Tournaments

 Playing-off 3 remaining players

 From: Gregg Cattanach Date: 23 April 2007 Subject: 3 player tournament Forum: GammOnLine

```Does anyone have a fair and efficient way to play off three players? Assume
that there is supposed to be two prizes, \$200 for first and \$100 for
second.

This might come up with a tournament that started with 12 players, and
instead of byes it played down to 3 'finalists'.

Any ideas are appreciated.

Gregg C.
```

 Chuck Bower  writes: ```The old fashioned way is to offer the players a chance to buy the bye. ```

 David Montgomery  writes: ```You could have round-robins with decreasing match lengths until one of the players wins both his matches. ```

 Jordan Lampe  writes: ```In Sumo they settle a 3 wrestler playoff with a "circle-fight". The three players draw lots: A, B, and Wait. First, A and B play. The winner plays Wait. If he wins, then the tournament is over. If he loses, then Wait plays the loser of the first match. If Wait wins, the tournament is over, but if Wait loses, then A and B play each other again. And you keep going on and on until somebody wins two in a row. Some math will show you this is somewhat unfair to the person who gets picked as "Wait" (assuming all 3 players are equal strength). The way to "fix" this is to redraw the lots if the first 3 matches end with all 3 players 1-1. ```

 Chuck Bower  writes: ```Total prize pool: \$300. Expectation per player = \$300/3 = \$100. So after the bracketing (but before any matches are played), each player's expectation should remain \$100. The player who gets the bye (by random draw) thus must have an expectation of \$100 and since he's 50-50 to win, 0.5*first + 0.5*second = \$100. ==> first+second = \$200. The remaining \$100 goes to the two semifinalists. Solution 1: Winner of semifinal match receives \$100. Winner of final match receives \$200. This is still mathematically equitable. Note that the player who gets the bye but fails to win a match gets nothing (and there seems to be considerable justice in this :). Solution 2: Everyone gets \$50 before play begins; winner of semifinal match gets (additional) \$50; winner of final match gets (addditional) \$100. Note that the bye-player gets \$50 + 0.5*(\$100) so his/her expectation is \$100, as it should be. By symmetry and conservation of \$, the other two players' expectations are also \$100 each so the division is equitable. Solution 3: Everyone gets \$25 before play begins; semifinal match winner gets (additional) \$75; final match winner gets (additional) \$150. (Again you can check that each player's expectation is \$100.) ```

 Maik Stiebler  writes: ```As much as I have embraced, Chuck's solutions, they still have the problem of giving the player with the bye a much greater chance of "winning" the tournament. That might be a real concern in either small stakes/amateurish or highly prestigious tournaments. A fair way of solving this would be handicapping the player who received the bye such that his chance of winning the final is 1/3. This can be done by giving the handicapped player a suitable score disadvantage. Alternatively, a series of final matches could be played. Once the handicapped player loses, he is out, if he wins, the handicap changes to the other player. In the Nth match, the score disadvantage method could be used. The fair payout plan to go with that is to pay the same amount to both losers, and more to the winner. ```

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