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78-458: Positional voting Cardinal voting Quota-remainder methods Approval-based committees Fractional social choice Semi-proportional representation By ballot type Pathological response Strategic voting Paradoxes of majority rule Positive results In social choice theory , the majority rule ( MR ) is a social choice rule which says that, when comparing two options (such as bills or candidates ),

156-413: A ⋅ x a − 1 ⋅ y b b ⋅ y b − 1 ⋅ x a = a y b x {\displaystyle MRS={\frac {a\cdot x^{a-1}\cdot y^{b}}{b\cdot y^{b-1}\cdot x^{a}}}={\frac {ay}{bx}}} . The MRS is the same for the function v ( x , y ) =

234-549: A ⋅ log ⁡ x + b ⋅ log ⁡ y {\displaystyle v(x,y)=a\cdot \log {x}+b\cdot \log {y}} . This is not a coincidence as these two functions represent the same preference relation – each one is an increasing monotone transformation of the other. In general, the MRS may be different at different points ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} . For example, it

312-515: A . For the Nauru system, the first preference a is worth one and the common difference d between adjacent denominators is also one. Numerous other harmonic sequences can also be used in positional voting. For example, setting a to 1 and d to 2 generates the reciprocals of all the odd numbers (1, 1/3, 1/5, 1/7, …) whereas letting a be 1/2 and d be 1/2 produces those of all the even numbers (1/2, 1/4, 1/6, 1/8, …). The harmonic variant used by

390-406: A = N , the number of candidates. The value of the first preference need not be N . It is sometimes set to N – 1 so that the last preference is worth zero. Although it is convenient for counting, the common difference need not be fixed at one since the overall ranking of the candidates is unaffected by its specific value. Hence, despite generating differing tallies, any value of a or d for

468-475: A Borda count election will result in identical candidate rankings. The consecutive Borda count weightings form an arithmetic progression . Common systems for evaluating preferences, other than Borda, are typically "top-heavy". In other words, the method focuses on how many voters consider a candidate one of their "favourites". Under first-preference plurality (FPP), the most-preferred option receives 1 point while all other options receive 0 points each. This

546-405: A bill. Mandatory referendums where the question is yes or no are also generally decided by majority rule. It is one of the basic rules of parliamentary procedure , as described in handbooks like Robert's Rules of Order . One alternative to the majority rule is the set of plurality rules , which includes ranked choice-runoff (RCV) , two-round plurality , or first-preference plurality . This

624-420: A diminishing marginal rate of substitution. It can be shown that consumer analysis with indifference curves (an ordinal approach) gives the same results as that based on cardinal utility theory — i.e., consumers will consume at the point where the marginal rate of substitution between any two goods equals the ratio of the prices of those goods (the equi-marginal principle). Revealed preference theory addresses

702-546: A function u such that: But critics of cardinal utility claim the only meaningful message of this function is the order u ( A ) > u ( B ) > u ( C ) {\displaystyle u(A)>u(B)>u(C)} ; the actual numbers are meaningless. Hence, George's preferences can also be represented by the following function v : The functions u and v are ordinally equivalent – they represent George's preferences equally well. Ordinal utility contrasts with cardinal utility theory:

780-403: A geometric progression with a common ratio of one-half ( r = 1/2). Such weightings are inherently valid for use in positional voting systems provided that a legitimate common ratio is employed. Using a common ratio of zero, this form of positional voting has weightings of 1, 0, 0, 0, … and so produces ranking outcomes identical to that for first-past-the-post or plurality voting . Alternatively,

858-415: A given rank position ( n ) is defined below; where the value of the first preference is a . w n = a 2 a + ( n − 1 ) d = a 1 + ( n − 1 ) d a , {\displaystyle w_{n}={\frac {a^{2}}{a+(n-1)d}}={\frac {a}{1+{\frac {(n-1)d}{a}}}},} where w 1 =

