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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 every voter an equal vote). Examples include publicly-traded companies (which typically grant stockholders one vote for each share they own), as well as the European council, where the number of votes of each member state is roughly proportional to the square root of the population.[1]
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Introduction to Weighted Voting
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Weighted Voting: The Banzhaf Power Index
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Weighted Voting: Coalitions and Critical Players
Transcription
- WELCOME TO INTRODUCTION TO WEIGHTED VOTING. IN THIS LESSON WE'LL DEFINE WEIGHTED VOTING, USE SHORTHAND FORM FOR WEIGHTED VOTING, AND ALSO DEFINE DICTATOR VETO POWER AND DUMMY IN A WEIGHTED VOTING SYSTEM. WEIGHTED VOTING GIVES SOME VOTERS DIFFERENT AMOUNTS OF INFLUENCE OR WEIGHT CONCERNING THE OUTCOME OF AN ELECTION. THIS IS OFTEN COMMON PRACTICE WHEN IT COMES TO CORPORATE SHAREHOLDER MEETINGS WHEN SHAREHOLDERS VOTING WEIGHT IS BASED UPON HOW MANY SHARES THEY OWN. THE MORE SHARES SOMEONE OWNS, THE MORE WEIGHT OF THEIR VOTE. THE CONDITIONS THAT MUST BE MET FOR A WINNER MIGHT BE A SIMPLE MAJORITY, BUT THERE COULD ALSO BE DIFFERENT REQUIREMENTS TO SELECT A WINNER. EACH INDIVIDUAL, OR ENTITY, CASTING A VOTE IS CALLED A PLAYER IN THE ELECTION. THEY'RE OFTEN NOTATED AS P SUB ONE, P SUB TWO THROUGH P SUB N, WHERE N IS THE TOTAL NUMBER OF VOTERS. EACH PLAYER IS GIVEN A WEIGHT, WHICH USUALLY REPRESENTS HOW MANY VOTES THEY GET. THE QUOTA IS THE MINIMUM WEIGHT NEEDED FOR THE VOTE TO PASS, OR WEIGHT NEEDED FOR THE PROPOSAL, TO BE APPROVED. A WEIGHTED VOTING SYSTEM WILL OFTEN BE REPRESENTED IN A SHORTHAND FORM AS WE SEE HERE, WHERE Q IS THE QUOTA, W SUB ONE IS THE WEIGHT OF PLAYER ONE, W SUB TWO IS THE WEIGHT OF PLAYER TWO AND SO ON. LET'S TAKE A LOOK AT AN EXAMPLE. WE WANT TO WRITE A POSSIBLE WEIGHTED VOTING SYSTEM FOR THE FOLLOWING SITUATION. A COMPANY HAS FIVE SHAREHOLDERS. THESE SHAREHOLDERS HOLD THE FOLLOWING OWNERSHIP STAKE IN THE COMPANY GIVEN HERE AS PERCENTAGES. THEY ARE TRYING TO DECIDE IF THEY SHOULD INVEST IN A NEW PRODUCT LINE TO GROW THE COMPANY. THE COMPANY BYLAWS STATE THAT A MAJORITY OF THE OWNERSHIP, MORE THAN 50%, HAS TO APPROVE THE DECISION, BECAUSE THE OWNERSHIP IS GIVEN AS PERCENTAGES AND THE QUOTA HAS TO BE MORE THAN 50%. WE'LL START WITH THE QUOTA, WHICH WE'LL START AT 51 AND THEN A COLON FOLLOWED BY THE WEIGHT OF EACH OWNERSHIP, WHICH WE CAN ALSO THINK OF AS THE NUMBER OF VOTES GIVEN TO A VOTER BASED UPON THEIR PERCENT OF OWNERSHIP. SO WE COULD LIST THE WEIGHT AS 30, 25, 20, 15, 10. THIS WOULD BE A POSSIBLE WEIGHTED VOTING SYSTEM FOR THIS SITUATION, BUT THERE ARE AN INFINITE NUMBER OF WEIGHTED VOTING SYSTEMS THAT WE COULD CREATE FOR THIS SITUATION. IN ORDER TO HAVE A MEANINGFUL WEIGHTED VOTING SYSTEM THOUGH, THERE MUST BE LIMITS ON THE QUOTA. NUMBER