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1
Game Reports / Duchess or Pirate ship?
« on: July 10, 2018, 03:05:48 pm »
I played a game IRL with the following kingdom:

Duchess, Storeroom, Sea Hag, Worker's Village, Noble Brigand, Pirate Ship, Royal Carriage, Pooka, Tribute, Crypt, Arena

The big thing here is Sea Hag without curse trashing.  We both opened Sea Hag/Royal Carriage, using Cursed Gold, so you might imagine that our decks got junked up at record speed.  Once that happened, it seemed like the best way forward was to stockpile Royal Carriages, which could be bought with Cursed Gold.  The Royal Carriages could be called on Duchess or Pirate Ship to buy Provinces.  I went for Duchess, and my partner went for Pirate Ship.  We tied.

What would you have done here?  Is Duchess or Pirate Ship better, or would you have gone for something else entirely?

I think Duchess has the edge because Pirate Ship is slower, and you're happy to trash your treasures, which just get in the way of drawing Royal Carriages.

2
Dominion General Discussion / Plotting cards with DATA
« on: May 24, 2018, 04:54:43 pm »
Recently I did some analysis of game logs in order to calculate the "impact factor" of each card.  This is based on 140k 2-player pro-rated games on Isotropic, in which at least one player was ranked in the top 100.  The data only includes expansions up to Guilds.

For my next analysis, I wanted not to rank cards by impact, but characterize/classify cards by type.  I applied Principal component Analysis (PCA), which basically uses a small number of dimensions to describe as much variance among the cards as possible.

The details

I'm going to start out by saying, if you don't understand any of this, and/or you think it's all bullshit, that's okay.  Really, it's just fun to look at the plots whether or not you think they reflect any deeper meaning.  Anyways I'm not going to go into every little detail.  You can ask me anything, or else you can pm me and I'll give you a github link.

In order to do this analysis, I defined two relations, which I call "promote" and "love".  Prom(X,Y) is the extent to which the gain percentage of Y increases when X is present.  Love(X,Y) is the extent to which the gain percentage of X increases relative to other cards, when Y is present.  For each card X, I calculated 412 features corresponding to Prom(X,Y) and Love(X,Y) for each card Y.  I also applied some normalization so that my analysis doesn't care so much about how much it promotes/loves other cards, but rather, which cards it promotes/loves.

Additionally, you might notice that each card is assigned a certain color.  These are based on a basic cluster analysis.  I created seeds for 9 clusters (villages, terminal draw, nonterminal draw, trashing, junking, gaining, payoff, victory), and I used some algorithm to infer classifications for the rest of the cards.  The clustering algorithm is not very good, don't ask me why Rats is a village.  Anyways, it's a guide to the eye.

Requests

Here I show a few sample plots, but I can generate more upon request.  Just ask for Component X vs Component Y.  You can also ask one or both of the axes to show the set of cards loved/promoted by component X.

If you would like to request I change my analysis completely, I'll consider it but I may not have time for that.

Plots

Without further ado, here are a few plots to start out.









Additional notes

By inspection, I believe the first 6 components correspond to:

1. Important terminal cards vs villages
2. 5-cost cards, vs things that help you get 5-costs
3. Thinners vs cards that are best in thin decks
4. Slog cards vs engine cards
5. Trashing vs draw
6. Cheap cards and defense cards vs strong attacks and gainers

After #6, I couldn't tell what the components were getting at.  Most components describe card types that are complementary to each other.  For instance villages tend to love/promote terminals.  Component 4 seems to be a major exception, in that engine cards seem to love/promote other engine cards.  So component 4 describes cards that prefer to be unmixed.

3
Dominion General Discussion / Data Mining: Card Impact Factor
« on: April 29, 2018, 02:46:42 am »
In another thread, I suggested that you can estimate card power by measuring how much the presence of the card impacts what cards you gain.  Well, ben_king kindly supplied some data, so I did it.

The Data

The data set consists of 140,000 game logs from Isotropic.  All games are 2 players, pro rated, no bots, and each game has at least one player in the top 100 Isotropish rankings.  It's the same data set ben_king used for this analysis, and this tooThis data predates Adventures, so cards only go up to Guilds.

The Math

ben_king's first analysis calculated "gain percentages", the probability of gaining a card given that it was in the supply.  What I'm calling the "impact factor" is a measure of how much the presence of a card changes the gain percentages of all supply piles (relative to their average gain percentage).  For instance, if the presence of Duke increases the gain percentage of Duchy by 0.2, then I add 0.2 to Duke's impact factor.

Tricky details: Contributions to impact factor are weighted by how likely it is the cards will show up together.  So Duchy makes a full contribution to Duke, because Duchy is in every game, but Count makes a much smaller contribution to Duke because Count is not in every game.  The gain percentage of Duke makes a contribution to its own impact factor, calculated as (gain percentage)*(probability Duke is not in supply).  Note that the whole calculation is based on supply piles, not kingdom cards, so I don't include Prizes or Black Market cards, but I do include Potion, Estate, Knights, etc.  Yes, the impact factor of Estate must be zero, just think about it.

I am not claiming that the "impact factor" is a measure of the "strength" of a card.  Maybe it is, maybe it isn't.  Tell me what you think.

Update: I posted another list that's calculated without any weighting, and without contribution from non-kingdom piles.

The Results

RankCardImpact         RankCardImpact
1Rebuild2.97110Throne Room1.03
2Mountebank2.55111Cellar1.02
3Goons2.46112Market1.01
4Cultist2.43113Stonemason1.01
5Ill-Gotten Gains2.38114Ghost Ship1.00
6Ambassador2.36115Remodel1.00
7Governor2.23116Oasis0.99
8University2.19117Vagrant0.99
9Scrying Pool2.17118Haven0.98
10Tournament2.16119Great Hall0.98
11Familiar2.01120Watchtower0.98
12Minion2.00121Trading Post0.96
13Swindler2.00122Doctor0.96
14JackOfAllTrades1.99123Moneylender0.95
15Chapel1.96124Lookout0.95
16Masquerade1.94125Silk Road0.95
17Colony1.94126Inn0.92
18Platinum1.94127Smugglers0.92
19Soothsayer1.87128Pawn0.91
20Fishing Village1.84129Forge0.91
21Fool's Gold1.81130Rabble0.89
22Wharf1.80131Envoy0.88
23Marauder1.80132Sage0.88
24Witch1.80133Feodum0.86
25Sea Hag1.75134Horn of Plenty0.86
26Torturer1.75135Death Cart0.85
27King's Court1.74136Vault0.85
28Black Market1.68137Trade Route0.84
29Border Village1.63138Smithy0.84
30Tunnel1.60139Farmland0.84
31Steward1.57140Expand0.82
32Forager1.57141Wishing Well0.81
33Market Square1.55142Moat0.80
34Plaza1.52143Mint0.80
35Remake1.51144Council Room0.79
36Urchin1.51145Harem0.79
37Gardens1.50146Storeroom0.78
38City1.49147Advisor0.78
39Highway1.48148Pearl Diver0.78
40Wandering Minstrel1.45149Baron0.78
41Hunting Party1.44150Hunting Grounds0.77
42Alchemist1.43151Taxman0.77
43Worker's Village1.41152Scavenger0.77
44Lighthouse1.41153Catacombs0.76
45Conspirator1.41154Journeyman0.76
46Grand Market1.40155Potion0.76
47Ruins1.40156Cartographer0.75
48Baker1.40157Nomad Camp0.75
49Peddler1.39158Treasury0.75
50Hamlet1.38159Procession0.74
51Squire1.36160Develop0.74
52Young Witch1.34161Feast0.74
53Ironmonger1.33162Loan0.73
54Herald1.31163Band of Misfits0.71
55Vineyard1.31164Mystic0.70
56Upgrade1.31165Beggar0.70
57Prince1.31166Library0.69
58Bridge1.31167Noble Brigand0.68
59Nobles1.30168Rats0.68
60Fortress1.30169Workshop0.68
61Jester1.30170Bank0.68
62Junk Dealer1.29171Spy0.67
63Festival1.28172Armory0.67
64Count1.27173Venture0.67
65Duke1.27174Duchess0.67
66Fairgrounds1.25175Poor House0.67
67Knights1.25176Cutpurse0.66
68Candlestick Maker1.24177Oracle0.65
69Butcher1.24178Rogue0.65
70Hermit1.24179Mine0.64
71Bishop1.23180Pillage0.63
72Bandit Camp1.23181Stash0.63
73Margrave1.21182Woodcutter0.61
74Walled Village1.21183Talisman0.60
75Caravan1.20184Masterpiece0.59
76Apprentice1.20185Outpost0.59
77Apothecary1.19186Secret Chamber0.58
78Hoard1.18187Merchant Ship0.58
79Bazaar1.18188Graverobber0.58
80Village1.17189Fortune Teller0.58
81Haggler1.17190Saboteur0.57
82Warehouse1.16191Tribute0.57
83Counterfeit1.15192Herbalist0.55
84Salvager1.15193Bureaucrat0.55
85Laboratory1.14194Treasure Map0.54
86Embassy1.14195Cache0.54
87Merchant Guild1.14196Philosopher's Stone0.53
88Menagerie1.14197Mandarin0.52
89Crossroads1.12198Explorer0.52
90Stables1.11199Navigator0.52
91Ironworks1.11200Thief0.51
92Mining Village1.11201Pirate Ship0.50
93Spice Merchant1.11202Adventurer0.50
94Possession1.11203Transmute0.49
95Quarry1.10204Royal Seal0.48
96Farming Village1.09205Chancellor0.47
97Golem1.08206Harvest0.45
98Shanty Town1.08207Scout0.45
99Tactician1.07208Contraband0.42
100Trader1.07209Coppersmith0.39
101Monument1.05210Counting House0.38
102Scheme1.05211Copper0.00
103Courtyard1.05212Duchy0.00
104Embargo1.05213Silver0.00
105Island1.04214Gold0.00
106Native Village1.04215Curse0.00
107Militia1.04216Province0.00
108Altar1.03217Estate0.00
109Horse Traders1.03

Edit: fixed Colony and Platinum, which were calculated incorrectly.

4
Dominion Articles / Salt the Earth and global effects
« on: March 10, 2018, 09:28:15 am »
If I can't have it, neither can you!

What do Salt the Earth and Embargo have in common?  Each one has a global effect, which in principle affects all players equally.  Salt the Earth can make the game shorter for everyone.  The Embargo token can curse anyone.

Global effects are everywhere in Dominion.  Trashing the last Gladiator lets everyone access Fortune.  Emptying a supply pile powers up everyone's Cities, and weakens everyone's Poachers.  Trashing an action with Lurker lets anyone gain the action from the trash.  Trashing actions gives everyone's necromancers more options.

We can divide these effects into three components: The direct effect (e.g. with Salt the Earth, spend $4 and get 1 VP), the global effect (e.g. a victory card in the supply is trashed), and the timing effect (the global effect begins on your turn, e.g. ending the game by trashing the last province).  Usually you get these effects because you want either the direct effect or the timing effect.  But when do you specifically want the global effect?

Global effects are most useful when you and your opponent(s) are in asymmetrical positions.  For example, if I build towards Provinces, and you build towards Duchy/Duke, I might want to embargo Dukes and you might want to embargo Provinces.

But global effects can have a profound impact on strategy even when nobody uses them.  Suppose that a kingdom has two strategies, X and Y.  In order to choose between them, we imagine a hypothetical game between a player who chooses X and a player who chooses Y.  It could be that Salt the Earth tips the scales towards strategy X, and therefore both players should choose X.  And since both players are choosing the same strategy, perhaps neither should buy Salt the Earth.

In games between expert players, global effects tend to appear weak, because experts tend to correctly identify (roughly) the best strategy.  This leads to "mirror" games where players have similar strategies.  But even when global effects should not be underestimated just because they rarely get used.

5
Dominion Articles / The Physics of Dominion
« on: February 25, 2018, 05:36:09 pm »
Preface: I wrote an earlier article about this, but from the outset I wanted to run some simulations and return with a cleaner article.  Here it is!  I have an associated GitHub repository but it's attached to my real name so I'll only share over pm.

