Back to "The Zare Formula" written by Douglas Zare

Regrets

by Douglas Zare
25 January 2003

Douglas Zare

"Think deeply of simple things."
- A. E. Ross

Even for experts, it is worthwhile to return periodically to the fundamentals. Just as musicians practice making a single note beautiful, backgammon players should benefit from asking, "What is the purpose of doubling?" Rather than looking for the right answer, I favour trying to understand this type of fundamental question in many ways

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Subject: Re: Regrets

From: DBrthrtn Member has a photo available

Date: 25 Jan 2003 12:33 EST

Over the board, I'm sure that I would not double the first position. I'd reason that it's generally bad to give your opponent a decent chance to win the match exactly on a big cube (i.e. no overage on the 4-cube), so the potential regret is large. Also, you will still be trailing in the match even if you win on a 4-cube, so your potential gain is small. So, I'd wait for a better opportunity.

The second position is just the opposite, I'm sure I would double here. Your regret is minimal since a loss on a 2- cube gets your opponent to the Crawford game anyway. Also, your potential gain is enormous since you can still gammon your opponent on the 4-cube by hitting everything and getting some "co-operation". (Maybe it's even a drop?!? So I guess Woolsey's rule says it must be a double!).

The third one seems in-between to me, both the potential risk and reward are sky-high! But you already said that the wrong decision is a huge blunder so it must require going though the math to get it right. White can drop and retain a 4-2/7 lead, I'd call that about 64% match equity. What's White's take point? Depends on the assumptions you make about gammons. To make the mental math simpler, I'd probably ignore my own gammon losses, and lower the result just a percent or two at the end to "compensate". I'd make a WAG and say that, when I win after a take, I win a gammon about 1 game in 3 (I think I remember that about 40% of your wins are gammons when you close out two, and I don't always close out two but I will most of the time, and 1 in 3 is easy to work with). So, when I win, White has 50% 2/3 of the time, and 0% 1/3 of the time, so call White's equity 33% when I win after a take. When I lose after a take, White has 100%. So White is risking 31% (64%-33%) to swing 67% (100%-33%), I'd call that about a 45% take point (I said I'd lower the result a bit at the end, and 45% is a round number). I'd say White has plenty, I miss 70% and although that's not quite gin for him it's hard to see him losing the other 25% somewhere down the road. The take seems too easy, so I guess I'd hold on to the cube and hope for a better opportunity later in the match. Also, if I hold on to the cube and win, then 1/3 of the time I'll be leading the match 6-5 (assuming 1/3 gammons if I win again). So while I certainly regret not doubling my wins, maybe it's not by as much as I intially thought.

I'm looking forward to seeing what the right answers are; I'm sure I'll learn something. Great column.

Subject: Re: Regrets

From: DBrthrtn Member has a photo available

Date: 25 Jan 2003 22:00 EST

Oops! In the third position my gammon losses are meaningless once the cube is on 4, doh! I don't think it changes anything though, I was pretty much ignoring them anyway (I just should have been *totally* ignoring them).

Subject: RE: Regrets

From: eskimo Member has a photo available

Date: 27 Jan 2003 05:42 EST

[Repost. Weird, this didn't appear even when the posting looked ok. Are these forum threads moderated?]

-------------------

Hey, I understood this one :-)! I'm usually not enough of a mathematical genius to get Zare's finer points. I do try though :-).

Pos 1: ND/T. Too easy for White with several high rolls solving all his problems for the match. Losing means 5-4 which is not that bad. Guess this is one of those lemon turkeys.

Pos 2:D/uh oh, T. Red can get gammoned in which case he loses anyway. Crawford is not a lot better so it's pretty much hit or be bust. In which case I rather get 4 points and the gammon possibility is nice enough too. I would even get a head-ache as White deciding about the take here. Wonder how often White loses the match? 20%? Maybe it depends a bit on my opponent but against a lower rated opponent I pretty probably would drop. Against a better opponent it could be that all eggs in one basket is a good idea. Maybe :-).

Pos 3:ND/T. As Red I do have 2 points so trailing 2away, 7away isn't a total disaster. Missing now pretty much means the match. For White the score should get to 5-6 if things go wrong. Not that bad considering it's the match for him. Ok, there's that gammon possibility. Red will not always pick up the second blot and saving the gammon with two men on the bar isn't that bad. I'd be weighing a ~70% match win against a ~30% gammon match loss?

Great positions (or at leats the middle one :-). Could only calc lemons for the first pos though (and did that after my initial judgement).

