This is a proof that any strategy is better than no strategy. No game is random.
Assume that every conceivable game is not random, and therefore that a strategy can be devised for each. Even if the assumption is wrong, then nothing is lost because any counter-example game to the assumption would be random or break-even regardless any strategy already devised for it.
Eg, suppose someone randomly picks two unequal real numbers from 1 to 10 and then reveals one of the numbers, either the higher one or the lower one. Bet what, that the other number is higher or lower than the one revealed?
A real number in base n=10 is any number of the form 123.456789... which comprises the digits 0 to n-1=9.
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