There is no answer to this question, not without involving some 'puzzling trickery'.
We have been given that G, A, M, E and S are all single digits.
Now, GAMES can be interpreted in one of 3 ways mathematically:
GAMES = 10000G + 1000A + 100M + 10E + S
GAMES = G + A + M + E + S
GAMES = G x A x M x E x S
Assume GAMES = 10000G + 1000A + 100M + 10E + S
Taking the first equation:
GAMES / G = 1
We can multiply both sides by G to arrive at:
GAMES = G
Now, the left hand side can be rewritten as shown below, similar to how the number 123 is actually 1x100 + 2x10 + 3x1:
(10000 x G) + (1000 x A) + (100 x M) + (10 x E) + (1 x S) = G
Which can be simplified to:
9999G + 1000A + 100M + 10E + S = 0
The sum of positive numbers (on the left side) cannot equal zero, and is therefore unsolvable.
Assume GAMES = G + A + M + E + S
Similar to the above, this will resolve to:
A + M + E + S = 0
Which again brings us to an impossible situation where the sum of positive digits equals zero.
Assume GAMES = G x A x M x E x S
Taking the first equation:
GAMES / G = 1
This then becomes:
A x M x E x S = 1
The product of individual digits 0-9 cannot equal 1, since we a constraint of the puzzle is that the digits are distinct.
Addendum - We know that G, A, M, E and S are all positive digits, because:
A, M, E and S all appear with a digit to their left, so these letters cannot be replaced by a negative digit, therefore must be positive or zero.
None of the 5 letters can be zero, as the division by that digit would result in an infinite result and not 1-5.
G must be a positive digit, because the division by A, M, E and S (which are known to be positive by the preceding two points) all result in a positive number, indicating that GAMES itself is positive.
