Scrabble tournament game

• Chew's score: 0
• Kaufman's score: 0
• Chew to play holding ACLOOPU.
• Chew plays 8d COPULA for 26 points.

8d COPULA and 8d CUPOLA are objectively equal according to computer simulation. (The top computer program for Scrabble plays as well as or better than the top human players. Furthermore, a computer can "simulate" a position by drawing random letters for both players and playing forward a few moves. If it plays ahead thousands of times and averages the scores, it can give a extremely accurate evaluation of the average value of each possible play in a given position. Therefore computer simulation provides a rock-solid foundation for analysis, to which expert human judgement can be added.)

COPULA leaves Kaufman fewer opportunities to exploit squares 7g and 9g, but provides more hooking options: COPULAE COPULAR COPULAS SCOPULAE SCOPULAS. The presence of many hooks makes the game more volatile, because it makes rack-emptying bonus plays easier to come by. It is generally easier to make a seven-letter word from the letters on one's rack than it is to form an eight-letter word or longer by playing through letters on the board, but a seven-letter word in one's rack can only be played if there is a hook for it.

Although it scores the same number of points as CUPOLA, Chew prefers COPULA on the grounds that it will leave more hooks, increase volatility, and therefore increase his chances of upsetting a stronger player. On a whimsical note, he retains an O and dreams about the 1/95,589 chance of drawing the letters for 8a PRECOPULATORY, scoring 119 points.

• Chew's score: 26
• Kaufman's score: 0
• Kaufman to play holding AHNNRV?.
• Kaufman plays 7g NAH for 25 points.

Playing j6 VAR (forming COPULAR) and 9g NAH (forming UN, LA, and AH) would each score 25 points as well. The difference is in the number of hooks each play leaves. Kaufman would like to leave more hooks, because his blank makes him more likely to be able to exploit them than Chew. Note that both players want to leave hooks, for different reasons. (If Chew knew Kaufman had a blank, he might think twice about opening the board.) Thus j6 VAR, which kills the front hooks from COPULA, is slightly worse for Kaufman than 7g NAH, while 9g NAH is slightly better by virtue of leaving the back hooks from COPULA open, and creating a hook at 7i for his N and R (NAH or RAH). Computer simulation values j6 VAR as one point worse and 9g NAH as one point better, on average, than the actual play.

• Chew's score: 26
• Kaufman's score: 25
• Chew to play holding AEIJOSX.
• Chew plays 9i JO for 38 points.

Playing 9i JIAO would score two more points and leave a balanced rack, but it is more important to keep SIX on hand for a 65 point play at 10h (forming HAJI and OX) next turn.

Simulation pegs the value of 9i JOE as four points higher than 9i JO, one point more this turn and an average of three points more next turn. It may seem counter-intuitive to relinquish an E while holding the X, but there are good reasons for it:

1. It creates a backup square for the X at 8l.
2. Playing an extra tile increases the odds of drawing a blank next turn.
3. Playing an extra tile increases the chance (from 29% to 40%) of drawing something which can be used in place of the S on Chew's big play next turn. For example 10h FIX would score 68 and keep the S for the following turn.
4. It creates a second hook for Chew's S (JOS is no good but JOES is).
5. No E's have been played, so the E is 35% likely to be replaced.

• Chew's score: 64
• Kaufman's score: 25
• Kaufman to play holding EENORV?.
• Kaufman plays e3 OVERdONE for 70 points.

Kaufman may have been looking for a double-double in column e (which would have quadrupled the word value), and therefore have been distracted from hooking onto JO (forming JOE) in column k for 8 extra points. k4 OVERNEW scores 78 points, edging out REWOVEN, OVERMEN, and VENOMER in simulation.

• Chew's score: 64
• Kaufman's score: 95
• Chew to play holding AEEILSX.
• Chew plays f8 PAX for 55 points.

If Kaufman had played k8 REWOVEN, Chew would now have played 13f ALEXINES for 98 points. As it is, he can choose among 10h SIX for 65 points, f8 PAX for 55, and 10j XI for 50.

