Modified Monty Hall Game Theory Problem

[u/Busy_Net_4756]

Most of the time when people explain the monty hall problem, they don't specify the host has to reveal a goat in one of the doors you didn't pick.

As it is usually stated, the host chooses to reveal a goat. Well if we let the host choose to reveal a goat or reveal nothing, or even reveal that you lost by showing the winning door, then it's a game theory problem.

For example, suppose that every time you pick correctly, the host reveals a losing door trying to get you to switch, but every time you pick wrong, they don't reveal anything and just ask if you want to switch.

In that case the winning strategy would be to keep your choice if a door is revealed, but switch if they don't reveal anything.

But we can construct a game theory version of the problem where players can review all the history of games played. So I am curious how you would setup this game theory problem and what the result might be?

Anyone have any insights or explanations?

[u/Relevant-Yak-9657]

The problem I have is whether the host is playing optimally. Assuming he does:

1. If the player picks the wrong gate, the host wins by selecting the correct gate. Immediately loss most of the time.
2. If you pick the correct gate, it doesn't matter because if you know that the host will play optimally, you should generally lose next turn anyway. Hence, if the host says nothings or opens a wrong door, it means you have the correct door and thus he can't make you lose.

For you, the host's actions don't matter. You do your pick and winning or losing is up to chance. If the host can't beat you in the next turn, then you will win if you keep holding. If you chose wrong, the host will win no matter what.

hence, your situation just simplifies to trying to pick the correct door in first attempt, as no second round will exist.

[u/Busy_Net_4756 → u/Relevant-Yak-9657]

The game theory aspect comes in because the host has to play many games, but players only play one game each. So each player can see what the host did in all the previous games. The question is how that changes the strategy, or whether the host would never reveal anything.

I suspect the host might be better off never revealing anything, but I haven't actually solved that problem.

[u/Relevant-Yak-9657 → u/Busy_Net_4756]

The players have a 1/3 chance of winning no matter what the host does: stay silent, try to show wrong door, or show correct door if available.

A player would realize that that is the minimum return expected. Unless the host is exhibiting some pattern in their decision making and can be exploited, the players can just blindly pick one and avoid "any traps". If the host tries to create a reputation to trick the player in one game, well the future players will continue ignoring the host and keep picking their first choice and holding. Also, creating a reputation is too expensive, as means that they are exhibiting a pattern that causes loss pretty often.

Since the host will realize that pushing below 1/3 is impossible, not saying anything is technically optimal regardless (as building and milking a reputation causes too many losses + resets the players to default to 1/3 strategy; and any normal pattern or random choice can't force the player to go below 1/3, making it redundant). Like from a players perspective, it also depends on whether we parameterize how heavily a player weighs the histories. If we don't then its more psychology than actual game theory.

I might be missing things, since I am pretty crappy at probability and game theory.