PhD Joanne Ryan
email: joanne.ryan@inserm.fr

Jacqueline Scali

MSc Isabelle Carrière

MD Pierre-Yves Scarabin

PhD Karen A Ritchie

PhD Marie-Laure Ancelin

Estrogen receptor gene variants are associated

Keywords: estrogen receptor ESR1 and ESR2, single nucleotide polymorphisms, anxiety, generalised anxiety disorder, phobia, elderly women, hormone treatment, association

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Summary

Background: Estrogen is thought to play a key role in anxiety, but it remains unknown whether genetic variants in the estrogen receptors (ERs) can influence the risk of anxiety. This study investigated whether ESR1 and ESR2 gene variants were associated with specific anxiety disorders in postmenopausal women and evaluated the potential modifying effect of hormone treatment (HT) on these associations.

Methods: One thousand and ninety-two community-dwelling women aged 65 years and older were recruited as part of the ESPRIT Study in Montpellier, France. Anxiety was assessed using the Mini-International Neuropsychiatry Interview (MINI), according to DSM-IV criteria. Two ESR1 and three ESR2 polymorphisms were genotyped.

Results:

The most common anxiety disorders were phobia (14.2%) and generalised anxiety disorder (GAD, 8%). The A allele of ESR2 rs1256049 was associated with an increased risk of GAD [OR: 2.06, 95% CI: 1.09-3.87], while both ESR1 polymorphisms were specifically associated with phobia. The C allele of ESR1 rs2234693 decreased the risk of phobia by 42% [95% CI: 0.41-0.83], and this remained significant even after Bonferroni correction. The G allele of ESR1 rs9340799 was associated with a 31% decreased phobia risk [95% CI: 0.49-0.96]. There was also evidence of a significant gene-environment interaction, where only women who were currently using HT had a reduced risk of phobia with these ESR1 gene variants.

Conclusions: This study confirms earlier findings of an association between ESR1 and global anxiety in older women, however these associations varied depending on the anxiety syndrome and the use of HT. The results also suggest that the ESR2 may contribute to the genetic vulnerability to GAD, but these findings require further confirmation.

Introduction

Anxiety is the most prevalent psychiatric disorder affecting one in four adults across their lifetime [START_REF] Kessler | Lifetime prevalence and age-of-onset distributions of DSM-IV disorders in the National Comorbidity Survey Replication[END_REF] and over 10% of individuals in later-life [START_REF] Ritchie | Prevalence of DSM-IV psychiatric disorder in the French elderly population[END_REF].

Female gender is a risk factor for later-life anxiety [START_REF] Kessler | Lifetime prevalence and age-of-onset distributions of DSM-IV disorders in the National Comorbidity Survey Replication[END_REF], and evidence suggests the involvement of estrogen. Estrogen can influence the levels of neurotransmitters which have been directly implicated in anxiety, such as γ-aminobutyric acid (GABA), the major inhibitory neurotransmitter in the brain, and serotonin (5-HT) (Zinder and Dar, 1999). Estrogen upregulates GABA receptor expression and ligand binding and regulates 5-HT1A receptor number and function, thus controlling the activity of serotonergic neurons. In vitro and in vivo experiments have shown that estrogen treatment increases basal and stimulated extracellular GABA concentrations and stimulates serotonergic activity (see for review [START_REF] Birzniece | Neuroactive steroid effects on cognitive functions with a focus on the serotonin and GABA systems[END_REF]).

Studies using murine models have also demonstrated estrogen's anxiolytic effects (Walf and Frye, 2006).

Given the potential aetiological role of estrogen in anxiety, genes for the estrogen receptors (ERs) are obvious candidates implicated in this disorder. Indeed, studies in knock-out mice deficient in ESR1 or ESR2 genes support the involvement of these receptors in social recognition and anxiety (Walf and Frye, 2006). Only a very small number of studies in humans however, have examined the possibility that genetic variations in the ERs may influence the risk of anxiety. A study of children and adolescents reported an association between ESR1 gene variants and anxiety-related traits [START_REF] Prichard | Association of polymorphisms of the estrogen receptor gene with anxiety-related traits in children and adolescents: a longitudinal study[END_REF], and ESR1 variants were recently associated with contamination obsessions and cleaning compulsions in young adult patients with obsessive-compulsive disorder [START_REF] Alonso | Variants in estrogen receptor alpha gene are associated with phenotypical expression of obsessive-compulsive disorder[END_REF]. In terms of late-life anxiety, the Rotterdam study reported that ESR1 haplotypes consisting of the T and A alleles of rs2234693 and rs9340799 respectively were associated with an increased risk of global anxiety level in 1335 older women (Tiemeier et al., 2005), however, to our knowledge, no subsequent study has sought to replicate these findings. Furthermore, prior studies have not examined ESR2, although evidence in mice suggests that ESR2 may be particularly important in modulating anxiety-like behavior (Walf and Frye, 2006).

Our study aimed to replicate and further the Rotterdam study's findings, by investigating the association between the same ESR1 gene variants and the most common anxiety disorders in this elderly female population (generalized anxiety-disorder [GAD] and phobia). Three ESR2 polymorphisms were also examined. Furthermore, as older postmenopausal women have low endogenous estrogen levels, we also investigated whether the use of estrogen-containing hormone treatment (HT) could modify the association between ER variants and anxiety.

Method

Study Population

Data were obtained from the ESPRIT study of psychiatric disorders in community-dwelling elderly [START_REF] Ritchie | Prevalence of DSM-IV psychiatric disorder in the French elderly population[END_REF]. Eligible participants (aged at least 65 years and noninstitutionalised) were randomly selected from the electoral rolls in Montpellier, France. Of the people initially drawn at random, 27.3% did not participate. All procedures were carried out with the adequate understanding and written consent of the participants. The national ethics committee approved the study. Of the 1277 non-demented women recruited to the study, 185

were excluded from this analysis as they did not provide samples for genotyping analysis and/or were not assessed for anxiety disorders. Table 1 shows the participant's characteristics.

Anxiety disorders

A standardized psychiatric examination validated in the general population, the Mini-International Neuropsychiatry Interview (MINI) (Sheehan et al., 1998), was used for the diagnosis of current and past anxiety disorders (GAD, phobia, obsessive compulsive disorder, panic and post-traumatic stress disorder), according to DSM-IV criteria. Cases detected by the MINI were reviewed by a panel of psychiatrists to validate the initial diagnosis [START_REF] Ritchie | Prevalence of DSM-IV psychiatric disorder in the French elderly population[END_REF]. Current use of anxiolytics, validated by presentation of the prescription or medication, was also recorded.

Estrogen receptor polymorphisms

DNA extracted from white blood cells (Puregene kit, Qiagen, France) was stored at -80°C.

Genotyping was performed by Kbiosciences (Hoddesdon Herts, UK) using their competitive allele-specific PCR Single-Nucleotide Polymorphism (SNP) genotyping system (KASPar) which has an error rate of <0.3%. The ESR1 polymorphisms examined (rs2234693 and rs9340799),

were the same as those analyzed previously in the Rotterdam study (Tiemeier et al., 2005). These

SNPs are located at position 397 and 351 of intron 1 respectively, and they appear to be functionally significant [START_REF] Maruyama | Lack of an association of estrogen receptor alpha gene polymorphisms and transcriptional activity with Alzheimer disease[END_REF][START_REF] Alonso | Variants in estrogen receptor alpha gene are associated with phenotypical expression of obsessive-compulsive disorder[END_REF]. Three ESR2 polymorphisms with unknown functional consequences but showing potential causal associations with other hormone-related health outcomes (Silvestri et al., 2006;[START_REF] Rexrode | Polymorphisms and haplotypes of the estrogen receptor-beta gene (ESR2) and cardiovascular disease in men and women[END_REF] were investigated: rs1256049 (position 1082 of exon 5), rs4986938 (position 1730 in the 3'-untranslated region of exon 8) and rs1271572 in the promoter region. Ninety-two percent of the population (n=1004) had full genotyping data for all polymorphisms.

Hormone treatment

Women stated whether they were current HT users, which was validated by presentation of the prescription or the medication itself. The majority of women (77.4%) used transdermal estradiol (16.5% unopposed, 22.6% combined with oral progesterone and 38.3% synthetic progestin), and 15.8% used oral estradiol (14.2% combined with synthetic progestin). Remaining women (6.8%) used other forms of HT. No women used ethinylestradiol or conjugated equine estradiol.

Statistical analysis

Genotype deviation from the Hardy-Weinberg equilibrium was evaluated with a χ 2 test, and pairwise linkage disequilibrium was estimated. Logistic regression models were used to estimate the association between the ER SNPs and anxiety disorders (GAD and phobia) assuming a dominant model (SAS v9.1, SAS Institute, Inc., NC). A first-order interaction between the SNPs and current HT use was also examined by including a product term in the models. When significant, subsequent analysis was stratified to determine independent group effects. Multivariate logistic models examined the potential confounding effect of age, education level, comorbidity, cognitive impairment or severe depression.

Results

Twenty-one percent of the population (n=232) had a current anxiety disorder. GAD and phobia were the most prevalent, affecting 87 (8%) and 155 (14.2%) of the women respectively. Only 18 women were diagnosed with comorbid phobia and GAD. Other anxiety disorders (i.e. panic, obsessive-compulsive disorder and post-traumatic stress disorder) were rare among this elderly population (n=8). Compared to non-anxious women, those with anxiety differed on a number of socio-demographic health and lifestyle characteristics (Table 1).

All of the SNPs were in Hardy-Weinberg equilibrium. The ESR1 SNPs were in strong linkage disequilibrium (|D'|=0.98), as were the three ESR2 SNPs (|D'|>0.90 for all pairwise comparisons). Both ESR1 SNPs were significantly associated with the risk of phobia, with the variant alleles C and G of rs2234693 and rs9340799 respectively, decreasing the risk by over 30% (Table 2 and supplementary Figure 1). None of the ESR2 polymorphisms were significantly associated with the risk of phobia, but the variant A allele of rs1256049 was associated with a twofold increased risk of GAD. When an interaction term between these SNPs and HT was included in these models, there was evidence that current HT use could modify the association between ESR1 SNPs and phobia. For rs2234693 (interaction p=0.032) women who were using HT had a reduced risk of phobia with the C allele [170 women, OR: 0.28, 95% CI: 0.12-0.64, p=0.0025], while for women who had never used HT, this allele did not significantly modify the risk of phobia [699 women, OR: 0.79, 95% CI: 0.49-1.28, p=0.34]. The findings were very similar for rs9340799 (interaction p=0.019), where the A allele was associated with a decreased risk of phobia for HT users only [HT users, OR: 0.29, 95% CI: 0.12-0.71, p=0.0064; never users, OR: 0.95, 95% CI: 0.61-1.49, p=0.83]. These associations were not confounded by age, education level, comorbidity, cognitive impairment or severe depression. If a Bonferroni correction for multiple comparisons was applied, this would lower the significance threshold to 0.005 (five SNPs and two anxiety disorders), but the association between rs2234693 and the risk of phobia would still remain significant. Such a correction would, however, be overly conservative and increase the risk of a type 2 error, given that the tests were not independent (i.e.

the SNPs are in strong linkage disequilibrium).

