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Topics - jonaskoelker

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1
Puzzles and Challenges / Trash all the things!!
« on: January 02, 2018, 06:16:20 pm »
Objective: put all the cards in the trash as fast as possible.

Pick a kingdom of your choice, satisfying the recommended rules. Solo play, the shuffles always favor you, 0 <= #Events + #Landmarks <= 2, Colony and Platinum are allowed if you have at least one randomizer card from Prosperity and required if all randomizers are from Prosperity; similarly for shelters.

I'm not sure that Black Market helps you; if you use it, some people like solutions that work online—bonus points for making them happy.

Hard mode: 2+ players, your opponents do nothing, you have trash all of their cards as well.

Rules reminder: if you trigger the game-end condition and buy Donate on the same turn, you never get to do the Donate thing. (That's a Dominion rule, not a this-particular-challenge rule.)

2
Some cards are described as swingy.

User rrenaud once compiled a sorted list, based on how much entropy was left in the win-when-in-kingdom rates once rating was factored out, and concluded that Goons, Ambassador and Tournament are high-skill cards, whereas Embassy is a high-randomness card.

What makes Embassy high-randomness is the fact that there is an Embassy-centered strategy (Embassy BM) which is fairly strong and easy to play correctly.

What makes Tournament high-skill, I think, is that it requires good judgment and game sense to know when to go for it, how best to go for it, and which prizes to pick when you connect it with a Province. Somehow, [joke]rrenaud's ironclad mathematical proof—proof, I tell you![/joke]—that Tournament is high-skill seems not to have changed the common perception of Tournament as swingy.

I think I have a different definition of swingy which classifies Tournament as swingy. It allows for a 2x2 matrix of swingy yes/no and high-skill yes/no; we could for example use rrenaud's measure to indicate whether a card is high-skill.

Whenever you make a play, you can compare you chances of winning (assuming optimal play) in the game state before and after making that play.

Weak cards are cards that reliably only improve your win chances by a small amount (or not at all, or even harm you). Strong cards are cards that usually do something better than weak cards for your win chances.

Swingy cards are cards that are usually ok-ish but sometimes have an unusually large impact on your win chance when played. That is, your win chance after is significantly better than your win chance before. Playing it causes a big swing.

I equivocated between "making a play" and "playing a card". I only used the latter phrase for convenience—Cultist can be highly swingy, in particular if exactly one player opens 5/2 (but also if it misses shuffles, or the first ruins hit vs. miss shuffles). In the case of opening 5/2, it's the dealing of the opening hands that's the swingy play. That play is made by RNGsus, so really the definition should read "the move greatly changes the win percentages of the players", without using 'you[r]'.

And considering this further, having Province in hand when you play Tournament (and being first to do so) is often largely attributable to shuffle luck. When the shuffle has been made in a way that favors you, actually making the swingy play is just playing optimally—so really, I think swingy is something like "some of the moves made by RNGsus have the capacity to greatly alter the win probabilities of the players".

The degree to which a card is swingy is then the capacity of RNGsus to impact win chances, averaged across all the kingdoms in which the cards occur. The inherent swinginess of Tournament is attributed a little bit to e.g. Smithy, because they can occur on the same kingdom, but Embassy's randomness is also attributed a little bit to Smithy by the same logic. I claim without evidence that each card's inherent swinginess will be attributed overwhelmingly to itself, and each card's measured swinginess will be its inherent swinginess, adjusted a little bit in the direction of the average inherent swinginess across all cards, but the ranking by measured swinginess and the ranking by inherent swinginess will be the same.

A small pedantic footnote: instead of win probabilities I should probably use "expected number of match points" (loss=0 and win,draw=1/#winners), such that the players' win probabilities expected number of match points sum to 1.

I skipped over some rigor in a few places, but I think my definition roughly approximates what people mean by swingy. I would love to hear whether people agree or disagree, and especially if you think you can improve this definition. I think of it as a 'first draft' :)

Edit: today I relearned that the internet is a poor medium for carrying tone of voice, which is often crucial to indicate when you're joking. To address this, I have inserted [joke] tags in the appropriate places.

3
Solo Challenges / Copper Rush: Drain the Copper pile as fast as possible
« on: December 18, 2017, 05:36:38 pm »
Here's the challenge: empty the Copper pile into your deck as fast as possible.

Normal mode: you have to gain a Copper n times, where n is the number of Copper in the pile at the start of the game, and the pile must be empty.

Hard mode: all the Copper must be in your deck (deck includes hand and discard pile).

