[This is an article that I thought was funny, in a math-nerd way.]
My wife, Kathleen Davis Barr, and I used to daydream a lot about playing together as partners
in a Doubles tournament and winning. We tried several times over the years. We entered some
Limited Doubles events for Intermediate/Advanced players and made it to the finals once, but
lost. In Open Doubles events, I don’t think we ever even got as far as the semifinals.
After some first round losses at a couple tournaments in a row, we put our Doubles dream on hold
until we improved.
In the meantime, we also had a chance to restrategize. Maybe it’s better if we don’t put all our
eggs in one basket in the Doubles tournaments. Maybe entering separately with other partners
is a better idea. Assuming that we split the entry fees with our partners, we would then each be
only paying half an entry fee. Of course, that’s the same amount as paying a full entry fee if we
teamed together, but separately we have two chances to win! As a married couple that shares
all their finances, it doesn’t matter which one of us wins because we’ll share the prize money.
(And when I say it doesn’t matter which one of us wins, I mean that strictly financially).
However, we also would only win half as much prize money compared to if we played together
since whoever wins would have to split the winnings with their partner. So really this is a
lower-risk, lower-reward strategy.
In this scenario, our daydream Doubles tournament would be if our teams played against each
other in the Doubles final. Not only would it be fun, but then we would both be winning prize
money! After thinking about this some more, it’s not hard to realize that if a spouse plays
another spouse in a Doubles final, and those spouses combine their winnings, then no matter
what the prize distribution percentage is between 1st and 2nd place, the two spouses will go
home with exactly half of the total prize money available to 1st and 2nd place. One way to think
about that is if we played each other one-on-one in a normal final, then we’d definitely take
home 100% of 1st and 2nd place money, but in a doubles final we would each share with our
partners, so we would take home 50% of the total pot. That’s also assuming that each pair of
doubles partners is splitting their winnings equally.
This “Spouse vs. Spouse Payout Theorem” turns out to be easy to prove mathematically and
can be generalized for any number of players per team, from 1 to n. Assume:
P = total prize pool for 1st and 2nd place, and
x = amount paid to 1st place, and
(P - x) = amount paid to 2nd place.
n = number of players on each team
Again, assuming that each team is splitting winnings equally, then each spouse gets 1/n of the
prize money that the team wins. So the spouse on the winning team wins (1/n) * x dollars and the
other spouse on the losing team wins (1/n) * (P - x) dollars. When the spouses combine those
winnings, together they have:
(1/n)x + (1/n)(P - x)
= (1/n)x + (1/n)P - (1/n)x
= (1/n)P
So in a Doubles tournament, the couple always takes home ½ of P.
There are a couple interesting corollaries to this proof too.
1) There is no point in either spouse being involved in any hedging decisions because they
will win the same amount regardless of the distribution percentages. Whether the split is
60/40, 80/20, etc., it won’t matter. That means any hedging decisions can be left entirely
up to the other partners.
2) In Doubles, if each spouse chooses for their partner one spouse of another married
couple (e.g, Barr/Rockwell vs. Barr/Rockwell) then no matter what the prize distribution,
each couple will win exactly half the pot.
So if David Rockwell and I are ever playing against Linda Rockwell and Kathleen in a Doubles
final, we’ll have to play for something else since the money won’t matter, like the losing team
buys the winning team dinner. Oh wait...never mind
Monday, July 24, 2017
Round Robin scenarios for 3 players or 4 players (or teams)
Normally, when we have a tournament, we award the winner with club points equal to the number of players who attend. The next runner(s) up get half those points, the next gets half again, etc. Any player with a win gets some points and players with no wins get 0 points. This works well when we use a double elimination bracket, which we do when we have 5 or more players.
Its common though towards the end of the year that we have less players, so sometimes we have a few nights of 3 or 4 players. On those nights, we do a round robin style tournament where each person plays each person once. The end result can have several ties and so its less clear how to award points. So finally, I documented each scenario and how many points we will award. Of course this can be applied to round robins in any sport with 3 or 4 teams, or other games, and the points can be multiplied or divided however desired.
