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[Awaclus]

I think everyone has an intuitive understanding of what makes a card strong, but it's much more difficult to put it in a formula. Here are some ways to determine card strength and reasons why they don't work:

1. How often (top?) players gain the card
This doesn't take into account a scenario in which nobody gains the card, but it makes an impact by merely being present in the kingdom (e.g. Salt the Earth). It also doesn't take into account a scenario in which a card is only very slightly weaker than many other similar cards (e.g. Walled Village), in which case the card in question is not gained in many kingdoms because a slightly stronger card is gained instead. It also can't tell the difference between a low-impact card that you buy almost every time because it has a low opportunity cost (e.g. Pearl Diver) and a card that you buy almost every time because it has a huge impact (e.g. Goons).

2. How often do (top?) players win with the card vs. without the card
When a card is very strong, top players might have a very high win rate without the card because the weaker opponent will buy the card every time, whereas the top player is better able to consider the rare situation in which the card is not worth buying. The reverse is true for very weak cards. On the other hand, if we consider the entire player base instead of just top players, strong cards that don't have a high skill requirement (e.g. Rebuild) will appear higher while strong cards that do have a high skill requirement (e.g. Bridge) will appear lower. Much like 1), this does not take into account cards that can make an impact by being present even if nobody buys them.

3. If two top players of similar strength play a match where they are and aren't allowed to gain a card, respectively, how big of an advantage will the former have
This addresses all of the previous problems, but introduces a new one: the player who is allowed to buy the card has access to the knowledge that the other player will surely not buy it, which gives them an inherent advantage, partially independently of the card's strength. This also makes self-synergistic cards seem stronger than they really are. In the case of a real match between two real players, as opposed to a thought experiment, it also has the problems that six games is a very low sample size which can result in kingdoms or shuffles favoring one player over the other, and the players might not be giving their best performance at the time of the match even if they are very closely ranked.

Personally, I think that given how difficult it is to objectively measure a card's strength, it might be more important to talk about when to buy the card, and when to consider the card for your kingdom analysis as separate things. How big of an impact it makes in both cases is not so important. For example, Pearl Diver isn't really a very "strong" card, but you can buy it whenever there's nothing better to do, and due to its low cost and spammability, it can make pileout endings easier which is something that you should consider during kingdom analysis. It doesn't make a very big impact in either case, but you're not going to have the thought process "oh Pearl Diver only helps me a tiny little bit here, I guess I'm not going to buy it".

What do you think? Are there any other ways of measuring card strength that actually work?

[JW]

3. If two top players of similar strength play a match where they are and aren't allowed to gain a card, respectively, how big of an advantage will the former have
This addresses all of the previous problems, but introduces a new one: the player who is allowed to buy the card has access to the knowledge that the other player will surely not buy it, which gives them an inherent advantage, partially independently of the card's strength. This also makes self-synergistic cards seem stronger than they really are. In the case of a real match between two real players, as opposed to a thought experiment, it also has the problems that six games is a very low sample size which can result in kingdoms or shuffles favoring one player over the other, and the players might not be giving their best performance at the time of the match even if they are very closely ranked.

This definition, but assume that the players play an infinite number of games (varying who is and isn't allowed to gain the card), play equally well on average regardless of whether they're allowed to gain the card or not gain the card, and that the player doesn't know that their opponent isn't allowed to gain the card. Also, gaining the card is probably always allowed for the purpose of ending the game on piles, and the card can't be the Obelisk Museum or Young Witch bane piles for obvious reasons. An actual cage match will be an approximation to that ideal.

Alternately, a strong card is one that Awaclus says is superior to Loan.

[samath]

In the case of a real match between two real players, as opposed to a thought experiment, it also has the problems that six games is a very low sample size which can result in kingdoms or shuffles favoring one player over the other, and the players might not be giving their best performance at the time of the match even if they are very closely ranked.

Two potential ways to resolve this:
1) Use Judgment Matches to compare two cards. One player is only allowed to gain one of them, and the other player is only allowed to gain the other one. At the beginning of each game, both players simultaneously reveal which card they want access to. If they pick the same card, count that as a win for that card and immediately move on to the next game. