And now… on with the main feature:
First up is Sea Hag.
Best result: 2 cards versus Bmu: 98%
Interesting: 1 card versus Bmu: 94%
This girl makes Witch look like Piper Halliwell from Charmed. And where Witch provides at least some utility to the player, Sea Hag’s sole purpose is to latch on to the opponent’s jugular vein. Given the strength of early Curses, this works like a… let’s just say it’s damn effective. So I wonder which one to choose when there is both Witch and Sea Hag available. And this is where things get really interesting:
1 Sea Hag versus 1 Witch: Sea Hag wins 52%
2 Sea Hags versus 2 Witches: Witch wins 61%
Sea Hag is slightly stronger against Bmu, and the one-on-one results reflect that. But when adding a second card, the added attack strength of the second Hag is far less effective than the added utility of the second Witch. And that means that the Witch comes out on top. My favorite must-buy card may have been been wounded, but is far from defeated.
But there is another strong contender to best attack card: the Mountebank.
Best result: 2 cardsversus Bmu: 91%
Interesting: 1 card versus Bmu: 87%
Almost… but no cigar! Mountebank is an awesome card by all means, but the fact that victims can escape by discarding a curse from a previous round impairs the Mountebanks effectiveness somewhat. If that bit was not part of Mountebank then 3 Mountebanks would devastate Bmu with a 98% win rate. But now it is slightly less wicked than Witch against Bmu. As for a direct confrontation between those heavyweights:
2 Mountebanks versus 2 Witches: Witches win 65%
Brutal! I was afraid that even though the Witch outperforms Mountebank against a non attacking strategy, Mountebank would wing it when the curses run out. But Witch reigns supreme. (Things get only slightly worse if we have them fight one on one, in which case the Witch wins 61% of the games.)
Best result: 3 cards versus Bmu: 90%
Interesting: 1 card versus Bmu: 82%
Time now for a more civilized and stately attack card. Don’t let the name fool you, Goons is the older, wiser and more successful brother of Militia. The +1 VP per buy combined with the +buy makes this a formidable card combining decent attack with a great VP boost. It turns out the VP ability of Goons needs to be fed, even with Coppers if nothing better can be afforded for the second buy. It also turns out that you can choose buy Estates over Coppers whenever you have enough coin or choose to ignore Estates for the second buy. The choice does not significantly alter the results. Apparently the extra VP from the estates are countered by the extra loss of buying power. Strategy Details here.
Good results, but not nearly good enough to beat the killers. 2 Witches versus 3 Goons yields a 76% win rate for the dark side.
Best result: 2 cards versus Bmu: 72%
Interesting: 1 card versus Bmu: 70%
The attack may seem weak, but its good to know that it does pack somewhat of a punch. If this is the only attack on the board, by all means use it. But compared to the other attacks, one of its better features is that it was easy to simulate.
Single Ambassador 1 versus Bmu: 69%
Simulating the Ambassador is a bit like simulating the Chapel. When is it a good idea to trash that Copper? When do you stop trashing your Estates? Single Chapel 1 provided a great base for my Ambassador strategy. But as we saw with Mountebank, spamming your opponent with Coppers is not as effective as you might hope. Making this attack not nearly as effective as other attacks.
Minion chaining 1 versus Bmu: 66%
Finally an attack card that has no trivial strategy to play. The basic idea is that you aim for getting a hand with 2+ Minions. Play all but the last one for money and use that one to get a new hand of cards. Now the problem with this is that you need a small army of Minions to pull that off. However, to fund that army you will need some treasure to get started. You don’t want that treasure, because it will be in the way when you are chaining your Minions. So you’d expect that buying an absolute minimum of treasure is the best strategy.
And that’s were it’s good to simulate these things. It turns out to be much better to buy a couple of Golds (max 3) and all the Silvers you can get. Once you reach 3 Golds you stop buying that and focus on Minions.
Playing minions is also tricky. When playing the last minion in your hand you have to weigh the value of your current hand +2 (from the Minion) against the potential value of four new cards. I found a simple heuristic that works like this:
if (current hand value +2) > 5 : use Minion for money
else: use Minion for cards
Once you have 2 minions, the target becomes to buy Provinces.
if (current hand value +2) > 8 : use Minion for money
else: use Minion for cards
All in all, the results from the single card point of view are disappointing. I expected Minion chaining to be more effective. Maybe the Minion strategy would really benefit if we were to add a trasher… like the Chapel…
Continuing the analysis of simple cards we currently can simulate, I have a couple of cards that are very decent by themselves. And some that are less than decent. The strategy details are listed here.
