Calculating the voting power of coalitions by using the voting polynomial

(Much of the material here is taken from Measuring Voting Power by Philip Straffin.)

In 1971 the Premiers of the ten provinces of Canada met in Victoria to negotiate an amendment procedure for the Canadian constitution. The history is interesting. The constitution of Canada was contained in the British North America Act of 1867, by which Britain granted independence to Canada. However, no procedure had been given by which Canadians could amend their constitution other than by petitioning the British Parliament to enact the amendment. This was a strange situation, and "patriation of the constitution", in other words, bringing the constitution under Canadian control, had been a patriotic issue in Canada since the 1920's. In order to patriate the constitution, the Canadian provinces had to agree on an amending procedure, and this problem was to be addressed by the Victoria Conference.

The problem is complicated by the diversity of the Canadian provinces, in size as well as in politics and culture. In 1970 the two largest provinces, Ontario and Quebec, contained 64% of the Canadian population, Any scheme which treated all provinces equally (as in the United States, where a constitutional amendment must be approved by 3/4 of the states, with large and small states treated equally) would surely be unfair to residents of these provinces.

The arnending procedure proposed by the Victoria Conference recognized provincial disparity. A constitutional amendment would have to be approved by

Notice that both Ontario and Quebec have veto power over constitutional amendments. British Columbia also seems to have considerable power. The "prairie provinces" are Alberta, Saskatchewan and Manitoba; the "Atlantic provinces" are New Brunswick, Nova Scotia, Prince Edward Island and Newfoundland. How fair is this scheme? Does the unequal power of the provinces rnirror, at least roughly their relative populations? To answer questions like these, we need to formalize and quantify the notion of the power of, a voter in a voting body. We will use elementary probability and calculus to develop such a power index. The index we will define has seen wide use in many areas of political science.

A mathematical model of a voting body strips away all personalities and ideologies, and considers only which groups of voters can pass bills (or constitutional amendments in the above case). Those subsets of voters which can pass bills axe called winning coalitions.

Perhaps the most common kind of voting situation is one in which each voter casts one vote, and a majority of votes is necessary to pass a bill. In other words, the winning coalitions are exactly those which contain more than half of the voters. However, there are voting bodies in which members cast different numbers of votes. Such a body is called a weighted voting body, and is described by the symbol [q; Wl, W2, ---, Wn]. Here there are n voters, the ith voter casts Wi votes, and a quota of q votes is needed to pass a bill. For example, the symbol [7; 4, 3, 2, 1] represents a body (1) in which there are four voters, (call them A, B, C, and D) casting 4, 3, 2, 1 votes respectively, and 7 votes are necessary to pass a bill.

Weighted voting bodies are fairly common. Most familiar to Americans is the United States Electoral College. Other classic examples include the Council of Ministers of the European Community, the World Bank, several United Nations organizations, and many county boards in New York state. A legislature in which there are representatives of several parties, who vote under strict party discipline, can also be thought of as a weighted voting body. In this interpretation, example 1 might represent a legislature of 10 members, with 4 belonging to Party A, 3 to Party B, 2 to Party C, and 1 to Party D, with the requirement that a 2/3 majority (7 votes of 10) is needed to pass a bill. The Victoria Scheme is not a weighted voting scheme. However, its rules do exactly specify the winning coalitions.

The naive way to think of the distribution of power in a weighted voting body like that of would be to suppose that power is in strict proportion to the number of votes. Thus, A has 40% of the votes and hence should have 40% of the power. A little reflection should convince you that this is not reasonable. For instance, note that in (1) A has veto power: even if B, C and D all favor a bill, it cannot pass without A's approval. This should lead us to believe that A might well have more than 40% of the voting power in this game. Two even more compelling examples are (2) [6; 7, 1, 1, 1] where the voters will be called A, B, C and D and (3) [6; 3, 3, 3, 11] where the voters will also be called A, B, C and D.

In (2) A has 70% of the votes, but she clearly has all of the power. A is a dictator, in the sense that a bill passes if and only if A votes for it. In (3) D has 10% of the votes, but no power. D's vote can never make any difference to the outcome, and D is called a durnmy. In (2) B, C and D are all dummies.

If voting power in a weighted voting body is not proportional to numbers of votes, how can we define and measure it precisely? We will start by thinking of the voting power of a voter as the probability that that voter's vote will make a difference to the outcome of a vote on a bill. In other words, voter i's power will be the probability that a bill will pass if voter i votes for it, but would fail if voter i votes against it.