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936-448: A high value and all the remaining options with a common lower value. The two validity criteria for a sequence of weightings are hence satisfied. For an N -candidate ranked ballot, let the permitted number of favoured candidates per ballot be F and the two weightings be one point for these favoured candidates and zero points for those not favoured. When analytically represented using positional voting, favoured candidates must be listed in

1014-485: A mathematical sequence such as an arithmetic progression ( Borda count ), a geometric one ( positional number system ) or a harmonic one ( Nauru/Dowdall method ). The set of weightings employed in an election heavily influences the rank ordering of the candidates. The steeper the initial decline in preference values with descending rank, the more polarised and less consensual the positional voting system becomes. Positional voting should be distinguished from score voting : in

1092-598: A method for converting sets of individual preferences (ranked ballots) into one collective and fully rank-ordered set. It is possible and legitimate for options to be tied in this resultant set; even in first place. Consider a positional voting election for choosing a single winner from three options A, B and C. No truncation or ties are permitted and a first, second and third preference is here worth 4, 2 and 1 point respectively. There are then six different ways in which each voter may rank order these options. The 100 voters cast their ranked ballots as follows: After voting closes,

1170-515: A minority needs its own supermajority to overturn a decision. To support the view that majority rule protects minority rights better than supermajority rules, McGann pointed to the cloture rule in the US Senate, which was used to prevent the extension of civil liberties to racial minorities. Saunders, while agreeing that majority rule may offer better protection than supermajority rules, argued that majority rule may nonetheless be of little help to

1248-534: A numeric function, an agent's preference relation can be represented graphically by indifference curves. This is especially useful when there are two kinds of goods, x and y . Then, each indifference curve shows a set of points ( x , y ) {\displaystyle (x,y)} such that, if ( x 1 , y 1 ) {\displaystyle (x_{1},y_{1})} and ( x 2 , y 2 ) {\displaystyle (x_{2},y_{2})} are on

1326-413: A preference relation on X {\displaystyle X} . It is common to mark the weak preference relation by ⪯ {\displaystyle \preceq } , so that A ⪯ B {\displaystyle A\preceq B} reads "the agent wants B at least as much as A". The symbol ∼ {\displaystyle \sim } is used as a shorthand to

1404-430: A right might be majoritarian , but it would not be legitimate, because it would violate the requirement for equal rights . Voting theorists claimed that cycling leads to debilitating instability. Buchanan and Tullock note that unanimity is the only decision rule that guarantees economic efficiency. McGann argued that majority rule helps to protect minority rights , at least in deliberative settings. The argument

1482-451: A second one is given 10 points. The next eight consecutive preferences are awarded 8, 7, 6, 5, 4, 3, 2 and 1 point. All remaining preferences receive zero points. In positional voting, the weightings ( w ) of consecutive preferences from first to last decline monotonically with rank position ( n ). However, the rate of decline varies according to the type of progression employed. Lower preferences are more influential in election outcomes where

1560-483: A worse-ranked candidate must receive fewer points than a better-ranked candidate. The classic example of a positional voting electoral system is the Borda count . Typically, for a single-winner election with N candidates, a first preference is worth N points, a second preference N – 1 points, a third preference N – 2 points and so on until the last ( N th) preference that is worth just 1 point. So, for example,

1638-464: Is any monotonically increasing function, then the functions v and v give rise to identical indifference curve mappings. This equivalence is succinctly described in the following way: In contrast, a cardinal utility function is unique up to increasing affine transformation . Every affine transformation is monotone; hence, if two functions are cardinally equivalent they are also ordinally equivalent, but not vice versa. Suppose, from now on, that

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1716-404: Is continuity . A preference relation is called continuous if, whenever B is preferred to A, small deviations from B or A will not reverse the ordering between them. Formally, a preference relation on a set X is called continuous if it satisfies one of the following equivalent conditions: If a preference relation is represented by a continuous utility function, then it is clearly continuous. By