The goal is to try to award close to the same number of total points in each case that has the same number of players. In a 3 player round robin with no ties, the points are 3 for winner, 1.5 for second, 0 for third because that person has no wins. That's 4.5 total points. So when 3 players have a 3-way tie, each gets 1.5 points so that the total is again 4.5. That works out perfectly but 4 players is a little trickier.
In a 4 player round robin with no ties, winner gets 4, second gets 2, third gets 1, fourth gets 0 (no wins). That's 7 points total. If we were to use a double elimination bracket for 4 players, which we do sometimes if we want to finish faster, then we end with a tie for 2nd. We don't make the winner of the consolation bracket play the loser of the main bracket, for the sake of time. In that case, we award 4 for winner, 2 each to the two players with 1 loss each, and 0 to the person with no wins. That's a total of 8 points. So the goal for the 4 player round robins where there are ties is to award either 7 points total or 8 points total, or somewhere in between.
Its common though towards the end of the year that we have less players, so sometimes we have a few nights of 3 or 4 players. On those nights, we do a round robin style tournament where each person plays each person once. The end result can have several ties and so its less clear how to award points. So finally, I documented each scenario and how many points we will award. Of course this can be applied to round robins in any sport with 3 or 4 teams, or other games, and the points can be multiplied or divided however desired.
The goal is to try to award close to the same number of total points in each case that has the same number of players. In a 3 player round robin with no ties, the points are 3 for winner, 1.5 for second, 0 for third because that person has no wins. That's 4.5 total points. So when 3 players have a 3-way tie, each gets 1.5 points so that the total is again 4.5. That works out perfectly but 4 players is a little trickier.
In a 4 player round robin with no ties, winner gets 4, second gets 2, third gets 1, fourth gets 0 (no wins). That's 7 points total. If we were to use a double elimination bracket for 4 players, which we do sometimes if we want to finish faster, then we end with a tie for 2nd. We don't make the winner of the consolation bracket play the loser of the main bracket, for the sake of time. In that case, we award 4 for winner, 2 each to the two players with 1 loss each, and 0 to the person with no wins. That's a total of 8 points. So the goal for the 4 player round robins where there are ties is to award either 7 points total or 8 points total, or somewhere in between.
3 PLAYER POSSIBLE SCENARIOS
--------------------
2-0, 1-1, 0-2 : 3 points to 1st, 1.5 points to 2nd, 0 points to 3rd (4.5 total)
1-1, 1-1, 1-1 : 1.5 points each (4.5 total)
4 PLAYER POSSIBLE SCENARIOS
--------------------
3-0, 2-1, 1-2, 0-3: 4 points, 2 points, 1 point, 0 points (7 total)
3-0, 1-2, 1-2, 1-2: There are no head-to-head tiebreakers, so 4 points to winner, 1 point for each 1-2 player (7 total)
2-1, 2-1, 2-1, 0-3: There are no head-to-head tiebreakers, so 2.5 points for each 2-1 player (7.5 total)
2-1, 2-1, 1-2, 1-2: For each of the ties, there will be a head-to-head tiebreaker, so clear 1st through 4th placings can be determined. 4 points, 2 points, 1 point, 0.5 points (since 4th place had a win) (7.5 total)Bracket Efficiency Recommendations for Casual Tournaments
Our club has our regular meetings on weeknights. We like to play a double elimination bracket of 7 point matches in the main round and 5 in consolation whenever possible, but those can take several hours even with 4 to 8 people. Although we're all competitive and want the draw for the bracket to be fair and random, we also want to get home at a reasonable hour on a weeknight. Therefore, since weeknight tournaments are more casual and laid-back than a weekend tournament (and especially than an ABT tournament) we developed a couple modifications for our brackets to speed things up.