Despite the great variation between different games of Dominion, there is one strategy concept that is particularly enduring: the dichotomy between decks that draw (here called "Engines"), and decks that do not (here called "Big Money").  In this article, I will not give strategy advice, but instead try to understand the dichotomy through the lens of physics.  I will show that the line between engines and big money is a phase transition, much like the transitions between water, liquid and gas.

Understanding phase transitions

Most people are familiar with the phase transitions of water. At a certain temperature, water will freeze; at another temperature, water will boil. The phase of water also depends on pressure.  We can draw a phase diagram which shows the state of water as a function of temperature and pressure.


Although water is the most familiar example, phase transitions occur everywhere in physics, and in pure math as well. 

It is particularly informative to consider the bond percolation model.  Suppose that we have a network of pipes, with hubs arranged in a 2-dimension grid. Some adjacent hubs are connected by pipe, and others are not. If the pipes are arranged randomly, what is the probability that water will be able to flow from point A to point B, in the limit as A and B are infinitely far apart?


The black lines represent pipes. Image credit: Wikipedia

We have two possible scenarios:
  • The hubs are broken up into a series of islands, each inaccessible from the others. Some islands might be very large, but they are still finite in size. As the distance between A and B goes to infinity, there is zero probability that they are connected on the same island.
  • There is at least one infinitely large network of hubs. Although there will still be some disconnected islands, there is now a nonzero probability that neither A nor B is on an island, and they are connected to each other.
If you increase the density of pipes, there will be a sudden transition from case 1 to case 2.  This is a phase transition.  Although the model is based on an infinite grid of pipes, a phase transition still occurs even in a finite grid--it just won't be quite as sudden.

Phase transitions in Dominion

In Dominion, we are not trying to traverse a network of pipes, we are trying to traverse our deck by drawing cards.  As with the network of pipes, we will consider the limit as the deck is infinitely large.


The simplest deck that we can consider is a deck with just Coppers and Labs.  In this deck, we can tune one parameter, the fraction of Labs (L). As a function of L, what is the expected payoff of a single turn?

Here's what the simulations show. For L < 1/2, the expected payoff is finite. For L > 1/2, the payoff is infinite (i.e. it gets larger and larger the longer the simulation runs). L = 1/2 is the phase transition.


At first, it would seem you can't do any better than infinity. However, what you can do is increase the probability of drawing infinitely on any given turn. I refer to this probability as Reliability.

Interestingly, at exactly L = 1/2, the expected payoff is infinite, but the reliability is zero. However, this only applies to an infinite deck. In a finite deck, the phase transition is not as sharp, and the payoff is capped by the total payoff in our deck. The phase transition also occurs slightly earlier because we draw 5 cards for free.


The Village/Smithy Phase Diagram


Now we'll consider an infinite deck with Villages, Smithies, and Copper.  Now we have two tuning parameters, the fraction of Villages (V) and fraction of Smithies (S)--and the fraction of Copper is just 1-S-V.  With two parameters, we can draw a two-dimensional phase diagram, just like the phase diagram of water.


For some values of S and V, the expected payoff is finite, which is shown with one color scale. For other values of S and V, there is a probability of infinite payoff; in these cases I show the reliability with a different color scale. The diagram is shaped like a triangle because the total number of Villages and Smithies must be less than the number of cards in the deck.

The diagram also shows that if you add a few Smithies, you can improve your deck, even in absence of any Villages.  This is the "Big Money plus terminal draw" strategy.  However, this strategy is not associated with any phase transition, and is fundamentally distinct from the Village/Smithy engine.

Conclusions

This analysis shows how Engine and Big Money strategies are fundamentally different.  It also shows why, if you go for an engine, you usually want to go all the way.  Moving towards the engine phase transition has some opportunity cost, and sometimes, the opportunity cost is worth it to see that big increase in payoff that occurs at the phase transition.  Sometimes, the opportunity cost is not worth it.  Rarely is it best to move towards the phase transition, only to stop short.

This analysis applies to a wide variety of Engines in Dominion, but it is also interesting to consider exceptions.  In sufficiently small decks, phase transitions become meaningless.  There is no phase transition in draw-to-X decks, such as the one enabled by Minion, and City Quarter evades analysis as well.  Some sifters actually generate multiple phase transitions, the first phase transition allowing you to draw through your deck, and the second allowing you to draw the cards you discarded.

Although a simple physics-based analysis cannot teach you how to play Dominion well, I hope that it has helped you appreciate some of its contours.

6
Other Games / Dominion clones
« on: February 07, 2018, 06:11:46 pm »
Have you played any deck builders besides Dominion?  Did you like them?  Why weren't they as good as Dominion?

I'll start.

Ascension and Star Realms are very similar deck builders, where instead of having supply piles, there's a tableau of randomized cards.  This leads to a much more tactical game than Dominion, which I think is less fun.  Between these two, Star Realms is much better, because it's faster and more balanced.  The card design in Ascension suggests that the creators didn't realize trashing was so powerful.

Puzzle Strike: Bag of Chips is so close to Dominion I would call it an outright ripoff.  Also I heard that the chips and chip artwork was shamelessly stolen from a Dominion fan.  That said, we had some fun with it, and the theme is neat (it's based on a fictional puzzle arcade game).  As for the balance, it kinda seems like the creator thinks the village idiot strategy is OP, so every kingdom is over-terminaled.

Rune Age is a game by Fantasy Flight, in the Rune Wars universe.  It's kinda like Dominion, except that each player chooses one of four factions, and has supply piles that are only available to them.  Also, there are three resource types--money, military, and influence.  And there are several different game modes, including one coop mode.  I liked it, although after playing it a while I started to think some factions were better than others.

Eminent Domain is easily my favorite non-Dominion deck builder, because it's so different.  A major part of the game is about laterally shifting your deck instead of building upwards.  There are five basic card types which comprise most of your deck, and throughout the game you want to shift the relative densities.  Also, there's unlimited saving cards between turns, which adds a whole level of strategic depth.  My one regret is that there's hardly any variable setup, so it tends to be similar from game to game.

7
Variants and Fan Cards / Some heirloom ideas
« on: December 13, 2017, 12:00:33 am »
I really like Embargo, and how it works best in non-mirror games, but unfortunately it's so weak because nobody wants to waste time buying it.  So... it's another problem solved by heirlooms.  I don't have any plans to perfect these cards, but those of you who like making fan cards, maybe there are some ideas for you.