Eskimo

Subject: Re: Regrets

From: plasticmillion

Date: 27 Jan 2003 12:46 EST

Trying to take a market loser vs. lemon approach:

1) Red's market losers just aren't big enough to justify doubling here, even as an underdog. Let's take a great roll by Red (say 44) followed by a mediocre roll by White (say 51), and then take the Thorpe count:

Red: 8 (pips) + 14 (checkers) + 6 (ace point) - 2 (points) = 26 White: 26 (pips) + 12 (checkers) - 5 (points) = 33

Obviously Red has lost his market but not by that much and is only cashing 10-15% of the time. Contrast this with the number of lemons (any average roll by Red and White's 20 rolls with 5 or 6, 44, 33 and 22 at least), and it is clear that Red should hold onto the cube.

2) Perfect underdog double. White is a big favorite with many gammons, but Red loses his market by a mile if he hits here (unless White responds very well) and will likely be too good to double. The lemons are pretty insignificant since Red will regret doubling only in 25- 35% of games where White wins a single game, and not by much since who wants to come back from 6-0/7 anyway? Better to kill White's gammons and make it to the cocktail bar earlier if you lose.

3) Unlike position 2 Red is very unlikely to be gammoned here. So his double is a lemon in almost all his losses, since 7-2/9 is not such a crushing deficit. Obviously his market losers are big, but the match situation is not decisive, as the difference between winning with the 2 and 4 cubes is only 4-5/9 vs. 6-5/9. In match equity terms (using Janowski's formula in my head here, so these are approximations!):

Hits 30% of the time: - With 2 cube MWC is 43% - With 4 cube MWC is 59%

Misses 70% of the time: - With 2 cube MWC is 17% - With 4 cube MWC is 0%

So Red gives up 17 x 70 = 13% MWC to gain 16 x 30 = 4.8% MWC. He should hold the cube.

It goes with out saying that White is a favorite in all these positions and shouldn't even consider passing!

Matt

Subject: Re: Regrets

From: Solucky

Date: 30 Jan 2003 07:41 EST

Nice article, but i miss sometimes a article how to decide if my opponent is much stronger or much weaker than me. Snowie expect "perfect play" from both sides after the cube. At sample if i am 60% favourite in a single game, the chance to win 2 in a row is 36% not 25 %. If i am 40% underdog only 16% to win 2 in a row. This at sample for position 1 against Magriel, Roberty, Meyburg... I MUST REDOUBLE. Against a absolute nobody I dont:-)

Wolfgang

Subject: Re: Regrets

From: zare

Date: 30 Jan 2003 22:15 EST

Thanks to all who posted comments. Keep them coming! I'll post solutions analyzed by the lemons vs. market losers method in a few days.

It is certainly worth considering if there is a huge disparity in playing strength, but I think it is vital to know how much equity you are giving up in theory. Otherwise, it can be hard to determine if the equity invested is worth it.

When opponents tell me that they doubled because I'm a stronger player, I sometimes think to myself, "And the reason I'm stronger is that I don't double like that."

That said, when considering erring intentionally because the skill is uneven, it is very important to consider how much play is left in the position. If you are a 60-40 favorite at DMP, then you might pass in a bearoff with 35% chances trailing 2-away 3-away. If your opponent erroneously doubles in the opening or in a blitz and you have 30% chances, you should take and recube, since you should expect to win more than 36% by outplaying your opponent.

If you have a good feel for how much of an advantage you have at various match scores and in different positions, you can still estimate the average magnitude of market losers and the average magnitude of lemons. It can be tough to do this because you have to factor in possible erroneous cube decisions by your opponent.

Douglas Zare

Subject: Re: Regrets

From: plasticmillion

Date: 31 Jan 2003 05:11 EST

One small clarification. My statement that "White is a favorite and therefore shouldn't consider passing" is inaccurate, since there are definitely positions where the leader should pass a double even though he is a money favorite (was reminded of this reading Kit Woosley's great How to Play Tournament Backgammon the other day).

Nonetheless, I don't believe that this is the case (or even close to the case) in position 2. Even if Red were to gammon every time he hits, this still leaves White with 70% MWC (assuming he covers and bears off safely if Red misses). Passing and going to 5-away 3-away gives less chance than this, and obviously Red is very far from a gammon even if he does hit. Money beaver and HUGE take at this score and cube level, I think.

Matt

Subject: Re: Regrets

From: zare

Date: 06 Feb 2003 13:35 EST

Position 1 analysis.

The first position is analogous to the position studied in article. A key thing to notice is that Red had 11 checkers, not 10, which makes Red's position a 6 roll position. By Walter Trice's 7n+1 formula, that means the effective pip count is close to 43.

