Pass the chalk.
In your limited example the numbers worked out to 33% popularity largely because of the tiny sample you used. Yes, I understand that in a larger sampling these trends tend to recur, but what is more likely to happen is this scenario:
1: A B C
2: B C A
3: C A B
4: A C B
5: B A C
6: B A C
7: B C A
8: A C B
9: B C A
10: B C A
11: B A C
12: A C B
13: A B C
14: C A B
15: A B C
16: C B A
17: C A B
18: B A C
Wherein B is the clear winner, being preferred to A and C, while A is preferred only to C. C is the clear loser, being preferred to neither A nor B.
However, the above scenario is NOT how we're running the tournament. It will not be A vs. B vs. C vs. D vs. E vs. F vs. G vs. H in the War bracket, A vs. B vs. C vs. D vs. E vs. F vs. G vs. H in the Politics bracket, and so on. Rather, it will resemble any other tournament bracket system, where A will take on H, B will take on G, C will take on F, and D will take on E. Winners will advance, so we can assume that B would face C in Round two (picking highest seeds to win) and A would take on D.
This is not a Battle Royale, but a head-to-head competition featuring seeds.
But thanks for the lesson all the same! Very enlightening!
All voting systems are flawed. Period. You missed my point and then made it for me at the same time.
The way you design the draws by seeding is simply crafting an electoral system which will produce a winner. But, there exists no such system for aggregating preferences that can't fall victim to the cycle problem. The three case example is just an example of how it can manifest. You should read Kenneth Arrow's Social Choice and Individual Values where he formalizes a proves this point using some formal logic if your interest is greater than simply tongue in cheek (read the 1963 version because there was a small error as originally published in 1951). Anyhow, Arrow proves the general case of the vote cycle. There's also some points on Condorcet efficiency that are worth reading - the probability with which a Condorcet winner will be selected if it exists and the probability that a CW actually exists.
If you're defining greatest as the one who is democratically preferred, you can't just make a system for determining winners and say, "this system is democratic." It must meet some objective criteria. Its simply one of several different systems that WILL produce a winner. There are a number of others, including dictatorship, random assignment, perverse democracy (where we choose the least preferred outcome), among many others. Will any of the winners have the positive democratic property of being the Condorcet winner? Maybe. But almost certainly not because with N>3 voters and n>3 possible outcomes, there's no such thing as a democratic outcome. Seeds or no seeds. Tournament or no tournament. The entire point of Arrow's proof (and the logical implication of Condorcet's work) was that it is impossible to aggregate individual preferences in any coherent way so that you can define a social choice. The point is that if we hypothetically designed the tournament differently with the same group of voters and the same group of outcomes, we would very possibly (and likely) get a completely different result. Ergo, you have designed a system that will produce a winner, but what does that mean? Its simply an exercise in futility. It doesn't even mean that the figure who is chosen as the winner is anything but an artifact of the way you designed the tournament. Not the candidate most preferred by those who participate in the election. Just the winner under a specific institutional setup.
Go back to my original example and trust me that it is the norm when you increase the number of voters AS WELL AS the number of outcome possibilities. The probability that any one will defeat every other one in pairwise competition is extremely small.
1: A B C
2: B C A
3: C A B
you want to impose a tournament system... so, say A is seeded 1, gets a bye and we have a pairwise competition.
B will defeat C, and then go to face A and A will win.
But imagine if you had seeded the tournament differently. Instead, I seed C as 1 and B and A go up against each other. A defeats B, and then goes up against C and C loses.
Which one is the democratic outcome, when democratic is defined as the one that satisfies the basic notion that democracy select the one preferred by most people. THAT OUTCOME DOES NOT EXIST. Therefore, the way you design the system determines the outcome. It doesn't matter how you design it to produce an outcome, that outcome will not have any positive democratic properties. It will simply be selected because the "agenda setter" (in the case of our tournament, you) designed the electoral institution instead of some other agenda setter (for instance, BrutallyHuge - whose system would simply be dictatorial - Larry Johnson, RB KC defeats all other nominees).
Now, if preferences over the 100 or so choices do end up as they do in your example ---- that there is one magical outcome that in each case a majority prefers to all the other possible outcomes, then alas, there is no cycle and we are spared this ugliness. But with as many nominations that we have, the probability of that is infinitesimal.