in computer science , a _ string _ ( or sequence ) of length is defined as a finite sequence of characters from a finite alphabet . a string is a data type used to represent and store information .words in a specific language , for example , are stored in a computer in terms of strings .even whole texts may be stored by means of strings .apart from fields such as information and text processing , strings arise , in particular , in the field of computational biology .this is because most of the genetic instructions involved in the growth , development , functioning and reproduction of living organisms are stored in _ deoxyribonucleic acid _ ( dna ) and _ ribonucleic acid _ ( rna ) molecules , which are double - stranded ( in the case of dna ) or single - stranded ( in the case of rna ) sequences of nucleotides .hereby , each nucleotide is composed of a nitrogenous base , a five - carbon sugar ( ribose or deoxyribose ) , and at least one phosphate group .nucleotides in the context of rna have one of four different nitrogenous bases : guanine ( g ) , uracil ( u ) , adenine ( a ) , and cytosine ( c ) .therefore , any rna molecule can be represented as a string of symbols from .such a string is called the _ primary structure _ of an rna molecule. however , rna molecules generally fold in space , and different nucleotides bind together , for example , by means of hydrogene bonds . in simplified terms, g can only bind with c and u can only bind with a. en example of the secondary structure of an rna molecule is shown in figure [ fig : example : a ] .  in computer science terms , the hydrogene bonds of the secondary structure of an rna sequence be represented by a so - called _ arc annotation set _ , which is an unordered set of pairs of positions of . henceforth , the positions of a string range from 1 to .each pair is called an _ arc annotation _ ( or simply an arc ) between positions and . as a convention, it must hold that .moreover , is called the _ left endpoint _ of arc , and is called the _right endpoint_. a pair is called an _ arc - annotated sequence _  .note that the secondary structure of an rna sequence can be described by an arc - annotated sequence .for an example see figure [ fig : example : b ] .in fact , arc - annotated sequences have been widely used for this purpose ( see , for example ,  ) .in particular , arc - annotated sequences have been useful in the context of the structural comparison of rna sequences . an important way of comparing two ( or more ) sequences consists in computing their _ longest common subsequence _ ( lcs ) . given a sequence over a finite alphabet , sequence is called a _ subsequence _ of , if can be produced from by deleting characters .given a set of input strings , the problem of finding the longest commons subsequence of all input strings is , in general , np - hard  .the best techniques available nowadays for solving this problem are based on beam search  .the longest common subsequence problem in the context of arc - annotated sequences the _ longest arc - preserving common subsequence _ ( lapcs ) problem has first been introduced in  . in particular ,in the same works it was shown that the most general case of the problem ( without any restrictions on the arcs ) is np - hard . in the meanwhile , five different variants of the problem that is , restrictions of the general problem have been studied in the related literature , and efficient algorithms were developed for three of these variants .. ] however , as far as we know , only one algorithm that is applicable to the most general case has been proposed so far ( see  ) . in this work ,we first phrase the lapcs problem in the form of an integer linear program ( ilp )  .then we make use of this ilp in the context of a simple evolutionary algorithm based on _ solution merging _ , where two or more solutions are merged and the best solution in this union is derived by means of an exact technique .aggarwal et al . originally suggested such an approach , labeled _optimized crossover _ , for the independent set problem .we provide an extensive experimental comparison of the proposed algorithm in comparison to the heuristic from the literature and to the application of a randomized multi - start heuristic .the remainder of this paper is structured as follows . in section [ sec :lapcs ] , a technical description of the tackled problem is provided . in subsequent sections see section [ sec : heuristic ] and section [ sec : ilp - model]we describe a heuristic from the literature and phrase an ilp model for the tackled problem .next , the proposed algorithm is outlined in section [ sec : algorithm ] .finally , an extensive experimental evaluation on artificial and real problem instances is provided in section [ sec : experiments ] , and an outlook to future work is given in section [ sec : conclusions ] .given two input sequences and , we define the set of possible assignments as the set of all  and that = y[j] ] . in case , where is a parameter of the algorithm , the variable with the smallest weight value is deterministically chosen . otherwise , a candidate list of size containing the options with the lowest weight valuesis generated and exactly one variable is then chosen uniformly at random and added to . note that is another parameter of the solution construction process .finally , the set of options for the next construction step is generated .being the last variable that was added to , contains for each letter the variable ( if any ) with the lowest weight value calculated as .the solution construction is finished when the set of options is empty .however , note that a solution constructed in the way as described above does not necesarrily respect all arc preservation constraints .therefore , the same _ repair mechanism _ is utilized as in the heuristic from section [ sec : heuristic ] in order to transform into a valid lapcs solution .summarizing , the following techniques are included in the experimental evaluation : ( 1 ) the heuristic described in section [ sec : heuristic ] ( heuristic ) , ( 2 ) the hybrid ea ( hyb - ea ) , and ( 3 ) hyb - ea without the application of solution merging .this last algorithm henceforth denoted by ms - heur is bascially a multi - start heuristic that constructs randomized solutions ( and applies the repair procedure to them ) until it runs out of computation time .comparing hyb - ea with ms - heur will enable us to measure the contribution of solution merging .the three above - mentioned algorithms were implemented in ansi c++ using gcc 4.7.3 , without the use of any external libraries .in addition , the ilp models in the context of hyb - ea were solved with the ilp solver ibm ilog cplex v12.6 in one - threaded mode .the experimental evaluation has been performed on a cluster of pcs with intel(r ) xeon(r ) cpu 5670 cpus of 12 nuclei of 2933 mhz and at least 40 gigabytes of ram .note that we also tried to apply cplex to the complete ilp models for each problem instance .however , the models were too large , even in the case of the smallest problem instances .the remainder of this section is organized as follows .first , the set of benchmark instances is described .second , the tuning experiments that were conducted in order to determine a proper setting for the parameters of hyb - ea are outlined .finally , an exhaustive experimental evaluation is presented .two sets of benchmark instances were generated .the first set , labelled set1 , consists of artificial problem instances .each of these instances consists of two artifically generated rna strings of length .the probability of each letter and each position was chosen to be . moreover , for each input string we randomly generated a number of . hereby , it was taken care that all arcs were different .for each combination of and we randomly generated 30 problem instances .this makes a total of 900 problem instances .  for the second benchmark set , labelled set2 , we downloaded arc - annotated rna sequences from the rnase p database  . in total we assembled 10 problem instances ,whose characteristics are described in table [ tab : instance - characteristics ] . moreover , the secondary structures of the rna sequences involved in instances real_1 and real_8 are exemplary shown in figure [ fig : real ] .the automatic configuration tool  was used for tuning the parameters of hyb - ea .the following parameters of hyb - ea were considered for tuning : ( ) the number of solution constructions per iteration ( ) the determinism rate , ( ) the candidate list size , and ( ) the maximum time in seconds allowed for solution merging ( at each call of the solution merging procedure ) .in particular , hyb - ea was tuned separately for each input string length , which after initial experiments seemed to have a greater influence on the behavior of the algorithm than the number of arcs . for each randomly generated two tuning instances for each of the three values of .this makes a total of six tuning instances for each . the tuning process for each given a budget of 1000 runs of hyb - ea , where each run was given a computation time limit of cpu seconds .finally , the following parameter value ranges were considered concerning the four parameters of hyb - ea :  * * , where a value of means that the selection of the assignment to be added to the partial solution under construction is always done randomly from the candidate list , while a value of means that solution constructions are nearly deterministic . * * ( in seconds ) .the tuning runs with produced the configurations of hyb - ea as shown in table [ tab : parameter - values : hyb - ea ] .the following trends can be observed .apart from , the number of solution constructions is always set to five .this is because the smaller , the smaller is the ilp model that has to be solved by cplex in the context of solution merging at each iteration of the algorithm .moreover , the smaller the ilp model , the more efficient is cplex in solving such a model .the values of are consistently between and , whereas the values of are consistently set to two or three .finally , the settings of seem somewhat erratic . however , this is due to the fact that the application of cplex in solution merging is very efficient and stops , most of times , much below 5 cpu seconds . therefore , it is only of importance that the value of is at least set to 5.0 .> m1cm|rrrr & & & & + 100 & 10 & 0.3 & 2 & 5.0 + 200 & 5 & 0.7 & 3 & 1.0 + 300 & 5 & 0.7 & 2 & 5.0 + 400 & 5 & 0.7 & 3 & 10.0 + 500 & 5 & 0.3 & 2 & 20.0 + 600 & 5 & 0.7 & 2 & 5.0 + 700 & 5 & 0.5 & 2 & 20.0 + 800 & 5 & 0.7 & 2 & 5.0 + 900 & 5 & 0.5 & 2 & 5.0 + 1000 & 5 & 0.7 & 2 & 5.0 +   the results concerning the artificial problem instances from set1 are presented in table [ tab : results : artificial ] .each row provides the results of heuristic , ms - heur and hyb - ea in terms of the average solution quality obtained for the 30 problem instances of the corresponding combination of and .all techniques were applied exactly once to each problem instance .the computation time limit used for ms - heur and hyb - ea was cpu seconds .the column with heading * time * shows the average computation time for the 30 problem instances in the case of heuristic , and the average time at which the best solution of a run was found , in the case of ms - heur and hyb - ea .the best result of each table row is marked by a lightgrey background .the following observations can be made :  * heuristic is very fast .its application to any of the problem instances requires less than one cpu second .moreover , the results of heuristic are always better than the results of ms - heur .* hyb - ea is , by far , the best algorithm in the comparison .it obtains the best result for each combination of and .in particular , hyb - ea always improves over ms - heur .this means that the solution merging component is an essential part of hyb - ea .this is remarkable as the application of cplex to the original problem instances was not viable at all .nevertheless , in the context of solution merging , mathematical programming plays an important role for the success of the hyb - ea algorithm .  