White's position is very efficient. It's more efficient than you can expect a position to be for this pip count. It wastes just under 7 pips on average, so the effective pip count is about 32+7=39 pips. The roll is worth about 4 pips, which means that Red's raw game winning chances are about 50%. (They are actually just under 49%.)

The take point for White is about 27%. (I have a table of racing take points according to Snowie's MET in my December 2001 column.) The only market losers are when Red rolls doubles, which are worth 14 pips, and then White rolls something average or below, which happens about 10% of the time.

If Red rolls doubles, and White rolls 3-2 (6/3 2/off), how large is the market loss? Red's winning chances would be up to 85%. This would be a market loser by 0.526 EMG or 5.36% mwc.

If Red does not roll doubles, and White rolls decently, how much of a lemon is that? This can be tough to analyze if you don't know whether to double in the initial position. If you know exactly when to double, the early-late ratio tells you how to relate this to the market losers. Anyway, if White rolls 5-4, then Red will be an underdog, winning only 36.5% of the time. This is a lemon by about 0.360 EMG or 3.7% mwc, and this parlay (nondouble followed by a decent roll) happens much more frequently than the market losers, so it is clear not to redouble.

Another way to look at this is that Red is only slightly in the doubling window. That means doubling now is almost the same as never doubling. It would take the volatility of the last roll of the game to make doubling correct. Here, the position is not that volatile yet, and doubling now just doubles too many losses.

Redoubling would be a blunder costing 0.180 EMG, and this is the closest decision of the three.

Douglas Zare

Subject: Re: Regrets

From: zare

Date: 06 Feb 2003 16:30 EST

Positions 2 and 3 analysis.

At first sight, these may seem similar. There is a single direct shot, and if one redoubles now and wins a gammon, one wins the match. However, there are some serious differences between the positions.

In position 2, if Red misses the shot, it is very likely that Red immediately gets closed out. After that, a gammon loss is likely. With those 6 extra crossovers needed to bear in and an imperfect board, my OTB guess was that I would get gammoned 50% of the time upon being closed out. So, if Red redoubles and is closed out, Red regrets only the difference between getting to Crawford 7-away (9% mwc) half of the time and not at all, for a lemon equity of about 4.5% mwc. Of course, not all dances lead to being closed out, but that's the right order of magnitude.

In position 3, a gammon is unlikely even if Red dances. Further, the missed opportunity is not Crawford 7-away, but 2-away 7-away, which gives the trailer many more chances. The lemon equity is much greater, more like 14% mwc.

Ok, how large are the market losers? It might appear initially that the market losers are larger in position 3, just because no one would take after a hit. However, even if both checkers are hit Red is an underdog to win a gammon, and hitting the first checker leaves Red only a favorite, not gin to hit the second checker. If Red buries the spares too deeply to hang back, this means fewer gammons and more losses whether the second checker is hit or not. In position 2, despite White's anchor, there are more gammon wins for Red than in position 3. It is gammonish to have several checkers sent back, period.

The match score is again an important consideration. Even though it means the match either way to hit and win a gammon, this is more costly at 3-away 4-away than at 4-away 7-away, since at 3-away 7-away the alternative was to be a strong favorite or even in the match.

After a hitting 5-3 in position 3, it would be a huge blunder to take a redouble to 4, costing about 25.7% mwc. However, that's not the amount that one should regret not doubling before hitting, as one is too good to redouble. The amount one should regret not doubling is only about 21.3% mwc. Regretting by 21.3% 11/36 of the time is much better than regretting by 14% 25/36 of the time, so it is a blunder to redouble in position 3. Redoubling costs about 2.9% mwc, or 0.225 EMG.

In the actual match, I didn't redouble, hit, picked up the second checker, and didn't win a gammon despite bearing off quite aggressively.

In position 2, hitting without getting hit back tends to produce large market losers. A hit with 4-2 (Bar/21* 6/4, diversifying, though 13/11 is also reasonable) followed by a dance also leaves Red too good to redouble, and Red will regret not doubling by about 22% mwc again. This is slightly larger than the regret from hitting in position 3, even though White retains a lot of winning chances after a hit in position 3!

Since the lemon equity is so much lower, it is right to redouble in position 2. Redoubling is correct by 3% mwc, or 0.235 EMG, even though Red is a huge underdog, getting gammoned about as much as Red wins.

When you take a 2-cube trailing 3-away 7-away, you have to be willing to recube to 4 in positions like position 2, or position 3. At 4-away 7-away, the regret from an early redouble is much larger, so you can't be as aggressive with the recubes to 4. To get a feel for when to double in a volatile position, I recommend using a bot to investigate the amount you regret after good and bad exchanges.

Douglas Zare

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