From club game last Wednesday night. I simulated it on a work PC for a week. Assume an approximately even score.
----------------------------- 1|= ' = ' =|1 2| - " " - |2 3| - W B - |3 4|' O D E ' - '|4 5| G A T - |5 6| " R " " " |6 7| ' T ' ' ' |7 8|= C L O U D K I V A|8 9| ' H I M A N T L E D |9 10| " E N " " L " |10 11| N A G - P I |11 12|' E T ' G Y M '|12 13| A X ' ' P |13 14| E T " " - |14 15|= ' = ' =|15 -----------------------------
AEOOORS
| | | | | | | | | | | | | | | | | | | | | | | | | V
In the game, I played -OOO. After my opponent played CARVE A11, I thought I should have blocked with ARSE A12, but that keeps 3 O's. The other idea was to block and rid an O with OAR A11. Or perhaps I should just score and move the O's elsewhere? Before peaking rank these ideas in order. Multi-ply simulation results below:
| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | V -OOO 11.3 ROOSE 14J 9.3 BOO 3F 9.1 OAR C2 6.4 TOO J9 6.1 OAR A11 2.5 ARSE A12 1.8So the two moves I preferred after getting spanked by CARVE were bad. A good example of the fallacy of the negative instance.
To which I received the following reply:
Date: Fri, 12 Dec 1997 11:43:17 -0500 (EST)
From: Shane W Frederick
To: Jaygee@primenet.com
Subject: fallacy of the negative instance
I received a forwarded posting regarding a certain OOOAERS rack and you used the term "fallacy of the negative instance." I'm a decision theorist and scrabble fanatic. I study all kinds of heuristics and biases, but I have never heard this term before. It sounds very interesting. Could you describe it to me, or give me some cites where I could read about it?Thanks, Shane Frederick.
p.s. How would you position the following words as opening plays. (Ignore leave in case 2).
BRONZER CREED
So I write:
Date: Fri, 12 Dec 1997
From: Jim Geary
To: Shane W Frederick
Subject: Re: fallacy of the negative instance
Hi,
My definition of "fallacy of the negative instance" is when
your thinking is improperly skewed by negative results.
I have no idea where I came up with this term, so perhaps I
coined it, but I doubt it. In the game, it occurred to me that
the best play was to pass the three O's, and I did so. After
my opponent played CARVE a11, I then thought "shit, I shoulda blocked."
Computer analysis later reveals that my instinct was correct, but
my thinking was warped by the bad thing that happened to me in the
given scenario.
I play a lot of poker, and see people make this mistake all the time. They see people draw out on them with crap cards and think, "gee, maybe crap cards aren't really crap." Their thinking is warped by the negative instances. They "forget" about all the time they are betting and their opponents fold (i.e. missing with the crap cards).
Regarding the opening racks, as a general rule of thumb, take points over placement on the first turn. There are very few exceptions. The extra 2 points from placing CREED at 8d as opposed to 8h more than compensate for the vowel next to the DLS and the prevention of 7-letter bingos. CREED h8 additionally has the drawback of putting an E -- the most "thru-able" letter -- in the double-double column.
The BRONZER rack placement is such a different points v spot problem that you can disregard most of the finesses and base your placement strictly on the negative value of the spot. Instinct/experience makes me want to say the spot is worth about 7 points. As long as the spot is worth less than 14 points it would be mathematically correct to hang the S-hook.
A more painstaking analysis: Opponent should have an S on his rack ~(4/93) x 7 (actually, a little less because of the times he gets 2 S's..) of the time and avail himself of the spot for an average of -say- 88 points. But, (65/93)*(4/86)*7 of the time you will pick an S without him having done so, and avail yourself of the spot. Now -- I'm going out on a limb here -- I'll guess that when this gets done decaying, that the opponent will be about a 7:5 favorite to hit the spot. The cost to you of opening the spot is the difference of the times he gets it times the value of the spot. Well, 7:5 is an edge of 17%. The value of the spot is the difference in the average score when hitting it and the opportunity cost elsewhere on the board. For a top-level expert, average progress per turn is about 35 points, but on this wide open board, it is a little higher. For tidiness, we'll say the difference is 50 points. 17% of 50 is only 8.5 points, so that would be the value of the spot, provided all my assumptions and cocktail-napkin math are correct. Meanwhile, your scoring an extra 14 points, so in a purely linear world, not making the risky placement would actually cost you 5.5 points.