Quote
Allowance - Treasure, Heirloom $0
$1
If you have at least 2 Actions in play, you may trash this.  If you do, put a Prime token on a supply pile.  For each Prime token on a pile, cards from that pile cost $1 more.
The Prime token idea comes from someone else, but I don't remember who.  The only purpose of requiring 2 Actions in play is to delay the effect until around shuffle 3, by which point you can tell what strategies people are trying.

Quote
Shady Dealer - Treasure, Heirloom, $0
$1
You may reveal a Province from your hand to trash this.  If you do, choose a kingdom card supply pile.  That pile is no longer in the supply.
The idea of removing piles from the supply also came from someone else.  But now it's permanent.  This could be devastating.  I am not good at coming up with names, but the idea is that when you get provinces, your kingdom becomes more respectable and you can banish your Shady Dealer, and their goods along with them.

Quote
Bounty - Treasure, Heirloom, $0
$1
At the beginning of your Night phase, put a coin token on a supply pile.  (Whenever a player gains a card, they take all coin tokens from its pile.)
Bounty creates this little mini game of predicting which cards that you'll gain before your opponent.

I also came up with some cards to go with the Heirlooms, but these are just some basic throwaway ideas.

Quote
Town Guard - Action, $5
+1 Action
+$2
Draw up to 4 cards in hand.
Heirloom: Allowance
I don't really know what to pair with Allowance, but the idea is that draw-to-X cards tend to lead to divergent strategies, and also it defends against Militia which could be important if prime tokens are around.

Quote
Aristocrat - Action, $5
+$4
Reveal the top 3 cards of your deck.  Discard one that the player on your left chooses.  Put the rest on top of your deck in any order.
Heirloom: Shady Dealer
Obviously this isn't very good, but it helps you spike that Province, which is probably pretty good.  Later Aristocrat will topdeck those provinces, or I guess discard them if your opponent is afraid of letting you trigger Shady Dealer.

Quote
Recruiter - Night, $2
Gain a card that you do not have in play costing up to $4.
Heirloom: Bounty
For a workshop variant, this isn't very good, but it helps you collect some bounty.

8
Dominion Articles / Dominion phase diagrams
« on: November 27, 2017, 12:07:44 pm »
In another thread, some people were discussing a theoretical way to define an engine by imagining an infinitely large deck.  Equivalently, we could imagine a deck where we draw cards with replacement--meaning that every time we draw a card, we shuffle a copy of that card into our deck.

I am a condensed matter physicist.  This problem interested me because the transition between big money and an engine is similar to physical phase transitions, such as that between water and ice.

I'd be interested in writing an article that is accessible to a broad audience, but this is not that article.  First I need to figure out what's going on.  And I know there's a lot of math expertise around f.ds, so maybe you can help.

The Laboratory engine

Suppose we have a deck with just Coppers and Laboratories.  Let's say the fraction of labs is D (for "draw") and the fraction of coppers is 1-D.  It's fairly easy to figure out the expected value of a turn.

Let x be the average value of a card.  The average value of a turn is simply 5*x.
x = 1-D + D*2*x
x = 1+D/(1-2D)

I drew this with a mouse so it's not to scale.

As you can see, the average value of a turn diverges as D approaches 1/2.  That's the phase transition.  When D is greater than 1/2, your deck has a nonzero probability of drawing forever.  Let's use "reliability" to refer to the probability that you draw forever.  I don't have an explicit expression for the reliability, but I know that the larger D is, the larger the reliability is.

When D is at the critical value of 1/2, this is a special case.  The reliability is zero, but the average value of a turn still diverges as you increase the size of your deck.  mad4math proved that the average value of a turn is grows as sqrt(N), where N is the size of your deck.

When D is somewhat below 1/2, we might say that this is "practically an engine".  In practice we only need to draw finite decks, and we even get to draw 5 cards for free at the start of turn.  So yeah, depending on the size of your deck, the phase transition actually occurs earlier than these calculations suggest.

The Village/Smithy engine

Next we consider a more complicated case, a Village/Smithy engine.  Let's say that the fraction of villages is A (for "action") and the fraction of smithies is D (for "draw"), and the fraction of coppers is 1-A-D.  I don't know the best way to calculate the average value of a hand, but here I give it my best shot.

Let x be the average value of a card, not including dead draw.
x = 1-A-D + Ax + min(A,D)*3*x
x = (1-A-D)/(1-A-3*min(A,D))

The engine phase transition occurs when the denominator is zero.

I am not sure how to add the value of dead draw, but it should be p*3*(1-A-D), where p is the probability of having at least one extra smithy in hand, with only one action remaining and no villages.  I believe this is most important when D > A and (1-A-D) is large.  Anyway, here's the phase diagram:

It's triangle-shaped because we have the inequality A+D < 1.  I didn't have the space to draw it, but there should still be a "practically an engine" region.

I was also thinking there might be a more elegant choice of variables.  Let V be the number of villages per copper, and S be the number of smithies per copper.  In this case, x = 1/(1+S-3min(S,V)).  I'm not sure if this is more or less intuitive, but here's the phase diagram:


Hmm... I wonder if you could get rid of the "practically an engine" region if you just define V and S to be the number of villages/smithies per copper past the first five coppers.

Questions
-I know the calculations aren't perfect, so can you think of any better way?
-How can I estimate the value of terminal draw?
-Is there any way to estimate reliability?
-If I wanted to make this broadly accessible, which parts are particularly confusing or need explanation?
-Any other thoughts?

9
Rules Questions / Exchange ghost for changeling
« on: November 16, 2017, 06:11:19 pm »
Changeling is clearly a favorite in the rules questions competition.

The rulebook says:
Quote
The card being exchanged is returned to its Supply pile, or non-Supply pile, and the card being exchanged for is taken and put into the player's discard pile.

Just to be clear, you can exchange a card for a Changeling as long as that card has a pile, either supply or non-supply?  So if I gain a Ghost, I can return Ghost to its pile, and put a Changeling in my discard pile?  However, if I gain a Cursed Gold with Rogue, I cannot exchange it, because Cursed Gold does not have a pile to be returned to?

