Please login or register.
1 Hour 1 Day 1 Week 1 Month Forever
Login with username, password and session length

This section allows you to view all posts made by this member. Note that you can only see posts made in areas you currently have access to.

1

Game Reports / Unlikely Comeback

« on: March 28, 2018, 11:19:12 pm »

Game #13063068

The board had the following things:

amulet, forager, urchin, quarry, tournament, cultist, wharf, villa, warehouse, young witch

triumphal arch, dominate

I opened quarry/amulet, my opponent opened urchin/forager. Both my opening buys missed the shuffle, so I picked up forager/quarry while my opponent got a 2nd urchin and quarry and trashed a bit. Things continued to miss the shuffle for me, and my opponent ended up colliding his quarries first and promptly bought all 10 villas before starting on the wharves.

At this point all reasonable chances for me were gone, so I started buying a bunch of tournaments and urchins, hoping to come out on the right end of a 3 pile. My opponent continued buying wharves while playing a bunch of villas; I eventually got enough trashing to be able to draw my deck and buy a province. After I hit province, he emptied the wharves and was like a dollar/buy short on emptying any of 2-3 other piles. On my next turn, down 7 points, I managed to (barely) collide tournament and province to pick up steed. I hoped to get another prize but sadly my province bottom-decked on the reshuffle. I drew it with my last tournament and realized that I needed to forage my province in order to hit 14 and 3 buys (with 2 quarries), just enough to buy dominate and the last 2 tournaments for a 2 point win.

Was not expecting that to work out.

2

Puzzles and Challenges / Best Asymptotic Point Scoring

« on: July 06, 2017, 01:18:32 am »

Suppose you're playing a game of dominion where the kingdom piles are infinite instead of their usual size (and therefore the game never ends). The puzzle is to come up with a strategy that yields the best score as a function (f) of n, the number of turns played so far. Since I believe it's possible to score an unbounded number of points in a single turn on some boards, I want to restrict to only boards where the number of points that can be scored on the nth turn is bounded by a function of n (i.e. no infinite points in a single turn). Ideally, I'd like solutions to work regardless of shuffle luck.

To give a simple example, suppose you opened monument/donate, and then just played monument every turn for the rest of forever. On the nth turn, you would have n-2 points, so in this case f(n) = n-2 = O(n).

To do slightly better, suppose that you trashed down to a golden deck of bishop, silver, silver, gold, province, and trashed the province, buying a new one each turn. This would yield f(n) ~= 5n

It's not too hard to do better than linear in n, but I'm not sure quite how well you can do. I'll add the best that I've come up with so far after not too much thought in spoiler tags below, and leave it up to anyone interested to go crazy with this.

I was thinking about solo games, but feel free to try with 2 players if you think that would help (maybe you can do something with possession).

See also Busy Beaver amount of Coin and How high can you go for similar puzzles.

Function:

f(n) =O(n!)

Strategy: (I don't think this board allows infinite points in a turn, but I haven't thought it through enough to be fully confident.)

Have a vineyard on your island mat, and have a deck containing the following:

2 schemes
2 treasuries
2 scrying pools
potion
k ironworks

turn:

1. start with 2 treasuries, 2 scrying pools, and something else in hand
2. play scrying pools to draw the deck
3. play k ironworks to gain k ironworks
4. play treasuries
5. play schemes
6. play potion
7. buy scrying pool
8. Topdeck 2 pools with schemes, and 2 treasuries

You had roughly k actions at the beginning of the turn, now you have roughly 2k actions, for twice as many vineyard points. Next turn, you'll be able to play the 2k ironworks to gain 2k ironworks, and then play the extra scrying pool to play those 2k ironworks to gain another 2k, giving you roughly 6k ironworks in total (and buying another scrying pool). Doing this for n turns will yield O(n!*k) points, hence the value of f(n).


Perhaps one can do (asymptotically) better on this board, I just wanted to come up with something that did reasonably well. Feel free to come up with something better.

3

Puzzles and Challenges / Most VP from 1 buy

« on: May 28, 2017, 10:28:00 pm »

Suppose you buy one card (resolving the buy and any effects triggered by that buy). What is the largest number of VP you can gain from that process? Here are the rules/constraints I'm imposing:

1. (at most) 4 player game following standard set up procedures (at most 2 events/landmarks, etc)
2. You can choose the game state as long as it's legal (i.e. the game hasn't ended before your turn, it's possible to reach that state from a standard start state, etc)
3. On some boards, the number of VP you can gain might be unbounded (e.g. Temple). I want to restrict to setups where this is not the case.

Not sure if I'm forgetting anything, if you think something is amiss, feel free to comment.

4

Puzzles and Challenges / Busy Beaver amount of Coin

« on: April 28, 2017, 09:41:30 pm »

Consider a modification of (1 player) dominion, where supply piles are infinite (or if you don't believe in infinity, arbitrarily large). Suppose also that you start your turn with an n card deck, and all of those n cards are in your hand.

On certain boards, it is possible to construct such a scenario where you are able to generate arbitrarily many coins on your turn (e.g. a board with highway and villa (and n >=4), since you can play 4 highways and then buy and play arbitrarily many villas).

I want to only consider boards on which, for any positive n, the number of coins you can generate is bounded by some (finite) function of n (call it f(n)).