You pick the kingdom, including Shelters and Colonies (if you have at least one Dark Ages/Prosperity card), and 0-2 events/landmarks. Assume perfect shuffle luck.

Bonus points if you don't use an infinite combo, i.e. if there exists some n such that in the turn where you satisfy the ordinary win condition, you couldn't satisfy the same condition if the Copper pile started with n Coppers.

Example solutions:
  • Buy Copper every turn. This takes 53 turns. (Hard mode)
  • Banquet for a Copper every turn. This takes 18 turns. (Hard mode)
  • Turn 1 Alms for a Villa, buy King's Court off 5xCopper, Villa, Baker token. Turn 2 buy Beggar. Turn 3 through 8: King's Court a Beggar, gaining 9 Copper and buying Donate, trashing everything except King's Court and Beggar. (Normal mode)

4
I recently played on this lovely kingdom (IRL, and won): Patrician, Settlers, Farmers' Market, Engineer, Sacrifice, Temple, Archive, Charm, Crown, City Quarter, Advance, Triumphal Arch.





Here's my own analysis of the board: you can build a trashed down engine that draws deck with City Quarter and payloads Crowned Farmers' Markets, sprinkled with one or two treasures if need be. If you run out the Farmers' Markets, or at an opportune moment, dig into the Emporium pile. Extremely late in the game, your +buy can come from Charm, but probably you 3-pile before that. With Engineer to gain cheap actions and Advance to turn them into Emporium and Crown, piles can run out in no time.

There are no attacks that disrupt the engine. All the alt-VP on the board favors the engine: Crown plus Farmers' Market gives you better control over it than any money strategy can ever have; thanks to Advance you can justify picking up Temple for the points; and Triumphal Arch only (really) rewards engines. So you're playing (more or less exactly) that engine.

Given that you can draw deck and there are no tricks with Forum discarding your Provinces and Archive setting them aside for temporary thinning, Archive seems not worth going for.

Charm is either a Silver with a +buy, or it gains; but what to gain? Buy Patrician, gain Settlers, meh? Buy Crown gain... more Charm? But you want actions rather than treasures to make City Quarter good. Buy Temple, gain Sacrifice, after you've already added a trasher to your deck? I think Charm is only good in your deck if its the only way you have of getting enough +buy.

Given that there's plenty of virtual money payload, to make City Quarter good you're going to want actions more than treasures; but if you have more Crowns than Farmers' Markets, adding in a single Gold, maybe two, can be okay.

Temple trashes more cards per play than Sacrifice. I think that makes it better—but after the Estates are out, Sacrifice provides more economy which impacts building. An building is exactly the part I'm completely uncertain about.

  • How do you put this engine together?
  • Do you get one or two trashers? Or even more? Which one(s)?
  • Do you just mass Farmers' Market until you add CQ, or do you mix Farmers' Market with cheap actions (Patrician, Settlers), advancing them into Crown? By massing Farmers' Market (e.g. by trashing Engineer) you can win the split and secure a greater multiple of 4 VP, if you have the time to cash them all in, but if done early it overterminals your deck quite a bit.
  • On your $2 turns, do you take Patrician, Settlers or Engineer? I think Settlers is good early, while you still have Copper, whereas Patrician will eventually be better than a blank cantrip when you have lots of Crowns and Emporiums; but by then you probably should be drawing deck with City Quarter.
  • When do you get City Quarter? When your action density crosses 1/2? Earlier, later?
  • Have I missed anything?

5
On http://wiki.dominionstrategy.com/index.php/List_of_cards there were some Nocturne cards that were not yet transcribed. I transcribed them. I caught two mistranscriptions by myself, one only after saving it. Given my fallibility, some proofreading would be nice.

The diff is at http://wiki.dominionstrategy.com/index.php?title=List_of_cards&diff=39268&oldid=38127.

The transcribed cards are:
  • Bard
  • Changeling
  • Cobbler
  • Conclave
  • Den of Sin
  • Guardian
  • Monastery
  • Night Watchman
  • Sacred Grove
  • Secret Cave
  • Tormentor
  • Tracker
  • Tragic Hero

6
Pick a kingdom of your choice. Make a sequence of plays, after which you know 0 or more of the cards on top of your deck. What's the longest sequence of known cards you can set up?

Example solution:

The kingdom is Baker, Vault, Native Village, Ratcatcher, Duchess.
Open Vault/Ratcatcher, and buy Native Village and Duchess later.
Trash all Estates and mat the Ratcatcher.
Put all but 5 cards on your NV mat.
On the megaturn, unmat all your cards with NV; discard 7xCopper to Vault; play Duchess, shuffle, look at a Copper and put it back.

