Gizem ÇEVİK Betül YENİ Parmis SHAHMALEKİ Serap GÖKSU

Gizem ÇEVİK Betül YENİ Parmis SHAHMALEKİ Serap GÖKSU
IMPLICIT MULTI ATTRIBUTE EVALUATION Gizem ÇEVİK Betül YENİ Parmis SHAHMALEKİ Serap GÖKSU SPRING - FIRST PROJECT GROUP PRESENTATION DATE: MARCH 10, 2016 END 604E - GROUP DECISION MAKING UNDER MULTIPLE CRITERIA

MULTI ATTRIBUTE EVALUATION
There are different classifications of multi criteria decision making problems and methods. A major distinction between MCDM problems is based on whether the solutions are explicitly or implicitly defined. In our daily lives, we usually weigh multiple criteria implicitly and we may be comfortable with the consequences of such decisions that are made based on only intuition. On the other hand, when stakes are high, it is important to properly structure the problem and explicitly evaluate multiple criteria.

IMPLICIT VS EXPLICIT q ∈ Q → where q is the vector of k criterion functions (objective functions) and Q is the feasible set, Q ⊆ Rk. If Q is defined explicitly (by a set of alternatives), the resulting problem is called a Multiple Criteria Evaluation problem. If Q is defined implicitly (by a set of constraints), the resulting problem is called a Multiple Criteria Design problem. The quotation marks are used to indicate that the maximization of a vector is not a well-defined mathematical operation. This corresponds to the argument that we will have to find a way to resolve the trade-off between criteria (typically based on the preferences of a decision maker) when a solution that performs well in all criteria does not exist.

THEORY CHOICE & SOCIAL CHOICE
Supposing that, even if two scientists agree on the features that a good theory should have, they will not necessarily be led to make the same choices for they may weight the features differently. This makes also their theory choices different. Carnap style inductive logic aims to devise an algorithm for inductive reasoning, in which a set of hypotheses and a body of data as input should be best confirmed. However this logic seems to impossible dream. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

The Problem The problem of theory choice (TC) is formally identically to an standard social choice (SC) problem. With this idea, our goals will be: To use formal results from Arrow’s social choice theory to assess Kuhn’s ‘no algorithm for theory choice’ theory To illustrate how techniques from theoretical economics can be applied to problems in epistemology. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

Kuhn’s Thesis Thomas Kuhn famously argued that there is ‘no neutral algorithm for theory choice’ in science. According to Kuhn, a good algorithm should include these 5 important poperties / criteria, namely: Accuracy Consistency Scope Simplicity Fruitfulness Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

Kuhn’s Thesis He argues that the criteria fail to uniquely determine theory choice for two reasons: Ambiguity Weighting Kuhn’s example: Ptolemy’s theory vs. Copernicus’ theory He says that Copernicus’ theory was simpler than Ptolemy’s in that it invoked more parsimonious mathematics, but was no simpler in that the computational labour required to predict planetary positions was the same for both. If this is correct, then simplicity, in this example, needs to be sub-divided into two criteria: mathematical parsimony and computational ease, neither of which is ambiguous. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

Kuhn’s ‘No Algorithm for TC’
Ih this case, simplicity needs to be sub-divided into these criteria and so these criteria should be weighted, which is more important. Sub-dividing an ambiguous criterion exacerbates the fundamental weigting problem. ‘Value judgements’ play an inavitable role in theory choice, and thus the ideal of ‘objectivity’ is unattainable. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

The Crucial Quesion View of author: no way of constructing an algorithm based on these five criteria, which meets minimal standards of acceptability. Thus, the main question is why there is no unique algorithm for theory choice. No good way to choose? No acceptable one? Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

Arrow’s Impossibility Theorem for SC
As we already know, the social choice focuses How good they are for society as a whole Ri: weak preferences Pi: strict preferences Ii: indifference of preferences Transitive, reflective, complete bineary relation over the alternatives. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

Arrow’s Impossibility Theorem for SC
A social choice rule (social welfare function) is a kind of algorithm for making social choices, based on information about individuals’ perefenrences. Arrow’s main point: ‘what conditions should the function satisfy?’ U: Unrestricted domain P: Weak Pareto I: Independence of irrelevant alternatives N: Non-dictatorship Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

TC Cast As a SC Problem Arrow and Kuhn’s common structure is to regard each criterion of theory choşce as an individual, with their own preferences order over the alternative theories. Simplicity can be defined reasonably precisely, enough to permit pair-wise comparisons between the theories that we wish to choose between. Then, we can define a bineary relation ‘is at least as simple as’, on the set of alternative theories, which will be a weak ordering, that is, reflexive, transitive and complete. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

TC Cast As a SC Problem (cont.)
Plausibly, one might take a theory’s ‘scope’ to be its total set of logical consequences, and the relation ‘T1 has at least as much scope as T2’ to mean that T2’s consequence class is a subset of T1’s. But this relation, though reﬂexive and transitive, need not be complete, for the consequence classes of a pair of theories may be non-nested, that is, the theories may be non-comparable for scope. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

The theory choice rule is deﬁned in exactly the same way as Arrow’s social choice rule. For instance; suppose we have four alternative theories, and three criteria: simplicity, accuracy, and scope. By assumption, we know how to rank-order the theories by each criterion. We feed this information into the theory choice rule, which then outputs an ‘overall’ ranking of the theories, from best to worst. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

When it comes to four condition of Arrow, we can briefly say that U, P, N, and I are conditions on reasonable theory choice, then it is obvious that an Arrovian impossibility result applies. So long as there are at least three alternative theories, there exists no theory choice rule that satisﬁes all four conditions. This spells bad news for the possibility of making ‘rational’ theory choices. Both Kuhn’s view and the Arrow-inspired view imply, obviously, that there is no single algorithm for theory choice, over three or more alternatives, which is rationally acceptable. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

Possible Escape Routes
One is simply that many real cases of theory choice are bineary, that is, involve just two alternatives. In that terms, because of the main condition of Arrow theorem, numerous algorithms become possible that satisfy conditions U, P, I and N. (so simple and not efficient) Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

To Reject One Condition
Another possibility for reconciling Arrow with Kuhn is simply to reject one or more of Arrow’s conditions. Kuhn would argue that an acceptable algorithm for rational theory choice need not respect Arrow’s conditions. (not plausible, but one possible argument against condition N) For example; a strong empiricist might well hold that the criterion of ‘ﬁt-with-the-data’ should be a dictator. Similarly, in a discussion of Kuhn’s five criteria, McMullan argued that ‘accuracy’ held a special role, for it is an end in itself while the others are only valuable in so far as they are reliable indicators of accuracy. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

Rejecting ‘N’ (2) An ordinary (not strong) dictatorship of ‘ﬁt with the data’ could use criteria such as simplicity to break ties, that is, to settle cases where the dictator is indifferent. This is known as a ‘serial’ or ‘lexicographic’ dictatorship, and represents a more moderate form of empiricism. Accepting a serial dictatorship of ‘ﬁt with the data’ is in principle a way out of the impossibility result, since this theory choice rule does satisfy conditions I, P, and U. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

Rejecting ‘N’ (cont.) However, even if one accepts the underlying empiricist motivation, there are two problems with this solution. Two problem: To make it work a specification of the order To contain ‘noise’ in the data invariably The problem of over-fitting constitutes a strong reason not to relax condition N in the manner mooted above, even if we are empiricists. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

To Modify The Goal (3) Instead of trying to rank-order the alternatives, as in Arrow’s formulation of the problem, suppose we instead try to pick the best. More precisely, we seek a ‘choice function’ which tells us, for any subset of the alternatives, which is (or are) the best. This escape route is thus thought unpromising by most social choice theorists, and seems equally unpromising as applied to theory choice. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

Domain Restriction (4) For example; if a gain in simplicity always means a loss of accuracy. Then, certain proﬁles would be impossible, and could be legitimately excluded from the domain of the theory choice rule. However, that such trade-offs always exist does not seem very plausible; and anyway there is no guarantee that the resulting domain restriction would be of the right sort to alleviate the Arrovian impossibility. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

Sen’s ‘Informational Basis’ Approach
to escape from Arrow, using an ‘enriched informational basis’ Arrow’s meagre input with these reasons: Purely ordinal – no information about intensity of preferences Do not permit interpersonal comprasions So Sen offers that we should use utility functions, one for each individual in society, not a profile of preferences orders. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

Sen’s ‘Informational Basis’ Approach
An individual’s utility function is required to represent their preference order, in the sense that xRiy iff ui(x) ≥ ui(y), for all alternatives x and y. Note that if ui represents Ri, then any increasing transformation of ui will also represent Ri. Thus there is a many–one relation between utility functions and the preference orders that they represent. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