1794-466: Is a ranked voting electoral system in which the options or candidates receive points based on their rank position on each ballot and the one with the most points overall wins. The lower-ranked preference in any adjacent pair is generally of less value than the higher-ranked one. Although it may sometimes be weighted the same, it is never worth more. A valid progression of points or weightings may be chosen at will ( Eurovision Song Contest ) or it may form

1872-518: Is a function representing the preferences of an agent on an ordinal scale . Ordinal utility theory claims that it is only meaningful to ask which option is better than the other, but it is meaningless to ask how much better it is or how good it is. All of the theory of consumer decision-making under conditions of certainty can be, and typically is, expressed in terms of ordinal utility. For example, suppose George tells us that "I prefer A to B and B to C". George's preferences can be represented by

1950-406: Is an additive function : There are several ways to check whether given preferences are representable by an additive utility function. If the preferences are additive then a simple arithmetic calculation shows that so this "double-cancellation" property is a necessary condition for additivity. Debreu (1960) showed that this property is also sufficient: i.e., if a preference relation satisfies

2028-562: Is in accordance with the fact, stated above, that these preferences cannot be represented by a utility function. For every utility function v , there is a unique preference relation represented by v . However, the opposite is not true: a preference relation may be represented by many different utility functions. The same preferences could be expressed as any utility function that is a monotonically increasing transformation of v . E.g., if where f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} }

2106-496: Is indifferent between this bundle and the bundle ( x 0 − λ ⋅ δ , y 0 + δ ) {\displaystyle (x_{0}-\lambda \cdot \delta ,y_{0}+\delta )} . This means that he is willing to give λ ⋅ δ {\displaystyle \lambda \cdot \delta } units of x to get δ {\displaystyle \delta } units of y. If this ratio

2184-400: Is kept as δ → 0 {\displaystyle \delta \to 0} , we say that λ {\displaystyle \lambda } is the marginal rate of substitution (MRS) between x and y at the point ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} . This definition of the MRS is based only on

2262-469: Is more favourable to candidates with many first preferences than the conventional Borda count. It has been described as a system "somewhere between plurality and the Borda count, but as veering more towards plurality". Simulations show that 30% of Nauru elections would produce different outcomes if counted using standard Borda rules. The Eurovision Song Contest uses a first preference worth 12 points, while

2340-529: Is often used in elections with more than two candidates. In these elections, the winning candidate is the one with the most votes after applying some voting procedure, even if a majority of voters would prefer some other candidate. The utilitarian rule , and cardinal social choice rules in general, take into account not just the number of voters who support each choice but also the intensity of their preferences . Philosophers critical of majority rule have often argued that majority rule does not take into account

2418-434: Is possible that at ( 9 , 1 ) {\displaystyle (9,1)} the MRS is low because the person has a lot of x and only one y , but at ( 9 , 9 ) {\displaystyle (9,9)} or ( 1 , 1 ) {\displaystyle (1,1)} the MRS is higher. Some special cases are described below. When the MRS of a certain preference relation does not depend on

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2496-402: Is preferentially-independent of the subset { z }, if for all x i , y i , z , z ′ {\displaystyle x_{i},y_{i},z,z'} : In this case, we can simply say that: Preferential independence makes sense in case of independent goods . For example, the preferences between bundles of apples and bananas are probably independent of

2574-456: Is represented by a utility function v ( x , y ) {\displaystyle v(x,y)} . Suppose the preference relation is monotonically increasing , which means that "more is always better": Then, both partial derivatives, if they exist, of v are positive. In short: Suppose a person has a bundle ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} and claims that he

2652-576: Is seen under proportional representation in the Netherlands , Austria , and Sweden , as empirical evidence of majority rule's stability. Positional voting Condorcet methods Positional voting Cardinal voting Quota-remainder methods Approval-based committees Fractional social choice Semi-proportional representation By ballot type Pathological response Strategic voting Paradoxes of majority rule Positive results Positional voting

