Door Guessing GAME

[fabiocbinbutter]

With the "reshuffle and keep going" mechanic, the optimal strategy for the MitD is to discard the rest of the deck each time, forcing a reshuffle every turn. Redcloak's odds are marginally better for a second card from the same deck (1/51) instead of the first card from a reshuffled deck (1/52), which is true every time the deck is reshuffled; and the MitD is the only player that can act on that information (to deprive Redcloak of the marginal improvement of odds).

At first, I thought this too, but then I realized that by picking a large number of cards on his first turn (for example 25), then MitD has a ~50% chance of discarding the gate, making it useful information. If he does discard the gate in this way, he can 100% safely guarantee another 25 turns using this information by not discarding anything and forcing Redcloack to pick duds for 25 turns.

We know that the expected value of the discard all strategy is (one plus) the mean of a negative binomial distribution (r=1,p=51/52), so 52 turns.

On the otherhand, if MitD's first action is to discard roughly half the deck (25 cards), he has a 25/51 chance to hit the gate and a (26/51) chance to miss the gate.

Mitd Hit: 25/51
MitD Miss: 26/51

Given that he misses the gate, Redcloak has one chance to get it at 1/26. He has a 25/26 chance to miss the gate (after which for simplicity of deriving the EV, we will say that MitD returns to the default 'discard all' strategy)

MitD miss & Redcloak hit: (26/51) * (1/26)
MitD miss & Redcloak miss: (26/51) * (25/26)

Going down the other branch, if MitD hit, redcloack has 100% miss chance:

MitD hit & Redcloack miss: (25/51) * 1

Now the EV's for each branch are:

MitD miss & Redcloak hit: (26/51) * (1/26) * 1 turn
MitD miss & Redcloak miss: (26/51) * (25/26) * (1+52) turns (because we just added one safe turn, then returned to the default EV52 strategy)
MitD hit & Redcloack miss: (25/51) * 1 * (26+52) turns (because we just added 26 safe turns, then returned to the default EV52 strategy)

Adding these all up we get an EV of
(26/51)*(1/26)*1 + (26/51)*(25/26)*(1+52) +(25/51) * 1 * (26+52) = 64.2
...in other words, a 28% improvement

And this is just modifying one turn with a not necessarily optimized initial discard number. If we were to apply this strategy each time the whole deck were shuffled rather than just the first play, the EV would be necessarily be higher.

Please let me know if I made any mistakes in logic or calculations...

(Actually, re-reading the rules, it's not clear whether the discard pile is revealed or not. I guess I just assumed so from playing so much Magic the Gathering, lol)