[Guest]

so this problem came up in a tournment setting, a couple players had upset victories, where their opponent just barely managed to snatch a victory from the jaws of defeat.

it was a win, but what a way to win! so, let's turn this into a puzzle:

you have a set number players (for the sake of ease let's say 8) each of a different rank.

the higher ranked players should generally win against the lower ranked ones. players of close rank it can go either way. never the less, you want to give the lower ranks a chance to win the tournament. you have up to n-1 rounds. (in this case 7.) what's the best way to ensure a fair tournament for everyone?

(i realize this has been studied to some detail. but i want your own work.)

[Guest]

Define fair.

How are the higher rankings determined? What sport/game are we talking about here? What defines a round? Seems you define it by a match (n players, n-1 rounds). Are we limiting ourselves to single elimination? What about double elimination? Round Robin?

[Molly Mae]

Agreed that the type of tournament is important to the solution, but I'll go out on a limb and try to define "fairness"

List all players from lowest rank to highest rank 1 through 8. Any player's rank is n. Assume all players have a unique rank and that the difference between any two consecutive ranks is constant. Absolute fairness might be achieved if n_x = (n_O)/r where n_O is the total rank of all opponents and r is number of rounds played.

Such a situation is not possible in this game (all games played by 1 and 8 bring them further from their average), but it's the best way I can assume fairness.

EDIT: Revised wording.

Edited September 28, 2011 by Molly Mae

[Guest]

define fair

fair

1. free from bias, dishonesty, or injustice: a fair decision; a fair judge.

2. legitimately sought, pursued, done, given, etc.; proper under the rules: a fair fight.

What sport/game are we talking about here?

for the sake of agrument, let's say it's baduk or go. (ranks are exponentially more difficult to obtain.)

How are the higher rankings determined?

ranks are determined before hand, assume each player has played in several tournaments before, and his rank was calculated from that.

What defines a round?

a match where every player participates, and plays agianst one other opponent. (or gets a bye.)

Are we limiting ourselves to single elimination? What about double elimination? Round Robin?

these are all possible, my question is which tournament type is best?

[Guest]

Again, which is best or which is most fair?

And the particulars matter. For example bowling does a gauntlet type tournament. Bowlers earn rankings via scores during the week, then 7 plays 8 with the winner playing 6, then 5, 4, 3, 2, 1. I like this style and would argue it's fair.

Btw I want to know what you consider fair for the sake of this question. Fair can still be determined several ways. I'm going to assume you mean each player has an equal chance of winning? I could also extend that to mean not taking ability into account. In Texas for example, home poker games are considered legal so long as each player has a fair opportunity to win. Ie no rakes, no other type of "house" advantage. But I guarantee you, not everyone in my home games has at least an equal chance of winning as I. Nonetheless, it is a fair game.

[Molly Mae]

This might sound a bit skewed, but number all players 1 to 8 from lowest ranked to highest ranked. Starting with number 1, let him choose his opponent. Continue down the list until 7. When you come to a player who has already been chosen by someone, skip him. The lower ranked player should always play black, as well.

This strays from my previous attempt to mathematically define fair, but if this were a real tournament (as it sounds) and I were hosting it, I would probably use this method for chess or go assuming that all players are ranked by the same organisation and that all rankings are current.

Edited September 28, 2011 by Molly Mae

[Guest]

for the sake of this question, i consider fair to be where an upset win; ie a match you probably should have won but didn't, doesn't neccarily mean you lose the tournament, and the chance of an upset win is unlikely. for example double elimination would probably be fair under this restriction, as would round robin. i'm more interested not in the tournament type however, but in the pairings. round robin would be the easiest to pair, everyone simply plays everyone, and players rotate upon thier match completing, no matter the outcome.

with double elimation, pairings are a bit tougher, but the advantage is fewer rounds.

we are assuming everyone plays by the rules, and the rules themselves are fair. but the tournement itself can have an implicit bias.

for example. in go it is common for tournments to be organised thusly. the top tier players play agianst eachother, the middle tier players play agianst eachother, and the bottom tier palyers play against eachother. for each game you win you play a tougher opponent, if you lose, you play weaker. the player with the most wins, or if there is a tie, then the player who's opponents he defeated won the most; wins.

its diffcult to put it into words.

let me do it with an example:

lets say both a low tier player, and high tier player, won 4 games. the four players the low tier defeated each won 2 games, where the four players the higher teir player defeated won 3, 3, 2, 2. the higher teir player would win the tournment.

now, this seems fair to me in most cases, but a crafty player could perhaps execute an upset victory by intentionally losing say his third game out of five if he won the first 2, then get an easy win for his fourth game and work hard to win the last. is there a way to prevent this?

[Molly Mae]

is there a way to prevent this?

Only admit selfish and proud go players into the tournament.

[Guest]

Without time constraints, the round robin would be fairest. With round robin, you must also consider the possibility of a tie and how to handle that situation. Second fairest is double direct elimination which helps protect a team from a fluke loss, especially in typically low scoring games like soccer. Double direct elimination would require half the number of games and would be almost as fair. The direct elimination which requires half the games of double direct would be least fair. I knew a very good high school basketball coach that would sometimes play second string in a game that the first string could win easily for the purpose of changing his rank for a better seed in the tournament. We also have the possibility of an injury during the season putting out better players that are in the tournaments which would also offset the rankings. I have been to early season soccer tournaments where rankings were determined by the previous season's team ranking even though half the players could be different. The usual seeding for direct elimination is 1-8; 2-7; 3-6; 4-5; which gives the higher ranks a better chance of staying in the tournament but also means that a bottom rank team must play teams with better rankings and reduces their chances of staying in the tournament. If you change the pairing of 1-2; 3-4; 5-6; 7-8; which is more fair for the lower teams but sets up to illiminate the second ranked team on the first round. If time limitations requires direct elimination, there is no way to be equally fair to everyone and the 1-8; 2-7; 3-6; 4-5; seeding would be the most fair since ranking is the best estimate we have of abilities. Another choice with 8 teams is have two groups of four do round robin games and move top two from these to have a 4 team final round robin. This would require 18 games increasing fairness better than double direct elimination (14 games) though not as fair as full round robin (28 games) significantly reduces games with little loss of fairness.