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Regarding the hidden information, consider this thought experiment:
Both players pick a word, then simultaneously reveal their choice to the opponent. A deterministic algorithm takes both words as input and outputs either a 0 or a 1, selecting one of the players as the winner. The algorithm is known to both players beforehand.
Let's say the deterministic algorithm is a hash function. Now the outcome of the game is essentially ("for all intents and purposes") random, and there is very little trace of skill left. Even though it's not random, it's actually 100% deterministic. It's just that you don't have a good way to predict the outcome of your choice.
So is hidden information really sufficient to argue that a game does not have an element of chance?
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