Solutions (due to discussions with my MIT friends, especially Paul Christiano) proposed to the question in the previous post:

What is the most number of points, in a solo Dominion game with infinite access to all cards, one can obtain on turn n, asymptotically?

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First of all, I’ll be leaving out many minor details, left for the reader to fill in.  For example, all these decks will take some setting up.  To do so, just trash everything you don’t want, buy everything you do want, and Island away whatever you used to trash.

1. Base Set

Let’s start with just the base set.  I claim we can get $\Omega(9^n)$ points by the $n$th turn.  For points, we will use Gardens.  Thus let’s first worry about the problem of gaining many cards quickly.

Consider the hand:  3x Throne Room, 2x Workshop, 1x Smithy.  Playing these 6 cards gains us 4 cards and draws back 6 cards (playing the Throne Rooms first, of course, to continue the chain and make actions irrelevant). If the 6 cards we draw back happens to be 3x Throne Room, 2x Workshop, 1x Smithy, we can do same thing again, continuing the Throne Room chain!  Of course, we should Workshop cards as needed, according to the ratio 3 Throne Rooms : 2 Workshops : 1 Smithy, so that we can draw them and use them later in the turn.

So if our deck started a turn as size $x$, it ends the turn as size $x + \frac{4}{6} x + (\frac{4}{6})^2 x + \ldots = 3x$.  Thus our deck grows by a multiplicative factor of 3 each turn, so that we have $\Omega(3^n)$ cards on the $n$th turn.  Now, on our last turn, we Workshop only Gardens.  A constant fraction of our deck is Workshops, so we obtain $\Omega(x)$ Gardens.  And since Gardens gives us points proportional to $x^2$, we can have $\Omega(9^n)$ points on the $n$th turn.

If you understood the above, the rest of the details should be easy.  Unfortunately, this solution (and the next one) requires infinitely good luck.

2. Base Set + Intrigue

We’ll use the same idea as above, but with better cards.

Consider the hand:  1x Throne Room, 5x Ironworks, 1x Library.  Throne Room an Ironworks, Ironworks 4 more times, then play Library, redrawing 7 cards.  This gains 6 cards and redraws to replace itself (and does not lose an action).  If we play a Bridge sometime early in the turn, we can Ironworks for Libraries, thus allowing us to get the cards needed to keep going.

With similar analysis as above, a deck with $x$ cards turns into a deck with $7x$ cards in one turn, and we end up with $\Omega(49^n)$ points on the $n$th turn.

At this point, you probably get the main idea, and I encourage you to stop and think about what Seaside cards will increase the base of the exponent even more!  It’s quite awesome!  And there’s conveniently a block of text in-between to make sure you don’t accidentally cheat.

An aside: Getting probability 1 of success

Notice that Library makes this require much less luck, since we can control exactly what cards we draw!  But we still need good luck to make sure we draw a Library at the beginning of each turn.  Using Courtyard doesn’t work, since with only Library guaranteed, we may still be action-starved.

Actually, fixing this is quite easy.  We use a trick which I have actually used in real Dominion games – discard and draw.  Basically, we should discard cards which we want to guarantee for next turn, and then draw cards to shuffle those onto the top of our draw.  (In real games, this trick is often useful for Scrying Pool decks.)

So for this specific application, suppose my deck also contains 4 Villages and a Secret Chamber (one of the Villages can be replaced by anything that draws a card, e.g. Spy).  Now, I play very carefully near the end of my turn, such that the last 7 cards remaining in my discard/draw, right before I play my final Library, are the following:  2 Libraries, 4 Villages, and a Secret Chamber.  I play a Village for the action, and then Secret Chamber, discarding 2 Libraries and 2 Villages.  My discard now consists of those 4 cards, and my draw is empty.  Then, I play my Village, drawing one of these 4 cards.  The remaining 3 are now at the top of my draw, and contain at least one Village and Library!  I end my turn, and draw them next turn.

This shows that we can in fact get $\Omega(49^n)$ with probability 1, with Base Set + Intrigue.  Actually, I cheated a bit.  We need to play a Bridge.  I think a modification of this, with a couple Laboratories and extra Villages thrown in, will work, at least with high probability.  But the details would get messier (feel free to let me know if you work them out).

