Evaluation and optimization of electoral systems


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In formal terms, we can identify partitions and coverings on the set of candidates and on the set of electors. In order to carry out the computational rules that form an electoral system, the set of voters V is usually not considered altogether. Mainly for organizational purposes, but also for the philosophical notion of representation underlying the different systems, the voters are divided into a certain number k of "small" or manageable groups of citizens belonging to the same geographic area, called political districts.

This means that each political district corresponds to a subset of voters, and each voter belongs only to one political district at a time. The subset of voters corresponding to the jth district can be denoted by Vj.

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The Hierarchy of Districts Many electoral systems adopt a hierarchy of political districts. In this case, the seats are assigned on the basis of successively larger groups of votes obtained by merging districts at higher and higher levels. Partition B is finer than partition A. The finest partition ED of voters the one with the largest number of classes consists of the electoral districts. Each electoral district can be thought of as an "atom," since it cannot be further decomposed.

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Then let ER be a partition obtained by aggregating the classes of SD, and let the new larger classes be called regions. Some electoral systems adopt only one level, namely, the basic partition SD of electoral districts; others adopt more than one but usually no more than three.

A rooted tree representing the hierarchy of districts. There is a strict relation between the set of candidates C and the set of parties I. As a matter of fact, each candidate competes for a unique party, and thus one can define a mapping from C to I which associates with each candidate the party he or she runs for.

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Since each party running in the elections is represented by at least one candidate, the above mapping is surjective. This kind of relationship is sketched in Figure 2. A surjective mapping from C to I. The result of such mapping is nothing more than a partition of the set of candidates in different classes made by the candidates belonging to the same party.


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As we have already mentioned, the parties partition the set of candidates into different lineups. In fact, what really happens in an election is a bit more complicated: each party selects a possibly empty subset of its candidates to present in the single electoral districts in which it will compete. This is an additional operation on the set of candidates belonging to the same party, but it does not necessarily give rise to another partition.

In fact, a party can decide 2. The set of parties I induces a partition on the set of candidates C. The assignment of candidates to electoral districts can be formally described by a covering K — Kt, K2, Since every candidate runs at least in one district, one has However, since a candidate may run in several districts, two subsets Kj and KI are not necessarily disjoint, but they may partially overlap see Figure 2.

The three electoral districts define a covering of the sets of candidates; e. A list of candidates. Party Alliances and Cartels In several countries, and usually at the higher levels of the district tree T, parties are allowed to form alliances or coalitions for the purpose of seat assignment. In such case, seats are assigned to cartels or groups of lists involving allied parties. Subsequently, all the seats won by each cartel are redistributed among the lists within the cartel. Usually the coalitions do not differ from node to node in T, but there are remarkable exceptions: for example, in the Italian political elections of , "Forza Italia," the party of the media tycoon Silvio Berlusconi, was allied in the northern regions with the separatist Northern League and in the southern ones with the right-wing party, National Alliance.

Thus, in order to provide a flexible description of an electoral system, one may define an alliance mapping x are the cartels associated with the partition 7rq. In order to provide a uniform description of the electoral engine, we shall make the assumption that, at the bottom level district level of the tree T, every cartel consists of a single list. Coalitions and corresponding cartels. Seat Apportionment The set S of seats to be assigned as a result of the vote is another basic element of the system. In some countries, seats are assigned to parties and eventually to candidates, altogether at the national level.

More often, seats are preliminarily allotted to the individual districts, with the provision that every seat assigned to a given district must go to some candidate running for that district. In some other cases, a fraction of seats is assigned to the districts and the remaining seats are allotted to groups of districts regions or the whole nation. In order to obtain a general definition of seat apportionment, recall the notion of a hierarchy of districts and its representation as a rooted tree T N, A.

Each node of T represents a class of some partition in the hierarchy, that is, a district, a region, the nation. In general, a seat apportionment may be defined as a mapping a from a set S of seats to the node set N of T: the mapping may assign a seat either to a district, or to a region, or to the whole nation.

Of course, some nodes may get no seats. The underlying rule is that every seat attributed to a district may be won only by a candidate competing there; if a seat is allocated to a region, the winner must run in at least one of the districts of the region. The districts are single-membered if only one seat is associated with the corresponding territorial district, and they are multimembered if there are more seats associated with the same territorial district. In case of multimember districts we talk about small or big districts according to the number of seats in that district, which can vary from two and three to, say, a hundred.

It should be emphasized that, while the actual vote takes place, as a rule, only at the district level, the assignment of seats to parties and to their candidates may occur also at other levels of the tree. Moreover, in some countries, such as Germany, Austria, Denmark, Iceland, and Sweden, supplementary seats may be granted on the basis of the vote outcome. Ballot Structure During the election, each voter casts a ballot. The information contained in the voters' ballots is processed by the electoral formula including tie-breaking rules, etc.

The ballot structure is an essential component of an electoral system and it may take several alternative forms. According to Rae a ballot is categorical when "the voter gives his mandate to [one or more candidates of] a single party" and ordinal when the voter "divides his [mandate] among parties or among candidates of different parties.

