Who wins at
Memory?
While playing Memory today, I thought about 1) the best strategy to win and 2) who wins statistically, the player to go first or the player to go second?
Let's assume that both players have perfect memory, so that all that matters is statistics. Let's also assume that you must reveal at least one new card on your turn, so we don't get stuck with both players only choosing known cards because revealing a new card decreases your chances of winning.
1) I have a good idea about the optimal strategy. When there are lots of cards left, it's is better to only reveal one new card, and then pick a card which is already known.
Reasoning: if you reveal a card which you don't know the match for, you should not reveal another card, because that increases the risk that your opponent can match that card with an already known card.
If there are only a few cards left, it can be better to switch to reveal two new cards, if the first one isn't a match to a known card. For example: 6 cards left (AABBCC), 2 cards known (AB). You reveal a card which is C. If you guess on one of the three remaining cards, you have 1/3 chance of winning. If you instead pick one of the known cards, you have 0 chance of winning, because all of the new cards which the opponent turns up will match a known card.
2) For a small number of pairs, calculating player 1's chance of winning is easy. For example, for 2 pairs is 1/3. Reveal one card, A. You now have 1/3 chance of guessing the matching card. Whoever gets the first pair also gets the second.
For 3 pairs, it is 1/15 (win directly) + 8/15 (win on second turn), or 3/5.
How about x number of pairs? Is there a way to generalize this?