This article originally appeared in the October 2000 issue of GammOnLine.
Thank you to Kit Woolsey for his kind permission to reproduce it here.

The Great White Unicorn

By Dorn Bishop
I love playing online. It is the backgammon equivalent of telecommuting. No need to shave, put clothes on, talk to people. Nope, I can pursue my chosen avocation while lounging around in my underwear, slurping coffee, and picking lint out of my navel. Not that I ever do that sort of thing. But if I did it wouldn't be that big of a deal.

Something else I love about playing online: I am never away from my wife and kids. Oh sure, I am secluded in my home office while, uh.... Honey, Pardner, and Sport are somewhere else in the house, doing non-backgammon kinds of things like conversing and bathing. My point, though, is that when I play online, I can delude myself into believing that I am Mr. Family Man -- a caring, sensitive, New Age Phil Donahue who spends quality time with his wife and kids.

Sometimes, though, things can get pretty boring online. Who knows why. Maybe the game decisions aren't exciting that day. Maybe I'm losing. Maybe I've slurped so much coffee that I start seeing unicorns in the garden outside my window.

It's at times like these that I start to get worried. I worry that I am missing out on something in life. I worry that my bank account is shrinking faster than my head is. I worry that my wife will judge me a booby for seeing a unicorn in the garden and will send me to the booby hatch.

And so it was during a recent August dusk. I stumbled into Club GamesGrid looking for a friendly money game. The establishment's proprietor knew a pigeon when he smelled one across the cyberroom. He invited me to play. I accepted. I asked the waitress for another cup of coffee. She slapped me. Ouch. I had some nerve, ordering my wife around like she was some kind of waitress or something. (Mental note: stop treating wife like waitress.)

Things went downhill from there. I was on Mr. Big's turf now. Roll after roll, cube after cube, game after game went against me. My mind started wandering. The room started spinning round and round like I walked into a play. Helen Reddy started singing in the background. Angie girl are you alright? Tell the radio goodnight. All alone again, Angie baby....

Then it happened: a unicorn sighting! Well, okay, that was the second thing that happened. The first thing that happened was that I slammed my radio against the wall because I really, really hate Helen Reddy. But then it happened: I saw a real, honest-to-goodness unicorn staring me in the face! It was standing right there in front of my eyes. I called my wife. No dice. She was still smarting about that waitress crack.

Now, dear reader, I thought that I knew what a unicorn looked like, but my perception of reality had been vastly skewed by erstwhile tales of wives and James Thurber. No, a real unicorn looks nothing like a horny horse. (Mental note: rephrase last simile in next draft of article.) A real unicorn is a much more delicate, ephemeral, and divine creature than second-hand accounts can convey. But they do exist. By the grace of Pipcelot and Snowie, I now know that they do exist!

Desperate to memorialize this extraordinary vision for fame, posterity, and evidence in my upcoming commitment proceeding, I searched in vain for my camera, only to be drawn away from that task at hand by a rude, incessant beeping coming from the playing area. Actually, it was more like a high-pitched binging. Bing, it went. Just like that, except with more g's, like bingggg. It seemed that Mr. Big was anxious to continue his winning streak, and was now so flush with success that he was actually cubing me from the bar in this position:





money game


Hmmm....unicorn, schmunicorn I thought. The figment of my imagination could wait another minute or two. We were talking about real money here. More importantly, we were talking about a golden opportunity for me to recoup some of the dusk's losses. Sure I might get gammoned: Mr. Big might hit one or two of my checkers; I might dance; he might scoop up the rest of my blots; the polar ice caps might melt; I might get backgammoned; and Menudo might make a comeback.

But I try to be a "glass is half full" kind of guy in these types of situations, especially when I am steaming anyway. First, the take was trivial. Mr. Big was going to dance 16 out of 36 times, giving me a pretty efficient redouble immediately, and he might well have a drop. Second, even if Mr. Big hit me, I might hit him back on the ace point and he might then dance. Or I might anchor on the 3 point and win from there. Or I might get closed out but water from the polar ice caps might flood the table during the bearoff causing a draw to be declared. Or the madcap yet insouciant music of Menudo might drive my opponent insane.

So, as I say, the take was clear. But remember that I had unicorns on my mind -- furry creatures with big teeth that look a lot like beavers up close. Could I talk myself into believing that I had positive equity holding the cube? Well, I already had almost 16 immediate wins out of 36 when Mr. Big danced. Since he probably still had a slight take after dancing, I gave myself 15 wins out of those 16 games. In the other 20 games, I figured I would hit Mr. Big back immediately in about 6 of them, and would win about 4 of those. In approximately 6 others, I would anchor up and win at least 2 of those. And in the other eight, I could probably still eke out 1 more win.