Discussion

Animal studies suggest that estrogen is anxiolytic and these effects can be mediated by the ERs (Walf and Frye, 2006). The estrogen-ER complex is capable of regulating the expression of hundreds of genes, including those involved in the synthesis and metabolism of various neurotransmitters in the brain. Genetic variation in the ER may therefore influence a person's risk of developing anxiety, but this has been insufficiently studied. The Rotterdam study of 1335 women aged over 55 years, found a significant association between ESR1 polymorphisms (rs2234693 and rs9340799) and global anxiety level (Tiemeier et al., 2005), but no subsequent study has attempted to replicate these findings or to examine whether this could vary across anxiety disorders. We report here that in older women these same ESR1 gene variants were specifically associated with the risk of phobia, but current HT use could modify this association.

We also found that ESR2 rs1256049 was associated with the risk of GAD. Our study thus contributes important new findings to this field.

Although anxiety disorders often co-occur, there is a considerable degree of heterogeneity among the disorders and prior familial studies suggest a differential genetic component for GAD and phobia [START_REF] Kendler | The structure of the genetic and environmental risk factors for six major psychiatric disorders in women. Phobia, generalized anxiety disorder, panic disorder, bulimia, major depression, and alcoholism[END_REF]. A large twin study also found that GAD, panic and agoraphobia shared common genetic risk factors that were distinct from those of other specific phobias. Social phobia shared some genetic components with both groups [START_REF] Hettema | The structure of genetic and environmental risk factors for anxiety disorders in men and women[END_REF]. Our study adds to this research, reporting for the first time differential ER associations in elderly women across anxiety disorder.

The exact reason why these SNPs would be differential associated with phobia and GAD is unknown. The ESR1 rs2234693 and rs9340799 appear to be functionally significant, with the C and G alleles respectively being associated with higher gene expression [START_REF] Maruyama | Lack of an association of estrogen receptor alpha gene polymorphisms and transcriptional activity with Alzheimer disease[END_REF][START_REF] Alonso | Variants in estrogen receptor alpha gene are associated with phenotypical expression of obsessive-compulsive disorder[END_REF], and more favourable estrogen-dependent outcomes (i.e. bone mineral density [START_REF] Ioannidis | Association of polymorphisms of the estrogen receptor alpha gene with bone mineral density and fracture risk in women: a metaanalysis[END_REF]). This could suggest that estrogen plays a important protective role against the risk of phobia in later life. The functional consequence of the ESR2 rs1256049

has not been established and it may be in linkage disequilibrium with other unidentified polymorphisms. Although ESR1 predominates in the amygdala and the hypothalamus (Walf and Frye, 2006), animal studies using ER subtype-specific compounds or animals deficient in one of the receptors suggest that ESR2 plays an important role in mediating estrogen's anxiolytic effects, and ESR1 may actually mediate estrogen's anxiogenic actions [START_REF] Lund | Novel actions of estrogen receptor-beta on anxiety-related behaviors[END_REF]Walf and Frye, 2006;Weiser et al., 2008). Our results contribute to a recent evidence indicating that ER polymorphisms can interact with HT to modify health outcomes, including bone mineral density (Silvestri et al., 2006), cholesterol levels (Herrington et al., 2002), and depression (Ryan et al., submitted manuscript). This is the first study to report that this gene-environment interaction may modify the risk of phobia. Despite reports that HT can improve psychological dysfunction, including anxiety symptoms in menopausal women, there is insufficient evidence to support a direct beneficial effect of HT in treating post-menopausal anxiety [START_REF] Scali | A prospective study of hormonal treatment and anxiety disorders in community-dwelling elderly women (the Esprit Study)[END_REF].

These current results however, suggest that genetically, some women may respond more beneficially to HT in terms of psychiatric outcomes.

This study is strengthened by its large population-based design and the use of a standardised psychiatric examination to assess specific anxiety disorders according to DSM-IV criteria, rather than just global anxiety like most previous studies. Our study is also one of the first to examine the association between ESR2 SNPs and anxiety. Study limitations include the possibility of population stratification which we could not control for because French law prohibits the collection of ethnic-related data. Genotype frequencies for both anxious and nonanxious participants were however, similar to those observed in Caucasian populations [START_REF] Ioannidis | Association of polymorphisms of the estrogen receptor alpha gene with bone mineral density and fracture risk in women: a metaanalysis[END_REF][START_REF] Rexrode | Polymorphisms and haplotypes of the estrogen receptor-beta gene (ESR2) and cardiovascular disease in men and women[END_REF]. Duplicate genotyping was not performed, however the genotyping system used does have a very low error rate (<0.3%). Replication of these results in other large population-based studies is needed.

Our findings provide further support for estrogen's anxiolytic effect, and indicate that certain women are genetically more susceptible to developing anxiety in later-life. This suggests the potential for developing specific ER-targeted hormone treatments for women most at risk of anxiety.

Table 1 .

 1 Sociodemographic, lifestyle and health characteristics of the participants according to their anxiety status. Unable to independently complete all tasks from the Instrumental Activities of Daily Living and the Activities of Daily Living scales. A history of vascular disease [angina pectoris, myocardial infarction, stroke, cardiovascular surgery, bradycardia or palpitations], more than 1 chronic illnesses [asthma, diabetes (fasting glucose≥7.0mmol/l or treatment), hypercholesterolemia (total cholesterol ≥6.2mmol/l), hypertension (resting blood pressure ≥160/95mm Hg or treatment) and thyroid problems] or a diagnoses of cancer within the last 2 years. of major depressive disorder using the MINI according to DSM-IV criteria or severe depressive symptoms assessed with the Centre for Epidemiology Studies Depression Scale (≥23).

	Characteristic	No anxiety	GAD	Phobia	Group difference
		(n= 850 )	(n= 87)	(n=155)	(anxious vs.
					non-anxious)
			Mean (SD)		t test	p
	Age (years)	73.2 (5.7)	71.9 (5.4)	72.0 (4.9)	1.87	0.06
	BMI (kg/m 2 )	24.6 (4.0)	24.6 (4.4)	24.8 (4.2)	-0.87	0.39
			N, (%)		χ 2 (df 1)	p
	≥ 12yrs schooling	272 (28.0%)	20 (23.0%) 25 (16.1%) 9.8	0.002
	Married or living with others	573 (59.1%)	49 (56.3%) 97 (62.6%) 0.09	0.77
	Heavy drinker (≥ 24g each day)	51 (5.4%)	4 (4.7%)	8 (5.3%)	0.12	0.73
	Heavy smoker (10 pack years)	40 (4.1%)	5 (5.8%)	8 (5.2%)	0.32	0.57
	Physical activity limitations a	51 (5.4%)	3 (3.6%)	9 (6.0%)	0.39	0.53
	Cognitive impairment b	296 (30.6%)	22 (25.3%) 59 (38.3%) 3.81	0.05
	Comorbidity c	138 (14.2%)	18 (20.7%) 26 (16.8%) 3.65	0.06
	Current use of hormone treatment 132 (13.9%)	18 (20.9%) 28 (18.5%) 1.86	0.39
	Severe current depression d	156 (16.2%)	25 (28.7%) 47 (30.5%) 25.8	<0.001
	Current use of antidepressants	72 (7.4%)	10 (11.5%) 13 (8.4%)	1.22	0.27
	Current use of anxiolytics	123 (12.7%)	12 (13.8%) 19 (12.3%) 1.09	0.30

a b Mini-Mental State Examination Score of less than 26.

c d A diagnosis

Table 2 .

 2 Logistic regression models for the association between ER polymorphisms and anxiety disorders in elderly women.

	Polymorphism	Genotype	N	GAD		Phobia
				OR [95% CI]	p	OR [95% CI]	p
	ESR1 rs2234693:	TT	305 1		1
		TC or CC 769 0.81 [0.50-1.29] 0.37	0.58 [0.41-0.83] 0.0029
	ESR1 rs9340799:	AA	443 1		1
		AG or GG 630 0.99 [0.64-1.55] 0.98	0.69 [0.49-0.96] 0.03
	ESR2 rs1256049:	GG	971 1		1
		GA or AA 91	2.06 [1.09-3.87] 0.02	0.62 [0.30-1.25] 0.18
	ESR2 rs4986938:	GG	368 1		1
		GA or AA 707 1.18 [0.73-1.90] 0.51	1.06 [0.74-1.53] 0.76
	ESR2 rs1271572:	GG	358 1		1
		GT or TT 717 0.67 [0.43-1.05] 0.08	1.05 [0.73-1.52] 0.78

Sheehan, D.V., Lecrubier, Y., Sheehan, K.H., Amorim, P., Janavs, J., Weiller, E., Hergueta, T., Baker, R., Dunbar, G.C., 1998. The Mini-International Neuropsychiatric Interview (M.I.N.I.): the development and validation of a structured diagnostic psychiatric interview for DSM-IV and ICD-10. J Clin Psychiatry. 59, 22-33;quiz 34-57. Silvestri, S., Thomsen, A.B., Gozzini, A., Bagger, Y., Christiansen, C., Brandi, M.L., 2006. Estrogen receptor alpha and beta polymorphisms: is there an association with bone mineral density, plasma lipids, and response to postmenopausal hormone therapy?

Menopause. 13, 451-61. Tiemeier, H., Schuit, S.C., den Heijer, T., van Meurs, J.B., van Tuijl, H.R., Hofman, A., Breteler, M.M., Pols, H.A., Uitterlinden, A.G., 2005. Estrogen receptor alpha gene polymorphisms and anxiety disorder in an elderly population. Mol Psychiatry. 10, 806-7. Walf, A.A., Frye, C.A., 2006. A review and update of mechanisms of estrogen in the hippocampus and amygdala for anxiety and depression behavior. Neuropsychopharmacology. 31, 1097-111. Weiser, M.J., Foradori, C.D., Handa, R.J., 2008. Estrogen receptor beta in the brain: from form to function. Brain Res Rev. 57, 309-20. Zinder, O., Dar, D.E., 1999. Neuroactive steroids: their mechanism of action and their function in the stress response. Acta Physiol Scand. 167, 181-8.


Matías Núñez
email: matias.nunez@u-cergy.fr.