in order to study the magnitude of the improvement of hyb - ea over ms - heur, we show the percentage improvement of hyb - ea over ms - heur by means of boxplots for each combination of and in the three graphics of figure [ fig : improvement ] .the x - axis of each graphic ranges from to .these grapics show , first , that the improvement of hyb - ea over ms - heur is generally around .moreover , it can be observed that the improvements become bigger with a growing number of arcs .+   +  in a second set of experiments we applied the three algorithms considered in this work to the set of 10 real problem instances ( set2 ) .the results are provided in table [ tab : results : real ] , in the following way .heuristic was applied exactly once to each problem instance .the corresponding result can be found in columns with headings * result * and * time*. both ms - heur and hyb - ea are applied 30 times to each problem instance , with a computation time limit of 30 seconds per application . in both caseswe provide the best result obtained over 30 applications ( column * best * ) , the average of the 30 results obtained ( column * avg . * ) , and the average time at which the best solution of each run was found ( column * time * ) .again , the best result for each of the 10 instances is marked by a lightgrey background .apart from one case ( real_4 ) , the results are consistent with our observations in the context of the artificial benchmark instances .in the case of real_4 , outperforms hyb - ea .this suggest that this problem instance has certain characteristics that are difficult to be discovered by the step - by - step way of constructing solutions as used in hyb - ea. remember that heuristic , in contrast , uses the dynamic programming algorithm by smith and waterman in order to generate the ( possibly invalid ) initial solution .in this paper we proposed a simple , hybrid evolutionary algorithm for solving the so - called longest arc - preserving common subsequence problem .the most important feature of this algorithm is a crossover component based on solution merging . at each iteration ,the best solution found so far is merged with randomly generated solutions , and a general purpose integer linear programming solver is used to find the best solution within the resulting set of assignments .the results show that the algorithm is superior to the only existing heuristic from the literature .moreover , we have shown that the solution merging component is an essential part of the algorithm .  in future workwe will try to replace the probabilistic way of constructing solutions by a probabilistic version of the smith and waterman algorithm . in this way it might be possible to avoid situations such as the one for real - life instance real_4 , where our algorithm was not able to outperform the existing heuristic .this work was funded by project tin2012 - 37930-c02 - 02 ( spanish ministry for economy and competitiveness , feder funds from the european union ) and project sgr 2014 - 1034 ( agaur , generalitat de catalunya ) .our experiments have been executed in the high performance computing environment managed by the rdlab at the technical university of barcelona ( ` http://rdlab.cs.upc.edu ` ) and we would like to thank them for their support .patricia a. evans . finding common subsequences with arcs and pseudoknots . in maxime crochemore and mike paterson , editors ,_ proceedings of cpm 1999 10th annual symposium on combinatorial pattern matching _ ,volume 1645 of _ lecture notes in computer science _ , pages 270280 .springer berlin heidelberg , 1999 .k. huang , c. yang , and k. tseng .fast algorithms for finding the common subsequences of multiple sequences . in _ proceedings of the 2004 international computer symposium _ , pages 10061011 .ieee press , 2004 .

Abstract: the longest arc - preserving common subsequence problem is an np - hard combinatorial optimization problem from the field of computational biology . this problem finds applications , in particular , in the comparison of arc - annotated ribonucleic acid ( rna ) sequences . in this work we propose a simple , hybrid evolutionary algorithm to tackle this problem . the most important feature of this algorithm concerns a crossover operator based on solution merging . in solution merging , two or more solutions to the problem are merged , and an exact technique is used to find the best solution within this union . it is experimentally shown that the proposed algorithm outperforms a heuristic from the literature .


notwithstanding the important role that selection has traditionally played in evolutionary theory , evolution is not possible if selection has not variation to act upon .thus mutations widely understood as imperfect replications are the fuel to evolutionary dynamics .but mutations act at the level of the _ genotype _ whereas selection acts at the level of the _ phenotype _ the physical manifestation of the genotype , and the translation from one to the other the so - called genotype - phenotype ( gp ) map is far from trivial  .most mutations have no effect on the phenotype ( they are neutral ) , whereas occasionally a mutation has a dramatic ( mostly deleterious but sometimes benefitial ) phenotypic effect .thus , evolutionary dynamics is critically affected by the structure of the gp map  .understanding the gp map is a challenge for the evolutionary community , overall because addressing this problem in real systems is of an overwhelming complexity . accordingly , several simplified models have been studied to gain insights into this difficult issue  .computationally tractable models incorporate only a few levels among those involved in an actual gp map . they have considered protein folding  or protein aggregation  at basic molecular levels , and gene - regulatory  or metabolic  networks at higher functional levels .recent models encompass different levels at the same time  : in contrast with simple sequence - structure gp maps , the inclusion of different levels from genotype to phenotype permits the emergence of properties such as environment - dependent molecular function .pioneer among those models was the folding of sequences of rna into their secondary structure taken as a proxy for function  , which likely represents the most studied gp map to date .folding is driven by the formation of hydrogen bonds betwen cg , au , and gu base pairs , and the secondary structure of the molecule is determined by its minimum free - energy configuration . despite its apparent simplicity and the inherent impossibility to capture all features of natural gp relationships , rna sequence - to - secondary structure maps have properties shared by all gp maps studied to date , as the relationship between the number of genotypes yielding the same phenotype and the neutrality of the latter  .an important question in characterising this gp map is how many different secondary structures an rna molecule base pairs long can form .that problem was solved long ago , with the help of recurrence equations and subsequent generating functions , for several variants of the model .asymptotic expressions were provided when is large under different constraints imposed to the secondary structure such as having a minimum number of unpaired nucleotides in hairpin loops or stems of a minimal given length . another relevant question , which represents a step forward in the relation between structure and function , is how many secondary structures present particular structural moieties  .a prominent example is that of short sequences with hairpin loops , which have been shown to act as ribozymes with ligase catalytic activity under general conditions  . this undemanding phenotype - to - function map could have been essential in the emergence of rna molecules with complex activity in a prebiotic rna world  .the number of different structural motifs ( stems and hairpin loops among others ) in the limit of large was shown to be normally distributed  .symbolic methods introduced in modern combinatorics  greatly simplified the solution of that problem . in his exhaustive work , reidys tackled in depth the properties of rna folded structures bearing a type of tertiary interactions known as pseudoknots .the functional form of the number of structures with pseudoknots as a function of sequence length is of the general form , with and their values depending on restrictions put on the folded structure .an important constraint is the complexity of pseudoknots , which conditions the mathematical description of the problem . specifically , folded rna molecules are first reduced to a core skeleton containing information only on the pseudoknot architecture of the fold .generating functions for the number of possible alternative core structures with the previous architecture are derived and , subsequently , full folds are recovered by reintroducing stems and unpaired nucleotides in all possible compatible positions through composition of suitably defined generating functions .eventually , the total number of structures with the required pseudoknot properties and other possible structural constraints is obtained .further details can be found in  .this tricky procedure for structures with pseudoknots is not necessary in the case of plain secondary structures , as we show here .application of symbolic combinatorics to the latter case serves as an introduction to the calculation of the number of secondary structures for circular rna sequences . as will be shown , particular properties of circular rna demand the introduction of combinatorial techniques beyond those needed to enumerate open rna sequences with or without pseudoknots .circular rnas form covalently closed continuous loops with specific properties that distinguish them from linear rnas . among others ,circular rnas are non - coding , and have higher resistance to exonuclease - mediated degradation and higher structural stability .viroids , first described half a century ago  , are a relevant example of circular rna . these pathogenic , naked rna molecules of a few hundred nucleotides in length infect plants , occasionally causing strong symptoms .the mechanisms implied in cell entry , replication and propagation are still partly unknown .viroids present secondary structures with highly conserved regions that fall within two structural classes : rod - like and branched folds .the secondary structure of viroids plays an essential role in chemical function  and acts as a buffer to control the structural effect of point mutations  .circular rnas encoded in animal genomes , on the other hand , are currently a hot topic  .indeed , recent studies report a previously unsuspected abundance of circular rnas , which awakes the hunch that they must play main functional roles in the cell  . while some of those circular rnas have gene regulatory activity, the function performed by thousand of others is as yet unknown  .therefore , a theoretical understanding of the structural diversity of secondary structures of circular rnas appears as a timely endeavour , further considering that closed rna sequences have folding restrictions different from those of their linear counterparts . as we demonstrate here , their particular properties entail a comparatively lower number of secondary structures and lead to different asymptotic behaviour .the paper is organised as follows .section [ sec : brief ] briefly introduces those aspects of the symbolic method  relevant for our study . in section [ sec : rna ] we derive the generating function for the number of secondary structures with stems of length at least and hairpins with at least unpaired nucleotides , and recover the known expressions in the limit .section [ sec : basepairs ] contains the calculation of the frequency of structures with a given number of base pairs and is followed by the simultaneous count of the number of hairpins in section [ sec:2elements ] .the method extends to multivariate analysis suitable for counting combinatorial structures with any number of constraints .section [ sec : rings ] introduces the main novelty of this work , that is , the enumeration of secondary structures in circular rnas , followed by a derivation of the distributions of base pairs and hairpins as a function of in section [ sec : elementsrings ] .we close with a brief discussion .a full account of symbolic methods in combinatorics can be found in part a of ref .however , for the sake of self - containedness , we provide a very brief account in this section .readers familiar with this method can safely skip this section .  