Best result: 2 cards versus Bmu: 83%
Interesting: 1 card versus Bmu: 77%
Garion implemented this one because he was curious of how good it was. And then he enthusiastically forced me to investigate as well. With good reason. This is the best non attacking single card we have so far. It even outshines Militia with respect to its win percentage against Bmu. Ooh! we can simulate a match between them ;). If you have both strategies buy at most one card, Monument wins 57% to 43% for Militia.
Now since Monument is a terminal action, the same reasoning applies as with the witch and the Militia. For this simple case the optimum lies at buying two Monuments.
For all my simulating, I actually haven’t played Dominion that many times. The classic learning curve of dominion has the following stages: “Village idiot”, “Big money”, “Single cards” and “Complex strategies”. I am still in phase 3, learning which cards are good by themselves. And I use the Simple Smithy strategy as the litmus test for more complex combos. I ask myself: is this complete strategy actually better than just buying a single Smithy card?
The Smithy is actually a well simulated card. And from other simulations I expected 2 smithies to do better than one. Especially if you buy the second Smithy after having shuffled your deck a few times. But I found that using Bmu as a base strategy, a single Smithy is slightly better than having 2 Smithies. If you buy the second Smithy in turn 5+ the difference do becomes very small, so this might be useful when creating combos.
Best result: 1 card versus Bmu: 74%
Council room is like a Smithy on steroids. You get an extra card and an extra buy. So why does it score worse than Smithy? Is it the extra card the opponent gets? Or is it the higher cost?
If we remove the extra buy, the win rate plummets to: 65%. Apparently with 8 effective cards the extra buy really packs a punch, even with my unoptimized extra buy implementation*. If we also change the price to 4, the win rate climbs to: 70%. So if the Council Room were just like a Smithy that draws an additional card but also gives an extra card to the opponent, it would still be worse than Smithy itself. Apparently without an extra buy the extra card helps your opponent more than it helps you. All in all I find this a fascinating card, for which I hope to find nice combos.
* Up until now the simulator did not support multiple buys. As a trivial implementation for all rule based strategies I just re-iterate the list of buy rules from top to bottom as long as we have buys left. In practice this means that for now the extra buy will mostly bring in additional Duchies and Estates near the end of the game. I’m sure there is optimizations to be done here.
Best result: 4+ cards versus Bmu: 62%
Interesting: 1 card versus Bmu: 56%
The Market is the best non-terminal action of the bunch. I tested with an increasing maximum number of markets that the simple strategy was allowed to buy. With higher maximums the win rate increased, until it stabilized at 4 Markets. Allowing the strategy to buy more than 4 Markets did not increase -but also not decrease- the win rate. This strategy only works if you favor Provinces and gold over the Market, so I guess you will not be buying more than 4 Markets anyway.
As an exercise I reduces the cost of Market to 4 to see how big the effect would be. Well, not that much actually. 4 Markets priced at 4 score 68% against Bmu, while a single Market priced 4 only raises the score by a 1 percent point to 57%.
Best result: 4+ cards versus Bmu: 61%
Interesting: 1 card versus Bmu: 55%
Without any other cards to back it up, the Laboratory is slightly outperformed by Market. (Which immediately makes me wonder if ChapelMarket is a viable alternative to Chapelab.) But first let’s see how good the Lab would be priced at 4. With 5 bargain Labs you score 70%, and a single el cheapo Lab scores 56%. Nice, but even at reduced price they still don’t outperform my litmus Smithy.
Best result: 3+ cards versus Bmu: 57%
Interesting: 1 card versus Bmu: 53%
In the absence of additional terminal actions to boost, Bazaar is a Market without the +buy.
Best result: 3+ cards versus Bmu: 56%
Interesting: 1 card versus Bmu: 54%
This cards is dubbed the “expensive silver” when bought in absence of other terminal. Its refreshing to learn that this is not the whole truth. The +buy may not be impressive, but it is something.
