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BGonline.org Forums
A fair spot for the undefeated player
Posted By: Rich Munitz In Response To: A fair spot for the undefeated player (Sam Pottle)
Date: Tuesday, 16 June 2009, at 9:45 p.m.
Of course the probability of "someone" chosen at random making the finals of the once-defeated bracket is 1 in 63.
The issue here is whether that person deserves to enter the finals on equal footing with the undefeated player (50/50), whether they should have to win 2 matches consecutively (25/75), or whether the fair settlement value of both finalists lies somewhere in between.
It has been suggested by some that the once-defeated player has a harder job to reach the finals than the undefeated player (they certainly have to play and win more matches) and that the player doing so will on average be a stronger player than the undefeated player. It has been stated that some very smart math wiz people have analyzed this format and have reached these conclusions. However, I have not seen this analysis and therefore cannot validate it. But based on these claims, it was proposed that the two finalists have been equally vetted by reaching their respective positions and should therefore face off as equals rather than one player having a 3:1 advantage.
What I have attempted to do is to quantify the difficulty of the achievements of reaching the finals through the various paths by determining the actual probabilities of obtaining exactly a 6-0, 7-1 or 8-1 record with a backgammon match simulated by a coin toss.
Conclusions? I conclude that it is indeed harder to reach the finals of the undefeated flight (to go 6-0) than to reach the finals of the once-defeated flight (to go 8-1). However, I also conclude that the undefeated player's task was more unlikely on average only by a 61:39 ratio, not the 75:25 that the original finals format would suggest, nor the 50:50 that the current format would suggest. The proponents of change, were justified in the sense that the win-once final (50%) is closer to fair value (39%) than the win-twice final (25%). Therefore, I suggest an adjustment in the current finals formula that would reward the undefeated player for their more difficult accomplishment, while retaining the single finals match which is preferable for scheduling purposes. A 2-3 point spot for the undefeated player seems about right, depending upon match length.
Where are the real mathematicians out there? Have I analyzed this correctly? Or did I make some major goof? As Matt ponders - is the assumption of "all players equally skilled" really so ridiculous so that all subsequent analysis is garbage? Or is it a reasonable simplification?
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