Now your deck has seven known cards on top, all of which are Copper.

7
Here's the simplest infinite money loop I'm aware of:

Setup:
  • 2xOverlord in hand.
  • 1xOverlord in your discard pile (and nothing else).
  • 0 cards in your deck.
  • You can play Overlord as King's Court, Mining Village and Lurker.
Play one Overlord as King's Court, then:
  • Play Overlord as Mining Village (drawing Overlord), trashing it for $2.
  • Play the same Overlord as Lurker, gaining itself back from the trash.
  • Play the same Overlord as King's Court.
Repeat the loop with the Overlord you drew, drawing the first Overlord back.

With 3xOverlord in hand and an empty discard pile, one iteration of the loop—i.e. Overlord as King's Court, then Overlord as Minining Village, Lurker and King's Court—gets you to the right game state, so really the setup is one tripled action, one Overlord in hand and one Overlord in your discard pile.

To bring King's Court into Overlord range, use Ferry or 2x(Overlord-as-)Highway. To have 0 cards in deck, use your favorite trasher, or Bonfire/Hireling/Lurker, or Native Village, or whatever else you like. You can payload other self-trashers than Mining Village, but be mindful that you need to draw the other Overlord. Teacher tokens can make a lot of things possible here. To make good use of $∞, add Travelling Fair to the board.

One weakness of this loop is that it's extremely restrictive; other loops let you play an arbitrary payload card of your choice infinitely often. The loop I'm presenting here needs to spend two out of three Overlord plays on Lurker and King's Court, so the first of the three has to be a self-trasher (or else all your cards eventually end up in play and the process stops).

Some loops without this restriction:

Kingdom: Overlord, Lurker, Raze, Watchtower, Crown, Mandarin

Kingdom, by cost:
2: Lurker
2p: Scrying Pool
3: Ambassador, Masquerade
4: Cutpurse, Thief, Pirate Ship, Villa
7: King's Court
Events/Landmarks: Bonfire, Travelling Fair

There are more loops in the same thread.

8
There's an old and very bad thread on the board which I don't want to Necromancer. In it, eHalcyon makes some claims about probabilities. Here I do the math to see if those claims are accurate.

If you had five 5 coppers and 5 estates, there is just as much chance of you drawing ccccc as there is of you drawing ccece, even if one "looks" more random to the human mind. 

If we put little stickers on the cards labeled "1" through "10", there would be 10! different orderings. But since we don't distinguish between the 5 estates nor the 5 coppers, there are only 10! / (5!*5!) = 252 = 2*2*3*3*7 orderings.

How many of those orderings start with "ccccc"? As many as there are (non-distinguished) ways of ordering the remaining cards, which is just 1 (that ordering being "eeeee").

The number of orderings that start with "ccece" equals the number of orderings which end with some reordering of "cceee". There are [5 choose 2] = 10 such orderings.

Hence, you're 10 times as likely to draw "ccece" than you are to draw "ccccc". You are even more likely to draw "3 coppers and 2 estates in any order" as you are to draw "ccece" in that specific order, because there are more orders (and all orderings are equally likely).

[D]rawing a hand of all actions, then a hand of all treasures followed by a hand of all greens seems really non-random to us.  But with perfect randomness, there is just as much chance of drawing these clumps as there is of getting three hands each with mixes of all three.

Restated, if you draw three batches each of five balls from a jar with five green, five white and five yellow balls, not putting the balls back between draws, you're equally likely to draw three monochromatic batches as you are to draw three rainbow batches (all colors represented).

The total number of orderings is 15! / (5!*5!*5!) = 756756.

There are six orderings of monochromatic batches: GWY, GYW, WGY, WYG, YGW, YWG. (This is shorthand for ggggg+wwwww+yyyyy etc.); 6 out of 756756 is a bit less than 1 in 100 000.

A rainbow batch has gwyXZ in some order, where X and Z are both members of {g, w, y}. X and Z can either be equal or different. This means that a rainbow batch either has three of one color and one of each of the rest, or two pairs and one loner.

Suppose the first batch is rainbow, with a color occurring thrice; let's say it's green, so it's gggwy in some order. There are 2g+4w+4y left. If all batches are rainbow, the remaining two batches are either [gwyyy+gwwwy], [gwwyy+gwwyy] or [gwwwy+gwyyy].

Note that gggwy has 5! / (3!*1!*1!) = 20 orderings, and gwwyy has 5! / (1! * 2! * 2!) = 30 orderings.

