x 0 , it means that , when k = 1:

v 1 (x 0 , x 0 ) = 1
2 and v 1 (x 0 , x) = 0 for all x = x 0 . If k > 1, it is clear that v 1 (x 0 , x 0 ) = 1, and v 1 (x 0 , x) = 0 as well for x = x 0 .

If the policy space is unidimensional, we have k * (f ) = 1 since there is always a Condorcet winner x 0 in dimension 1. (x 0 is defined by

x 0 -∞ f (a)da = 1
2 ). Let us assume moreover that f is a uniform distribution on an compact interval X, say X = [0; 1]. Then we can easily show that X

* 1 =]1 -( 1+k 4 ); k+1 4 [ if 1 < k ≤ 3, and X * 1 = X if k > 3.
It means that for a uniform distribution on an interval, we have k * * (f ) = 3.

Getting an explicit form for k * (f ) and k * * (f ) remains an open question when the dimension of the policy space is higher than 1.

A model with status quo

We examine an alternative model, where valence measures the capacity to efficiently implement a policy change [START_REF] Carrillo | Platform Divergence, Political Efficiency and the Median Voter Theorem[END_REF] . More precisely, we assume that there exists a status quo q, with q ∈ R n . If the elected candidate j applies a policy platform x j different from the status quo q, this creates temporal disturbances, disorder, inconveniences, even if finally x j is preferred by most voters. The utility of a voter i, if candidate j is elected, can be specified the following way:

u(θ j , x j , a i , q) = -x j -a i - 1 θ j x j -q (10)
where x j , θ j are respectively the policy platform and the valence of candidate j, a i is the bliss point of voter i, and q is the status quo. In Equation ( 10), x j -a i is [START_REF] Carrillo | Platform Divergence, Political Efficiency and the Median Voter Theorem[END_REF] We would like to thank an anonymous referee for suggesting this alternative specification.

the classical downsian distance between the bliss point of the voter and the applied policy, 1 θ j x j -q measures the pure inconvenience coming from a policy change. It increases with the importance of change x j -q , but is lower if candidate j's valence θ j is high, which means that he is able to implement efficiently a policy change.

The support of candidate 2 becomes:

Ω 2 (x 1 , x 2 , θ 1 , θ 2 , q) = {a ∈ R n ; u(θ 2 , x 2 , a, q) > u(θ 1 , x 1 , a, q)} = a ∈ R n ; -x 2 -a -1 θ 2 x 2 -q > -x 1 -a -1 θ 1 x 1 -q = {a ∈ R n ; x 1 -a > x 2 -a + b} = Ω ad 2 (x 1 , x 2 , b
) where b = 1 θ 2 x 2 -q -1 θ 1 x 1 -q Ω 2 (x 1 , x 2 , θ 1 , θ 2 , q) is then equal to the support Ω ad 2 (x 1 , x 2 , b) of candidate 2 in the additive case, with the additive valence bias b. A similar result is true for candidate 1. Proposition 1 becomes:

Proposition 7 (i) Ω 2 (x 1 , x 2 , θ 1 , θ 2 , q) = ∅ if b ≥ x 2 -x 1 (ii) Ω 2 (x 1 , x 2 , θ 1 , θ 2 , q) and Ω 2 (x 1 , x 2 , θ 1 , θ 2 , q) are unbounded if -x 2 -x 1 < b < x 2 -x 1 (iii) Ω 2 (x 1 , x 2 , θ 1 , θ 2 , q) = R n if b ≤ -x 2 -x 1
where b = 1 θ 2 x 2 -q -1 θ 1 x 1 -q Proof. Straightforward, applying Proposition 1 to Ω ad 2 (x 1 , x 2 , b).

If we assume that candidate 1 has a valence advantage, i.e. that θ 1 > θ 2 , then we can define the set X * 1 as in Equation ( 9). Proposition 4 is still valid in this model with status quo, since v 1 (x 2 , x 2 ) = 1 again.