Jean-François Laslier

Francis Bloch

Francesco De Sinopoli

Santiago Oliveros

Eduardo Perez-Richet

Overstating: A tale of two cities *

Keywords:

If voters vote strategically, is it useful to offer them the possibility of expressing nuanced opinions? Say that a ballot is overstating if it is neither abstention-like nor can be expressed as a mixture of the available ballots. The paper shows that when two additive voting rules share the same (up to an affine transformation) set of overstating ballots, they are strategically equivalent in large elections. It also characterizes "robust" rules, whose set of voting equilibria remains unaltered by adding any finite number of ballots: a rule is robust if and only if it is strategically equivalent to Approval Voting. These results do not hold for small electorates.

Introduction

Shouting and voting are believed to be activities of a different sort. Whereas the latter often goes hand in hand with the idea of finding an agreement (democracy and peace), the former is recurrently associated with the lack of agreement (conflict and wars). Indeed, to put an end to a dispute, one might suggest to switch from shouting to voting. However, even if such a distinction seems to be common sense, both activities have been remarkably close to each other in the past. For instance, Spartans shouted to elect senators to the Gerousia, Sparta's Council of Elders. Each "voter" was allowed to shout as much as he wanted for each of the different candidates. The candidate who had "the most and loudest" acclamations was elected senator, see [START_REF] Girard | Acclamation Voting in Sparta: An early use of Approval Voting[END_REF] [START_REF] Girard | Acclamation Voting in Sparta: An early use of Approval Voting[END_REF] for an account of the Spartan voting method.

The Spartan shout is close to Evaluative Voting (EV ). Under EV , the voter evaluates each candidate independently on the same numerical scale; the grades are added and the candidate with the largest total is elected. [START_REF] Baujard | Framed field experiments on approval voting: lessons from the 2002 and 2007 french presidential elections[END_REF] [START_REF] Baujard | Framed field experiments on approval voting: lessons from the 2002 and 2007 french presidential elections[END_REF] report on an experimental work on EV with the scale {0, 1, 2}. This rule, also called Range Voting, is obviously related to Utilitarianism; see [START_REF] Karni | Impartiality: Definition and Representation[END_REF] [START_REF] Karni | Impartiality: Definition and Representation[END_REF], [START_REF] Dhillon | Relative Utilitarianism[END_REF] [START_REF] Dhillon | Relative Utilitarianism[END_REF], [START_REF] Segal | Let's Agree that All Dictatorships are Equally Bad[END_REF] [START_REF] Segal | Let's Agree that All Dictatorships are Equally Bad[END_REF], d'Aspremont and Gevers (2002) [START_REF] Gevers | Social Welfare Functionals and Interpersonal Comparability[END_REF] and [START_REF] Gaertner | General scoring rules[END_REF] [START_REF] Gaertner | General scoring rules[END_REF] for axiomatic analysis. A related procedure was in use for several centuries by Venetian oligarchs to elect their Dogi. Instituted in 1268 and used until 1789, the Venetian system allowed voters to express their opinion about each candidate. A voter was given three balls to indicate approval, disapproval or dubbio 1 . Another variation of the Spartan method is simply the modern voting rule known as Approval Voting (AV ), often advocated for its flexibility since it emerged in the literature in the mid 70s. Approval Voting2 is Evaluative Voting with the scale {0, 1}.

Is shouting so different than voting? The previous question can be rephrased in formal voting theory by asking whether extending the set of ballots available under a given voting rule modifies the set of voting equilibria? A voting rule V is an extension of the voting rule V if all the ballots available under V are available under V . For instance, Evaluative Voting (the Spartan shout) is an extension of AV . Assuming strategic voting, one might suspect that the set of voting equilibria should not be too altered by such an extension. However, this need not be the case for any extension: a well-known extension is known to modify the set of equilibria in a convenient manner. Indeed, AV is an extension of Plurality voting (henceforth P V3 ) and, as shown by [START_REF] Myerson | A Theory of Voting Equilibria[END_REF] [START_REF] Myerson | A Theory of Voting Equilibria[END_REF], AV improves the aggregation of preferences when compared with P V in the noteworthy divided majority situation.

We focus on additive voting rules, in which a ballot is a list of points that the voter is affording to the candidates, and where points for each candidate are simply added. (A formal definition of this family of voting rules is provided in the next section.) We analyze such an issue in the context of strategic voting, that is assuming that voters strategically cast their votes in order to maximize their (expected) utility (abstention is allowed). We study equilibria and consider that two voting rules are strategically equivalent if they have the same equilibrium outcomes.

To study small electorates, we use a standard refinement of Nash equilibrium and provide examples that show that voters need not overstate at equilibrium.

To tackle the problem on large elections, we focus on the first and simplest model in this direction, proposed by [START_REF] Myerson | A Theory of Voting Equilibria[END_REF] [START_REF] Myerson | A Theory of Voting Equilibria[END_REF]. In such a model, for any pair of candidates, the voter considers that there is a positive probability that her vote is pivotal on this pair, but some of these probabilities are vanishingly small compared to others. We first define the notion of strategically equivalent voting rules. Two rules are strategically equivalent whenever there exists an equilibrium under both of them under which the candidates get the same expected scores (up to a linear transformation) and the pivot probabilities satisfy the same ordering.

We then derive a sufficient condition for the strategic equivalence of voting rules. The condition is simple. Say that a ballot is "overstating" if it is not abstention-like (the ballot does not treat all candidates alike) and if it cannot be expressed as a mixture of other available ballots. If two voting rules offer the same set of overstating ballots, up to an affine transformation, then they are strategically equivalent.

The use of this sufficient condition is fairly straightforward, implying several interesting consequences.

The first consequence is that, in our terms, shouting is voting. The rules used in two cities lead to the same set of voting equilibria: in other words, Approval Voting and Evaluative Voting (EV ) are strategically equivalent. The second consequence concerns a different family of voting rules in which no restriction is given over how the grades should be allocated between the different candidates. In an election held under Cumulative Voting (CV ), a natural extension of Plurality Voting, a voter is endowed with a finite number of points, and he is allowed to distribute them freely between the different candidates. Different authors 4 have discussed such a method as it gives a high degree of flexibility to the voter. With such a voting rule, voters have the possibility of overstating their vote: that is to give the highest possible amount of points to only one of the candidates. We prove that this is indeed the case in equilibrium, implying that P V and CV are strategically equivalent. We hence prove that for both P V and AV , there exist extensions that do not modify the set of voting equilibria. But, up to now, we have left unaddressed the question of whether there exists voting rules which set of voting equilibria remains unaltered by any finite extension; a voting rule satisfying such a definition is robust 5 . The answer is positive and surprising. Without loss of generality, we work with normalized rules in which the maximum score of a candidate in a ballot is 1. We prove that a voting rule is robust if and only if it is strategically equivalent to AV . To do so, we first show that AV (or any voting rule that contains all the AV ballots) is robust. The reason is simple: any normalized ballot can be expressed as a strict convex combination of AV ballots, and hence our sufficient condition for strategic equivalence applies. Furthermore, any other normalized voting rule which is extended by adding the ballots of AV is strategically equivalent to AV . Hence, any robust voting rule must contain all the AV ballots and hence it is strategically equivalent to AV .

The described equivalence between voting equilibria described is valid along the lines of the theory of large elections proposed by [START_REF] Myerson | A Theory of Voting Equilibria[END_REF] [START_REF] Myerson | A Theory of Voting Equilibria[END_REF]. Nevertheless, small elections (that is elections with few voters) raise new questions. For instance, the information available to voters might be much more detailed in a small election than in a mass election, implying that such theory is of scant interest in the former case. In order to investigate whether the previous claims still hold in environments with few voters, we discuss two voting situations in the case of Evaluative voting. The first example is a pure strategy equilibrium in which there exists a sequence of trembles à la [START_REF] Selten | A Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games[END_REF] [START_REF] Selten | A Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games[END_REF] which induce a voter's unique best response not to be overstating for any arbitrarily small (even though positive) perturbation. In the second one, the strategy combination is a mixedstrategy equilibrium in which the unique pure strategy best response for a voter is not overstating. Indeed, one of the voters of the election mixes between his undominated strategies making uncertain the final electoral outcome for the rest of the voters. The "mixing" removes the weak preference for overstatements. This situation is stronger that the first one. The situations studied prove that the lack of overstatement can be a best response even in equilibria that satisfy different equilibria refinements. The refinement (trembling-hand perfection) used in this work is among the most classical ways of obtaining equilibria as a limit of games with uncertainty (i.e. perturbed games). However, it is not too difficult to generalize the results to settings in which the uncertainty comes from other sources. For instance, Bayesian games with some uncertainty about voters' types or common values' settings with imperfect information about the true state of nature are good candidates for models in which overstating is not a best response for a strategic voter. Finally, we discuss how the set of possible winners in equilibrium is altered by an extension of the set of ballots. Even though we cannot fully characterize the set of perfect equilibria (due to the lack of structure of voters' mistakes) we are able to show that the equilibria under which "unappealing" candidates win the election both exist under a voting rule and its extension. In other words, the extension of the voting rule does not seem to refine the set of possible winners of the election.

This paper is organized as follows. Section 2 presents the basics of the model. Section 2 to 5 are devoted to large elections: Section 3 describes the equilibrium concept, Section 4 states the sufficient condition for strategic equivalence, Section 5 presents the strategic equivalence between the above-mentioned voting rules, and Section 6 contains the results on the robustness of a voting rule. Section 7 presents the results concerning the environments with few voters, and Section 8 provides some concluding comments.

The setting

There are N voters in the election. Each voter has a type t that determines his strict preferences over the set of candidates K = {1, 2, . . . , K}. The preferences of a voter with type t (a t-voter) is denoted by u t = (u t (k)) k∈K , in which u t (k) denotes the utility a t-voter gets if candidate k wins the election. All types t belong to a finite set of types T . The distribution of types is denoted by r = (r(t)) t∈T with Within this work, we stick to the comparison of additive rules: a ballot is a vector b = (b 1 , b 2 , . . . , b K ) where b k is the number of points given to candidate k, to be added to elect the candidate with the largest score. Each voter must choose a ballot b from a finite set of possible ballots denoted by B.

For instance, in an election held under Plurality Voting (P V ) voters can abstain or give one point to at most one candidate. Formally: B P V = {Any permutation of (1, 0, . . . , 0)} ∪ {0, 0, ..., 0)}.

Similarly, an Approval Voting (AV ) ballot consists of a vector that lists whether each candidate has been approved or not: for each j ∈ K , b j ∈ {0, 1}. Hence:

B AV = {0, 1} K . Definition 1. A voting rule V is an extension of the voting rule V if all ballots in V are available in V , i.e. B V ⊂ B V .
For instance, Approval Voting is an extension of Plurality Voting as B P V ⊂ B AV .