So there are 20 * (20*20 + 30*30 + 20*20) = 34000 orderings that start with gggwy. But picking green was arbitrary; there are just as many orderings that start with wwwgy or yyygw. (Having picked green, we shouldn't count the swap of white and yellow as separate, because that would double-count the gwyyy+gwwwy and gwwwy+gwyyy follow-ups. It would also double-count gwwyy+gwwyy, because it isn't different from gyyww+gyyww.)

So that's 3*34000 = 102000 orderings that start with a rainbow batch with one color repeated thrice.

If instead it starts with two pairs, let's say gwwyy in some order, the rainbow follow-ups are [gwwyy+gggwy], [ggwwy+ggwyy], [ggwyy+ggwwy] and [gggwy+gwwyy], where all hands can be reordered.

That's 30 * (30*20 + 30*30 + 30*30 + 20*30) = 90000 orderings. But this again is with green arbitrarily picked to be the starting loner, counting those with a white or yellow loner first (and the other two colors paired) gives us 3*90000 = 270000 orderings.

In total, that's 102 000 + 270 000 = 372 000 orderings out of 756 756, or approximately half.

So rainbow batches are about 1 in 2 compared to monochromatic batches which are about 1 in 100 000, so rainbow batches are about 50 000 times as common.



Conclusion

I think eHalcyon's quoted claims are incorrect. Here are some similar claims which I think are correct:
  • The ordering ccccc eeeee is just a likely as the ordering ccece eceec.
  • The ordering ggggg wwwww yyyyy is just as likely as the ordering wggyg wgwyw ywyyg.
I agree that monochromatic hands feel 'less random', whatever the hell that means, to most people (including me). I think eHalcyon's point is accurate, but slightly misstated, which in turns means that this whole thread is a bunch of pedantic nitpicking.

Am I doing f.ds'ing right? ;)

9
As the title says, the Poor House challenge is this: play poor house in a kingdom of your design, such as to maximize [the amount of money you have before playing Poor House] minus [the amount of money you have after playing Poor House]. That is, maximize the net loss.

This is not quite the same as maximizing the amount of treasures you reveal: if you have $0, play Poor House and reveal 5 treasures, you have $0 afterwards as well, for a net loss of $0 rather than 1. If you had $1+ and made the same play, you would lose $1. To lose money, you must first have that money (modulo the +$4 from Poor House).

I'm highly confident that the worst possible score is -5: if your +$1 token is on Poor House and you reveal no treasures, you gain $5, which is the same as losing -$5.

I'm even more confident that the best score is greater than 0 ;)

Here are some variables you might want to tweak when picking a kingdom:
  • The number of players in the game (2-6).
  • Whether or not to use Colonies.
  • Whether or not to use Shelters.
  • Whether to have 0-2 or a greater number of events plus landmarks.
  • The size of the Black Market deck (from 0 up to the number of randomizers not in the kingdom).
Pick whichever set of variables you like the best. Colonies without Prosperity randomizers is cool too, likewise for Shelters and Dark Ages. If your kingdom is within the limitations of recommended play and/or what the online implementation allows, you win double the bragging rights ;)

Your opponents play as follows: they never play any cards. They never buy anything. When one of them has to make a decision not covered by these rules, they all resign and the game ends.

10
Recently I had the following 2p kingdom come up. How would you play it? Catapult/Rocks, Chariot Race, Gladiator/Fortune, Castles, Temple, Archive, Legionary, Wild Hunt, Overlord, Royal Blacksmith — Palace, Ritual.

[kingdom]Catapult/Rocks, Chariot Race, Gladiator/Fortune, Castles, Temple, Archive, Legionary, Wild Hunt, Overlord, Royal Blacksmith, Palace, Ritual[/kingdom]

No villages, the only +buy is Fortune, the only non-terminals are Archive and Chariot Race. That makes the board rather slow.