Perhaps this is as good a time as any to point out another advantage of playing online, albeit an advantage that many players frown upon: the use of mechanical aids to assist during play. Yes, I am talking about... the dreaded Texas Instruments calculator (gasp!). I confess that I have fallen prey to the siren calls of this scientific wonder on occasion, and did so again here. I quickly punched in the numbers: 15 + 4 + 2 + 1. Then I hit the "=" key (for those of you who are computerially disinclined, "=" means the equal sign). The number that appeared on the screen was "22". 22?!?! I was going to win 22 out of 36 games holding the cube?!? It sure looked like a furry woodland creature from that vantage point.

"Easy big fella," I said to myself. "Remember that Mr. Big is going to gammon you a lot of the time in the 14 games that he wins." (Mental note: Talk to therapist about stopping voices inside of head.) True enough, I thought, but I did not think that he would gammon me more than half of the time that he hit. And since a gammon is worth half as much as a win, that meant I only needed to subtract a maximum of 3.5 games from 22, leaving me with 18.5 wins out of 36 games -- making me the prohibitive favorite! Kind of!

So I duly beavered, silently pitying Mr. Big for his poor judgment in doubling while publicly apologizing to him for my insolence. Mr. Big took my beaver with characteristic aplomb, especially after he rolled a 3-1 and I danced. Reeling from the trauma, I mustered all of my skill and energy to will the dice to let me enter and then to roll multiple doubles in order to escape getting gammoned. Baron Vernon Ball would have been proud.

As I say, the game was lengthy and required all of my rapt attention and telekinetic energy. By the time it ended, I looked again for the unicorn but, alas, it had vanished. If, indeed, it had ever really been there. After a few more games, I bid my host goodnight and retired for the evening.

The next day I looked at the previous day's position on Snowie. Just then, my eyes starting playing tricks on me again. A 3-ply evaluation analyzed the position as "Double, beaver" -- as if Mr. Big and I had both handled the cube properly! Could it be? Could this really be a position where my opponent increased his own equity by activating gammons through the Jacoby Rule, but still had a negative equity after doing so? And, even more remarkably, could this be a position where my opponent was still better off playing for twice that negative equity after I beavered him than if he had never doubled at all?

I had read about such positions being theoretically possible, but had never actually seen one in real play. In fact, I regarded them as a myth since Kit Woolsey needed to go to these extraordinary lengths to contrive such a "double, beaver" position for use in Bill Robertie's Advanced Backgmamon treatise:





money game


If Blue rolls a six, he will gammon White. If he does not, White will double Blue and Blue will have to pass. Even though Blue will roll a six in only 11 out of 36 games -- about 30.5% of the time -- he does better to turn the cube now in this "kill or be killed" situation. That is because, under the Jacoby Rule, Blue can only win a gammon if the cube has been turned.

If he does not double, Blue's net equity will be (30.5 x 1 pt.) - (69.5 x 1 pt.) = - 0.39 points. If he doubles, however, Blue's net equity will be (30.5 x 4 pts.) - (69.5 x 2 pts) = 122 - 139 = - 0.17 points. So Blue is better off by doubling.

At the same time, White is clearly better off by beavering Blue's double. With a positive equity of 0.17 points per game holding a 2 cube, White should not hesitate to crank up the stakes to 4. That will double his equity to 0.34 points per game.

Does that mean that Blue should not double after all? Hardly. A negative equity of -0.34 points per game is still better than a negative equity of -0.39 points per game, which would be Blue's equity if he chose not to double.

Returning, then, to the 3-ply evaluation of my position with Mr. Big, Snowie provided this remarkable evaluation:

3-ply             Money equity .211
1.4%  26.5%  51.4%  48.5%  9.3%  0.5%

1.  Double, take  -0.046
2.  No double  -0.054  (-.008)
3.  Double, pass  1.000 (+1.046)

Double, beaver
Upon closer examination of these figures, I realized that I was, indeed, looking at a figment of my imagination. To be sure, using Snowie's 3-ply numbers, White is .008 points better off by doubling than by not doubling. White should double, however, only if he knows that Blue will not beaver. That is because the hairy beast will turn White's expected equity of -0.046 into twice that amount, or -0.092 -- an equity that is significantly worse than the -0.054 points that White can obtain by not doubling at all.

So Snowie was correct that White increases his equity by doubling. And Snowie was correct that Blue increases his equity by beavering. However, Snowie failed to understand that a correct beaver by Blue would place White in a worse position than if he never doubled. Therefore, even though Snowie analyzes the position as "Double, beaver," that conclusion is facially invalid based on the 3-ply evaluation's estimated results. The position should actually be evaluated as "No double, beaver" or, more accurately, "Double, but only if you know your opponent will not beaver."

