How
large
of an advantage does Player #2 have in OSKI ?
by 

OSKI is a word game
recently implemented at the
Little
Golem turnbased site. Briefly, players alternate turns by
placing any letter of their choice on an empty cell, and
forming a new word. The number of letters
in the length of the new word formed is the number of points the player
earned for that move. Since the
initial seed word always consists of three letters, and with exactly 19
cells on the board, all games last just eight moves each. The player with the most points at the end of the
game is the winner. After playing a number of games, I
suspected the player moving second (Player #2) had a distinct advantage.
Discussions in the Little Golem forum with other players all agreed with this.
I was curious to know how much of an advantage Player #2 had.
How often did Player #2 win? On the average, what was the average
margin of victory? How much would this margin of victory
change based upon the vocabulary of the players? Catherine is capable of analyzing any
OSKI
board position and returning all of the possible plays, up to 12
letters in length or less. Catherine can also play games against
itself, uninterrupted, for any number of games desired. If I look for words that are just 8 letters or less in
length, Catherine can play a full game in just two minutes.
When playing games against itself, Catherine begins the game by placing
a random 3letter word in the center position. As per the rules, the
3letter word must contain three different letters. There are 1,153
different threeletter words that fit this description in the
SOWPODS
word list and Catherine chooses one at random for each game. (SOWPODS is the word list used for this and other word
games at the Little Golem website.)

9letter words  8letter words  7letter words  6letter words  5letter words  
Number of Games Played  262  800  1,650  3,500  4,500 
Number of wins by Player #1  17 (6.49%)  35 (4.38%)  29 (1.76%)  16 (0.46%)  3 (0.07%) 
Number of wins by Player #2  209 (79.77%)  677 (84.62%)  1,492 (90.42%)  3,197 (91.34%)  3,819 (84.87%) 
Number of Draws  36 (13.74%)  88 (11.00%)  129 (7.82%)  287 (8.20%)  678 (15.07%) 
Average Final Score of Player #1  55.06  53.55  50.45  45.28  38.87 
Average Final Score of Player #2  57.09  55.57  52.19  46.52  39.76 
Number of Halftime Wins by Player #1  3 (1.15%)  8 (1.00%)  13 (0.79%)  10 (0.29%)  2 (0.04%) 
Number of Halftime Wins by Player #2  244 (93.13%)  749 (93.62%)  1,554 (94.18%)  3,224 (92.11%)  3,821 (84.91%) 
Number of Halftime Draws  15 (5.73%)  43 (5.38%)  83 (5.03%)  266 (7.60%)  677 (15.04%) 
Average Halftime Score by Player #1  23.40  23.15  22.69  21.30  18.87 
Average Halftime Score by Player #2  25.46  25.20  24.47  22.54  19.76 
Number of wins by Player #1 by exactly 1 point  10 (58.82%)  24 (68.57%)  21 (72.41 %)  13 (81.25%)  2 (66.67%) 
Number of wins by Player #1 by exactly 2 points  5 (29.41%)  10 (28.57%)  5 (17.24 %)  3 (18.75%)  1 (33.33%) 
Number of wins by Player #1 by exactly 3 points  1 (5.88%)  1 (2.86%)  3 (10.34 %)  0 (0.00%)  0 (0.00%) 
Number of wins by Player #1 by exactly 4 points  0 (0.00%)  0 (0.00%)  0 (0.00%)  0 (0.00%)  0 (0.00%) 
Number of wins by Player #1 by 5 or more points  1 (5.88%)  0 (0.00%)  0 (0.00%)  0 (0.00%)  0 (0.00%) 
Number of wins by Player #2 by exactly 1 point  55 (26.32%)  160 (23.63%)  445 (29.83 %)  2,149 (67.22%)  3,653 (95.65%) 
Number of wins by Player #2 by exactly 2 points  45 (21.53%)  225 (33.23%)  743 (49.80 %)  957 (29.93%)  165 (4.32%) 
Number of wins by Player #2 by exactly 3 points  55 (26.32%)  170 (25.11%)  246 (16.49 %)  84 (2.63%)  1 (0.03%) 
Number of wins by Player #2 by exactly 4 points  28 (13.40%)  84 (12.41%)  50 (3.35 %)  7 (0.22%)  0 (0.00%) 
Number of wins by Player #2 by 5 or more points  26 (12.44%)  38 (5.61 %)  8 (0.54 %)  0 (0.00%)  0 (0.00%) 
Number of games Player #1 did better on final move  75 (29.63%)  183 (22.88%)  134 (8.12%)  38 (1.09%)  2 (0.04%) 
Number of games Player #2 did better on final move  66 (25.19%)  129 (16.12%)  70 (4.24%)  20 (0.57%)  0 (0.00%) 
Number of games final move of each player was equal  121 (46.18%)  488 (61.00%)  1,446 (87.64%)  3,442 (98.45)  4,498 (99.96%) 
Observations: Against two players of equal strength, who always play a word that is at least equal to the longest possible word on for that move, (and who both both do not look ahead more than the given board position), Player #2 does indeed have a very large advantage. Player #2 wins 79% to 91% of the games outright, no matter what the vocabulary. The average margin of victory for Player #2 increases with the vocabulary. With a full vocabulary of 9letter or 8letter words or less, the average margin of victory is about two points. However, note that about 37%  39% of the time, Player #2 wins by exactly three or four points. The frequency that Player #2 is leading the game at halftime is always larger than the final Player #2 win percentage! For example, with a complete knowledge of all 8letter and less words, Player #2 wins the game 84.62% of the time, but Player #2 is leading at halftime a full 93.62% of the time! When Player #1 does win, 90% of the time it is by just one or two letters. Only one time out of all 10,712 games did Player #1 win by four points or more. Player #1 is often able to make up a very tiny bit of ground on his final move of each game. For example, with a vocabulary of all 8letter and less words, 22.88% of the time Player #1 is able to play a move that is worth more than Player #2's final move. 16.12% of the time Player #2 played a word that was worth more than Player #1s final move. Note: When Player #2 does better on the final move, this means, of course, that this move only became possible as a result of Player #1's last move. (Otherwise, Player #1 would have played it when it was his turn.) A one move lookahead feature will only benefit Player #1 in this case. 
Frequency
of 9Letter
Words
After the 262, 9letter or less games were
played, I later determined the total number of 9letter words that were
formed in all of those games. I was surprised by the number
 it was slightly higher than I initially would have suspected.
If you're curious, there were nine games (of 262) that DID indeed start with a 9letter word on Black's third move, the first opportunity to see such a word:
1) sap, pass, pases, peases, pleases, pleasers,
pleasures, pleasured, reassured, treasured, reassures, resamples,
tressures, simplesse, masterly, sesamoid, promisers In the game immediately above, Player #2 scored a total of 66 points... the most points possible with a vocabulary of just 9letter words. (5 points on Move #2, 7 points on Move #4, and 9 points on EACH of her other moves.) This was the highest scoring game of all 10,712 games I played. 
Interesting
Games
Game # 253:
Catherine formed the longest move available, FICHU, leaving a choice of nothing but several 4letter words for Player #2 on the final move. (Catherine randomly chose FASH.)
This is a position from one of the two games out of 4,500 that Player #1 was ahead at halftime, with a vocabulary of nothing but 5letter words: Game # 524: Player #1 just formed the word 5letter word WUXIA. It's Player # 2's turn to play and there are no other 5letter words possible. Player #2 randomly picked FIAR. For the rest of the game, nothing but 5letter words were played, and thus Player #1 went on to win the game by one point.
The Most Interesting Game Of All The one game out of all 10,712 games, in which Player #1 won by five points, looked like this: Game # 88: 
Update:
A small change to the program was made and the 8letter vocabulary test
was rerun. 
8letter words  8letter words but Player #1 chose the LEAST popular letter 