Sen’s ‘Informational Basis’ Approach (cont.)
The concept of a social welfare functional (SWFL) This is a function that takes as input a proﬁle of utility functions, and yields as output a social ranking of the alternatives. An SWFL is analogous to an Arrovian social choice rule, in that both yield the same output; however, the former takes a proﬁle of utility functions, rather than preference orders, as input. Potentially, this allows more information to be taken into account. Arrow’s conditions on the SWFL will be denoted U', P', I', and N'; they are motivated by arguments similar to those that motivate the Arrovian originals. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

Sen’s ‘Informational Basis’ Approach (cont.)
Arrow’s impossibility result can be derived in Sen’s framework, but it requires an additional condition, capturing the fact that Arrow uses purely ordinal, non-interpersonally comparable information. This condition is called ‘invariance with respect to ordinal, non-comparable information’ or ONC. Arrow’s theorem can now be stated in Sen’s framework: for three or more social alternatives, no SWFL can satisfy conditions ONC, U', P', I', and N‘. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

ONC If the ONC condition is imposed on the SWFL, this implies that interpersonal comparison of utility is deemed impossible (or meaningless). There are two ways it can be relaxed: drop the assumption that utility is purely ordinal permit interpersonal comparisons. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

ONC (cont.) Scale Form Condition Result Ordinal vi Interpersonel comparisons cannot be made. Cardinal vi = aui + b a > 0 Utility differences bacome meaningful. Ratio vi = aui Absolute Actual utility numbers are meaningful. If utility is non-comparable, then each individual can apply a transformation (from the permissible class) independently of others. If utility is fully comparable, then each individual must apply the same transformation. Depending on the utility scale, a form of partial comparability may also be possible. With cardinal utility, if utility is unit comparable, then individuals’ positive linear transformations must all have the same slope, but can have different intercepts. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

ONC (cont.) Numerous alternatives to Arrow’s ONC condition are now possible. Cardinal-scale utility with no comparability (CNC) Cardinal-scale utility with full comparability (CFC) Ratio-scale utility with full comparability (RFC) Ratio-scale utility with no comparability (RNC) Absolute-scale utility with full comparability (AFC) ONC is the strongest condition—for the classes of proﬁles that it treats as informationally equivalent are very large, and thus the restriction on the SWFL considerable. By contrast, AFC is the weakest condition—it places each proﬁle into a singleton class of its own, which implies no restriction on the SWFL. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

ONC (cont.) With RNC, percentage increases in utility can be meaningfully compared, that is, statements such as ‘in moving from alternative x to y, individual 1’s percentage gain is greater than individual 2’s’, are meaningful. It is easy to verify that the truth-value of this statement will be unaltered if the two individuals apply different ratio-scale transforms to their utility functions. Sen’s work demonstrates clearly that Arrow’s imposibility results is in large part a consequence of yhe impoverished information he feeds intı his social choice rule. Enriching the informational basis, while retaining Arrow’s four conditions is sufficient to avoid the impossibility. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

TC: The informational basis
Just as Sen replaced Arrow’s social choice rule with a social welfare functional, so we need to replace our theory choice rule with a ‘theory choice functional’. So instead of starting with a proﬁle of ‘preference orders’, one for each criterion of theory choice, we start with a proﬁle of ‘utility functions’, that is, real-valued representations of those orders. In principle, this allows an enrichment of the informational basis. After that, we need to consider both measurement scales and ‘inter-criterion’ comparability. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

TC: The informational basis (cont.)
Examples: Fruitfulness: It is hard to believe that differences in fruitfulness can be compared; a statement such as ‘the difference in fruitfulness between T1 and T2 exceeds the difference between T2 and T3’ hardly seems meaningful. Simplicity: The hypothesis ‘y =ax+b’ is simpler than ‘y =ax2+bx+c’ because the former contains two free parameters, the latter three. So in this case, simplicity is measured on an absolute scale—the actual numbers are meaningful, so only the identity transformation preserves information. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

Different scales may be appropriate for different criteria, and may depend on the inferential techniques that we are using. It may be that for the ‘large scale’ theory choices that Kuhn was interested in, ordinal comparisons are all that can be achieved. Finally, note that the situation for theory choice is more complicated than for social choice. In social choice, one normally assumes a single type of measurement scale for all utility functions. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

Inter-criterion comparability is needed to avoid theimpossibility result, as we know, the prospects for escaping the Arrovian predicament by enriching the informational basis of theory choice may seem dim. However; If all criteria are absolutely measurable, then interpersonal comparability follows immediately. If the ‘utility’ functions that represent the simplicity and accuracy orderings cannot be transformed without loss of information, then statements such as ‘the accuracy of T1 is less than the simplicity of T2’ automatically become meaningful. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

If the criteria of theory choice are each measured on their own ratio-scale (i.e. RNC), then this: (i) permits a limited form of inter-criterion comparability, and (ii) avoids Arrovian impossibility so long as all ‘utilities’ are non-negative. Ratio-scale measurability is fairly plausible in certain inferential contexts. As regards point (ii), the restriction to non-negative ‘utilities’ seems unproblematic; if ‘scope’ has a natural zero point, why demand that the theory choice functional be able to deal with proﬁles in which some theories (per impossible) have negative scope? So there is a potential escape route from Arrow here too. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

The general moral is that enriching the informational basis of theory choice does permit an escape from Arrow; though which enrichments are defensible must be answered on a case-by-case basis. For replacing ONC with an alternative condition (such as AFC), while retaining Arrow’s four conditions, does not narrow down the class of permissible theory choice functionals to a single one. So we escape Arrow’s predicament only to enter Kuhn’s: many acceptable algorithms, and no way to select between them. In the theory choice case, escaping Arrovian impossibility by enriching the informational basis seems to lead us straight to Kuhn’s ‘no algorithm’ thesis. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

Illustration: Bayesianism
On the Bayesian view, we use two criteria to choose between the hypotheses: prior probability P(Ti), and likelihood, P(E/Ti). Bayesians argue that the right way to do it is to multiply the prior by the likelihood, that is, to consider the quantity [P(Ti) x P(E/Ti)]. This quantity can be used to generate an overall ranking of the theories, from best to worst. P': If theory T1 has a higher prior and a higher likelihood than theory T2, that is, P(T1)>P(T2) and P(E/T1)>P(E/T2), then T1 will obviously be ranked higher than T2 by the BCF. (Satisfy) I': Whether T1 or T2 is ranked higher by the BCF is entirely determined by the priors and likelihoods of those two theories; no other information is relevant. (Satisfy) Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

Illustration: Bayesianism (cont.)
N': Neither criterion (prior or likelihood) is able to dictate over the other—it is not true that if T1 has a higher prior than T2 then it must be ranked higher, and similarly for likelihood. (Satisfy) U': can only take on values in the unit interval [0,1]; moreover, it is required that PP(Ti)1. Thus there are two restrictions on the permissible values of the functions we feed into the Bayesian theory choice functional. (Not satisfy) ONC: It is quite possible to have two profiles <P(Ti), P(E/Ti)> and <Q(Ti), Q(E/Ti)>, where the two prior functions P(Ti) and Q(Ti) rank the theories identically, and the two likelihood functions P(E/Ti) and Q(E/Ti) also rank them identically, and yet the overall rankings generated by the BCF are different in the two cases. (Not Satisfy) Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

Illustration: Bayesianism (cont.)
AFC: Given a profile <P(Ti), P(E/Ti)>, applying any transformation to it other than the identity transformation will alter its informational content—for the actual probability numbers are meaningful. The BCF is by no means the only theory choice functional that satisfies AFC and the Arrovian conditions; so there is potential for a Kuhnian ‘no unique algorithm’ argument. However, such an argument would have to counter the Bayesians’ argument for why their theory choice functional is the ‘correct’ one. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

Illustration: Statistical Model Selection
Two criteria are used: simplicity and fit-with-the-data. The two will often exhibit a trade-off: improving fit means sacrificing simplicity. Clearly, we have here the ingredients for a Kuhnian ‘no unique algorithm’ claim—there are many conceivable ways of combining fit and simplicity into a single decision rule. Families Form Condition Simplicity Best-fitting LIN y = 𝑎 +𝑏𝑥 a, b ∈ R simpler L(LIN) PAR y = 𝑎 +𝑏𝑥 + c 𝑥 2 a, b, c ∈ R L(PAR) EXP y = 𝑎 𝑥 a ∈ R L(EXP) The highest Akaike score is chosen to find the best fitting, which is defined as [log-likelihood H–k]. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

