K P Jablonski

C Fretter

L Carron

T Forné

M.-T Hütt

A Lesne
email: lesne@lptmc.jussieu.fr

Genome Supranucleosomal Organization and Genetic Susceptibility to Diseases

The notion of disease-associated single-nucleotide polymorphisms (da-SNP), as determined in genome-wide association studies (GWAS), is relevant for many complex pathologies, including cancers. It appeared that da-SNPs are not only markers of causal genetic variation but may contribute to the disease development through an influence on gene expression levels. We argue that understanding this possible functional role of da-SNPs requires to consider their embedding in the tridimensional (3D) multi-scale organization of the human genome. We then focus on the potential impact of da-SNPs on chromatin loops and recently observed topologically associating domains (TADs). We show that for some diseases and cancer types, da-SNPs are over-represented in the borders of these topological domains, in a way that cannot be explained by an increased exon density. This analysis of the distribution of da-SNPs within the 3D genome organization suggests candidate loci for further experimental investigation of the mechanisms underlying genetic susceptibility to diseases, in particular cancer.

INTRODUCTION

In recent years, genome-wide association studies (GWAS) have been performed by international scientific consortiums on large cohorts of patients and healthy people in order to disentangle the genetic components from other factors, typically environmental factors, in the etiology of complex diseases, including cancer [1,2]. These studies statistically relate single-nucleotide polymorphisms (SNPs) with disease development, but provide no immediate explanation of the role of the genetic variants in the biological mechanisms responsible for the disease [3,4]. The need to go beyond statistical association and unravel functional aspects of cancer risk loci has been rapidly underlined [5]. While da-SNPs have originally been considered only as markers of the genetic variations causally involved in the disease, their possible direct implication in the pathologies, mainly through a dysregulation of gene expression levels, is now considered. On the other hand, recent experimental advances dramatically improved the knowledge of 3-dimensional genome organization and its functional role in transcriptional regulation [6]. Joining these two research domains opens a novel research direction addressing the role of genome architecture and its modifications in understanding the genetic risk to diseases [7]. We here briefly review the state-of-the-art about these questions, then provide statistical evidences showing how insights on the functional consequences of da-SNPs can be gained by considering their surrounding 3D genome organization as well as distal effects of genetic variations in regulatory elements. number of SNPs count While the results of these studies are available from the public GWAS catalog [17], the limits of GWAS and the difficulties in the interpretation of da-SNPs have been pointed out [18]. We observe that the number of SNPs associated to the same disease in the GWAS catalog, e.g. a cancer in a specific organ, is in the range of one hundred (depending on the thresholds on p-value and effect size in the definition of a da-SNP) (Figure 1). This indicates that a single da-SNP is not sufficient to determine the appearance (or non-appearance) of a disease, but only participates, together with numerous other genetic and non-genetic factors, in the biological mechanisms leading to the pathology. Therefore, more mechanistic understanding of the disease association is required, in particular for the development of personalized medicine [19] and in oncology [5,20]. Thanks to the 1000 Genome Project [15,16] and similar works, the knowledge of the genome of individuals became available, allowing a better characterization of da-SNPs, and opening the way towards experimental functional investigations. Globally, a novel view thus emerged according to which SNPs can have functional consequences [5,[21][22][23] and directly increase the propensity to develop a pathology [24]. One can note that the modification of the expression of a gene due to the variant form of a SNP can be further amplified by the gene regulatory network, thus leading to a major functional disturbance.

Different relationships to disease

A basic distinction lies in the dysfunction involved in the pathology, which can be due either to a defective biological factor (change in the sequence of a protein or non-coding functional RNA), or to a dysregulation of the amount of some relevant factors (typically their expression levels). This distinction reflects in the possible functional consequences of a SNP, as follows.

A first class, coding SNPs, corresponds to da-SNPs located in exons hence affecting the sequence of a protein [25]. In this case, the protein structure and presumably the protein function are affected, and the causal role of the SNP is thus straightforward. This class can be extended including pseudo-coding SNPs, located in genomic regions that are transcribed into various non-coding RNAs (tRNA, microRNA, siRNA) [26]. Another situation of SNPs having a direct impact on a protein structure are mutations disrupting the proper splicing events or affecting protein translation when located in 3' or 5' UTR [27].

The second class is composed of non-coding da-SNPs, located in introns or intergenic regions. Their association to diseases is expected to lie in the modification, in individuals harboring the variant allele, of some levels of gene expression, and as such they are generally termed regulatory SNPs [21,28]. It actually appeared that most da-SNPs (more than 90%) belong to this class [29][30][31]. A simple sub-class of regulatory SNPs is composed of da-SNPs located in the binding site of a regulatory protein, typically a transcription factor, so that the mutation (i.e. the variant allele) directly disturbs the transcription initiation of the gene regulated by this transcription factor, and accordingly its expression level [9,32]. We will focus on another, more recently evidenced, sub-class composed of architectural SNPs, where the SNP variation first reflects in a local variation of the genome spatial conformation, which in turn affects some gene expression levels [7].

Previous approaches on cancer-associated SNPs typically used gene expression data to infer which SNPs may affect gene expression levels (eQTLs) and to delineate the associated pathways [9,33]. We here adopt a complementary viewpoint and investigate da-SNPs, and in particular cancer-associated SNPs, in the context of the tri-dimensional (3D) genome organization.

3D GENOME ORGANIZATION AND GENETIC RISK 3D genome organization and its functional role

Recently developed techniques of chromosome conformation capture combine chemical crosslinking and sequencing to identify genomic loci contacting each other in vivo. They have shown that the mammalian genome displays three main architectural features at the large-scale level (supranucleosomal level, beyond the kb scale), nested in a hierarchical way (Figure 2): chromatin loops, topologically associating domains (TADs) of larger size exhibiting more internal contacts than contacts between domains [34,35], and a segregation in active and inactive compartments [36]. [34]) evidence the existence of disjoint self-contacting domains, known as topologically associating domains (TADs). The full contact map (i.e. of all chromosomes, resolution 160kb) for human embryonic stem cells is displayed on panel A, and a zoom on chromosome 5 on panel B (resolution 80kb). The color indicates the contact frequency (the darker the more frequent). On panel C, the partition of the genome in TADs is illustrated for a part of chromosome 5, with an additional zoom (resolution 40kb): TADs appear in the form of disjoint dark triangles along the diagonal of the contact map. One of them is underlined in black as an example. Fuzzy sub-structures of smaller size within TADs may feature chromatin loops

TADs existence and distribution, as well as the genome segregation into active and inactive compartments have been unraveled using genome-wide Hi-C. TADs organize the genome into a modular and presumably functional structure at the sub-megabase scale [37]. Chromatin loops and TADs have also been investigated experimentally using local quantitative and high-resolution chromosome conformation capture technique (3C-qPCR) [38,39]. Such experiments using 3C-qPCR clarified the distinction between the TADs as originally evidenced in [34] and [35], of size 200 kb to 1Mb (median size about 800 kb) and structures of various sizes termed chromosome contact domains [40], among which the smallest ones rather correspond to chromatin loops embedding a single gene and its proximal regulatory sequences [41].

The above-mentioned architectural features are closely involved in the regulation of gene expression [6], and architectural changes are observed in pathologies. Significant changes in genomic contacts have been observed in bladder cancer and lymphomas [42], as well as in breast cancer [43,44], revealing altered chromatin architecture. Another emblematic example is the observed disruption of chromatin enhancer-promoter loops at specific loci in βthalassemia (β-globin gene, [7]) and more generally erythroid pathologies (MYB gene, [45]). Gene-specific multienhancer contacts playing a key role in this 3D genome spatial organization are similarly likely to be mutationsensitive loci [46]. More generally, bioinformatic analyses show that a large number of da-SNPs (over all diseases) are located at enhancers inside TADs [31].

Changes in genome architecture, in particular TAD disruption, have also been evidenced in various cancers [47,[START_REF] Naimark | AIP Conference Proceedings[END_REF]. The changes in long-range chromatin interactions associated with either the fusion of adjacent TADs or the creation of new TADs are potentially related to cancer progression, due to the ensuing dysregulation of oncogenes and tumor suppressors [START_REF] Valton | [END_REF]. Specifically, TAD boundaries have been shown to be altered (microdeletion, mutations or epimutations in the binding sites of the architectural protein CTCF) in T-cell acute lymphoblastic leukemia (T-ALL) [50] and in gliomas [51]. More generally, variations of chromatin architectural patterns have been observed between individuals [52]. This prompts to investigate their possible relationships with genetic variations, possibly mediated by variation in the epigenetic marks [START_REF] Naimark | AIP Conference Proceedings[END_REF], and their functional correlates.

Architectural SNPs

The understanding of the functional role of the 3D genome organization triggers an increasing interest in noncoding da-SNPs, whose effect on gene regulation is mediated by a change in chromatin 3D organization [7]. Such a scenario represents an instance of chromatin allostery, where the chromatin itself behave as an allosteric object [53]; here the allosteric transition would be triggered by the SNP and the effector would be some event necessary for transcription, e.g. TF binding or the formation of a promoter-enhancer loop [54].

These architectural SNPs can have an impact at multiple scales. A basic instance is a SNP in the binding site of the architectural protein CTCF (CCCTC-binding Factor), which have been shown to affect both loops and TAD boundaries [55], with stronger binding sites at TAD boundaries [56]. Another frequent instance is a SNP located in an enhancer, and whose variation affects the proper formation of enhancer-promoter loop [31,57]. For instance, some risk loci for epithelial cancers, including colon, breast, and prostate cancers, have been found at enhancers forming a long-range chromatin loop with the MYC proto-oncogene in a tissue-specific way [START_REF] Ahmadiyeh | Proc. Nat. Acad. Sci. USA[END_REF]. It is has been shown experimentally, using chromosome conformation capture techniques, that a single SNP can actually disrupt an enhancer-promoter loop and have deleterious effects through a dysregulation of the associated gene, potentially leading to disease, e.g. autoimmune diseases [START_REF] Verlaan | [END_REF], asthma [60] or erythroid pathologies [45]. More generally, certain SNPs located in regulatory regions may affect not only the expression of nearby genes, e.g. through a modification of epigenetic marks, but also the expression of distant genes located in cis (that is, on the same chromosome, as opposed to trans acting factors) in the same TAD. This has been shown for SNPs associated with autoimmune diseases, with distal molecular coordination effects of range about 50 kb, within the same TAD [61].

We will here investigate the distribution of da-SNPs with respect to TADs and more specifically TAD borders. We here call TAD border the region limiting a TAD and across which very few physical contacts occur, thus corresponding to a marked insulation (slightly different from and more general than what is called TAD boundary in [34], namely the region separating two TADs along the genome). These borders can be observed using the 3C-qPCR technique (see e.g. the case of the HoxD locus in [41]). Certain TAD borders are involved in gene regulation events. The emblematic example is the case of Hox genes, where a displacement of a TAD border at the HoxD locus induces a switch of the Hoxd genes (from Hoxd11 to Hoxd8) from a telomeric TAD to a more centromeric TAD thus changing their expression levels [62]. Therefore, regulation of gene expression can be altered by the presence of a SNP variant as soon as this mutation is able to disrupt the border. Engineered genetic alteration of TAD borders using CRISPR/Cas9 at the WNT6/IHH/EPHA4/PAX3 locus induces limb malformation in mice; specifically, mutations in the binding sites of the architectural protein CTCF have been shown to cause the appearance of longrange interactions between some promoters of a TAD and enhancers located in the adjacent one, reflecting topologically in the fusion of adjacent TADs and functionally in the abnormal expression of these genes [63].

STATISTICAL INVESTIGATION

We performed a statistical investigation of the relationship between disease association (that is, genetic risk) and location with respect to TAD borders, considering first all da-SNPs then focusing the analysis to cancer-associated SNPs (that is, SNPs whose variant form is statistically associated with an increased risk of developing a cancer, not to be confused with somatic mutations accumulating in cancer cells during cancer development). We aim at a novel annotation of da-SNPs based on their potential impact on the functional 3D chromatin architecture, which could suggest candidate loci for a further experimental and mechanistic study.

