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326

Dominion Articles / Money density math: Gold/2xSilver/Cache, Silver/Peddlers, trashing estates

« on: May 23, 2017, 09:12:39 am »

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.

327

Rules Questions / Re: Effect of Outpost in a Possession'ed turn?

« on: May 12, 2017, 01:47:57 pm »

Outpost is being reworded in the 2nd edition - see Shuffle it's implementation.  Bob takes the 3-card turn, then the normal turn.

Is there somewhere I can access the new wording, given that I'm not a paying subscriber of dominion.games? If not, could you post the wording here?

Bob would first take a 3-card Outpost turn, as part of the resolution of the Outpost played while he was being Possessed. That concludes the effects of the Possession turn; it then moves on to Bob's normal turn.

Is there some line of reasoning that leads to this conclusion?  Some set of principles being applied? Since your conclusion is different from mine, assuming you're right, my assumptions and/or deductions must be wrong; how/where?

However, if you had throne'd Possession then played Outpost at the end of the second one, it wouldn't do anything because you had already taken consecutive turns (through Possession).

I assume the first 'you' means Alice and the second 'you' means Bob, since it's Bob who's taking the Possession'ed turns; if so, then what you say matches my understanding of the rules and cards.

Here's the way that I think of it: each player has an extra turn queue.  Every card that gives extra turns just add to the extra turn queue.  At the end of the turn, you look at the extra turn queue and do any of them in player order.  If there are multiple, the player gets to choose.

So, some implications of that: if Alice plays both Possession, Outpost and Mission on her turn, Alice chooses whether the Mission or Outpost turn happens next; in either case she will have a 3-card starting hand in that turn (because the drawing happens in the modified cleanup phase of the turn in which P+O+M are being played).

Then... I'm still not sure how Outpost works, so I'll say that if there's a second extra turn, it must be taken before the Possession'ed turn. Then the Possession'ed turn happens, then Bob's normal turn (if no more extra turns are introduced.)

Is that right?

Also, choosing between multiple extra turns is a thing that makes sense, because some of them might have restrictions attached to them (if they're Mission turns).  I think Mission-turn-or-not is the only thing that characterizes a turn before it happens, so instead of a queue you might as well have two counters and choose which to decrement, agree?

Well, actually, this gives me an interpretation of Outpost that matches Jack Rudd's claims: it adds a turn to the extra-turn-queue of the player who played it, if and only if said extra-turn-queue is empty.

Implication: if you play Outpost before Mission you get to take three consecutive turns, but if you play Mission before Outpost (I think Villa might be necessary) then the Outpost does nothing (except cost you cards, depending on edition). Agree?

328

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?

329

Help! / Re: GoodStuff best on Cel/Merch/Harbi/Vil/Mil/TR/MLender/CRoom/Festi/Artisan?

« on: May 01, 2017, 02:05:30 pm »

More experience reports:

I recently played on the following kingdom: Cellar, Vassal, Village, Gardens, Militia, Bandit, Council Room, Laboratory, Mine, Artisan.





ISTR losing a few games playing Big Money vs. Big Money; the one time I won, it was with an engine.  I think it had ~4 CRs, one Militia, a few labs, a Mine and almost all the villages (~8 or so).  My opponent played BMish with Bandit, Artisan, a few Labs, a Cellar, a few Villages; nothing that looks too coherent to me.  I ended up with 7 Provinces, 3 Duchies etc.

It took a while before I reliably drew deck, but once I got there I (almost?) never dudded. I could single-province much more reliably than my opponent; I never made it to double-provincing, mainly because I never added any treasure beyond an initial Silver.  With no trashing other than Mine + my opponent's Bandit, I think this was the right call if I were to have any hope of drawing deck.  (Maybe one or two Golds would've been okay, perhaps, but I want the likelihood of having village plus draw in the opening-hand-plus-cantrip-depth to be high.)

Based on this experience, I'm a believer :-)

The delta from OP kingdom to this is: remove Merchant, Harbinger, Throne Room, Moneylender, Festival, add Vassal, Gardens, Bandit, Laboratory, Mine.

So less trashing, no cantrips of note, but Lab is there to supplement Village/CR.  And there's Alt-VP, but I never bought any Gardens.  If the engine works on a board with less trashing, and therefore more stop cards, surely it should work on the OP board, I think.  Wrinkle: if the OP has stronger Money enablers than this board, the conclusion doesn't necessarily follow, but I don't think that's true; Merchant/BM with Moneylender support sounds like... meh; I'll happily contest the Merchants if I think my opponent is going for this, and be happy with my outlook.