Illustration: Statistical Model Selection (cont.)
Arrow conditions P', I', and N' are satisfied by Akaike choice functional. However U' is not satisfied—for the log-likelihood function only takes negative values and the free parameter function only takes positive integer values, which implies a domain restriction. Also, ONC is not satisfied. Again, the appropriate assumption is AFC since the actual numbers assigned by the two functions are meaningful, any transformations will change their informational content. Repeatly, escaping the Arrovian predicament may land us in the Kuhnian one. However, there are also arguments for why the Akaike criterion is the uniquely ‘correct’ one. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

Theory Choice and Social Choice Kuhn vs Arrow - Conclusion
To sum up, for Kuhn’s claim is that there are many algorithms, all equally acceptable, while Arrow’s claim is that no algorithm meets minimum standards of acceptability. After identifying Kuhn’s five criteria with Arrow’s individuals, it can be easily observed that theory and social choice problems have the same structure. In the light of all, Arrowian impossibility only can be avoided by opening the door to many different algorithms. However, it is not seen possible to find a unique method. Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011)

Schulze Method The Schulze method is a voting system developed in 1997 by Markus Schulze. It selects a single winner using votes that express preferences. The method can also be used to create a sorted list of winners. It is a Condorcet Method. The Schulze method is a Condorcet method, which means the following: if there is a candidate who is preferred by a majority over every other candidate in pairwise comparisons, then this candidate will be the winner when the Schulze method is applied. A new monotonic, clone-independent, reversal symmetric, and condorcet - consistent single-winner election method - Markus Schulze (2011)

Schulze Method (cont.) The input to the Schulze method is the same as for other ranked single-winner election methods: each voter must furnish an ordered preference list on candidates where ties are allowed. The output of the Schulze method gives an ordering of candidates. Therefore, if several positions are available, the method can be used for this purpose without modification, by letting the k top-ranked candidates win the k available seats are available. A new monotonic, clone-independent, reversal symmetric, and condorcet - consistent single-winner election method - Markus Schulze (2011)

Schulze Method (cont.) The Schulze method satisfies a lot of voting criteria. Because of these reasons, already several private organizations have adopted this method. The Schulze method is used by several organizations; Wikimedia organization – Debian – Ubuntu – Gentoo – Software in the Public Interest – Free Software Foundation Europe – Pirate Party political parties.. A new monotonic, clone-independent, reversal symmetric, and condorcet - consistent single-winner election method - Markus Schulze (2011)

Schulze Method – Satisfies and Rejections
Satisfied Criteria Failed Criteria Transitivity Independence of Irrelevant Alternatives Resolvability Participation Pareto Consistency Reversal Symmetry Monotonicity Independence of Clones Smith MinMax Set Prudence A new monotonic, clone-independent, reversal symmetric, and condorcet - consistent single-winner election method - Markus Schulze (2011)

Schulze Method – Resolvability
Resolvability basically says that usually there is a unique winner. There are two different versions of the resolvability criterion. Version 1: An election method satisfies the first version of the resolvability criterion if (for every given number of alternatives) the proportion of profiles without a unique winner tends to zero as the number of voters in the profile tends to infinity. A new monotonic, clone-independent, reversal symmetric, and condorcet - consistent single-winner election method - Markus Schulze (2011)

Schulze Method – Resolvability
Version 2: The second version of the resolvability criterion says that, when there is more than one winner, then, for every alternative a ∈ S, it is sufficient to add a single ballot w so that alternative a becomes the unique winner. A new monotonic, clone-independent, reversal symmetric, and condorcet - consistent single-winner election method - Markus Schulze (2011)

Schulze Method – Pareto
The Pareto criterion says that the election method must respect unanimous opinions. There are two different versions of the Pareto criterion. Version 1: This version addresses situations when every voter strictly prefers alternative a to alternative b, then alternative a must perform better than alternative b. Version 2: The second version says that, when no voter strictly prefers alternative b to alternative a, then alternative b must not perform better than alternative a A new monotonic, clone-independent, reversal symmetric, and condorcet - consistent single-winner election method - Markus Schulze (2011)

Schulze Method – Reversal Symmetry
Reversal symmetry basically means that, when there is a vote on the best alternatives and then there is a vote on the worst alternatives and when in both cases the same alternatives are chosen, then this questions the logic of the underlying heuristic of the used election method. In other words, reversal symmetry is a voting system criterion which requires that if candidate A is the unique winner, and each voter's individual preferences are inverted, then A must not be elected. A new monotonic, clone-independent, reversal symmetric, and condorcet - consistent single-winner election method - Markus Schulze (2011)

Schulze Method – Smith The Smith criterion defined such that its satisfaction by a voting system occurs when the system always elects a candidate that is in the Smith set, which is the smallest non-empty subset of the candidates such that every candidate in the subset is majority-preferred over every candidate not in the subset. (A candidate X is said to be majority-preferred over another candidate Y if, in a one-on-one competition between X & Y, the number of voters who prefer X over Y exceeds the number of voters who prefer Y over X.) A new monotonic, clone-independent, reversal symmetric, and condorcet - consistent single-winner election method - Markus Schulze (2011)

Schulze Method – Independence of Clones
Independence of clones says that running a large number of similar alternatives, so-called clones, must not have any impact on the result of the elections. This criterion is very desirable especially for referendums because, while it might be difficult to find several candidates who are simultaneously sufficiently popular to campaign with them and sufficiently similar to misuse them for this strategy, it is usually very simple to formulate a large number of almost identical proposals. A new monotonic, clone-independent, reversal symmetric, and condorcet - consistent single-winner election method - Markus Schulze (2011)

Schulze Method – MinMax Set
Minimax selects the candidate for whom the greatest pairwise score for another candidate against him is the least such score among all candidates. Formally, let score (X,Y) denote the pairwise score for X against Y. Then the candidate, W selected by minimax (aka the winner) is given by: W = min_X ( max_Y score(Y, X)) A new monotonic, clone-independent, reversal symmetric, and condorcet - consistent single-winner election method - Markus Schulze (2011)

Schulze Method – Prudence
Prudence as a criterion for single-winner election methods has been popularized mainly by Arrow and Raynaud (1986). This criterion says that the strength λD of the strongest link ab, that is not supported by the binary relation O, should be as small as possible. So λD : = maxD { (N[a,b],N[b,a]) | ab ∉ O } should be minimized. A new monotonic, clone-independent, reversal symmetric, and condorcet - consistent single-winner election method - Markus Schulze (2011)

Schulze Method – Basic Definition
At the first step of Schulze method, the pairwise matrix of the alternatives should be created. Let we suppose, N[A,B] is the number of voters who strictly prefer alternative A to alternative B. We presume that the strenght of the link AB depends only on N[A,B] and N[B,A]. Therefore, the strength of the link AB can be denoted (N[A,B], N[B,A]). Then, the digraph which shows the graph theoretic interpretation of pairwise election should be drawn. A new monotonic, clone-independent, reversal symmetric, and condorcet - consistent single-winner election method - Markus Schulze (2011)

Schulze Method – Basic Definition (cont.)
If N[A,B] > N[B,A], then there will be a link from vertex A to vertex B of strength (N[A,B], N[B,A]). At the third step, all the possible paths between alternatives should be calculate and strongest paths should be defined. Then a table showing the lists of strongest paths should be created. After that, the critical links of the strongest paths can be found. A new monotonic, clone-independent, reversal symmetric, and condorcet - consistent single-winner election method - Markus Schulze (2011)

Schulze Method – Example
Let we say 21 voters and 4 alternatives candidate as a, b, c and d. A new monotonic, clone-independent, reversal symmetric, and condorcet - consistent single-winner election method - Markus Schulze (2011)

Schulze Method – Example (cont.)
Our pairwise matrix looks as follows; If N[i, j] > N[j,i], then there is a link from vertex i to vertex j of strength (N[i, j], N[j,i]) A new monotonic, clone-independent, reversal symmetric, and condorcet - consistent single-winner election method - Markus Schulze (2011)

The digraph will be used to determine the strengths of the strongest paths. A new monotonic, clone-independent, reversal symmetric, and condorcet - consistent single-winner election method - Markus Schulze (2011)

A new monotonic, clone-independent, reversal symmetric, and condorcet - consistent single-winner election method - Markus Schulze (2011)

ij ∈ O if and only if PD[i,j] >D PD[j,i]. So in this example we get O = {ab, ac, cb, da, db, dc}. i ∈ S if and only if ji ∈/ O for all j ∈ A\{i}. So in example 1, we get S = {d}. A new monotonic, clone-independent, reversal symmetric, and condorcet - consistent single-winner election method - Markus Schulze

