Depends on how you look at it (or, I guess, whether you look at it). If, after the first player cuts, you were to look through the remaining cards to see what the second player had left to pick from, you could calculate that second player’s odds. Those odds would depend on which cards were eliminated; they would be different than if the second player could pick any card except for the one chosen by the first player.
But if you don’t look, and thus don’t know which cards are left to choose from, the second player still has an equal chance of getting any of the cards except the one card chosen by the first player.
Thus, it seems like the act of looking changes the odds. Hence @md-2000’s reference to “Schroedinger’s draw.”
(If you’re familiar with the game Deal or No Deal, imagine how the game would be different if the eliminated suitcases were never opened and no one knew what was inside them.)
This would be bad. If the first guy shows AH then all future players now know exactly where they need to aim. They may get it, they may not, but they now have a target. If you do this, you need to remove the bottom card before putting it back.
But that’s irrelevant here. The player who makes the first cut certainly does not have this information before he makes his cut, and that’s what matters. If he did have that information, he would have an unfair advantage.
If an independent party were to look at what cards remain available after the first cut and update the probabilities, that might be interesting information on the players’ prospects, but it is not information that anyone can act on to gain any advantage.
Yes, it’s a critical part of the Monty Hall problem that Monty knows what’s behind each door. This must be true, since the door he opens and eliminates is always a goat, he’s not opening a door at random (and this is not always emphasized adequately when the problem is described). But what’s quite different in that situation is that the player gains this information via Monty’s action and can then act on it. I think the best way to understand the two-thirds probability of success if you switch in the Monty Hall problem is to realize that if (say) you originally chose door 1, when Monty inspects what’s behind doors 2 & 3 and eliminates a goat, he’s effectively offering you the chance to switch to the best of doors 2 & 3.
I assume knowing what the previous cutter(s) got affects the perception of odds and hence perception of fairness. Similarly, cutting deep is a dick move - it doesn’t affect the odds, but affects the perception of odds and fairness, if the subsequent cutters think it does.
The logic is something like - “if I can pick from a whole deck, there are 4 aces I could land on; with 1/5 the deck, good chance there’s no ace in there.” Flawed logic about a properly shuffled deck, but a conception in the minds of some players.
So the question then is - how do you decide who cuts first? I think cut, return all cards, shuffle the whole deck, cut again, is the perceived as most fair technique.