First, we play "first available" in the consolation round. In other words, we don't use the "Loser goes to A", etc. designations of the bracket. If you lose in the first round, you're placed in the highest open spot of the consolation bracket. The big advantage of this is that as soon as two people have lost in the main round, they can start playing immediately. Using normal placement, the two first losers could be on opposite sides of the bracket and end up each sitting around waiting for the next loser on their side of the bracket. another advantage is that in cases of uneven players, like 7, the last person to lose will get a bye in the consolation round. So the slowest part of the bracket gets a little speed boost at that point. With "first available", there is much less sitting and waiting. The only disadvantage is that two people can end up playing each other a second time in the consolation round even before getting to the consolation final. That's not a big deal to us since most of us in the club have played each other countless times anyway. And even using the normal placements, people can meet up a second time in the consolation finals.
One other trick we use is when we have 6 players. We use an 8 player bracket, but instead of using the "seed" placement numbers on the main bracket, we just place players in first top 6 slots of the bracket. That way all 6 players start playing immediately instead of 2 people on opposite sides of the bracket both getting byes and waiting. The side-effect is that for the players in the 5th and 6th spots, whoever wins will get a bye to the final. But that's really no extra advantage over normal placement. In both cases, there are 2 players who only need one win to get to finals. But with the modification, you play first and get a bye when you win. With regular placements, you get the bye first. the same concept can be applied to 12 players in a 16 player bracket too, just place them all in the first 12 spots so that you don't have FOUR people with byes sitting around waiting.
A few other ideas for 9, 10, and 11 players in a 16 player bracket:
9 players: Losers of starting 4 matches go to first available consolation spots A, B, G, and H. Semi-final losers go to first available consolation spot of M or N. C, D, E, F, I, J, and K are byes
10 players: Losers of starting 4 matches go to first available consolation spots A, B, G, and H. Semi-final losers go to first available consolation spot of M or N. C, D, E, F, J and K are byes
11 players: Losers of starting 4 matches go to first available consolation spots A, B, G, and H. Semi-final losers go to first available consolation spot of M or N. C, D, E, F, and K are byes
First, we play "first available" in the consolation round. In other words, we don't use the "Loser goes to A", etc. designations of the bracket. If you lose in the first round, you're placed in the highest open spot of the consolation bracket. The big advantage of this is that as soon as two people have lost in the main round, they can start playing immediately. Using normal placement, the two first losers could be on opposite sides of the bracket and end up each sitting around waiting for the next loser on their side of the bracket. another advantage is that in cases of uneven players, like 7, the last person to lose will get a bye in the consolation round. So the slowest part of the bracket gets a little speed boost at that point. With "first available", there is much less sitting and waiting. The only disadvantage is that two people can end up playing each other a second time in the consolation round even before getting to the consolation final. That's not a big deal to us since most of us in the club have played each other countless times anyway. And even using the normal placements, people can meet up a second time in the consolation finals.
One other trick we use is when we have 6 players. We use an 8 player bracket, but instead of using the "seed" placement numbers on the main bracket, we just place players in first top 6 slots of the bracket. That way all 6 players start playing immediately instead of 2 people on opposite sides of the bracket both getting byes and waiting. The side-effect is that for the players in the 5th and 6th spots, whoever wins will get a bye to the final. But that's really no extra advantage over normal placement. In both cases, there are 2 players who only need one win to get to finals. But with the modification, you play first and get a bye when you win. With regular placements, you get the bye first. the same concept can be applied to 12 players in a 16 player bracket too, just place them all in the first 12 spots so that you don't have FOUR people with byes sitting around waiting.
A few other ideas for 9, 10, and 11 players in a 16 player bracket:
10 players: Losers of starting 4 matches go to first available consolation spots A, B, G, and H. Semi-final losers go to first available consolation spot of M or N. C, D, E, F, J and K are byes
11 players: Losers of starting 4 matches go to first available consolation spots A, B, G, and H. Semi-final losers go to first available consolation spot of M or N. C, D, E, F, and K are byes
Notes on Backgammon
Hello! My name is Eric Barr. I am the director of the Kansas City Backgammon club. I've been the director (with lots of help) since 2004. Over the years, I've written several things about backgammon that I thought I would compile into one place. None of them are any strategies that I've made up, I'm not that good. So far, they're mostly articles about directing, with a few other topics here and there. I'll post the old ones in more or less chronological order before I start writing new posts.
Thanks for reading!
Thanks for reading!
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