So I think my experience with this new kingdom transfers well to the previously listed kingdom; the thinning of Moneylender should make the case for an engine even better.

... Which makes me wonder what I did wrong in the OP-kingdom games for which I don't have my final deck listing available. 

Except for the cost, Festival seems better than Silver.  Agree/disagree?

Well, yeah. But the fact that it does cost $5 and not $3 is super important for big money.

Is that because the opportunity cost of Festival is a Duchy?  If so, it seems like there should be a sweet spot in the mid-game where you still want to build, but it's not so early that $5 is out of reach, where you might get a Festival over Silver and have the price not be an issue.  Except that this window is so narrow that the resulting Festival/terminal-draw ratio will make it likely that you will draw it dead, in which case you should've had a Silver.

And I just thought of the possibility that you might be playing BM with no terminal draw.  If so, and if I'm right about why Festival costing $5 is a big deal, the case for Festival on your $5 turns becomes a little stronger, since you won't ever draw them dead.  But maybe I'm wrong.  Comments are always welcome :-)

330

Help! / Re: GoodStuff best on Cel/Merch/Harbi/Vil/Mil/TR/MLender/CRoom/Festi/Artisan?

« on: April 30, 2017, 01:22:26 pm »

A strawman deck consisting of 10xVillage, 10xMerchant, 1xSilver [...] Is that it?

You're narrowing in on the key insight.

Experince suggests that the straw man deck, suitably un-strawed, is more relevant than I thought. I recently played the following board: Chapel, Harbinger, Merchant, Moneylender, Remodel, Throne Room, Council Room, Market, Sentry, Artisan.





After a few losses with me playing CRoom/BM and whatnot, I put together the following winning deck: 10xMerchant, 5xMarket, 5xSentry, 1xArtisan, 3xSilver, 6xProvince, 1xDuchy, 1xEstate.  My experience while playing it suggested to me that Merchant as a source of money is probably a fine idea.  Maybe the reason I discounted it at first was that I underestimated how quickly you can pile up on those when they only cost $3 (and you have some +buy); I probably undervalue cantrip coppers a bit.  IIRC, I added at least one Sentry relatively early, which helped me stay thin and cycle a lot.

Which leads me to think that maybe the right way to go on the titular kingdom is Moneylender/Silver, then get Merchants very aggressively, enough Villages for all my terminals, an Artisan on my first $6 turn, a CR on my first $5, Militia on my first $4 or when drawing deck, then more Merchants, throwing in a Cellar at an opportune moment.  Maybe one extra Village (above my terminal needs) in case I want to Artisan-gain a terminal mid-turn?

I'm not sure I like Harbinger as a trashing accelerator: I need to draw it in the right order vis-a-vis Moneylender, and once I draw deck reliably it becomes a do-nothing cantrip, except it can play tricks with Cellar (discard something useful, then pick it up later), but that seems very marginal.  I think Cellar would be a better trashing accelerator: play all available cantrips, then mulligan if Moneylender is not in hand; I think that would trash Coppers faster.

In the limit, the stop cards should be Moneylender, Militia, Artisan, Silver, 3xEstate, plus whatever greening I have done. Of those, I want to play Silver, Artisan and Militia; the rest can be discarded to Cellar.  Spitballing, that sounds like two Council Rooms should be sufficient in the limit, perhaps with the aid of a Throne Room and 2xCellar?

Comments are more than welcome.

(Discussion about why my deck was suboptimal on this new kingdom is also welcome, if you feel like it.)

However, a starting hand with only Throne Rooms, Festivals, treasures, and green is a dud.

All those plurals means you start with 7+ cards in your opening hand in my Good Stuff deck, TR+Festival plus any two treasures other than Copper+Silver (5/6 chance of getting something else) gives me a province.  Hardly a dud turn for a deck which is strategically equivalent to Big Money, I would say 

But of course it doesn't kick off an engine, which is your point, and that I agree with

perhaps [they] have more in common than they have differences.

The thing is, the elements they have in common are strategically relevant and the difference they have is superficial.

I like the qualitative distinction (strategic relevance) better than my vague quantitative distinction, "more".

