The "Condorcet winner" of an election is the candidate that is preferred by a majority of the voters over any other single candidate. The basic idea is that each voter has some order of preference of the available candidates, and you can tell which candidate they prefer out of any pair by comparing their ranks in that list.
A well-known problem is that, if there are more than two candidates and more than four voters, there may not be a Condorcet winner in a particular election. The case with three candidates ABC and five voters is as follows: ABC, ABC, CAB, CAB, BCA. A is preferred to B by four out of five; C is preferred to A by three out of five; and B is preferred to C by three out of five. Such examples demonstrate that the idea of a Condorcet winner does not solve the problem of collective choice; it's an interesting question how probable the existence of a Condorcet winner is.
So here's a little experimental environment. You can set the number of voters and the number of candidates and randomly generate the preferences for each voter, and see the Condorcet results....
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A Condorcet winner will appear in the matrix of pairwise matches with their entire row and their entire column a single solid color: their color.
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In first-past-the-post (used, for instance, in the US presidential election and most other US elections) each voter votes for exactly one candidate, and the candidate with the most votes wins. If we assume "sincere voting" --- each voter votes for their first choice --- we can deduce the FPTP results from the preference orders displayed above; the winner is not yet computed. Obviously this is a terrible system, and every other voting system proposed so far has been proposed because its proponents think it is better.
In the Borda count, your last choice gets 1 point; your next-to-last choice gets 2 points; and so forth, until your first choice gets as many points as there are candidates. The Borda count winner of this election is not yet computed.
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