Conclusion

Further research may deepen our understanding of the logic of valence advantages and the role they play in elections. The multiplicative model yields a fresh look at the existing literature on spatial models of voting. It highlights the role of the extremes. Since extreme voters are far from the center, they are mainly concerned with the valence dimension. Even if extreme voters have very different position on the policy space, they have almost identical preferences concerning candidates. On the contrary, for more centrist voters, the impact of announced platforms dominates the valence effect. It then appears that moving from the additive to the multiplicative case has dramatic consequences on the behavior of the electorate.

  different platform. Valence advantages then account for differentiation in the spatial model. The literature on valence-issues has grown significantly in the last decade offering a better understanding of the strategic consequences of the introduction of a valence advantage (Groseclose(2001), Schofield (2003), Aragones and Palfrey(2002), Ansolabehere and Snyder(2000), Dix and Santore (2002) among others). Recent contributions proposed more sophisticated models, that help understanding the origins of valence advantages. A valence advantage could arise from a greater capacity to commit to a precise platform (Egan 2007), party support (Wiseman 2006) or campaign spending (Herrera et al. 2007, Ashworth and Bueno de Mesquita 2007). Explicitly modeling the process of valence formation allows us to extend the Downsian framework to take into account an endogenous determination of valence advantages. Candidates can choose a level of effort that (stochastically) increase their valence (Carrilo and Castanheira, 2002 and 2006 and Meirowitz 2006), a level of campaign spending (Erikson and Palfrey 2000, Sahuget and Persico 2006, Zakharov 2005).

Figure 2 :

 2 Figure 2: dim 1

  enough, where M = sup a∈X |f (a)| and C n > 0 i.e. candidate 2 loses the election if x 1 -x 2 ≤ 2η, and k sufficiently high .

See Poole and Rosenthal, 1984.

Among others, see[START_REF] Wittman | Candidate with policy preferences: A dynamic model[END_REF] 1990, and Osborne 1995 for a survey

The origin of the term "valence" deserves comment. Stokes indicates that he borrowed the term from the German psychologist Kurt Lewin. In psychology indeed, valence refers to the attraction or repulsion for an object. Note that psychology itself borrowed this notion from chemistry, where valence measures the potential number of chemical links of an atom.

The median hypersurface of x 1 and x 2 is the set of points in IR n that are equidistant from x 1 and x 2 .

Appendix

Proof of Proposition 2.

Recalling the property of the Euclidean inner product, x 1 -a, x 1 -a = x 1 2 -2 x 1 , a + a 2 , we obtain:

Thus Ω 2 is a ball of center c, radius r. Moreover we have

Finally:

k This concludes the proof.

Proof of Proposition 3.

-First, we prove that

and as S 2 must be uniformly continuous on the compact set X × X:

We have Ω 2 (x 1 ,x 2 ,k) f (a)da < 1 2 -γ for every x 2 ∈ X.

-Let us now turn to the convexity problem.

Consider x ∈ X * 1 and y ∈ X * 1 , and a point z ∈ [x, y]. We want to show that z ∈ X * 1 , i.e. that S 2 (z, x 2 ) < 1 2 , for any x 2 ∈ X. Let x 2 ∈ X be given.

and of radius

2 ) and Ω 2 (y, x" 2 ) have the same center, and have radius r 0 , r ′ and r" respectively. t → t + βz -(1 + β)x 2 is a convex function, so r 0 ≤ max(r ′ , r"), i.e. Ω 2 (z, x 2 ) ⊆ Ω 2 (x, x ′ 2 ) or Ω 2 (z, x 2 ) ⊆ Ω 2 (y, x" 2 ). thus S 2 (z, x 2 ) ≤ max (S 2 (x, x ′ 2 ), S 2 (y, x" 2 )) < 1 2 . This is true for any x 2 ∈ X, so that z ∈ X * 1