Schulze Method – Conclusion
Schulze method satisfies the most common voting method criteria. Most of the election methods only generate a set of winners and don’t generate a binary relation, Schulze method generates both. Schulze Method – Conclusion A new monotonic, clone-independent, reversal symmetric, and condorcet - consistent single-winner election method - Markus Schulze (2011)

Schulze Method – Conclusion
Suppose the MinMax score of a set X of candidates is the strength of the strongest pairwise win of a candidate A ∉ X against a candidate B ∈ X. Then the Schulze method guarantees that the winner is always a candidate of the set with minimum MinMax score. This property is the most characteristic property of the Schulze method, since this is the first time that an election method with this property is proposed. Schulze Method – Conclusion A new monotonic, clone-independent, reversal symmetric, and condorcet - consistent single-winner election method - Markus Schulze (2011)

Voting Paradoxes Monotonicity:
“if 𝑥 is the unique collectively best alternative for a given profile π of individual preference rankings, and if profile 𝜋 ′ is obtained from π by moving x up in some of the rankings, leaving all else unchanged, then 𝑥 should be one of the collectively best alternatives for profile 𝜋 ′ ”(Fishburn1982, p. 119) if a voting method violates the monotonicity axiom therefore regarded as susceptible to a paradox called the More-is-Less Paradox (Fishburnand Brams1983,p.208). Varieties of failure of monotonicity and participation under five voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

Voting Paradoxes Participation:
“a voter never loses by joining the electorate and reporting (sincerely) his preferences” A voting rule that fails to satisfy the participation axiom is said to be susceptible to the No-Show paradox (Fishburn and Brams 1983, p. 207) Varieties of failure of monotonicity and participation under ﬁve voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

Voting Methods we concentrate on the single-winner voting methods that are susceptible to both paradoxes. The voting methods are: - Plurality with Runoff (P-R) - Alternative Vote (AV; aka Instant Runoff Voting) - The Coombs method - The Dodgson method - The Nanson method Varieties of failure of monotonicity and participation under ﬁve voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

More-is-Less paradox (non-monotonicity)
The examples of the More-is-Less paradox are distinguished from examples of the No-Show paradox by the fact that the examples of the More-is-Less paradox involve a ﬁxed number of voters. In such examples, there are always some voters whose reported rankings change as the example unfolds. We call these voters dynamic voters. It is assumed that all dynamic voters have the same initial rankings of the candidates and all change their rankings in the same way. Varieties of failure of monotonicity and participation under ﬁve voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

Sub-types of non-monotonicity
The four sub-types of non-monotonicity discussed in this section are: [M+COND+B]: The example is a failure of monotonicity [M], a Condorcet winner [COND] exists, and the dynamic voters are better off [B] from changing their reported rankings, treating their original rankings as their true rankings. [M+COND+W]: As in sub-type (1) but the dynamic voters are worse off (W) from changing their rankings, treating their original rankings as their true rankings. [M+CYC+B]:As in sub-type(1)but the initial majority relation method contains a top cycle (CYC). [M+CYC+W]:As in sub type (2) but the initial majority relation method contains a top cycle (CYC). Varieties of failure of monotonicity and participation under ﬁve voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

More-is-less Paradox Under P-R&AV 1. An example of sub-type [M+COND+B]
Suppose there are 43 voters whose rankings of three candidates, a, b, and c, are as follows: Condorcet winner is c but a is a ultimate winner. 5 out of the 14 voters whose ranking is b > c > a decide to change their ranking (strategically) to a > b > c So b eliminated in first round and c will beat a in second round. C is winner. More-is-less Paradox Under P-R&AV 1. An example of sub-type [M+COND+B] Varieties of failure of monotonicity and participation under ﬁve voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

2. An example of sub-type [M+COND+W]
Suppose there are 17 voters whose rankings of three candidates, a,b, and c, are as follows: a is the Condorcet winner. Under P-R (as well as under AV) a will be eliminated after the first round and b will beat c in the second round and hence b becomes the ultimate winner. The two voters whose ranking is c > b> a change it to b > c > a, thereby increasing b’s support. As a result of this change c is eliminated in the first round, and a beats b in the second round and becomes the ultimate winner. Varieties of failure of monotonicity and participation under five voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

3. An example of sub-type [M+CYC+B]
Suppose there are 43 voters whose Rankings of three candidates, a, b, and c, are as follows: Here the majority relation method is cyclical a > b > c > a. Under P-R &AV c will be eliminated after the first round and a will beat b in the second round and hence a will become the ultimate winner. Two of the voters whose ranking is b > c > a change it to a > b > c thereby increasing a’s support. As a result of this change b is eliminated in the first round, and c beats a in the second round and becomes the ultimate winner. Varieties of failure of monotonicity and participation under five voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

4. An example of sub-type [M+CYC+W]
Suppose there are 17 voters whose rankings of three candidates, a, b, and c, are as follows: The majority relation method contains a top cycle a > b > c > a. In the first round c is eliminated, and a beats b in the second round and becomes the ultimate winner. Now suppose that the two voters with ranking b > a > c change their ranking to a > b > c. As a result of this change candidate b will be eliminated in the first round and c will beat a in the second round and become the ultimate winner Varieties of failure of monotonicity and participation under five voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

More-is-less Paradox Under Coombs Method
There are no [W] examples under the Coombs method Varieties of failure of monotonicity and participation under ﬁve voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

1. An example of sub-type [M+COND+B]
45 voters have to elect under the Coombs method one out of three candidates, a, b, or c, and that their rankings of these 3 candidates are as follows: b is the Condorcet winner. After the deletion of b, candidate c is ranked first by an absolute majority of the voters and is elected. Now suppose that 11 voters whose ranking is b> a > c are motivated to increase c’s support by changing their ranking to b > c > a. Candidate b is still the Condorcet winner but as a result of this change, a will be eliminated first under the Coombs method, and there after b will be elected. Varieties of failure of monotonicity and participation under five voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

Suppose that 100 voters have to elect under the Coombs method one out of three candidates, a, b,or c and that their rankings of these three candidates are as follows: The majority relation method is cyclical a > b > c > a. After c is deleted a beats b and thereby becomes the ultimate winner. Now suppose that the 15 voters whose ranking is c > b > a change it to c > a > b. As a result b will be eliminated under the Coombs method and thereafter c will be elected thus making the dynamic voters better off. Varieties of failure of monotonicity and participation under ﬁve voting methods- Dan S. Felsenthal · Nicolaus Tideman

Non-monotonicity Under The Nanson Method
As the Nanson method is Condorcet-efﬁcient there are no [COND] examples under this method. Varieties of failure of monotonicity and participation under ﬁve voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

Suppose there are 36 voters who must elect one out of four candidates, a, b ,c or d under the Nanson method and whose rankings of these candidates, as well as the resultant Borda scores of the four candidates, are as follows: Varieties of failure of monotonicity and participation under ﬁve voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

The Borda scores of the candidates can be computed as the sums of the rows of the matrix:
The sum of Borda scores of all four candidates is 216 hence the average Borda score is 54, that is, 216/4. Candidate d is eliminated after the ﬁrst round. So in the second counting round, we have: Varieties of failure of monotonicity and participation under ﬁve voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

Here the sum of Borda scores of all three candidates is 108, hence their average Borda score is 36, that is, 108/3. So according to the Nanson method one eliminates at the end of the second counting round both candidates a and b thus candidate c becomes the ultimate winner The voter whose ranking is a > b > c > d changing his ranking to a > c > b > d. So now both candidate b and candidate d are eliminated after the ﬁrst counting round. In the second counting round the (revised) Borda scores of candidates a and c are 19 and 17, respectively, so candidate a becomes the ultimate winner thus the voter who changed his ranking is better off. Varieties of failure of monotonicity and participation under ﬁve voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

Suppose there are 100 voters whose rankings of four candidates, a,b,c, and d, are as follows: The majority relation method is cyclical a > b > c > a > d. The sum of Borda scores of all four candidates is 600, and the average score is 600/4, or 150. So according to the Nanson method, one eliminates at the end of the ﬁrst counting round candidate d, and the recomputed Borda scores for candidates a,b, and c, are 116, 86, and 98, respectively. Since the average of these scores is 100, a wins. Varieties of failure of monotonicity and participation under ﬁve voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

Now suppose that the 12 voters with ranking b > a > c > d change it to a > b > c > d thereby increasing a’s support. candidates b and d are eliminated in the ﬁrst round and thereafter c wins hence the dynamic voters are worse off. Varieties of failure of monotonicity and participation under ﬁve voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