To calculate this probability, we will need to remember the following properties of probability:

Our idea of voting power as the probability that a voter's vote will make a difference assures, by the first property, that a dictator will have voting power 1, a dummy will have voting power 0, and all other voters will have power between 0 and 1. To go farther, we need to make some assumptions about how voters vote. Let us suppose that each voter will vote "yes" on a bill with some probability p between 0 and 1, independently of how other voters vote. We can then use the last three properties to calculate each voter's voting power, as a function of p. Let's start with a simple example: [3; 2, 1, 1]. Recall that this means that there three voter, A, B and C, with A having 2 votes and B and C both having 1 vote each, and, 3 votes are required to pass a bill. A's vote will make a difference to the outcome of a vote by this body if either B or C, or both, vote yes. (If both B and C vote no, the bill will fail regardless of how A votes.) If all voters vote yes with probability p, the probability that this will happen is
The first term corresponds to B voting "yes" and C voting "no", the second term corresponds to B voting "no" and C voting "yes" and the third term corresponds to B and C both voting "yes". Similarly, B's vote will make a difference to the outcome if A votes yes and C votes no. (If A votes no, the bill will fail regardless of how B votes; if A and C both vote yes, the bill will pass regardless of how B votes.) Thus
By symmetry, we also have tex2html_wrap_inline17. For a general voting body, the polynomial tex2html_wrap_inline19 is called the power polynomial for voter i, and it contains interesting information about the ability of voter i to influence the outcome of a vote. However, it might be more useful to have a single number as our measure of voter i's power. One reasonable way to get such a number would be to take the average value of over all values of p between 0 and 1. Of course, this is where calculus enters the picture, since you will remember from calculus that the average value of a function f(x) on the interval [a,b] is defined to be

If voter i has power polynomial tex2html_wrap_inline19 then the power of voter i is simply defined to be the average value of tex2html_wrap_inline19 on the interval [0,1].

The voting power as just defined can be used to answer such questions as: "If Canada were to have a 100 seat senate, how many votes should each province have in this body?" Using power polynomials to calculate voting power, determine a fair division of votes in this senate. the table below gives population data for the provinces.

Table 1: Canada's Provinces

ProvincePopulationMajor Cities
Newfoundland568,474St. John's
Prince Edward Island129,765Charlottetown
Nova Scotia899,942Halifax, Dartmouth, Sydney
New Brunswick723,900Fredericton, Moncton, Saint John
Québec6,895,963Montréal, Laval, Québec, Trois Rivières
Ontario10,084,885Toronto, Ottawa, Hamilton, London
Saskatchewan988,928Saskatoon, Regina
Alberta2,545,553Calgary, Edmonton
British Columbia3,282,061Vancouver, Victoria
Yukon Territory27,797Whitehorse
Northwest Territories57,649Yellowknife

However, the examples already discussed should make it clear that it is dangerous to simply give each province the same number of votes as its percentage of the total population of Canada. Instead, the voting power should be proportional to this percentage.

As an example, let us say that the following symbol represents a proposed votig scheme for the senate:
[58; 8, 7, 3, 5, 21, 24, 5, 7, 9, 11]

where the provinces are listed in the following order: Newfoundland, Nova Scotia, Prince Edward Island, New Brunswick, Quebec, Ontario, Manitoba, Saskatchewan, Albera, British Columbia. How would one calculate the voting power of any one province? Take Quebec as an example. The first step is to calculate Quebec's power polynomial. This is done by considering all possible ways that the remaining provinces can vote "yes" or "no". As an example take one such combination

Example 1 of a vote outcome

Nova Scotiayes
Prince Edward Islandyes
New Brunswickyes
British Columbiano

With this vote the total of the "yes" votes, remembering to include Quebec's 21 votes, is 8 + 7 + 3 + 5 + 21 + 7 = 51. Since this is less than the required 58 votes, this term corresponding to this vote is not included in Quebec's power polynomial. On the other hand, in the following vote

Example 2 of a vote outcome

Nova Scotiayes
Prince Edward Islandno
New Brunswickyes
British Columbiano

the total vote is 7 + 5 + 21 + 24 + 7 = 64 which greater than the required 58 votes. Hence the term (1- p)p(1- p) p p(1- p) p(1- p) (1- p). This analysis would have to performed for the remaining 514 possible voting outcomes and, for each case in which the total vote is greater than 58, the resulting term must be added to the power polynomial. After all 516 possibilities have been considered, the result is Quebec's power polynomial. Obviously, this tedious work should not be done by hand but, Maple should be put to work instead. Maple's for ... from ... to ... do looping ability is one way to handle this. One such loop should be used for each province; so there would be 9 nested loops in all. At the centre of the loop the if .. fi command should be used to determine whether or not to add a term to the polynomial obtained so far. Once the power polynomial has be calculated, it is an easy matter to have Maple evaluate the required definite integral.

Once you have set up a way of doing this (you might find it useful to write a procedure to handle the work for you) you can evaluate different possible voting schemes for the senate and compare the voting power of the provinces with the actual population of the province. The question of what constitutes a "good fit" is another matter. Your conclusion should have some comment on this.


Juris Steprans
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Department of Mathematics and Statistics.
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