What's the largest achievable f(n)? Clearly you can get at least f(n)=5n (just have a hand with n platinums). However, you can certainly do much better than this. I'll break this up into 2 parts:

1. What's the best O() complexity achievable? Is it O(n^2), O(2^n), O(2^2^n)?

2. Can we also figure out the optimal constant?

I have some thoughts on 1, but probably one can do better than the best I've come up with. Feel free to let me know if you find any issues with my formulation.

5

Dominion General Discussion / Relative strengths of 2 card combos

« on: October 26, 2016, 12:45:37 pm »

So the Neat and potentially useful card interactions thread is great, but one thing I often find myself asking is how good various combos are relative to each other.  For instance, we know that warehouse/treasure map and hermit/market square are both “potentially useful” combos, but is one of them significantly better than the other?  A lot of times, the answer to such a question is “depends on the board”, but I’m interested in what happens when it doesn’t.  I’d like to ask the question “If you have two combos (A and B) and neither has any board support, which wins the majority of the time when pitted against each other?”

I’m not sure if something like this has been done before.  If it has, I’d be interested to see the results.  If not, here is my attempt to do so.  Feel free to skip to the bottom where I actually start listing combos.

To formalize the game I want to play, I’m going to start with a list of 2 card (or possibly “uh, card-shaped thing”) combos and attempt to rank them.  Here are my rules:

Say we want to compare combo A and combo B.  Construct a board with only the base cards (no colonies/shelters or anything) and the components of these two combos.
-   Play a 2 player game where one player plays combo A and the other player plays combo B.
-   The combo A player can only buy base cards or components from combo A.
-   The combo B player can only buy base cards or components from combo B.
-   Both players are attempting to play optimally subject to these constraints (i.e. maximizing their probability of winning).
-   Randomize starting player.
Combo A beats combo B iff combo A wins the majority of the time.

Some qualifiers to note:
1.   Determining the winner of a matchup seems like it could be pretty difficult to do.  “Playing optimally” might be hard for more difficult combos (e.g. upgrade/rats), and determining who wins the majority of the time with confidence might also be hard for two closely matched combos.
2.   There is no guarantee that the results are transitive.  It would be nice to still have a ranking that at least closely conforms to the results, maybe with notes as to which relations are violated.
3.   Due to the constraints of the game, some notoriously good combos will not do well.  For example, king’s court/bridge is great, but it would do terribly in this game, since it has no trashing.
4.   More generally, the constraint of only getting two kingdom cards seems to be fairly restrictive.  I can think of a lot of strong deck paradigms (goons engine, horn megaturn) that wouldn’t work here.  I guess that means that dominion is actually a pretty interesting game.
5.   Along the same vein, this idea feels somewhat like a generalization of the question “What’s the best big money strategy?” since that question is more or less the one card version of my question.
6.   I’m not sure if there are things like Donate that ruin this game.  It would be less fun if the top 13 combos were Donate/X or something.  If that ends up being the case, I want to veto Donate (or whatever the offending “uh, card-shaped thing” is) for this game.  This is obviously subjective.

--------------------------------------------------------------------------------------------------------------------------------------------------------

With all this setup, I’m going to provide a starting point to build off.  I’ll list a bunch of combos and provide a preliminary guess as to how they rank.  I make no claims as to the completeness of my list (There are obviously a lot of good combos that I’ve missed/not bothered to include), or the correctness of my ranks (I’m sure a lot of my ranks are off, some are probably way off).

1. Counting House/Travelling Fair
2. Ferry/Rebuild
3. Royal Carriage/Bridge
4. Villa/Jack of all Trades
5. Masterpiece/Feodum
6. Dungeon/Tunnel
7. Ferry/Governor
8. Hermit/Market Square
9. Scavenger/Stash
10. Ferry/Cultist
11. Lurker/Cultist
12. Courtyard/Delve
13. Witch/Delve
14. Mountebank/Delve
15. Cultist/Delve
16. Cultist/Dominate
17. Crossroads/Gear
18. Inheritance/Ironmonger
19. Bonfire/Jack of all Trades
20. Hunting Party/Tournament
21. Apprentice/Market Square
22. Dungeon/Treasure Map
23. Ferry/Mountebank
24. Wharf/Fool's Gold
25. Magpie/Pathfinding
26. Storeroom/Encampment
27. Jack of all Trades/Counterfeit
28. Gear/Treasure Trove
29. Forager/Highway
30. Ranger/Lost Arts
31. Native Village/Bridge
32. Wharf/Coin of the Realm
33. Forager/Minion
34. Fishing Village/Wharf
35. Beggar/Gardens
36. Courtyard/Quest
37. Ironworks/Gardens
38. Gear/Counterfeit
39. Inheritance/Magpie
40. Forum/Fool's Gold
41. Counterfeit/Capital
42. Beggar/Triumph
43. Urchin/Save
44. Masquerade/Lost Arts
45. Artificer/Storyteller
46. Duplicate/Duke
47. Forum/Duke
48. Hermit/Artificer
49. Embassy/Tunnel
50. Upgrade/Rats
51. Gear/Lost Arts
52. Alchemist/Dominate
53. Native Village/Apothecary
54. Treasure Trove/Gardens
55. Horse Traders/Duke
56. Forager/Peddler
57. Loan/Minion
58. Junk Dealer/Graverobber

Please comment if you want to include any combos, or if you find one of the current rankings to be off (or if you have any other questions or comments).  If you want to include a combo, it would be helpful to also figure out where it ranks.  I can keep this list updated if anyone else besides me actually cares about this at all.  Also, I'm not going to include any combo that loses to straight money, even if it is good with other support (e.g. kc/bridge).