Number of Games Played  800  1,300 
Number of wins by Player #1  35 (4.38%)  199 (15.31 %) 
Number of wins by Player #2  677 (84.62%)  846 (65.08 %) 
Number of Draws  88 (11.00%)  255 (19.62) % 
Average Final Score of Player #1  53.55  50.98 
Average Final Score of Player #2  55.57  52.09 
Number of Halftime Wins by Player #1  8 (1.00%)  27 (2.08%) 
Number of Halftime Wins by Player #2  749 (93.62%)  1,073 (82.54%) 
Number of Halftime Draws  43 (5.38%)  200 (15.38%) 
Average Halftime Score by Player #1  23.15  22.13 
Average Halftime Score by Player #2  25.20  23.59 
Number of wins by Player #1 by exactly 1 point  24 (68.57%)  122 (61.31 %) 
Number of wins by Player #1 by exactly 2 points  10 (28.57%)  55 (27.64 %) 
Number of wins by Player #1 by exactly 3 points  1 (2.86%)  16 (8.04 %) 
Number of wins by Player #1 by exactly 4 points  0 (0.00%)  5 (2.51 %) 
Number of wins by Player #1 by 5 or more points  0 (0.00%)  1 (0.50 %) 
Number of wins by Player #2 by exactly 1 point  160 (23.63%)  328 (38.77 %) 
Number of wins by Player #2 by exactly 2 points  225 (33.23%)  265 (31.32 %) 
Number of wins by Player #2 by exactly 3 points  170 (25.11%)  155 (18.32 %) 
Number of wins by Player #2 by exactly 4 points  84 (12.41%)  70 (8.27 %) 
Number of wins by Player #2 by 5 or more points  38 (5.61 %)  28 (3.31 %) 
Number of games Player #1 did better on final move  183 (22.88%)  389 (29.92%) 
Number of games Player #2 did better on final move  129 (16.12%)  176 (13.54%) 
Number of games final move of each player was equal  488 (61.00%)  735 (56.54%) 
Average Number of Moves Available  Move 1  unknown  188.78 
Average Number of Moves Available  Move 2  unknown  268.90 
Average Number of Moves Available  Move 3  unknown  404.20 
Average Number of Moves Available  Move 4  unknown  477.41 
Average Number of Moves Available  Move 5  unknown  595.25 
Average Number of Moves Available  Move 6  unknown  641.47 
Average Number of Moves Available  Move 7  unknown  711.54 
Average Number of Moves Available  Move 8  unknown  690.51 
Average Number of Moves Available  Move 9  unknown  723.39 
Average Number of Moves Available  Move 10  unknown  666.96 
Average Number of Moves Available  Move 11  unknown  649.40 
Average Number of Moves Available  Move 12  unknown  564.39 
Average Number of Moves Available  Move 13  unknown  524.81 
Average Number of Moves Available  Move 14  unknown  411.27 
Average Number of Moves Available  Move 15  unknown  340.90 
Average Number of Moves Available  Move 16  unknown  199.50 
As you can see,
the program now also keeps track of the number of possible moves
(words) available during each turn. As you can imagine,
these numbers for both sides will be higher when Player #1 is not
using the 'less frequently chosen letter' strategy. With more time (and more of an interest) there are lots of other types of data and totals and percentages I could keep track of. And as mentioned above, someday I'd like to include a larger sample of runs for 9letter words, as well as a sample of runs with the possibility of 10 and 11letter words. MAYBE I will do this in the future, but I've already moved on to another programming project, so don't look for any further updates any time soon.  Ed 