Voting means: There are several options available, and each voter ranks the options in a (linear) list. Ties between options are allowed, but omitted here for simplicity. Then we need a voting scheme producing a winner. Some schemes even produce a complete ranking of the options. We will see that outcomes of elections are dependent on the voting scheme (procedure), and that there is no perfect scheme.
Note that even that requirement of every voter having a ranking may not be realistic. Individual rankings could be intransitive. They could also change during the different steps of the election, which is also something we cannot model.
Two obvious requirements for a democratic voting scheme would be:
Universality: the voting scheme should be deterministic (no randomness),
and all rankings of the voters are accepted by the scheme.
Non-imposition or citizen sovereignty: Every ranking of the alternatives should be possible as outcome.
An option that is number one preference for the majority of voters, is called the majority solution.
Obviously, in most elections we don't have a majority solution, but if we have one, we have only one. Common sense dictates:
(1) Majority criterion: If there is a majority solution (being number one preference for the majority (more than n/2) of voters), then that option should be the winner.
An option that would beat every other option in a head-to-head two-option plurality vote (where every voter just has one vote and the option with most votes wins) is called a Condorcet solution. Every majority solution is also a Condorcet solution, but not converely. Still, there are many situations without a Condorcet solutions. These situations are called "Condorcet Paradox", although there is not really anything paradox in it. It just states that the relation "... beats ... in a head-to-head plurality vote" is not transitive. Compare this situation with the "balanced spinner" situation, where there is also no best spinner. An example of a Condorcet paradox situation is when two voters have preference list A, B, C, two voters have C, A, B, and two voters have B, C, A. Then A beats B, B beats C, and C beats A. (By the way, the above is a Condorcet paradox even if two, two, and three voters have the above mention preference lists). If there is a Condorcet solution, again there is only one.
(2) Condorcet criterion: If there is a Condorcet solution, then that option should be the winner.
Solution and paradox are named after the Marquis de Condorcet (1743-1794)
Every voter has one vote and the option with the most votes wins. We can even get a ranking according to the number of votes. If there is a majority solution, the plurality method declares it the winner, so the method obeys the majority criterion. It doesn't obey the Condorcet criterion, as can be seen by the example of 3 voters having a ranking of A,B,C, 2 voters having a ranking of B, C, A, and 2 voters having a ranking of C, B, A. A wins with the plurality method, but the Condorcet solution is B.
Assume there are three options A, B, and C. There are 6 possible rankings of these options. Fill in the numbers representing how many voters would have the corresponding ranking, and see what the different voting methods produce;
The most obvious disadvantage of the plurality method is that it disregards the voter's preferences except their first preferences. This is tried to cure with the Borda count method. If there are n alternatives, for each voter the first alternative receives n points, second receives n-1, and so on. This is done for every voter and the numbers are added. The result ranking reflects the number of points given. Jean-Charles de Borda (1733-1799) proposed this system in 1770.
As the plurality method, Borda count also violates the Condorcet principle--- it even violates the weaker majority criterion: Assume 16 voters have preferences A, B, C, 8 voters have preferences B, C, A, and 7 voters have preferences C, B, A. B wins the Borda count with a score of 70, although the majority did vote for A.
A method that tries to achieve the majority for some option in rounds is the Hare method, proposed by Thomas Hare (1806-1891) in 1861. In each round each voter has one vote and the votes are added for the options like in the plurality method. In the first round, if one option receives the majority of first places, then it is elected, otherwise the option with the fewest first places is eliminated, and the process is repeated until we get a majority. (But of course, this is not a majority with all initial options considered.) This method is used for electing the Australian House of Representatives and also for electing the President of Ireland, for instance.
As an example, assume 120 voters have preference ranking D, A, B, C, 100 have A, C, B, D, 90 have B, C, A, D, 80 have C, B, D, A, and 45 have C, B, A, D. There is no majority solution, but the Condercet solution would be A, since it wins against B, C, D in two-option votes. However, the winner of the plurality vote is C, C also wins the Borda count, and C also wins the Hare method.
Before continuing with four more voting schemes, we discussed the 6th grade class example (or in a more serious context, a society that is divided somehow by race, religion, etc into two parts). The assumption is that every member of one part will prefer every member of his/her own part over all members of the other part. Let's stick with the 6th grade class example and call the two parts the boys and girls, and let's assume that we have slightly more girls than boys, let's say 15 boys and 16 girls. The boys only vote for the boys and the girls only for the girls. The candidates consist of 1 boy and 2 girls. In a plurality vote the boy is most likely to win, since the girls divide their votes over two candidates, whereas in a Borda Count the most popular girl is the winner. We will come back to this type of examples in the following voting schemes discussed next.
In each round each voter has one vote against who they think is the worst option. This is repeated until only one option---the winner---remains. This method was introduced by Clyde Coombs (1912-1988). We took the same boys/girls example with 2 boys and 3 girls as candidates and applied the Inverse Hare to the situation. Let's assume that Beth gets all girls' first votes but all boys' last votes. In this case Beth is eliminated in the first round (in the very likely situation where the other two girls both get at least two third place votes from the girls). Yet Beth is the majority solution. Therefore the Inverse Hare method does not obey Condorcet and Majority criteria. Secondly, if one boy doesn't run at all, a girl is elected. Throwing out one option, even one that woudn't have won, shuffles the result ranking. This situation ideally we should have in a good voting scheme, so let us formulate it as our third principle:
(3) Independence of irrelevant alternatives: If one alternative is removed, then the voting system should still create the same ordering of the remaining alternatives.
Unfortunately, Plurality method, Borda count, and Hare method all violate this principle.
Next we talked about Runoff. The first round uses plurality method, and if itís a majority, we take it, but otherwise we make a second round and throw out all but the leading two. These remaining two options face each other in a runoff election. Runoff doesn't produce a ranking of the options, just a winner.
Some problems of this method can be explained with our boys/girls example, where the candidates are 2 boys and 3 girld. If the first votes for the two boys are 7 and 8 (from the boys) and for the girls 6. 5. 5 for the girls (from the girls), then the runoff round will be between the two boys and one boy will win. If the stronger boy, however, overdoes campaigning and attracts even more votes and gets 11 votes from the boys, leaving 4 for the other boy, then he competes against a girl in the runoff and loses. This looks again paradox: You can get a worse result by gaining sympathies. Let's formulate this as another principle a good voting method should have:
(4) Monotonicity: ("better is better") If one voter promotes on alternative, then the ranking of that alternative in the result should at least not drop.
Plurality and Hare method also violate this principle, but Borda count obviously obeys it. Higher ranking of an option for some voter means that this option gets more points.
All options fight against each other one-to-one with plurality method and only these two options. The winner gets one point. The option with the most points wins. Obviously this method obeys the Condorcet principle, and therefore also the Majority principle.
Each voter gives one point to as many options he/she approves. Then the votes are counted and ordered accordingly. This method is different to the six discussed above, since it is the only one where the voter's preferences alone do not determine the result ranking---each voter also makes a decision of how many of his/her preferences to grant a point.
Here is an interesting link on voting and elections.
Please apply different voting procedures to the following example: There are 5 options, called A, B, C, D, and E:
Thus in close cases, different voting methods may obtain different winners---in our example, all 5 options are obtained.