Of course, Scrabble isn't like a poker game that never ends. The idea is to be up after 100 tiles have been run through. Therefor, when you're winning you want to avoid risks and when you're losing, you want to create volatility. Many people would say that because of this, on the opening rack in question, avoid setting up the big spot. This early in the game, however, I feel that it is a little premature to start playing it close to the vest. Take the Equity!
The only time I would hedge would be if I was playing someone 400 points lower rated than me. In that case, I figure the only way they are gonna beat me is with a healthy dose of luck, so I certainly might avoid giving them opportunities to be lucky.
Since I started writing this letter, I've been simulating the BRONZER rack on a different computer here. It so far thinks that the 8h placement is about 8 points better. :) Oh, well nobody's perfect.
A further response from Shane:
Date: Fri, 12 Dec 1997
From: Shane W Frederick
To: Jim Geary
Subject: Re: fallacy of the negative instance
Excerpts from mail: 12-Dec-97 Re: fallacy of the negative.. by Jim Geary
>>My definition of "fallacy of the negative instance" is when... That is what I thought you meant. The phrase has a nice ring to it. This brings up a distinction between regret and recrimination. It isn't really a fallacy to regret your play given a negative outcome ex post, but it is inappropriate to feel recrimination -- that is, when your ex post perspective leads you to the erroneous belief that you made a bad play ex ante. I see what I think is another fallacy among scrabble players, especially lower ranked ones. It seems that the opponents don't take into consideration lost opportunities for themselves, or the likely effect on opponents rack when deciding whether to challenge a word suspected of being a phony. Thus, this leads people to be overly averse to challenging low scoring phonies. For example, say someone plays a (non)word like OARIE onto a triple for 17 points, the opponent may be dubious, but often reasons (and sometimes verbalises) "well I'm not going to challenge for 17 points", even it reduces their play by 15 points (not to mention, the benefit of sticking the opponent back with all the vowels they are trying to dump). Now, I'd guess that a (non)word like HOYDIN scoring 32, but not affecting the persons play WOULD draw a challenge. (Obviously, this proposition is hard to test, because you would have to hold constant the person's subjective probability that the sequence of letters the opponent has laid down is a legitimate word). Still, I've noticed that people are VERY reluctant to challenge low scoring phonies. Obviously, the situation doesn't come up as much at the highest level of play because most of the shorter, and lower scoring words are known. Do/did you find this to be true? >>Regarding the opening racks, as a general rule of thumb, take points over placement on the first turn... I am very humble regarding my intuitions, but it seems really difficult to believe that the 8d positioning is worth the 2 points. First, the incidence of double doubles seems very quite low. Second, the simulations would find more such plays than even a good human would. Third, words starting H,M,W,Y, and so on make such good return plays-- decent scores with good defense, that allowing them just seems too costly. (Although, if the person chooses not to use them, they are still there for you -- I always have a hard time accounting for the two move ahead possibilities). Fourth, of course, is blocking parallel bingos, which the play at h8 must do 5-10% of the time-- surely more often than a double double is played. (S hook bingos are probably not an issue here, since the word hooks on both ends). >> Since I started writing this letter, I've been simulating the... Regarding the simulations. Are they to be trusted? Ignoring for the moment that certain types of plays (especially through plays) will be unrepresentative of human performance, how do we even know that the equity or whatever it is called is computed correctly by the guys who wrote the program. The computer certainly plays well, but what is the gold standard? One last question. On average, what proportion of 5 letter words do the top, say, 50 players, know? Thanks for your previous (and hopefully future) responses.
And in response to everything:
Date: Fri, 12 Dec 1997
From: Jim Geary
To: Shane W Frederick
Subject: More stuff
On Fri, 12 Dec 1997, Shane W Frederick wrote:
I see what I think is another fallacy among scrabble players, especially lower ranked ones. It seems that the opponents don't take into consideration lost opportunities for themselves, or the likely effect on opponents rack when deciding whether to challenge a word suspected of being a phony. Thus, this leads people to be overly averse to challenging low scoring phonies.
As a matter of gaming, though, it may be correct not to challenge when the opponent's score is low. This of course means that if it hurts your score, you should be more likely to challenge. I wrote an article on "When is it correct to challenge" for my old Scrabble theory rag, the JGNewsletter. I may someday transcribe it for my web page.