Methods

We used the compilation of GWAS data freely available in the GWAS catalog (www.ebi.ac.uk/gwas/, version r2017-06-26) [17]. We translated the genomic coordinates given in the GWAS catalog (Genome assembly GRCh38.p10, in short hg38 reference) into the hg19 reference. We used both directly observed and imputed da-SNPs, where imputation is based on the belonging to the same haplotype block than an observed association. We constructed the list of da-SNPs and the subset of cancer-associated SNPs by filtering the EFO label (Experimental Factor Ontology) indicating the trait(s) or disease(s) to which each SNP of the catalog has been significantly associated. Only diseases (including cancers) displaying at least 10 associations are considered. The da-SNPs of the list are moreover annotated with their genomic location in exons, introns or intergenic regions.

TAD coordinates have been determined in [34] for human foetal lung fibroblasts (IMR90 cells) and human embryonic stem cells (hESC), from two replicates, with only minor variations between the two cell types. We used the data corresponding to hESC (human embryonic stem cells), for which more annotations are available). To account for the topological characterization of borders seen as regions across which the contact frequency displays a marked decrease [41], we defined TAD borders as zones of 20kb located inside the TAD at the limits of this TAD (recall that the median size of a human gene is 23kb [START_REF] Scherer | Guide to the Human Genome[END_REF]). Results remain stable under variation of the border size.

Enrichment in da-SNPs of a given genomic region (e.g. TAD borders) has been assessed for each disease using a uniformly random null model, described by a binomial distribution of parameters the fraction of base pairs in the given genomic region and the total number of SNPs associated to the disease (this distribution corresponds to the large-size limit of the currently used hyper-geometric distribution, given that the total number of SNPs in the human genome is of order of tens of millions --the number is evolving each year as new results accumulate [15,16]). Enrichment p-value is then computed as the probability (cumulative binomial distribution) to get by chance at least the observed number of da-SNPs in the considered genomic region. Multiple testing (over the different diseases in the GWAS catalog, considering independently cancer types and non-cancer diseases) was corrected for using the Benjamini-Hochberg procedure [START_REF] Benjamini | [END_REF]66]. A (corrected) p-value smaller than 0.05 assesses the statistical significance of the enrichment. The number of da-SNPs present in the TAD borders gives a measure of the effect size.

The active compartment has been determined at a resolution of 40kb, by an analysis of the principal eigenvector of the contact correlation map that refines the method in [36] (data from [34], normalized according to [67]). The enrichment p-value has been computed using a uniform random null model and a cumulative binomial distribution.

Location of genetic risk loci with respect to topological domain borders

We first observed in the GWAS catalog (for all diseases investigated in GWAS) that only 8% of da-SNPs are located in exons, in agreement with the evaluation given in [7] and [31]. This number confirms that most da-SNPs are non-coding and their potential effect lies in a dysregulation of gene expression. As such, these da-SNPs belong to the class of expression quantitative traits loci (eQTLs, [9,52]). It is to note that introns may contain enhancers for another gene [68], so that intronic da-SNPs may participate to the (dys)regulation of another gene. We then performed a statistical investigation of the location of da-SNPs with respect to genomic architectural features, considering first all diseases, then more specifically cancers. To detect a possible impact of some da-SNPs on TAD delimitation, we considered TAD borders, namely regions surrounding the limits of the TADs, and across which a step-wise decrease of the contact frequency occurs [41] (see section Methods). We found that for a small fraction of diseases, da-SNPs are over-represented in TAD borders (Figure 3). We more specifically compared the set of cancers and the set of other diseases (Figure 3). Both sets contain diseases for which TAD border enrichment is observed, and which can be identified in our analysis. This result supports a role of TAD borders and their modifications in these particular diseases, and suggests further investigation of the corresponding da-SNPs and surrounding loci. Noticeably, our results show that a significant enrichment of TAD borders in da-SNPs is present only for specific diseases and cancers, and thus represents a nontrivial feature. Noticeably, cancers dominate among the diseases displaying such TAD-border enrichment. The corresponding cancer types are listed in Table 2, together with the number of associated SNPs, while the list of the non-cancer diseases can be found at http://www.lptl.jussieu.fr/user/lesne/disease_list1.txt . It has been observed in [34] that inter-TAD regions (termed TAD boundaries) are enriched in housekeeping genes. However, this fact cannot explain the association of embedded SNPs to a specific disease, since a flaw in a housekeeping gene would produce far more defects than a specific pathology. Our result about TAD borders enrichment in da-SNPs for only a small subset of diseases points at the existence of more specific mechanisms. The cancer-associated SNPs located in TAD borders, for cancer types displaying TAD-border enrichment, appear to be preferentially located in the active compartment (Table 3). Moreover, the enrichment of TAD borders is mainly due to non-exonic SNPs (Figure 4). Over-representation in TAD borders of exonic SNPs is observed for only two cancer types (gastric and ovarian cancers). Interestingly, exonic and non-exonic enrichments exclude one another, which points at two distinct mechanisms. Overall, these results support a functional interpretation of these SNPs as being involved in the deregulation of gene expression, presumably through a change in the TAD borders, and promote further experimental investigation to specify and validate this interpretation. For each cancer type, described by its EFO label (Experimental Factor Ontology), the TAD border enrichment p-value is computed separately for the subsets of exonic and non-exonic da-SNPs. Enrichment in exonic SNPs is observed only for two cancer types (gastric and ovarian cancers).

da-SNPs in exons in introns intergenic

EFO term

cancer associated SNPs (for cancer types in

Non-exonic enrichment is observed for breast, oral cavity and testicular germ cell cancer, chronic lymphocytic leukemia, and hepatocellular carcinoma. The dotted red lines indicate the significance threshold at p=0.05. The dotted blue line indicates equal p-values for exonic and non -exonic subsets It has recently been shown that the very existence of TADs (but not the compartmentalization into active and inactive domains) depends on the CTCF protein [69], which suggests analyzing the relationship between cancerassociated SNPs and CTCF binding sites. The link with cohesin binding sites is also a promising direction [31,70,71]. It is however to note that CTCF binding sites determined by sequence analysis have not the same strength (occupancy rate), and that only part of them are involved as insulators at the TAD boundaries [56] so that the analysis cannot be done in a straightforward way.

CONCLUSION: CANDIDATE LOCI FOR EXPERIMENTAL STUDIES

The present approach proposes to extend GWAS results interpretation by including in the analyses some aspects of the 3D genome architecture. We here focused on the large-scale sub-megabase organization of the genome into topological domains, with the working hypothesis that modifications in their borders may mediate a functional role of some disease-associated single-nucleotide polymorphisms.

Our main result is a list of diseases, including some cancer types, and a list of cancer-associated SNPs whose variant forms could contribute to a dysregulation of gene expression at a TAD border. This result delineates candidate loci for experimental studies, for instance 3C-qPCR experiments measuring contact frequencies in a quantitative way and thus able to assess a displacement or a disruption of a TAD border [41]. Genome-wide Hi-C investigation can also been envisioned, as done in a study of breast cancer genetic risk focusing on the long-range interactions between putative regulatory elements, harboring cancer-associated SNPs, and distal target genes [72]. Chromatin immuno-precipitation (ChIP-qPCR) can then be exploited to find which protein binding, if any, is affected at these SNPs and elucidate the basic step of their functional role and diseases association.

FIGURE 1 .

 1 FIGURE 1. Histogram of the number of da-SNPs per disease. It is drawn over all diseases considered in the GWAS catalog

FIGURE 2 .

 2 FIGURE 2. Topologically associating domains (TADs) in the human genome as derived from Hi-C data. The figure sketches how genome-wide chromosome conformational capture data (Hi-C contact maps, data from[34]) evidence the existence of disjoint self-contacting domains, known as topologically associating domains (TADs). The full contact map (i.e. of all chromosomes, resolution 160kb) for human embryonic stem cells is displayed on panel A, and a zoom on chromosome 5 on panel B (resolution 80kb). The color indicates the contact frequency (the darker the more frequent). On panel C, the partition of the genome in TADs is illustrated for a part of chromosome 5, with an additional zoom (resolution 40kb): TADs appear in the form of disjoint dark triangles along the diagonal of the contact map. One of them is underlined in black as an example. Fuzzy sub-structures of smaller size within TADs may feature chromatin loops

FIGURE 3 .

 3 FIGURE 3. TAD border enrichment in disease-associated SNPs. The normalized histograms of (minus the logarithm of) pvalues assessing TAD border enrichment for all non-cancer diseases (blue) and all cancer types (yellow) are compared (TAD border of size 20kb inwards). The grey color corresponds to the overlap of the two histograms. The vertical line indicates the significance threshold at p=0.05. It means that for each disease and each cancer type contributing to the right part of the histograms, the associated genetic risk loci display a significant over-representation in TAD borders

FIGURE 4 :

 4 FIGURE 4: Enrichment of TAD borders in exonic and non-exonic cancer-associated SNPs. For each cancer type, described by its EFO label (Experimental Factor Ontology), the TAD border enrichment p-value is computed separately for the subsets of exonic and non-exonic da-SNPs. Enrichment in exonic SNPs is observed only for two cancer types (gastric and ovarian cancers).Non-exonic enrichment is observed for breast, oral cavity and testicular germ cell cancer, chronic lymphocytic leukemia, and hepatocellular carcinoma. The dotted red lines indicate the significance threshold at p=0.05. The dotted blue line indicates equal p-values for exonic and non -exonic subsets

4 TAD

 4 border non-exonic SNPs (log p) TAD border exonic SNPs (-log p)

TABLE 1 .

 1

Distribution of da-SNPs. The table indicates the location of da-SNPs in the different genomic compartments, according to the data gathered in the GWAS catalog (over all diseases).

TABLE 2 . Cancer types displaying a significant enrichment of TAD borders in da-SNPs.

 2 For each of these 12 cancer types, the total number of da-SNPs and the number of da-SNPs belonging to a TAD border are indicated. The list of cancer-associated SNPs located in TAD borders for these cancer type, is given in Annex, Table4.

		Cancer type	p-value (TAD	number of	number of da-SNPs
			border enrichment)	da-SNPs	in TAD borders
	EFO_0000095	Chronic lymphocytic leukemia	0.0043	124	13
	EFO_0000178	Gastric cancer	0.0157	18	4
	EFO_0000182	Hepatocellular carcinoma in	0.0145	10	1
		hepatitis B infection
	EFO_0000305	Breast cancer	0.0055	244	22
	EFO_0000571	Lung adenocarcinoma	0.0013	25	1
	EFO_0000756	Melanoma	0.0055	55	13
	EFO_0001071	Lung cancer	0.0034	50	7
	EFO_0001075	Ovarian cancer	0.0002	35	8
	EFO_0005088	Testicular germ cells cancer	0.0157	41	7
	EFO_0005570	Oral cavity cancer	0.0145	9	3
	EFO_0005842	Colorectal cancer	0.0112	181	12
	EFO_1000650	Breast cancer	0.0004	27	3
		(estrogen-receptor negative)

Table 2 ) total number number in the active compartment % p-value of the enrichment

 2

	73	47	64,4%	7.21 10 -6

TABLE 3 . Preferential location of cancer-associated SNPs in the active compartment.

 3 For cancer types displaying a TADborder enrichment (Table2), the genetic risk loci located in TAD borders (Annex, Table4) are over-represented in the active compartment (which represents only 39.9% of the genome).

ACKNOWLEDGMENTS

This work has been supported by the "Mission for Interdisciplinarity" of the French National Center for Scientific Research (CNRS), program InFIniTI 2017, project 3D-SNPs, Grant 232647 (to A.L.) and by the "Agence Nationale de la Recherche", project CHRODYT, Grant ANR-16-CE15-0018-04 (to T.F.). The present paper is based on a talk given by AL during the international conference "Physics of Cancer: Interdisciplinary Problems and Clinical Applications", Tomsk, May 23-26, 2017. AL thanks the organizers of this conference and ISPSM SB RAS (Tomsk) for hospitality. MTH thanks LPTMC (Paris) for hospitality and CNRS funding of his stay. MTH also acknowledges financial support from the German Ministery for Education and Research (sysINFLAME project within the e:med program, grant 01ZX1306D).

The study reported here was conducted according to accepted ethical guidelines involving research in humans and/or animals and was approved by an appropriate institution or national research organization. The study is compliant with the ethical standards as currently outlined in the Declaration of Helsinki. All individual participants discussed in this study, or for whom any identifying information or image has been presented, have freely given their informed written consent for such information and/or image to be included in the published article.