Alas, it’s much harder to do this with Base Set only, and the base of the exponent decreases.  For one thing, we need Library to draw, which involves using Remodel on the items we Workshop, since we can’t Bridge.  Secondly, we can’t use Secret Chamber, so Library is the best way to do the discarding for the trick (Spy may work but it’d be much harder), and this involves much craftier deck control.   I claim you can alternate between hands of:  [3x Throne Room, 1x Workshop, 1x Village, 1x Library, 1x Remodel] and [3x Throne Room, 3x Workshop, 1x Library].  This gets you 8 cards using 14, and so we end up with $\Omega((\frac{49}{9})^n)$ points.

3. Base Set + Intrigue + Seaside

We’re going to use the same hand:  1x Throne Room, 5x Ironworks, 1x Library.  But we can throw in two more tricks which help us do much better.

The first trick is Outpost.  Outpost essentially gives us an extra turn, so that instead of a 7-fold increase, we get 49-fold increase in number of cards!  So we have $\Omega(49^n)$ points on turn $n$.

The second trick is sheer awesome:  Embargo.  The only pre-Prosperity card which can act as more than one buy.  In our last couple turns, we gain and play $\Omega(49^n)$ embargoes, embargoing Gardens (for convenience’s sake).  Then, we gain cards which give us buys (Ironworks for Markets, perhaps), and then buy $\Omega(49^n)$ Gardens.  We should Bridge a couple times so money isn’t an issue.

This gives us $\Omega((49^n)^2)$ curses!   Each curse takes away a point, but also adds $\Omega(\frac{49^n}{10})$ points through Gardens.  So we end up with $\Omega((49^n)^3) = \Omega(117649^n)$ points!

Again, you can work out the details, if you’d like.  But it would be more fun to now stop and think about how Alchemy cards will let us do way better (if you haven’t already).  It’s quite neat!

An aside: Part 2

With Seaside, we now no longer need anything clever to smooth out any variance, thanks to Haven!   With 16 Havens (8 played each turn), we can guarantee 4 Villages, 3 Bridges, and a Library, at the start of each turn.  There is a slight subtlety, in that our turns may start with enough cards so that Library can actually draw enough cards that we need to get going.

Also, Fishing Village + Courtyard may work instead of Haven, but it’s a bit less clean. In fact, having $\Omega(\log^2 n)$ Caravans/Wharves probably works too, but requires doing analysis.

4. Base Set + Intrigue + Seaside + Alchemy (without self-Possession)

Okay, first of all, if one is allowed to play Possession and repeatedly and recursively possess oneself on the same turn (which I believe the actual rules allow for), then getting infinite points on one turn is easy.  Many things work.  Here’s a simple way:  Have enough Caravans that you always draw your whole deck (they should be slightly more than 2/3s of your deck, and half of them should always be out).  Play some cards for actions and money, and play an Island and Possession.  Buy back the Island (as well as what you Island-ed, if anything (Gardens, perhaps).  I believe the Possession goes back in your deck, so you don’t need to re-buy it).  Notice your deck never changes.

So we limit ourselves to not having Possession.  There is still a game-breaking option on the table:  Scrying Pool.  Scrying Pool lets us draw our entire deck each time we play it!  So if we have $x$ cards at the beginning of the turn, we can imagine one is a Scrying Pool and the rest are Ironworks.  We Ironworks for $x-2$ more Ironworks and one more Scrying Pool, and then Scrying Pool to draw everything again!  Repeating this process gives us $x + (x - 1) + \ldots + 1 = \Omega(x^2)$ cards!

There is just one problem:  We can’t Ironworks (or Workshop or University) for Scrying Pool, or anything with a Potion cost.  But there is exactly one way out:  Transmute!  Transmuting a Treasure card gains us another Transmute, which can then be Remodeled (or Upgraded, if we’ve Bridged exactly once) into Scrying Pool.  But actually, we want another Transmute to play on the same turn, and we can’t use Transmute to get it, so we’ll use Throne Room too.

Here is a turn, flushed out:  Scrying Pool to draw your deck, which consists of a Scrying Pool, 3 Throne Rooms, a Remodel, a Transmute, 2 Villages, and a ton of Ironworks.  Throne Room a Throne Room a Throne Room.  First play Ironworks, gaining two Silvers, then Village, drawing them, and lastly Transmute, turning both into Transmutes.  Village to draw one of these new Transmutes, and Remodel it into a Scrying Pool.  Then Ironworks 3 Throne Rooms, 2 Villages, a Remodel, and a ton more Ironworks!  Scrying Pool to draw everything you just gained.  You now have the exact same hand, but with 7 less Ironworks.  Repeat!