Large Elections

We assume that each voter maximizes his expected utility to determine which ballot in the set B he will cast. In this model, his vote has an impact in his payoff if it changes the winner of the election. Therefore, a voter needs to estimate the probability of these situations: the pivot outcomes. We say that two candidates are tied if their vote totals are equal. Furthermore, let H denote the set of all unordered pairs of candidates. We denote a pair {i, j} in H as ij with ij = ji. For each pair of candidates i and j, the ij-pivot probability p ij is the probability of the outcome perceived by the voters that candidates i and j will be tied for first place in the election. A voter perceives that the probability that he will change the winner of the election from candidate i to candidate j by casting ballot b with b i ≥ b j to be linearly proportional to b i -b j , and that the constant of proportionality (the ij-pivot probability) is the same for the perceived chance of changing the winner from j to i if b j ≥ b i6 .

A vector listing the pivot probabilities for all pairs of candidates is denoted by p = (p ij ) ij∈H . This vector p is assumed to be identical and common knowledge for all voters in the election. A voter with ij-pivot probability p ij anticipates that submitting the ballot b can change the winner of the election from candidate j to candidate i to be p ij max{b i -b j , 0}.

Let E t [b] denote the expected utility gain of a t-voter from casting ballot b when p is the common vector of pivot probabilities:

E t [b] = ij∈H (b i -b j ) • p ij • [u t (i) -u t (j)]. (1)
The expected utility gain from casting ballot b equals the expected utility of casting ballot b minus the expected utility of abstaining. Focusing on utility gains simplifies notation.

A (voting) strategy is a probability distribution σ over the set B that summarizes the voting behavior of voters of each type. For any ballot b and any type t, σ(b | t) is the probability that a t-voter casts ballot b. Therefore,

τ (b) = t∈T r(t)σ(b | t)
is the share of the electorate who cast ballot b. Hence, the (expected) score of candidate k is

S(k) = b∈B b k τ (b).
The set of likely winners of the election contains the candidates whose expected score S(k) is maximal given the strategy σ.

Myerson and Weber (1993) [START_REF] Myerson | A Theory of Voting Equilibria[END_REF] assume that voters expect candidates with lower expected scores to be less likely serious contenders for first place than candidates with higher expected scores. In other words, if the expected score for some candidate l is strictly higher than the expected score for some candidate k, then the voters would perceive that candidate l's being tied with any third candidate m is much more likely than candidate k's being tied for first place with candidate m 7 . j (and both candidates above the rest of the candidates), which is in turn the same one as the probability of candidate j being in first place one vote ahead candidate i. [START_REF] Myerson | A Theory of Voting Equilibria[END_REF] [START_REF] Myerson | A Theory of Voting Equilibria[END_REF] justify this assumption by arguing that it seems reasonable when the electorate is large enough. This is not verified in Poisson games, a formal model of large elections in which the pivot probabilities are derived endogenously from the structure of the game. 7 This assumption is needed in order to ensure the existence of equilibrium. The results presented here do not lie on the ordering of the pivot probabilities.

Definition 2. Given a (voting) strategy σ and any 0 < ε < 1, a pivot probability vector p satisfies the ordering condition for ε (with respect to σ) if, for every three distinct candidates i, j and k:

S(i) > S(j) =⇒ p jk ≤ εp ik .
Besides, [START_REF] Myerson | A Theory of Voting Equilibria[END_REF] [START_REF] Myerson | A Theory of Voting Equilibria[END_REF] assume that the probability of three (or more) candidates being tied for first place is infinitesimal in comparison to the probability of a two-candidate tie.

Definition 3. The strategy σ is a (voting) equilibrium of the game if and only if, for every positive number ε, there exists some vector p of positive pivot probabilities that satisfies the ordering condition and such that, for each ballot b and for each type

t, σ(b | t) > 0 =⇒ b ∈ arg max d∈B E t [d].
It should be stressed that, in this definition, the pivot probabilities p ij are supposed to be constant when the voter contemplates casting one ballot or the other. This point will play an important role in the next section. It is justified when the number of voters is large for, in that case, the voter cannot change with his single vote the order of magnitude of these probabilities. It can be shown that the set of equilibria is non-empty 8 .

Finally, an important concept in our model should be defined: the equivalence between equilibria under different voting rules. Definition 4. An equilibrium σ U of an election held under a voting rule U is equivalent to an equilibrium σ V of the same election held under V if and only if 1. the pivot probabilities satisfy the same ordering and 2. the scores of the candidates coincide, up to an affine transformation.

The sets of voting equilibria of an election held under two voting rules U and V are equivalent if for any voting equilibrium of the election held under U (resp. V ), there exists an equivalent voting equilibrium of the election held under V (resp. U ) Definition 5. Two voting rules are strategically equivalent if and only if their set of voting equilibria are equivalent.

We will pay special attention to the set of possible winners W V that arise under a voting rule V . A possible winner is a candidate who wins the election in equilibrium with positive probability. The set of possible winners of an election held under the voting rule V is

W V = {k ∈ K | There exists an equilibrium σ in which S(k) is maximal}.
If two voting rules are strategically equivalent, then they have the same set of possible winners. However, the converse need not be true; for instance, the rankings of other candidates may differ. It is noteworthy that the definition of strategic equivalence used is rather demanding. It requires more than the set of possible winners being the same under two voting rules. This demanding definition reinforces our results as we show that this strong version of equivalence holds in the Myerson-Weber setting.

A sufficient condition for strategic equivalence

We now introduce some categorizations of the ballots that will be useful throughout.

An abstention ballot is a ballot with all the coordinates alike ; the set of such ballots is denoted by Abs(B).

An interior ballot b is a ballot which is not an abstention ballot and that can be expressed as a strict convex combination of other ballots in B, i.e. there exist ballots

b 1 , b 2 , . . . , b m ∈ B with b = i α i b i with α i ∈ (0, 1) and i α i = 1.
An overstating ballot is a ballot which is neither an interior nor an abstention ballot. Given the set of ballots B, the set of interior and overstating ballots are respectively denoted by Int(B) and Ove(B) with,

Ove(B) = B \ {Int(B) ∪ Abs(B)}. Remark 1.
Casting an abstention ballot or a ballot which is a convex combination of at least one abstention ballot is dominated for every voter.

To see this, let b j denote an abstention ballot with b j = (x, . . . , x). By formula

(1), E[b j ] = ij∈H (x -x) • p ij • [u t (i) -u t (j)] = 0 so that any abstention ballot is dominated. Let c = i α i b i denote the convex combination of at least one abstention ballot so that E t [c] = i =j α i E t [b i ] as E t [b j ] = 0. If E t [c] ≤ 0, then casting ballot c is dominated. If E t [c] > 0, then i =j α i E t [b i ] > 0 so that there must exist b i with i = j with E t [b i ] > 0.
Then, casting ballot c is dominated by the mixed strategy that casts ballot c with probability 1 -α j and ballot b i with probability α j .

In other words, neither an abstention ballot nor a ballot which is a convex combination of at least one abstention ballot are cast with positive probability in equilibrium.

Remark 2.

The definition of interior ballot implies that if a ballot c is interior then it is the strict convex combination of overstating and abstention ballots:

c = b∈B α b • b for some B with B ⊂ Ove(B) ∪ Abs(B) with α b ∈ (0, 1) and b∈B α b = 1. Remark 3.
The ballot set B is finite and hence not every ballot can be expressed by a convex combination of other ballots in B.

Remark 4.

The set Ove(B) of overstating ballots is non-empty for any non-trivial voting rule. To see this, let us suppose that Ove(B) = ∅ for some voting rule with ballot set B. By definition, B = Ove(B)∪Int(B)∪Abs(B). By Remark 3, not every ballot can be expressed by a convex combination of other ballots in B, so that B = Int(B). As we have assumed that Ove(B) = ∅, we must have that Abs(B) = ∅, i.e. every ballot which is not interior is an abstention ballot. Hence, every ballot of such a rule is an abstention ballot as any interior ballot is a strict convex combination of other ballots. Thus, such a voting rule can be labeled as trivial as it elects for any preference profile the whole set of candidates.

Example 1.

Let us consider a three-candidates election held under Cumulative Voting. We assume that a voter is endowed with at most two points that can be freely distributed among the different candidates. The set of allowed ballots B CV is: B CV = {(0, 0, 0),(2, 0, 0), (0, 2, 0), (0, 0, 2), (1, 1, 0), (1, 0, 1), (0, 1, 1), (1, 0, 0), (0, 1, 0), (0, 0, 1)}.

The interior ballots are the following ones:

Int(B CV ) = {(1, 1, 0), (1, 0, 1), (0, 1, 1), (1, 0, 0), (0, 1, 0), (0, 0, 1)}, whereas there is a unique abstention ballot:

Abs(B CV ) = {(0, 0, 0)}.
(To see why, it is enough to write that for instance the interior ballot (1, 1, 0) equals the convex combination 1/2(2, 0, 0) + 1/2(0, 2, 0).) Finally, the set of overstating ballots is:

Ove(B CV ) = {(2, 0, 0), (0, 2, 0), (0, 0, 2)}.

Overstating

By Remark 1, an abstention ballot is not cast with positive probability in equilibrium so that there are two types of equilibria: interior and overstating. We refer to an interior equilibrium whenever a voter's best response includes an interior ballot. On the contrary, an overstating equilibrium is an equilibrium in which every voter's best response uniquely includes overstating ballots.

Proposition 1.

[Overstating] For any interior equilibrium, there exists an equivalent overstating equilibrium.

Proof. Let σ be an interior equilibrium such that some t-voter's best response satisfies σ(c | t) = 1 with c an interior ballot. Therefore we can write c = b∈B α b • b for some subset B of the set of overstating ballots as c is cast with positive probability at equilibrium (Remarks 1 and 2). Formula (1) implies that

E t [c] = E t [ b∈B α b • b] = b∈B α b • E t [b].
In other words, the t-voter is indifferent between casting ballot c and playing a mixed strategy over the set B which mimics the convex combination that defines ballot c. Formally, the voter t is indifferent between strategy σ as defined 9 and strategy σ

* with σ * (b | t) = α b for all b ∈ B and σ * (•| t ) = σ(•| t ) for voters with type t = t.
Besides, for every ε > 0, the pivot probability vector p that justifies the strategy σ also justifies the strategy σ * as the scores of candidates coincide under both strategies, implying that σ * is an equilibrium.

All in all, both σ and σ * are justified by the same pivot probability vector and under both of them, the expected scores of the candidates coincide. Hence, for any interior equilibrium σ, there exists an equivalent overstating equilibrium σ * .

Strategic equivalence

Proposition 1 proves that interior equilibria are not informative in the sense that they do not add any information regarding the equilibria that can be attained under a voting rule. Building on such a result, we now give a simple sufficient condition to ensure the strategic equivalence of two voting rules. Indeed, we show that if there exists a bijective affine transformation between the overstating ballots of two voting rules, then both rules are strategically equivalent.

Theorem 1. Whenever there exists a bijective affine transformation between the overstating ballots Ove(B U ) and Ove(B V ) of two voting rules U and V , U and V are strategically equivalent.

