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
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:
.
For a general voting body, the polynomial
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
then the power of voter i is simply defined to
be the average value of 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 |
| Province | Population | Major Cities |
| Newfoundland | 568,474 | St. John's |
| Prince Edward Island | 129,765 | Charlottetown |
| Nova Scotia | 899,942 | Halifax, Dartmouth, Sydney |
| New Brunswick | 723,900 | Fredericton, Moncton, Saint John |
| Québec | 6,895,963 | Montréal, Laval, Québec, Trois Rivières |
| Ontario | 10,084,885 | Toronto, Ottawa, Hamilton, London |
| Manitoba | 1,091,942 | Winnipeg |
| Saskatchewan | 988,928 | Saskatoon, Regina |
| Alberta | 2,545,553 | Calgary, Edmonton |
| British Columbia | 3,282,061 | Vancouver, Victoria |
| Yukon Territory | 27,797 | Whitehorse |
| Northwest Territories | 57,649 | Yellowknife |
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 |
| Province | Population |
| Newfoundland | yes |
| Nova Scotia | yes |
| Prince Edward Island | yes |
| New Brunswick | yes |
| Ontario | no |
| Manitoba | no |
| Saskatchewan | yes |
| Alberta, | no |
| British Columbia | no |
| Example 2 of a vote outcome |
| Province | Population |
| Newfoundland | no |
| Nova Scotia | yes |
| Prince Edward Island | no |
| New Brunswick | yes |
| Ontario | yes |
| Manitoba | no |
| Saskatchewan | yes |
| Alberta, | no |
| British Columbia | no |
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.