ONE, THE QUOTA MUST BE MORE THEN HALF THE TOTAL NUMBER OF VOTES, BECAUSE NOTICE HOW IF IT WASN'T THERE COULD BE MORE WEIGHT AGAINST A PROPOSAL AND IT COULD STILL PASS. AND NUMBER TWO, THE QUOTA CAN'T BE LARGER THAN THE TOTAL NUMBER OF VOTES BECAUSE IF IT WAS NO PROPOSAL WOULD EVER PASS. A PLAYER WILL BE A DICTATOR IF THEIR WEIGHT IS EQUAL TO OR GREATER THAN THE QUOTA. THE DICTATOR CAN ALSO BLOCK ANY PROPOSAL FROM PASSING. THE OTHER PLAYERS CANNOT REACH QUOTA WITHOUT THE DICTATOR. LOOKING AT THIS WEIGHTED VOTING SYSTEM HERE, NOTICE HOW PLAYER ONE HAS A WEIGHT OF 21, WHICH IS MORE THAN QUOTA AND THEREFORE PLAYER ONE IS A DICTATOR. ALSO NOTICE, PLAYER ONE HAS THE ABILITY TO BLOCK ANY PROPOSAL BECAUSE IF PLAYER TWO AND PLAYER THREE FORM A GROUP, THEY STILL ONLY HAVE A WEIGHT OF 6 + 3 OR 9, WAY BELOW QUOTA. SO WE CAN SEE, AGAIN, PLAYER ONE IS A DICTATOR. NEXT, A PLAYER HAS VETO POWER IF THEIR SUPPORT IS NECESSARY FOR THE QUOTA TO BE REACHED. IT IS POSSIBLE FOR MORE THAN ONE PLAYER TO HAVE VETO POWER OR FOR NO PLAYER TO HAVE VETO POWER. LOOKING AT THIS FIRST EXAMPLE HERE, NOTICE HOW PLAYER ONE IS NOT A DICTATOR, BUT IF PLAYER TWO AND THREE FORM A GROUP THEY ONLY HAVE A WEIGHT OF 26, WHICH IS BELOW QUOTA. AND THEREFORE, PLAYER ONE IS REQUIRED TO REACH QUOTA. AND THEREFORE, PLAYER ONE HAS VETO POWER, BUT NOTICE HOW IF PLAYER ONE AND PLAYER THREE FORM A GROUP THEY HAVE A WEIGHT OF 30, WHICH DOES REACH QUOTA. SO PLAYER TWO IS NOT REQUIRED TO REACH QUOTA AND IF PLAYER ONE AND PLAYER TWO FORM A GROUP THEY HAVE A WEIGHT OF 34, ONCE AGAIN MORE THAN QUOTA, SO PLAYER THREE IS NOT REQUIRED. SO, AGAIN, WE CAN SAY THAT PLAYER ONE HAS VETO POWER. LOOKING AT OUR SECOND EXAMPLE HERE, NOTICE THAT OTHER PLAYERS ARE DICTATORS AND IF PLAYER ONE AND PLAYER TWO FORM A GROUP, THEY REACH QUOTA. IF PLAYER ONE AND PLAYER THREE FORM A GROUP, THEY REACH QUOTA. AND ALSO IF PLAYER TWO AND PLAYER THREE FORM A GROUP, THEY REACH QUOTA. SO NO SINGLE PLAYER'S REQUIRED TO REACH QUOTA AND THEREFORE, NO PLAYER HAS VETO POWER. NEXT, A PLAYER IS A DUMMY IF THEIR VOTE IS NEVER ESSENTIAL FOR A GROUP TO REACH QUOTA. SO LOOKING AT THIS SYSTEM, NOTICE HOW PLAYER THREE IS NEVER REQUIRED, PLAYER ONE AND PLAYER TWO CAN MAKE QUOTA, BUT PLAYER ONE AND PLAYER THREE CANNOT REACH QUOTA BECAUSE THEY'D HAVE A WEIGHT OF 14. IF PLAYER TWO AND PLAYER THREE FORM A GROUP THEY HAVE A WEIGHT OF 12, AGAIN, WHICH IS BELOW QUOTA. AND THEREFORE, PLAYER THREE IS NEVER ESSENTIAL TO REACH QUOTA SO WE CAN SAY PLAYER THREE IS A DUMMY PLAYER. NOW, LETS TAKE A LOOK AT SOME EXAMPLES TO REVIEW ALL THESE CONCEPTS. WE WANT TO CONSIDER THE FOLLOWING WEIGHTED VOTING SYSTEM. WE'RE FIRST ASKED HOW MANY PLAYERS THERE ARE. THERE WOULD BE ONE, TWO, THREE, FOUR PLAYERS. NEXT, WHAT IS THE TOTAL NUMBER OF VOTES OR WEIGHT? WELL, THAT WOULD BE 8 + 8 + 8 + 2 = 26, SO WE'LL SAY THERE'S A TOTAL OF 26 VOTES. THE QUOTA WOULD BE 18. NOW WE WANT TO IDENTIFY ANY DICTATORS, BUT NOTICE HOW NONE OF THE WEIGHTS ARE GREATER THAN OR EQUAL TO THE