Under the Dodgson method
Non-monotonicity Under the Dodgson method As the Dodgson method is Condorcet-efﬁcient, there are no [COND] examples under this method. It is not possible to cause a Condorcet winner to lose by moving that candidate up in the voters’ rankings. There are also no [B] examples. Varieties of failure of monotonicity and participation under ﬁve voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

Suppose there are 100 voters who are divided into four groups, who must elect one out of ﬁve candidates a,b,c,d,e, under the Dodgson method, and whose rankings of the candidates are as follows: The majority relation method has a top cycle:[b > a > e > b] > c > d. It can be depicted in the following matrix of paired comparisons: Varieties of failure of monotonicity and participation under ﬁve voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

As the number of inversions needed to make a the Condorcet winner is smallest, a is elected under the Dodgson method. Now suppose that,the 11 voters in group G4 increase their support of candidate a by changing their rankings from e > a > b > d > c to a > e > b > d > c. The results of this change are depicted in the following paired comparisons matrix: So as the number of changes needed for b to become the Condorcet winner is smallest, b would be elected under the Dodgson method thereby making the members of group G4 worse off. Varieties of failure of monotonicity and participation under ﬁve voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

Type [P-BOT] Paradoxes
The paradox of this section arises if one of the candidates, say candidate c, who has not been elected originally, may be elected if the electorate is increased as a result of additional voters whose bottom-ranked candidate is c join the electorate, and consequently these additional voters are worse off. Varieties of failure of monotonicity and participation under ﬁve voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

Suppose there are 19 voters whose rankings of three candidates a, b, and c, are as follows:
Type [P-BOT] Paradox Under P-R & AV 1. An example of sub-type [P-BOT+COND+W] Varieties of failure of monotonicity and participation under ﬁve voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

The ranking by majority relation method is b > c > a ,i. e
The ranking by majority relation method is b > c > a ,i.e.,b is a Condorcet winner. Under the P-R and AV methods, candidate a is eliminated after the first round and b is elected in the second round. As candidate c has not been elected, suppose now that, two additional voters whose ranking is a > b > c join the electorate (thereby further downgrading c). As a result b is eliminated in the first round, and c is elected in the second round in violation of the participation axiom. Varieties of failure of monotonicity and participation under five voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

2. An example of sub-type [P-BOT+CYC+W]
Suppose there are 102 voters whose rankings of three candidates, a, b, and c, are as follows The majority relation method is cyclical (a > b > c > a). Under the P-R & AV methods candidate a is eliminated after the first round and b is elected in the second round. As candidate c has not been elected, suppose now that, two additional voters whose ranking is a > b > c join the electorate .As a result b is eliminated in the first round and c is elected in the second round in violation of the participation axiom. Varieties of failure of monotonicity and participation under five voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

Type [P-TOP] Paradoxes
If a candidate, say candidate x, has been elected initially, then it is possible that another candidate, y, will be elected if, additional voters whose top-ranked candidate is x join the electorate. Varieties of failure of monotonicity and participation under ﬁve voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

The [P-TOP] Paradox Under The Coombs Method 1
The [P-TOP] Paradox Under The Coombs Method 1. An example of sub-type [P-TOP+COND+W] Suppose there are 42 voters who must elect one out of four candidates, a,b,c, or d, under the Coombs method, and that their rankings of the candidates are as follows: Varieties of failure of monotonicity and participation under ﬁve voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

Here a is the Condorcet winner
Here a is the Condorcet winner. Since none of the candidates is ranked first by an absolute majority of the voters, candidate c is eliminated in the first round under the Coombs method, candidate b is eliminated in the second round, and thereafter candidate a is elected. Now suppose that, three additional voters join the electorate whose ranking is a > c > b > d. still none of the candidates is ranked first by an absolute majority of the voters. So according to the Coombs method candidate d is eliminated in the first counting round, candidate a is eliminated in the second counting round, whereupon candidate b is elected. Varieties of failure of monotonicity and participation under five voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

2. An example of sub-type [P-TOP+CYC+W]
In the first part of Example (p.76), candidate a was elected under the Coombs method. Now suppose that, a group of 11 voters with ranking a > c > b joins the electorate. Although the number of voters who rank a first has now increased, still none of the candidates is ranked first by an absolute majority of the voters. So according to the Coombs method candidate b is eliminated, where upon candidate c is elected. Thus not only is candidate a harmed by receiving additional top-rank support, but also the voters who provided this additional support are harmed. Varieties of failure of monotonicity and participation under five voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

Suppose there are 18 voters who must elect one out off our candidates,a,b,c,and d,under the Nanson method, and that their rankings of the candidates are as follows: The majority relation method contains a top cycle [a > b > c > a] > d. The [P-TOP] Paradox Under The Nanson Method 1. An example of sub-type [P-TOP+CYC+W] Varieties of failure of monotonicity and participation under ﬁve voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

The Borda scores of candidates a,b,c, and d, are 28, 29, 34, and 17, respectively, and the average Borda score is 27. Hence according to the Nanson method, candidate d is eliminated in the first count. The Borda scores of candidates a,b, and c, in the second counting round are 16, 17, and 21, respectively, and the average Borda score is 18. Consequently both candidates a and b are eliminated and therefore candidate c wins. Now suppose that, ceteris paribus, one additional voter whose ranking is c > b > d > a joins the electorate.The Borda scores of candidates a,b,c, and d, are now 28, 31, 37, and 18, respectively, and the average Borda score is Hence according to the Nanson method, candidates a and d are eliminated in the first count, and candidate b beats c in the second counting round and is elected. Varieties of failure of monotonicity and participation under five voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

The [P-TOP] Paradox Under The Dodgson Method 1
The [P-TOP] Paradox Under The Dodgson Method 1. An example of sub-type [P-TOP+CYC+W] In the ﬁrst part of Example (p.84) candidate a was elected under the Dodgson method. Now suppose that, a group (G5) of 10 voters with ranking a > b > e > c > d joins the electorate. As a result, we obtain the following paired comparison matrix: Varieties of failure of monotonicity and participation under ﬁve voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

1. An example of sub-type [P-TOP+CYC+W] (cont.)
From this matrix, it is possible to see that despite the increase in a’s support it would still take at least 9 voters from group G1 to invert in their rankings b > a to a > b in order for a to become the Condorcet winner, whereas now for b to become the Condorcet winner only 4 voters in group G4 would have to invert e > a to a > e in their rankings, and thereafter to invert e > b to b > e i.e., a total of 8 inversions. So as the number of inversions needed for b to become the Condorcet winner is smallest, b would be elected under the Dodgson method thereby making the (additional) members of group G5 worse off. Varieties of failure of monotonicity and participation under ﬁve voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

Voting paradoxes-Summary
Table1summarizes the types of examples provided under each of the investigated ﬁve voting methods. ‘Yes’ means we provided an example, and ‘No’ means we explained why no such example exists. Varieties of failure of monotonicity and participation under ﬁve voting methods- Dan S. Felsenthal · Nicolaus Tideman (2012)

ELECTION INVERSIONS Since the 1950s mathematicians and social scientists have been researching the possibilities of counterintuitive discrepancies between the preferences of individual decision makers and the collective choices they make. Most famous is Condorcet’s Paradox. Condorcet’s Paradox: the phenomenon where preferences may be ‘cyclical’ so that there may not be any alternative that cannot be beaten by at least one other alternative when compared in pair-wise contests. Election Inversions, Coalitions and Proportional Representation: Examples of Voting Paradoxes in Danish Government Formations - Peter Kurrild-Klitgaard (2013)

ELECTION INVERSIONS (cont.)
More recent work has also turned attention to the types of voting paradoxes that may occur in systems of proportional representation. Two versions of a particular kind of voting paradox. The ‘compound majority paradox’ that may occur in both first-past-the-post electoral systems (FPTP) and Proportional representation systems (PR) that are perhaps mostly overlooked in the latter. Election Inversions, Coalitions and Proportional Representation: Examples of Voting Paradoxes in Danish Government Formations - Peter Kurrild-Klitgaard (2013)

VOTING PARADOXES AND ELECTION INVERSIONS
One broad type of voting paradox is what has been called ‘compound majority paradoxes’. Various types of compound majority paradoxes may occur when decisions take place in two (↑) rounds, and where the preferences of the relevant decision makers in the one round do not correspond to the preferences of the relevant decision makers in the other rounds. Such questions of how to aggregate ‘parts’ into ‘wholes’, and how the properties of the whole may be different from a summation of the parts. Election Inversions, Coalitions and Proportional Representation: Examples of Voting Paradoxes in Danish Government Formations - Peter Kurrild-Klitgaard (2013)