I suppose the only time this matters is with Ghost, Wish, or gaining Travellers from the trash.

10
Dominion Articles / Poacher
« on: September 19, 2017, 03:45:24 pm »
You want these if you can get away with it.

In Dominion, there are many cards that give +1 Card, +1 Action, +, or some similar bonus.  These are called Peddler variants, named after a card from Prosperity.  Poacher is the simplest such card, especially when none of the piles empty except for Provinces.

In this article, we will first consider Poacher in the case where no piles empty, and then in the case where piles empty.  Then we will discuss how to predict whether piles will empty, and show a few examples.

Poacher without discard

In many cases, you will be deciding between Poacher and Silver, since they cost nearly the same and serve similar roles.  Poacher... is just better than Silver.  In almost all cases.  Poacher gives you , plus the value of one average card from your deck.  For most of the game, in most games, the average card in your deck is better than Copper (if not in terms of economy, by providing some other utility such as trashing or gaining).  Therefore Poacher is better than a Silver.

Another way of thinking of it is that Poacher + Copper = Silver.  Therefore Poacher = Silver - Copper.  Buying Silver instead of Poacher is a bit like buying Copper instead of nothing.

Another thing is that Silvers anti-synergize with each other by getting in each other's way.  They slow down your deck cycling, making you see important cards less often. By contrast, Poachers do not get in the way of each other and do not slow down cycling.

Some reasons you might get Silver instead of Poacher:
-You have only .
-There's something that cares about the card names (e.g. Merchant).
-There's something that cares about card type (e.g. Smithy without extra Actions).
-You're playing a very junked deck where your average card is worse than Copper.
-You're drawing your whole deck either way.
-You're worried that a pile will empty.

There are often other things you want to prioritize over Poachers, but as far as building your economy goes, a stack of Poachers is great.  That is, as long as you don't get the whole stack...

When piles empty

If a single pile empties, it's not the end of the world!  Poacher is still as good as Oasis (from Hinterlands), which is an okay card costing .

In a few cases, discarding cards is actively helpful.  The most notable case is when you have draw-to-X cards such as Library.  You can also discard cards with the intention of trashing them with Sentry.  However, even in absence of such interactions, discarding cards can be mostly harmless.

More specifically, it's mostly harmless in games where you have some cards which are worse than copper.  For example, if you never trash Estates.  Or if the pile that emptied are the Curses or Duchies, you probably have some Curses/Duchies around to discard.  Even if you're forced to discard Copper, Poacher is still as good as the average card in your deck, and had also provided benefit earlier in the game.

But there are two caveats.  First, be mindful of mid-turn shuffles, which will include the junk cards that you discarded.  Second, there is an antisynergy with other discard effects (e.g. Cellar, opponent's Militia, other Poachers), because you only have so much junk to discard.  There's also an antisynergy with cards that trash your junk, although you can usually use the same cards to trash your Poachers if you don't like them anymore.

If more than one pile empties, then Poacher usually becomes garbage.

How to predict when piles will empty

-Cards that give +Card and +Action (e.g. Merchant, Laboratory)
-A draw engine (e.g. Village, Smithy, and Market)
-Victory cards that compete with Provinces (e.g. Gardens)
-Cards that hand out Curses
-Gainers and a pile worth gaining (e.g. Workshop)
-Bonus tokens from Adventures

These signs only matter if they're good enough to go for.  For instance, Gardens obviously won't cause any piles to empty if Gardens is so bad that nobody goes for it.  Of course, if you go really heavy on Poachers, your opponent might decide that Gardens are good after all because it messes up your strategy.  It's not an all or nothing decision.  If you think there's a moderate risk of piles emptying, you might get just a few Poachers, as much as you can get away with.

Another obvious point is that piles are more likely to empty the more players you have, and they empty quicker too.  With 4 players you may find that even terminal actions get depleted because everyone wants 2-3 copies.

Some special consideration needs to be given to the Poacher pile itself.  Obviously, you should never get 9 Poachers, because your opponent can pick up just one and ruin your deck.  On the other hand, if you get 6, it usually isn't worth it to your opponent to pick up the last 4, because that's expensive and ruins their own deck almost as much.  In 2-player games, you and your opponent often end up with a similar number of Poachers, because nobody wants to get so far ahead that their opponent can counter by emptying a pile.

Example Games

These examples are all sourced from Burning Skull's How to Base Dominion series on YouTube.

Example 1
Discarding with Poacher is good here because of its interaction with Library.  BS takes 6 Poachers to opponent's 2, and wins.

Example 2
This is a Chapel game where no piles are in danger of emptying.  Because of Chapel, the average card is a lot better than Copper, so Poacher is way better than Silver.  I'd like to note that in turn 5, instead of buying a Poacher, BS opts to trash an extra Copper and buy a Silver.  This is a clear, if minor, mistake. Recall that Poacher + Copper = Silver, except that Poacher + Copper is better because you can trash the Copper later.

Example 3
In this game, Merchant/Workshop is good and leads to an empty pile.  However, by the time it has happened, the Poachers have already done a lot of work.  Also, both players have 2-3 Poachers so it hurts them about equally.


IMO this article is "long" and I would appreciate suggestions for things to cut.

11
Rules Questions / Duration + Procession
« on: September 08, 2017, 12:05:22 pm »
This was brought up in another thread.  What happens if you play a Duration with Procession?  Does Procession stay in play?

It seems that the answer used to be "yes", but had been overruled to be "no".  But the wiki is inconsistent (1, 2, 3), and so is Jeebus' rule guide (p 22 of v3.1).  So, it would be nice to get a final confirmation and then to correct these resources.

12
Dominion Articles / Shuffle skipping probabilities
« on: May 26, 2017, 01:12:14 pm »
When you shuffle a deck without a card (usually because it's in play or in your hand), the card is said to skip the shuffle.  Generally you want bad cards to miss the shuffle, and good cards not to miss the shuffle, but it's unclear just how much you should go out of your way for it.

In this article I'm going to answer some really basic questions.  How likely is a card to skip a shuffle?  Are cantrips more likely to skip the shuffle?  What's better, peddler, or lab+copper?