QUOTA OF 18 AND THEREFORE, NO PLAYERS ARE DICTATORS. WE ALSO WANT TO IDENTIFY WHETHER ANY PLAYERS HAVE VETO POWER, WHICH MEANS THEIR SUPPORT IS REQUIRED TO REACH QUOTA. FOR EXAMPLE, NOTICE HOW IF WE CONSIDER PLAYER TWO, THREE AND FOUR 8 + 8 + 2 = 18, WHICH DOES REACH QUOTA AND THEREFORE, PLAYER ONE IS NOT REQUIRED. AND WE COULD SAY THE SAME FOR PLAYER TWO AND PLAYER THREE. AND NOTICE THAT IF PLAYER ONE, TWO AND THREE FORM A GROUP, OF COURSE, THEY REACH QUOTA SO PLAYER FOUR IS NOT ESSENTIAL. AND THEREFORE, NO PLAYERS HAVE VETO POWER. WE ALSO WANT TO IDENTIFY ANY DUMMY PLAYERS. A DUMMY PLAYER IS A PLAYER THAT'S NEVER ESSENTIAL TO REACH QUOTA, BUT WE JUST SHOWED THAT IF PLAYER TWO, THREE, AND FOUR FORM A GROUP, OR A COALITION, NOTICE 8 + 8 = 16 AND THEREFORE, PLAYER FOUR IS REQUIRED TO REACH QUOTA. AND THEREFORE, PLAYER TWO IS NOT A DUMMY PLAYER. SO IN THIS CASE THERE ARE NO DUMMY PLAYERS. LET'S TAKE A LOOK AT ANOTHER EXAMPLE WITH THE SAME QUESTIONS, BUT A DIFFERENT WEIGHTED VOTING SYSTEM. AGAIN, THE FIRST QUESTION IS HOW MANY PLAYERS ARE THERE? THERE ARE ONE, TWO, THREE, FOUR, FIVE PLAYERS. THE TOTAL NUMBER OF VOTES, OR THE WEIGHT, WOULD BE 18 + 5 + 3 + 3 + 1 = 30. SO WE'LL SAY THE TOTAL NUMBER OF VOTES IS 30. THE QUOTA IS HERE TO THE LEFT OFF THE COLON. SO THE QUOTA IS 16. NOTICE IN THIS SYSTEM WE DO HAVE A DICTATOR. NOTICE PLAYER ONE HAS A WEIGHT THAT'S GREATER THAN OR EQUAL TO THE QUOTA OF 16 SO PLAYER ONE IS A DICTATOR. WE ALSO WANT TO IDENTIFY ANY PLAYERS THAT HAVE VETO POWER, WHICH MEANS THEIR VOTES ARE REQUIRED TO REACH QUOTA. IF WE CONSIDER PLAYERS TWO, THREE, FOUR AND FIVE, THEIR TOTAL WEIGHT WOULD BE 5 + 3 + 3 + 1 = 12, WHICH IS BELOW QUOTA. AND THEREFORE, NOTICE PLAYER ONE WOULD ALWAYS BE REQUIRED TO REACH QUOTA. AND THEREFORE, PLAYER ONE HAS VETO POWER. WE ALSO WANT TO IDENTIFY ANY DUMMY PLAYERS. REMEMBER, A DUMMY PLAYER IS NEVER ESSENTIAL TO REACH QUOTA. NOTICE IF PLAYERS TWO, THREE, FOUR AND FIVE FORM A GROUP, OR A COALITION, THEY STILL ONLY HAVE A WEIGHT OF 12, WHICH IS BELOW QUOTA. THE ONLY WAY TO REACH QUOTA IS WITH PLAYER ONE'S SUPPORT. AND THEREFORE, PLAYER TWO, THREE, FOUR AND FIVE ARE ALL DUMMY PLAYERS SINCE NONE OF THEM ARE ESSENTIAL TO REACH QUOTA. LET'S TAKE A LOOK AT ONE MORE EXAMPLE. HERE WE DON'T KNOW THE QUOTA, BUT WE'RE GIVEN THE WEIGHT OF THE FIVE PLAYERS. THE FIRST QUESTION IS WHAT IS THE SMALLEST VALUE THAT THE QUOTA Q CAN TAKE? REMEMBER, THE QUOTA HAS TO BE MORE THAN HALF THE VOTES SO WE'LL FIND THE SUM OF THESE VOTES, OR THESE WEIGHTS, AND DIVIDE BY TWO. AND THEN Q HAS TO BE GREATER THAN THAT VALUE. SO WE'D HAVE 14 + 10 + 6 + 2 + 2 DIVIDED BY 2 = 34 DIVIDED BY 2 = 17. REMEMBER, THE QUOTA HAS TO BE MORE THAN HALF AND SINCE HALF WOULD BE 17 VOTES, OR A WEIGHT OF 17, THAT MEANS THE SMALLEST VALUE OF Q WOULD HAVE TO BE ONE MORE THAN 17 OR 18 SINCE WE WOULDN'T USE A FRACTION OF A VOTE OR A FRACTION OF A WEIGHT. SO THE SMALLEST VALUE OF Q IS 18, THE LARGEST VALUE WOULD BE THE TOTAL NUMBER OF VOTES, WHICH WE JUST FOUND WAS 34. 14 + 10 + 6 + 2 + 2. SO THE LARGEST VALUE OF Q IS 34. AND THEN FINALLY, WHAT IS THE VALUE OF Q, THE QUOTA, IF AT LEAST 2/3 OF THE VOTES ARE REQUIRED TO PASS A MOTION? WELL, THAT WOULD BE 2/3 x 34, 34/1 THAT WOULD BE 68/3, WHICH COMES OUT TO APPROXIMATELY 22.7. SO WE'D HAVE TO ROUND UP TO 23. SO IF AT LEAST 2/3 OF THE VOTES ARE REQUIRED, Q WOULD BE 23. I HOPE YOU FOUND THIS HELPFUL.