Except for the cost, Festival seems better than Silver.  Agree/disagree?

Well, yeah. But the fact that it does cost $5 and not $3 is super important for big money.

Is that because the opportunity cost of Festival is a Duchy?  If so, it seems like there should be a sweet spot in the mid-game where you still want to build, but it's not so early that $5 is out of reach, where you might get a Festival over Silver and have the price not be an issue.  Except that this window is so narrow that the resulting Festival/terminal-draw ratio will make it likely that you will draw it dead, in which case you should've had a Silver.

Is that approximately it?

Just CR sounds pretty good. [...] Yes to everything.

Cool, I appreciate the confirmation :-)

Also, why is the engine best here?

Mostly because you get to play a Militia every turn and that almost kills big money [...] outright.
 [...] Militia makes you want to ask "is there any way I can avoid playing big money here".

Oh yeah, there's that 

Once again I appreciate everyone's thoughtful responses :-)

331

Help! / Re: GoodStuff best on Cel/Merch/Harbi/Vil/Mil/TR/MLender/CRoom/Festi/Artisan?

« on: April 29, 2017, 08:51:25 am »

["good stuff" exactly equals big money: y]ou buy cards that are good on their own, avoid antisynergies, and green at big money speed. [Treasures vs. Actions: doesn't matter].

Cool, I think you've identified both the elements in common between Good Stuff and Big Money, and the feature that sets them apart: one buys (more) actions, the other buys (more) treasures. Based on the difference, I won't call them exactly the same, but they perhaps have more in common than they have differences.

Also, by that definition I would classify my deck as a Good Stuff deck (matching my intention).  I think it began greening a little later than Smithy/BM would, but in compensation greened a little faster (on average).

The problem with incorporating Festival into that strategy here is that it isn't a very good card on its own.

Okay; why is that?

Here are my thoughts:

Except for the cost, it looks better than Silver to me, mostly for letting me pack more terminals (the +buy is only relevant if I draw more money off of CR, which has its own +buy).  The only way in which Festival isn't strictly better than Silver in this kingdom is that you can draw or gain it dead with Council Room or Artisan, and it doesn't power up Merchant (of which I didn't have any).  With enough of them, the probability of having a Festival conditional on having a Council Room and/or Artisan is pretty good, so the dead Festival issue becomes very marginal.  (And with Artisan I might just gain a Silver instead of Festival, if it's the one terminal I play that turn.)

So except for the cost, Festival seems better than Silver.  Agree/disagree?

In a Good Stuff/BM deck, I take it Gold would be a better payload, but at the cost of limiting the terminal space.  Terminals have anti-synergy in the form of collision, so I would cut down on those too, at which point I should also throw out Throne Room due to the high likelihood of drawing it without any other action.  At which point I'm left with Artisan, Harbinger, Militia, Moneylender and Council Room.  It's not clear what I would want to gain repeatedly with Artisan; Silver, Duchy, Harbinger maybe?  I definitely don't want more than one Moneylender, probably not more than one Militia, maaaaybe two.  And not too many actions that Council Room can draw dead.  And enough draw that I can play lots of treasures to actually grab some provinces.  So chuck almost all the actions and just CR, maybe?

So if I take "don't play Festival in a Good Stuff deck" to heart and follow it to its logical conclusion, I think I end up at BM-CRoom, or something that looks very much like it. Do you agree with this analysis?  If Festival should be taken out, and taking it out leads to BM-CR, the implication seems to be that BM-CR should beat my deck.  Do you think that's the case? (I take it you agree with the consensus that engine beats BM on this board?)

Also, why is the engine best here?  Because it will reliably be able to double-province at least twice while remaining reliable, and because its +buy offers better end-game control, and these outweigh the drawback associated with beginning to green later (than GoodStuff/CR-BM)?

332

Help! / Re: GoodStuff best on Cel/Merch/Harbi/Vil/Mil/TR/MLender/CRoom/Festi/Artisan?

« on: April 28, 2017, 06:12:01 pm »

Thanks for your responses.

Consensus seems to be that the engine is best; noted. I don't recall whether I included Cellar in my engine attempt, but I'll definitely include it in my next attempt.

I should make clear that the deck I listed was not my attempt at building an engine.  I was deliberately aiming for a deck that sometimes would just "play $8, buy Province" and once in a while have a bigger turn, sprinkled with a Militia attack or a $5 gain here and there.  Does that qualify as the "Good Stuff" archetype?

