Yet Another Connection Conundrum

Alice and Bob are playing a game on a $4$ x $4$ board. The players alternate between turns. On each turn, the player must place exactly one of their markers on an unoccupied square. In order to place a marker on the square located on the $i$-th column and $j$-th row of the board, the player must pay a cost $Cij$. This process continues until the board is completely filled (i.e. no more moves can