Now, we square the number of cards each turn.  But Outpost still lets us play an extra turn.  Thus each turn, we can raise the number of cards to the fourth power!

For smoothing, we can now use Courtyard instead of Haven, since we only need a single Scrying Pool on top the next turn, to guarantee going off.

Number of cards is now $2^{\Omega(4^n)}$.  This grows really quickly – so quickly that the Embargo trick, as well as buying multiple Gardens, no longer helps!  They have the effect, essentially, of taking a couple extra turns.  And it’s rather meaningless to talk about the constant in the exponent – after all, it depends on the number of turns it took to set up this whole machinery.  So all the glory may as well come from a single Gardens (or Vineyard!).

5. Base Set + Intrigue + Seaside + Prosperity

This is a very easy infinite, and there are many ways to get it to work.  One way is to have 3x King’s Court, 3x Ironworks, 1 Library.  This gains 9 cards and redraws itself, with some King’s Courts plays to spare. You’ll need to have Bridged at some point, since we need to gain King’s Courts. Bottom line: Arbitrarily many cards leads to arbitrarily many points, with even a single Gardens.

Of course, Ironworks can be replaced by any card gainer (Workshop is just as good, and a bit of extra effort makes Feast work), and Library by any sufficiently good card-drawer (King’s Courted Council Rooms are even better, but many things work). Bridge can be replaced by Remodel/Upgrade as well, but it just makes life harder. Even Gardens can be replaced, with Duke + Duchy. (This was true in the Intrigue version too… but not in the Seaside version!)

Basically, this puzzle is under-constrained and silly thanks to the degeneracy of King’s Court! (But the Feast + Duke version should be used for maximum style!)

6. Conclusion

Overall, we have achieved the following:

 Sets Allowed Points Obtained Key Cards Worst Case Success Base Set $\Omega(9^n)$ Throne Room, Smithy, Workshop, Gardens Using different combo, gets worse exponent Base Set + Intrigue $\Omega(49^n)$ Throne Room, Ironworks, Library, Bridge, Gardens Yes (Library, Laboratory, Secret Chamber, Village) Base Set + Intrigue + Seaside $\Omega(117649^n)$ Throne Room, Ironworks, Library, Bridge, Embargo, Outpost, Gardens Yes (Haven) Base Set + Intrigue + Seaside + Alchemy $2^{\Omega(4^n)}$ Ironworks, Scrying Pool, Transmute, Remodel, Outpost, Village, Gardens Yes (Courtyard/Haven) Base Set + Intrigue + Seaside + Prosperity $\infty$ (arbitrarily many) King’s Court, card gainer, card drawer, Gardens/Duke Yes (trivially)

I don’t care so much about variations to what I proposed above. I’ve mentioned many (Duke/Vineyards, Workshop, Upgrade, etc.), I’m aware of many more, and I’m sure there are even more beyond that. But are any of my solutions non-optimal?  Pretty likely… and I’m extremely interested in actual improvements! You should suggest them if you see any!

You should also play around with other combinations of sets.  It’s may be interesting, perhaps, to try Prosperity without King’s Court (and without Alchemy). Talisman (and maybe Hoard and Counting House, though unlikely) probably becomes very important!  Amusingly, when paired with Alchemy, Minting Philosopher’s Stone becomes an alternative to Transmute, for gaining Scrying Pool (the only other way, as far as I know).  And I haven’t even touched the Promotional Cards, Cornucopia, and Hinterlands.  Black Market could be useful for putting Treasures in play, though the Black Market deck itself would be unlikely to be useful (barring it having Possession), since it would be finite. Hinterlands seems especially likely to be interesting, with Haggler and Tunnel, as well as Crossroads, Scheme, and maybe Border Village and Trader.

Edit:  With Dark Ages, it’s now possible to get $\infty$ without King’s Court or Possession.  Check out:  http://forum.dominionstrategy.com/index.php?topic=4050.0. It might also be possible without this trick, perhaps using Procession.

Edit:  It’s pointed out to me that there are ways to get infinity as of Hinterlands (including Promo), even without King’s Court/Possession. The key cards are Black Market, Horn of Plenty, and Mandarin. With Dark Ages, Rats becomes a hilarious simple way. See http://forum.dominionstrategy.com/index.php?topic=1741.0 and http://forum.dominionstrategy.com/index.php?topic=4229 for more.