This means that they have equal power, even though player one has five more votes than player two. How many coalitions are there? From the last few examples, we know that if there are three players in a weighted voting system, then there are seven possible coalitions. How about when there are four players? So when there are four players, it turns out that there are 15 coalitions. When there are five players, there are 31 coalitions there are too many to list, so take my word for it.
In fact, seven is one less than , 15 is one less than , and 31 is one less than. So it appears that the number of coalitions for N players is. Find the Banzhaf power index for each player. Since there are five players, there are 31 coalitions. This is too many to write out, but if we are careful, we can just write out the winning coalitions.
No player can win alone, so we can ignore all of the coalitions with one player. Also, no two-player coalition can win either. So we can start with the three player coalitions. So player one is critical eight times, player two is critical six times, player three is critical six times, player four is critical four times, and player five is critical two times. Every player has some power. Player one has the most power with No one has veto power, since no player is in every winning coalition.
Shapely-Shubik takes a different approach to calculating the power. Instead of just looking at which players can form coalitions, Shapely-Shubik decided that all players form a coalition together, but the order that players join a coalition is important. This is called a sequential coalition. Instead of looking at a player leaving a coalition, this method examines what happens when a player joins a coalition.
If when a player joins the coalition, the coalition changes from a losing to a winning coalition, then that player is known as a pivotal player. Now we count up how many times each player is pivotal, and then divide by the number of sequential coalitions. Note, that in reality when coalitions are formed for passing a motion, not all players will join the coalition. The sequential coalition is used only to figure out the power each player possess. As an example, suppose you have the weighted voting system of.
One of the sequential coalitions is which means that P 1 joins the coalition first, followed by P 2 joining the coalition, and finally, P 3 joins the coalition.
When player one joins the coalition, the coalition is a losing coalition with only 12 votes. Thus, player two is the pivotal player for this coalition.
Another sequential coalition is. Then player three joins but the coalition is still a losing coalition with only 15 votes. Then player two joins and the coalition is now a winning coalition with 22 votes. In a corporate setting , a stockholder with more shares is able to make more of an impact on company policy and decision-making than a stockholder with fewer shares.
On a condo or HOA board , someone who owns multiple residences in the building totaling up to 10, square feet is able to influence decisions more significantly than the person with one 1, square feet apartment.
Are weighted votes "fair"? Determining the amount of influence Relative voting power A person may be allotted a set amount of votes. Calculating relative voting power By taking into account the number of votes and amount of power per person, organizations can determine a relative voting power , expressed as a percentage.
On quotas and quorum A quorum refers to the minimum number of votes needed to get a resolution passed. These two factors will determine when a quorum is reached. The model employed seeks to ensure that both new members and long-standing members have their voices heard.
Voting weights are periodically re-calculated. They are always updated within one month of any event requiring a vote of the full membership. A Voting Power VP for each institution is calculated by taking the square root of the member's total financial contributions fees to date, plus the square root of the total volumes deposited to date. Each member's VP value is divided by the sum total of all member VPs and converted to a percentage. View this table as a spreadsheet with formulas.
This means player 5 is a dummy, as we noted earlier. There will be a lot of coalitions in this system. How can you find them all if presented with a problem like this? A good strategy is to list coalitions beginning with the largest and decreasing by weight and number of players.
The advantage of this method is that by finding the largest coalitions first, you should be able to identify the winning coalitions before having to sort through very many losing coalitions. With this in mind, the winning coalitions are listed below with the critical players underlined. However, we still have to identify winning two-player coalitions.
Counting up times that each player is critical gives us the Banzhaf score and Banzhaf index for each player:. Interestingly, even though the Liberal Democrats have only one less representative than the Conservatives, and 14 more than the Scottish Green Party, their Banzhaf power index is the same as the Scottish Green Party's.
In parliamentary governments, forming coalitions is an essential part of getting results, and a party's ability to help a coalition reach quota defines its influence. The voting system tells us that the quota is 36, that Player 1 has 20 votes or equivalently, has a weight of 20 , Player 2 has 17 votes, Player 3 has 16 votes, and Player 4 has 3 votes.
The winning coalitionsthose witb at least 36 votesare listed below with their weights. The critical players in each coalition are underlined.
Banzhaf used this index to argue that the weighted voting system used in the Nassau County Board of Supervisors in New York was unfair. The county was divided up into 6 districts, each getting voting weight proportional to the population in the district, as shown below. Calculate the power index for each district. The total number of votes in this system is , so if the quota is a simple majority, we have the following weighted voting system:. This system has many winning coalitions, listed below using the abbreviations from the table above.
As usual, the critical players in each coalition are underlined. It turns out that the three smaller districts are dummies.
Any winning coalition requires two of the larger districts. The weighted voting system that Americans are most familiar with is the Electoral College system used to elect the President. Most states give all their electoral votes to the candidate that wins a majority in their state, turning the Electoral College into a weighted voting system, in which the states are the players.
As I'm sure you can imagine, there are billions of possible winning coalitions, so the power index for the Electoral College has to be computed by a computer using approximation techniques. Section 6.
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