Here are some strategies:
  • BMU: buy the most expensive of [Province, Gold, Silver] that you can afford until you run out the provinces (tactically bump up Palace points near the endgame, and/or buy Duchy, maybe). Has a hard time with Legionary (I guess); maybe Archive can help a little? But this deck gets thick very quickly (whatever "quickly" means on this board), so you won't have Archive often, or many of them simultaneously, unless you delay ramping up Palace points.
  • BM+Archive+Legionary: trash a bit, get some Archives, get a Legionary and a few Golds, attack a lot, buy Provinces and maybe sprinkle in some Rituals. Perhaps Overlord plays here?
  • Ritual golden(ish) deck: trash down, get some Archives, get a Fortune, repeatedly trash Curse, trash a Gold for VP, buy a replacement Gold. Transition into Provinces when the Curses are out.
  • Jam your deck full of Chariot Races and expensive cards (Gold and Province seems good), some Archives maybe, milk CR for VP. Sounds not terribly awful, but in my experience with Chariot Race it has been... lackluster.
  • Play one Wild Hunt per turn, gain 40 VP and an estate around turn 45 ??? sounds terrible, quite easy to contest, at which point... question mark?
I think Castles can fit into BM/Archive/Legionary, maybe in Ritual/golden if you have the time once the Curses run out. I haven't tried putting them into BMU, but I think it likes Province better; BMU's VP/turn rate drops off pretty sharply once it hits 7 Gold/Silver/Copper triples. But in general they seem like an expensive and slow VP supplement, probably not worth going for unless you put effort into getting the two scaling ones ($3 Humble and $10 King's).

I lean towards thinking BM/Archive/Legionary is strongest. The draw is limited, so the discard attack is strong, it can green reasonably soon and use Archive as temporary pseudo-thinning of green cards, so it should be possible to build to a stage where in can semi-almost-reliably attack and buy Province for a small handful of turns. My very lovely opponent never bought Legionary (we played ~5 games on this board) which I think helped me a lot; she swore a bit when I played mine ;D

What do you guys think? What's best here? Which cards never play?

11
Motivation

I have $6 and two buys; when should I get Gold vs. Cache vs. 2xSilver? I'm playing BM on a board with no terminal draw; should I go for a cantrip copper over Silver? I'm playing a wonky board with nothing going on; is Hermit or Ironworks a better BM enabler?

Here I'll do some money density math to give partial answers to such questions.

Comparing purchases, the generic math

Let m be the total money in your deck and c be the number of cards, not counting cantrips (because they draw a replacement card when you play them). Then your money density, d, is m/c.

If you add a bundle of c1 cards which in total produce m1 money, your new money density d_1 is (m+m1)/(c+c1). If you add another bundle of c2 cards which produce m2 money, your new money density d_2 is (m+m2)/(c+c2).

The first bundle is better, in the sense of increasing your money density the most (or equivalently, in the sense of having the highest post-change money density) when d_1 > d_2, that is, when (m+m1)/(c+c1) > (m+m2)/(c+c2). Rewriting,
Code: [Select]
    (m+m1)/(c+c1) > (m+m2)/(c+c2)
<=> (m+m1)(c+c2) > (m+m2)(c+c1)
<=> m*c + m1*c + m*c2 + m1*c2 > m*c + m2*c + m*c1 + m2*c1  [distributive law]
<=> m*(c2-c1) + m1*c2 > c*(m2-m1) + m2*c1                  [subtract (m*c + m*c1 + c*m1) on both sides]

It seems obvious that if you add the same number of cards, the bundle with the most money is better. Let's sanity-check the math, letting c1=c2:
Code: [Select]
    m*(c2-c1) + m1*c2 > c*(m2-m1) + m2*c1
<=> m*0 + m1*c1 > c*(m2-m1) + m2*c1  [apply c1=c2]
<=> c*(m1-m2) + c1*(m1-m2) > 0       [move stuff around]
<=> (c+c1)(m1-m2) > 0
<=> (m1-m2) > 0                      [assuming c+c1 > 0]
<=> m1 > m2
In words, bundle 1 is better when it has the most money. Obvious truth is obvious, and the math passed a simple sanity check.

Next, let's assume (WLOG) that the second bundle has more cards; c2=c1+dc (dc > 0);
Code: [Select]
    m*(c2-c1) + m1*c2 > c*(m2-m1) + m2*c1
<=> m*(c1+dc-c1) + m1*(c1+dc) > c*(m2-m1) + m2*c1
<=> m*dc + m1*dc > c*(m2-m1) + m2*c1 - m1*c1
<=> (m+m1)*dc > (c+c1)(m2-m1)
(Actually I never use the assumption that dc > 0, so this formula also works when c2 < c1.)

Comparing some specific treasure bundles

Let's use this to compare Gold vs. 2xSilver, i.e. m1=3,c1=1 vs. m2=4,c2=2 and dc=c2-c1=1
Code: [Select]
    (m+m1)*dc > (c+c1)(m2-m1)
<=> (m+3)*1 > (c+1)*(4-3)  [insert values of variables]
<=> m > c - 2
So Gold is better (in terms of money density) when total money exceeds the number of cards minus two; in that case, your money density m/c is greater than (c-2)/c = 1-(2/c). This threshold is always less than 1, it approaches 1 as your deck grows larger, and it's 0.8 on turn one; your actual money density is 0.7, so if you can muster $6 and two buys on turn one, go for 2xSilver over Gold (if all you care about is money density right then and there; there are of course other considerations).