SIX and PAX are objectively equal. The ten points sacrificed this turn are balanced by greater future scoring opportunities from retaining the S and a more balanced rack. Kaufman's play after PAX averages three points more, while Chew's next play averages eleven points more, and Chew's next rack after that averages two points better: 65-55 = -3+11+2

Chew prefers PAX, not for the increased equity, but for the increased volatility. PAX gives him a 30% chance of opening up a 60-point lead by next turn compared to only 20% for SIX. Against a stronger opponent, he wants the swings as wild as possible in the middle game. Against a weaker opponent, SIX or XI would be preferable for the opposite reason. XI would close column k, and despite the immediate sacrifice of 15 points, XI simulates only two points worse than SIX because it retains the S.

• Chew's score: 119
• Kaufman's score: 95
• Kaufman to play holding AAPQRSW.
• Kaufman plays 10j WAP for 29 points.

WARP and WRAP also deserve consideration, because drawing an extra tile increases the chances that the Q will be playable next turn. Also, playing down from k9 sets up an extra hook for the S. k9 WARP leaves the hotspot on 10j for the opponent, so k9 WRAP is preferable even though it leaves a hook for a T. In simulation k9 WRAP for 31 points is 3 points more valuable than the actual play of 10j WAP for 29.

• Chew's score: 119
• Kaufman's score: 124
• Chew to play holding EEEIILS.
• Chew plays 11i LEI for 16 points.

Chew's play is four points worse than 11j LEI for 20 (forming OWL, AE, and PI), but best of all would have been 4b LEIVE for 16. Dumping an extra vowel to improve rack balance (i.e. having an equal number consonants and vowels) more than offsets the immediate sacrifice of points and the creation of an extra S hook. Chew's next play after LIEVE averages 9 more points than after 11j LEI.

• Chew's score: 135
• Kaufman's score: 124
• Kaufman to play holding AELQRSU.
• Kaufman plays 5b SQUEALER for 68 points.

Clearly best. Neither player noticed at the time, but Kaufman accidentally overscored SQUEALER by one point. It should score only 67.

• Chew's score: 135
• Kaufman's score: 192
• Chew to play holding EEINOST.
• Chew plays b4 ESSONITE for 62 points.

b3 ESSONITE also scores 62 points but fares one point worse on subsequent plays in simulation, so the actual play is best.

• Chew's score: 197
• Kaufman's score: 192
• Kaufman to play holding EFIINTY
• Kaufman plays a1 TINY for 42 points.

Kaufman misses c9 FINITE for 39 points, which sims four points better despite sacrificing three points this turn. With EEEEEIIIIIF still unseen, keeping EIF is just asking to get stuck with duplicates on the next draw of tiles. In contrast, keeping the Y for A4 and turning over six tiles looking for SS? gives an average of seven more points next turn.

• Chew's score: 197
• Kaufman's score: 234
• Chew to play holding BDHNTUV.
• Chew plays a7 HUB for 39 points.

Chew considered the relative merits of a7 HUB vs a11 BUNDT for 31 points. Computer simulation says he chose poorly, because BUNDT averages two points better in the long run. Playing HUB and keeping DNTV is too risky, unless the board opens up. In contrast, after playing BUNDT, the H and V will play well at D10 or H4 given a reasonable draw.

• Chew's score: 236
• Kaufman's score: 234
• Kaufman to play holding AEEFILW.
• Kaufman plays 4i ALEWIFE for 80 points.

Best.

• Chew's score: 236
• Kaufman's score: 314
• Chew to play holding DEINRTV.
• Chew plays o1 INVERTED for 158 points.

This is a huge swing which illustrates why Chew was strategically angling for a wild game. Yet Kaufman's previous play of ALEWIFE can't be criticized, even though it opened up the triple-triple. One has to take the 80 points and hope the opponent doesn't get lucky.

• Chew's score: 394
• Kaufman's score: 314
• Kaufman to play holding MOSTTU?.
• Kaufman plays 12f UTMOSTs for 84 points.

Kaufman's play is at least four points better than either m3 TIMEOUTS for 81 or 1a TETOTUMS for 80.

• Chew's score: 394
• Kaufman's score: 398
• Chew to play holding DEFGIRY.
• Chew plays a11 DIRGY for 36 points.
• Kaufman challenges DIRGY. It is not acceptable.

Chew was not trying to bluff; he truly thought DIRGY was acceptable. Perhaps he spends too much time reading unabridged dictionaries.