2730-402: Is straightforward. All the preferences cast by voters are awarded the points associated with their rank position. Then, all the points for each option are tallied and the one with the most points is the winner. Where a few winners ( W ) are instead required following the count, the W highest-ranked options are selected. Positional voting is not only a means of identifying a single winner but also

2808-420: Is that cycling ensures that parties that lose to a majority have an interest to remain part of the group's process, because any decision can easily be overturned by another majority. Furthermore, suppose a minority wishes to overturn a decision. In that case, under majority rule it just needs to form a coalition that has more than half of the officials involved and that will give it power. Under supermajority rules,

2886-400: Is the most top-heavy positional voting system. An alternative mathematical sequence known as a geometric progression may also be used in positional voting. Here, there is instead a common ratio r between adjacent weightings. In order to satisfy the two validity conditions, the value of r must be less than one so that weightings decrease as preferences descend in rank. Where the value of

2964-467: Is the only decision rule that has the following properties: If voter's preferences are defined over a multidimensional option space, then choosing options using pairwise majority rule is unstable. In most cases, there will be no Condorcet winner and any option can be chosen through a sequence of votes, regardless of the original option. This means that adding more options and changing the order of votes ("agenda manipulation") can be used to arbitrarily pick

3042-516: The intensity of preference for different voters, and as a result "two voters who are casually interested in doing something" can defeat one voter who has "dire opposition" to the proposal of the two, leading to poor deliberative practice or even to "an aggressive culture and conflict"; however, the median voter theorem guarantees that majority-rule will tend to elect "compromise" or "consensus" candidates in many situations, unlike plurality-rules (see center squeeze ). Parliamentary rules may prescribe

3120-449: The MRS is a function λ ( y ) {\displaystyle \lambda (y)} , a possible function v Y {\displaystyle v_{Y}} can be calculated as an integral of λ ( y ) {\displaystyle \lambda (y)} : In this case, all the indifference curves are parallel – they are horizontal transfers of each other. A more general type of utility function

3198-586: The bare minimum required to "win" because of the likelihood that they would soon be reversed. Within this atmosphere of compromise, a minority faction may accept proposals that it dislikes in order to build a coalition for a proposal that it deems of greater moment. In that way, majority rule differentiates weak and strong preferences. McGann argued that such situations encourage minorities to participate, because majority rule does not typically create permanent losers, encouraging systemic stability. He pointed to governments that use largely unchecked majority rule, such as

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3276-461: The binary number system, a common ratio greater than one-half must be employed. The higher the value of r , the slower the decrease in weightings with descending rank. Although not categorised as positional voting electoral systems, some non-ranking methods can nevertheless be analysed mathematically as if they were by allocating points appropriately. Given the absence of strict monotonic ranking here, all favoured options are weighted identically with

3354-441: The binary, ternary, octal and decimal number systems use a radix R of 2, 3, 8 and 10 respectively. The value R is also the common ratio of the geometric progression going up in rank order while r is the complementary common ratio descending in rank. Therefore, r is the reciprocal of R and the r ratios are respectively 1/2, 1/3, 1/8 and 1/10 for these positional number systems when employed in positional voting. As it has

3432-581: The bundle, i.e., the MRS is the same for all ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} , the indifference curves are linear and of the form: and the preference relation can be represented by a linear function: (Of course, the same relation can be represented by many other non-linear functions, such as x + λ y {\displaystyle {\sqrt {x+\lambda y}}} or ( x + λ y ) 2 {\displaystyle (x+\lambda y)^{2}} , but

3510-407: The chosen progression employs a sequence of weightings that descend relatively slowly with rank position. The more slowly weightings decline, the more consensual and less polarising positional voting becomes. This figure illustrates such declines over ten preferences for the following four positional voting electoral systems: To aid comparison, the actual weightings have been normalised; namely that