I'm surprised that people recommend against Festival.  That doesn't mean you're wrong, I just didn't predict that. I'll happily concede that my mix of Festival vs. CR is not the right one for building a deck-drawing engine (which I wasn't trying to). Ideally you want enough +cards to draw your deck, plus any mid-turn gains, minus the initial 5, but plus any cards lost to Militia, and I was nowhere near that.

Let me try and make a case for Festival, though, and then invite counterarguments. First, Festival vs. Village.

Comparing Festival+CR vs. Village+CR, one draws an extra card, the other gives +$2, +1 buy.  If I kick off in the first place I'll have enough +buy from CR, so the main benefit of Festival over Village is the +$2. This means that the extra card from Village should be something better than a Silver; otherwise Festival at $5 and 1 buy looks better to me than Village+Silver at $6 and 2 buys.  (Said another way: using Festival as a village means the village doubles as payload, whereas the extra card from Village is sometimes a payload stop-card.)

So let's say the extra card from Village is a Gold.  Playing 3xFestival+CR is net +0 cards and $6 (and +2 actions) for $20 and 4 buys, whereas Village+CR+3xGold gives $9 for net +0 cards, but costing $26 and 5 buys. 50% more money for less than a 50% price or #buy increase; sure, that's more efficient. It is more sensitive to ordering issues, though: with Village+CR+treasures, you optimally want all your villages first, then draw, then treasures, though any mixture of not too many stop cards first will do.  With Festival+CR, only two kinds of things need to be appropriately ordered. My gut claims (without proof) that this causes higher reliability. Agree, disagree?  (Also, the money from Festival helps you in the early turns much more than the cycling from Village, since your money density is much less than $2/card, yes?)

Is the advantage of Village+Gold vs. Festival that you can have a smaller density of stop cards for equivalent (money) payload?  Couldn't you compensate for that by adding more Council Rooms?

"If even 4 of those 6 Festivals had been Villages instead, that's a whole Council Room's worth of drawing" but $8 less payload.  3xGold makes up for that, so it's only one more card for approximately equivalent payload, but costing more $ and buys: 3 more buys and 3x6-4x(5-3)=10 more money; or $7 more for 2xGold+Silver and equivalent payload. So maybe Village+Gold is better than mass Festival, but slower to build, but still better despite being slower to build?  Is it more reliable for having a smaller proportion of stop cards?  But then, won't the faster buildup of the Festival engine mean it can green sooner, and compensate for its lower reliability by spending more time greening?

I'll try out a game where I focus more on Village+CR, but in the abstract it's not obvious that Festival is the worse choice. Could you pinpoint the reason why?

I read a part of 4est's post as suggesting payloading a Militia and not more than 2 Gold.  In my head that sums to $8, enough to single Province if you successfully kick off. Wouldn't you want to double Province, or at least Province/Duchy at $13? Is that off of 6 Merchants, one Silver, two Gold and a Militia?

Speaking of Merchants, and more math, 2xMerchant adds two (conditional) cantrip coppers at $6 and 2 buys, whereas Festival adds the same money and more terminal space, but costing a card to play, for $5 and 1 buy. 3xFestival+CR adds $6 and net +0 cards, for $20 and 4 buys, versus Merchant adding $6 and net +0 cards, for $18 and 6 buys. Merchant is more conditional, CR wants a Militia finisher (but Merchant likes that as well).

Festival seems much more helpful during the early turns, where I'm much less likely to cycle around to my Silver and where I'm often bit short on +buy, though it's of course easier to get to $3 than it is to get to $5.  (Early dud $3 turns are not wasted, I'll just pick up a Harbinger instead of a Festival if Festival is my plan.) Also, a late-game dud turn with Festival means I play $8 and buy a Province (or $6, Duchy). With Merchant, I either never draw them, or I cycle a bunch of cards but don't find my Silver and have no money.  Is the comparative utility of dud turns outweighed by the comparative (un-)likelihood of those turns happening?

It's once again not obvious that Merchant is better than Festival. What am I missing here?

It seems to me that the limitations of Festival, both as a village and as money payload, can be overcome by adding in an extra Council Room or two, compared to Festival's competitors.  Am I wrong here?  Or is an even CR split the bottleneck which makes this a bad plan? Or... ?