Some Compound Majority Paradoxes
Ostrogorski’s Paradox The most well-known compound majority paradox. This shows that for a given set of voter preferences it is possible for one party to be the majority winner in a two-party contest while the other party is preferred by a majority of the voters on a majority of the issues. Election Inversions, Coalitions and Proportional Representation: Examples of Voting Paradoxes in Danish Government Formations - Peter Kurrild-Klitgaard (2013)

Some Compound Majority Paradoxes (cont.)
Anscombe’s Paradox Majority of the voters may be frustrated on a majority of issues, and where a majority of the group may lose on a majority of the issues considered. (e.g., a group of voters faced with choice of voting ‘yes’ or ‘no’ to a number of issues) Election Inversions, Coalitions and Proportional Representation: Examples of Voting Paradoxes in Danish Government Formations - Peter Kurrild-Klitgaard (2013)

‘Paradox of Multiple Elections’ An outcome consisting of majority winners on several issues may not coincide with any voter’s opinion. Election Inversions, Coalitions and Proportional Representation: Examples of Voting Paradoxes in Danish Government Formations - Peter Kurrild-Klitgaard (2013)

‘Referendum Paradox’ Deals with the fact that in many democratic systems direct and indirect forms of voting are intertwined. – for example, where voters (first) elect one group of representatives who then proceed to make policy choices so that a discrepancy emerges between the latter and the majority view of the voters. Election Inversions, Coalitions and Proportional Representation: Examples of Voting Paradoxes in Danish Government Formations - Peter Kurrild-Klitgaard (2013)

Table 1. Referendum Paradox: Hypothetical Example Nurmi üç milyon oy verenin 200 MP’yi seçtiği ve basit bir şekilde her bir MP’nin tam olarak oy vereni temsil ettiği, ve ‘evet’ ve ‘hayır’ arasında basit bir ikili seçimin olduğu Referandum Paradoksunun (Tablo 1) örneklemesini kullanmıştır. Eğer tercihler ve oylar örnekte olduğu gibi ise, öneri yüzde 45 karşı oya, üç milyon oyun yüzde 55’i ile bir referandumdan geçecek . Bununla birlikte eğer MPler kendi seçmenlerinin görüşlerini izlerse 5/6sından (200’den 167’si) azı olmamak üzere ‘hayır’ oyu kullanacaktır. Bu paradoksal sonuç gerçekte iyi bilinen doğru olmayan sonuçların makro-özelliklerle (bütünler) ilgili gözlemlere dayanan mikro-özelliklere (parçalar) ulaşmak üzere olması vasıtasıyla ‘ekolojik yanılgı’ ile ilişkilidir. Election Inversions, Coalitions and Proportional Representation: Examples of Voting Paradoxes in Danish Government Formations - Peter Kurrild-Klitgaard (2013)

Election Inversions and Their Origins
In practice, there are several possible sources for the occurrence of election inversions. There are faithless decision-makers. Most obviously, there is the risk that electors/parliamentarians may behave differently at a later round than they had promised at the earlier round, or that the promises they made were not feasible. There are wasted votes. In PR systems in the case of thresholds – that is, where either formal or ‘effective’ thresholds require that a certain share of the total number of votes is obtained in order to achieve representation and where one or more parties running in the election fail to pass that threshold. A large number of PR systems have formal thresholds – Denmark (%2), Ukraine (%3). It may be due to a failure to redraw electoral district borders and reallocate seats proportionally to fit the demographic fundamentals. Election Inversions, Coalitions and Proportional Representation: Examples of Voting Paradoxes in Danish Government Formations - Peter Kurrild-Klitgaard (2013)

Coalitions and Compound Majorities
Most studies of election inversions and compound majority paradoxes have focused on the potential discrepancy between the preferences of the voters in a majority of electoral districts and the preferences of a majority of the voters at a national level. It is, as indicated, easy to identify such examples in FPTP systems, but less so in PR systems Election Inversions, Coalitions and Proportional Representation: Examples of Voting Paradoxes in Danish Government Formations - Peter Kurrild-Klitgaard (2013)

Coalitions and Compound Majorities (cont.)
Some types of election inversions may occur under PR systems too, specifically if elections become a choice between parties forming governing coalitions but where the choice confronting the voters on the ballot is not between competing coalitions but competing parties, and where every vote cast may not weigh equally heavy in the collective decision. In PR systems it is rare that a single party obtains enough votes to form a majority government alone, and accordingly governments tend to be coalition governments based on the seats allocated to two or more parties. Election Inversions, Coalitions and Proportional Representation: Examples of Voting Paradoxes in Danish Government Formations - Peter Kurrild-Klitgaard (2013)

Table 2. Threshold Paradox: Hypothetical Example Öncelikle seçim barajları sebebiyle oylar heba edildiğinde ne olabileceğine göz atalım, orantılı olarak dağıtılan 101 koltuk için yarışan ve yüzde 5 barajın olduğu ve A ve B’nin X bir koalisyonda işbirliği yaptığı, C partisinin başka bir alternatif Y hükümeti temsil ettiği, A, B ve C partili bir parlemento için bir seçimin varsayıldığı Tablo 2’yi dikkate alın. Bu kuramsal senaryoda, X koalisyonundaki iki parti (A ve B), birlikte yüzde 51’lik oy elde ediyor, bu sırada rakipleri C yüzde 49 alıyor. Bununla birlikte, parti B’nin sadece yüzde 4 alması dikkate alındığında ve böylece barajı geçemediğinde, parti C oyunların çoğunluğuna dayanarak koltukların kesin çoğunluğunu elde eder. Dolayısıyla, C büyük çoğunluk başka bir alternatif koalisyon için oy vermiş olsa bile koltukların çoğunluğu ile bir hükümet oluşturabilir. Ortada olan fakat mantıksız paradoks oyların çoğu bir alternatif için olurken karşı tarafın tam tersini elde etmesidir. Baraj Paradoksu temelde oyların çoğunluğunu alan bir partinin koltukların çoğunluğunu alamadığı bütün durumlarda görünür. Bu mantıklı olarak tüm seçim sistemlerinde bir olasılıktır ve muhtemelen FPTP seçim sistemlerinde değil ama nispeten yaygın bir olgudur. Election Inversions, Coalitions and Proportional Representation: Examples of Voting Paradoxes in Danish Government Formations - Peter Kurrild-Klitgaard (2013)

Both voting paradoxes identified here may be seen as variations of the Referendum Paradox. There is a discrepancy between two different ‘tiers’ one where the voters’ intentions are imputed into the process and one where the MPs form a governing majority coalition. The difference vis-à-vis the Referendum Paradox is that the cause of the potentially paradoxical outcomes are the institutional arrangements (threshold leading to wasted votes; uneven representation of geographical areas). Election Inversions, Coalitions and Proportional Representation: Examples of Voting Paradoxes in Danish Government Formations - Peter Kurrild-Klitgaard (2013)

Table 3. Federal Paradox: Hypothetical Example Asimetrik temsil konusu ile ilgili alternatif bir senaryoyu dikkate alalım. Özellikle, seçim bölgelerinin iki farklı tür seçimsel birleşimi olan politika sistemi olan bir ülke varsayalım: koltuk/oy veren oranı bakımından aşırı temsil edilen bir bölge, diğeri yetersiz temsil edilen. Tablo 3 ‘Federal Paradoks’ denilebilen daha iyi bir tanımı yoksun olan kuramsal bir örnek verir. –‘federal’dir çünkü bölgeler üstü yüksek bir seviyede farklı birimlerin temsillerini içerir. Örnekte, iki parti, A ve B, parlementodaki 101 koltuk için yarışır fakat 96 koltuk ülkenin bir parçasında orantılı olarak dağıtılırken, beş koltuk diğer bir bölgede orantılı olarak dağıtılır. Parti A genel toplamın çoğunluğu olan 1,951,000 (% 50.02) oy alır fakat iki bölge arasındaki koltuk/oy veren oranındaki asimetri sebebiyle, A 101 koltuktan sadece 49’unu kazanır. Parti B, diğer bir deyişle, oy verenlerin azınlığı tarafından desteklenmesine rağmen (%49.97) koltukların çoğunluğunu kontrol eder. Paradoks yine bir koalisyonun daha çok oy aldığı fakat diğer koalisyonun daha çok koltuk kazandığı mantıksız bir sonuç verir. Election Inversions, Coalitions and Proportional Representation: Examples of Voting Paradoxes in Danish Government Formations - Peter Kurrild-Klitgaard (2013)

