Play a Perfect Game of 4x4 Hex
Marbelous, 973b

This is a naive implementation of the hinted strategy in the question. It expects the board to be provided as 16 commandline/mainboard parameters like hex.mbl B W E E E W E E E B E E E E E E and it will output the zero-indexed position of white's next move.

00 }1 }0
&amp;G B? W!
&amp;0 &amp;G &amp;G
!!
00 }0 }1
&amp;H B? W!
&amp;1 &amp;H &amp;H
!!
.. }D .. }9 }9 }D }E }A }D }9 }A .. }E }F }5
}9 B! }E E? W? E? .. E? B? B? W? .. E? .. E?
B! ?0 B! &amp;M &amp;N &amp;O E? &amp;I \\ .. &amp;J .. &amp;K E? &amp;5
?0 ++ ?0 &amp;9 .. &amp;D &amp;P &amp;A &amp;I /\ &amp;J .. &amp;E &amp;L !!
++ .. ++ !! .. !! &amp;E !! \/ &amp;L /\ &amp;K !! &amp;F
\\ .. // .. .. .. !! .. .. \/ .. \/ .. !!
.. =3 \/
&amp;M /\ &amp;N
\/ &amp;O /\ &amp;P
}0 }1 }6 .. }6 .. }7 &amp;7 }2 }3 }A }A }B }E }F
..
..
..
..
..
..
..
..
..
.. .. .. .. .. .. .. .. .. .. .. .. .. B? W!
.. .. .. .. .. .. .. .. .. .. .. .. .. &amp;Q &amp;Q
.. .. .. .. B? .. E? W? E? .. E? B? E? &amp;F \/
.. .. .. &amp;S /\ &amp;T &amp;S &amp;T &amp;U E? &amp;A &amp;R &amp;R !!
.. .. .. &amp;7 .. .. \/ .. &amp;2 &amp;V !! &amp;B \/
.. .. .. !! .. .. &amp;U /\ &amp;V &amp;3 .. !!
.. .. .. .. .. .. .. .. .. !!
.. .. ..
.. .. E?
E? .. &amp;6
&amp;X E? !!
!! &amp;Y
.. !!
}4 }8 }C
\/ \/ \/
30 30 31 31 32 33 35 36 37 39 31 30 31 31 31 33 31 34 31 35
&amp;0 &amp;X &amp;1 &amp;Y &amp;2 &amp;3 &amp;5 &amp;6 &amp;7 &amp;9 &amp;A &amp;A &amp;B &amp;B &amp;D &amp;D &amp;E &amp;E &amp;F &amp;F
:W?
}0
^4
=1
{0
:B?
}0
^0
=0
{0
:E?
}0
^1
=0
{0
:W!
}0
^4
=0
{0
:B!
}0
^0
&gt;0
{0


I think I can probably golf about 200 characters out of this with better branching and elimination of code re-use.