Each elector is granted a "stock" of at most r preferences, which may be arbitrarily distributed among no more than r candidates in Kj. Preference ballots are further categorized with respect both to the number of preferences and to the possibility of panachage. If it is required that all the chosen candidates belong to the same party then the ballot is called homogeneous; otherwise when panachage is permitted the ballot is called heterogeneous. Classification of the possible ballot types.

Evaluation and Optimization of Electoral Systems

The set of all possible ballots in district j will be denoted by flj. Single or Multiple Ballots The next feature we want to analyze is the number of ballots. In fact, some systems in particular systems using single-membered districts expect the voters to vote more than once to select the winning candidates. The idea underlying such methods is that the final selection can be made by successive elimination of the less voted candidates.

In other words, the first ballot is used to cancel a set of candidates so that at the next ballot a smaller set is considered, just as if several subelections are performed each time, until only one candidate is left. Methods that adopt more than one successive ballot will rank the candidates into three categories on the basis of the partial result obtained each time.

In fact, after each ballot the candidates can be distinguished into elected candidates, candidates who are deferred to the next ballot, and candidates who are definitively eliminated from the election. Vote and Vote Counts In every district j, each elector expresses his or her preferences for the lists Lij and possibly for the candidates in Kj by choosing a specific ballot in the set flj of all possible ballots.

On the other hand, the principles of democracy and the reasons of anonymity require that the electors be indistinguishable. Hence, what matters in practice is not the individual vote cast by each elector, but the vote count, which summarizes the outcome of the individual votes and constitutes the input for the electoral engine, whose final output is the assignment of seats to candidates.

Seat Assignment The basic task of an electoral system is the assignment of seats to elected candidates. Notice that what matters is not only who is elected but also which seat goes to each elected candidate. As a matter of fact seats are usually apportioned to specific districts or to specific regions, and thus they have strong territorial features. On this basis, one may formally define a seat assignment as follows.

Then a seat assignment is defined as a mapping a from E to 5. This mapping specifies, for each elected candidate c, the actual seat cr c assigned to c.


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  7. Since different candidates get different seats, the mapping cr is injective see Figure 2. On the other hand, a might not be surjective, since after the current ballot there might still be some unassigned seats. Electoral Engine Once the electors have cast their votes, these are fed into a mechanism whose eventual output is the assignment of seats to candidates.

    In all electoral rounds but the final one, a decision must also be made about which candidates should be admitted to the next round. Formally, for each round there is an electoral 2. Injective mapping from the set of elected candidates to the seats.

    With reference to Figure 2. The votes are attributed either to lists or to candidates or to rankings, depending on the ballot structure, as discussed above. Then a filter selects the active votes that are actually used to assign seats at the level of hierarchy under consideration. In most cases, the active votes coincide with the total inflow uq, but in some cases, the active votes represent only a part of the total inflow we refer, in particular, to the detraction rules used in many mixed or hybrid systems as further described in Chapter 8.

    Subsequently, a converter device transforms votes attributed to lists into seats assigned to cartels this is what is usually referred to as electoral formula. The total number of assigned seats must be equal to the number Sq of seats apportioned to node q.

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    The filter also outputs an outflow vq of votes going to the next higher level, namely, into the father of node q the only node of T from which there is an arc going into q. Since the active votes might be "burnt" because of detractions, exclusion thresholds, and other special devices in the vote-to-seat conversion, one has vq Putting Everything Together: The General Electoral System After having discussed all the basic ingredients of an electoral system, we are back to the initial question: what is an electoral system?

    Electoral engine in a node of the district tree. Since for most of the examples given in the following pages the vote-to-seat transformation is independent from the specific district under consideration, for the sake of readability, we will drop the suffix j when it is superfluous. However, in case such notation may be misleading, the alphabet adopted is changed.

    Plurality Systems Among the electoral systems currently adopted in democratic countries, the majority systems are, in fact, a minority. Nevertheless, the first-past-the-post system is largely used in Anglo-Saxon countries, such as Great Britain, the United States, and New Zealand, as well as in Italy; the double ballot is adopted in France and was also used in Italy between and except for a short period ; the alternative transferable vote is adopted in Australia.

    First-Past-the-Post The philosophy of this system is "winner takes all" and nothing is left to the losers. It is the most drastic electoral system, and perhaps the most ancient, based on the principle of territorial representation.

    Evaluation and optimization of electoral systems Evaluation and optimization of electoral systems
    Evaluation and optimization of electoral systems Evaluation and optimization of electoral systems
    Evaluation and optimization of electoral systems Evaluation and optimization of electoral systems
    Evaluation and optimization of electoral systems Evaluation and optimization of electoral systems
    Evaluation and optimization of electoral systems Evaluation and optimization of electoral systems
    Evaluation and optimization of electoral systems Evaluation and optimization of electoral systems
    Evaluation and optimization of electoral systems Evaluation and optimization of electoral systems

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