ANNEX: LIST OF CANDIDATE ARCHITECTURAL DA-SNPS

EFO term Cancer type da-SNPs in TAD borders

 1). Note that the number of SNPs associated to a cancer may be different from Table 2, in which the cancer-associated SNPs observed in independent studies were considered as different entries, like it is done in the GWAS catalog. The data presented in the above table can be downloaded at http://www.lptl.jussieu.fr/user/lesne/SNP_list2.txt


Sebastien Rouquette
email: sebastien.rouquette@univ-montp2.fr

Parameter estimation of gaussian heat flux : investigation of stationnary Gas Tungsten Arc Welding (GTAW) operation

Keywords: gradient conjugate, adjoin problem, inverse heat transfer problem. 1

A mathematical modelling has been established in order to predict the weld pool shape evolution in the frame of stationary gas tungsten arc welding experiment. In this mathematical modelling, the heat flux was modelled with a Gaussian function. This heat flux represents the energy generated by the electrical arc and absorbed into the workpiece. An inverse approach is used for estimating the parameters describing this heat flux from the "measured" temperatures. The Gradient Conjugate Method is used for solving the inverse problem. An adjoin problem is also solved in order to compute the gradient of the objective function. The aim of this communication is to check numerically the feasibility of the inverse problem and its robustness to random measurement errors.

Introduction

Welding processes are widely employed in manufacturing industries such automobiles, airplanes, petrochemical plants or again nuclear power stations. These structures require high level of safety what means high quality of joint properties (mechanical and microstructure). Few manufacturing welding processes reach such level of performance when its control is well handled. Gas tungsten arc welding (GTAW) is one of the most widely employed welding processes that is applied with success for welding of stainless steels and non-ferrous materials. In this process, a tungsten electrode is shielded with a flow of either pure inert gas such as argon or helium or a mixture of both gases (some nitrogen or hydrogen can be added). The assembly of work pieces is obtained through a local fusion due to the heat generated from an electrical arc.

The analysis of the physical phenomenon (electromagnetism, fluid flow, heat transfer …) that takes place in the process is crucial to understand, for example, weld pool formation [START_REF] Fan | Heat transfer and fluid flow in a partially or fully penetrated weld pool in gas tungsten arc welding[END_REF][START_REF] Traidia | Effect of Helium-Argon Mixtures on the Heat Transfer and Fluid Flow in Gas Tungsten Arc Welding[END_REF] as well as microstructure changes in the thermally affected base metal [START_REF] Zhang | Modeling and real time mapping of phases during GTA welding of 1005 steel[END_REF] and residual stress distribution in the joint [START_REF] Chang | Numerical and experimental investigations on the residual stresses of the butt-welded joints[END_REF]. The heat flux absorbed within the work piece must be known accurately in order to better understand all the mechanical and metallurgical consequences of the welding operation.

In order to perform a reliable multi-physics simulation of the work piece, it is extremely important to know correctly the heat input absorbed into the work piece. Inverse heat conduction problem represents an alternative approach of obtaining the heat flux incoming into to the work piece. This present work is also a preliminary study on the heat flux estimation of gta welding spot. The main goal of this work is to validate numerically the estimation of the unknown parameters (gtaw efficiency and heat flux distribution) with temperature measurements. Gtaw operation involves several phenomena: heat transfer, electromagnetism, fluid flow, phase transformation, free surface, deformation … these overall phenomena are represented in figure 1. The weld pool (molten metal) is driven through a combination of forces: Buoyancy (Boussinesq), Lorentz (electromagnetic), Marangoni (surface tension) and Impigning plasma. The fusion of the metallic work piece is due to the high thermal energy generated by the electrical arc between the electrode tip and work piece surface. There is also some volumetric heat generated within the work piece due to the electric current also called Joule effect.

2.

GTAW multiphysic problem

GTAW assumptions

The following assumptions are made in order to simplify the gtaw model:


The study is restricted to stationary spot welding so the problem is axisymmetric.



The flow is laminar and incompressible due to the small size of the weld pool.



The thermo-physical properties of the 304L stainless steel are temperature dependent as well as the surface tension coefficient.



The magnetic field is not considered as the Lorentz force is known to be a minor one for intensities below 200A [START_REF] Fan | Heat transfer and fluid flow in a partially or fully penetrated weld pool in gas tungsten arc welding[END_REF] 

The cylindrical work piece is immobile and it is kept up with four rods in order to limit any heat transfer by conduction. The radius and thickness are respectively mm 40 and mm 4 .



The arc plasma is not modelled as well as the free surface of the weld pool. Based on these above assumptions, the governing partial differential equations are expressed in the   z r, coordinate system.

Comparison of numerical results with experimental data

The reader will find the details of the governing equations (heat transfer and fluid flow) in [START_REF] Unnikrishnakurup | Multiphysic modelling of GTAW Process and experimental validation for investigating weld pool behaviour[END_REF] as well as the values of the constant used in the simulation except for the thermal conductivity, specific heat, mass density and dynamic viscosity are chosen as temperature dependent and their evolution are taken from [START_REF] Kenneth | recommended thermo-physical values for commercial alloys[END_REF].

The finite element software used for this simulation is Elmer ( http://www.csc.fi/english/pages/elmer ). Elmer is a finish Open Source Finite Element Software for Multiphysical Problems developed by the CSC -IT Center for Science. The mesh was realised with Gmsh software (http://geuz.org/gmsh/ ). The mesh consisted of 840 nodes and 1564 triangle finite elements. The order of the finite element was set to 1 for the temperature and the fluid velocity and 0 for the pressure. The experimental conditions for the stationary gtaw are summarized in table 1. The average welding intensity and tension are respectively 150A and 10.4V. The values used in the simulation are the ones measured on the welding generator. Welding intensity and tension are simultaneously measured with temperature measurements as well as the top surface evolution of the weld pool is recorded with a high speed camera. These data are reported in figure 2 and3. The temperatures were measured at three different positions on the backside of the stainless steel disc. The positions were respectively: 0mm (Tmes 1), 4mm (Tmes 2) and 8mm (Tmes 3). The maximum temperature of the disc back side is about 1100°C for sensor 1, see figure 3. On the backside of the disc, it can be evaluated that the thermal gradient is about 62.5°C/mm while it was about 150°C/mm on the symmetry axis (the weld pool width was estimated to 1.5mm from a macrograph not presented here). These thermal gradients are quite classic during welding operations. The final radius of the weld pool was about 5.5 mm for a 5.8s arc duration.

The established gtaw model overestimates the measured temperatures at the three sensors, see figure 2, as well as it underestimate the weld pool radius, figure 3.

The comparison between experimental and calculated temperatures and weld pool radius shows some important discrepancies. In this simulation gtaw efficiency and gaussian radius were evaluated from the literature and they are constant parameters. These two parameters must be known accurately as they influence the heat flux distribution. Then, it is proposed to estimate these parameters with an inverse heat transfer technique in the next chapter. In addition, the gtaw efficiency is supposed to vary with the time in order to capture some transient phenomena between the arc plasma and the steel disc.

Inverse heat transfer problem (ihtp)

In order to simplify the stated direct problem in the previous chapter, this chapter will deal with an inverse heat conduction problem only. Once the basis of this simplified inverse problem is made, it will be extended to the thermo-convective gtaw case. The aim of this inverse problem is to estimate a time varying gtaw efficiency and a constant gaussian radius parameter.

Energy conservation with no convective transport:

I in 0 1                            z T z r T r r r t T C eq p    (1)
with

T f H C C L f p eq p   
and L f is the liquid fraction:

             L L S S L S L T T if T T T if T T T T T T if f 0 ; ; (2)
With the following boundary conditions are:

      I on e R t I t U t r z T AB R r b W W b                   2 2 1 2 . 2 1 ,    (3)     I on T T T T h n T DE CD BC              4 0 4 0    ( 4
) I on r T FA EF         0  (5)
The initial condition is:

     in 0 , 0 T t r T (6)   t r, 
is the heat flux exchanged between the argon arc plasma and the top surface workpiece. The heat flux distribution is modelled with a Gaussian distribution [START_REF] Unnikrishnakurup | Multiphysic modelling of GTAW Process and experimental validation for investigating weld pool behaviour[END_REF]. Inverse heat transfer problems are known to be ill posed [START_REF] Tikhonov | Solutions of Ill-Posed Problems[END_REF] in contrast to the direct heat transfer problems which are well posed. In this latter, the solution exists, the solution is unique and the solution is stable to small changes in the input data.

A variety of numerical techniques have been proposed for the solution of inverse problems dealing with heat transfer problems. As examples, Murio [START_REF] Murio | The Mollification Method and the Numerical Solution of Ill-Posed Problems[END_REF] developed the mollification method, Alifanov [START_REF] Alifanov | Extreme Methods for Solving Ill-Posed Problems with Applications to Inverse Heat Transfer Problems[END_REF] with co-workers suggested and developed the iterative regularization method. In this approach an optimization problem is solved, using a method such as the conjugate gradient, being the gradient equation determined by an adjoint problem [START_REF] Jarny | A general optimization method using adjoint equation for solving multidimensional inverse heat conduction[END_REF]. This latter technique is used in this work for the estimation of the time varying gtaw efficiency and its gaussian radius distribution.

Statement of the inverse heat transfer problem

The IHTP general formulation is written as [START_REF] Necati | Inverse heat transfer, fundamentals and applications[END_REF]: The gradient conjugate algorithm is presented in figure 4. This method requires first to solve the mathematical model which describes the investigated process. Then, calculated temperatures are compared to measured temperatures on the real process. Then, the objective function as defined in ( 7) is computed. If the objective function is greater then a defined threshold, say 10-3 , the iterative process starts in order to estimate the unknown parameters. The unknown parameters are: a time dependent efficiency: η(t) and the gaussian radius distribution: RB. The iterative process consists in solving the adjoin problem in order to get the gradient of the objective function. Hence, the sensitivity problem is solved in the aim to compute the descent depth. Once, the gradient and the descent depth are computed, the new set of unknown parameters are evaluated. The mathematical problem is solved again with the updated unknown parameters until the objective function reaches the predefined threshold or a given number of iterations.

Gradient conjugate method with the adjoin and sensitivity problems

The adjoin and sensitivity problems are quickly defined in the paragraphs below. The reader can refer to [START_REF] Necati | Inverse heat transfer, fundamentals and applications[END_REF] to understand the details of their computation. Then, the gradient expressions of the objective function are also given without any details. The reader can refer to [START_REF] Necati | Inverse heat transfer, fundamentals and applications[END_REF] for the expression of the descent depth.

The adjoin problem:

The adjoin equations are obtained by solving the following minimization problem defined by the cost function under the constraint of the heat conduction equation:

                         T dtd T t T Cp p S p T L     , , (8)
where a Lagrange function   z t r ; ,  is introduced. Adjoin equations are obtained when the following conditions is checked:

0      L (for a fixed value   z t x ; , 
). The adjoin equations are defined by the equations ( 9) to ( 12):

I in p) t; , ( 1                            i eq p r E z z r r r r t C       (9)
With the following boundary conditions ( 16)-( 17) and final condition (18):

  I on T h n DE CD BC AB                  3 4  ( 10
)
I on r FA EF         0   (11)      in 0 , end t t r  (12) Remark:                 d x x t T p t r T p t r E i meas i i  ; , ; , .

The sensitivity problem:

The sensitivity equations are defined as follows:

        p t r T p p t r T T ; , ; , lim 0     ( 13
)
Where T  is the sensitivity function which results of a small variation

  B R p    , 
of the unknown parameters of the heat flux. Applying (13) leads to the sensitivity equations ( 14) to (18):

I in 0 1                            z T z r T r r r t T C eq p     (14)
Associated to the following boundary conditions and initial condition:

    I R r R R t R t I U z T AB B B B B W W                                2 3 2 5 . 0 exp 2 . 2       (15)   I T T h r T DE CD BC                3 4
(16)

I on r T FA EF         0   (17)      in 0 0 , t r T  (18)
Gradient expressions:

The gradient of the objective function is deduced from the adjoin function

  z t x ; ,  when:   p p T S p p L dL   , ,      with     p p T S p T S     , , and 0    T T L  .
Consequently, the gradient of the objective function according to gtaw efficiency is:

                         T AB B B W W AB dtd z t x R r R I U S ; , 5 . 0 exp . 2 2 2    (19)
While the gradient with respect to gaussian radius RB is:

                            T AB B B W W R AB B dtd z t x R r R t I U S ; , 5 . 0 exp . 2 3    (20)

Numerical inverse analysis with exact data and noised data

Three cases are investigated here. The goals of this numerical work are double:  First, we verify that the inverse heat transfer problem converges towards the exact parameters whatever is the initial set of parameters (cases "Exact 1" and "Exact 2");  Secondly, we want to evaluate the robustness of the inverse algorithm to some noise measurement on the temperature used. Then, a random error is added to the reference temperature with a maximum standard deviation  

      t t T t T REF NOISE    ).
For this numerical study, reference temperatures are required for solving the ihtp. These reference temperatures have been obtained with a given values for the gtaw efficiency and gaussian radius. The time variation of gtaw efficiency is shown in figure 6 as the red curve. The gaussian radius RB was set to 3.2 mm. The duration of the simulation was 5s. For the cases "Exact 1" and "Noised", the parameters were initialized to 0.1 for the efficiency and 10mm for RB. For case "Exact 2", the parameters were initialized to 0.9 for the efficiency and 1mm for RB.