Without other cards, Woodcutter is effectively a terminal Festival variant. Indeed it scores identical to buying a single Festival.
And nobody likes the Woodcutter anyway… At the very least they could have named it Lumberjack for a few cheap grins.
Best result: 1 card versus Bmu: 54%
Now here is the biggest surprise so far: a Moneylender by itself is just as good as a Woodcutter. Say what? That seems very wrong. That warrants some further investigation. To be continued…
Best result: 1 card versus Bmu: 42%
I haven’t found any good strategies for victory/action or victory/treasure cards yet besides Nobles+Nobles. And this result shows that blindly buying the Great Hall at the start of the game is not the way to use this card at all.
Best result: 1 card versus Bmu: 32%
Now I understand that Village does not bring anything to the table in the simple card strategy. No money, no cards, no +buy, nothing. It’s best ability is that its self replacing. However what is interesting is just how bad it scores. By buying a single useless card at the start of the game you almost halve your chance to win!
Go Village idiot go!
[Edit: I realized that card names should be capitalized. And I cleaned up some typos while I was at it.]
Today I take a detour from hard core Chapel bashing. (And unfortunately also from nice graphs.)
Currently our simulator only implements about 20 kingdom cards. And since I was doing cleaning yesterday, I had a lot of incentive to take small breaks to run batches of simulations on these cards. I wanted to see how good each of these action cards is, when you are only allowed to buy of that card type. This gives some very basic insight into the strength of the card on its own. And it will be vital when evaluating the strength of cards in combos.
I used the (very) Simple Card Strategy, so I can only simulated cards that do not require any additional logic when actually playing the card. See here for details. Now since I have simulated 15 cards, I’ll post the results in a few topics, allowing me to give some additional info next to the raw numbers. And we start with three attack cards.
Witch is the Big Bad Dark Jedi of dominion classic. It is by far the strongest card our humble simulator has. In fact Witch is the only action card (that we can simulate) that is a better buy than Gold.
I am curious how the Witch stacks up against other curse dealing cards, especially Mountebank. Some non-optimized simulations indicated that if the Witch did not deal Curses (only have the +2 cards) it would score about 50% against Bmu. So it really is the Curses alone that beat our poor Bmu into submission.
Witch is a terminal action, so when buying your second Witch you introduce the risk of getting a hand with a Witch that you cannot play. But the benefit of putting more Curses into your opponent’s deck far outweighs the occasional inconvenience. Buying three Witches is still better than buying one Witch, but two Witches is optimal.
Note that Bmu [d/e]=5/2 is not particularly optimized against attacks. Optimizing against attacks will be a topic in its own right.
Best result: 2 cards versus Bmu: 79%
interesting: 1 card versus Bmu: 73%
Another attack, one that will actually cause lots of problems later on. At some point we have to define how strategies will discard their cards, and this will make everything more complex. Also the simulator does not have a mechanism in place yet that allows one strategy to react to cards the strategy plays. (I actually look forward to the latter problem.) For now I use a basic strategy that discards the lowest value cards first: victory cards, Coppers, Silvers and finally Golds.
The results are impressive: two Militias secured 4/5 victories against Bmu. Since Militia doubles as a Silver, it does not hamper the economic growth too much. Three Militias perform just as good as two, which is different from the Witch. I guess that with three Witches the risk that one Witch will draw another Witch becomes too big. This is a problem the Militia does not have.
The last attack card we can simulate is more of an annoyance than a threat. For attacker and defender alike. The attack part is not totally worthless, because if we disable it the win rate of this strategy drops to 51%. And remember that a win rate of 65% means that the attacker wins two thirds of all games.
But then that 51% is how good its Silver gaining ability is. Which is surprisingly bad, if you compare buying a Bureaucrat to a buying a Silver. Having a Silver in hand immediately raises money to spend in that turn by 2, while the Bureaucrat raises the money in your next hand by two, minus the card that you won’t draw next turn. In early game the expected value of a card is lower than 1, so having a Silver instead is good. But the total money value of both turns combined will -at best- be as good as having had a Silver instead of the (value 0) Bureaucrat. The benefit of the added Silver only shows up later by slightly increasing your average deck value.
