39 percent of the votes = 54 percent of the seats = 100 percent of the power.

This line of thought has been used to criticize Canada’s democracy. These specific values pertain to the 2015 Canadian election however the underlying point applies to most of Canada’s past majorities. A minority of voters are able to elect their party to a majority of the seats. Having a majority allows the government to pass legislation through the House of Commons without any support from the opposition parties. Obviously, there is more to power than having a majority in the House of Commons. Canada has a senate and Supreme Court. Furthermore, a government cannot make constitutional changes without provincial support. But the central idea sheds a light on an interesting feature of legislative bodies and their division of power.

It has been known for a long time that a political party or bloc’s power is not proportional to its number of seats (or its votes). So what is it proportional to? How do we best describe a party’s relative ability to pass or prevent passage of legislation? There are two major power indexes that have been devised in order to answer these questions. They are the Banzhaf power index and the Shapley–Shubik power index.

The Banzhaf power index, created in 1946, measures a party’s power based on how often it can be a critical party in a winning coalition. This can be measured by first writing out all winning coalitions. For example:

Party A has 6 seats

Party B has 4 seats

Party C has 2 seat

Party D has 1 seat

7 votes are needed to pass legislation

Therefore the winning coalitions are: AB, AC, AD, ABC, ABD, ACD, ABCD, and BCD

The next step is to identify critical parties. These are parties whose removal from a coalition will cause it to be a losing coalition.

This leaves us with: AB, AC, AD, A, A, A, and BCD

This final list allows calculation of the values. The values are calculated using frequency over total:

Party A has 6/12 or ½ the power

Parties B, C, and D all have 2/12 or 1/6 of the power each

The Shapley–Shubik power index, created in 1954, measures a party’s power based on how many times it is the pivot vote. This can be found by listing all orders of the parties:

ABCD ABDC ACBD ACDB ADBC ADCB

BACD BADC BCAD BCDA BDAC BDCA

CABD CADB CBAD CBDA CDAB CDBA

DABC DACB DBAC DBCA DCAB DCBA

Once this is done, for each order look at what would happen if the parties voted yes going from left to right. Make a note of which party puts the vote over the required 7 vote threshold. For example, ABCD would have B as the pivot vote (A has 6, add B’s 4 and that brings you to 10). Once this is done:

B B C C D D

A A A D A C

A A A D A B

A A A C A B

Once again, calculate the power by using frequency divided by the total:

Party A has 12/24 or ½ the power

Parties B, C, and D have 4/24 or 1/6 the power each

The two power indexes usually arrive at the same values. There are occasions where they differ however. For example:

Party A has 3 votes

Party B has 2 votes

Party C has 2 votes

5 votes are needed to pass legislation (note: this is more than a simple majority)

In this example the values would be:

Party A: 3/5 for Banzhaf and 2/3 for Shapley–Shubik

Party B: 1/5 for Banzhaf and 1/6 for Shapley–Shubik

Party C: 1/5 for Banzhaf and 1/6 for Shapley–Shubik

In general though, the two systems usually agree on what the power distribution is. And they have been used in the past in attempts to prove the unfairness of municipality subdivisions. Specifically they highlight the fact that if seats are given to various city boroughs in blocs p[proportional to population the end result could be certain blocks having 0 power. Thus, they are essentially disenfranchised.

This can also be true for national legislatures. Small parties can potentially have zero power if there is no situation in which there vote could make a difference. Ultimately, there is no clear solution and any attempts to solve this problem would take a major reform in the way legislatures function.