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Asymptotic Dominion

July 8, 2012

This post will only be accessible to those who play Dominion, and those with some background in asymptotics.

We study the following question.  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?

Solutions given in my next post.

———-

It’s always fun to imagine the sort of deck that is guaranteed to sustainably get a certain number of points every turn, if piles never ran out.  Here are two examples of decks (among many) which do this, which have very different flavor.

  • University, Bishop, Bishop, 4/5 cost action, 2 cost (5 points a turn):
    University for a 4 or 5 cost, Bishop the two non-Bishop cards in hand, buy a 2 cost.  I’ve played this deck before.
  • King’s Court, Scheme, Chancellor, 5x Stash + anything ($12 a turn):
    King’s Court the Scheme, and play Chancellor.  Play 5 Stashes and buy.  Return the three action cards to the top, and put the 5 Stashes right after.

In this latter scenario, one can buy a Colony each turn indefinitely.  So on the nth turn, one can have about 11n points, guaranteed!  But actually, this is not very good – instead of buying Colony, we can just buy Gardens (we then only need one stash).  Then, the number of points we have on turn n is about \frac{n^2}{ 10}, which is far more for large n!

This leads to a natural question:  What is the most number of points, in a solo dominion game with infinite access to all cards, one can obtain on turn n?  Of course, obtaining the exact number is not very enlightening or fun, so instead we ask for asymptotics.  Asymptotically, how many points is it possible to have by turn n?  (Note that it’s fine to get all your points on the last turn.) Essentially, we’re playing resource-unbounded Dominion, and trying to build the ultimate engine deck!

It turns out, this is a very fun exercise.  Even with only the base set, it’s possible to do much better than O(n^3) or even O(n^{1000}).   If you’d like to think about it yourself, don’t read this next post, where I post my solutions to the puzzle for each of the following sets (of sets of cards):

  1. Base set only
  2. Base set + Intrigue
  3. Base set + Intrigue + Seaside
  4. Base set + Intrigue + Seaside + Alchemy (disallowing self-Possession)
  5. Base set + Intrigue + Seaside + Prosperity
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