Example: No draw

Suppose we have deck with c non-drawing cards (where c > 5).  If c = k mod 5, then k cards will skip the shuffle.  Thus the probability that any given card skips the shuffle is k/c.  As a function of c, this looks like a sawtooth function, as k cycles through 0, 1, 2, 3, 4.  It's useful to smooth out the sawtooth part, by letting k be its average value of 2.  So on average, a given card skips the shuffle with probability 2/c.

Complication: a card will never skip more than one shuffle in a row.  2/c is the average probability per shuffle in the long run, but if we know it didn't skip the previous shuffle, then the probability is 2/(c-2).

Example: One cantrip

Suppose we have a deck with non-drawing cards and 1 cantrip.  It turns out the cantrip is about 50% more likely to skip!

Let's just consider the first shuffle.  The probability of skipping is still k/c, unless k=0, in which case it's 5/c.  Taking the average over k, we get 3/c, which is 50% more than the base probability of 2/c.

Okay, but complications:  That was only the probability in the first shuffle, what about later shuffles?  Also, the presence of a cantrip may affect the probability that other cards skip the shuffle too.  To figure this out, I ran an elementary simulation with 30-34 cards and 10k shuffles.  In this table, p_cantrip is the probability that the cantrip will skip, and p_stop is the probability that any other card will skip.

cp_cantripp_stop
304.803/c4.13793/c
310.992/c0
321.984/c1.03226/c
332.8512/c2.0625/c
343.8352/c3.09091/c

If we do some averaging, the probability that the cantrip will skip is 2.9/c, and the probability of anything else skipping is 2.06/c.  But this is still nearly a 50% difference.  This appears to invalidate some of jonaskoelker's math (although I think it is a reasonable approximation).

Example: One lab

What's the general mathematical statement?  The probability of skipping is about (2+d)/c, where d is the number of cards you draw while it's in play.  So for a lab, the probability of skipping is 4/c.  The simulation bears this out, with the probability coming out to 3.8/c for 30-34 cards.

So what's better, Peddler, or Lab + Copper?  A peddler increases the value of your deck by about 1-3/c.  Lab+copper increases the value by 1-2/c, while increasing your deck size by (1-2/c)-(1-4/c) = 2/c.  As far as money density goes, it turns out Peddler is better whenever your money density is greater than 0.5.

But of course, it's a very small improvement, on the order of 1/c.  And this is only really valid, by the way, for a low density of labs.  If you have lots of labs, they statistically tend to clump together, because a turn where you play a lab is a turn where you're more likely to find other labs.  Also, they make your average hand size larger, which makes other cards skip more often too.

Strategic comments

At this point it might be obvious that I'm more interested in the math than the strategy.  But here's some strategy takeaway:

-Statistically, if you play everything, drawing cards are a lot more likely to skip the shuffle than stop cards.
-Any cards that skip the shuffle are in most cases guaranteed to make it into the next shuffle.
-If you opt not to play a card, it's kind of like instead of skipping this shuffle, it skipped the previous shuffle and got replaced with a dead card.  Generally, this is only good if you're also preventing other good cards from skipping the shuffle.

I didn't really talk about engines, but feel free to talk about them in this thread.

13
Dominion Articles / Engine capacity math
« on: January 31, 2017, 02:21:54 pm »
Here is an article I wrote.  I think it's too long, but it's been sitting around for a while, so you can finally have it. :P

There are two major ways to improve your deck: (a) improve the average value of any given starting hand, or (b) improve the total value of your deck while maintaining the ability to make use of its total value.  The most common way to accomplish (b) is to use components (splitters, terminal draw, non-terminal draw) to draw and play all your stop cards (terminal payoff, non-terminal payoff, and junk).

Basic question: What is the capacity of this deck, in the best case scenario?  In absence of any drawing components or attacks, the capacity is 5 stop cards, one of which may be terminal.  To increase the capacity beyond that, you need to add components.  I suggest that a useful way to characterize an engine is a quantity I call the component-draw ratio: the number of components you need to draw one additional stop card.  I will calculate this for a bunch of examples.

Basic examples

  • Play 1 Lab, draw 1 additional card.  The component-draw ratio is 1.
  • Play 1 Village, 1 Moat, draw 1 additional card.  The component-draw ratio is 2.
  • Play 1 Festival, 1 Smithy, draw 1 additional card.  The component-draw ratio is 2.
  • Play 2 Villages, 2 Rangers, draw 3 additional cards.  The component-draw ratio is 4/3.
  • Play 1 Bustling Village, 2 Hunting Grounds, draw 6 additional cards.  The component-draw ratio is 1/2.

Easy so far, right?

Durations

To make calculations for durations (or Coin of the Realm), suppose that you're drawing your deck every turn.  Most duration cards stay out an extra turn, so you need twice the number of these components.  Thus, even though Wharf and Fishing Village can lead to greater reliability, they sometimes have higher component-draw ratios.

  • Duration Caravan, play Caravan, draw 1 additional card.  The component-draw ratio is 2.
  • Duration Wharf, play Village, play Wharf, draw 3 additional cards.  The component-draw ratio is 1.
  • Duration Fishing Village, play Fishing Village, play 2 Smithies, draw 3 additional cards.  The component-draw ratio is 4/3.
  • Duration Fishing Village, duration 2 Sorceresses, play Fishing Village, play 2 Sorceresses, draw 1 additional card.  The component-draw ratio is 6.

The Fishing Village/Sorceress component-draw ratio is one of the worst in the game, and you probably wouldn't bother with it at all.

(Math note: I calculate all these ratios by solving a simple system of equations.  To input into these calculations, I say Village nets +1 action, and Smithy nets -1 action and +2 cards.  For duration cards, you simply take this net value and divide by the number of turns that the card stays out.  For example, Wharf nets -1/2 action and +3/2 cards.)

Terminal payoff

Not all stop cards are equal, and some require more or fewer components.  There are three major varieties of stop cards: nonterminals, terminals, and junk.  Gold is a nonterminal, and you just need to draw it and keep in your hand.  An Estate is junk, because you need to draw it, but don't need to keep it in hand.  Bridge is a terminal, and you need to draw it and spend an action to play it.