a combinatorial class will be a set of elements on which a _ size _ function is defined .the counting problem is to obtain , the number of elements such that .a related problem is to obtain the generating function ( runs on all possible sizes ) whose coefficients yield the sequence .the second writing for turns out to be very useful when thinking about these problems , because it means that every element of contributes to the sum defining with as many factors as its size .if a second function is defined on the elements of , namely ( representing any other feature of ) , we can introduce the bivariate generating function clearly counts the number of elements in of size and feature value , and the second writing can be interpreted as every element adding to the generating function besides the factor as many factors as the value of the feature .we can combine combinatorial classes to obtain new combinatorial classes .we first have the combinatorial product , which is the set made of the ` composite objects ' , where and ( notice that and are in general different objects ) .the size of the set is defined as ( the size of the composite object is the sum of the sizes of the components ) .accordingly ,   another operation is the combinatorial sum , , also referred to as disjoint union . is the union of and provided the elements of these two sets are distinguishable ( in other words , it is as if we paint the elements of these two sets with two different colors and then make the union of them both ) .therefore is either an element of or an element of and inherits the corresponding size .hence ,   there are further more complex operations with combinatorial classes .thus where , the class made of the _ null _ element alone ( ) , is referred to as the _ sequence _ of , i.e. , the combinatorial class made of the null element , plus all elements of , plus all pairs of elements of , and so on . by applying the transformation rules for the sum and the product   sequencescan be constrained to have composite elements just of certain specific compositions .for instance , ( times ) is restricted to sequences made of exactly elements of its generating function being .likewise define sequences containg at least and less than elements of respectively .then are their corresponding generating functions .other interesting operations with combinatorial classes are powersets ( ) , multisets ( ) , and cycles ( ) . is the class whose members are made of subsets of elements of .thus and therefore ( the last step follows by writing the product as the exponential of a sum of logarithms and then taylor - expanding those logarithms . )   is the class whose members are made of sequences of arbitrary length of elements of .thus and therefore   is the class whose members are made of circular sequences of arbitrary length of elements of .the derivation of the generating function of is more involved ( * ? ? ?* a.4 ) , but can be written in terms of euler s totient function as , and if is the prime factorisation of .thus , , , , etc . ].\ ] ]  one last class we will need is , whose members are pairs of elements of regardless of the order ( when the order matters the class is ) .there are many ways to obtain its corresponding generating function , but perhaps the easiest is to first introduce , the class of pairs of identical elements of .its corresponding generating function is because it contains one element per element of , but its size is double .then ,  ] stands for the sequence of from the root of the tree to the first branching point .this sequence must have at least  , but its length is otherwise unlimited hence the operator .what one can find at the first branching event is described by the next factor -\mathcal{b}- \mathsf{seq}_{<m}[\{\circ\}] ] excludes branchings that are not allowed : there can be neither a single branch that would mean extending the previous stem nor less than  and nothing else that would mean a hairpin loop with less than unpaired bases .let be the generating function of , the number of different -long secondary structures starting with a stem .since every ( unbonded base ) in contributes to and every ( pair of bonded bases ) contributes to , we can translate as where is the generating function of  ] , so we have to separate them out in and reintroduce them with a mark . in other words , we need to replace  ] .since the former gives rise to the term in , this operation amounts to replacing by in this and subsequent equations .now , interpreting as the genating function of the bivariate polynomials  being the number of rna secondary structures with base pairs and hairpins , we can obtain the asymptotic behaviour of the probability distribution through that of its characteristic function following the procedure explained in the previous section we find with  being the singularity of with smallest absolute value , and defined as in , , with replaced by defined in eq . .if we now identify we obtain the mean vector and covariance matrix of a bivariate normal distribution .for instance , setting , we get   thus , asymptotically ,   obtaining the marginal distribution of base pairs amounts to setting in .one can easily check that it correspond to the distribution .likewise , the marginal distribution of hairpins follows from setting in .it turns out to be a normal distribution with mean and variance .new structural elements can be counted in a similar vein , and their corresponding asymptotic distribution will be multivariate normal distributions whose parameters can be determined as we have done in this section .let now denote the combinatorial class containing all secondary structures of circular rnas . as for open sequences ,counting is better done using the tree representation of fig .[ fig : rnafold ] .if secondary structures of linear sequences are encoded in rooted trees , those of circular sequences would correspond to unrooted trees .there is an ambiguity though when passing from rooted to unrooted trees .the rules to transform structures into trees are directional , as illustrated in fig .[ fig : rnaring ] . to avoid that we introduce a new type of node , a square , to mark the extremes of all stems meeting at a hairpin , a multiloop , or a buldge .the square is understood to represent a base pair for each stem meeting at it . with this new representation each secondary structure of a circular rna uniquelydetermines a tree with two types of inner nodes filled circles and squares and empty circles for leaves , regardless of the direction we choose to read the structure .we will need a new combinatorial class to obtain , namely \times \left(\mathsf{seq}\big[\{\circ\}+\mathcal{b}\big]-\mathcal{b}- \mathsf{seq}_{<m}[\{\circ\}]\right ) , \label{eq : bk}\ ] ] the class of secondary rna structures starting with a stem of exactly base pairs .notice that implies that , and it follows from and that   counting unrooted trees is a more complicated issue than counting rooted trees . as a matter of fact , the strategy to do it is to reduce the problem to counting rooted trees .this is achieved thanks to a so - called _ dissymetry theorem _ that relates both classes of trees .if denotes a class of rooted trees and denotes that of their corresponding unrooted trees , then where denotes the class of unrooted trees with a marked node , and denotes the class of unrooted trees with a marked link . in our case , stands for , the class we want to count . as for ,an analysis of the proof of the theorem reveals that the involved arise as a result of removing links in trees of .thus , for the kind of trees we aim at counting we need to adapt this result , because links in are part of a stem , and stems must have at least base pairs . also ,as leaves ( empty circles ) are never the root of a tree , the argument can focus on inner nodes and inner links .consider .removing an inner link in yields two trees , one belonging to and another one belonging to , such that and .therefore let us now mark a link of to tranform it into an element of .two rooted trees from and with the same index contraints hang from both sides of the marked link . sincethe order of these two trees is irrelevant , using the idea behind the definition of ( sec .[ sec : brief ] ) . finally , if we mark a node as root , the two hanging branches are one tree from and another one from , such that and ; but if we mark a node as root , the resulting tree is formed by a ring from which either leaves ( ) or trees hang . thus \\ & -\mathcal{b}\times \mathsf{seq}_{<m}\big[\{\circ\}\big]-\mathsf{cyc}_2[\mathcal{b } ] , \end{split}\ ] ] where the two last terms stand for the removal of hairpins not allowed by the constraints (  ] ) which would be indistinguishable from longer stems .summarising , \\ & -\mathcal{b}\times \mathsf{seq}_{<m}\big[\{\circ\}\big]-\mathsf{cyc}_2[\mathcal{b } ] .\end{split}\ ] ]  now , , \end{split}\ ] ] and similarly ,\ ] ] so the generating function of is - \frac{b(z)^2}{z^{2s}}\left[s-1-(s-2)z^2\right]= -\frac{b(z)^2}{z^{2s}}\big(1-z^2\big).\ ] ] on the other hand , and so the generating function of is if we take into account that the generating function of $ ] is ,\ ] ] we finally obtain \\ & -\sum_{k=1}^{\infty}\frac{\varphi(k)}{k}\log\left[1-z^k - b\big(z^k\big )\right]-b(z)t_m(z ) , \end{split}\ ] ] or using , \\ & + \sum_{k=1}^{\infty}\frac{\varphi(k)}{k}\log r\big(z^k\big)-b(z)t_m(z ) .\end{split } \label{eq : vz}\ ] ] incidentally , is derived straight away from as   cccccc & # struct . & & # struct . & & # struct .+ 10 & 1 & 20 & 105 & 30 & 20423 + 11 & 1 & 21 & 166 & 31 & 35091 + 12 & 3 & 22 & 287 & 32 & 60838 + 13 & 3 & 23 & 486 & 33 & 105169 + 14 & 6 & 24 & 816 & 34 & 182728 + 15 & 7 & 25 & 1364 & 35 & 317068 + 16 & 14 & 26 & 2368 & 36 & 552059 + 17 & 20 & 27 & 4011 & 37 & 961008 + 18 & 38 & 28 & 6972 & 38 & 1677222 + 19 & 59 & 29 & 11811 & 39 & 2928607 +  table [ tab:1 ] lists the coefficients of up to discounting 1 for the unfolded chain . for long chains we can obtain an asymptotic formula out of . despite its appearance especially because of the presence of an infinite series , finding the singularity closest to the origin of is an easy task .that singularity is to be found in the functions and , as a root of .we know because all coefficients in the power series are larger than ( as a matter of fact , for , we already found ) .this means that the corresponding root of terms of the form , with , will be . in other words , all terms and with are analytic at .the only possibly competing singularity would come from a root of in .but implies , whose solutions for are and therefore their modulus is larger than .  from this discussionwe conclude that the singular terms of that will contribute to the asymptotic behaviour of its coefficients are those containing , and .accordingly , can be written , when , as \delta(z)^{1/2}}{4z^{2s}(1-z^2+z^{2s } ) } + \frac{t_m(z)\delta(z)^{1/2}}{2(1-z^2+z^{2s } ) } \\ & -\frac{\delta(z)^{1/2}}{(1-z)(1-z^2+z^{2s})+z^{2s}t_m(z ) } -\frac{\delta(z)^{3/2}}{3\big[(1-z)(1-z^2+z^{2s})+z^{2s}t_m(z)\big]^3 } \\ & + o\left(\delta(z)^{5/2}\right ) , \\ = & \ , \zeta(z)+\frac{\delta(z)^{3/2}}{4z^{2s}(1-z^2+z^{2s } ) \big[(1-z)(1-z^2+z^{2s})+z^{2s}t_m(z)\big ] } \\ & -\frac{\delta(z)^{3/2}}{3\big[(1-z)(1-z^2+z^{2s})+z^{2s}t_m(z)\big]^3 } + o\left(\delta(z)^{5/2}\right),\end{aligned}\ ] ] where is an analytic function in a circle containing .now , since follows from the very definition of , the expression above simplifies to   as in sec .