Figure 5 presents the evolution of the three objective functions along the iterative process. The two objective functions for cases "exact 1 & 2" decreased monotonously to low values: 25.4 and 503.1 respectively. The "noised" case decreased for 20 iterations before stabilizing to an average value of 1.9e4. This behaviour is classically observed with data including noise measurement. Figures 6 and7 present the estimated parameters at the final iteration of the ihtp (so the 95 th iteration here). For the two exact cases, both the efficiency and gaussian radius are accurately estimated. It can be noticed that the efficiency value for the final time is incorrectly estimated due to the final condition of the adjoin problem, see equation [START_REF] Necati | Inverse heat transfer, fundamentals and applications[END_REF]. The efficiency value could have been imposed to zero at the final time as experimentally the arc will be switched off. With noise measurement, the efficiency is still estimated but some discrepancies are noticeable in comparison to the exact value. The estimated efficiency is still in good agreement with the exact one despite a noise measurement. The gaussian radius is very well estimated for the three investigated cases, figure 7. The noise measurement finally affected mostly the efficiency parameter. This due to the assumption done on the efficiency parameter: this parameter is defined for each step time of the simulation so for a simulation of 5s with a step time of 0.05s, it consisted of 100 parameters to estimate for the efficiency. While the gaussian radius was defined with one parameter for all the simulation. The temperature residuals for the "noised" cases is proposed in figure 8. For the two other cases, this results are not relevant as this temperature difference was lower than 2°C. For the "noised" case, the temperature difference is large due to the noise added to the reference data. Despite, this important noise, the gradient conjugate method was robust and led to a good estimation of the efficiency and gaussian radius parameters.

Conclusions

A gtaw modelling have been established in a previous work [19]. The comparison of the numerical results with the experimental data exhibited lots of discrepancies between the calculated and measured temperatures on one hand and between the calculated and measured weld pool radius on the other hand. Assuming that the gtaw stated modelling was correct, it occurred that the heat flux exchanged between the arc plasma and the steel disc was not known correctly. The heat flux absorbed by the work piece is supposed to be Gaussian and defined with two main parameters: process efficiency and gaussian radius. An inverse heat transfer problem is developed in order to estimate these unknown parameters from "experimental" data. The optimization method used for solving the inverse problem is the gradient conjugate method.

The inverse heat transfer problem was investigated through few numerical cases in order to be sure that it gives the right data on one hand and it was robust to noise measurement on the other hand. The inverse problem gave successful results with the reference temperature (without noise) and it was robust to the errors added on measurement data. Most of the noise measurement affected the estimation of the efficiency. Despite that, the efficiency was correctly estimated.

The inverse problem established in this communication was applied to a heat conduction problem. As said in the first chapters of this communication, the convective transport during welding is not negligible. As a consequence, this inverse heat transfer problem will consider soon the convective transport in order to give interesting and reliable results.

Figure 1 :

 1 Figure1: physical phenomena involved in gta welding[START_REF] Traidia | Optimal parameters for pulsed gas tungsten arc welding in partially and fully penetrated weld pools[END_REF].

Figure 2 :

 2 Figure 2: comparison between measured and simulated temperatures.

Figure 3 :

 3 Figure 3: comparison between measured and simulated weld pool radii.

  welding tension and intensity,   t is the gtaw efficiency and B R is the Gaussian radius.

Find the vector p which minimizes the cost

Figure 4 :

 4 Figure 4: Gradient conjugate algorithm with adjoin problem.

Figure 5 :

 5 Figure 5: evolution of the objective function along the iteration process.

Figure 6 :

 6 Figure 6: estimated efficiency along the iteration process.

Figure 7 :

 7 Figure 7: estimated gaussian radii along the iteration process.

Figure 8 :

 8 Figure 8: evolution of the temperature residual with respect to the time.

Table 1 :

 1 GTAW experimental conditions

	W I ~150A	W U ~10.4V
	Shielding gas	argon	Arc length	2mm
	Disc thickness	6mm	Disc radius	50mm
	Electrode type	tungsten	Electrode diameter	2.4mm


Thanh-Trung Vo

Patrick Mutabaruka

Saeid Nezamabadi

Jean-Yves Delenne

Farhang Radjai

Farhang Radjai 3d

3D simulations of the granulation of wet granular materials

come   L'archive ouverte pluridisciplinaire

3D simulations of the granulation of wet granular materials

Thanh-Trung Vo 1,2 , Patrick Mutabaruka 3 , Saeid Nezamabadi 1 , Jean-Yves Delenne 4 , Farhang Radjai 1,3

1 Laboratoire de Mécanique et Génie Civil, Université de Montpellier, CNRS, Montpellier, France.

2 Bridge and Road Department, Danang Architecture University, Danang, Vietnam. ). The model also assumes that the liquid is transported by the primary particles modeled as agglomerates of finer particles. We find that this model is able to simulate the agglomeration of particles in a rotating drum in which a given amount of liquid is homogeneously distributed. Our simulation show that the granule size increases exponentially with the number of drum rotations and in proportion to the amount of liquid. We investigate the effects of process parameters such as particle size distribution, friction coefficient between the primary particles and liquid viscosity. We also consider the effect of cohesion on different flow regimes as a function of Froude number.

Figure 1 Figure 2

 12 Figure 1 Industrial and numerical granulation drum.

Figure 3 Figure 4

 34 Figure 3The evolution of the size of the granule as a function of the number of rotations for different values of the size ratio.


William V Gehrlein

Dominique Lepelley
email: dominique.lepelley@univ-reunion.fr

The Significant Impact of Abstentions on Election Outcomes

Two approaches are considered in an attempt to mollify previous observations regarding the very negative impact that voter abstentions can have on Plurality Rule, Negative Plurality Rule and Borda Rule when voters' preferences are assumed to be statistically independent. The first option is to add a degree of dependence among voters' preferences, and the second is to consider extensions of these three voting rules by using each as the basis for a two-stage elimination election procedure. The end result in both cases is that improved results are indeed observed for both Condorcet Efficiency and the probability that a Borda Paradox is observed for larger values of voter participation rates. The very disturbing result is that both of these options actually tend to make things worse with voter participation rates of 40% or less, and voter participation rates are observed in practice at this level. * 6 𝑖=1

, so that the voter participation rate is 𝑛 * 𝑛 . An observed voting situation is then defined by any combination of 𝑛 𝑖 * terms that sum to 𝑛 * . Candidate A is the Observed Condorcet Winner (OCW) based on the preference rankings of the participating voters if 𝐴𝑴 * 𝐵 and 𝐴𝑴 * 𝐶, with: 𝑛 1 * + 𝑛 2 * + 𝑛 4 * > 𝑛 3 * + 𝑛 5 * + 𝑛 6 * [𝐴𝑴 * 𝐵] (3) 𝑛 1 * + 𝑛 2 * + 𝑛 3 * > 𝑛 4 * + 𝑛 5 * + 𝑛 6 * [𝐴𝑴 * 𝐶].

The Observed Condorcet Loser (OCL) is also defined in the obvious manner.

Gehrlein and Lepelley (2017a) perform an analysis to conclude that the likelihood that some very bad election outcomes might be observed is significantly increased when voter participation rates are low. For example, depending upon which voters choose to abstain in any particular case, the ACW and the OCW do not necessarily have to coincide, and this current study begins by considering the probability that the ACW and OCW will be the same candidate as a function of the voter participation rate.

This overall analysis is then significantly extended to determine just how serious the impact of abstentions might be on a number of other very negative election outcomes. The second phase of our study evaluates some commonly considered voting rules on the basis of their Condorcet Efficiency, which measures the conditional probability that each of these rules will elect the ACW given that one exists, based on the available results from observed voting situations after abstention takes place. The voting rules that we initially consider are the single-stage voting rules: Plurality Rule (PR), Negative Plurality Rule (NPR) and Borda Rule (BR). The potentially extreme negative impact of voter abstention is obvious in this case, since all voting rules effectively become random choosers for the winner of an election, with Condorcet Efficiencies * > 𝑛 3 * + 𝑛 5 * [𝐴𝑷𝐵] (13) 𝑛 1 * + 𝑛 2 * > 𝑛 4 * + 𝑛 6 * [𝐴𝑷𝐶]. (14)

Introduction

We consider the impact that voter abstention can have on elections with three candidates {𝐴, 𝐵, 𝐶} when there are n possible voters in the electorate. To begin, we define the preferences of these voters by using 𝐴 ≻ 𝐵 to denote the fact that any given voter prefers Candidate A to Candidate B. There are six possible linear voter preference rankings on these candidates that are transitive and have no voter indifference between candidates, as shown in Figure 1.

A A B C B C B C A A C B C B C B A A 𝑛 1 𝑛 2 𝑛 3 𝑛 4 𝑛 5 𝑛 6
Figure 1. Six possible linear preference rankings on three candidates.

Here, for example, 𝑛 1 represents the number of possible voters with the preference ranking 𝐴 ≻ 𝐵 ≻ 𝐶, with 𝐴 ≻ 𝐶 being required by transitivity. The total number of possible voters is n with

𝑛 = ∑ 𝑛 𝑖
Much of what follows is based on the notion of using the preference rankings of all possible voters in an actual voting situation to perform majority rule comparisons on pairs of candidates, such that Candidate A beats B by Pairwise Majority Rule [𝐴𝑴𝐵] if 𝐴 ≻ 𝐵 more times than 𝐵 ≻ 𝐴 in the preference rankings of the possible voters. Candidate A is defined as the Actual Condorcet Winner (ACW) if both 𝐴𝑴𝐵 and 𝐴𝑴𝐶, with:

𝑛 1 + 𝑛 2 + 𝑛 4 > 𝑛 3 + 𝑛 5 + 𝑛 6 [𝐴𝑴𝐵] (1) 𝑛 1 + 𝑛 2 + 𝑛 3 > 𝑛 4 + 𝑛 5 + 𝑛 6 [𝐴𝑴𝐶].

(2)

It is well known that an ACW does not necessarily exist [START_REF] Condorcet | An essay on the application of probability theory to plurality decision making[END_REF]], but such a candidate would clearly be a very good choice for selection as the winner of an election whenever there is one, since a majority of the possible voters would oppose the choice of either of the other two candidates. The Actual Condorcet Loser (ACL) is then defined in the obvious manner, and such a candidate would be a terrible choice for selection as the winner.

We proceed to consider election outcomes when abstentions are allowed, so that some of the possible voters can choose not to participate for any reason. Let 𝑛 𝑖 * denote the number of voters with the associated preference ranking in Figure 1 who actually choose to participate in the election, with 0 ≤ 𝑛 𝑖 * ≤ 𝑛 𝑖 , for 𝑖 = 1, 2, 3, 4, 5, 6. The total number of voters who participate is defined by 𝑛 * = ∑ 𝑛 𝑖 of 33.3%, when voter participation rates are near zero! The point of interest is how quickly voting rules recover to achieve acceptable Condorcet Efficiency values as voter participation rates increase from near-zero.

After the initial Condorcet Efficiency component of this analysis is completed, the probability that these voting rules will perform in a very poor manner by electing the ACL is considered as a function of the voter participation rate. We then proceed to extend this same overall evaluation to consider common two-stage voting rules: Plurality Elimination Rule (PER), Negative Plurality Elimination Rule (NPER) and Borda Elimination Rule (BER). As in the case of Condorcet Efficiency values, the likelihood that voting rules will select the ACL also approaches 33.3% for near-zero voter participation rates. We limit our attention that the number of possible voters is very large as 𝑛 → ∞, and the overall results of our analysis typically present a somewhat pessimistic outlook for the performance of voting rules for scenarios in which voter participation rates are as low as those that can be observed in actual elections.

Probability of ACW and OCW Coincidence

Candidate A will be both the ACW and OCW whenever the actual and observed voting situations have preference rankings for voters that are simultaneously consistent with (1), ( 2), ( 3) and ( 4).