Anyway, to play additional terminals, you need additional components.  So now I'll calculate the component-action ratio, the number of components needed to gain +1 Action without changing hand size.  Here are a couple examples:

  • Play 1 Village, getting +1 action.  The component-action ratio is 1.
  • Play 2 Festivals, 1 Smithy, getting +1 action.  The component-action ratio is 3.

Notice that even though Village/Moat and Festival/Smithy have identical component-draw ratios, they have different component-action ratios.  This shows how Festival/Smithy can struggle more when the payoff is terminal.

Of course, for things like Lost City, it's impossible to get +Actions without also drawing cards.  In these cases, you can't calculate component-action and component-draw ratios separately.

Sifting

There are two major uses of sifting in a deck drawing engine.  You might use it only for reliability, to ensure that you have the right components at the right time.  Or you use might it for draw, that is, when you have junk stop cards that don't need to be kept in hand.  If you're not sure whether you're using sifting for draw, ask yourself: Do I want to reshuffle and redraw all the cards I discarded, or would I rather leave them in the discard?

In the cases where sifting is being used for draw, we can compute component-draw ratios.  Some of this draw is "true" draw, which increases your hand size, and some of it is "sifting" draw, which sets aside or discards the extra cards.

  • Play 1 Forum, sift two additional cards.  The component-draw ratio is 1/2.
  • Play one Wandering Minstrel, discarding 1 stop card on average, and play Smithy.  You've drawn 2 additional cards, and sifted 1.  The component-draw ratio is 2/3.
  • Duration Gear to get 2 cards, play Village, play Gear, setting aside 2 cards.  You've drawn 1 additional card, and sifted 2.  The component-draw ratio is 1.
  • Duration 2 Archives, play an Archive.  You've drawn 2 additional cards, and sifted 3 cards by setting them aside.  The component-draw ratio is 3/5.

I find that these calculations can be quite dependent on your deck.  For example, the component-draw ratio of a Cartographer is directly proportional to the density of junk cards in your deck.  Also, many sifters are constrained such that it's impossible to sift all the junk, and only the junk.  For example, Wandering Minstrel treats all non-Actions as junk, and won't sift junk already in hand.  Archive sifts cards by setting them aside, but the cards you sift are the same ones you draw on later turns.

Throne Room and other oddballs

There are many cards I can't analyze very well, because their power is too variable (City Quarter, draw-to-X, Scrying Pool, Magpie), or they're limited to once per turn (Crossroads, double Tactician), or are not intended to draw consistently (Madman).  Yet it is still possible to apply these ideas to at least a few oddballs.

-Throne Room.  Throne Room is like a copy of another Action card in your hand, only with +1 action.  If you chain N Throne Rooms together, it will copy N other Action cards, and provide +(2N-1) actions.  So, without increasing the number of components at all, it provides +1 action for almost every component you have. Basically, in terms of best-case engine capacity, Throne Room has the same effect on your deck as Champion.

-King's Court.  If you chain N KCs together, you get to play 2N-1 Action cards a total of 3 times each.  In the end, you're using 3N-1 cards to play 6N-3 components.  King's Court is like Champion, and with enough of them it may halve your component ratios.

-Expedition.  How much this helps to draw your deck depends on how easily you can pay for it.  For instance, suppose you paid for it with a Silver and Market.  Silver and Market are usually payoff cards, but could be treated here as draw components.  The component-draw ratio is 2.

-Storyteller.  Similar to Expedition, you can consider treasures as draw components.  If on average you play 3 Silvers with Storyteller, the component-draw ratio is 4/3.

Strategic impact

Engines with different component ratios face different sorts of challenges.

Engines with low component ratios usually have more expensive components, and they may struggle to hit the high price points.  And since stop cards may be a larger fraction of the deck, they have more trouble having the right components in hand when needed.  Here, sifting might be used more for reliability than for draw.

Engines with high component ratios struggle to gain lots of components, and may hit a ceiling when piles run out.  Since each stop card makes a difference, thinning helps a lot, and green/junk hurt a lot.  Handsize attacks also hurt.  Here, sifting might be used more for draw than for reliability.

So for example, engines using Ghost Ship for draw are particularly difficult because of the high component ratios and handsize attack.  Engines using Margrave are much easier because of the lower component ratios, and an attack that actually improves reliability.

Of course, component ratios are only one aspect to consider.  Some components are easier or harder to gain (e.g. Port vs Village), and some provide additional payoff (e.g. Festival/Smithy vs Village/Moat) or reliability (e.g. Labs vs Village/Smithy).  This is just a starting point.

14
Variants and Fan Cards / Attacks that attack yourself
« on: August 04, 2016, 04:18:51 pm »
Inspired by the thread about debt attacks, I came up with a few cards that attack yourself.  I thought I'd share just for discussion and inspiration.

Quote
Toll Road
$5 Action-Attack-Duration
Until the end of your next turn, on each player's Buy phase, cards cost <1> more.
At the beginning of your next turn, +4 cards, +1 Buy.
This hurts opponents for one turn, and hurts yourself for two turns.  But otherwise, the card is strong enough to balance it out.

Quote
Cobbler
$4 Action-Attack
Each player reveals the top three cards of their deck, puts the Copper back, and discards the rest.
+3 cards
Top-decking copper is weak, especially when you're also doing it to yourself.  But if Cobbler is your last action, Copper might not be so bad.

Quote
Sacred Urn
$4 Treasure-Attack
When you play this, it's worth $2 if you have no other Treasure cards in play.
Each player with at least 4 cards in hand places one on top of their deck.
Top-decking a card from hand can be good (Courtyard) or bad (Ghost Ship).  Can play this in such a way that it's good for you and bad for your opponents?  Or perhaps just play it when you have fewer than 4 cards left in your hand.

15
Variants and Fan Cards / Empires speculation
« on: May 12, 2016, 08:49:10 pm »
Between the Empires previews and Empires release, we only have two weeks for wild mass guessing!  What will the $14 event do?  How will we be able to bid? etc.

Post your most wild-eyed ideas here.  If you were correct, you will earn bragging rights, and if you were incorrect we'll just pretend it was a fan card all along, well it is the fan card forum.  Obviously playtesters can't participate.

