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If we start with a move into cell (2,2) then 8 moves are left (if there is no win before), which gives us a theoretical amount of 8! = 40320 many different possible continuations. But if we look to a concrete continuation then only a small fraction does make 'sens' for the game.
-->pwx1=[1/9]
Probability for beginner 'X' to take the middle field to win
-->BOARD(2,2)='X'
BOARD =
!_ _ _ !
! !
!_ X _ !
! !
!_ _ _ !
-->pwo=[4/8]
Probability of the opponent 'X' to take one of the corners
-->BOARD(1,1)='O'
BOARD =
!O _ _ !
! !
!_ X _ !
! !
!_ _ _ !
-->pwx1=[1/9 6/7]
We count only those fields as 'possible' which can lead
to a possible goal
-->BOARD(3,2)='X'
BOARD =
!O _ _ !
! !
!_ X _ !
! !
!_ X _ !
-->pwo=[4/8 4/6]
Theoretically opponent 'O' has 4 possible cells for a goal,
but practicall he has to chose (1,2) to hinder a win by 'X'
-->BOARD(1,2)='O'
BOARD =
!O O _ !
! !
!_ X _ !
! !
!_ X _ !
-->pwx1=[1/9 6/7 5/5]
Theoretically beginner 'X' has 5 cells to complete a goal, but
he has to select (1,3) to hinder 'O' to win.
-->BOARD(1,3)='X'
BOARD =
!O O X !
! !
!_ X _ !
! !
!_ X _ !
-->pwo=[4/8 4/6 2/4]
Theoretically opponent 'O' has 2 possible cells for a goal,
but practicall he has to chose (3,1) to hinder a win by 'X'
-->BOARD(3,1)='O'
BOARD =
!O O X !
! !
!_ X _ !
! !
!O X _ !
-->pwx1=[1/9 6/7 5/5 3/3]
Theoretically beginner 'X' has 3 cells to complete a goal, but
he has to select (2,1) to hinder 'O' to win.
-->BOARD(2,1)='X'
BOARD =
!O O X !
! !
!X X _ !
! !
!O X _ !
-->pwo=[4/8 4/6 2/4 2/2]
Opponent has no more a chance to win but he has to defend himself
because beginner 'X' can still win. He has to select (2,3)
-->BOARD(2,3)='O'
BOARD =
!O O X !
! !
!X X O !
! !
!O X _ !
-->pwx1=[1/9 6/7 5/5 3/3 1/1]
No player can win any longer. Only 1 cell is left which does not matter any way.
-->BOARD(3,3)='X'
BOARD =
!O O X !
! !
!X X O !
! !
!O X X !
-->proda=1, for i=1:4, proda=proda*pwx1(1,i),end
proda = 0.0952381
-->prodb=1, for i=1:4, prodb=prodb*pwo(1,i),end
-->proda*prodb
ans =
0.0158730
Overall probability to generate such game is 1.5%
-->com1=[1 4 6 1 1 1 2 1]
This is the 'real' number of possible alternatives.
-->fac_goal=1, for i=1:length(com1), fac_goal=fac_goal*com1(1,i), end
fac_goal =
48.
-->fac_th=1, for i=1:8, fac_th=fac_th*i, end
fac_th =
40320.
-->fac_prac=fac_goal/(fac_th/100)
fac_prac =
0.1190476
Thus only 0.12% of the theoretical possible games are making 'sense'.
What we can see here is that there is no real win chance for the beginner, even if he starts in the middle (2,2) of the board. But this assumes an 'ideal' player taking the 'best' options. Assuming this we see that there are only 48 of 40320 possible games to play to approach a goal state, this are 0.12% of the theoretical possible cases. Furthermore there have been 5 of 9 moves which have been caused by the need to block the other wo win, 3 of 5 possible wins for the beginner and 2 of 5 possible wins of the opponent, each time 'after' the beginner. If the number of '48' possible games for 'case 1 with corners for opponent' is some measure for 'case 1 with middle fields for opponent', then we could assume about '96' possible games for one field and then - working hypothesis - about 'sensful' 864 games for all cases.
This has to be analyzed further.
If it is true that the beginner can not win if the opponent responds 'ideally' then we can not measure a LCS system with his wins; we can only measure how close it is to zero losses. Every deviation of a zero loss rate would indicate that the system is sub-optimal. If it would moreover win and not lose then we would have an indicator that the system is playing 'ideally' and at the same time the opponent is sub-optimal. In this sense it is possible to use an ideal LCS as a measure for the capabilities of its opponent.