The coincidence probability for the ACW and OCW is obviously driven both by the probability that various actual voting situations are observed and by the mechanism that determines the subset of possible voters who choose to participate in the election. Two standard assumptions from the literature are used as a basis for models to consider these two components, and each will be seen to have its own interpretation of how the voter participation rate 𝛼 𝑉𝑅 is defined. The first of these models is 𝐼𝐶(𝛼 𝑉𝑅 ) which is based on the Impartial Culture Condition (IC) that assumes complete independence between voters' preferences. The second model is 𝐼𝐴𝐶(𝛼 𝑉𝑅 ) which is based on the Impartial Anonymous Culture Condition (IAC) that inherently assumes some degree of dependence among voters' preferences. Probability representations for the coincidence of the ACW and OCW will be considered in turn with these two models.

ACW and OCW Coincidence Results with IC

When 𝐼𝐶(𝛼 𝑉𝑅 ) is considered, we focus on the probability that a randomly selected voter from the set of possible voters will have each of the associated linear preference rankings in Figure 1, where 𝑝 𝑖 is the probability that the associated 𝑖 𝑡ℎ raking will be observed on a random selection with ∑ 𝑝 𝑖 6 𝑖=1

= 1. The basis of the IC requires that 𝑝 𝑖 = 1 6 for all 𝑖 = 1,2, … ,6, so that each possible voter is equally likely to have any of the six linear rankings. Then, [START_REF] Gehrlein | Probabilities of election outcomes for large electorates[END_REF] developed an extension of IC, such that 𝐼𝐶(𝛼 𝑉𝑅 ) further assumes that each possible voter will independently have a probability 𝛼 𝑉𝑅 of participating in the election.

The basic IC assumption was used by [START_REF] Guilbaud | Les theories de l'intérêt general et le probleme logique de l'agregation[END_REF] to develop a representation for the limiting probability 𝑃 𝐴𝐶𝑊 (𝐼𝐶, ∞) that an ACW exists to begin with as 𝑛 → ∞, and

𝑃 𝐴𝐶𝑊 (𝐼𝐶, ∞) = 3 4 + 3 2𝜋 𝑆𝑖𝑛 -1 ( 1 3 ) ≈ .91226. (5)
The development of a representation for the limiting probability that the ACW and OCW coincide begins with the definitions of four variables {𝑋 1 , 𝑋 2 , 𝑋 3 , 𝑋 4 } that have different values that are based on the linear preference ranking that is associated with a randomly selected voter and on whether, or not, that voter participates in the election. These variables are defined in Table 1.

Table 1. Definitions of 𝑋 1 , 𝑋 2 , 𝑋 3 and 𝑋 4 .

Ranking 𝑋 1 [𝐴𝑴𝐵] 𝑋 2 [𝐴𝑴𝐶] 𝑋 3 [𝐴𝑴 * 𝐵] 𝑋 4 [𝐴𝑴 * 𝐶] 𝐴 ≻ 𝐵 ≻ 𝐶 (𝑝 1 ) +1 +1 +1 +1 𝐴 ≻ 𝐶 ≻ 𝐵 (𝑝 2 ) +1 +1 +1 +1 𝐵 ≻ 𝐴 ≻ 𝐶 (𝑝 3 ) -1 +1 -1 +1 𝐶 ≻ 𝐴 ≻ 𝐵 (𝑝 4 ) +1 -1 +1 -1 𝐵 ≻ 𝐶 ≻ 𝐴 (𝑝 5 ) -1 -1 -1 -1 𝐶 ≻ 𝐵 ≻ 𝐴 (𝑝 6 ) -1 -1 -1 -1 Abstention --- --- 0 0
The entries from Table 1 show that the value of 𝑋 1 for the linear preference ranking of a randomly selected voter is +1 whenever 𝐴 ≻ 𝐵 and it is -1 if 𝐵 ≻ 𝐴, so that AMB for n voters if the average value of 𝑋 1 has 𝑋 ̅ 1 > 0. In the same fashion AMC if 𝑋 ̅ 2 > 0, so Candidate A is the ACW whenever both 𝑋 ̅ 1 > 0 and 𝑋 ̅ 2 > 0. It then follows from the same logic that A will be the OCW for participating voters when both 𝑋 ̅ 3 > 0 and 𝑋 ̅ 4 > 0.

The expected values of these four variables with 𝐼𝐶(𝛼 𝑉𝑅 ) are given by: 𝐸(𝑋 1 ) = +1𝑝 1 + 1𝑝 The variance and covariance terms of the 𝑋 𝑖 variables in Table 1 with 𝐼𝐶(𝛼 𝑉𝑅 ) are given by:

𝐸(𝑋 1 2 ) = 𝐸(𝑋 2 2 ) = 1 and 𝐸(𝑋 3 2 ) = 𝐸(𝑋 4 2 ) = 𝛼 𝑉𝑅 𝐸(𝑋 1 𝑋 2 ) = (+1)(+1)𝑝 1 + (+1)(+1)𝑝 2 + (-1)(+1)𝑝 3 + (+1)(-1)𝑝 4 + (-1)(-1)𝑝 5 + (-1)(-1)𝑝 6 = 1 3 𝐸(𝑋 1 𝑋 3 ) = (+1)(+1)𝑝 1 𝛼 𝑉𝑅 + (+1)(+1)𝑝 2 𝛼 𝑉𝑅 + (-1)(-1)𝑝 3 𝛼 𝑉𝑅 + (+1)(+1)𝑝 4 𝛼 𝑉𝑅 + (-1)(-1)𝑝 5 𝛼 𝑉𝑅 + (-1)(-1)𝑝 6 𝛼 𝑉𝑅 + 0(1 -𝛼 𝑉𝑅 ) = 𝛼 𝑉𝑅 𝐸(𝑋 1 𝑋 4 ) = (+1)(+1)𝑝 1 𝛼 𝑉𝑅 + (+1)(+1)𝑝 2 𝛼 𝑉𝑅 + (-1)(+1)𝑝 3 𝛼 𝑉𝑅 + (+1)(-1)𝑝 4 𝛼 𝑉𝑅 + (-1)(-1)𝑝 5 𝛼 𝑉𝑅 + (-1)(-1)𝑝 6 𝛼 𝑉𝑅 + 0(1 -𝛼 𝑉𝑅 ) = 𝛼 𝑉𝑅 3 𝐸(𝑋 2 𝑋 3 ) = (+1)(+1)𝑝 1 𝛼 𝑉𝑅 + (+1)(+1)𝑝 2 𝛼 𝑉𝑅 + (+1)(-1)𝑝 3 𝛼 𝑉𝑅 + (-1)(+1)𝑝 4 𝛼 𝑉𝑅 + (-1)(-1)𝑝 5 𝛼 𝑉𝑅 + (-1)(-1)𝑝 6 𝛼 𝑉𝑅 + 0(1 -𝛼 𝑉𝑅 ) = 𝛼 𝑉𝑅 3 𝐸(𝑋 2 𝑋 4 ) = (+1)(+1)𝑝 1 𝛼 𝑉𝑅 + (+1)(+1)𝑝 2 𝛼 𝑉𝑅 + (+1)(+1)𝑝 3 𝛼 𝑉𝑅 + (-1)(-1)𝑝 4 𝛼 𝑉𝑅 + (-1)(-1)𝑝 5 𝛼 𝑉𝑅 + (-1)(-1)𝑝 6 𝛼 𝑉𝑅 + 0(1 -𝛼 𝑉𝑅 ) = 𝛼 𝑉𝑅 𝐸(𝑋 3 𝑋 4 ) = (+1)(+1)𝑝 1 𝛼 𝑉𝑅 + (+1)(+1)𝑝 2 𝛼 𝑉𝑅 + (-1)(+1)𝑝 3 𝛼 𝑉𝑅 + (+1)(-1)𝑝 4 𝛼 𝑉𝑅 + (-1)(-1)𝑝 5 𝛼 𝑉𝑅 + (-1)(-1)𝑝 6 𝛼 𝑉𝑅 + 0(1 -𝛼 𝑉𝑅 ) = 𝛼 𝑉𝑅 3
.

The resulting correlation matrix for these four variables is then given by 𝑹 𝟏 , with:

𝑹 𝟏 = [ 1 1 3 √𝛼 𝑉𝑅 √𝛼 𝑉𝑅 3 1 √𝛼 𝑉𝑅 3 √𝛼 𝑉𝑅 1 1 3 1 ]
.

The probability that any 𝑋 𝑖 ̅ √𝑛 takes on a specific value, including 𝐸(𝑋 𝑖 ̅ √𝑛), is zero in any continuous distribution. So, the limiting probability that Candidate A is both the ACW and OCW is given by the multivariate normal positive orthant probability Ф 4 (𝑹 𝟏 ) that 𝑋 𝑖 ̅ √𝑛 ≥ 𝐸(𝑋 𝑖 ̅ √𝑛),

for 𝑖 = {1,2,3,4}. The symmetry of 𝐼𝐶(𝛼 𝑉𝑅 ) with respect to the three candidates leads to the conclusion that the conditional limiting probability 𝑃 𝐴𝐶𝑊 𝑂𝐶𝑊 (𝐼𝐶(𝛼 𝑉𝑅 ), ∞) that the ACW and OCW coincide, given that an ACW exists, is then directly obtained from

𝑃 𝐴𝐶𝑊 𝑂𝐶𝑊 (𝐼𝐶(𝛼 𝑉𝑅 ), ∞) = 3Ф 4 (𝑹 𝟏 ) 𝑃 𝐴𝐶𝑊 (𝐼𝐶,∞) . ( 6
)
Closed form representations for these positive orthant probabilities only exist for special cases, and 𝑹 𝟏 is one such case from [START_REF] Cheng | The orthant probability of four Gaussian variables[END_REF]. Using that result with ( 5) and ( 6) leads to

𝑃 𝐴𝐶𝑊 𝑂𝐶𝑊 (𝐼𝐶(𝛼 𝑉𝑅 ), ∞) = [ 3 16 + 3 4𝜋 {𝑆𝑖𝑛 -1 ( 1 3 )+𝑆𝑖𝑛 -1 (√𝛼 𝑉𝑅 )+𝑆𝑖𝑛 -1 ( √𝛼 𝑉𝑅 3 )} + 3 4𝜋 2 {(𝑆𝑖𝑛 -1 ( 1 3 )) 2 +(𝑆𝑖𝑛 -1 (√𝛼 𝑉𝑅 )) 2 -(𝑆𝑖𝑛 -1 ( √𝛼 𝑉𝑅 3 )) 2 } ] 3 4 + 3 2𝜋 𝑆𝑖𝑛 -1 ( 1 3 ) . ( 7
)
This representation corresponds to the result in [START_REF] Gehrlein | Probabilities of election outcomes for large electorates[END_REF], and the derivation has been developed in detail here as an example, since the same basic procedure will be used later to develop additional new probability representations for other election outcomes.

There is no election output to evaluate without any voter participation when 𝛼 𝑉𝑅 = 0, so Table 2 lists values of 𝑃 𝐴𝐶𝑊 𝑂𝐶𝑊 (𝐼𝐶(𝛼 𝑉𝑅 ), ∞) from ( 7) for 𝛼 𝑉𝑅 → 0 and for each 𝛼 𝑉𝑅 = .1(. 1)1.0. chance of coincidence when 𝛼 𝑉𝑅 = .5. Low voter participation rates can clearly have a huge negative impact on election outcomes with the independent voter 𝐼𝐶(𝛼 𝑉𝑅 ) scenario.