EMPIRICAL ILLUSTRATIONS FROM DANISH ELECTIONS
The two paradoxes identified here may occur in a number of political systems, and in the case of the Danish political system they certainly both occur, even simultaneously. Election Inversions, Coalitions and Proportional Representation: Examples of Voting Paradoxes in Danish Government Formations - Peter Kurrild-Klitgaard (2013)

EMPIRICAL ILLUSTRATIONS FROM DANISH ELECTIONS (cont.)
The Danish Folketing is composed of a total of 179 seats, where 175 seats are allocated proportionally in a two-tier system and with a %2 formal threshold being the primary hurdle that needs to be passed in order to obtain representation. These 175 seats are allocated (using Hare’s Quota – i.e., the Largest Remainder) with: 135 seats at the one tier in multimember constituencies; and 40 seats at the other (national) tier as compensatory seats. All seats are reallocated between the electoral districts on the basis of population every five years. Election Inversions, Coalitions and Proportional Representation: Examples of Voting Paradoxes in Danish Government Formations - Peter Kurrild-Klitgaard (2013)

The latter are allocated at the national level ‘on top’ of the seats of the multi-member districts so as to achieve a proportional allocation, using Sainte-Lagüe’s formula, for all parties that either pass %2 of the votes on the national level, or wins a constituency seat or wins at least as many votes in two of three regions as the average number of valid ballots cast divided by the number of constituency seats. These features produce an electoral systems with the smallest amount of disproportionality between votes and seats. The remaining 4 seats are, independently of the underlying population size, allocated to the two autonomous regions within the Kingdom of Denmark, Greenland and the Faroe Islands, with two each (allocated using d’Hondt’s Formula). Election Inversions, Coalitions and Proportional Representation: Examples of Voting Paradoxes in Danish Government Formations - Peter Kurrild-Klitgaard (2013)

It is fairly easy to identify the presence or absence of the Threshold Paradox. All that is needed is information about what parties supported what coalitions at the time of government formation, their vote share and their number of seats. The former is the most difficult, given that coalitions to some extent are endogenous to the process of government formation. This has occasionally meant lengthy coalition negotiations with the small social-liberal, centrist party, the Radicals, as the frequent (pivotal) median party, trying to broker governments other than to a simple left- or right-leaning government. Election Inversions, Coalitions and Proportional Representation: Examples of Voting Paradoxes in Danish Government Formations - Peter Kurrild-Klitgaard (2013)

The Federal Paradox is harder to detect in the real world given that the political parties running in Greenland and on the Faroe Islands are not identical to one another, and neither are they formally identical to the parties standing for elections in ‘continental’ Denmark. For that reason, judgments need to be made as to what parliamentary coalitions the four North Atlantic seats would likely caucus or collaborate with. Election Inversions, Coalitions and Proportional Representation: Examples of Voting Paradoxes in Danish Government Formations - Peter Kurrild-Klitgaard (2013)

A Case of the Federal Paradox
Table 4. Danish General Election Result, 21 November (Votes Excluding the Faroe Islands and Greenland) Bu ön bilgiler doğrultusunda, ele alınan paradokslardan birinin veya her ikisinin oluştuğu bazı genel seçimlerini ele alabiliriz. İlk olarak, “kıtasal” Danimarka’da 2,882,981 seçmenin oy kullandığı 1971 genel seçimlerini ele alalım (Tablo 4). Bunlardan dar bir çoğunluk (1,453,162 [ yüzde 50.4]) Hilmar Baunsgaard (Radikal) önderliğinde görevdeki merkez-sağ hükümeti destekleyen (veya temsil edilmeleri halinde desteklemeleri öngörülen) partilere oy vermişken 1,423,076 seçmen ( yüzde 49.4) önceki Başbakan Jens Otto Krag (Sosyal Demokrat görüşlü) önderliğinde bir hükümeti destekleyen sosyalist partilere oy verdi. Sonuç merkez-sağ için 88 koltuk, sol için 87 koltuk şeklinde oluştu, ve böylece Baraj Paradoks örneği oluşmadı. Ancak, “kıtasal” Danimarkalıların oylarını kullanma şekline bağlı olarak seçim tersine döndü: Mr Krag her iki Grönland parlamenterinin (içlerinden biri kabinede bakan olacak şekilde) ve Faroe Adaları’ndan iki parlamenterden birinin destediğini aldığı için sonuçta hükümeti kurabildi. Bu, Krag’a 179 parlamenterden 79’unun kendisini desteklemesiyle bir hükümet verdi. Başka deyişle “kıtasal” Danimarkalı seçmenlerin çoğunluğunun bir merkez-sağ hükümetine oy vermesine rağmen bir Sosyal Demokrat hükümet kuruldu. Bu, Kuzey Atlantik oylarının nasıl eklendiği sorusunu doğurdu. Faroe Adaları’nda 5,897 oy Danimarkalı merkez-sağ ile uyumlu partilere giderken 4,170 oy solu destekleyen partilere gitti ve 3,058 oy temsil edilmeyen veya favori hükümetleri belirsiz gözüken adaylara gitti. Grönland’dan gelen oylara aynı hesaplama yapılamaz çünkü 1977 öncesinde parlamenterler siyasi partilerden değil, bağımsız olarak seçiliyordu. Faroe Adaları oyları “kıtasal” sonuçlara eklense, bir Baunsgaard hükümeti toplamda 1,459,059 seçmeni toplayan partiler tarafından destekleniyordu (Danimarka ve Faroe Adaları oylarının toplamının yüzde 50.4’ü) ancak bir Krag hükümeti 1,427,246 oy (yüzde 49.3) alan partiler tarafından destekleniyordu. Danimarka ve Faroe Adaları’ndan toplam 9,801 seçmen (yüzde 0.3) bir veya diğer koalisyon için sayılamadı. 1970’lerde Grönland’daki seçmen dağılımının hiçbir zaman 20,000 oydan fazla olmadığı, çoğunlukla 10,000’ yakın oy alabildiği düşünülürse Krag’ı destekleyen adayların hiçbir şekilde bir Baunsgaard hükümetini geçecek derecede oy oranına sahip olması mümkün değil. Başka bir deyişle, 1971 seçimi bir Federal Paradoks örneğidir. Election Inversions, Coalitions and Proportional Representation: Examples of Voting Paradoxes in Danish Government Formations - Peter Kurrild-Klitgaard (2013)

A Case of the Federal Paradox (cont.)
Table 5. Danish General Election Result, 12 December (Votes Excluding the Faroe Islands and Greenland) 1990 Danimarka seçimleri baraj paradoksunun oluşumuna açık bir örnektir (partiler için oy ve koltuk dağılımı Tablo 5’te verilmiştir). Seçimde Sosyal Demokratların sağlam kazançlarının görülmesine ve oyların çoğunluğunun sol kanada (1,630,219 oy) gitmesine rağmen çok fazla sol parti yüzde 2 barajını aşmayı başaramadı; buna bağlı olarak tam iki parti için toplam “heba oylar” – halkçı Common Course ve (komünist) Unity List – kabaca altı koltuğa denk gelen oyların yüzde 3.5’una ulaştı. Görevdeki Başbakan Poul Schlüter (Muhafazakar) hükümetini destekleyen merkez-sağ partiler toplamda yalnızca 1,609,219 (yüzde 49.7) oy topladılar fakat 175 sandalyenin çoğunluğunu (91) kazandılar. Dört Kuzey Atlantik sandalyesi sol ve sağ arasında eşit dağıldı ve ne koltuk ne de oy olarak bir etkisi oldu. Başka bir deyişle, Federal Paradoks değil fakat Baraj Paradoksu kendisini gösterdi. Election Inversions, Coalitions and Proportional Representation: Examples of Voting Paradoxes in Danish Government Formations - Peter Kurrild-Klitgaard (2013)

Political systems based on PR are usually seen as being more ‘fair’ in the sense that they are more likely to produce an outcome preferred by a majority of the voters. Two types of compound majority voting paradoxes that may occur not only in FPTP systems, but also in PR systems: the Threshold Paradox and the Federal Paradox. Both identify situations where a majority of the voters cast their ballots for parties supporting one government alternative but where a majority of the seats are allocated to parties supporting another. Conclusion Election Inversions, Coalitions and Proportional Representation: Examples of Voting Paradoxes in Danish Government Formations - Peter Kurrild-Klitgaard (2013)

Both instances of the phenomenon are familiar flaws of many electoral systems, but viewed as cases of voting paradoxes they emphasize that political systems with PR and parliamentarism, such as those found in much of the West, are not immune to the types of logical problems found more generally by social choice theory and more often associated with, say, FPTP electoral systems. Election Inversions, Coalitions and Proportional Representation: Examples of Voting Paradoxes in Danish Government Formations - Peter Kurrild-Klitgaard (2013)