16
Variants and Fan Cards / Mechanics inspired by passing
« on: December 04, 2015, 11:19:11 pm »
1. I was thinking about the mechanic of passing cards.  For example, LastFootNote's Wanderer.

Quote
Wanderer: Action, $3
+4 Cards. The player to your right gains this card.

One of the problems with this is that it's political.  If you buy one, you most help the player on your right.  Ideally, you'd pass a Wanderer to all players, but then that's just a recipe to pile out Wanderers.  So the idea is you have a token to indicate who "really" has the wanderer.  For example, one implementation would be

Quote
Wanderer(2): Action, $3
If you have your Wandering token, then spend it, +4 Cards, and each other player takes their Wandering token.
Otherwise, +1 card, +1 action.

When you buy this, take your Wandering token.

This card is interesting, but not that much like the original, and it loses all the flavor.  So, um, maybe it wasn't a great idea.

2. Earlier, I had suggested another self-passing card, one where you get a bonus for buying it, but otherwise it's a dead card.

Quote
Tedious Tale: Action-Victory, $6
The player on your left gains this card.

When you buy this, +3 VP chips.

So let's try to do that with tokens:

Quote
Tedious Tale(2): Action-Victory, $6
If you have the Tale token, +1 action, +1 card.  Otherwise, take the Tale token.

3 VP

The idea here is that there is only one Tale token that can only be held by one player at a time. Taking the token represents "passing" the card.  But still, it's rather different from the original Tedious Tale, since you can't give your opponent dead cards that they didn't buy themselves.

At this point I'm thinking, forget passing!  This token idea opens up so much more design space.  You could have a card that does something different depending on whether it's contested or not.  For instance:

Quote
Smithing village: Action, $4
If you have the smithy token, +3 cards.  Otherwise, +1 card, +2 actions, and take the smithy token.
Smithing village is probably a terrible idea.  But you see how very simple effects could be combined into a complex card.

3. One more idea!  I wanted to design a card (White Elephant here) which is like Masquerade, but which incentivizes swapping more expensive cards.  I was unsatisfied with it, and it didn't seem to work no matter what I changed.  So I thought, why swap cards between neighbors?  Just swap cards with the last person to play the card.

Quote
White Elephant: Action, $5
+1 action
Reveal a card from your hand.  Have it switch places with the current Gift card.
If you received a card costing less than the one you revealed, then +$3, and you may trash a card from your hand.
------------------------------
Setup: Set aside a silver as the initial Gift card.  The Gift card is not in the supply.

I'm not sure that's balanced.  But you get the idea.

Can you think of any well-balanced cards using these mechanics?

17
Variants and Fan Cards / Card ideas for gift
« on: October 28, 2015, 05:16:14 pm »
Hi, I'm a (former) lurker.  My boyfriend joked that what he wanted for Christmas is a new Dominion expansion.  So as a fun arts and crafts project, I'm creating fan cards.  I was trying to make things that were more amusing than functional, but maybe a few of them are more functional than amusing.  There's sort of a theme of unwanted gifts, which is totally a coincidence and not self-referential at all.  Tell me what you think!


Update: newest versions are here.


Ghost Town
$3 Action
+1 card
+2 actions
Reveal your hand and discard all action cards.  For each action discarded, look at the top two cards of your deck, discard one and put the other into your hand.

This is the village that gets rid of the cards that you'd want to pair with village.

Supreme Court
$3 Action-duration
You may put a card in your hand on top of your deck.
At the beginning of your next turn, you may choose an action card in your hand and play it twice.

This is the version of throne room that's okay to open with.  Somewhere in there is a political joke.  My boyfriend has a degree in law, that's the joke.

Retirement Fund
$3 Action-Victory
+1 action
Put 1 VP on your tavern mat.  VP on your tavern mat do not count towards your final score.

At the end of the game, for each Retirement Fund in your deck, you may take up to 3 VP on your tavern mat.

I like how alt-victory cards encourage all new strategies.  This one encourages you to buy some early, and pick up more in the late game.

Deciduous Forest Hag
$4 Action-Attack
+1 coin
+1 buy
Each other player with at least 5 cards in hand trashes a card from their hand that is not a curse.  Each other player gains a curse.

A lesser-known cousin of the swamp and sea hags, this is the result of an attempt to attach the drawback of Bishop to a different card.  So now it's on a curser, and the trashing is forced.

Uncharted village
$4 Action-Reserve
+2 cards
+2 actions
Put this on your tavern mat

If this is on you tavern mat, at the beginning of your turn, you may call this.  If you do, discard two cards.

We all like lost city, so now this is a lost city that has a different drawback.  You can skip a turn to find all your uncharted villages again.

Captain
$5 Action
Each other player may reveal this card (they may look through your discard pile to decide).  If anyone does, and this card is a Captain, then every other player takes their -1 card token.
If this card is a Captain, +2 coins, and every other player takes their -1 coin token.

In games with this card, as an action you may play any card face down.  That card is a Captain until it is revealed.

Obviously this is inspired by Coup.  Even if it doesn't work, we will laugh about it.

Garbage Processor
$5 Action
+1 action
The player to your left looks through your discard pile and chooses three cards.  Discard or trash one of them.  Put the rest into your hand.

I thought this might be too strong, like a lab with trashing. But then I thought, sometimes it will give you two copper and trash a copper.  Sometimes your discard pile will be empty.  So maybe it's balanced?

Awkward gift
$5 Action
If you have at least three actions in play, +2 actions and gain two gold.  Otherwise, +3 cards and gain an action card costing 4 or less.

The idea behind this card is to have a strong effect which is hampered by awkward placement.  If you have smithy-BM, you didn't really want that action, but okay, let's make the best of it.  If you have an engine, you didn't really want that gold, but gold is nice I guess.

White Elephant
$5 Action
+3 cards
Each player sets aside a card from their hand.  Set aside cards are revealed.  Each player gains a card costing up to 1 coin more than their revealed card, and puts the revealed card in the hand of the player to their left.  You may trash a card from your hand.

Masquerade is fun, but there's no incentive to pass good cards.  This variant encourages people to pass better cards, sometimes.  Also, pile control.

Tedious Tale
$6 Action-Victory
The player on your left gains this card.

When you buy this, +3 VP chips.

This tale is only really interesting if you were there.

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