It would appear that a11 DEIFY for 42 points is the best play, but in fact c9 DIRGE is slightly better despite scoring only 30. DIRGE blocks the scorching Z spot at c9, which otherwise contributes 10 points to Kaufman's expected final equity. (Chew doesn't know that Kaufman doesn't have the Z.) Also DIRGE blocks the -EX hook at d10, while setting up an even better 15-point hotspot for Chew's F at a11.

The important thing to realise in this situation is that with two more full copies of DIRGE unseen and excellent places to play the F and the Y, it's natural to look for ways to play DIRGE to good effect and not waste the F and the Y on a mere triple score. Note further that in simulations, DIRGE doesn't catch up to DEIFY until the third subsequent turn, since it needs time for the FY to get played off.

• Chew's score: 394
• Kaufman's score: 398
• Kaufman to play holding ACEGNOT.
• Kaufman plays a11 OCTAN for 32 points.

Objectively 3i GENOA for 35 points is the better play. Kaufman knows that Chew holds RIDGY, which can be played at a11 or d10, but that isn't such a huge threat. He should take the points and run.

On the other hand, Kaufman had only three minutes left on his clock, and he spent some of that deciding whether to challenge DIRGY. After his play he has only one minute remaining. OCTAN scores reasonably well, keeps a vowel and a consonant for flexibility, and leaves the board open to make it easier to find a good move quickly, whereas GENOA would kill the top of the board, blocking in particular the K spot at h4, and keep two consonants that might end up being hard to play.

• Chew's score: 394
• Kaufman's score: 430
• Chew to play holding DEFGIRY.
• Chew plays 14a AERIFY for 40 points.

AERIFY is superior to d10 DEIFY for 35 points or d10 RIDGY for 30. With so many scoring opportunities on the board, opening the bottom row isn't a significant additional risk: opponent will reply with an average 38 points after AERIFY or 37 after DEIFY or RIDGY. The problem with DEIFY is that it sets up a new little (4-point) hotspot for Kaufman at c13, while eliminating small hotspots in Chew's favour at b2, k5 and m5. In other words, DEIFY scores only eight points for the F while giving Kaufman a chance at just as many for it, instead of saving the tile for a play in which it can earn 12 or 16 points. The problem with RIDGY is of course that even though it plays off the troublesome, potentially doubled DG, it scores too few points in doing so and wastes the Y.

Computer simulation shows that AERIFY wins 21% of the time compared to 16% for DEIFY and 17% for RIDGY. This late in the game, it's important to run simulations all the way to the end of the game, to correctly account for issues of endgame timing, specific utility of tiles, and hotspot values.

One should also calculate and compare winning percentages rather than total expected equity, if one is interested in winning the game rather than maximising spread.

• Chew's score: 434
• Kaufman's score: 430
• Kaufman to play holding ABDEEGO.
• Kaufman plays 15f AGO for 17 points.

With less than a minute on his clock, Kaufman makes a mistake under time pressure. (Players are penalized ten points for every minute they exceed their allotted time.) Immediately after the game, he regretted not having played 15f ADOBE for 29 points. He feared that it might set up a big play in row 14, and he wanted to get rid of the G and O that he thought might hinder his endgame, but that is not enough compensation for the loss of points.

In unlimited-depth simulations with DG known to be on the opponent's rack, Maven favours 15f ABODE for 29 points (wins 76% of the time) over ADOBE (75%) and d8 CADGE for 32 (72%). The important issues here are endgame timing and opponent rack information.

ABODE and ADOBE are better than CADGE even though CADGE scores three more points this turn and five more next turn, because they leave one less tile in the bag, shortening the game by an average of half a play and improving the chances of playing out in two moves from 45% to 80%. This is especially important when there's a danger of running overtime: playing CADGE means a 16% chance of having to spend time finding a third move, compared to 7% for ABODE or ADOBE. If Kaufman needs to make up a ten-point time penalty, his chances of winning drop to about 60%.

On the other hand, if it isn't known that Chew still holds DG, CADGE becomes a better play because it plays off a D and a G with two more of each to come. In this case, each of ABODE, ADOBE and CADGE win about 68% of the time with no time penalty, or 61% with a ten-point penalty.

One might also consider blocking Chew's hotspot at H1 with either 3g DOGE for 18 or 3h EGAD for 19. This would be a good idea if tournament conditions punished a wide loss significantly more than a narrow loss. But if Kaufman is playing to win rather than playing to avoid a big loss, the ten-point sacrifice isn't worth the seven-point risk. There's only a 10% chance of Chew making a 70-point play at H1.