Proof. Let U and V denote two voting rules such that there exists a bijective affine transformation f : b -→ f (b) = α + β • b, from Ove(B U ) onto Ove(B V ), for some reals α and β.

Let σ be a strategy in an election held under U in which every voter only casts ballots in the set Ove(B U ). Let σ * denote a strategy in the same election held under Let us now prove that the strategy σ is an overstating equilibrium of the election held under U if and only if the strategy σ * is an overstating equilibrium of the election held under V with both σ and σ * being equivalent.

V that satisfies σ * (b * | t) = σ(b | t), (2)
First of all, an equilibrium is overstating if and only if the unique ballots that are cast with positive probability are overstating. Thus, if we show that σ and σ * are equilibria under U and V respectively, then both equilibria will be overstating.

Let us now prove that casting ballot b is a best response given σ if and only if casting ballot b * is a best response given σ * :

b ∈ arg max d∈ Ove(B U ) E t [d] ⇐⇒ E t [b] ≥ E t [d] ∀ d ∈ Ove(B U ) ⇐⇒ α + βE t [b] ≥ α + βE t [d] ∀ d ∈ Ove(B U ) ⇐⇒ E t [b * ] ≥ E t [d * ] ∀ d * ∈ Ove(B V ) ⇐⇒ b * ∈ arg max d * ∈ Ove(B V ) E t [d * ].
Besides, given that the strategy σ * satisfies (2), the scores of the candidates S(•) given σ and S * (•) given σ * satisfy

S * (k) = α + βS(k) ∀ k ∈ K,
and whence the scores of candidates coincide up to an affine transformation under both strategies.

In order to prove the equivalence between σ and σ * , it remains to be checked that pivot probabilities satisfy the same ordering with both strategies σ and σ * . However, as the scores of candidates coincide up to an affine transformation with both strategies, a pivot probabilities vector p satisfies the ordering condition under σ if and only if it satisfies the ordering condition under σ * .

We have proved so far that σ is an overstating equilibrium under U if and only if there exists an equivalent overstating equilibrium σ * under V . In other words, if there exists a bijective affine transformation between Ove(B U ) and Ove(B V ), the set of overstating equilibria under both U and V are equivalent. But the previous equivalence finishes the proof as by Proposition 1, any interior equilibrium under a voting rule is equivalent to an overstating equilibrium under the same voting rule.

Theorem 1 has the advantage of being extremely simple to use: indeed, as will be shown the next section, almost no computation is needed to check the strategic equivalence of two voting rules.

Applications

Two applications of Theorem 1 are now described. The main interest of such a theorem is that it allows to "simplify" voting rules, in which the term simplify has been coined by the recent literature on mechanism simplification 10 . In this literature, a mechanism is simplified by reducing the message space of the agents, while no new equilibria are created as a consequence of this reduction. When the number of voters becomes large enough, adding or removing interior ballots to a voting rule does not modify the set of voting equilibria. Our results hence prove that when the number of voters is large enough, many voting rules can be simplified.

Evaluative Voting: One man, Many extended votes

An AV ballot consists of a vector that lists whether each candidate has been approved or not: ∀j ∈ K, b j ∈ {0, 1}. Hence it is simple to see that Abs(B AV ) = {(0, . . . , 0), (1, . . . , 1)}.

Under Evaluative Voting, a voter can assign up to m points to each candidate for some positive m. Hence,

b is an EV ballot if ∀ j ∈ K, b j ∈ {0, 1, . . . , m},

Cumulative Voting: One man, One extended vote

In an election held under P V , voters can give one point to at most one candidate. Formally, we say that b is a P V ballot if for every j ∈ K, b j ∈ {0, 1} and there is at most one b j = 0.

In an election held under Cumulative Voting, a voter can assign up to m points to each candidate for some positive m with the restriction that the sum of the points he can assign to each of the candidates is at most m. Hence,

B CV = b j ∈ {0, 1, . . . , m} ∀ j ∈ K and k∈K b k ≤ m .
Theorem 3. CV and PV are strategically equivalent.

Proof. Given the set of ballots under P V and CV , one obtains that Ove(B P V ) = {Any permutation of (1, 0, . . . , 0)}.

and that

Ove(B CV ) = {Any permutation of (m, 0, . . . , 0)}.

which, by Theorem 1, concludes the proof.

Robust Voting rules

Within this section, we introduce the notion of robust voting rule and show that a voting rule is robust if and only if it is strategically equivalent to AV . Definition 6. A voting rule V is robust if any extension of V is strategically equivalent to V .

Robustness requires that the set of electoral outcomes is not modified by allowing voters to choose from a wider set of ballots.

A voting rule is normalized if the maximum score of a candidate in a ballot is 1. Remark that any ballot of a normalized rule can be expressed as the mixture (convex combination) of AV ballots. Hence, for any normalized voting rule V which extends AV , we have Ove(B AV ) = Ove(B V ), implying, by Theorem 1, the strategic equivalence of V and AV as summarized by the next result. Furthermore, extending any voting rule V by "adding" the AV ballots modifies the set of overstating ballots and hence leads to a voting rule which is strategically equivalent to AV . We hence can state without proof the following characterization of robust voting rules. Theorem 4. A voting rule is robust if and only if it is strategically equivalent to AV .

Small Elections

The results previously presented are a consequence of the model used in which voters' perceptions over the impact of their ballots in switching the winner of the election have a very specific shape. Such a theory fits particularly well the study of mass elections. Indeed, as shown by further developments of the theory 11 , more formal models give, roughly speaking, similar predictions depending on whether the ordering condition is satisfied. However, it seems that the specific shape of expected utility is not particularly relevant for studying voting in committees (that is voting with few voters). Indeed, in a committee, the information a voter knows can be much more detailed than in a large election.

When switching to an environment with few voters, we prove two results:

• Not overstating might be the unique best response for a strategic voter.

• The set of winners of an election is not significantly altered by allowing voters to express an intensity of preference.

Overstating need not be optimal

In order to prove that not overstating may be a unique best response in a voting game with few voters, we present two examples. The first one presents a pure strategy equilibrium whereas the second one is a mixed strategy one. In order to test the robustness of the examples, we focus on trembling-hand perfection à la Selten. The definition of perfection is as follows:

Definition 7. A completely mixed strategy profile σ ε N is an ε-perfect equilibrium in an N -voters game if ∀i ∈ N , ∀b i , bi ∈ B, if U i (b i , σ ε N \{i} ) > U i ( bi , σ ε N \{i} ), with σ ε ( bi ) ≤ ε,
in which U i (b) denotes the payoff of voter i given the strategy combination b. We refer to the strategy combination σ N as a perfect equilibrium if there exists a sequence

{σ ε N } of ε-perfect equilibria converging (for ε → 0) to σ N .
Example 1:

There are three candidates K = {1, 2, 3} and three different types T = {a, b, c}, with cardinal utilities given by: u a = (3, 1, 0), u b = (0, 3, 1) and u c = (0, 1, 3).

There are seven voters in the electorate. Voters 1 and 2 have type a, voters 3 and 4 have type b and voters 5, 6 and 7 have type c.

We consider Evaluative Voting in which voters can give up to two points to each of the candidates.

We let f denote the strategy combination with f = ((2, 0, 0), (2, 0, 0), (0, 2, 1), (0, 2, 1), (0, 0, 2), (0, 2), (0, 0, 2)).

It is simple to see that voters 3 and 4 do not overstate. Besides, f is an equilibrium in undominated strategies in which candidate 3 wins the election by more than two votes. Proposition 2. In Example 1, f is a perfect equilibrium in which some voters' best responses are not overstating.

This proposition, the proof of which is included in the appendix, shows that not overstating might be a voter's best response in a perfect equilibrium. Indeed, a perfect equilibrium is the limit of completely mixed strategies of the voters that arise as a consequence of uncorrelated mistakes of the voters. Hence, voters' expected utility is not anymore "smooth" as it is by assumption in the large elections model.

Example 2:

There are three candidates K = {1, 2, 3} and four different types T = {a, b, c, d}, with cardinal utilities given by: u a = (6, 1, 0), u b = (0, 6, 1), u c = (0, 1, 6) and u d = (0, 3, 6).

There are seven voters in the electorate. Voters 1,2 and 3 have type a, voters 4 and 5 have type b, voter 6 has type c and voter 7 has type d .

We consider Evaluative Voting in which voters can give up to two points to each of the candidates.

We let g denote the strategy combination g = ((2, 0, 0), (2, 0, 0), (2, 0, 0), (0, 2, 1), (0, 2, 1), (0, 0, 2), g 7 ).

in which g 7 stands for the mixed strategy 1/3(0, 0, 2) + 1/3(0, 1, 2) + (1/3)(0, 2, 2) of voter 7. Every voter plays an undominated strategy in the strategy combination g. It is easy to check that g is a mixed-strategy equilibrium of the election in which voters 1 to 6 are playing a unique best response.

Proposition 3. In Example 2, g is a perfect equilibrium in which some voters' unique best responses are not overstating.

Remark 1: The source of the non overstating behavior in the second example is clearly the uncertainty faced by voters 4 and 5, as a consequence of the mixing of voter 7. The same logic applies in a (Bayesian) game of incomplete information in which voters are not sure of the type of their opponents.

Remark 2: The utility vectors in both examples are consistent with single-peaked preferences.

Possible Winners remain unchanged

We now address the issue of the set of possible winners in an election with a small number of voters. To do so, we give a proposition which extends a previous result of De [START_REF] Sinopoli | Sophisticated vote and Equilibrium Refinements under Plurality Rule[END_REF] [START_REF] Sinopoli | Sophisticated vote and Equilibrium Refinements under Plurality Rule[END_REF] (which focused in Plurality Voting). We show that any candidate who is not a Condorcet loser can win the election under Plurality Voting and Cumulative Voting.

Prior to stating it, we need the definition of Condorcet loser.

Definition 8. Candidate k' is a Condorcet loser if #{i ∈ N | u i (k) > u i (k )} > #{i ∈ N | u i (k ) > u i (k)} ∀ k ∈ K \ k .
Proposition 4. In an election held under either P V or CV with at least 4 voters, for every candidate k who is not a Condorcet loser there exists a perfect equilibrium in which k wins the election.

