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The example is a coin flip but it could be any random based game by using a different algorithm than "number mod 2". You're correct that it can't act on real world data.

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More specifically xor 2 initially hidden integers and mod X + 1 gives you a random number from 1 to X.

PS: Adding numbers mod X will provide a fair random number as long as at least one side provides a fair random number to start with. Which is a stronger guarantee if your making a real money wager.

... with a bias if X is not a power of 2.

Depends on the maximum range provided. If both part's pick a X bit number then you need a power of 2. If they pick a number 1 to X then can use X.

You're right of course. Just remember to use addition mod X instead of xor in that case to combine the choices.

Could this be modified to support, say, a chess game?

Chess doesn't have any random element so it wouldn't need this kind of system. You could implement card games and board games that do have randomness like candyland.

You could implement chess with only movement commitments and no atomic fair random number generation.

Some aspects of it would be interesting for non-random games too, e.g. the payout if one player stops submitting moves, verifying who won, ..

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