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[matt979]

I won a game today because my opponent possessed me.

With one Province left and less than a Province of score difference, he possessed my Bishop-S-C-C-Province; my next hand was G-S-C-C-C.

If he knew both my hand and my deck, he clearly wouldn't possess me. Maybe even knowing my hand would be sufficient not to possess me.  So even though possessing me was obviously the odds-on right thing to do, if it makes any sense at all to repurpose the Fundamental Theory of Poker to Dominion then his play happened to be FTOD-incorrect.

This is vanishingly unlikely to have wider applications -- I can't think of situations where your play (basically limited to reactions) could successfully deceive your opponent about your hand, or even be worth doing.  But maybe a different analogy is better, to the bridge (the game, not the card :-)) skill of keeping track of cards and understanding what's (nost likely) actually in your opponent's hand.

Just some slapdash thoughts for now...

[DStu]

Just for someone who ran about this FTOP for the first time:

It says that, when you don't know everything about your opponent, your performance is the best if you (by chance) behave as if you knew everything about him?

Sounds really trivial to me...

[luv2breformed]

Sklansky's FTOP is misleading anyway. The reason is because you can never know your opponent's exact hand (barring some sort of ridiculous tell). What you actually know is your opponent's RANGE of hands. You CANNOT play optimally against his exact hand. You can only play optimally against his range of hands.

Suppose you know for certain that your opponent will go all-in preflop only with AA and AK. Given his range (and a rakeless environment), it is always mathematically correct to call him with 22 because there are only 6 ways he can make AA, and 16 ways he can make AK(of course you may pass up on this slight edge if you know you are a much better player than him and may get a better edge in the future). FTOP says if you knew he had AA, you would fold, and if you knew he had AK you would call. But again, you CAN'T know that information.

Applying this proper understanding to Dominion, we can actually use it to help us make meaningful decisions. Suppose you have been counting your opponents cards and your own. You are considering breaking the PPR as there are 2 provinces left, and if you get both you know you will win. However, you will lose if he can buy the last 1 on his turn. Suppose you have 8 this turn, and given your deck composition you know you will have 8 next turn. He has 10 cards left in his deck, consisting of Gold, Gold, Estate, Province, Silver, Silver, Curse, Curse, Province, Province. Do you break the PPR?

All that must be done is that you must analyze your opponent's RANGE of possible hands. How many ways can he make 8? Only 2 ways, with both Golds and one of the silvers, or both Golds and the other silver (obviously he can make more than 8 by adding to these combinations, but that is what he needs at minimum). Compare this to the many combinations of 3 cards that will NOT make 8. Clearly, here you SHOULD break the PPR.

Now, if your opponent proceeds to buy the last province on his turn, did you play incorrectly? Absolutely not. You made the best possible play against his RANGE, and it is mathematically correct every time regardless of what actually happens.

This example may seem rather contrived, but it illustrates the point. And rather than keeping track of exact information like that, you can easily keep rough track of your opponent's deck composition. Has he already seen his ONLY Gold this shuffle? That can inform your future decisions. Did he already play both his tournaments this shuffle without revealing either of his provinces? That can inform your future decisions. Hope this was helpful.

[DStu]

Seem's like I've understood it correctly. On first glance it's trivial and absolutely useless in its stated form, it's often important to just formulate trivial things to use them when you want to reason further. I like to remember that's the case with many central theorems, but looking over the list of "fundamental theorems" on wikipedia only shows one other which could be called trivial, and that the one of linear programming.

Nevertheless they are important fundaments for the theory build upon them, and maybe that's the case here to, as luv2breformed demonstrated. Because it's best when for you when you behave as if you knew what's the hand of your opponent, you better find out as much about it as you can. That might be still trivial, but not as much as the first statement, and definitely it's not useless...