Proof. Let 1 and 2 be two candidates who are not Condorcet losers. Let us divide the voters in two groups: the voters who prefer candidate 1 to candidate 2,

V (1, 2) = {i ∈ N | u i (1) > u i (2)}, and the remaining ones V (2, 1) = {i ∈ N | u i (2) > u i (1)}.
Under CV , a voter can assign up to m points to a single candidate. Under P V , the proof remains unchanged with the constraint that m = 1. Consider the mixed strategy d ε such that for every voter i ∈ V (1, 2), where η i denotes the mixed strategy of voter that assigns equal probability to all his pure strategies with obvious notations,

d ε i = (1 -ε -ε 2
)(m, 0, . . . , 0) + ε(0, m, 0 . . . , 0) + ε 2 η i , and such that for every voter i ∈ V (2, 1),

d ε i = (1 -ε -ε 2 )(0, m, . . . , 0) + ε(m, 0, 0 . . . , 0) + ε 2 η i .
For each voter, the pivot event which becomes infinitely more likely as ε tends towards zero is one in which candidates 1 and 2 are involved. 1 Hence, each voter plays his best response with probability higher than ε in the sequence of mixed strategies d ε . Besides, as ε approaches zero, every voter in the set V (1, 2) votes for candidate 1, and every other voter votes for candidate 2, which implies that either candidate 1 or candidate 2 wins the election, proving the claim.

The previous result implies that extending the set of available grades in the case of P V does not refine in a relevant way the set of possible winners of elections with few voters.

The reason why the equilibrium depicted by the Proposition 4 can be constructed is simple. For any pair of candidates 1 and 2 (who are not Condorcet losers), we split the electorate in two blocs: the ones who prefer candidate 1 to candidate 2 (the partisans of candidate 1) and the ones who prefer candidate 2 to candidate 1 (the partisans of candidate 2). Let us assume that partisans of candidate 1 assign her the maximum number of points whereas partisans of candidate 2 behave in the same manner with respect to candidate 2. Each of the two blocs is homogenous in the sense that each voter makes the same mistakes. Hence, when casting his ballot, a voter knows almost surely that, provided being pivotal, his vote will break the close race between candidates 1 and 2. Therefore, it is a best response for the partisans of a candidate to assign her the maximum number of points, proving that this is an equilibrium. Both voting rules analyzed in the Proposition 4 share the feature that a voter can assign the total number of points to a single candidate, leading to the construction of this "almost-everything-can-happen" type of result.

The Majority Preferred Candidate In order to conclude our investigation in the case of a reduced number of voters, we focus on the majority preferred candidate situation, in a similar spirit to the one depicted by Nuñez (2010) [START_REF] Núñez | Condorcet Consistency of Approval Voting: A Counter Example on Large Poisson Games[END_REF]. Let us consider a voting game held under Evaluative Voting. There are three types of voters in the electorate:

u a = (3, 0, 1), u b = (1, 3, 0) and u c = (1, 0, 3), with voters 1,2 being of type a, voters 3 to 7 being of type b and voters 8 to 10 of the third type. We will refer to candidate 2 as the majority preferred candidate as 5 voters over 10 rank him first. Candidate 1 is only ranked as a first option by two over ten voters in the election but it can nevertheless be elected at equilibrium in elections held under EV and AV as shown by next result.

Proposition 5. There exists a perfect equilibrium in which candidate 1 is the unique winner of the election held under both EV and AV.

Proof. Under EV , a voter can assign up to m points to a single candidate. Consider the mixed strategy e ε where η i denotes the mixed strategy of a voter that assigns equal probability to all his pure strategies,

e ε i = (1 -ε -ε 2 )(m, 0, 0) + ε 2 η i with i = 1, 2 e ε i = (1 -ε -ε 2 )(m, m, 0) + ε(0, 0, m) + ε 2 η i with i = 3, . . . , 7, e ε i = (1 -ε -ε 2 )(0, 0, m) + ε 2 η i with i = 8, . . . , 10.
For each voter i = 1, 2, the pivot event which becomes infinitely more likely as ε tends towards zero is (4m, 3m, 5m) so that it is a strict best response to vote only for his first-ranked candidate. Similarly, for each voter i = 3, . . . , 7, the pivot event which becomes infinitely more likely as ε tends towards zero is (4m, 2m, 5m) so that it is a strict best response to vote for his first-ranked and his second-ranked candidate. Finally, the event that determines voters i = 8, 9, 10's best responses is (5m, 3m, 4m) and hence their unique best response is to cast ballot (0, 0, m). Besides, as ε approaches zero, candidate 1 wins the election as every voter who votes for candidate 2 also votes for candidate 1, proving the claim.

The bottom-line of this example is that even if we do not provide a characterization of possible winners under Evaluative Voting, enlarging the set of possible grades does not remove the coordination problems already present under Approval Voting. Hence, one can intuitively think that the set of possible winners should not be too refined by EV (when compared to AV ), if at all. Similar coordination problems as the ones illustrated by Proposition 5 have been already identified by Nuñez (2010) [START_REF] Núñez | Condorcet Consistency of Approval Voting: A Counter Example on Large Poisson Games[END_REF] in the case of AV . The logic of this unattractive equilibrium boils down to voters' anticipations. In a certain manner, AV performs better than P V in preference aggregation as, with the former voting rule the voter does not face the classical trade-off between voting for his preferred candidate and voting for his preferred likely winner (the wasted-vote effect). However, this property of AV (and of EV ) may not be enough to ensure a correct preference aggregation in every election. If the majority of voters anticipate that their preferred candidate is not included in the most probable pivot outcome, this may lead to the election of an unappealing candidate. Indeed, due to their anticipations, the majority of voters favors their preferred likely winner by assigning her the maximum number of points and at the same time vote for their preferred candidate, leading to the election of the former candidate.

Conclusion

Building on the theory of strategic voting in large elections, we have derived a sufficient condition for the strategic equivalence of voting rules that simply depends on the ballots available to the voters. The condition says that whenever two voting rules share the same set of overstating ballots (up to an affine transformation), then they are strategically equivalent. Hence, such a condition helps us to draw some conclusions over how adding ballots to a given voting rule modifies the set of voting equilibria. First, we prove that it is possible to add ballots to both Plurality Voting and Approval Voting without modifying the set of voting equilibria. In the case of Approval voting, there is no difference between shouting (the Spartan Shout) and voting (AV ) when voters act strategically. As far as Plurality voting (P V ) is concerned, Cumulative voting extends P V while being strategically equivalent to P V . We then characterize robust voting rules, voting rules which set of voting equilibria is not modified by adding any finite number of ballots. We show that any robust voting rule is strategically equivalent to AV . As has been shown, the previous results do not extend to a context with a reduced number of voters.

We have very few observations to back up, or to invalidate, these theoretical results. Laslier and Van der Straeten (2004) [START_REF] Laslier | Une expérience de vote par assentiment lors de l'élection présidentielle française de 2002[END_REF] report on an experiment comparing EV with the 0 to 10 scale and AV, and Baujard and Igersheim (2010) [START_REF] Baujard | Framed field experiments on approval voting: lessons from the 2002 and 2007 french presidential elections[END_REF] report on an experiment comparing EV with the 0-1-2 scale and AV. In both cases it is observed that the outcome of the election (the elected candidate) is the same under the two systems, even if it is not observed that voters concentrate on extreme grades.

An interesting extension of the present work would be to understand whether similar results apply under proportional representation or in multi-seat elections in which voters have to distribute their votes. Finally, it is noteworthy to underline that our sufficient condition for strategic equivalence remains silent over the different rank scoring rules as characterized by [START_REF] Young | Social choice scoring functions[END_REF] [START_REF] Young | Social choice scoring functions[END_REF]. Indeed, there are no interior ballots in a rank scoring rule and the different scoring rules do not share the same set of overstating ballots.

A Appendix: Proof of Proposition 2

In an election held under Evaluative Voting in which voters can give up to two points to each of the three candidates, voters have three undominated strategies: to give two points to their favorite candidate, no points to their least preferred candidate and zero, one or two points to their middle ranked candidate. The previous observation is important, as in a perfect equilibrium voters only choose undominated strategies 12 . It is easy to see that the strategy combination f = ((2, 0, 0), (2, 0, 0), (0, 2, 1), (0, 2, 1), (0, 0, 2), (0, 0, 2), (0, 0, 2)) is an undominated equilibrium in which b-voters do not overstate and in which candidate 3 wins the election. Consider the following completely mixed strategy combination f ε , where η i denotes the mixed strategy of voter i which assigns equal probability to all his pure strategies.

i = 1, 2 f ε i = (1 -27ε 2 )(2, 0, 0) + 27ε 2 η i i = 3, 4 f ε i = (1 -27ε 2 )(0, 2, 1) + 27ε 2 η i i = 5, 6, 7 f ε i = (1 -ε 1 -ε 2 -25ε 2 )(0, 0, 2) + (ε 1 -ε 2 )(2, 0, 0) + (ε 2 -ε 2 )(2, 2, 0) + 25ε 2 η i , in which ε 1 = 1/3(ε + ε 2 ) and ε 2 = 1/3(2ε -ε 2 ).
It is easy to see that, for ε sufficiently close to zero, this is an ε-perfect equilibrium. Suppose all voters other than i choose the strategies prescribed by f . Then, the three undominated strategies of voter i are equivalent. Since for ε going to zero, the probability of voter 5 (the same statement is valid for voters 6 or 7) to tremble towards (2, 0, 2) or (2, 2, 0) is infinitely greater than the probability of any other mistake, due to the trembling of one or several voters, it is enough to check that in both of these events the limiting strategy is preferred to the other undominated strategy.

For voters 1 and 2, the relevant contingencies which allow them to discriminate between their three undominated strategies is when the behavior of the others is summarized by the vectors [START_REF] Brams | Game Theory and Politics[END_REF][START_REF] Brams | Game Theory and Politics[END_REF][START_REF] Gevers | Social Welfare Functionals and Interpersonal Comparability[END_REF] and [START_REF] Brams | Game Theory and Politics[END_REF][START_REF] Gevers | Social Welfare Functionals and Interpersonal Comparability[END_REF][START_REF] Gevers | Social Welfare Functionals and Interpersonal Comparability[END_REF] Hence, (2,0,0) is the best reply to f ε -i . The same statement is true for voter 2. For voters 3 and 4, the relevant contingencies can be summarized by the vectors (6, 2, 5) and [START_REF] Gevers | Social Welfare Functionals and Interpersonal Comparability[END_REF][START_REF] Brams | Game Theory and Politics[END_REF][START_REF] Cox | Centripetal and Centrifugal Incentives in Electoral Systems[END_REF]. Let us denote their probabilities by p((6, 2, 5) | f ε -i ) and p((6, 4, 5) | f ε -i ). Furthermore, given voter's best responses, we can write that 2p((6, 2, 5) | f ε -i ) = p((6, 4, 5) | f ε -i ). Since the non-overstating strategy is the best reply to f ε -i . The same statement applies for voter 4.

