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Maven Simulation Analysis: 1999 12 12

Joe Edley posted the position being discussed earlier this week, leading to a flurry of discussion on CGP. In what follows, I refer to two Maven simulations, an unrestricted one and one that makes inferences about opponent's rack. (Click on the phrases in the previous sentence to see the simulation reports.)

Here's what I said on CGP.

Two long Maven 2.1 simulation runs later, for a total of 53546 iterations
at a depth of three turns, here's what I have.

  11.2 KHI
  10.1 EVE
   8.9 HEINIE
   8.7 VEEP
   8.6 VINEAL
   7.6 VEINAL
   7.0 ALIVE
   6.7 KILO

There is a 95% chance that each of the above figures is within 0.4 points
of the correct expected equity as calculated by Maven in an infinite 
simulation.  The usual statistical report can be found at

  http://www.poslarchive.com/math/scrabble/analyses/19991212.html

It shows the following.

KHI and EVE are good plays because they significantly reduce opp's next
turn scoring chances: opp is kept to 30.3 +- 0.2 points next turn compared
to 37.4, 34.2, 37.7, 37.5, 35.1 and 32.1 for the other moves.  KHI gives
us a reasonable expected score for our own next turn too: 33.7 +- 0.2,
compared to 29.5, 31.0, 28.1, 34.6, 33.9, 35.1 and 32.8 for the rest.
With KHI, we come back with a bingo 14.5% of the time, almost as much
as our typical chances later on in the game, and get in a solid 5-letter
play almost half the time (we've just set up NIEVE M9 if nothing else).
EVE gives us only a 5.5% chance of a bingo, leaves us having to trade
tiles 2.1% of the time, and most often leads to a 4-letter play for us.
KHI and EVE are both defensive, but we shoot ourselves in the foot with
EVE because we've kept IIKN with no place to play any of it for decent
points.

HEINIE is nearsighted.  We take an extra eight points up front but
give opp back seven points right away (vs. KHI or EVE) and sacrifice
another three points the next turn (vs. KHI).  VEEP is similar.  An
extra eight points up front, but lose four the next turn and 5.5 the
turn after that.  ALIVE (my original choice) and KILO are even worse,
sacrificing a point now to give opp more points next turn and in the
case of KILO giving up another point on our own next turn.  This is
definitely a situation in which to apply the 'reduce next opp score' 
heuristic rather than the 'maximise my score this turn' heuristic.

VINEAL and VEINAL are the most generous to opp, giving over seven
points, without significant return the following turn compared to KHI.
VINEAL is a full point better than VEINAL.  Why?  This turns out to be an
interesting question.  Opp's reply score doesn't differ significantly
(37.7 +- 0.2 vs 37.5 +- 0.2 in fact favouring VEINAL), so it doesn't seem
to have anything to do with bingo exposure through the E or I.  On the other
hand, the expected value of our own next score is 0.65 +- 0.15 higher with
VINEAL.  Look then at the square values and hit probabilities on our next
turn.  With VINEAL, there's a significant increase on the left side of row 6,
and in the upper right quadrant of the board.  I believe this is because of 
the way our leave (EIK) works with each board.   We are fairly likely to be
able to play down from the H with our K.  If we can't do, what are we going
to do with the K and the I?  Placing an N at 6H gives us a good chance of 
placing the K on the TLS at 6F on our next turn, as long as opp doesn't block
us by playing along row 5.  If opp does play along row 5, then opp may well
give us access to DWS in the upper right quadrant.  I think it's reasonable
to value this at about the half point Maven suggests.


What can we infer about opponent's leave after GLOM?  Assuming a reasonably
strong opp...

9H GOLEM (28) suggests opp did not have an E.
9G GLIM (19) suggests opp did not have an I.  (If opp had been trying to 
  split Os, 9J LIMO (20) or 7J MOOL (20) looks better.)
9I ALAMO (25), L3 MOOLAH (23), L8 HAULM (20),
  L8 HOMOLOG (26), L4 LOUGH (18) and L4 LOUGH (18)
  respectively suggest opp did not have AA AO AU OO OU UU,
  i.e., opp kept at most one vowel.
K6 COPALM (24), L3 MOLOCH (26) and L3 CLOUGH (24) suggest opp did not 
  have CA CO CU, i.e., opp did not have a C.
M3 GLOAMS (31), M3 GLOOMS (31) and M3 MOGULS (31) suggest opp did not
  have SA SO SU, i.e., opp did not have an S.
9H GOX (29) suggests opp did not have an X.
L8 HOAGY (24), 7C GLOOMY (22) and K4 GLUMPY (28) suggest opp did not
  have YA YO YU, i.e., opp did not have a Y.

In summary, opp's leave probably consisted of either

1) three of BDFGH JKLMN PQRTV WZ, or
2) one of AOU and two of BDFGH JKLMN PQRTV WZ.

This rules out about 10% of opp's racks, and we can see what that does
to the simulation.  Still with a 95% confidence of +- 0.4, we have:

   old   new  play
  11.2  10.9  KHI
  10.1   9.8  EVE
   8.9   8.2  HEINIE
   8.7   8.2  VEEP
   8.6   8.0  VINEAL
   7.6   6.8  VEINAL
   7.0   6.6  ALIVE
   6.7   6.5  KILO

As we would expect, plays which put out an open vowel (HEINIE, VEEP, VINEAL,
VEINAL and ALIVE) suffer compared to the defensive top choices of KHI and EVE.
Looking at the breakdown by turn, we see a clear increase in opponent scores
for the open plays. Our next turn score is unaffected, except when we play VEEP.  
In this case, we lose almost three points, presumably because opp uses the
valuable E in VEEP 23.4% of the time instead of 21.7%, leaving us with fewer
scoring opportunities for our IIKN.

In conclusion, KHI is clearly the best play and it looks even better when 
you start thinking about what opponent's last rack must have been.