Given that he knows Chew holds DG, Kaufman might also play 15f OBE for 20 points, saving the possibility of playing CADGE on the next turn. The board doesn't merit the delay though, as after CADGE Kaufman can expect to average 36 points, compared to only 32 points after OBE

• Chew's score: 434
• Kaufman's score: 447
• Chew to play holding DDGIRRZ.
• Chew plays d10 RID for 18 points.

Chew still has about 15 minutes on his clock and Kaufman is already 20 seconds overtime. Chew decides this isn't the key play and slaps down a poor play right away, to score a few points, split duplicate tiles for endgame flexibility, and put Kaufman back under time pressure.

This was a costly error, because 3g ZIG for 32 points would win 83% of the time no matter how quickly Kaufman plays. ZIG scores best while blocking h1 plays and forcing Kaufman to keep any I's he has if he wants to play the K at h4.

RID wins only 50% of the time. The additional time pressure is not likely to compensate for the lost points. Kaufman has to go two more minutes overtime for RID's winning percentage to hit 80%.

If Chew had spent another few seconds on this play, he might have seen m3 RIGID for 18, which wins 60% of the time but only needs Kaufman to go overtime by one more minute to win close to 80% of the time. Even better is d11 RIGID, though it requires Chew to spend a few more seconds scanning the rest of the board to find it.

• Chew's score: 452
• Kaufman's score: 447
• Kaufman to play holding BDEEIIK.
• Kaufman plays 14h BIKIE for 25 points.

Kaufman takes 39 seconds to play, leaving his clock at 59 seconds overtime when he draws the last two tiles out of the bag. If Kaufman can make the rest of his plays in one second, BIKIE wins 22 times in 36, while 3g KIBE for 28 (keeping the other I to play at K3) wins 25/36. If his clock ticks over another minute, BIKIE wins 18/36 to KIBE's 15.5/36. Assuming Kaufman can make one play but not two in his remaining second, BIKIE and KIBE both win 21/36. (For example, if he plays BIKIE and draws AO, he can win only by taking two turns to set up and play at 1h, and the margin doesn't offset the extra ten-point penalty.) Given that it's significantly easier to play four tiles in one second than five tiles, he's better off taking his chances with BIKIE.

Note that h2 BIKE 38 or h2 DIKE 37 score better this turn but lead to longer endgames and win only 16.5/36 and 15.5/36 respectively.

• Chew's score: 452
• Kaufman's score: 472
• Chew to play holding ADGMRSZ.
• Chew plays 3c GROSZ for 50 points.

Chew is fuming for having missed the put-away on his last turn, and resolves to spend as long as it takes to analyse this position. He has plenty of time, and tells Kaufman (who looks a little tense) to sit back and relax. A few minutes into it, Kaufman knocks a pencil off the table. Seconds tick by before he announces correctly that he won't be needing the pencil before game end, and will therefore not pick it up. After twelve minutes, Chew finally persuades himself that GROSZ is indeed optimal, even though it gives him a one-point loss if Kaufman plays out in one second. (Recall that with no more tiles hidden in the bag, each player can calculate exactly what tiles the other has.)

• Chew's score: 502
• Kaufman's score: 472
• Kaufman to play holding DEGO.
• Kaufman plays 15l DOGE for 29 points.
• Kaufman scores 12 points for the ADM in Chew's rack.
• Kaufman loses 10 points for overstepping time control by 1:00.

Kaufman plays out in one second, and wins 503-502! However, the scoring must be double-checked on such a close game. After a harrowing recount, the players discover that SQUEALER was overscored by one, so the game ends in a 502-502 tie. This game sets a record for the highest tied score in sanctioned tournament play.

Final Position

This article is based on an account (http://www.math.toronto.edu/jjchew/scrabble/analysis/19970615) cowritten by Toronto Scrabble Club (http://www.math.toronto.edu/jjchew/scrabble/club3/) director John Chew (mailto:jjchew@math.utoronto.ca) and 1997 World Scrabble Champion Joel Sherman (http://www.scrabbleassociation.com/cgi-bin/player.pl?surname=sherman&given), and appears here with their consent.