With more practical relevance: favor 2xSilver only when money density applies and you have low money density (i.e. a slog), and then probably only some of the time, depending on the particular kingdom.

(In engines, my choices are determined more by engine capacity math than by money density considerations; money density stops being a useful concept when you draw a constant fraction of your deck, e.g. 1, rather than a constant number of cards.)

Let's compare Gold to Cache: m1=3,c1=1 vs. m2=5,c2=3 and dc=3-1=2:
Code: [Select]
    (m+3)*2 > (c+1)*(5-3)
<=> 2m + 6 > 2c + 2
<=> m > c - 2
Oh look, it's the same condition. Geronimoo's simulation of a Duke/Duchy slog comes to the conclusion that you want Cache over Gold. It seems plausible that money density would be around 1 (which is around 1-(2/c)) if you green early and often (and you don't need much more in Duchy/Duke), so I consider this a confirmation of what the math predicts. (In slogs you draw a constant number of cards rather than a constant fraction of your deck, so the concept of money density is sensible in the first place.)

Just for completeness, let's compare 2xSilver to Cache, i.e. m1=4,c1=2 vs. m2=5,c2=3 and dc=3-2=1:
Code: [Select]
    (m+m1)*dc > (c+c1)(m2-m1)
<=> (m+4)*1 > (c+2)(5-4)
<=> m + 4 > c + 2
<=> m > c - 2
Hey, what do you know; it's the same condition. So whenever m > c - 2 (mediocre-or-better money density) we have Gold > 2xSilver > Cache, and whenever m < c - 2 (very low money density) we have Cache > 2xSilver > Gold.

Comparing treasures and cantrip money

Next, let's use the generic formula to compare silver and some flavor of cantrip copper. Intuitively, a silver in hand gives you $2 where a cantrip copper gives you $1 plus your money density (in expectation), so we should expect cantrip coppers to be better when money density exceeds 1. Let's do the math, and remember that we don't count cantrips as cards because they draw replacements when you play them; for cantrip copper, m1=1,c1=0 and for silver, m2=2,c2=1 giving dc=1-0. Cantrip copper is best when:
Code: [Select]
    (m+m1)*dc > (c+c1)(m2-m1)
<=> (m+1)*1 > (c+0)(2-1)
<=> m > c - 1
When m is an integer, m > c - 1 is the same as m >= c, in which case money density is at least one. So the money density analysis matches the payload-when-in-hand analysis perfectly.

Of course, there are considerations other than money density: if you're playing BM with terminal draw, drawing cantrips dead is worse than drawing live treasures. And no two cantrip coppers are the same; what they do for you depends on what else your deck is doing.

Let's also consider Grand Market (m1=2,c1=0), compared to Gold (m2=3,c2=1). Intuitively it should be the same, GM is better when 2+density > 3, but let's check when GM is best:
Code: [Select]
    (m+m1)*dc > (c+c1)(m2-m1)
<=> (m+2)*1 > (c+0)(3-2)
<=> m > c - 2
That's not the same result, though. Let's do a thought experiment: you deck is 20 copper, plus either a Gold or a Grand Market. On the turn where you draw your non-copper, your payload will be 1x2+5x1 or 1x3+4x1, in both cases $7; but because GM draws you a card, you will see the higher payload ever so slightly more often. So it makes sense that the cutoff shouldn't be m/c=1.

But then, why is the threshold different for Peddler/Silver vs. GM/Gold? Because one Gold can outweigh two estates (in terms of hitting m/c = 1) where a silver can only outweigh one. *Vigorous handwaving*

Just for fun, let's compare cantrip copper (m1=1,c1=0) to Gold (m2=3,c2=1 and dc=1-0=1):
Code: [Select]
    (m+m1)*dc > (c+c1)(m2-m1)
<=> (m+1)*1 > (c+0)(3-1)
<=> m > 2c - 1
So cantrip copper is best when money density is at least $2/card, way above what you need for Province, close to enough to what you need for a Colony. (Reasonable, since intuitively $1 plus a card is better than $3 when a card is better than $2.)

Specific money densities and the importance of trashing estates

So, that was some math to help you evaluate particular money bundles vs. each other, not assuming anything about the rest of the deck, except the applicability of money density in the first place. Now I want to consider some specific (somewhat contrived) decks.