Combining Village over Festival with Merchant for money seems extra counterintuitive to me: either the Merchants become unreliable for not having very many Silvers, or the extra card from Village will often be a Silver, which seems worse than just playing a Festival (more money and buys needed).

On the other hand, a strawman deck consisting of 10xVillage, 10xMerchant, 1xSilver and 4 other stop cards will always be capable of drawing the 4 stop cards and have at least one action (worst case is drawing all five stop cards). So maybe there's something to having a high cantrip density, some of which are villages, plus a small number of stop cards, some of which are Council Rooms. Is that it?

I welcome all your counterarguments :-)

Faust says "You can add roughly 1 Throne Room per 3 other actions"; why 3? To minimize the risk of drawing TR dead? Or is that some "experience has shown this to be best" thing?

333

Help! / GoodStuff best on Cel/Merch/Harbi/Vil/Mil/TR/MLender/CRoom/Festi/Artisan?

« on: April 28, 2017, 10:57:36 am »

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?

334

Help! / Re: How to engine with Vil/Smith/Lib/Lab/Festi/CouncilR/MoLender?

« on: April 28, 2017, 10:16:26 am »

[putting-a-cookie-into-a-fax-machine.jpg] [standing_ovation.jpg] +1like(y)

Your pronunciation of my name is pretty spot on; I have no idea how to pronounce it in English, but I have a sense of how to pronounce it in English-with-a-vaguely-Germanic-accent (I'm Danish myself), which is less than half an epsilon different from how you do it ;-)

Also, I appreciate your experience report on F/L vs. V/S, and CRoom vs. no CRoom in V/S vs. BM.

Cool job, bro' :-)

335

Help! / Re: How to engine with Vil/Smith/Lib/Lab/Festi/CouncilR/MoLender?

« on: April 21, 2017, 02:22:30 pm »

I'll think about it.

If you do it, I'll fax you a cookie ;-) (and/or appreciate it very much. Probably the latter.)

[If you Bandit every turn, BM] just buys Golds more often than he usually would to replace the ones you trash, gains Provinces at a slower rate as a result, and still doesn't clog at all. But the fact that he gains Provinces at a slower rate is good enough for your purposes.

If BM has a deck full of fictitious Bronze treasures, costing $6 and providing $1.2 each, I trash one and BM buys a replacement, leaving BM at a steady state.  But if BM buys Gold instead, average money density should go up, since 3 > 1.2.  Letting k=#Gold, n=#cards, if 3k/n = 1.2 then k/n = 0.4, putting the average hand at two Golds and three greens for money density to be at steady-state in expectation.  This will likely never happen, since some Golds plus that many greens likely means the game has already ended; two treasures and three greens sounds representative of my Gardens games. Also, this analysis doesn't take Smithy into account, and shuffle unevenness means sometimes BM will have three Golds and buy Province, sometimes one and buy Silver.  Meh.

So Bandit every turn slows down BM, as you said, but the Gold-minting effect is likely the more important.  (Yes?)

That's true, and Festival doesn't draw so that's another factor.

Right, the even more generic statement is that F/L in general has a lower density of actions with +cards.

This is the bit where I'm not certain I understand the reasoning.

It is situation-specific, but it's a bit more specific than that. [Festival+anything: OK to good/great, terminal collision and skipping Moneylender: bad.  This is even more true the more of each you add. You should approximately maximize expected goodness]

Those are some good points. Thanks for stepping through your reasoning.

336

Dominion Articles / Re: Golden Decks

« on: April 18, 2017, 10:10:48 am »

my favorite one that i made that didn't work was village/mine/mine/IGG

What did you want it to do, and why does it fail?  What I see that you can do is mine an IGG into another IGG, twice, yes?  You pile out eventually, though; is that the problem?  Does that violate the definition of "Golden"?

If you make the deck 2xVillage, 3xMine, 1xIGG, you can even do it three times.  (You start with five out of six cards in hand; the last one can't be two villages, so draw the last card off a Village, then play a second Village, then Mine IGG into IGG three times.)  With 2xKing's Court, 2xMine, 1xIGG, you can even do it six times.  (KC a KC, KC'ing 2xMine, doing IGG-to-IGG six times.)