ACW and OCW Coincidence Results with IAC

It is well known that the introduction of a degree of dependence among voters' preferences generally reduces the probability that most paradoxical election outcomes will be observed, relative to the case of complete independence with IC [see for example [START_REF] Gehrlein | Voting paradoxes and group coherence: The Condorcet efficiency of voting rules[END_REF]]. [START_REF] Gehrlein | The impact of abstentions on election outcomes when voters have dependent preferences[END_REF] investigated this impact on the conditional probability for AOW and OCW coincidence with the assumption of 𝐼𝐴𝐶(𝛼 𝑉𝑅 ), such that all actual and observed voting situations are equally likely to be observed, given that the voter participation rate is fixed at

𝛼 𝑉𝑅 = 𝑛 * 𝑛 .
To begin developing this representation for the coincidence probability of ACW and OCW with 𝐼𝐴𝐶(𝛼 𝑉𝑅 ), we note that this requires that (1), ( 2), ( 3) and (4) must hold, along with

𝑛 1 + 𝑛 2 + 𝑛 3 + 𝑛 4 + 𝑛 5 + 𝑛 6 = 𝑛 (8) 𝑛 1 * + 𝑛 2 * + 𝑛 3 * + 𝑛 4 * + 𝑛 5 * + 𝑛 6 * = 𝑛 * (9) 0 ≤ 𝑛 𝑖 * ≤ 𝑛 𝑖 , for 𝑖 = 1, 2, 3, 4, 5, 6 (10)
We would start by obtaining a representation for the total number of voting situations for which the restrictions (1), ( 2), ( 3), ( 4), ( 8), ( 9) and ( 10) simultaneously apply as a function of n and 𝑛 * , and then divide that by the total number of voting situations for which (1), ( 2), ( 8), ( 9) and ( 10) simultaneously apply as a function of n and 𝑛 * . The final representation could then be expressed as a function of n and 𝛼 𝑉𝑅 .

In the limit as 𝑛 → ∞, this process of developing representations to count the number of voting situations that meet the specified restrictions reduces to computing volumes of subspaces. All observed voting situations with the same value of 𝛼 𝑉𝑅 are then assumed to be equally likely to be observed with 𝐼𝐴𝐶(𝛼 𝑉𝑅 ), but it is not assumed that all 𝛼 𝑉𝑅 are equally likely to be observed. This general procedure has been used many times to develop limiting IAC-based probability representations in the literature, and we rely throughout this study on a particular method for doing this that uses the multi-parameter version of Barvinok's Algorithm that is described in detail in [START_REF] Lepelley | On Ehrhart polynomials and probability calculations in voting theory[END_REF]. The resulting representation for the limiting conditional probability that the ACW and OCW coincide with 𝐼𝐴𝐶(α VR ) is denoted 𝑃 𝑂𝐶𝑊 𝐴𝐶𝑊 (𝐼𝐴𝐶(α VR ), ∞), and it is given by:

𝑃 𝑂𝐶𝑊 𝐴𝐶𝑊 (𝐼𝐴𝐶(α VR ), ∞) = 444𝛼 𝑉𝑅 5 -4376𝛼 𝑉𝑅 4 +11817𝛼 𝑉𝑅 3 -15576𝛼 𝑉𝑅 2 +10080𝛼 𝑉𝑅 -2520 128(42𝛼 𝑉𝑅 5 -274𝛼 𝑉𝑅 4 +603𝛼 𝑉𝑅 3 -624𝛼 𝑉𝑅 2 +315𝛼 𝑉𝑅 -63) , 0 ≤ 𝛼 𝑉𝑅 ≤ 1 2 (11) = 10812𝛼 𝑉𝑅 5 -364𝛼 𝑉𝑅 4 -3947𝛼 𝑉𝑅 3 +1761𝛼 𝑉𝑅 2 -185𝛼 𝑉𝑅 -13 128(42𝛼 𝑉𝑅 5 +64𝛼 𝑉𝑅 4 -73𝛼 𝑉𝑅 3 +39𝛼 𝑉𝑅 2 -10𝛼 𝑉𝑅 +1) , 1 2 ≤ 𝛼 𝑉𝑅 ≤ 1.
Computed values of 𝑃 𝑂𝐶𝑊 𝐴𝐶𝑊 (𝐼𝐴𝐶(α VR ), ∞) from ( 11) are listed in Table 2 for 𝛼 𝑉𝑅 → 0 and for each 𝛼 𝑉𝑅 = .1(. 1)1.0, to show some very surprising results. The introduction of a degree of dependence with 𝐼𝐴𝐶(α VR ) can indeed increase the probability of ACW and OCW coincidence relative to the case of complete voter independence with 𝐼𝐶(α VR ). But, this only occurs for α VR ≥ .5, and the coincidence probability is actually reduced by introducing dependence for most cases with α VR ≤ .5! When a large proportion of voters choose not to participate in an election, it is now quite evident that very bad things really can happen with the resulting election outcomes and adding a degree of dependence can make things even worse, so we proceed to investigate just how bad things can get in such cases.

Actual Condorcet Efficiency of Single-Stage Voting Rules

As mentioned above, the Condorcet Efficiency of a voting rule measures the conditional probability that the voting rule will elect the ACW, based on the results of an observed voting situation. A single-stage voting rule determines the winner of an election in a single step from the information on the voters' ballots. A weighted scoring rule 𝑊𝑆𝑅(𝜆) requires voters to rank the three candidates and a weight of one is assigned to each voter's most preferred candidate, a weight of zero to the least preferred candidate and a weight of 𝜆 to the middle-ranked candidate.

The candidate who receives the greatest total score from the observed voting situation is then declared as the winner. Such a voting rule reduces to PR when 𝜆 = 0, so that voters must only report their most preferred candidate on a ballot. When 𝜆 = 1, voters must only report their two more-preferred candidates, which defines NPR since it is the same as having each voter cast a ballot against their least-preferred candidate. Voters obviously do not really have to report a ranking on the candidates when either PR or NPR is employed, but they must do so for BR which uses 𝜆 = 1/2.

Actual Single-Stage Rule Efficiency with IC

The development of a representation for the Condorcet Efficiency 𝐶𝐸 𝑊𝑆𝑅(𝜆) (𝐼𝐶(𝛼 𝑉𝑅 ), ∞) of 𝑊𝑆𝑅(𝜆) as 𝑛 → ∞ with the assumption of 𝐼𝐶(𝛼 𝑉𝑅 ) begins by defining four variables that are based on the likelihood that given voter preference rankings are observed, as shown in Table 3.

Table 3. Definitions of 𝑌 1 , 𝑌 2 , 𝑌 3 and 𝑌 4 .

Ranking 𝑌 1 [𝐴𝑴𝐵] 𝑌 2 [𝐴𝑴𝐶] 𝑌 3 [𝐴𝑾𝐵] 𝑌 4 [𝐴𝑾𝐶] 𝐴 ≻ 𝐵 ≻ 𝐶 (𝑝 1 ) +1 +1 1 -𝜆 +1 𝐴 ≻ 𝐶 ≻ 𝐵 (𝑝 2 ) +1 +1 +1 1 -𝜆 𝐵 ≻ 𝐴 ≻ 𝐶 (𝑝 3 ) -1 +1 𝜆 -1 +𝜆 𝐶 ≻ 𝐴 ≻ 𝐵 (𝑝 4 ) +1 -1 +𝜆 𝜆 -1 𝐵 ≻ 𝐶 ≻ 𝐴 (𝑝 5 ) -1 -1 -1 -𝜆 𝐶 ≻ 𝐵 ≻ 𝐴 (𝑝 6 ) -1 -1 -𝜆 -1 Abstention --- --- 0 0
Following the logic of development of the representation for 𝑃 𝐴𝐶𝑊 𝑂𝐶𝑊 (𝐼𝐶(𝛼 𝑉𝑅 ), ∞), 𝑌 1 and 𝑌 2 are identical to 𝑋 1 and 𝑋 2 respectively, so that Candidate A is the ACW if 𝑌 1 ̅ > 0 and 𝑌 2 ̅ > 0.

Variables 𝑌 3 and 𝑌 4 define the difference in scores obtained by Candidate A over each of B and C respectively in the possible voter preference rankings, so that A will be the winner by 𝑊𝑆𝑅(𝜆) in an observed voting situation whenever both 𝑌 3 ̅ > 0 and 𝑌 4 ̅ > 0.

By utilizing the methodology that led to (6), it follows that

𝐶𝐸 𝑊𝑆𝑅(𝜆) (𝐼𝐶(𝛼 𝑉𝑅 ), ∞) = 3Ф 4 (𝑹 𝟐 ) 𝑃 𝐴𝐶𝑊 (𝐼𝐶,∞) . ( 12
)
Based on the representation in (12), computations with the 𝑌 𝑖 definitions from Table 3 lead to:

𝑹 𝟐 = [ 1 1 3 √ 2 3𝑧 √ 1 6𝑧 1 √ 1 6𝑧 √ 2 3𝑧 1 1 2 1 ] , with 𝑧 = 1-𝜆(1-𝜆) 𝛼 𝑉𝑅 .
Since z is symmetric about 𝜆 = 1/2 for all 𝛼 𝑉𝑅 , it follows that PR and NPR must have the same limiting Condorcet Efficiency as 𝑛 → ∞ for any given 𝛼 𝑉𝑅 . The form of 𝑹 𝟐 unfortunately does not meet the conditions for any special case with a closed form representation for Ф 4 (𝑹 𝟐 ), as we observed above for Ф 4 (𝑹 𝟏 ) in the development of (7). [START_REF] Gehrlein | Effects of abstentions on voting procedures in three-candidate elections[END_REF] therefore used a representation from [START_REF] Gehrlein | A Representation for quadrivariate normal positive orthant probabilities[END_REF] to obtain values of Ф 4 (𝑹 𝟐 ) by using numerical integration over a single variable. Resulting values of 𝐶𝐸 𝑊𝑆𝑅(𝜆) (𝐼𝐶(𝛼 𝑉𝑅 ), ∞) for PR, NPR and BR from (12) are listed in Table 4 for the case that 𝛼 𝑉𝑅 → 0 and for each 𝛼 𝑉𝑅 = .1(. 1)1.0. We see the expected result that Condorcet Efficiency values approach 1/3 to make all voting rules equivalent to random selection procedures when 𝛼 𝑉𝑅 → 0. We also note that BR dominates PR and NPR for all non-zero participation rates. The efficiencies decrease rapidly as voter participation rates decrease, and when this rate is 40% or less, all voting rules have Condorcet Efficiency less than 61%. This level of efficiency is quite disappointing, since voter participation rates as low as 40% are definitely observed in practice. For example, voter participation rates in the 2014 US elections [see [START_REF] Mcdonald | United States Election Project[END_REF] or other sources] range from 28.7% in Indiana to 58.7% in Maine, with an overall national participation rate of only 36.7%. While these participation rates do increase for elections during years when a president is being chosen, they are even lower during the primary elections to select final candidates. So, we certainly hope that the introduction of a degree of dependence among voters' preferences will improve this situation.

Actual Single-Stage Rule Efficiency with IAC

The impact that the inclusion of some degree of dependence among voters' preferences might have on this dreary expected result is considered by using the assumption of 𝐼𝐴𝐶(𝛼 𝑉𝑅 ) to obtain representations for 𝐶𝐸 𝑅𝑢𝑙𝑒 (𝐼𝐴𝐶(𝛼 𝑉𝑅 ), ∞). It is not feasible to obtain such a representation for a generalized 𝑊𝑆𝑅(𝜆) with a reasonable effort, as we did with 𝐶𝐸 𝑊𝑆𝑅(𝜆) (𝐼𝐶(𝛼 𝑉𝑅 ), ∞), but we can obtain a representation for each specified voting rule 𝑅𝑢𝑙𝑒 ∈ {𝑃𝑅, 𝑁𝑃𝑅, 𝐵𝑅}.

Based on the definitions of PR, NPR and BR that were given above, the following restrictions on observed voting situations apply:

Candidate A is the PR winner if

𝑛 1 * + 𝑛 2 11 = -(187353𝛼 𝑉𝑅 5 -987625𝛼 𝑉𝑅 4 +527980𝛼 𝑉𝑅 3 +221880𝛼 𝑉𝑅 2 -253805𝛼 𝑉𝑅 +56249) 4320(42𝛼 𝑉𝑅 5 +64𝛼 𝑉𝑅 4 -73𝛼 𝑉𝑅 3 +39𝛼 𝑉𝑅 2 -10𝛼 𝑉𝑅 +1) , 3 4 ≤ 𝛼 𝑉𝑅 ≤ 1.
The representations in ( 19), ( 20) and ( 21) are then used to obtain the 𝐶𝐸 𝑅𝑢𝑙𝑒 (𝐼𝐴𝐶(𝛼 𝑉𝑅 ), ∞) values with each Rule = PR, NPR and BR for values of 𝛼 𝑉𝑅 → 0 and for each 𝛼 𝑉𝑅 = .1(. 1)1.