Similarly, one can deduce that for voters i = 5, 6, 7 casting ballot (0,0,2) is a best response against f ε . Indeed, for voters 5, 6 and 7, the relevant contingencies are summarized by the vectors [START_REF] Gevers | Social Welfare Functionals and Interpersonal Comparability[END_REF][START_REF] Brams | Game Theory and Politics[END_REF][START_REF] Brams | Game Theory and Politics[END_REF] and [START_REF] Gevers | Social Welfare Functionals and Interpersonal Comparability[END_REF][START_REF] Gevers | Social Welfare Functionals and Interpersonal Comparability[END_REF][START_REF] Brams | Game Theory and Politics[END_REF]. Let us denote their probabilities by p((6, 4, 4) | f ε -i ) and p((6, 6, 4) | f ε -i ). Furthermore, given voter's best responses, we can write that 2p((6, 4, 4) | f ε -i ] = p((6, 6, 4) | f ε -i ). Since U 5 (0, 0, 2) = 3/2 p((6, 4, 4) | f ε -i ) + 4/3 p((6, 6, 4) |f ε -i ) = 25/12 p((6, 6, 4) | f ε -i ) > U 5 (0, 1, 2), U 5 (0, 2, 2).

The most probable pivot outcomes faced by voter 6 are as follows p((6, 4, 4) | g -6 ) = 1/3 p((6, 5, 4) | g -6 ) = 1/3 p((6, 6, 4) | g -6 ) = 1/3 From the pivot probabilities previously described, we have U 6 (0, 0, 2) = 25/9 U 6 (0, 1, 2) = 19/9 U 6 (0, 2, 2) = 13/9. implying that (0, 0, 2) is the unique best response for voter 6.

Voter 7

The most probable pivot outcome faced by voter 7 is the event [START_REF] Gevers | Social Welfare Functionals and Interpersonal Comparability[END_REF][START_REF] Brams | Game Theory and Politics[END_REF][START_REF] Brams | Game Theory and Politics[END_REF]. Due to her utility profile, voter 7 strictly prefers to use an undominated strategy and is indifferent among all of them: that is (0, 0, 2), (0, 1, 2), (0, 2, 2). Hence, the mixed strategy g 7 is a best response.

The second step of the proof consists in showing that g is a perfect equilibrium. To do so, consider the following completely mixed strategy combination g ε , where η i denotes the mixed strategy of voter i which assigns equal probability to all his pure strategies. i = 1, 2, 3 g ε i = (1 -27ε 2 )(2, 0, 0) + 27ε 2 η i i = 4, 5 g ε i = (1 -27ε 2 )(0, 2, 1) + 27ε 2 η i i = 6 g ε i = (1 -ε -27ε 2 )(0, 0, 2) + ε(0, 1, 2) + 27ε 2 η i , i = 7 g ε i = g 7 + 27ε 2 η i .

It is easy to see that, for ε sufficiently close to zero, this is an ε-perfect equilibrium. Suppose all voters other than i choose the strategies prescribed by g. Since for ε going to zero, the probability of voter 6 to tremble towards (0, 1, 2) is infinitely greater than the probability of any other mistake, it is enough to check that the limiting strategy is preferred to the other undominated strategy when either this mistake or no mistake at all occurs.

For voters 1 to 5, the relevant contingency is the one described by the limiting strategy g. Indeed, as has been shown, their unique best response is the one depicted by g as when the trembles tends towards, they have a unique best response. For voter 6, the same argument applies.

Finally, one can deduce that for voter i = 7 casting the mixed strategy ballot g 7 is a best response, against g ε . Indeed, for voter 7, the relevant contingency are summarized by the vectors [START_REF] Gevers | Social Welfare Functionals and Interpersonal Comparability[END_REF][START_REF] Brams | Game Theory and Politics[END_REF][START_REF] Brams | Game Theory and Politics[END_REF] and [START_REF] Gevers | Social Welfare Functionals and Interpersonal Comparability[END_REF][START_REF] Cox | Centripetal and Centrifugal Incentives in Electoral Systems[END_REF][START_REF] Brams | Game Theory and Politics[END_REF]. Let us denote their probabilities by p((6, 4, 4) | f ε -i ) and p((6, 6, 4) | f ε -i ). Since U 7 (0, 0, 2) = 3p((6, 4, 4) | f ε -i ) + 3p((6, 5, 4) |f ε -i ) = U 7 (0, 1, 2), U 7 (0, 2, 2). the mixed strategy g 7 is a best reply to g ε -i . Hence, {g ε } is a sequence of ε-perfect equilibria. Since g is the limit of g ε , it is a perfect equilibrium in which voters' best responses are not overstating.

  in which each ballot b * satisfies b * = f (b) = α + βb.

  with abstention ballots Abs(B EV ) = {(0, . . . , 0), (m, . . . , m)}. Theorem 2. EV and AV are strategically equivalent. Proof. The set of overstating ballots of AV satisfies Ove(B AV ) = {0, 1} K \ {(0, . . . , 0), (1, . . . , 1)}, and the set of overstating ballots of EV equals Ove(B EV ) = {0, m} K \ {(0, . . . , 0), (m, . . . , m)}, so that, by Theorem 1, EV and AV are strategically equivalent.

U 1

 1 (2, 0, 0) = 3/2 p((4, 4, 6) | f ε -i ) + 4/3 p((4, 6, 6) | f ε -i ) = 25/12 p((4, 6, 6) | f ε -i ) > U 1 (2, 1, 0), U 1 (2, 2, 0).

U 3 (

 3 0, 2, 1) = 1/2 p((6, 2, 5) | f ε -i ) + 4/3 p((6, 4, 5) |f ε -i ) = 19/12 p((6, 4, 5) | f ε -i ) > U 3 (0, 2, 0) = U 3 (0, 2, 2).

  . Let us denote their probabilities given voter's best responses by p((4, 4, 6) | f ε -i ) and p((4, 6, 6) | f ε -i ). Furthermore, given voter's best responses, we can write that 2p((4, 4, 6) | f ε -i ) = p((4, 6, 6) | f ε -i ). Since

The italian dubbio corresponds to the English doubt. According to[START_REF] Lines | Approval Voting and Strategy Analysis: A Venetian Example[END_REF] [START_REF] Lines | Approval Voting and Strategy Analysis: A Venetian Example[END_REF], this doubt is roughly equivalent to an abstention as "a doubt vote, if it ever did exist in doge elections, would essentially be a no vote".

There exists a whole literature on AV . See Laslier and Sanver (2010)[START_REF] Laslier | The basic approval voting game[END_REF].

In an election held under P V , a voter is allowed to give at most one point to at most one candidate. The candidate with the most votes wins the election. The most used rule for presidential elections is Plurality with a Runoff[START_REF] Blais | Direct Presidential Elections: A World Summary[END_REF] [START_REF] Blais | Direct Presidential Elections: A World Summary[END_REF]), but we here restrict attention to one-round voting systems.

See Sawyer and McRae (1962) [START_REF] Sawyer | Game Theory and Cumulative Voting in Illinois: 1902-1954[END_REF],[START_REF] Brams | Game Theory and Politics[END_REF] [START_REF] Brams | Game Theory and Politics[END_REF],[START_REF] Nitzan | The Vulnerability of Point-Voting Schemes to Preference Variation and Strategic Manipulation[END_REF] [START_REF] Nitzan | The Vulnerability of Point-Voting Schemes to Preference Variation and Strategic Manipulation[END_REF],[START_REF] Cox | Centripetal and Centrifugal Incentives in Electoral Systems[END_REF] [START_REF] Cox | Centripetal and Centrifugal Incentives in Electoral Systems[END_REF] and Gerber et. al (1998)[START_REF] Gerber | Minority Representation in Multimember Districts[END_REF].

Robustness requires that the set of electoral outcomes is not modified by allowing voters to choose from a wider set of ballots.

t r(t) = 1: in other words, r(t) represents the share of t-voters.

This is roughly equivalent to assume that the probability of candidates i and j being tied for first place is the same as the probability of candidate i being in first place one vote ahead candidate

See Theorem 1, page 105 in[START_REF] Myerson | A Theory of Voting Equilibria[END_REF] [START_REF] Myerson | A Theory of Voting Equilibria[END_REF].

Throughout the proof, t-voters play in pure strategies in the equilibrium σ. Similar arguments can be used to extend the proof whenever σ involves that some t-voters play in mixed strategies.

See Milgrom (2009, 2010) [START_REF] Milgrom | Assignment Messages and Exchanges[END_REF][START_REF] Milgrom | Simplified Mechanisms with an Application to Sponsored-search Auctions[END_REF] and Perez-Richet (2011)[START_REF] Perez-Richet | A Note on The Tight Simplification of Mechanisms[END_REF].

See[START_REF] Myerson | Comparison of scoring rules in poisson voting games[END_REF] [START_REF] Myerson | Comparison of scoring rules in poisson voting games[END_REF],[START_REF] Laslier | Strategic Approval Voting in a Large Electorate[END_REF] [START_REF] Laslier | Strategic Approval Voting in a Large Electorate[END_REF],Núñez (2009) [START_REF] Núñez | Sincere Scoring Rules[END_REF], Bouton and Castanheira (2010)[START_REF] Bouton | One Person, Many Votes: Divided Majority and Information Aggregation[END_REF],[START_REF] Goertz | On the Informational Efficiency of Simple Scoring Rules[END_REF] [START_REF] Goertz | On the Informational Efficiency of Simple Scoring Rules[END_REF], and Nuñez (2010)[START_REF] Núñez | Condorcet Consistency of Approval Voting: A Counter Example on Large Poisson Games[END_REF].

See Corollary 2.2.6, page 29 in van Damme (1996)[START_REF] Van Damme | Stability and Perfection of Nash equilibria[END_REF].

the non-overstating strategy is the best reply to f ε -i and similarly for voters 6 and 7.

Hence, {f ε } is a sequence of ε-perfect equilibria. Since f is the limit of f ε , it is a perfect equilibrium in which voters' best responses are not overstating.

B Appendix: Proof of Proposition 3

The first step of the proof consists in showing that g is a mixed strategy equilibrium. To do so, we compute the probability, under g, of each pivot outcome a player can face and, from these probabilities, the expected utility derived from each undominated strategy.

Voters 1,2,3

Even though the best responses are explained for the voter 1, the reasoning is analogous for voters 2 and 3.

From the pivot probabilities previously described, we have U 1 (2, 0, 0) = 25/9 U 1 (2, 1, 0) = 19/9 U 1 (2, 2, 0) = 13/9. which entails that (2, 0, 0) is the unique best response for voter 1.

Voters 4,5

Voter 4's best responses are analyzed, the reasoning being analogous for the voter 5. p((6, 2, 5) | g -4 ) = 1/3 p((6, 3, 5) | g -4 ) = 1/3 p((6, 4, 5) | g -4 ) = 1/3. From the pivot probabilities previously described, we have U 4 (0, 2, 0) = 1 U 4 (0, 2, 1) = 10/9 U 4 (0, 2, 2) = 1. implying that (0, 2, 1) is the unique best response for voter 4.