I am very humble regarding my intuitions, but it seems really difficult to believe that the 8d positioning is worth the 2 points. First, the incidence of double doubles seems very quite low. Second, the simulations would find more such plays than even a good human would. Third, words starting H,M,W,Y, and so on make such good return plays-- decent scores with good defense, that allowing them just seems too costly. (Although, if the person chooses not to use them, they are still there for you -- I always have a hard time accounting for the two move ahead possibilities). Fourth, of course, is blocking parallel bingos, which the play at h8 must do 5-10% of the time-- surely more often than a double double is played. (S hook bingos are probably not an issue here, since the word hooks on both ends).
Well, certainly no one has accused me of being humble, so I'll state
that I think that the non-strongest players overrate these considerations
you mention. To verify I'm simulating it on another computer while I
write this. The only factor that might matter is the blocking of
7-letter bingos. However CDER are great through-letters for 8's.
I would think that for most opening racks that have CREED as an option,
CREED is not in fact the best play. For just testing the placement
notion, I've simulated two cases: one that retains an AE, so as not
to be able to take much advantage of the DLS, and one that keeps EW,
to give the opener a fair shot at the DLS's/counterplay. The results
are:
CREED + EW: 8D 21.7 8H 19.4 CREED + AE: 8D 23.2 8H 21.0
As you can see, it's about a 2 point difference whether your holding a counter or not. The reason is because those DLS's just aren't that significant. Top players (and computers) can find many other ways to score without simple overlaps. Players whose games consist of much more overlapping (check out the difference between experts' and intermediate players' boards sometime) are prone to overrating the significance of vowels next to DLS's on the opening play. This is probably due to a subset of the FOTNI, in that experts are getting negative reinforcement all over the board by their expert opponents, but intermediates' counterpunches are along more conventional lines. This isn't meant as a personal jab at you, but it is true.
>> Since I started writing this letter, I've been simulating the >> BRONZER rack on a different computer here. It so far thinks that >> the 8h placement is about 8 points better. :) Oh, well nobody's perfect.After several hours of multi-ply simulation, the results apparently regressed to the mean some and it ultimately ended up being a 5 point difference. Therefore, the spot is worth 9. Hmm. Higher than I'd think.
Regarding the simulations. Are they to be trusted? Ignoring for the moment that certain types of plays (especially through plays) will be unrepresentative of human performance, how do we even know that the equity or whatever it is called is computed correctly by the guys who wrote the program. The computer certainly plays well, but what is the gold standard?This simulator I use was written by someone who is among the top ten strategists in the world. He takes into account things that 90% of players aren't even aware of. There are a few situations that simulation will not properly handle, but you have to be a very good player to see it and to adjust the results accordingly.
One last question. On average, what proportion of 5 letter words do the top, say, 50 players, know?
I can't speak for #50, but for top 10 players, they should know all of them if they are on top of their game. As I have recommended previously on cgp (see my website for some archived posts), learning the 5's is crucial for an aspiring player. Once you've mastered 2's, 3's and super-high probability bingos, start studying the 5s.
Best Wishes,
Jim Geary
On your web page http://www.primenet.com/~jaygee/FALLACY.HTM,
you mention the BRONZER rack to start, and consider whether it
should be placed at 8d or 8h. You conclude that equity-wise, it
should be about 5 pts better to slap it at 8h. My first thought
was that win%-wise a 8d positioning might be better. As for any use
you make of the spot will just super-increase your lead, while the
opponent's use of it allows him/her to mount a comeback.
Data from 10000 SOWPODS sims with BobBOT, infinite depth, with
average scores and rack values being noted for the first four ply:
OPP1 YOU1 OPP2 YOU2 ORAK YRAK EQUITY WIN%
BRONZER 8h 106 61.88 55.76 48.77 46.01 2.23 2.49 97.38 75.16
BRONZER 8d 92 43.02 44.72 44.99 44.30 3.19 3.32 93.15 77.01
As suspected, 8h wins the equity battle, but 8d wins the win% war.
After 5 plays, the 8h placement had a lead of 50 or more 72.7% of
the time, while the 8d placement had a lead of 50 or more 82.4% of
the time.
I would be fairly certain that OSPD data would be similar. The only
danger in interpretation I see is that the BOT doesn't necessarily
shut the board down when ahead, but that should affect each move
about the same.
===
Mark Watkins
mwatkins@math.uga.edu
I stand corrected. -- JG
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