But in the short term the Bureaucrat delays your Gold buys, which delays your overall economy. Only in the long term will you reap the benefits of the added Silvers, as your deck become more resistant to devaluation due to victory cards. All this is nicely illustrated by the average money graph. Remember that for this graph the Bureaucrat has its attack function disabled.
If you look at turn 3 and 4, you can see the effect of the Bureaucrat very clearly.
Because I just can’t leave it: there is a topic on BGG strategy forum where some dude describes how his Chapel strategy was beaten by a Bureaucrat. Now he was not playing Single Chapel 1, and she was not playing only a Bureaucrat. But for now let’s pretend they did, so that we can see if these results of strategies playing Bmu actually mean something. Given the respective win rates of Single Chapel 1 (55%) and Bureaucrat (65%) the latter should win marginally. And luckily the simulation confirms this:
Single Chapel 1: 40%
Simple Card < Gold: (Bureaucrat): 60%
Of course this does not prove that win percentages playing Bmu can always be measured against each other, but for starters I’m not disappointed.
[Edit: I realized that card names should be capitalized. And I cleaned up some typos while I was at it.]
The strategy works as follows:
- Open with Chapel and Remodel
- trash away until you have 5 cards left
- Either buy or upgrade yourself a Throne Room
When you have 5 cards left, these will be [Throne Room, Remodel, Chapel,x, x]. Now you use Throne Room Remodel 2 cards each turn to upgrade the Chapel and the other two cards into Provinces. Once you have Provinces you Remodel these into new Provinces in order to quickly end the game. You will score 18 points, and your goal is to end before your opponent scores more.
Creating an optimal Turbo Remodel strategy is not trivial. There are a lot of small decisions and each affects its efficiency. For instance it turns out to be much better to initially trash both Estates and Coppers. At first I thought you would want to end up with [Throne Room, Remodel, Chapel, Estate, Estate] but saving your estates really slows down your trashing too much.
I am sad to report that for all my tweaking the performance of Turbo Remodel remains plain ugly.
Turbo Remodel 1 wins: 2345 (27%)
Big Money Ultimate wins: 6293 (73%)
I’m sure one can find tweaks to make it perform slightly better, but I doubt this strategy can ever become competitive in its current form. However, this strategy does make for a very pretty graph. The average card count graph showcases all that is going on in a single image.
The Chapel saga continues, but sadly without Turbo Remodel.
[Edit: I realized that card names should be capitalized.]
We continue to evaluating the Chapel, but now concentrate results rather than how to analysis.
Previously we saw that using a single Chapel does not yield great results when not buying Duchies and Estates. But what if you allow both the Chapel and the competing Big Money strategy to do so? Finding the optimal strategy is not easy, and I’m open to suggestions to make it better. My current best is Single Chapel 1, and is very much a combination of Single Chapel canonical 1 and Bmu. Optimal [d/e] values for the Chapel strategy seems to be [3/2] but [3/1] gives almost identical results. Single Chapel strategies favor buying Duchies and Estates later than Big Money. Which makes sense to me.
But how good is it?
Single Chapel 1 wins 5319 (55%)
Big Money Ultimate wins 4306 (45%)
Hmmm, not impressive. But on the bright side I finally found a graph that makes it easier to understand what is going on: the average change in VP (delta VP), weighted with the percentage of games that is running. Say that Bmu -on average- gains 3 VP in turn 18, but only half the games are still playing in turn 18 then the graph will display a value of 1.5.
The result is a sort of relevant average VP gain in a given turn. The area under the graph is a measure for the total score of the strategy over all games. From this graph it is possible to compare how well strategies are contributing to certain stages in the game, as well as do a general overall comparison.
The simple conclusion is that the single Chapel strategy just does not cut it, barely beating Big Money. The graph shows the Chapel strategy outperforms Bmu from turn 13 onwards. But the negative scoring in the first turns seems to roughly match the better performance in the later turns. So even with Duchies and Estates in play, the trashing of the Estates still hurts the Chapel strategy a lot.