The results are listed in Table 5. The results in Table 5 show that the addition of dependence improved efficiency results for PR and BR with voter participation of 50% or more. However, the addition of dependence had a negative impact on NPR efficiency at all non-zero levels of participation. The overall efficiency results for 𝐼𝐴𝐶(𝛼 𝑉𝑅 ) in Table 5 are even more discouraging with lower participation rates for all voting rules than the results for 𝐼𝐶(𝛼 𝑉𝑅 ) from Table 4. BR still outperforms PR and NPR for all non-zero participation rates, but the addition of a degree of dependence among voters' preferences has led to even lower efficiencies for the voting rules when 𝛼 𝑉𝑅 is small. The efficiencies of all voting rules are now less than 56% when participation rates are 40% or less. Adding dependence among voters' preferences has made things worse when participation rates are less than 50%!

Borda Paradox Probabilities

The occurrence of a Borda Paradox is much more disconcerting than having a voting rule fail to elect the ACW. The worst possible outcome occurs when a Borda Paradox is observed, such that a voting rule elects the ACL. Given the potentially significant negative impact that voter abstentions have been seen to have on the Condorcet Efficiency of voting rules, it is natural to wonder how serious the impact of voter abstentions might be for observing examples of this even more dramatic Borda Paradox.

Borda Paradox Probabilities for Single-Stage Rules with IC

It is easy to develop a representation for the limiting probability 𝐵𝑃 𝑊𝑆𝑅(𝜆) (𝐼𝐶(𝛼 𝑉𝑅 ), ∞) that a Borda Paradox is observed when using 𝑊𝑆𝑅(𝜆) as 𝑛 → ∞ with the assumption of 𝐼𝐶(𝛼 𝑉𝑅 ) by mirroring the development of the representation for 𝐶𝐸 𝑊𝑆𝑅(𝜆) (𝐼𝐶(𝛼 𝑉𝑅 ), ∞) in ( 12). All that has to be done is to reverse the signs of the 𝑌 1 and 𝑌 2 variable values in Table 3 to make Candidate A the ACL and the winner by 𝑊𝑆𝑅(𝜆). It obviously follows from the logic that led to (12) and the definition of 𝑹 𝟐 that

𝐵𝑃 𝑊𝑆𝑅(𝜆) (𝐼𝐶(𝛼 𝑉𝑅 ), ∞) = 3Ф 4 (𝑹 𝟑 ) 𝑃 𝐴𝐶𝑊 (𝐼𝐶,∞) , (22)
where

𝑹 𝟑 = [ 1 1 3 -√ 2 3𝑧 -√ 1 6𝑧 1 -√ 1 6𝑧 -√ 2 3𝑧 1 1 2 1 ] , with 𝑧 = 1-𝜆(1-𝜆) 𝛼 𝑉𝑅 .
Since z remains symmetric about 𝜆 = 1/2 for all 𝛼 𝑉𝑅 , it follows as in the case of Condorcet Efficiencies, that PR and NPR have the same limiting probability of exhibiting a Borda Paradox for any specified value of 𝛼 𝑉𝑅 . The form of 𝑹 𝟑 is not a special case for which Ф 4 (𝑹 𝟑 ) has a closed form representation, so the representation from [START_REF] Gehrlein | A Representation for quadrivariate normal positive orthant probabilities[END_REF] is used to obtain values of Ф 4 (𝑹 𝟑 ). Computed 𝐵𝑃 𝑊𝑆𝑅(𝜆) (𝐼𝐶(𝛼 𝑉𝑅 ), ∞) values from ( 22) are listed in Table 6 for 𝛼 𝑉𝑅 → 0 and for each 𝛼 𝑉𝑅 = .1(. 1)1.0 The expected result in Table 6 is that the Borda Paradox Probabilities approach 1/3 as 𝛼 𝑉𝑅 → 0, since all voting rules are equivalent to random selection procedures in this particular case, and that BR has a lower Borda Paradox probability than PR and NPR for all non-zero participation rates. A well-known result is that BR cannot elect the OCL [START_REF] Fishburn | Borda's rule, positional voting, and Condorcet's simple majority principle[END_REF]], so that the ACL cannot be elected by BR if all voters participate. However, this result is not true when some voters abstain so that the OCL and ACL do not necessarily coincide. The probability of observing a Borda Paradox consistently increases as voter participation rates decrease, and stage. We consider Plurality Elimination Rule (PER), Negative Plurality Elimination Rule (NPER) and Borda Elimination Rule (BER). It is definitely of interest to determine if these twostage voting rules can exhibit better performance than the single-stage rules on the basis of Condorcet Efficiency and the probability that Borda's Paradox is observed with lower levels of voter participation rates.

Condorcet Efficiency of Two-stage Voting Rules with IC

The development of the representation for 𝐶𝐸 𝑊𝑆𝑅(𝜆) (𝐼𝐶(𝛼 𝑉𝑅 ), ∞) in ( 12) was based on the definitions of the four 𝑌 𝑖 variables in Table 3 had 𝑌 𝑖 ̅ > 0 for 𝑖 = 1,2,3,4. When we consider instead the more complex two-stage elimination rules 𝑊𝑆𝑅𝐸(𝜆), the resulting development of a representation for 𝐶𝐸 𝑊𝑆𝑅𝐸(𝜆) (𝐼𝐶(𝛼 𝑉𝑅 ), ∞) requires the use of five variables that are denoted as 𝑍 𝑖 for 𝑖 = 1,2,3,4,5 in Table 8.

Table 8. Definitions of 𝑍 1 , 𝑍 2 , 𝑍 3 , 𝑍 4 and 𝑍 5 .

Ranking 𝑍 1 [𝐴𝑴𝐵] 𝑍 2 [𝐴𝑴𝐶] 𝑍 3 [𝐴𝑾𝐵] 𝑍 4 [𝐶𝑾𝐵] 𝑍 5 [𝐴𝑴 * 𝐶] 𝐴 ≻ 𝐵 ≻ 𝐶 (𝑝 1 ) +1 +1 1 -𝜆 -𝜆 +1 𝐴 ≻ 𝐶 ≻ 𝐵 (𝑝 2 ) +1 +1 +1 𝜆 +1 𝐵 ≻ 𝐴 ≻ 𝐶 (𝑝 3 ) -1 +1 𝜆 -1 -1 +1 𝐶 ≻ 𝐴 ≻ 𝐵 (𝑝 4 ) +1 -1 𝜆 +1 -1 𝐵 ≻ 𝐶 ≻ 𝐴 (𝑝 5 ) -1 -1 -1 𝜆 -1 -1 𝐶 ≻ 𝐵 ≻ 𝐴 (𝑝 6 ) -1 -1 -𝜆 1 -𝜆 -1 Abstention --- --- 0 0 0
By comparing Table 8 to Table 3, we see that 𝑍 1 and 𝑍 2 are identical to 𝑌 1 and 𝑌 2 , so that Candidate A will be the ACW if 𝑍 ̅ 1 > 0 and 𝑍 ̅ 2 > 0. It is also true that 𝑍 3 is identical to 𝑌 3 , so that A beats B by 𝑊𝑆𝑅(𝜆) if 𝑍 ̅ 3 > 0. Variable 𝑍 4 defines the relative margin by which C beats B in each ranking under 𝑊𝑆𝑅(𝜆). So, B will be the lowest ranked candidate by 𝑊𝑆𝑅(𝜆) when 𝑍 ̅ 3 > 0 and 𝑍 ̅ 4 > 0, and it therefore will be eliminated under 𝑊𝑆𝑅𝐸(𝜆). Variable 𝑍 5 accounts for the fact that the ACW Candidate A will then be the majority rule winner over C for participating voters in the second stage of 𝑊𝑆𝑅𝐸(𝜆) if 𝑍 ̅ 5 > 0. Candidate A will therefore be the ACW and the winner by 𝑊𝑆𝑅𝐸(𝜆) when 𝑍 ̅ 𝑖 > 0 for 𝑖 = 1,2,3,4,5.

The correlation matrix between these five variables is found to be given by 𝑹 𝟓 :

𝑹 𝟓 = [ 1 1 3 √ 2 3𝑧 √ 1 6𝑧 √𝛼 𝑉𝑅 3 1 √ 1 6𝑧 -√ 1 6𝑧 √𝛼 𝑉𝑅 1 1 2 √ 1 6𝑧𝛼 𝑉𝑅 1 -√ 1 6𝑧𝛼 𝑉𝑅 1 ] , with 𝑧 = 1-𝜆(1-𝜆) 𝛼 𝑉𝑅 .
There are three candidates that could be the ACW and there are two remaining candidates that could be eliminated in the second stage of 𝑊𝑆𝑅𝐸(𝜆), so a representation for limiting Condorcet Efficiency is obtained from

𝐶𝐸 𝑊𝑆𝑅𝐸(𝜆) (𝐼𝐶(𝛼 𝑉𝑅 ), ∞) = 6Ф 5 (𝑹 𝟓 ) 𝑃 𝐴𝐶𝑊 (𝐼𝐶,∞) . ( 28
)
As we have observed before, z is symmetric about 𝑧 = 1/2, so the Condorcet Efficiency values of PER and NPER are identical for 𝐼𝐶(𝛼 𝑉𝑅 ) for any given 𝛼 𝑉𝑅 . The process of obtaining values of multivariate-normal positive orthant probabilities becomes much more complicated for the case of five variables, but they can still be obtained by using numerical techniques over a series of integrals on a single variable, as described in [START_REF] Gehrlein | Computing multivariate normal positive orthant probabilities with 4 and 5 variables[END_REF]. That procedure is used here to obtain values of Ф 5 (𝑹 𝟓 ) for use in ( 28), to obtain values of 𝐶𝐸 𝑊𝑆𝑅𝐸(𝜆) (𝐼𝐶(𝛼 𝑉𝑅 ), ∞) for PER, NPER and BER that are listed in Table 9 for 𝛼 𝑉𝑅 → 0 and for each 𝛼 𝑉𝑅 = .1(. 1)1.0. The results in Table 9 produce some interesting outcomes. First of all, it is well known from [START_REF] Daunou | A paper on elections by ballot[END_REF] that BR cannot rank the OCW in last place, so that it cannot be eliminated in the first stage with BER and the OCW must then win in the second stage. The Actual Condorcet Efficiency of BER is therefore 100% when all voters participate so that the OCW and ACW must coincide. However, this is not true when some voters abstain and the OCW and ACW do not necessarily coincide.

When we compare the Condorcet Efficiency results for the two-stage rules with 𝐼𝐶(𝛼 𝑉𝑅 ) in Table 9 to their counterpart single-stage rules in Table 4. BER dominates PER and NPER for all non-zero voter participation rates, just as BR dominated PR and NPR. However, the degree of this domination is significantly dampened with the two-stage rules, since BER has a less than 1% advantage over PER and NPER for voter participation rates of 60% or less. It seems logical to expect increased Condorcet Efficiency results with two-stage rules, and this is true for PER and NPER. But, it is a very surprising outcome to observe that BER efficiencies are actually It follows from the definition of two-stage elimination rules that the OCL cannot win with any of these rules, so that Borda's Paradox cannot be observed if all voters participate so that the ACL and OCL coincide. However, this observation is not true when some voters abstain, so that the OCW and ACW might not be the same.

By comparing the results of Table 11 to Table 6, very different behaviors are observed for the two-stage voting rules when they are compared to their single-stage rule counterparts. The Borda Paradox probabilities are reduced at all levels of non-zero voter participation for PER and NPER compared to PR and NPR. However, the BER Borda Paradox probabilities all increase from their corresponding BR values for all 0 < 𝛼 𝑉𝑅 < 1. BER still has lower Borda Paradox probabilities than PER and NPER over the range of voter participation rates, but the differences are less than 0.3% in all cases. The particularly disturbing result from Table 11 is that when the voter participation rates is 40% or less, all two-stage voting rules now have a Borda Paradox probability of greater than about 13%, compared to the 11% that was observed in Table 6 for the single-stage rules!

  computed values of 𝐵𝑃 𝑊𝑆𝑅𝐸(𝜆) (𝐼𝐶(𝛼 𝑉𝑅 ), ∞) from (32) for each PER, NPER and BER as 𝛼 𝑉𝑅 → 0 and for each 𝛼 𝑉𝑅 = .1(. 1)1.0.

Table 2 .

