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[Titandrake]

I was thinking about endgame buy decisions, came up with a simple setup, and spent a few minutes trying to solve it. I didn't get a finished solution so I'm passing it on to you.

Alice and Bob are both playing an engine that's capable of drawing the entire deck every turn. Both of their decks start by producing $13 each turn. They both have a reasonable amount of buys: at most 4 buys a turn. (Enough to do most things you want, but not enough to empty the Estates in one turn.)

Neither has bought any VP yet. There are no ways to gain cards mid-turn. The only way to get more money is by buying Treasures. You're guaranteed to draw all Treasures you buy. To make this better model a real game, you are not allowed to buy Copper, you must buy Gold instead of 2 Silver, and you cannot buy Estate unless there are 6 or fewer Provinces in the supply.

Alice goes first. What is Alice's optimal strategy? She's guaranteed to either win or tie by a strategy-stealing argument, but I'm not sure what the right buy-path is.

[luser]

Strategy stealing doesn't work, first player loses when he ties score.

[Cuzz]

Strategy stealing doesn't work, first player loses when he ties score.

And empty piles can mean that mirroring is not always possible anyway.

[Drab Emordnilap]

Strategy stealing doesn't work, first player loses when he ties score.

And empty piles can mean that mirroring is not always possible anyway.

In this context, does "strategy stealing" mean copying the actions of the player before you? I understood it to mean that if we suppose there is a winning strategy for player to, player one can execute the same winning strategy have a turn sooner.

[hypercube]

I don't think Bob can win or draw in this scenario if Alice buys gold T1 and T2 and then 3 Provinces. At that point Alice is certain that the game will end on her turn unless Bob is willing to lose the Province split 6-2, and if she ties the Provinces 4-4 she has enough economy to buy 3 Duchies with her Province T4 (vs. the 4 Provinces and 1 Duchy Bob can have at the end of T3).

[trivialknot]

I would posit that the correct strategy is to buy N gold, and then green.  Greening could be quite complicated, because it's a form of Nim.

For example, here's the optimal strategy I found for p1 taking the 0-gold strategy, and p2 taking the 1-gold strategy:
P1: province, duchy
P2: gold, duchy
P1: province, duchy
P2: 2 province
P1: province, duchy
P2: 3 duchy
P1: province, duchy
P2: 2 province

It ends in a tie.  The next thing to do is make a table for the N-gold strategy vs the M-gold strategy.  Hopefully one of the strategies is dominant.  If not, that means you need to look at your opponent to determine how many gold you should buy.

[trivialknot]

I tried playing around with it and I believe that if the 1st player takes the 1-gold strategy, they always win.  On the other hand, if the 2nd player takes the 1-gold strategy, the 1st player can win by buying 1-5 gold.

Sounds like the dominant strategy is to go first.

[Titandrake]

Strategy stealing doesn't work, first player loses when he ties score.

And empty piles can mean that mirroring is not always possible anyway.

I don't think either of those matters.

By "first player wins or ties", I didn't mean 1st player ends the game with the same number of points. I meant that if 1st player cannot win, they should be able to put the game in a state where 2nd player has to end the game in a tie.

Suppose 2nd player can get more points than the 1st player, given optimal play. Then the best strategy for player 1 is to buy nothing on their 1st turn, then copy the 2nd player's strategy. If both players follow this strategy, they'll both buy nothing for eternity, which was so absurd that it felt it had to be wrong.