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Deferred free drop
Posted By: Nack Ballard
Date: Wednesday, 9 December 2009, at 9:34 p.m.
In Response To: DMP favorite after opening 41? (Joe Russell)
Nack: But instead of 2 away, if opp is 4 away or 6 away, etc., then I'm taking after 52S so that I might still exercise the free drop in a subsequent game.
Joe: How far do you think you can extend this strategy if the opponent is even farther away? At 8 away or 10 away, do you think it might be right to take any position that is less than .03 so that you will save the option of dropping against a stronger opening later?
Nack: That's too much to give up, unless a match is played to an extraordinary length. If you meant .003, then I think your breakeven guess of 8 away to 10 away is about right.
Let's try a crude estimate for the value of the deferred free drop at 14. Having no rollouts for 31P, 42P, 61P and 65R, I'll use the respective DMP evals of .1080, .0720, .0680 and .0490. The other DMP equities (for 53P, 54S, 21$, 62S, 63S, 64R, 41S, 32S, 43Z, 52S and 51S, rolled out) are .0288, .0143, .0137, .0134, .0123, .0065, .0043, .0032, .0013, .0002 and .0028. The average of all fifteen is .0061.
For the reply equities, a starting point would be .0061 x 1.414 (the square route of 2). One sixth of the time doublets are rolled, thereby increasing volatility, though a slight offset is that your opponent can tweak his play to reduce the value of your free drop (i.e., not slotting with opening 21). On balance, I'll guess the multiplier is increased from 1.414 to 1.6, making the average reply equity .0061 x 1.6 = .0098.
You are equally likely to go first or second, so the overall advantage of one of the players at the moment the cube is given, on average, is (.0061 + .0098) / 2 = .0080. Half the time, that will be your opponent, so your average startover equity is around .004.
To get from 14 to 12, you have to lose the game and not get gammoned (or bg'd). I'll say that happens 36.5% of the time. Multiplying that by your gain of .004 yields .0015. In other words, if my math is right and my assumptions reasonable, at 14 you should drop not only with openings 51 and 52 but also 43, and any reply scenarios where your equity is less than .0015.
Let's move on to 16. If the deferred free drop breakeven is something like .0021, that doesn't add any opening plays but it increases the number of reply scenarios. You shouldn't drop with any opening rolls other than 51, 52 and 43 until the breakeven gets to .0032 (the equity of 32S), which is probably not until the trailer is at least 10 away. It is probably wrong to drop 41S (equity of .0043) unless the trailer is at least 12 away, and 64R (.0065) isn't on the chopping block until perhaps 16 away.
If the trailer needs needs a centillion points, the leader should defer his free drop until he inevitably reaches his worst scenario, which I think is 31P54D with an equity of about .1680 (the breakeven cap). But I've never heard of a match played to that number of points. I suppose the score could get that lopsided if the TD decided the fairest penalty for fastrolling, illegal plays, overshuffling and looking ugly was to dock many points each time from the offender's score.
Nack

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