The first will be the silver flood: it has c coppers, e estates and s silvers. Its total money m is c+2s, and the total number of cards # = c+e+s, so money density is (c+2s)/(c+e+s). If you want to hit a money density of $1.6 so you can buy a province, how many silvers do you need?
Code: [Select]
    (c+2s) / (c+e+s) >= 1.6
<=> c + 2s >= 1.6c + 1.6e + 1.6s
<=> 0.4s >= 0.6c + 1.6e
<=> s >= (10/4*6/10)c + (10/4*16/10)e
<=> s >= 1.5c + 4e
So for every two coppers you need three silvers (that's a province hand right there: SSSCC); for every blank you need four silvers (that's also a province hand: SSSSE). Said another way: for every estate you trash, that's four fewer silvers you need to buy. Also, trashing a copper moves you closer to $1.6/card more than adding a silver does.

Let's re-check that statement with the math from earlier: when is trashing a copper (m1=-1, c1=-1) better than adding a silver (m2=2, c2=1 with dc=1-(-1)=2):
Code: [Select]
    (m+m1)*dc > (c+c1)(m2-m1)
<=> (m-1)*2 > (c-1)(2-(-1))
<=> 2m - 2 > 3c - 3
<=> 2m > 3c - 1
<=> m > 1.5c - 0.5
So the above statement needs a caveat: trashing a copper is only better than adding a silver when you're already quite close to $1.6.

Let's compare a silver flood to a gold flood; same $1.6 target:
Code: [Select]
    (c+3g) / (c+e+g) >= 1.6
<=> c+3g >= 1.6c + 1.6e + 1.6g
<=> 1.4g >= 0.6c + 1.6e
<=> g >= (10/14*6/10)c + (10/14*16/10)e
<=> g >= 3/7c + 8/7e
So you need one gold (and a bit) per estate, and one gold per two-and-a-bit coppers. A province hand would be GGCCE, and we have 2 = 2*3/7 + 8/7, so the math survives a province-hand sanity check. Every estate you trash is one less gold you need to buy.

This suggests that if you're playing Big Money, trashing your estates is good; it means you can start greening that much sooner.  Of course, -3VP also means you'll have to green more, i.e. longer, but ramping up your money density sooner also means you'll hit $1.2 sooner, which is the Gold breakpoint; the upshot: trashing estates should help you more than the linear formula suggests. To know exactly how much, and whether it outweighs the -3VP, running simulations is probably useful.

The wiki entry on DoubleJack suggests that Jack is fast because it gains you silvers. But Ironworks can gain you a silver too; where Jack draws you a card, most often a copper the first few times, Ironworks "draws" you a virtual copper when you gain a silver. So why isn't DoubleIronworks a BM strategy to be reckoned with, only slightly less good than DoubleJack (ignoring attacks of course)? My money, and my math, is on the fact that trashing one estate brings you four silvers (or one gold) closer to the $1.6 threshold, and Ironworks doesn't trash estates.

This also suggests that on some wonky boards, Hermit can be a poor man's Jack in BM: it gains you Silver and trashes your estates; it fails to draw you cards, but maybe you can transition via Madman into a vanilla terminal draw BM build once Hermit has cleared out your estates, if there's some decent enough terminal draw on the board. Is the +$1 of Ironworks outweighed by trashing estates? Based on my math I would think so. Todo/future work: running simulations to compare various silver gainers and estate trashers (e.g. Hermit vs. Ironworks), to better understand the importance of each aspect of Jack.

TL;DR Buy many small treasures when treasure density is low, buy cantrip +$x over a $y treasure when money density exceeds y-x, and trashing estates is really frigging good. While this might not be news, I like having the math to back up more word-heavy arguments.

12
Rules Questions / Effect of Outpost in a Possession'ed turn?
« on: May 11, 2017, 06:39:34 am »
Setup: two-player game between Alice and Bob.



Alice plays Possession on her turn. Next turn is Bob's, possessed by Alice. On that turn, Bob/Alice plays Outpost. What happens next?

My guess:

If Outpost didn't say "This can’t cause you to take more than two consecutive turns", then Bob would take a 3-card turn followed by Bob taking a normal turn.  (Then, barring any turn-order manipulation it would be Alice's turn, then Bob again, etc.)

However, this would mean that Bob takes three consecutive turns: one possessed by Alice, one introduced by his own Outpost, and one normal turn.

Since Outpost rules out this particular outcome, one (or more?!) of the turns must be omitted.

I think logic would disappear in a puff of divinity if one were to skip the possessed turn, since the need to skip a turn only arises during that turn---general rule: turns that have already begun can not be skipped?