I think 2xKing's Court, 3xWitch is more effective at dealing out curses (9/turn).  Or 10xFamiliar. Or 10xFamiliar, 4xKing's Court. Throw in some double Tactician or Scheme/Scrying Pool stuff to support even more King's Courts, to pile out the curses even more mid-turn ;-)

337

Help! / Re: How to engine with Vil/Smith/Lib/Lab/Festi/CouncilR/MoLender?

« on: April 17, 2017, 11:45:13 am »

I don't know what's up with the "More cards required than available" error

Would it be easy for you to record a video of you playing back the two games, and posting it somewhere?  It'd be a shame for the work you put into creating the games to go to waste, and I don't know how to progress beyond the error message.

Bandit every turn [is good, but not that good]

Sure, I agree with your more nuanced statement.  I was engaging in a bit of hyperbole :-) I think in the limit, as the number of turns approaches infinity, my stated effect will happen with probability one, but then the Bandit deck will be flooded with Gold and not be able to play Bandit every turn, and piles will run out, and... infinity is not relevant for Dominion, you just buy up all the provinces :-)

Cellar also helps Library more because the engine isn't as reliable to begin with

Interesting.  I guess the reason is that you have fewer Libraries (per total number of cards) than V/S has Smithy, so you're more likely to have an opening hand with no draw in it.

[Lab engine is sometimes good]

Yeah; I figure that's mainly when the other draw and/or village cards are expensive and/or weak?  I guess Lab is trivially the best when everything else is weaker; is Lab-engine worse than the median village plus the median draw (for being more expensive)?

Yes, you do want more Festivals than Libraries in a Festival/Library game, and you might even want to overbuy Festivals early on just so that you can win (or at least, not lose) the split. [...] In F/L, I think you want way more Festivals than one before getting the second Library. Something like 3-4 at the very least. If you don't manage to hit $5, it's fine to buy a Village or two, but Merchants are also pretty good since you already have that Silver anyway.

This is the bit where I'm not certain I understand the reasoning. Well, I understand winning the split, but let's say Festival is uncontested; you still want several Festivals early, and a second Library later, I take it. Why?

I can think of some reasons:

• Economy -- Festival gives you money, which makes it more likely to hit $5, which is critical for buying more engine pieces.  I guess the underlying rule would then be "buy whatever maximizes the expected ratio of turns in which you hit $5 in your next shuffle (over the total number of turns in that shuffle)", and Festival is (almost?) always be the card that satisfies this criterion. Even more generically, buy whatever makes further engine development most successful, in expectation.
• It's the payload -- I'm thinking it sometimes makes sense to get the payload of an engine first, if it does something interesting on its own, and the connective parts later; either if the payload kinda' sorta' works without the connective parts, or if it does something interesting on its own. I don't think that's the reason here, though, except perhaps that it does economy, in which case see the previous point.
• Situation-specific: with only one Festival, a second Library doesn't have anything useful to draw, except on the freakishly rare occasion where you pull of a Festival+Library+Moneylender turn.  Also, for Library action draws to be useful, you have to play some kind of village first, so you want to have plenty of Festivals to make it likely that the Libraries collide with them.
• Here's a (bad?) rule of thumb: in the first k turns, you should buy engine components in proportion to how you want the end product to look like, for almost all values of k. I would ask: why? That seems pretty arbitrary.  In case of Village/Smithy, I would lean towards a little bit more Smithy early, for economy reasons (drawing copper). But maybe that's bad, because it clutters your deck with colliding terminals and not enough money density to buy even a Village.

Also, welcome to f.ds!

Thanks :-)

338

Help! / Re: How to engine with Vil/Smith/Lib/Lab/Festi/CouncilR/MoLender?

« on: April 15, 2017, 07:12:47 am »

Thanks for all the replies.  I am in fact new to Dominion Online as well---so new that I created my account after reading your replies---and thanks for the guidance.  I get the error "More cards required than available" when I try to load the games.  Any idea why, and what to do about it? I haven't purchased anything on DO, but I hear you get to play the base set for free. I played a game against a bot, and that game had Bandit, so it's not like I can't play 2nd ed. base set I would think...

Here are my thoughts in response to all the great comments:

"mix[ing] two different strategies", I think I did that in my post. I don't think I did that in my game.  I think I built a fairly well-running Village/Smithy engine, supplemented by a bunch of Laboratories, even though that wasn't my initial plan.  I think I did it waaaay too slowly, though.