3588-412: The consumer is equally satisfied with. The further a curve is from the origin, the greater is the level of utility. The slope of the curve (the negative of the marginal rate of substitution of X for Y) at any point shows the rate at which the individual is willing to trade off good X against good Y maintaining the same level of utility. The curve is convex to the origin as shown assuming the consumer has

3666-472: The decimal point are employed rather than fractions. (This system should not be confused with the use of sequential divisors in proportional systems such as proportional approval voting , an unrelated method.) A similar system of weighting lower-preference votes was used in the 1925 Oklahoma primary electoral system . For a four-candidate election, the Dowdall point distribution would be this: This method

3744-486: The denominators of the above fractional weightings could form an arithmetic progression instead; namely 1/1, 1/2, 1/3, 1/4 and so on down to 1/ N . This further mathematical sequence is an example of a harmonic progression . These particular descending rank-order weightings are in fact used in N -candidate positional voting elections to the Nauru parliament . For such electoral systems, the weighting ( w n ) allocated to

3822-409: The double-cancellation property then it can be represented by an additive utility function. If the preferences are represented by an additive function, then a simple arithmetic calculation shows that so this "corresponding tradeoffs" property is a necessary condition for additivity. This condition is also sufficient. When there are three or more commodities, the condition for the additivity of

3900-421: The existence of a representing function: When these conditions are met and the set X {\displaystyle X} is finite, it is easy to create a function u {\displaystyle u} which represents ≺ {\displaystyle \prec } by just assigning an appropriate number to each element of X {\displaystyle X} , as exemplified in

3978-429: The first preference is a , the weighting ( w n ) awarded to a given rank position ( n ) is defined below. w n = a r n − 1 , 0 ≤ r < 1 {\displaystyle w_{n}=ar^{n-1},\qquad 0\leq r<1} For example, the sequence of consecutively halved weightings of 1, 1/2, 1/4, 1/8, … as used in the binary number system constitutes

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4056-418: The first preference is set at one and the other weightings in the particular sequence are scaled by the same factor of 1/ a . The relative decline of weightings in any arithmetic progression is constant as it is not a function of the common difference d . In other words, the relative difference between adjacent weightings is fixed at 1/ N . In contrast, the value of d in a harmonic progression does affect

4134-401: The former, the score that each voter gives to each candidate is uniquely determined by the candidate's rank; in the latter, each voter is free to give any score to any candidate. In positional voting, voters complete a ranked ballot by expressing their preferences in rank order. The rank position of each voter preference is allotted a specific fixed weighting. Typically, the higher the rank of

4212-650: The group, while under supermajoritarian rules participants might only need to persuade a minority (to prevent a change). Where large changes in seats held by a party may arise from only relatively slight change in votes cast (such as under FPTP), and a simple majority is all that is required to wield power (most legislatures in democratic countries), governments may repeatedly fall into and out of power. This may cause polarization and policy lurch, or it may encourage compromise, depending on other aspects of political culture. McGann argued that such cycling encourages participants to compromise, rather than pass resolutions that have

4290-630: The indifference relation: A ∼ B ⟺ ( A ⪯ B ∧ B ⪯ A ) {\displaystyle A\sim B\iff (A\preceq B\land B\preceq A)} , which reads "The agent is indifferent between B and A". The symbol ≺ {\displaystyle \prec } is used as a shorthand to the strong preference relation: A ≺ B ⟺ ( A ⪯ B ∧ B ⪯ ̸ A ) {\displaystyle A\prec B\iff (A\preceq B\land B\not \preceq A)} if: Instead of defining

4368-479: The island nation of Nauru is called the Dowdall system as it was devised by Nauru's Secretary for Justice (Desmond Dowdall) in 1971. Here, each voter awards the first-ranked candidate with 1 point, while the 2nd-ranked candidate receives 1 ⁄ 2 a point, the 3rd-ranked candidate receives 1 ⁄ 3 of a point, etc. When counting candidate tallies in Nauru, decimal numbers rounded to three places after

