[pzhon]

Recursive Mathematica code:

Clear[gamevalue];
gamevalue[black_, white_] :=
gamevalue[black, white] =
If[black == white == 0 || black < 0 || white < 0, 0,
Max[0, black/(black + white) (-1 + gamevalue[black - 1, white]) +
white/(black + white) (1 + gamevalue[black, white - 1])]]

Badly formatted output:

Table[gamevalue[i, j], {i, 0, 10}, {j, 0, 10}]

0 1 2 3 4 5 6 7 8 9 10
0 1/2 4/3 9/4 16/5 25/6 36/7 49/8 64/9 81/10 100/11
0 0 2/3 3/2 12/5 10/3 30/7 21/4 56/9 36/5 90/11
0 0 1/5 17/20 58/35 71/28 145/42 527/120 294/55 347/55 1041/143
0 0 0 12/35 1 113/63 93/35 235/66 202/45 777/143 492/77
0 0 0 0 4/9 47/42 21/11 199/72 428/117 705/154 182/33
0 0 0 0 1/15 127/231 284/231 3457/1716 8594/3003 5356/143 1333/286
0 0 0 0 0 61/396 563/858 4583/3432 13606/6435 8449/2860 9319/2431
0 0 0 0 0 0 697/3003 49/65 18457/12870 53683/24310 133097/43758
0 0 0 0 0 0 0 311/1040 10224/12155 74131/48620 106045/46189
0 0 0 0 0 0 0 0 7793/21879 84855/92378 26995/16796