Remember To Take The Upside

by Gerry Tansey IMG_4602[1]

I’m a fan of “Jeopardy!”  However, based on some of the wagering I see, I must conclude that some of the contestants don’t want to win the game.  Let me give an example.  Suppose that Alice has $20,000, Bob has $8,000, and Charlie has -$,2000 (and his fourth Bloody Mary).  Bob hits a Daily Double on the final clue of the Double Jeopardy round.  Bob bets $3,000, sealing his doom with less than half of Alice’s total if he is wrong, but only reaching $11,000 if he is right.

The point here is that a $3,000 bet makes no sense.  It’s much better for Bob to bet everything in this situation.  Bob has no chance of winning if he misses the question, no matter what he wagers, but he is in a much stronger betting position in Final Jeopardy with $16,000 than with $11,000.  Bob could win the game without even answering the Final Jeopardy clue correctly if he has $16,000, whereas he must rely on the parlay “Alice answers incorrectly, Bob answers correctly” in order to win with $11,000.

How does all of this apply to backgammon?  Well, if you take a cube that is on the take/pass borderline, you must know when to redouble when things go your way.  If you don’t maximize your upside, then it is very wrong to take the downside risk in the first place.

Here is a trivial example that came up recently.  I was up 3-0 in a match to 7, and I owned the cube as White in the following position.



If I redouble here and Black takes, I will win the match if I win this game.  So if Black decides to take, he should redouble immediately.  The downside risk for Black (loss of the match) is the same, but by redoubling to 8 he maximizes his upside when things go well, namely winning the match rather than just taking a 4-3 lead.

But should Black take this recube if it is offered?  Well if Black passes, he will have to win the match from a 5-0 deficit.  A modern match equity table reveals that Black has about a 15.8 percent chance to do this, provided he is an equal player to White.  So Black should take if he thinks he has a greater than 15.8 percent chance to win.

Black should realize that he is quite a bit worse than he would be in a pure 3-roll vs 3-roll position, where he would have slightly greater than 21% winning chances.  All of White’s doubles except 22 work quite nicely.  White’s worst number is 31, which is only really bad if White follows it up with consecutive aces.  Meanwhile Black’s 22 and 11 do not save a roll.  Basically Black will have to roll an unanswered medium-sized set of doubles to win the game.



Analyzed in Rollout No redouble Redouble/Take
Player Winning Chances: 84.03% (G:0.00% B:0.00%) 84.03% (G:0.00% B:0.00%)
Opponent Winning Chances: 15.97% (G:0.00% B:0.00%) 15.97% (G:0.00% B:0.00%)
Cubeless Equities +0.681 +1.502
Cubeful Equities
No redouble: +0.834 (-0.151) ±0.000 (+0.834..+0.834)
Redouble/Take: +0.985 ±0.000 (+0.985..+0.985)
Redouble/Pass: +1.000 (+0.015)
Best Cube action: Redouble / Take

This is a huge redouble, and it is a borderline take.  If you struggled with the take/pass decision, congratulations!  You got this problem right.  If Black feels that he is a stronger player than White, he should pass, but it is really impossible for Black to make a mistake with this position…or so I thought.  In real life I redoubled, Black took, I rolled an average number…and Black didn’t redouble!  Black ended up winning this game for 4 points, and then subsequently the match, one… point… at… a… time.

Here is a less trivial example of a failure to maximize the upside of a take.  White is leading 4-2 in a match to 11 and holding a 2-cube.

the one

Don’t be too enthralled by the close raw pip count.  The reason the pip count is close is the burial of a lot of Black’s checkers on low points.  White is on roll and needs three crossovers before he can start bearing off.  Black is on the bar against a 5-point board.  If Black comes in on his first shake, he will then need three more crossovers before he starts bearing off, assuming he avoids getting hit by a White straggler.  But Black usually needs more than one turn to come in, and White will have already started bearing off if Black fans for a while.  It is easy to imagine scenarios in which Black wins this race, but they simply do not occur often enough.  Even though Black is trailing, this is a huge redouble and a whopper-sized pass.

the one


Analyzed in Rollout No redouble Redouble/Take
Player Winning Chances: 82.24% (G:1.05% B:0.01%) 82.31% (G:1.05% B:0.02%)
Opponent Winning Chances: 17.76% (G:0.05% B:0.00%) 17.69% (G:0.06% B:0.00%)
Cubeless Equities +0.655 +1.300
Cubeful Equities
No redouble: +0.863 (-0.137) ±0.003 (+0.860..+0.866)
Redouble/Take: +1.118 (+0.118) ±0.006 (+1.113..+1.124)
Redouble/Pass: +1.000
Best Cube action: Redouble / Pass

In the actual match, Black took this redouble.  White entered both of his checkers in two turns, while Black entered with a 65 after fanning on his first turn.  Black’s third roll was 55, and soon a position similar to the following position was reached, with Black on roll.

Now this one

Analyzed in Rollout No redouble Redouble/Take
Player Winning Chances: 61.99% (G:0.00% B:0.00%) 61.98% (G:0.00% B:0.00%)
Opponent Winning Chances: 38.01% (G:0.00% B:0.00%) 38.02% (G:0.00% B:0.00%)
Cubeless Equities +0.240 +0.725
Cubeful Equities
No redouble: +0.603 (-0.123) ±0.002 (+0.601..+0.605)
Redouble/Take: +0.725 ±0.000 (+0.725..+0.725)
Redouble/Pass: +1.000 (+0.275)
Best Cube action: Redouble / Take

For money, a recube would be a double whopper, but Black simply must make White eat an 8-cube at the score.  If Black loses with the cube on 4, he will have to win the match from a 9-away/3-away score, which he only does about 16 percent of the time.  But it is a lot better for Black to win on an 8 cube than on a 4 cube, as his match winning chances will be nearly 91 percent (at 10-4 Crawford) rather than just shy of 63 percent (at 6-4).  Black is in serious danger of losing his market.  If Black rolls an unanswered set, or if White rolls a missing 5, White will have a huge pass of a recube next turn.

In the actual match, Black did not redouble.  White eventually rolled a 5 that missed.  Black still did not redouble.  Black let the position become a 3-roll vs. 3-roll position before recubing, and White quickly passed.  White went on to win the match at DMP.

Remember, Black’s take of White’s 4-cube was bad in the first place.  But XG issued its judgment of that take assuming perfect play by both sides thereafter.  Black’s misunderstanding of the subsequent cube action made that decision even worse in retrospect.  If you’re going to make a bad take, make sure you maximize the upside.

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