Ok, moving on to a more competitive strategy: Chapelab. In this strategy you combine the trashing power of Chapel with the non-terminal drawing power of Laboratory. The idea is that you create a deck that can draw all cards each turn, guaranteing a Province draw each turn. Matt Sargent posted an elegant set of rules for a Chapelab strategy in his article on Board Game Geek. It only buys 2 Golds and at most 4 Labs and scores 61% against the Bmu. We can do even better if we take the basic buy rules from single Chapel 1, and the trash rules from Matt’s strategy to form a hybrid strategy.
Chapelab 1 wins: 6356 (66%)
Big Money Ultimate wins: 3224 (34%)
The graph shows that with the addition of Labs the Chapel starts to seriously out buy the Bmu in turn 9, which is 4 turns earlier than without the Labs. From turn 10 the influx of Provinces hurts the buying power, but in turn 14 the buying power of the Chapelab increases again. I guess that is due to the extra Golds being bought when Provinces were to expensive.
Thoughts so far
Given the reputation of Chapel, I’m not impressed with the performance of real Chapel strategies. It’s actually nothing new if you have read Matt’s simulation results, but I don’t share his conclusions that the Chapel is a good card until I actually see it shine. From his data I would guess that Chapel performs better as a counter to opposing curses, which was what it was intended for in the first place. I will have to investigate. But not before looking into a very specific Chapel based strategy.
Up next is Turbo Remodel.
[Edit: I decided that card names should be capitalized.]
This article explores ways to gather insight from strategies. Why does a particular strategy do well? I’m developing statistics analyzers for the Dominator, and want to put them in action in an actual case. To make it interesting, I examine a Chapel strategy competing with Big Money. To be precise: I run Single Chapel canonical 1 against Big Money canonical. Now I know both are not competitive strategies, but they are equal in the sense that they both don’t buy Duchies and Estates. Including these would complicate things too much for this early analysis. I will investigate real Chapel strategies later.
But first: pop quiz
If you play “Single Chapel canonical 1″ against “Big Money canonical”, what are your win chances?
A) Playing Chapel gives you roughly an 80% win chance.
B) Playing Chapel gives you roughly a 65% win chance.
C) Playing Chapel gives you roughly a 50% win chance.
The Chapel has been hyped a lot, even having been called the most powerful card in the game on more than one occasion. For instance on dominionstrategy.wordpress.com. My guess was that it is one of those cards that is very strong on its own, not needing any other kingdom cards to make it work.
So I ran the simulation and the results were not what I expected at all. Garion even didn’t believe them, so I wanted to fully understand them before writing about them. And in order to do this I added some victory points statistics logging. I’m recording the amount of victory points each player has in each turn, to get an average number of victory points for each strategy per turn. This resulted in the following graph:
What does this graph tell us? There is the expected drop in VP of the Chapel strategy in the early turns as the Estates get trashed. Then at turn 12 the Chapel overtakes the big money in average VP. Since only 1 game in my 10,000 runs ended in turn 12, this graph tells you that the Chapel strategy must win most of the games.
What about the actual results?
Chapel wins: 4886 (49%)
BigMoneyCanonical wins: 5103 (51%)
First of all I’ll bet good money that very few people actually choose answer C off the bat. Playing a single Chapel does not improve your odds?
And if the simulation is right -which I believe it is, because I checked against using another simulator- then my initial average VP graph must be flawed. Well indeed I did not tell the whole story about the way the average is calculated. Calculating that average the big question is: how do you combine the data from games that do not last the same number of turns. Say that you have a normal game that lasts 16 turns and one that lasts 17 turns. Everything is easy for calculating the average number of points for turn 16, but what to do with the data for turn 17. Consider two games where game 1 ends in turn 16, and game 2 last until turn 17:
|turn||chapel strategy score game 1||chapel strategy score game 2|
I thought it would be acceptable to extrapolate that the first game has 30 points in turn 17 as well.
Average points in turn 16: 30 + 26 / 2 = 28 points
Average points in turn 17: 30 + 26 / 2 = 28 points.
But as we have seen the resulting averages are misleading and of no use. So I tried again, this time not extrapolating scores from games. I needed to keep track of how many games ended in each turn, and this allowed me to calculate the average scores using only the games that provided data for it.
Average points in turn 16: 30 + 26 / 2 = 28 points
Average points in turn 17: 0 + 26 / 1 = 26 points.