Historical examples
Ancient Rome
The Roman assemblies provided for weighted voting after the person's tribal affiliation and social class (i.e. wealth). Rather than counting one vote per citizen, the assemblies convened in blocs (tribes or centuries), with the plurality of voters in each bloc deciding the vote of the bloc as an entity (which candidate to support or whether to favor or reject a law, for instance). Men of certain tribes and a higher social standing convened in smaller blocs, thus giving their individual vote the effect of many poor citizens' votes. In 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 Sweden and pre-unitary Germany, weighted voting preceded equal and universal suffrage, as well as women's suffrage, to different extents. In Sweden, universal and equal male suffrage to the lower house (Andra kammaren) was introduced by Arvid Lindman's first cabinet, while voting for city and county councils, which indirectly decided the composition of the upper house (Första kammaren), was graded along a 40-degree scale. Certain corporations also had votes of their own, thus multiplying the political strength of its owners. Weighted voting was abolished in 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 segregationist weighted voting system, in which the A roll (electing 50 seats) was reserved for Europeans, Coloureds and Asians meeting higher property and education requirements, and the B roll (electing eight seats) reserved for Africans meeting lower property and education requirements. In 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
- Corporatism
- Demeny voting
- Electoral college
- Preference voting
- Plural voting
- Prussian three-class franchise
- Vicente Blanco Gaspar
References
- ^ "Qualified majority – consilium". www.consilium.europa.eu/. EU. Retrieved October 8, 2015.
- ^ Blanco Gaspar, Vicente (1981). El voto ponderado. Instituto Hispano-Luso-Americano de Derecho Internacional. Madrid: Instituto Hispano-Luso-Americano de Derecho Internacional. ISBN 84-600-2197-1. OCLC 8776691.
- ^ Blanco Gaspar, Vicente. "El voto ponderado a nivel internacional" (PDF).
- ^ a b c d Tannenbaum, Peter. Excursions in Modern Mathematics. 6th ed. Upper Saddle River: Prentice Hall, 2006. 48–83.
- ^ Bowen, Larry. "Weighted Voting Systems." Introduction to Contemporary Mathematics. 1 Jan. 2001. Center for Teaching and Learning, University of Alabama. [1].
- ^ Daubechies, Ingrid. "Weighted Voting Systems." Voting and Social Choice. 26 Jan. 2002. Math Alive, Princeton University. [2].
This page was last edited on 16 March 2024, at 01:28