The data presented demonstrates that the two paradoxes have manifested themselves in at least two Danish elections formations over the last 40 years (in 1971 and 1990) in approximately one in eight government formations of the period. The Danish political system is frequently praised in such ‘populist’ terms as being one of the most fair in the world due to a high degree of proportionality between votes cast and seats won. However, the present analysis has demonstrated that even the Danish system may suffer from the defect that while a majority of the voters may vote for one outcome, the allocation of seats may produce the exact opposite outcome. Election Inversions, Coalitions and Proportional Representation: Examples of Voting Paradoxes in Danish Government Formations - Peter Kurrild-Klitgaard (2013)

ON THE MANIPULABILITY OF VOTING RULES
The manipulability problem in voting is that a voter can improve the social decision for herself by purposely misrepresenting her sincere preferences. The manipulability is probable if the range of non-dictatorial social choice rule has at least three alternatives. A tie-breaking order over alternatives is assumed to analyse single-valued versions of social choice rules (SCR). Breaking the symmetry between candidates leads to the risk of distortion of the computational results. A random tie-breaking rule preserves neutrality among alternatived. On the manipulability of voting rules: The case of 4 and 5 alternatives - Fuad Aleskerov a,b,∗, Daniel Karabekyan b, M. Remzi Sanver c, Vyacheslav Yakubaa (2012)

On The Manipulability Of Voting Rules (cont.)
After that, the degree of manipulability of several multi-valued SCR is explored by extending the manipulability indices defined for single-valued SCR to the multi-valued case. With this purpose, an extension of preferences over alternatives is implemented to reach the sets of alternatives by using two alternative methods for seven SCR in an environment of four and five alternatives. On the manipulability of voting rules: The case of 4 and 5 alternatives - Fuad Aleskerov a,b,∗, Daniel Karabekyan b, M. Remzi Sanver c, Vyacheslav Yakubaa (2012)

The Framework a finite set A m alternatives (m > 2) A′ = 2 𝐴 \ {∅}
Each agent from a finite set N = {1, , n} , n > 1, a preference Pi ∈ L over alternatives L is the set of linear orders over A. Profile, ⃗𝑃 A group decision is made by a SCR based on ⃗P and is considered to be an element of A′ . Thus we define a social choice rule as a mapping C : 𝐿 𝑛 → A′ . Every agent i is assumed to have an extended preference EPi over A′ which is induced by her preference Pi over A. On the manipulability of voting rules: The case of 4 and 5 alternatives - Fuad Aleskerov a,b,∗, Daniel Karabekyan b, M. Remzi Sanver c, Vyacheslav Yakubaa (2012)

The Framework (cont.) Under the leximax extension, any two sets are compared according to their best elements. The leximax extended preference EPi is defined as follows: On the manipulability of voting rules: The case of 4 and 5 alternatives - Fuad Aleskerov a,b,∗, Daniel Karabekyan b, M. Remzi Sanver c, Vyacheslav Yakubaa (2012)

The Framework (cont.) The concept of the leximin extension is defined similarly while it is based on ordering two sets according to a lexicographic comparison of their worst elements. The leximax extended preference EPi is defined as follows: On the manipulability of voting rules: The case of 4 and 5 alternatives - Fuad Aleskerov a,b,∗, Daniel Karabekyan b, M. Remzi Sanver c, Vyacheslav Yakubaa (2012)

The Framework (cont.) Moreover, under both methods, EPi is a linear order over A′. Manipulation in the case of multiple choice as follows. be a profile of sincere preferences, while C : 𝐿 𝑛 → A′ is manipulable by i at ⃗P if C(⃗P−i)EPiC(⃗P) for some ⃗P−i, where EPi is the extended preference of Pi. In other words, we suppose that outcome when the i-th agent deviates from her true preference is more preferable according to her extended preference over sets (with respect to her true preference over alternatives) than in the case when she reveals her sincere preference. On the manipulability of voting rules: The case of 4 and 5 alternatives - Fuad Aleskerov a,b,∗, Daniel Karabekyan b, M. Remzi Sanver c, Vyacheslav Yakubaa (2012)

Incides of Manipulability
To measure the degree of manipulability of Nitzan-Kelly’s index On the manipulability of voting rules: The case of 4 and 5 alternatives - Fuad Aleskerov a,b,∗, Daniel Karabekyan b, M. Remzi Sanver c, Vyacheslav Yakubaa (2012)

SCR Seven social choice rules are considered; Plurality rule
Q-approval Borda’s rule Black’s prodecure Threshold rule Uncovered set-I Strongest q-pareto simple majority On the manipulability of voting rules: The case of 4 and 5 alternatives - Fuad Aleskerov a,b,∗, Daniel Karabekyan b, M. Remzi Sanver c, Vyacheslav Yakubaa (2012)

Computation Scheme and Results
On the manipulability of voting rules: The case of 4 and 5 alternatives - Fuad Aleskerov a,b,∗, Daniel Karabekyan b, M. Remzi Sanver c, Vyacheslav Yakubaa (2012)

Computation Scheme and Results (cont.)
After all calculations, the last figure shows us that Black’s procedure is the best rule from a freedom of manipulation point of view. On the manipulability of voting rules: The case of 4 and 5 alternatives - Fuad Aleskerov a,b,∗, Daniel Karabekyan b, M. Remzi Sanver c, Vyacheslav Yakubaa (2012)

On the manipulability of voting rules: The case of 4 and 5 alternatives - Fuad Aleskerov a,b,∗, Daniel Karabekyan b, M. Remzi Sanver c, Vyacheslav Yakubaa (2012)

One can see that for some rules this index has larger values and forsome rules the values are smaller. But it is important to see which rule is the least manipulable for a given number of voters and alternatives and probability assumption. One can see that IAC (Impartial Anonymous Culture) does not influence too much the least manipulable rules for these numbers of voters. The only change is for the case of 5 alternatives, 4 voters and leximin extension. In the IC model the least manipulable rule is q-Approval rule q = 2, but still the degree of manipulability for this rule in IAC is among one of the lowest, as well as the Black’s procedure and Strongest q-Pareto Simple Majority rule. On the manipulability of voting rules: The case of 4 and 5 alternatives - Fuad Aleskerov a,b,∗, Daniel Karabekyan b, M. Remzi Sanver c, Vyacheslav Yakubaa (2012)

Results The answer on the question which rule is less manipulable depends on the method of preferences extension. If the number of voters is small the Threshold rule is less manipulable than the Borda rule. But when the number of voters is large enough the Borda rule is better in NK sense. The exact number of voters when this switch is revealed depends on the extension method used. The Black’s procedure, Uncovered Set I and Strongest q-Pareto Simple Majority rules are the least manipulable rules almost for any number of voters. For Leximax extension method the Black’s procedure is the least manipulable rule for odd number of voters and Strongest q-Pareto Simple Majority is the least manipulable rule for even number of voters. On the manipulability of voting rules: The case of 4 and 5 alternatives - Fuad Aleskerov a,b,∗, Daniel Karabekyan b, M. Remzi Sanver c, Vyacheslav Yakubaa (2012)

Results (cont.) For Leximin extension method the least manipulable rule is the Strongest q-Pareto Simple Majority rule for even, and Uncovered Set I is the least manipulable rule for odd number of voters. For small number of voters the least manipulable rules under IAC and IC are the same. Thus, the degree of manipulability of the rules depends on the number of alternatives, the number of voters, the method of preference extension and the index used in evaluation. Although there is no rule which is minimally manipulable for all cases, we can always find a rule which is best suited for a specific situation. But an interesting result is that the Black’s procedure is well-performing for many combinations of voters, alternatives, extension methods and indices of manipulability. On the manipulability of voting rules: The case of 4 and 5 alternatives - Fuad Aleskerov a,b,∗, Daniel Karabekyan b, M. Remzi Sanver c, Vyacheslav Yakubaa (2012)

REFERENCES Theory Choice and Social Choice: Kuhn versus Arrow - Samir Okasha (2011) A new monotonic, clone-independent, reversal symmetric, and condorcet - consistent single-winner election method - Markus Schulze (2011) Varieties of failure of monotonicity and participation under ﬁve voting Dan S. Felsenthal · Nicolaus Tideman (2012) Election Inversions, Coalitions and Proportional Representation: Examples of Voting Paradoxes in Danish Government Formations - Peter Kurrild-Klitgaard (2013) On the manipulability of voting rules: The case of 4 and 5 alternatives - Fuad Aleskerov a,b,∗, Daniel Karabekyan b, M. Remzi Sanver c, Vyacheslav Yakubaa (2012)

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