 2 Limiting ACW and OCW Coincidence Probabilities.

	𝛼 𝑉𝑅	𝐼𝐶(𝛼 𝑉𝑅 )	𝐼𝐴𝐶(𝛼 𝑉𝑅 )
	≈ 0	.3041	5/16 = .3125
	.1	.4236	.3482
	.2	.4806	.3949
	.3	.5290	.4564
	.4	.5742	.5357
	.5	.6186	.6310
	.6	.6640	.7291
	.7	.7126	.8160
	.8	.7675	.8887
	.9	.8365	.9492
	1.0	1.0000	1.0000
	Some results from the 𝑃 𝐴𝐶𝑊 𝑂𝐶𝑊 (𝐼𝐶(𝛼

𝑉𝑅 ), ∞) values in Table

2

are predictable. In particular, the coincidence probability is one when all voters participate with 𝛼 𝑉𝑅 = 1, and the coincidence probability decreases as this participation rate declines. When almost all voters independently abstain as 𝛼 𝑉𝑅 → 0, the candidate that becomes the OCW is effectively selected at random if there is an OCW, so the coincidence probability goes to 1 3 𝑃 𝐴𝐶𝑊 (𝐼𝐶, ∞). What is surprising is the very steep rate of decline in 𝑃 𝐴𝐶𝑊 𝑂𝐶𝑊 (𝐼𝐶(𝛼 𝑉𝑅 ), ∞) values as 𝛼 𝑉𝑅 decreases, with only about a 62%

Table 4 .

 4 Computed Limiting Values for Condorcet Efficiency with 𝐼𝐶(𝛼 𝑉𝑅 ).

		Voting Rule
	𝛼 𝑉𝑅	PR & NPR	BR
	≈ 0	1/3 = .3333	1/3 = .3333
	.1	.4399	.4576
	.2	.4882	.5151
	.3	.5277	.5630
	.4	.5630	.6068
	.5	.5961	.6490
	.6	.6280	.6911
	.7	.6595	.7345
	.8	.6911	.7810
	.9	.7234	.8337
	1.0	.7572	.9012

Table 5 .

 5 Computed Limiting Values for Condorcet Efficiency with 𝐼𝐴𝐶(𝛼 𝑉𝑅 ).

	Voting Rule

Table 6 .

 6 Computed Limiting Borda Paradox Probabilities with 𝐼𝐶(𝛼 𝑉𝑅 ).

		Voting Rule
	𝛼 𝑉𝑅	PR & NPR	BR
	≈ 0	1/3 = .3333	1/3 = .3333
	.1	.2355	.2208
	.2	.1965	.1761
	.3	.1670	.1423
	.4	.1423	.1140
	.5	.1207	.0892
	.6	.1012	.0669
	.7	.0834	.0466
	.8	.0669	.0280
	.9	.0515	.0115
	1.0	.0371	0

Table 9 .

 9 Computed Limiting Values for Two-Stage Efficiencies with 𝐼𝐶(𝛼 𝑉𝑅 ).

		Voting Rule
	𝛼 𝑉𝑅	PER & NPER	BER
	≈ 0	1/3 = .3333	1/3 = .3333
	.1	.4516	.4537
	.2	.5075	.5109
	.3	.5546	.5593
	.4	.5982	.6043
	.5	.6407	.6484
	.6	.6836	.6934
	.7	.7288	.7411
	.8	.7786	.7944
	.9	.8387	.8597
	1.0	.9629	1

Table 11 .

 11 Limiting Borda Paradox Probability for Two-stage Rules with 𝐼𝐶(𝛼 𝑉𝑅 ).

		Voting Rule
	𝛼 𝑉𝑅	PER & NPER	BER
	≈ 0	1/3 = .3333	1/3 = .3333
	.1	.2291	.2278
	.2	.1883	.1866
	.3	.1575	.1556
	.4	.1318	.1297
	.5	.1091	.1070
	.6	.0885	.0863
	.7	.0691	.0668
	.8	.0503	.0481
	.9	.0310	.0288
	1.0	0	0

𝑖=1 , and an actual voting situation defines any combination of 𝑛 𝑖 terms that sum to n.

NPR winner if 𝑛 5 * + 𝑛 6 * < 𝑛 Candidate A is the BR

The same procedure that was used to obtain the limiting representation for 𝑃 𝑂𝐶𝑊 𝐴𝐶𝑊 (𝐼𝐴𝐶(𝛼 𝑉𝑅 ), ∞) in ( 11) from the restrictions in (1), ( 2), (3), ( 4), ( 8), ( 9) and ( 10) is used to obtain representations for each 𝐶𝐸 𝑅𝑢𝑙𝑒 (𝐼𝐴𝐶(𝛼 𝑉𝑅 ), ∞). This is done by replacing (3) and ( 4) with ( 13) and ( 14) for PR to obtain: ≤ 𝛼 𝑉𝑅 ≤ 1.

A limiting representation is obtained for the Condorcet Efficiency of NPR by replacing ( 3) and ( 4) with ( 15) and ( 16):

A limiting representation for the Condorcet Efficiency of BR is obtained by replacing (3) and ( 4) with ( 17) and ( 18):

when this rate is 40% or less, all voting rules have a Borda Paradox probability of greater than 11%! That is a shockingly large probability for such a very negative outcome, given that voter participation rates as low as 40% are actually observed in practice.

Borda Paradox Probabilities for Single-Stage Rules with IAC

We consider the impact that the introduction of some degree of dependence among voters' preferences will have on the probability that Borda's Paradox will be observed by extending this analysis with the assumption of 𝐼𝐴𝐶(𝛼 𝑉𝑅 ). This begins by noting that Candidate A is the ACL whenever:

The representation for 𝐶𝐸 𝑃𝑅 (𝐼𝐴𝐶(𝛼 𝑉𝑅 ), ∞) in ( 19) was obtained by considering actual and observed voting situations for which (1), ( 2), ( 13), ( 14), ( 8), ( 9) and (10) held simultaneously.

Here, (1) and ( 2) respectively required that AMB and AMC to make Candidate A the ACW. This process is then repeated by replacing (1) and ( 2) with ( 23) and ( 24) to obtain a representation for 𝐵𝑃 𝑃𝑅 (𝐼𝐴𝐶(𝛼 𝑉𝑅 ), ∞), with This process is then repeated for both NPR and BR, and the resulting limiting representations for observing Borda's Paradox are given by:

14

Associated values of 𝐵𝑃 𝑃𝑅 (𝐼𝐴𝐶(𝛼 𝑉𝑅 ), ∞) from ( 25), 𝐵𝑃 𝑁𝑃𝑅 (𝐼𝐴𝐶(𝛼 𝑉𝑅 ), ∞) from ( 26) and 𝐵𝑃 𝐵𝑅 (𝐼𝐴𝐶(𝛼 𝑉𝑅 ), ∞) from ( 27) are listed in Table 7 for 𝛼 𝑉𝑅 → 0 and for each 𝛼 𝑉𝑅 = .1(. 1)1.0. 6 and7 shows the same general outcome that was observed when the Condorcet Efficiency of voting rules was analyzed earlier with 𝐼𝐶(𝛼 𝑉𝑅 ) and 𝐼𝐴𝐶(𝛼 𝑉𝑅 ). That is, the addition of a degree of dependence among voters' preferences with 𝐼𝐴𝐶(𝛼 𝑉𝑅 ) actually makes things worse for lower levels of voter participation. We now see that the probability of observing a Borda Paradox is greater than 15% for all levels of voter participation at 40% or less! The results for BR are uniformly the best of these options for all positive levels of voter participation, but BR still performs very poorly for lower levels of voter participation. If this very poor overall performance of single-stage voting rules at lower levels of voter participation is to be improved, the likely approach would be to consider the use of some other forms of voting rules.

Two-stage Voting Rules

Two-stage voting rules are sequential elimination procedures that take place in two steps. In the first stage, a voting rule is used to determine the candidate that is ranked as the worst of the three candidates. That candidate is eliminated from consideration in the second stage, where majority rule is used to determine the ultimate winner from the remaining two candidates. All that changes with the two-stage rules is the voting rule that is used to eliminate the loser in the first marginally smaller than BR efficiencies for . 1 ≤ 𝛼 𝑉𝑅 ≤ .5 and the Condorcet Efficiency values for two-stage rules remain less than 61% for voter participation rates that are 40% or less.

Condorcet Efficiency of Two-stage Voting Rules with IAC

The same type of two-stage voting rule Condorcet Efficiency analysis is extended with the assumption of 𝐼𝐴𝐶(𝛼 𝑉𝑅 ) and the resulting representations for PER, NPER and BER are shown respectively in ( 29), ( 30) and (31).

𝐶𝐸 𝑃𝐸𝑅 (𝐼𝐴𝐶(𝛼 𝑉𝑅 ), ∞) = (29) , for

Computed values of 𝐶𝐸 𝑅𝑢𝑙𝑒 (𝐼𝐴𝐶(𝛼 𝑉𝑅 ), ∞) for PER, NPER and BER from ( 29), ( 30) and ( 31) are listed in Table 10 for 𝛼 𝑉𝑅 → 0 and for each 𝛼 𝑉𝑅 = .1(. 1)1.0. 10 to their counterpart single-stage rules in Table 5 produces very similar results to those observed immediately above with 𝐼𝐶(𝛼 𝑉𝑅 ). BER typically dominates PER and NPER for all non-zero voter participation rates, but NPER has a very small advantage over BER for 𝛼 𝑉𝑅 values of .1 and .2. All differences in efficiencies are significantly dampened with the two-stage rules, with a less than 1% difference between the efficiencies for voter participation rates of 50% or less. Increased Condorcet Efficiency results are observed for PER and NPER over the range of non-zero voter participation rates, but BER efficiencies are marginally smaller than BR efficiencies for . 1 ≤ 𝛼 𝑉𝑅 ≤ .5. Condorcet Efficiency values for two-stage rules are now less than 56% for voter participation rates that are 40% or less, to make things worse than we observed for the independent voter case with 𝐼𝐶(𝛼 𝑉𝑅 ) with two-stage voting rules in Table 9.

Borda Paradox Probabilities for Two-stage Voting Rules with IC

The development of a representation for the limiting probability that Borda's Paradox will be observed for two-stage voting rules with 𝐼𝐶(𝛼 𝑉𝑅 ) directly follows the process that led to the representation for 𝐶𝐸 𝑊𝑆𝑅𝐸(𝜆) (𝐼𝐶(𝛼 𝑉𝑅 ), ∞) in (28). The only difference is that the signs for the variables 𝑍 1 and 𝑍 2 in Table 8 are reversed to make Candidate A the ACL, which is then elected as the ultimate winner by 𝑊𝑆𝑅𝐸(𝜆). It follows directly that

, where

Borda Paradox Probabilities for Two-stage Voting Rules with IAC

Representations are also obtained for the limiting probabilities 𝐵𝑃 𝑃𝐸𝑅 (𝐼𝐴𝐶(𝛼 𝑉𝑅 ), ∞), 𝐵𝑃 𝑁𝑃𝐸𝑅 (𝐼𝐴𝐶(𝛼 𝑉𝑅 ), ∞) and 𝐵𝑃 𝐵𝐸𝑅 (𝐼𝐴𝐶(𝛼 𝑉𝑅 ), ∞) by using the same process that lead respectively to the Condorcet Efficiency results in ( 29), ( 30) and ( 31). These resulting limiting probability representations are given by: ,

Computed values for the Borda Paradox probabilities in (33), ( 34) and ( 35) for each PER, NPER and BER respectively are listed in Table 12 as 𝛼 𝑉𝑅 → 0 and for each 𝛼 𝑉𝑅 = .1(. 1)1.0. 11 indicate that the addition of voter dependence with 𝐼𝐴𝐶(𝛼 𝑉𝑅 ) again causes an increase in the probability of observing a Borda Paradox for lower levels of voter participation with 𝛼 𝑉𝑅 values of 40% or less.

Conclusion

An initial analysis was performed to consider the probability that the ACW and OCW would coincide for the limiting case of voters with independent voters' preferences when voters have the option to abstain. This indicated that the probability of non-coincidence becomes relatively high as voter participation rates decline. Rather pessimistic results were also found for the Condorcet Efficiency and the probability that a Borda Paradox is observed with PR, NPR and BR under the same independent voter scenario. Two options were considered to improve this negative result. The first, was to add a degree of dependence among voters' preferences. The second was to use more complex versions of these three voting rules by using each as the basis for a two-stage elimination election procedure. The end result is that both of these options tended to make things better for both Condorcet Efficiency and the probability that a Borda Paradox is observed for larger values of voter participation rates. However, both options tended to make things worse for voter participation rates of 40% or less. The results indicate that low levels of voter participation that are consistent with some actual observed elections result in very poor performance for standard voting rules when voters have independent preferences. And, simple attempts to try to improve this dreary observation typically act to make things worse when voter participation rates are 40% or less.