"Library [...] antisynergizes with Treasures and Smithies." True, and a good point.  It also anti-synergizes with green, but almost everything does.  Is Library or Smithy worse with green?  Not on the board, but does Cellar help Library or Smithy more?  I'm thinking green is worse for Library, because each green in hand makes each subsequent Library worse, but that's not true about Smithy (and that's true in general for +cards vs. draw-up-to-n).  I'm thinking Cellar helps Library more, because the net -1 card doesn't hurt Library, and delivering its payload mid-turn (Festival, maybe some terminals) makes the next Library a stronger draw.  Agree, with both conclusion and reasoning)?  Nice point regarding Council Room only giving Library what it would have gotten by itself.  (Unless of course you help someone into a kickoff they didn't have otherwise).

"[in Village/Smithy you don't want too many Festivals]", right, it's a Silver/Village in one, with relative -1 card and +1 buy. The +1 card of Village helps the pieces connect better, which is more important during kickoff, the +$2 is not an amazing payload, and +1 buy should be enough both for building and greening. So you only want one.  Maaaaaaybe two.  Yes?  And if your opponent is playing Library, go for Council Room over Smithy for more cards with +1 buy as a side benefit, and you won't even need the Festival.  (I get why you skip CR vs. Big Money; in that case, playing Bandit every turn should grind your opponents deck into Smithy/Copper/green which loses, yes?)

"[play Moneylender often]" In general, thinning helps your engine pieces connect better, that's why you want to do lots of it, yes?  Losing copper doesn't harm your money base too much during build-up on Village/Smithy because the parts are cheap and you have Workshop, nor on Festival/Library because Festival carries money which compensates for the lost copper.

I really like Laboratory. No one talked about it.  I bought lots of them in my trainwrecked game.  Here's a bit of math: Village+Smithy gives +2 actions, +4 cards, the same as 2xLaboratory, in both cases by spending two cards and actions.  To kick off with Laboratory you just need enough of them. To kick off with Village/Smithy you need enough of them *and* in the right mixture (more Smithy than Village makes you run out of actions), which I guess you could model as some kind of random walk that should never cross the zero point, maybe.  So I guess Laboratory can be a good supplement to Village/Smithy, but not so much Festival/Library, for many of the same reasons that Smithy is bad with Library.  Also, 2xLaboratory costs $10, whereas Village+Smithy only costs $7, and with a lower threshold (3/4 vs. 5) when buys are spread out among several turns.

Could a Laboratory-based engine work here?  Open Smithy/Silver, get Laboratory whenever you can, a few villages on dud turns to support the Smithy, plus Moneylender and Bandit later?  Maybe Festival instead of Village if you never dud, for the +buy/+$?  Or is opening Moneylender/Silver better here?  Drawing and shuffling often with Smithy is a virtual thinness, of sorts, I think, maybe, and the Laboratory will only amplify this.  Once Smithy draws Laboratory dead often, you should also collide Laboratory and Smithy in your opening 5 often, so this will probably not be a problem?  Will the cheapness and extra gain from Workshop in the cheaper engine outrace this?

Soo much question.  Maybe I should simulate some of this if I want to know what's best :-)

(Dylan) "You need [just?] enough villages to play all your stuff, so [add Village/drawing ~50/50]", yep, you don't want to over-plus-action'ate if you don't have a use for it.  I think maybe in the Festival/Library strategy, you want more Festivals than Libraries, (a) because Festival is your payload, and (b) because playing multiple Festivals per Library makes the Library better.  Agree/disagree?

So, for building V/S, open Moneylender/Workshop, then trash/gain/buy as trash/gain/buy can.  In the end, I want balance between Village and Smithy, plus extra Villages for Moneylender and maybe Bandit.  Should I aim for balance during the opening (before drawing the deck)?  Or buy expensive parts whenever possible and cheap parts when impossible?  How important is faster shuffling due to more draw from Smithy?

When building F/L, open Moneylender/Silver, then... grab as grab can?  I'm thinking early on, a turn consisting of "Library, then buy a $5'er" is not so bad, and more drawing and shuffling is good, so perhaps the first $5'er should be a Library, then a Festival, then a second Library, then pile on the Festivals?  Turns with just "Library, then buy" get progressively worse as Moneylender does more of his job, though, but by then you should have more of your engine together, so those turns will also become a lot rarer.

Finally, thanks to everyone for your thoughts, and DrHandsomeface for your praise of my post.

339

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)