Voter 6


Virginie Wiktor

Philippe Grosseau

René Guyonnet

Eric Garcia-Diaz

Biodeterioration of cementeous matrix by fungi // Biodeterioration d'une matrice cimentaire par les champignons

come   L'archive ouverte pluridisciplinaire

Introduction

Micro-organisms -bacteria and cyanobacteria, fungi, algae, and lichens -are liable to grow on building materials: concrete, wall plastering, stone…. They are liable to degrade material properties: from an alteration of aesthetic aspect to a reduction of its service life. Biodeterioration mechanisms are complex and not yet absolutely understood. The biodeterioration is enhanced by environmental factors such as freeze/thaw cycles or pollution [START_REF] Th | Biodeterioration of stone: a review[END_REF].

Fungi are among the most harmful organisms associated to biodeterioration of organic and inorganic materials. Their occurrence on the stones is reported to be combined not only with aesthetical spoiling of the monuments, due to color changes and black spots, but also with a strong evidence that these organisms are causing crater shaped lesions, chipping and exfoliation of the rock surface combined with the loss of materials (Urzi et al. 1995, Urzi et al. 2000b). The black fungi such as Alternaria have the capacity to settle on the surface of rocks, to attach firmly and to penetrate deeper into the rock. The major way of attack and penetration of this group of rock infecting fungi is physical [START_REF] Diakumaku | Black fungi in marble and limestones -an aesthetical, chemical and physical problem for the conservation of monuments[END_REF]. Acidogenic fungi such as Aspergillus, Penicillium produce different organic acids, such as oxalic, gluconic, malic, citric, etc., in low and normal glucose concentrations (de la [START_REF] De La | Fungi in weathered sandstone from salamanca cathedral (Spain)[END_REF]. These acids form calcium salts or act as chelating agents of mineral cations favouring the biodeterioration process [START_REF] Berthelin | Microbial weathering processes[END_REF].

To control and to act efficiently against deterioration by fungi, it is necessary to have a better understanding of those mechanisms.

The aim of this paper is to present different stages taken into account for the setting up of a laboratory accelerated test of biodeterioration by fungi. Indeed in natural environment, biodeterioration takes place after several years. The most complex stage is to accelerate fungi development on specimens in order to reduce this time to observe biodeterioration. Two fungal strains were chosen for the study: (i) Alternaria alternata, responsible of physical deterioration, and (ii) Aspergillus niger, an acidogenic fungi responsible of chemical deterioration. Several parameters are studied: firstly physico-chemical parameters which accelerate fungi growth are determined. Then, relating to the matrix accelerated carbonation and leaching operation are performed to reduce matrix pH. Different ways of incubation are also studied.

Materials and methods

Matrix preparation

We focus on a model system: a pure cement paste. This study was conducted with white Portland cement CEM I 52.5 R. The water/cement mass ratio is 0.55. Hardened cement paste samples are parallelepipedic shaped, 1 × 2.5 × 8.5 cm. Samples were demolded 24 h after elaboration and stored 28 days at 100% relative humidity, at room temperature. Samples are sterilized by γ-radiation (30kGy) after 28 days conservation cure or after carbonation.

Accelerated carbonation operation

Carbonation operation is carried out in conditioning chamber under pure CO 2 flow. Relative humidity is maintained at 65±5% with salt solution of K 2 CO 3 (RP Normapur). In order to monitor the depth of carbonation, freshly broken samples of hydrated cement paste were sprayed with phenolphthalein indicator solution, which changes from colourless below a pH value of approximately 9.2 to purple above this pH value (RILEM 1989). Some samples are dried 24 h at 40°C before carbonation.

Leaching operation

Leaching operation is performed in 10 litres containers. Reactor solution is continuously replaced with deionized water. Flow rate is 7 ml/min. 3 carbonated specimens are dived into the reactor. Samplings of reactor solution are performed in order to measure calcium and carbonates concentration by ionic chromatography. pH is continuously measured.

Culture condition

Alternaria alternata is cultured at 24 ± 1°C, and 90 ± 5 % relative humidity. Some cultures are exposed to the light of horticultural neon tubes 10h/day in order to increase sporulation. Aspergillus niger is cultured at 24 ± 1°C, and 75 ± 5 % relative humidity.

Spore suspension

Spore suspension is performed with spores harvested 21 days for Alternaria alternata and 10 days for Aspergillus niger. Spores were harvested by flooding the plates with distilled saline solution (9g/l) and lightly scraping the surface. The resulting spore suspension was filtered through cheesecloth.

Incubation

Two ways of incubation are tested. Specimens are disposed in sterilized boxes (9.5 x 9.5 x 9.5 cm) as shown in figure 1. In the first way, specimens are covered by a thin agar-agar film whereas, in the second one, specimens are soaked with liquid nutritive medium. Then specimens are inoculated with spore suspension. Non carbonated, carbonated and carbonated leached specimens were inoculated.

Figure 1. Inoculation

Boxes are kept in culture chamber at 24±1°C and 75±5% or 90±5% HR, respectively for Aspergillus niger and Alternaria alternata.

CONSEC'07 Tours, France Wiktor 524

Results and discussion

Accelerated carbonation operation

Carbonation rate is maximum for relatives humidity comprised between 50 and 65% [START_REF] Chaussadent | Etat des lieux et reflexions sur la carbonatation du béton armé[END_REF]. So, to optimize carbonation, experiment is performed with K 2 CO 3 solution. Relative humidity was maintained at 65±5%. K 2 CO 3 appears to be a suitable salt to control relative humidity during carbonation.

After 48 hours of carbonation, phenolphthalein test revealed a carbonated zone of less than 1 mm for specimens not dried. Specimens dried have been carbonated on 2-3 mm thick (figure 2). However, it must be noticed that carbonation zone is not homogeneous : carbonation is most important on laitance face. Our aim is to reduce pH surface only, so carbonation depth obtained after 48 hours is appropriated.

Leaching operation

We can decompose the pH curve in 3 parts : (ii) 1.5 days to 23 days : we observe a slow and constant decrease in pH, from 9.8 to 8.5. We can assume that once soluble surface compounds are leached, another mechanism takes CONSEC'07 Tours, France

Wiktor 525 place : alkaline species released stem from pore solution. This is due to a concentration gradient between pore solution and leaching solution.

(iii) From 23 days : pH seems to reach a plateau around 8.5. Carbonates present in the leaching solution with calcium released involve calcite precipitation in pore matrix. This phenomenon acts as barrier which limits exchange between pore solution and leaching solution. Equilibrium of dissolution/precipitation of calcite is established, so that we can't obtain a lower pH.

Incubation

Incubation test for Alternaria alternata

First we notice that Alternaria growth on agar-agar medium is slow and less dense than in plate culture. Then we don't observe any development of the strain all over the test time for incubation with soaked specimens. Regarding to specimens covered with agar-agar film, no development is observed on non-carbonated and carbonated specimens. On the other hand, 10 days after inoculation, we notice a development on carbonated leached specimens, triggered by the presence of agar-agar medium (figure 4.). After 3 months of incubation, no more development than after 10 days is observed. So those short time tests of incubation stress 3 important points : (i) firstly, we can see that development is possible only on leached specimens, that is to say pH surface is a key parameter to develop our accelerated biodeterioration test. (ii) Then, way of incubation (soaked specimens or with thin agar-agar film) is fundamental to trigger strain growth. (iii) finally, Alternaria alternata is a strain very sensitive to experimental parameters including pH, so it isn't the more effective strain to set up rapidly our laboratory test. Concerning carbonated specimens, no growth on soaked specimen is noticed, but for 2/3 of samples development started after 10 days of incubation on agar-agar film. Most interesting result is obtained for carbonated leached specimen. For 1/3 of soaked specimens strain growth is observed after 10 days of incubation. For all specimens with agar-agar film development is very well triggered after 10 days (figure 6). This strain is less sensitive to pH surface than Alternaria alternata, that results in a better development on specimens. Absence of growth on non carbonated specimens confirms that pH surface is the key parameter for test set up. These results must be confirmed at 3 months.

Discussion

Weathering matrix by carbonation and leaching appears as a key parameter for the setting up. Then, we can assume that Aspergillus niger biodeterioration will be faster than Alternaria alternata's one. Hence, it grows faster and chemical deterioration by acid production necessitates less time to be noticed.

Conclusion

Our aim is to set up a laboratory accelerated test in order to have a better understanding of biodeterioration mechanisms. The more complex stage is to obtain in some months the same effects produced in some years in natural environment, that is to say in term of matrix weathering and fungal development. Incubation tests performed on non weathered and weathered specimens (figures 5-6) well underline the importance of pH surface on fungi development.

Once pH matrix resolved, we focus on incubation conditions. It appears that the best way to trigger fungi development is to cover carbonated and leached specimens (figures 4-5) with agar-agar film. We see that further settings are needed for slow development strain as Alternaria alternata.

According to preliminary results we can conclude that pH remains the main parameter to control. Best solution for the moment appears to be carbonation and leaching of specimens for matrix weathering and inoculation with spores suspension of specimen covered by a thin agar-agar film.

Figure 2 .

 2 Figure 2. Revelation of carbonated zone (in white) by phenolphthalein test. (a) Specimen not dried. (b) Specimen dried.

Figure 3 .

 3 Figure 3. pH evolution of leaching solution

Figure 4 .Figure 5 .

 45 Figure 4. Observations of incubation test with agar-agar film for Alternaria alternata at (a) 10 days and (b) 3 months of incubation on leached specimen

  Alternaria alternata Fries von keissler is obtained from DSMZ collection. Cultures are done on V8 juice agar medium (CaCO3 3g/L, Agar 20g/l, V8 juice 200ml/l), and Czapek-

		CONSEC'07 Tours, France
	2.5. Fungi
	2.5.1. Culture media
	Dox modified medium (cellulose powder 30g/l, KNO3 3 g:L, MgSO4,7 H2O 500 mg/l, KCl
	500 mg/l, FeSO4, 7 H2O 10 mg/l, K2HPO4 1 g/l, agar 20 g/l), to promote sporulation.
	Aspergillus niger strain is DSM N°823, obtained from DSMZ collection. Culture are
	performed on Potatoes Dextrose Agar medium (potatoes dextrose agar broth at 39 g/l).
	pH is adjusted at 7.2 with NaOH or HCl 0.1M solutions. Media are autoclaved 20 min at
	121°C.
	Wiktor	523

  We can assume that probably a longer carbonation may avoid leaching operation which necessitate about one month of experiment. Incubation tests with two different strains show us the importance of pH matrix for the test. Whereas some strains like Aspergillus niger are less sensitive with respect to pH, they still can't growth on basic surface (pH 12-13) such as cement.

		CONSEC'07 Tours, France
	analyses.
	Wiktor	527

As shown on incubation tests, carbonation isn't sufficient to trigger a good development of strains. XRD analyses show that carbonated specimens contain ettringite and portlandite, which mean that carbonation isn't complete. After leaching operation, we characterize only crystalline phases aragonite and calcite (polymorphs of calcium carbonate) by DRX