I also think it would be inconsistent with "do as much as you can" to skip the extra turn introduced by Outpost.

Hence, by process of elimination I conclude that Bob's normal turn would be skipped (and only that turn).

Do people agree/disagree with my conclusion? With my way of reasoning?

13
I recently played several games on the following kingdom:




(Cellar, Merchant, Harbinger, Village, Militia, Throne Room, Moneylender, Council Room, Festival, Artisan)

Here's my analysis: there's Village and Festival for +action, Council Room for +cards and +buy and Moneylender for thinning Coppers, so an engine is definitely possible. Militia (semi-)nullifies the benefit to my opponents of Council Room, and seriously hobbles Big Money.  The only support for Big Money is Council Room, with no good way of following it up with a Militia, so... engine is probably likely to beat BM.

I recall one game going all-out for the engine, and another game going almost-all-out BM (with a few Festivals as a $5 Silver, and perhaps one or two terminals; probably an opening Militia.)  I lost both games.

In the only game I won, my final deck consisted of Throne Room, Harbinger, Artisan, Militia, Moneylender, 2xCouncil Room and 6xFestival, 2xGold, 1xSilver, 1xCopper plus a big bunch of green. I remember playing multiple terminals in a few turns (3 or 4 maybe), drawing many cards in slightly fewer turns (2 or 3), perhaps with a single double-Province turn, but I never got close to drawing deck.

If I understand the "Good Stuff" category correctly, it sounds like my winning deck belongs there.

My questions to the community: which strategy do you think is best in this kingdom, and why?  Is it the Good Stuff deck?  What are the opening 3-6 buys for the best strategy, what's it's final composition, and what are the priorities in building and playing it? Any noteworthy subtleties?

If Good Stuff is best, what's the reason? I guess I know the technically correct answer, "the combined effect of all the variables makes it so", but I would guess there's a lesson which transfers to other kingdoms.  Is it the high cost of the drawer and gainer which makes Good Stuff outperform the engine, and the presence of Militia, and the ability to play it almost reliably, which makes it outperform Big Money?  How important is the +$2 from Festival?

My sense is that packing a lot of Festivals made me able to have a bunch of "play $8, buy Province" turns plus a few "play a lot of actions, Militia you, gain a Festival with Artisan, buy a Province and a Dunchy" turns; playing Big Money only allows for the Province turns, and putting in more villages and terminal draw didn't buy me very much in my experience.  Is that an indication that I misplayed stuff, or is this roughly how you would expect things to play out?

14
Help! / How to engine with Vil/Smith/Lib/Lab/Festi/CouncilR/MoLender?
« on: April 14, 2017, 03:17:26 pm »
I played the following kingdom (2nd ed. base set): Merchant, Village, Workshop, Moneylender, Smithy, Bandit, Council Room, Festival, Laboratory, Library.





I opened Moneylender/Workshop (seemed sane at the time -_-), trashed one or two coppers, gained some Villages and Smithies because I couldn't afford the good parts.  By the time my deck could draw itself it was about 30 cards thick, and my turns were super long, contrary to my expectations.  My opponent suggested I should sleep on the couch for taking all that time ;-)

I won, but I think I would've lost to a half-way competent Smithy/BM bot, so there's something I'm totally failing to understand about Dominion. Hence, "help!"

My own thoughts about the board: the only thing that isn't +action/cards/money/buy is the gain from Workshop (which is ~= -1 action -1 card +$4 +1 buy), and the trashing from Bandit. The engine parts are great, so the obvious play seems to be an engine that payloads a bunch of money and buys all the green cards.  Moneylender thins out the copper, sadly the Estates stay.  Festival is clearly the best village, Library is awesome drawing along with it, perhaps supplemented by some Smithies, yes?  Throw in a Bandit for the Gold, or does it make kicking off too hard?  Do it for the attack, but only if the opponent is playing something treasure-heavy, e.g. Smithy/BM?

Once up and running, I think the engine plays itself; what I would like to get better at is putting it together.  I think what went wrong in my game is that I trashed too much money to Moneylender, which made me never get to five until waaaaaay too late.  So don't do that.  Instead get more money by drawing it with Smithy and playing Festival for a while, yes?  Open Smithy/Silver, then 60/40 of Festival/Library ASAP?

I ramble.  To bullet-list my main questions:
  • What should the end-product look like?  Which mix of drawing cards?  Do you include a Bandit?
     Only if opponent is Smithy/BM-ish?
  • How do you get there?  In particular, what should your first, say, six to ten buys be?  When do you get Moneylender? On the first $4 turn after the opening?

(IRL game, so no log)

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