Subfields Case 3: Between Corners

Here we look to the case 3.1 where the beginner 'X' starts not in the middle field but in one of the fields between the corners (1,2),(2,1),(3,2),(2,3) ; then the opponent moves into the middle field.

-->BOARD=W3
 BOARD  =
 
!_  _  _  !
!         !
!_  _  _  !
!         !
!_  _  _  !
 
-->pwx31=[4/9]

-->BOARD(1,2)='X'
 BOARD  =
 
!_  X  _  !
!         !
!_  _  _  !
!         !
!_  _  _  !
 
-->pwo=[1/8] 
 
-->BOARD(2,2)='O'
 BOARD  =
 
!_  X  _  !
!         !
!_  O  _  !
!         !
!_  _  _  !
 
-->pwx31=[4/9 2/7]

To extend the actual piece to a goal the beginner has two options.

 
-->BOARD(1,1)='X'
 BOARD  =
 
!X  X  _  !
!         !
!_  O  _  !
!         !
!_  _  _  !
 
-->pwo=[1/8 4/7]
-->potwinx=[1]

The opponent has 4 options to extend the actual piece to a goal, but 
there is the immediate need to hinder the beginner to complete his goal.
 
-->BOARD(1,3)='O'
 BOARD  =
 
!X  X  O  !
!         !
!_  O  _  !
!         !
!_  _  _  !
 
-->pwx31=[4/9 2/7 2/5]
 -->potwino=[1] 

The beginner has 2 options to extend the actual iece to a goal, but 
there is thenimmediate necessity to hinder the opponent to win
 
-->BOARD(3,1)='X'
 BOARD  =
 
!X  X  O  !
!         !
!_  O  _  !
!         !
!X  _  _  !
 
-->pwo=[1/8 4/7 3/4]
 -->potwinx=[1 1]

The opponent has  3 options to extend the actual pieces to a goal, but 
there is the immediate need to hinder the beginner to complete his goal

-->BOARD(2,1)='O'
 BOARD  =
 
!X  X  O  !
!         !
!O  O  _  !
!         !
!X  _  _  !
 
-->pwx31=[4/9 2/7 2/5 2/3]
-->potwino=[1 1]

The beginner has 2 options to extend the actual piece to a goal, but 
there is the immediate need to hinder the beginner to complete his goal

-->BOARD(2,3)='X'
 BOARD  =
 
!X  X  O  !
!         !
!O  O  X  !
!         !
!X  _  _  !
 
-->pwo=[1/8 4/7 3/4 2/2]  
 
-->BOARD(3,2)='O'
 BOARD  =
 
!X  X  O  !
!         !
!O  O  X  !
!         !
!X  O  _  !
 
-->pwx31=[4/9 2/7 2/5 2/3 1/1]

-->BOARD(3,3)='X'
 BOARD  =
 
!X  X  O  !
!         !
!O  O  X  !
!         !
!X  O  X  !

-->prodall=proda*prodb  =  0.0011467 

Probability for this game to happen is 0.1%! 

-->com1=[1 1 1 2 4 2 1 1 1]

Possible alternatives per turn with regard of the necessity to hinder 
the other player to win.  
 
-->fac_goal=1, for i=1:length(com1), fac_goal=fac_goal*com1(1,i), end

 fac_goal  =
 
    16.  
 
-->fac_th=1, for i=1:7, fac_th=fac_th*i, end
  
 fac_th  =
 
    5040.

Again there is no real win chance for the beginner (Presupposing an 'ideal' player taking the 'best' options. Assuming this we see that there are only 16 of 5040 possible games to play to approach a goal state, this are 0.38% of the theoretical possible cases. Furthermore there have been 4 of 9 moves which have been caused by the need to block the other wo win, 2 of 5 possible wins for the beginner and 2 of 5 possible wins of the opponent, each time 'after' the beginner. If the combinatorics of the game above can be transferred to the other 3 options then the number of all possible cases for 3.1 would be 16 x 4 = 64 possible games.

This has to be analyzed further.

Again 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 thre 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.

For further experiments the following working hypotheses will betested:

1. Analyze and improve the version LCS_v0.18.sce until the win rate of a random beginner reaches 0%
2. Analyze and improve the version LCS_v0.18.sce until the win rate of two classifier systems as beginner and opponent simultaneously reaches 0%
3. Analyze and improve the version LCS_v0.18.sce until the win rate of an ideal beginner reaches 0%