[ sec : rna ] we can write , so near  and then darboux s theorem yields , \qquad k_s:=\frac{q(z_*)^{3/2}}{12z_*^{3s-\frac{3}{2 } } ( 1-z_*^2+z_*^{2s})^{3/2}}.\ ] ] for , we obtain , so we find the asymptotic estimate for the number of structures of circular rna sequences .we can introduce , the number of circular rnas with base pairs and hairpins , and , the generating function of the bivariate polynomials this generating function can be obtained , following the steps in sections [ sec : basepairs ] and [ sec:2elements ] , to be \\ & + \sum_{k=1}^{\infty}\frac{\varphi(k)}{k}\log r\big(z^k , w^k , u^k\big ) -b(z , w , u)t_m(z , u ) .\end{split } \label{eq : vzwu}\ ] ] it follows from this equation and the asymptotic analysis in the previous section that the characteristic function of the probability distribution is asymptotically given by where ^{3/2}.\ ] ] as expected , the leading term is the same as in .  identifying this expression with the expansion we obtain , for , , the probability distribution with symbolic method can be extended to the case of circular rnas in order to calculate the total number of closed secondary structures for sequences of lenght and the asymptotic distributions of the number of structures with specific moieties .circularization of rna eliminates some degrees of freedom that translate into a number of secondary structures -fold lower , as compared to the open linear counterpart .the exponent also appears in the enumeration of unrooted trees  , of which circular rnas are a particular case .the relationship between structure and function in circular rnas has to be stronger than in linear rnas , due at least to the non - coding nature of the former . from an evolutionary viewpoint , circularization of rnasmight be a low - cost procedure to seek new molecular functions .closed structures differ in essential ways from their open counterparts in their stability properties , and may as well bind different molecules due , for instance , to the sequences brought together when open ends are covalently closed  . at the same time , the number of available folds decreases under circularization by essentially a factor .this severe decrease in structural repertoire with respect to the open molecule implies that , on average , there are times more sequences that fold into a closed structure than into an open structure of the same length .the mutational robustness of closed structures is therefore very much enhanced .the enumeration of circular rna structures with pseudoknots is an open problem with relevance , among others , to better understand the _ in vivo _conformations adopted by viroids  and other circular rnas encoded in genomes , and the identification of their hypothetical interacting sites . a combination of the symbolic method and the additional techniques here used forcircular rna might facilitate the achievement of that goal .greenbury , s.f . ,ahnert , s.e . : the organization of biological sequences into constrained and unconstrained parts determines fundamental properties of genotype - phenotype maps . j. r. soc .interface * 12 * , 20150,724 ( 2015 )       memczak , s. , jens , m. , elefsinioti , a. , torti , f. , krueger , j. , rybak , a. , maier , l. , mackowiak , s.d . , gregersen , l.h . ,munschauer , m. , loewer , a. , ziebold , u. , landthaler , m. , kocks , c. , le noble , f. , rajewsky , n. : circular rnas are a large class of animal rnas with regulatory potency .nature * 495 * , 333342 ( 2013 )

Abstract: a quantitative characterization of the relationship between molecular sequence and structure is essential to improve our understanding of how function emerges . this particular genotype - phenotype map has been often studied in the context of rna sequences , with the folded configurations standing as a proxy for the phenotype . here , we count the secondary structures of circular rnas of length and calculate the asymptotic distributions of different structural moieties , such as stems or hairpin loops , by means of symbolic combinatorics . circular rnas differ in essential ways from their linear counterparts . from the mathematical viewpoint , the enumeration of the corresponding secondary structures demands the use of combinatorial techniques additional to those used for linear rnas . the asymptotic number of secondary structures for circular rnas grows as , with depending on particular constraints applied to the secondary structure . the abundance of any structural moiety is normally distributed in the limit , with a mean and a variance that increase linearly with .


analytical estimates for particle dynamics in electric and magnetic rings with and without focusing have been given in a variety of papers and notes .we have used these high precision estimates to provide benchmarks to test the accuracy of precision particle tracking programs .the term `` focusing '' in this paper is used to denote `` weak vertical focusing '' unless otherwise indicated .horizontal focusing is defined by the vertical focusing plus the geometry of the ring , always conforming with maxwell s equations .these benchmarks include :  * pitch correction to particle precession frequency in a uniform b - field with and without focusing . * vertical oscillations and energy oscillations in a uniform b - field with no focusing , electric focusing , and magnetic focusing . * radial and vertical oscillations and energy oscillations in an all - electric ring with and without weak focusing . * synchrotron oscillations and momentum storage with a radio frequency cavity ( rf ) in a uniform b - field . * an edm signal and systematic error with an rf wien filter in a magnetic ring .  in the analytical estimates that follow, we define as the lorentz factor of the design particle in the ring .the vertical pitch angle of a particle is defined such that where in cylindrical coordinates .the field focusing index is , with a number with range .these analytical estimates provide a means of benchmarking particle tracking programs . a precision tracking method that successfully passes all benchmarkscan provide a baseline to benchmark faster programs .we study the developed benchmarks in the context of runge - kutta / predictor - corrector integration .the validity of the developed benchmarks extends to any precision tracking program .precision experiments such as the muon ( ) and storage ring electric dipole moment ( edm ) experiments require measurements of sub - part per million ( ppm ) accuracy . in the case of a proton or deuteron storage ring edm experiment , a tracking program of extraordinary precisionis required to estimate the spin coherence time of the particle distribution and various lattice parameters , as well as to estimate the values of systematic errors associated with the experiment .many commonly used beam and spin dynamics programs ignore , or erroneously account for , second and higher - order effects .tracking in an electric storage ring poses the additional challenge of conforming with total - energy conservation while accounting for higher - order effects .  numerical integration with a sufficiently small step size allowed to run fora sufficiently long time may reproduce the analytical result with high accuracy .a tracking program to be used for estimates and investigations in precision experiments must be optimized to be as fast and accurate as possible .this calls for a well - tested and robust procedure to benchmark the accuracy of tracking programs in situations relevant to the experiments .  we summarize and derive analytical solutions to the equations of motion of a particle in various electric and magnetic rings .in several cases , comparison of the analytical estimate with precision tracking results identified discrepancies and indicated the need to improve the estimates . in this wayit was determined that the total correction due to vertical particle oscillations , the so - called pitch effect , can be significantly reduced .these analytical estimates provide individual benchmarks for tracking programs .a program well - benchmarked against these estimates would provide additional , more flexible means to benchmark faster programs in different conditions .we benchmarked a program based on runge - kutta / predictor - corrector method against the developed analytical estimates .runge - kutta / predictor - corrector integration should reproduce the analytical estimates to sub - ppm accuracy on a time scale on the order of hours , in order to be a feasible candidate for benchmarking faster programs .together , the analytical estimates and a program based on runge - kutta / predictor - corrector integration provide a powerful tool for benchmarking precision tracking programs for muon , storage ring edm , and other precision physics experiments requiring high precision beam and spin dynamics simulation .for a particle of mass and charge , there are two differential equations that govern particle and spin dynamics . for particle velocity and rest spin in external fields , the equations are : ,\ ] ] and the t - bmt equation , with an anomalous magnetic moment of the particle : .\end{aligned}\ ] ]  the predictor - corrector integration was used with a step size of to numerically solve the two differential equations with the corresponding initial conditions .although the method is relatively slow , it is very simple and accurate .this method uses the runge - kutta method to start the integration process and we refer to it as runge - kutta / predictor - corrector in this document .a magnetic ring consists of a uniform magnetic field , taken to be in the vertical direction .the correction to the precession frequency due to a vertical pitch is defined by , where is the ( ) frequency for a particle with anomalous magnetic moment .the predicted correction is : with , where is the vertical ( pitch ) oscillation frequency .   when there is no focusing or when , the correction from equation [ mrnf1 ] becomes :   where for linear oscillations , , where is the maximum pitch angle of the particle trajectory .for a particle with and a constant 1.0mrad vertical pitch as shown in figure [ pitch1 ] , the simulated correction to the ( ) precession frequency of is in very good agreement with the analytically predicted value of using equation [ mrnf2 ] .[ .65]the particle path in cartesian coordinates in a uniform b - field with pitch angle , for a ring with a 5 m radius.,title="fig : " ]  checking over several values of confirms that the analytic expression and the pitch correction in the tracking simulation agree for small , as expected .when there is magnetic focusing and when , the correction from equation [ mrnf1 ] becomes :    the analytical estimate for the average particle radial deviation from the ideal orbit with radius , with weak magnetic focusing index , takes the form : for a vertical pitch frequency significantly greater than the ( ) precession frequency of the particle , where is the momentum compaction factor.  equation [ mrmf1 ] predicts an average radial deviation of using 1mrad and a field index , consistent with the tracking results shown in figure [ drmrmf ] to sub - part per billion ( ppb ) level .the dependence of on the field index is shown to hold over a range of values in figure [ bfdr ] .[ .29]the particle deviation from the ideal radial position over time , modulo 50 .the simulations used a maximum pitch angle of and magnetic focusing with field index .,title="fig : " ]  [ 1.25]the average versus the field focusing index .the solid line represents the predicted values while the points are the results of tracking .the simulation used a maximum pitch angle of .,title="fig : " ]  in a continuous storage ring with weak focusing , field strength , and ring radius , the vertical and horizontal magnetic field components around the ideal trajectory can be expressed to second - order in the vertical position as : where the nonlinearity arises from the application of maxwell s equations in cylindrical coordinates .the horizontal and vertical tunes are given by and respectively .we make use of the relations from equation [ mrmf1 ] : and , where is the maximum vertical excursion . from thiswe see that :   by considering the time - averaged relative b - field change , calling the modification due to the different b - field encountered by the particle , we find that the correction to the ( ) precession frequency in a magnetic storage ring with weak focusing , i.e. , is given by the expression : where we see that several terms of the inhomogeneous b - field correction and the correction in equation [ mrnf3 ] cancel , leaving a small correction .the necessity of including the second - order inhomogeneous magnetic field contributions was overlooked by previous authors .investigations with precision tracking identified the discrepancy and motivated the improvement of the analytical estimate .this holds for a vertical pitch frequency much greater than the ( ) precession frequency of the particle , which for a weak focusing ring means .equation [ mrmf2 ] implies that the pitch effect can , in principle , be made to vanish for , but the condition makes it rather difficult to achieve . to test the tracking program we introduced a particle with 10 times the muon mass , with same value as the muon , stored in a ring radius of 7.112 m .the program indeed showed that the pitch correction vanishes with an uncertainty at the part per billion ( ppb ) level when was used .  for realistic muon parameters ,the observed ( ) frequency is off from its correct value by + 0.109ppm , for a vertical maximum pitch angle , and , consistent with the offset shown in figure [ pitch2a ] to sub - ppb level .[ .3]the pitch correction to the ( ) frequency with angle , , and , is 0.109ppm , consistent with equation [ mrmf2 ] to sub - ppb level ., title="fig : " ]  a resonance of the pitch effect correction occurs when the vertical pitch frequency is equal to the ( ) precession of the stored particles .the correction approaches equation [ mrnf2 ] for and equation [ mrnf3 ] for .the full range of pitch corrections over a range of index values is shown in figure [ pitch2 ] . when all the fields are taken properly into account , as shown above ,the tracking results reproduce the same curve to sub - ppb level for .[ 0.68]parameter from equation [ mrnf1 ] for the pitch correction to the ( ) frequency over a range of values using and .a resonance occurs when , at , as expected ., title="fig : " ]  a comparison of the frequency shift predicted by equation [ mrmf2 ] and the results from tracking is given in table [ mrmft1 ] .the analytical estimates of the pitch correction and the tracking results are in very good agreement , better than ppb level .this level of precision is adequate for the muon experiments currently underway  , both aiming for better than 0.1ppm total systematic error ..comparison of the frequency shift estimated using equation [ mrmf2 ] and the tracking results .the tracking results assume a muon with , stored in a magnetic ring with magnetic focusing and a radius m .the vertical angle used is .the observed frequency is shifted higher by the small factors given below depending on the -value used . [ cols="^,^,^ " , ]   [ prot ]  the wien filter provides another benchmark for testing the accuracy of the analytical estimates and the runge - kutta / predictor - corrector tracking method .we again see very good agreement between the analytically predicted values and those calculated by tracking .we have determined an array of analytical estimates for benchmarking tracking programs for precision storage ring experiments .the benchmarks form a robust test for electric and magnetic rings , with and without focusing , as well as rf cavities and wien filters .the inclusion of a high - order contribution to the weak magnetic focusing estimate resulted in the discovery of a method to reduce or eliminate the pitch effect , which was overlooked by other authors . togetherthese analytical estimates give a powerful tool to benchmark programs for studying particle motion and spin dynamics in a variety of storage ring configurations .the runge - kutta / predictor - corrector integration appears to be an accurate , albeit slow , tool for precision tracking .it has passed all the benchmarks that it was tested against , often to the part per billion level .the tracking program was able to successfully simulate particle dynamics in electric and magnetic rings with and without weak focusing , in agreement with analytical estimation .consequently , we conclude that the runge - kutta / predictor - corrector method can be used to benchmark faster tracking programs .other tracking programs are also capable of providing high accuracy at least under certain conditions , e.g. ,  .what we have presented here is an array of high precision analytical estimates that can be used to benchmark candidate simulation programs .we would like to thank the department of energy and brookhaven national laboratory for their continued support of the high school and supplemental undergraduate research programs .we especially thank the storage ring edm collaboration .doe partially supported this project under bnl contract no .de - sc0012704 .ibs - korea partially supported this project under system code ibs - r017-d1 - 2014-a00 .9 j.h . field and g.fiorentini , `` corrections to the frequency in weak focusing storage devices due to betatron oscillations . ''_ il nuovo cimento _ 21a ( 1974 ) 297 - 328 .morse , y.f .orlov , and y.k .semertzidis , `` rf wien filter in an electric dipole moment storage ring : the ` partially frozen spin ' effect . ''_ physical review special topics - accelerators and beams _ 16 ( 2013 ) 114001 .s.hacimerolu and y.k .`` results of precision particle simulations in an all - electric ring lattice using fourth - order runge - kutta integration . ''_ nuclear instruments and methods in physics research section a _ 743 ( 2014 ) 96 - 102 .mane , `` comment on ` results of precision particle simulations in an all - electric ring lattice using fourth - order runge - kutta integration'.''__nuclear instruments and methods in physics research a _ _ 769 ( 2015 ) : 26 - 31 .

Abstract: a set of analytical benchmarks for tracking programs is required for precision storage ring experiments . to determine the accuracy of precision tracking programs in electric and magnetic rings , a variety of analytical estimates of particle and spin dynamics in the rings were developed and compared to the numerical results of tracking simulations . initial discrepancies in the comparisons indicated the need for improvement of several of the analytical estimates . as an example , we found that the fourth - order runge - kutta / predictor - corrector method was slow but accurate , and that it passed all the benchmarks it was tested against , often to the sub - part per billion level . thus , high precision analytical estimates and tracking programs based on fourth - order runge - kutta / predictor - corrector integration can be used to benchmark faster tracking programs for accuracy .   analytical benchmarking , precision particle tracking , electric and magnetic storage rings , runge - kutta , predictor - corrector


many important properties in the theory of social choice concern the consistency of aggregation functions under varying parameters . what happens if two electorates are merged ?how should an aggregation function deal with components of similar alternatives? how should choices from overlapping agendas be related to each other ?these considerations have led to a number of consistency axioms that these functions should ideally satisfy .unfortunately , social choice theory is rife with impossibility results which have revealed the incompatibility of many of these properties . , for example , have pointed out that every social choice function that satisfies condorcet - consistency violates consistency with respect to variable electorates . on the other hand, it follows from results by and that all pareto - optimal social choice functions that are consistent with respect to variable electorates are inconsistent with respect to components of similar alternatives .the main result of this paper is that , in the context of _ probabilistic _ social choice , consistency with respect to variable electorates and consistency with respect to components of similar alternatives uniquely characterize an appealing probabilistic social choice function , which furthermore satisfies condorcet - consistency .probabilistic social choice functions yield lotteries over alternatives ( rather than sets of alternatives ) and were first formally studied by , , and .perhaps one of the best known results in this context is s characterization of strategyproof probabilistic social choice functions .an important corollary of gibbard s characterization , attributed to hugo sonnenschein , is that _random dictatorships _ are the only strategyproof and _ ex post _ efficient probabilistic social choice functions . in random dictatorships ,one of the voters is picked at random and his most preferred alternative is implemented as the social choice . while gibbard s theorem might seem as an extension of classic negative results on strategyproof non - probabilistic social choice functions , it is in fact much more positive ( see also * ? ? ?in contrast to deterministic dictatorships , the uniform random dictatorship ( henceforth , random dictatorship ) , in which every voter is picked with the same probability , enjoys a high degree of fairness and is in fact used in many subdomains of social choice that are concerned with the fair assignment of objects to agents ( see , * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?* ) .   in this paper, we consider two consistency axioms , non - probabilistic versions of which have been widely studied in the literature .the first one , _ population - consistency _ , requires that , whenever two disjoint electorates agree on a lottery , this lottery should also be chosen by the union of both electorates .the second axiom , _ composition - consistency _ , requires that the probability that an alternative receives is unaffected by introducing new variants of another alternative .alternatives are variants of each other if they form a component , they bear the same relationship to all other alternatives . on top of that , composition - consistency prescribes that the probability of an alternative within a component should be directly proportional to the probability that the alternative receives when the component is considered in isolation .apart from their intuitive appeal , these axioms can be motivated by the desire to prevent a central planner from strategically partitioning the electorate into subelectorates or by deliberately introducing similar variants of alternatives , respectively .our first result shows that there is no non - probabilistic social choice function that satisfies both axioms simultaneously .we then move to probabilistic social choice and prove that the only probabilistic social choice function satisfying these properties is the function that returns all _ maximal lotteries _ for a given preference profile .maximal lotteries , which were proposed independently by and other authors , are equivalent to mixed maximin strategies of the symmetric zero - sum game given by the pairwise majority margins .whenever there is an alternative that is preferred to any other alternative by some majority of voters ( a so - called condorcet winner ) , the lottery that assigns probability 1 to this alternative is the unique maximal lottery . in other words ,maximal lotteries satisfy _ condorcet - consistency_. at the same time , maximal lotteries satisfy population - consistency which has been identified by , , , and others as _ the _ defining property of borda s scoring rule . as such, the characterization can be seen as one possible resolution of the historic dispute between the founding fathers of social choice theory , the chevalier de borda and the marquis de condorcet , which dates back to the 18th century .random dictatorship , the most prevalent probabilistic social choice function , satisfies population - consistency , but fails to satisfy composition - consistency .for this reason , we also consider a weaker version of composition - consistency called cloning - consistency , which is satisfied by random dictatorship , and provide an alternative characterization of maximal lotteries using population - consistency , cloning - consistency , and condorcet - consistency ( see ).   clearly , allowing lotteries as outcomes for high - stakes political elections such as those for the u.s .presidency would be highly controversial and considered by many a failure of deliberative democracy .if , on the other hand , a small group of people repeatedly votes on where to hold their next meeting , randomization would likely be more acceptable and perhaps even desirable .the use of lotteries for the selection of officials interestingly goes back to the world s first democracy in athens where it was widely regarded as a principal characteristic of democracy .it has also been early observed in the social choice literature that `` unattractive social choices may result whenever lotteries are not allowed to compete . [ ] refusal to entertain lotteries on alternatives can lead to outcomes that to many appear to be inequitable and perhaps even inefficient '' . in contemporary research ,probabilistic social choice has gained increasing interest in both social choice ( see , * ? ? ?* ; * ? ? ?* ; * ? ? ?* ) and political science ( see , * ? ? ?* ; * ? ? ?* ; * ? ? ?whether lotteries are socially acceptable depends on many factors , only some of which are based on formal arguments . in our view , two important factors are _( i ) the effective degree of randomness _ and _ ( ii ) risk aversion on behalf of the voters_.  as to _ ( i ) _ , it is easily seen that certain cases call for randomization or other means of tie - breaking .for example , if there are two alternatives , and , and exactly half of the voters prefer while the other half prefers , there is no deterministic way of selecting a single alternative without violating basic fairness conditions .there are several possibilities to extend the notion of a tied outcome to three or more alternatives .an important question in this context is whether ties are a rare exception or a common phenomenon. a particularly rigorous and influential extension due to condorcet declares a tie in the absence of a pairwise majority winner . according to condorcet, it is the intransitivity of social preferences , as exhibited in the condorcet paradox , that leads to situations in which there is no unequivocal winner . as it turns out , maximal lotteries are degenerate if and only if there is a condorcet winner .our main result thus establishes that the degree of randomness entailed by our axioms is precisely in line with condorcet s view of equivocality .interestingly , there is strong empirical and experimental evidence that most real - world preference profiles for _ political _ elections do admit a condorcet winner ( see , * ? ? ?* ; * ? ? ?* ; * ? ? ?maximal lotteries only randomize in the less likely case of cyclical majorities . analyzed the support of maximal lotteries and found that the average support size is less than four under various distributional assumptions and up to 30 alternatives .by contrast , random dictatorship randomizes over all alternatives in almost all elections .  as to _( ii ) _ , risk aversion is strongly related to the frequency of preference aggregation .if an aggregation procedure is not frequently repeated , the law of large numbers does not apply and risk - averse voters might prefer a sure outcome to a lottery whose expectation they actually prefer to the sure outcome . hence , probabilistic social choice seems particularly suitable for novel preference aggregation settings that have been made possible by technological advance .the internet , in particular , allows for much more frequent preference aggregation than traditional paper - and - pencil elections . in recurring randomized elections with a fixed set of alternatives ,voters need not resubmit their preferences for every election ; rather preferences can be stored and only changed if desired .for example , maximal lotteries could help a group of coworkers with the daily decision of where to have lunch without requiring them to submit their preferences every day .another example are automatic music broadcasting systems , such as internet radio stations or software djs , that decide which song should be played next based on the preferences of the listeners . in contrast to traditional deterministic solutions to these problems such as sequential dictatorships ,the repeated execution of lotteries is a memoryless process that guarantees _ ex ante _ fairness after any number of elections .finally , it should be noted that the lotteries returned by probabilistic social choice functions do not necessarily have to be interpreted as probability distributions .they can , for instance , also be seen as fractional allocations of divisible objects such as time shares or monetary budgets .the axioms considered in this paper are equally natural for these interpretations as they are for the probabilistic interpretation .let be an infinite universal set of alternatives .the set of _ agendas _ from which alternatives are to be chosen is the set of finite and non - empty subsets of  , denoted by  .the set of all linear ( complete , transitive , and antisymmetric ) _ preference relations _ over some set will be denoted by .  for some finite set , we denote by the set of all probability distributions with rational values over . a ( fractional ) preference profile for a given agenda is an element of , which can be associated with the -dimensional ( rational ) unit simplex .we interpret as the fraction of voters with preference relation .preference profiles are depicted by tables in which each column represents a preference relation with .the table below shows an example profile on three alternatives . set of all preference profiles for a fixed agenda is denoted by and is defined as the set of all preference profiles , . for and , is the restriction of to alternatives in , for all ,    for all , is the fraction of voters who prefer to ( the set represents the preference relation on two alternatives with ) . in example 1 , .  elements of are called _ lotteries _ and will be written as convex combinations of alternatives . if is a lottery , is the probability that assigns to alternative .  a _ probabilistic social choice function ( pscf ) _ is an ( upper hemi- ) continuous function that , for any agenda , maps a preference profile to a convex subset of .a pscf is thus a collection of mappings from high - dimensional simplices to low - dimensional simplices .two further properties that we demand from any pscf are _ unanimity _ and _ decisiveness_. unanimity states that in the case of one voter and two alternatives , the preferred alternative should be chosen with probability 1 .since we only consider fractional preference profiles , this amounts to for all and , in other words , for every preference profile that yields multiple lotteries , there is an arbitrarily close preference profile that only yields a single lottery .probabilistic social choice functions considered in the literature usually satisfy these conditions and are therefore well - defined pscfs . for example , consider _ random dictatorship ( ) _ , in which one voter is picked uniformly at random and his most - preferred alternative is returned .formally , returns the unique lottery , which is determined by multiplying fractions of voters with their respective top choices , for all and , where denotes the unique alternative such that for all .for the preference profile given in example 1 ,  is single - valued and therefore trivially decisive and convex - valued .it is also easily seen that satisfies unanimity and continuity and thus constitutes a pscf .a useful feature of our definition of pscfs is that traditional _ set - valued _ social choice functions ( scfs ) ( also known as _ social choice correspondences _ ) can be seen as a special case , namely as those pscfs that map every preference profile on some agenda to for some .such pscfs will be called _( non - probabilistic ) scfs_.   as stated in , is an infinite set of alternatives . for conveniencewe will assume that .for ,  ] . for two sets and ,let denote the set of all bijections from to ( where if ) .let be the set of all permutations of .we will frequently work with profiles in which alternatives are renamed according to some bijection from one set of alternatives to another .for all , , and , let and , for , let such that .a well - known symmetry condition for pscfs is _ neutrality _ , which requires that all alternatives are treated equally in the sense that renaming alternatives is appropriately reflected in the outcome .formally , a pscf is neutral if     let be a composition - consistent pscf , , , and .we have to show that . to this end ,let .first , choose and such that for all  ] .thus , we have that .we now apply composition - consistency to and the components for all  ] . applying composition - consistency analogously to and for all  ] .finally , let such that and is a component in for all  ] .notice that and . since by construction of , we have .hence , .the fact that follows from application of the above to and .the following notation is required for our proofs .for some set , denotes the _ uniform distribution _ over .in particular , for , is the uniform lottery over , .the _ support _ of a lottery is the set of all alternatives to which assigns positive probability , .the _ -norm _ of is denoted by , . for , the _ convex hull _ is the set of all convex combinations of elements of ,  is _ convex _ if . the _ affine hull is the set of all affine combinations of elements of ,  is an _ affine subspace _ if .we say that are _ affinely independent _ if , for all with , implies for all  ] ,    population - consistency is arguably one of the most natural axioms for variable electorates and is usually considered in a slightly stronger version , known as _ reinforcement _ or simply _ consistency _ , where the inclusion in the equation above is replaced with equality whenever the left - hand side is non - empty ( see also ) . note that population - consistency is merely a statement about abstract sets of outcomes , which makes no reference to lotteries whatsoever .it was first considered independently by , , and and features prominently in the characterization of scoring rules by and .population - consistency and its variants have found widespread acceptance in the social choice literature ( see , * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?the second axiom prescribes how pscfs should deal with _decomposable _ preference profiles . for two agendas , is a component in if the alternatives in are _ adjacent _ in all preference relations that appear in , for all and , if and only if for all with .intuitively , the alternatives in can be seen as variants or clones of the same alternative because they have exactly the same relationship to all alternatives that are not in .for example , consider the following preference profile in which constitutes a component . \multicolumn{3}{c}{r}\\ \end{array } \qquad\qquad \begin{array}{cc } \nicefrac{1}{2 } & \nicefrac{1}{2 } \\\midrule a & b \\ b & a \\\\[2ex ] \multicolumn{2}{c}{r|_{a'}}\\ \end{array } \qquad\qquad \begin{array}{cc } \nicefrac{1}{3 } & \nicefrac{2}{3 } \\\midrule b ' & b \\ b & b ' \\ \\[2ex ] \multicolumn{2}{c}{r|_b}\\  \end{array } \tag{example 3}\ ] ] the ` essence ' of is captured by , where contains only one of the cloned alternatives .it seems reasonable to demand that a pscf should assign the same probability to ( say , ) independently of the number of clones of and the internal relationship between these clones . this condition will be called cloning - consistency and was first proposed by ( see also * ? ? ?its origins can be traced back to earlier , more general , decision - theoretic work by and where it is called _ deletion of repetitious states _ as well as early work on majoritarian scfs by . for a formal definition of cloning - consistency , let and such that . then , a pscf satisfies cloning - consistency if , for all such that is a component in , when having a second look at example 3 , it may appear strange that cloning - consistency does not impose any restrictions on the probabilities that assigns to the clones . while clones behave completely identical with respect to uncloned alternatives , they are not indistinguishable from _ each other_. it seems that the relationships between clones ( ) should be taken into account as well . for example, one would expect that assigns more probability to than to because two thirds of the voters prefer to .an elegant and mathematically appealing way to formalize this intuition is to require that the probabilities of the clones and are directly proportional to the probabilities that assigns to these alternatives when restricting the preference profile to the component .this condition , known as _ composition - consistency _ , is due to and was studied in detail for majoritarian scfs ( see , * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?* ) .  for a formal definition of composition - consistency , let and and define the operator is extended to sets of lotteries and by applying it to all pairs of lotteries in , .then , a pscf satisfies composition - consistency if for all such that is a component in , in example 3 above , , , and composition - consistency would imply that .in the context of scfs ( pscfs that only return the convex hull of degenerate lotteries ) , there is some friction between population - consistency and composition - consistency .in fact , the conflict between these notions can be traced back to the well - documented dispute between the pioneers of social choice theory , the chevalier de borda and the marquis de condorcet ( see , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) .borda proposed a score - based voting rule_borda s rule_that can be axiomatically characterized using population - consistency .it then turned out that the entire class of scoring rules ( which apart from borda s rule also includes plurality rule ) is characterized by population - consistency .condorcet , on the other hand , advocated _ condorcet - consistency _ , which requires that an scf selects a condorcet winner whenever one exists .as condorcet already pointed out , borda s rule fails to be condorcet - consistent .worse , even showed that no condorcet - consistent scf satisfies population - consistency ( the defining property of scoring rules ) . , on the other hand , showed that no pareto - optimal rank - based rule a generalization of scoring rules satisfies composition - consistency while this property is satisfied by various condorcet - consistent scfs .one of the few scfs that satisfies both properties is the rather indecisive pareto rule ( which returns all alternatives that are not pareto - dominated ) . since our definition of pscfs already incorporates a certain degree of decisiveness , we obtain the following impossibility .( the proofs of all theorems are deferred to the appendix . )notice that and thus , .neutrality implies that .since is a component in , we have .since is also a component in , composition - consistency implies that .observe that and hence .  applying this argument repeatedly ,we get for all .since for and is continuous , we get . similarly , it follows that .the fact that is convex - valued implies that .the characterization of for agendas of size 2 proceeds along the following lines . by unanimity , neutrality , and ,we know which lotteries have to be returned by every composition - consistent pscf for three particular profiles .then population - consistency implies that every such pscf has to return all maximal lotteries .last , we again use population - consistency to show that the function is not decisive if it additionally returns lotteries that are not maximal .   first , note that is fully determined by .let be the profile such that .since , there is  ] .similarly , for all with  ] .but then \}\subseteq \{r\in\mathcal{r}|_a\colon |f(r)|>1\} ] , constraints are needed to ensure skew - symmetry of on \times[n'] ] , , and constraints are needed to ensure that the first rows ( and hence also the columns ) of sum up to , for all  ] be the set of all permutations that are cyclic on and coincide with the identity permutation outside of .) ] for some  ] where the weight of the edge from to is .if , there exists a cycle along edges with positive weight of length at least 3 in the subgraph induced by  ] for some . if then .hence , the induction hypothesis is trivial .if , , we can find  ] such that  and . note that defines a cycle of length at least 3 in the graph that corresponds to .we define by letting let \text { and } m^1(i , j ) > 0\} ] for some . by construction of , we have that with .leverages lemmas [ lem : cancel ] , [ lem : str - regular ] , [ lem : skew - dim ] , and [ lem : hamilton ] to show two statements .first , it identifies the dimension of the space of all profiles that are regular on  .second , it proves that there is a full - dimensional subset of the space of all profiles that are regular on  for which every pscf that satisfies population - consistency and composition - consistency returns the uniform lottery over .[ lem : regular ] let be a pscf that satisfies population - consistency and composition - consistency and . then, there is of dimension such that for every .  to simplify notation, we assume without loss of generality that  ] . for and , let be the matrix that results from by permuting the rows and columns of according to , .from we know that we can find a set }^{[n']} ] , it follows from that )\in f(s) ] such that )\in f(r) ] with and ) ] and be defined as follows : if for some .note that is regular on  ] and ) ] are components in . for , we have by construction that .hence , it follows from that and .moreover , neutrality , convexity , and composition - consistency imply that by the symmetry of with respect to .now let .observe that is a component in and .thus , composition - consistency and imply that furthermore , is a component in and .as before , we get also is a component in and thus , \setminus b})\text.\ ] ] as is a component in and , we get then , it follows from convexity of that ) = \frac{k}{n'}{\mathrm{uni}}(b ) + \frac{1}{n'}\sum_{a_i\in [ n']\setminus b } a_i\in f(r_b^\pi)\text,\ ] ] since and for every  ] , since the s are linearly independent .hence , and , which implies that for all  ] for every  ] finishes the proof .we now consider pscfs that may return a lottery that is _ not _ maximal .the following lemma shows that for every such pscf there is a set of profiles with a strict condorcet winner for which it returns the uniform lottery over a fixed subset of alternatives if we additionally require population - consistency and composition - consistency .furthermore , this set of profiles has only one regular profile in its linear hull .later this statement is leveraged to show that every population - consistent and composition - consistent pscf returns a subset of maximal lotteries .[ lem : cond - dim ] let be a pscf that satisfies population - consistency and composition - consistency . if , there are and of dimension such that for every and .  if , there are , , and such that .since is not a maximal lottery , by definition , there is such that .linearity of matrix multiplication implies that there is such that , where is the entry of corresponding to .we first use composition - consistency to `` blow up '' alternatives such that the resulting lottery is uniform .let be the greatest common divisor of , .for every , let such that , , and all are pairwise disjoint .moreover , let .now , choose such that , is a component in for every , and for every .hence , for all as is neutral and is convex . to simplify notation ,let . by composition - consistency, it follows that .observe that   we now construct a profile such that is a strict condorcet winner in and . to this end , let be the uniform mixture of all profiles that arise from by permuting all alternatives in , then , for all . neutrality and population - consistency imply that .let and define , for  ] .now , let such that for all and for all , . for  ] and such that .furthermore , if there is ) ] , then .for all  ] be the profile such that if and is a component in and otherwise . by construction, we have that and for all .let and .then , we have that .the second part of the lemma follows from the symmetry of the construction .  for profiles which admit a unique maximal lottery, it follows from that .it turns out that every maximal lottery that is a vertex of the set of maximal lotteries in one of the remaining profiles is the limit point of a sequence of maximal lotteries of a sequence of profiles with a unique maximal lottery converging to the original profile .the proof of heavily relies on the continuity of .let be a pscf that satisfies population - consistency and composition - consistency , , and .if follows from that . by neutrality , we can assume without loss of generality that  ] .we first consider the case where is odd . by , there are and such that intuitively , defines a weighted cycle on  ] , we define and .population - consistency implies that for all  ] .hence , for every maximin strategy in , it follows that for every  ] .it follows from basic linear algebra that }\right ) = c^{k-1}\prod_{i = 1}^{k-2 } \left(\frac{1}{p_ip_{i+1}}\right)^2\neq 0\text,\ ] ] and hence , } ] has rank , since skew - symmetric matrices of odd size can not have full rank.a skew - symmetric matrix of odd size can not have full rank , since and , hence , . ]furthermore , }) ] has rank for all .in particular , if for all  ] .then ,  ] . for , this follows from the definition of .now , let .then , where the second equality follows from the induction hypothesis .now , where the third equality follows from the fact that .  for  ] , furthermore , it follows from that as , and , for  ] .a _ ( weak ) condorcet winner _ is an alternative that is _ maximal _ in according to in the sense that for all .if for all , is called a _ strict condorcet winner_. a pscf is _ condorcet - consistent _ if whenever is a condorcet winner in .it is well known from the condorcet paradox that maximal elements may fail to exist .as shown by and , this drawback can , however , be remedied by considering lotteries over alternatives . for two lotteries ,the majority margin can be extended to its bilinear form , the _ expected _ majority margin .the set of maximal lotteries is then defined as the set of `` probabilistic condorcet winners . ''formally , for all and , before computing maximal lotteries .this is , for example , the case for , , , and .maximal lotteries as defined in this paper were considered by , , and . allowed the application of any odd function to the entries of , which covers both variants as special cases.[fn : c1c2 ] ] as an example , consider the preference profile given in example 1 of .alternative is a strict condorcet winner and .this is in contrast to , which puts positive probability on any first - ranked alternative no matter how small the corresponding fraction of voters .the minimax theorem implies that is non - empty for all  .in fact , can be interpreted as the payoff matrix of a symmetric zero - sum game and maximal lotteries as the mixed maximin strategies ( or nash equilibrium strategies ) of this game .hence , maximal lotteries can be efficiently computed via linear programming .interestingly , is a singleton in almost all cases . in particular , this holds if there is an odd number of voters .moreover , we point out in appendix [ sec : mlsat ] that the set of preference profiles that yield a unique maximal lottery is open and dense , which implies that the set of profiles with multiple maximal lotteries is nowhere dense and thus negligible . as a consequence , satisfies decisiveness as well as the other properties we demand from a pscf ( such as unanimity , continuity , and convex - valuedness ) and therefore constitutes a well - defined pscf .in contrast to the non - probabilistic case where majority rule is known as the only reasonable and fair scf on two alternatives , there is an infinite number of such pscfs even when restricting attention to only two alternatives ( see , * ? ? ?* ) . within our framework of fractional preference profiles , a pscf on two alternativescan be seen as a convex - valued continuous correspondence from the unit interval to itself .unanimity fixes the function values at the endpoints of the unit interval , decisiveness requires that the points where the function is multi - valued are isolated , and population - consistency implies that the function is monotonic .two natural extreme cases of functions that meet these requirements are a probabilistic version of simple majority rule and the proportional lottery ( ) .interestingly , these two extreme points are taken by maximal lotteries and random dictatorship as for all and ,   when considering up to three alternatives and additionally taking composition - consistency into account , any such pscf coincides with majority rule on two - element agendas . violates composition - consistency , which can be seen in example 3 in .however , does satisfy cloning - consistency ( see ) .] when allowing an arbitrary number of alternatives , the axioms completely characterize .the proof of is rather involved yet quite instructive as it rests on a number of lemmas that might be of independent interest ( see appendix [ sec : fml ] ) .the high - level structure is as follows .the fact that satisfies population - consistency and composition - consistency follows relatively easily from basic linear algebra .  for the converse direction, we first show that population - consistency and composition - consistency characterize on two - element agendas . for agendas of more than two alternatives, we assume that is a population - consistent and composition - consistent pscf and then show that and .the first statement takes up the bulk of the proof and is shown by assuming for contradiction that there is a preference profile for which yields a lottery that is _ not _ maximal .we then identify a set of preference profiles with full dimension for which returns the uniform lottery over a fixed subset of at least two alternatives and which has the uniform profile , the preference profile in which every preference relation is assigned the same fraction of voters , in its interior . along the way we show that has to be condorcet - consistent for all preference profiles that are close to the uniform profile .it follows that there has to be an -ball of profiles around some strict condorcet profile ( close to the uniform profile ) , for which returns the uniform lottery over a non - singleton subset of alternatives as well as the lottery with probability 1 on the condorcet winner .this contradicts decisiveness .for the inclusion of in , we take an arbitrary preference profile and an arbitrary vertex of the set of maximal lotteries for this profile and then construct a sequence of preference profiles that converges to the original profile and whose maximal lotteries converge to the specified maximal lottery . from and continuity , we obtain that has to select this lottery in the original preference profile .finally , convexity implies that .[ rem : ind ] the axioms used in are independent from each other . satisfies population - consistency , but violates composition - consistency ( see also ) .the same is true for borda s rule .when defining via the third power of majority margins , satisfies composition - consistency , but violates population - consistency .were already considered by .see also footnote [ fn : c1c2 ] . ] also the properties implicit in the definition of pscfs are independent .the pscf that returns all maximal lotteries for the profile in which all preference relations are reversed violates unanimity but satisfies decisiveness , population - consistency , and composition - consistency .when not requiring decisiveness , returning all _ ex post _ efficient lotteries is consistent with the remaining axioms .satisfies all remaining axioms .whether convexity is required is open . ]the proof of exploits the infinity of the universe .as stated in the previous remark , satisfies composition - consistency and violates population - consistency .however , does satisfy population - consistency when there are only up to three alternatives .this implies that the statement of requires a universe that contains at least four alternatives .[ rem : unique ] the set of profiles in which is not single - valued is negligible in the sense specified in the definition of pscfs . when extending the set of fractional profiles to the reals , it can also be shown that maximal lotteries are almost always unique by using an argument similar to that of .[ rem : pcons ] does not satisfy the stronger version of population - consistency in which the set inclusion is replaced with equality ( see ) .this can be seen by observing that _ every _ lottery is maximal for the union of any two electorates whose preferences are completely opposed to each other .when there are at least three alternatives , it is possible to find two such preference profiles which yield the same unique maximal lottery and strong population - consistency is violated .however , whenever is single - valued ( which is almost always the case ) , strong population - consistency is equivalent to population - consistency and therefore satisfied by .[ rem : cloning ] requiring cloning - consistency instead of composition - consistency suffices for the proof of when additionally demanding condorcet - consistency .it is therefore possible to alternatively characterize using population - consistency , cloning - consistency , and condorcet - consistency .as above , the axioms are independent from each other . , as defined in , satisfies all axioms except population - consistency .the pscf that is identical to for agendas of size 3 and otherwise identical to satisfies all axioms except cloning - consistency . satisfies all axioms except condorcet - consistency . also satisfies _ agenda - consistency _ , which requires that the set of all lotteries that are chosen from two overlapping agendas should be identical to the set of lotteries that are chosen from the union of both agendas ( and whose support is contained in both agendas ) . the inclusion from left to rightis known as sen s or _ expansion _ , whereas the inclusion from right to left is sen s or _ contraction _ .actually goes back to and , where it is called _ independence of irrelevant alternatives _ ( not to be confused with arrow s iia ) .we refer to for more details . ]numerous impossibility results , including arrow s well - known theorem , have revealed that agenda - consistency is prohibitive in non - probabilistic social choice when paired with minimal further assumptions such as non - dictatorship and pareto - optimality .that is stronger than ours . ]in contrast to , which at least requires that every voter has a unique top choice , does not require the asymmetry , completeness , or even transitivity of individual preferences ( and still satisfies population - consistency and composition - consistency in these more general domains ) .in the restricted domains of matching and house allocation , on the other hand , maximal lotteries are known as _ popular mixed matchings _ or _ popular random assignments _ .it has already been observed by that is _ ex post _ efficient , pareto - dominated alternatives always receive probability zero in all maximal lotteries . strengthened this statement by showing that even satisfies -efficiency ( also known as ordinal efficiency ) as well as the even stronger notion of -efficiency ( see * ? ? ? * ) .while also satisfies -efficiency , random serial dictatorship ( the canonical generalization of to weak preferences ) violates -efficiency . is the only _ ex post _ efficient strategyproof pscf .however , and violate _group_-strategyproofness ( in fact , there exists no _ ex post _ efficient group - strategyproof pscf ) . and satisfy the significantly weaker notion of -group - strategyproofness , which is violated by probabilistic extensions of most common voting rules .a very useful property of is that it can not be manipulated in all preference profiles that admit a strict condorcet winner ( see * ? ? ?this material is based on work supported by the deutsche forschungsgemeinschaft under grants br 2312/7 - 2 and br 2312/10 - 1 .early versions of this paper were presented at the 6th seminar on ordered structures in games and decisions in paris ( november , 2012 ) , the workshop on mathematics of electoral systems in budapest ( october , 2013 ) , the 12th meeting of the society for social choice and welfare in boston ( june , 2014 ) , the micro - theory seminar at the university of glasgow ( february , 2015 ) , and the algorithms seminar at the university of oxford ( november , 2015 ) .the authors thank bhaskar dutta , christian geist , paul harrenstein , johannes hofbauer , olivier hudry , vincent merlin , bernard monjardet , herv moulin , klaus nehring , clemens puppe , peyton young , and william s. zwicker for helpful comments .the authors are furthermore indebted to the editor and anonymous referees for their constructive feedback .h. aziz , f. brandt , and m. brill . on the tradeoff between economic efficiency and strategyproofness in randomized social choice . in _ proceedings of the 12th international conference on autonomous agents and multi - 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parameter tractability of composition - consistent tournament solutions . in _ proceedings of the 22nd international joint conference on artificial intelligence ( ijcai ) _ ,pages 8590 .aaai press , 2011 .d. e. campbell and j. s. kelly .impossibility theorems in the arrovian framework . in k.j. arrow , a. k. sen , and k. suzumura , editors , _ handbook of social choice and welfare _ , volume 1 , chapter 1 .elsevier , 2002 .g. kreweras .aggregation of preference orderings . in_ mathematics and social sciences i : proceedings of the seminars of menthon - saint - bernard , france ( 127 july 1960 ) and of gsing , austria ( 327 july 1962 ) _ , pages 7379 , 1965 .r. l. rivest and e. shen .an optimal single - winner preferential voting system based on game theory . in _ proceedings of the 3rd international workshop on computational social choice _ , pages 399410 , 2010 .m. a. satterthwaite .strategy - proofness and arrow s conditions : existence and correspondence theorems for voting procedures and social welfare functions ._ journal of economic theory _ , 100 ( 2):0 187217 , 1975 .

Abstract: two fundamental axioms in social choice theory are consistency with respect to a variable electorate and consistency with respect to components of similar alternatives . in the context of traditional non - probabilistic social choice , these axioms are incompatible with each other . we show that in the context of _ probabilistic _ social choice , these axioms uniquely characterize a function proposed by fishburn ( rev . econ . stud . , 51(4 ) , 683692 , 1984 ) . fishburn s function returns so - called _ maximal lotteries _ , lotteries that correspond to optimal mixed strategies in the symmetric zero - sum game induced by the pairwise majority margins . maximal lotteries are guaranteed to exist due to von neumann s minimax theorem , are almost always unique , and can be efficiently computed using linear programming .