Weighted voting systems are voting systems based on the idea that not all voters are equal. Instead, it can be desirable to recognize differences by giving voters different amounts of say (weights) concerning the outcome of an election. This is in contrast to normal parliamentary procedure, which assumes that each member's vote carries equal weight.
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 plebs 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).
In several Western democracies, such as Sweden and pre-unitary Germany, weighted voting preceded equal and universal suffrage, as well as womens' 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.
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.
The mathematics of weighted voting systems
A weighted voting system is characterized by three things — the players, the weights and the quota. The voters are the players (P1 , P2, . . ., PN). N denotes the total number of players. 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]. The weights are always listed in numerical order, starting with the highest.
The notion of power
When considering motions, all reasonable voting methods 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? A player's power is defined as that player's ability to influence decisions.
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.
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 his 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. 
A dummy is any player, regardless of his 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).
Measuring a player's power
A player's weight is not always an accurate depiction of that player's power. Sometimes, a player with several votes can have little power. For example, consider the weighted voting system [20: 10, 10, 9]. Although P3 has almost as many votes as the other players, his votes will never affect the outcome. Conversely, a player with just a few votes may hold quite a bit of power. Take the weighted voting system [7: 4, 2, 1] for example. No motion can be passed without the unanimous support of all the players. Thus, P3 holds just as much power as P1.
It is more accurate to measure a player's power using either the Banzhaf power index or the Shapley–Shubik power index. The two power indexes often come up with different measures of power for each player yet neither one in necessarily a more accurate depiction. Thus, which method is best for measuring power is based on which assumption best fits the situation.The Banzhaf measure of power is based on the idea that players are free to come and go from coalitions, negotiating their allegiance. The Shapley–Shubik measure centers on the assumption that a player makes a commitment to stay upon joining a coalition.
- 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. .
- Daubechies, Ingrid. "Weighted Voting Systems." Voting and Social Choice. 26 Jan. 2002. Math Alive, Princeton University. .