But was Snowie's 3-ply evaluation of this position really correct? Rollouts strongly suggested that it was not, but unfortunately not in a way that soothed my ego:

Rollout:  95% Confidence Interval:
 -money cubeless eq.:  0.231 +/- 0.008
 -live cube no double:  -0.049 +/- 0.016
 -live cube double take:  0.001 +/- 0.024
Rollout Settings
 Truncated rollout, depth 5
 453 games (equiv. 103857 games),
 played 3-ply (medium, 50%), cube 3-ply
 settlement 0.550 at 8 pts.
 random seed, with race database

Rollout  Money Equity .231
1.9%  27.7%  51.1%  48.9%  8.5%  .3%

1.  Double, take  0.009
2.  No double  -0.031 (-0.040)
3.  Double, pass  1.000 (+.991)
According to these numbers, Mr. Big correctly had doubled and I had incorrectly beavered. "Oh sure," I rationalized to myself, "My beaver was undoubtedly correct as a practical matter anyway. After all, if I had won, Mr. Big would undoubtedly have started steaming. If I had won, Mr. Big's loss of equity from even one ill-advised checker play or cube decision would more than offset the equity I `lost' by beavering."

If I had won, if I had won....

Well, I had not won and now I was the one who was feeling steamed. "Must...control...fist...of...death," I thought. "Must ...not...make....ill...advised....checker..." To calm myself down, I looked outside my window into the garden. There, placidly chewing on my wife's blue potato bush, was a unicorn. Lurching back in my chair, I knocked over my cup of coffee onto the mouse and keyboard. I rushed to get a rag.

As I wiped up the coffee, I may have inadvertently moved a checker or two or three around on Snowie. When order was restored, I again looked out my window. To my amazement, there was still a unicorn staring me in the face, although now it looked slightly different from the one I had seen earlier. Fortunately, this time I had my camera handy. This is what it looked like:





money game


Rollout Money equity 0.219
1.8%  28.2%  50.9%  49.1%  9.5%  0.4%

95% confidence interval:
  - money cubeless eq.:  0.219 +/- 0.007
  - live cube no double:  -0.064 +/- 0.013
  - live cube double take:  -0.028 +/- 0.020

Rollout settings:
Truncated rollout, depth 5
1000 games (equiv. 166192 games)
played 3-ply (medium, 50%), cube 3-ply
settlement 0.550 at 8 pts,
random seed, with race database

1. Double, take  -0.016
2. No double  -0.050 (-0.034)
3. Double, pass  1.000 (+1.016)
Proper cube action:  Double, beaver

Live cube
1. Double, take  -0.028
2. No double  -0.064 (-0.036)
3. Double, pass  1.000 (+1.028)
Proper cube action:  Double, take
The above position differs from my position with Mr. Big in three small respects. First, Blue's blot is on his 5 point instead of his 4 point. Second, one of Blue's outfield checkers has been moved from the 11 point to the 9 point. And third, White's spare checker on his 6 point has been moved to his 5 point. Repeated rollouts have confirmed that all three of these changes are essential to create a true "Double, beaver" position. Make only one or two of these changes, and the position remains "Double, take" because White's net equity after doubling remains positive. On the other hand, move White's spare from his 5 point to his 4 point, and White's equity suddenly drops so far into negative territory as to turn the position into a non-double.

Curiously, and apparently glitchly, Snowie evaluates the proper cube action for "Live cube" rollouts as "Double, take" even though Blue clearly does better by beavering White's proper cube, doubling his equity from 0.028 to 0.056. White is still better off doubling, however, since his equity of -0.056 after getting beavered compares favorably to his equity of -0.064 by not doubling.

Skeptics may correctly point out that Snowie's margin of error leaves open the possibility that this is not truly a "Double, beaver" position, even after the equivalent of a 166,000 game rollout. Of course, skeptics said similar naysayings about the light bulb, the Loch Ness monster, the Apollo space mission, and New Coke. They were wrong then (except about the Apollo conspiracy, of course) and they are wrong now.

Frankly, I am not sure that a definitive answer can ever be provided in these types of positions, especially where repeated tough decisions will often be needed, and any one of those could result in a significant equity blunder (for example, deciding to cover a home board blot instead of scooping up an additional checker). Indeed, the answer may vary depending on the skill or mechanical nature of your opponent. Personally, and to paraphrase Dorothy Parker, I would rather have a frontal lobotomy than a bot in front of me.

Besides, like I said before, unicorns are very delicate and ephemeral creatures. They vanish just as quickly as they appear, leaving us to ponder whether they were ever really there in the first place. Now, if you will excuse me, I am late for my Loyal Order of Sasquatches meeting.

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