4446-426: The latter assumes that the differences between preferences are also important. In u the difference between A and B is much smaller than between B and C, while in v the opposite is true. Hence, u and v are not cardinally equivalent. The ordinal utility concept was first introduced by Pareto in 1906. Suppose the set of all states of the world is X {\displaystyle X} and an agent has

4524-579: The least minorities. Under some circumstances, the legal rights of one person cannot be guaranteed without unjustly imposing on someone else. McGann wrote, "one man's right to property in the antebellum South was another man's slavery." Amartya Sen has noted the existence of the liberal paradox , which shows that permitting assigning a very small number of rights to individuals may make everyone worse off. Saunders argued that deliberative democracy flourishes under majority rule and that under majority rule, participants always have to convince more than half

4602-418: The linear function is simplest.) When the MRS depends on y 0 {\displaystyle y_{0}} but not on x 0 {\displaystyle x_{0}} , the preference relation can be represented by a quasilinear utility function, of the form where v Y {\displaystyle v_{Y}} is a certain monotonically increasing function. Because

4680-448: The minority, making it stronger (at least through its veto) than the majority. McGann argued that when only one of multiple minorities is protected by the super-majority rule (same as seen in simple plurality elections systems), so the protection is for the status quo, rather than for the faction that supports it. Another possible way to prevent tyranny is to elevate certain rights as inalienable . Thereafter, any decision that targets such

4758-531: The opening paragraph. The same is true when X is countably infinite . Moreover, it is possible to inductively construct a representing utility function whose values are in the range ( − 1 , 1 ) {\displaystyle (-1,1)} . When X {\displaystyle X} is infinite, these conditions are insufficient. For example, lexicographic preferences are transitive and complete, but they cannot be represented by any utility function. The additional condition required

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4836-481: The option preferred by more than half of the voters (a majority ) should win. In political philosophy , the majority rule is one of two major competing notions of democracy . The most common alternative is given by the utilitarian rule (or other welfarist rules), which identify the spirit of liberal democracy with the equal consideration of interests . Although the two rules can disagree in theory, political philosophers beginning with James Mill have argued

4914-493: The ordinal preference relation – it does not depend on a numeric utility function. If the preference relation is represented by a utility function and the function is differentiable, then the MRS can be calculated from the derivatives of that function: For example, if the preference relation is represented by v ( x , y ) = x a ⋅ y b {\displaystyle v(x,y)=x^{a}\cdot y^{b}} then M R S =

4992-420: The points are respectively 4, 3, 2 and 1 for a four-candidate election. Mathematically, the point value or weighting ( w n ) associated with a given rank position ( n ) is defined below; where the weighting of the first preference is a and the common difference is d . w n = a − ( n − 1 ) d {\displaystyle w_{n}=a-(n-1)d} where

5070-428: The points awarded by the voters are then tallied and the options ranked according to the points total. Therefore, having the highest tally, option A is the winner here. Note that the election result also generates a full ranking of all the options. For positional voting, any distribution of points to the rank positions is valid, so long as the points are weakly decreasing in the rank of each candidate. In other words,

5148-439: The preference, the more points it is worth. Occasionally, it may share the same weighting as a lower-ranked preference but it is never worth fewer points. Usually, every voter is required to express a unique ordinal preference for each option on the ballot in strict descending rank order. However, a particular positional voting system may permit voters to truncate their preferences after expressing one or more of them and to leave

5226-400: The problem of how to observe ordinal preference relations in the real world. The challenge of revealed preference theory lies in part in determining what goods bundles were foregone, on the basis of them being less liked, when individuals are observed choosing particular bundles of goods. Some conditions on ⪯ {\displaystyle \preceq } are necessary to guarantee

5304-501: The rate of its decline. The higher its value, the faster the weightings descend. Whereas the lower the value of the common ratio r for a geometric progression, the faster its weightings decline. The weightings of the digit positions in the binary number system were chosen here to highlight an example of a geometric progression in positional voting. In fact, the consecutive weightings of any digital number system can be employed since they all constitute geometric progressions. For example,

5382-541: The remaining options unranked and consequently worthless. Similarly, some other systems may limit the number of preferences that can be expressed. For example, in the Eurovision Song Contest only their top ten preferences are ranked by each country although many more than ten songs compete in the contest. Again, unranked preferences have no value. In positional voting, ranked ballots with tied options are normally considered as invalid. The counting process

5460-418: The same curve, then ( x 1 , y 1 ) ∼ ( x 2 , y 2 ) {\displaystyle (x_{1},y_{1})\sim (x_{2},y_{2})} . An example indifference curve is shown below: [REDACTED] Each indifference curve is a set of points, each representing a combination of quantities of two goods or services, all of which combinations

5538-449: The set X {\displaystyle X} is the set of all non-negative real two-dimensional vectors. So an element of X {\displaystyle X} is a pair ( x , y ) {\displaystyle (x,y)} that represents the amounts consumed from two products, e.g., apples and bananas. Then under certain circumstances a preference relation ⪯ {\displaystyle \preceq }

5616-428: The smallest radix, the rate of decline in preference weightings is slowest when using the binary number system. Although the radix R (the number of unique digits used in the number system) has to be an integer, the common ratio r for positional voting does not have to be the reciprocal of such an integer. Any value between zero and just less than one is valid. For a slower descent of weightings than that generated using

5694-483: The theorems of Debreu (1954) , the opposite is also true: Note that the lexicographic preferences are not continuous. For example, ( 5 , 0 ) ≺ ( 5 , 1 ) {\displaystyle (5,0)\prec (5,1)} , but in every ball around (5,1) there are points with x < 5 {\displaystyle x<5} and these points are inferior to ( 5 , 0 ) {\displaystyle (5,0)} . This

5772-506: The top F rank positions in any order on each ranked ballot and the other candidates in the bottom N - F rank positions. This is essential as the weighting of each rank position is fixed and common to each and every ballot in positional voting. Unranked single-winner methods that can be analysed as positional voting electoral systems include: And unranked methods for multiple-winner elections (with W winners) include: Ordinal utility In economics , an ordinal utility function

5850-489: The two can be reconciled in practice, with majority rule being a valid approximation to the utilitarian rule whenever voters share similarly-strong preferences. This position has found strong support in many social choice models, where the socially-optimal winner and the majority-preferred winner often overlap. Majority rule is the most common social choice rule worldwide, being heavily used in deliberative assemblies for dichotomous decisions, e.g. whether or not to pass

5928-621: The use of a supermajoritarian rule under certain circumstances, such as the 60% filibuster rule to close debate in the US Senate . However such requirement means that 41 percent of the members or more could prevent debate from being closed, an example where the majority will would be blocked by a minority. Kenneth May proved that the simple majority rule is the only "fair" ordinal decision rule, in that majority rule does not let some votes count more than others or privilege an alternative by requiring fewer votes to pass. Formally, majority rule

6006-502: The utility function is surprisingly simpler than for two commodities. This is an outcome of Theorem 3 of Debreu (1960) . The condition required for additivity is preferential independence . A subset A of commodities is said to be preferentially independent of a subset B of commodities, if the preference relation in subset A, given constant values for subset B, is independent of these constant values. For example, suppose there are three commodities: x y and z . The subset { x , y }

6084-417: The winner. In group decision-making voting paradoxes can form. It is possible that alternatives a, b, and c exist such that a majority prefers a to b, another majority prefers b to c, and yet another majority prefers c to a. Because majority rule requires an alternative to have majority support to pass, majority rule is vulnerable to rejecting the majority's decision. A super-majority rule actually empowers

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