Weighted voting

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Weighted voting refers to voting rules that grant some voters a greater influence than others (which contrasts with rules that assign

European council, where the number of votes of each member state is roughly proportional to the square root of the population.[1]

Historical examples

Ancient Rome

The

Plebeian Council, where only the plebians could participate, these effects were somewhat relaxed, thus making the decision to grant its decisions (called plebiscites) the full force of law controversial (Lex Hortensia in 287 BC).[2]

Central Europe

In several Western democracies, such as

Nils Eden's reforms of 1918-19, when female suffrage was also introduced.[3]

French colonies

After 1946 and the

Brazzaville Conference of 1944, French colonial authorities set up a system of double collège where the local population would be divided in two electoral colleges, both returning the same numbers of delegates, the first being composed by French citizens and évolués
and the second by natives with indigenous status.

This system was also used in French Algeria until 1958.

This system was abolished on 1958 with the

Loi Cadre Defferre
.

Southern Rhodesia

Under its

1961 Constitution, the British colony of Southern Rhodesia
provided for a special form of weighted voting called cross-voting. Essentially, voters were rounded up in two voters' rolls, with the A roll bearing requirements generally reached by the European-descended population, but only in a few cases by Africans. The B roll provided for many Africans and a few Europeans, but not all the adult population. Despite its limited size in terms of voters, the A roll played the major influence in electing the 65 members of parliament, which was further bolstered by the lack of support to sign up for the B roll, and its much lower turnout.

In 1969, cross-voting was abolished altogether in favor of a de jure

its 1970 general election
, about 50,000 A roll voters (essentially all white) elected 50 parliamentary seats, a little more than 1000 tribal chiefs elected eight special seats, whereas the rest of the population were to be content with the remaining eight seats.

Hong Kong

The Hong Kong legislature elects 30 out of 90 of its members through so-called ’Functional Constituencies’, which in effect represent local business interests in a corporatist manner.[citation needed][further explanation needed]

Weighted voting games

A weighted voting game is characterized by the players, the weights, and the quota. A player's weight (w) is the number of votes he controls. The quota (q) is the minimum number of votes required to pass a motion. Any integer is a possible choice for the quota as long as it is more than 50% of the total number of votes but is no more than 100% of the total number of votes. Each weighted voting system can be described using the generic form [q : w1, w2, . . ., wN].[4]

The notion of power

When considering motions, all reasonable electoral systems will have the same outcome as majority rules. Thus, the mathematics of weighted voting systems looks at the notion of power: who has it and how much do they have?[5] A player's power is defined as that player's ability to influence decisions.[6]

Consider the voting system [6: 5, 3, 2]. Notice that a motion can only be passed with the support of P1. In this situation, P1 has veto power. A player is said to have veto power if a motion cannot pass without the support of that player. This does not mean a motion is guaranteed to pass with the support of that player.[4]

Now let us look at the weighted voting system [10: 11, 6, 3]. With 11 votes, P1 is called a dictator. A player is typically considered a dictator if their weight is equal to or greater than the quota. The difference between a dictator and a player with veto power is that a motion is guaranteed to pass if the dictator votes in favor of it.[4]

A dummy is any player, regardless of their weight, who has no say in the outcome of the election. A player without any say in the outcome is a player without power. Consider the weighted voting system [8: 4, 4, 2, 1]. In this voting system, the voter with weight 2 seems like he has more power than the voter with weight 1, however the reality is that both voters have no power whatsoever (neither can affect the passing of a motion). Dummies always appear in weighted voting systems that have a dictator but also occur in other weighted voting systems (the example above).[4]

Measuring power

A player's weight is not always an accurate depiction of that player's power. Sometimes, a player with a large weight votes can have very little power, or vice-versa. For example, in a weighted voting system where one voter has 51% of the weight, this voter holds all the power, even if there is another voter who theoretically has 49% of the weight.

The Banzhaf power index and the Shapley–Shubik power index provide more accurate measures of voting power, by estimating the probability that an individual voter's ballot will be decisive. Such indices often give counterintuitive results. For example, commentators often mistakenly assume the United States Electoral College is weighted in favor of smaller states (because it assigns every state 2 additional electoral votes). However, more detailed analysis typically finds that larger states have more power than implied by their number of electors, making the system as a whole biased towards larger states (unlike a simple popular vote).

See also

References

  1. ^ "Qualified majority – consilium". www.consilium.europa.eu/. EU. Retrieved October 8, 2015.
  2. OCLC 8776691
    .
  3. ^ Blanco Gaspar, Vicente (2015). "El voto ponderado a nivel internacional" (PDF).
  4. ^ a b c d Tannenbaum, Peter. Excursions in Modern Mathematics. 6th ed. Upper Saddle River: Prentice Hall, 2006. 48–83.
  5. ^ Bowen, Larry. "Weighted Voting Systems." Introduction to Contemporary Mathematics. 1 Jan. 2001. Center for Teaching and Learning, University of Alabama. [1].
  6. ^ Daubechies, Ingrid. "Weighted Voting Systems." Voting and Social Choice. 26 Jan. 2002. Math Alive, Princeton University. [2].
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