Removing "paradox" from "Monty Hall paradox"

Brief rundown of what the Monty Hall paradox is, just for the sake of completeness: you must choose one of three doors on a TV show, only one of which has a valuable prize behind it. After you make a choice, host of the show opens another door and offers you to switch. Switching turns out to be preferrable course of action, because the probability of the prize being behind another door is 2/3 instead of 1/2 that one might think.

I've heard of Monty Hall paradox many times, and heard it once more today. I've finally figured out what's paradoxical about it, at least for me. When I hear about TV show shenanigans — I assume that the host has an ability to trick show contestant because they probably have an incentive to keep the contestant from prize. So, if the rules of the show is that the host has to offer the choice, then using this choice is, for me, intuitively right.

Now, assume that it is not necessarily the case, that is, host can open another door and offer a choice to switch, but it's at their discretion whether to actually do it. Let's try to figure out what the optimal strategy is going to be in this situation.

Assume the following model:

• If the host knows that you've made the right choice, they offer you to switch with probability p₁ (with probability 1 − p₁ you get your prize instantly and there are nothing you can do about it)
• If the host knows that you've made the wrong choice, they offer you to switch with different probability, p₂.

Now, you've found yourself in a situation when host actually offers you a choice. Using Bayes' theorem, we can determine that the probability of you already selecting the prize is p₁ / (p₁ + 2p₂).

The correct strategy in this case seems to be to compare that probability with 1/2 and switch if it is less than that, which turns to be exactly when p₁ < 2p₂. Note that when the both probabilities are equal to 1, which corresponds to TV host having to offer the switch, this model still gives "switch" as an optimal strategy, so at least it's consistent with prior art.

Now, if you have some existing data about the show, you can probably estimate these parameters. Assuming uniform distribution for both p₁ and p₂ the answer is still choosing to accept the offer to switch. That's already too much of a complicated model for me to have intuitive feelings about it, so it doesn't feel paradoxical. Might be not the case for you though!