You immediately see that this can have strange consequences: the average number of points drops. To me this was counter intuitive because (apart from trashing Estates) the actual number of points only rises during games. Still, this method of averaging gives a much less misleading graph:
Here we see that the Chapel strategy is only superior in games that end between turn 12 and 16. If the Chapel strategy take longer than that, then -on average- the big money strategy is superior. It may be that in those cases the Chapel strategy is hurt by the extra Provinces.
Anyway if we look at the histogram of the number of turns played we see that the whole story starts to look plausible:
These totals also add up nicely to the win percentages of the strategies:
The total numbers for games ending up to turn 16 is: 4943
The total numbers for games ending after to turn 16 is: 5057
OK, some more graphs to see if we really understand why the Chapel does so poorly compared to expectations. Let’s look at the average value of each card. The basic idea of this version of the Chapel strategy is to raise the average value by trashing low value cards.
It is nice to see that the graph for Chapel drops flat after turn 10. From that point onwards the influx of Provinces balances the influx of money. I mentioned previously that I expected that once the Chapel deck start buying Provinces it would starts degrading. This does not seem to happen, so there must be another reason that the Chapel strategy is better in shorter games.
Let’s take a quick side step. One theory suggests that you need to have at least one Gold before buying your first Province. This should be very visible in the average card value graph, and so it is:
Now we do see a drop in average value after turn 9. Unfortunately the extra Gold does not actually improve the win-rate of this strategy, which stays at roughly 50%. So trading a Province for a Gold does as much damage as it does good.
But I’m still a bit at a loss. We saw the average VP of the Chapel strategy decline for longer games, but we also see the buying power steadily increase, especially for longer games. Maybe that average buying power does not translate into turn where you can actually buy a Province. One turn with 12 money plus one with 6 money only gives you one Province, even though the average is a whopping 9 money. The graph below shows the percentage of the time that a player had 8 or more money to spend:
This graph is final proof that the Chapel strategy is capable of outbuying Big Money when it comes to buying Provinces. So how on earth can it be that the Big Money strategy still is just as good?
My last stab at solving the mystery is by thinking about the victory conditions. A Chapel deck will always have less Estates than Big Money. So once Big Money buys its 4th Province, the Chapel deck can no longer win. Nonetheless my Chapel strategy implementation will never buy a card that will cost it the game. And it is easy to count how many times that happens for the 10k games simulated. The numbers:
Games where the Chapel deck did not buy a Province: 1801
Total number of Provinces not bought: 3498
Since there is no other way to gain victory points, in those games the Chapel deck will still lose, just a couple of turns later, judging by the numbers by about 2 turns. Now back to the win chance. Given that the Chapel deck will need 5 Provinces to win, and the big money only needs 4, we can relate the actual number of Provinces bought by these targets:
Chapel buys 42973 Provinces for an adjusted number of 42973 / 5 = 8594
Big Money buys 37027 Provinces for an adjusted number of 37027 / 4 = 9256
(Sanity check: 42973 + 37027 = 80,000 Provinces sold in total.)
In this respect Big Money is actually doing better than the Chapel strategy. So although the Chapel is generating more buying power than Big Money, it just is not enough…
The Chapel will increase the buying power, but in this simple form it also increases the number Provinces it needs to win. These effects compensate for each other, making this simple single Chapel strategy no better than canonical Big Money.
Next up is adding Duchies and Estates into the mix. And to give you a sneak preview: my best Chapel strategy so far (that does not use any other kingdom cards) scores 54% against Bmu. More on the esteemed Chapel later…
[Edit: I decided that card names should be capitalized. And thx Willie for pointing out a very basic calculation error.]
Simulating dominion is something that proved to be much more fun than I would have thought. That’s why I choose to start a blog about it.
If you have not yet found it: the Board Game Geek Strategy forum is a great place to discuss actual dominion game strategy. So if you want to discuss how to improve on the strategy side of things, please use that forum. Unfortunately the BGG forum is not suitable for posting sticky strategies, nor graphical charts for that matter. So that’s also where this blog comes in.
This blog has lived briefly at another url, but this is where it is going to stay.
So welcome, enjoy your stay.
A final notice: WordPress asks for your email address when you post a comment. I have no specific interest in collecting email addresses. So if you don’t want me to have it, just leave it empty. WordPress will process your comment regardless.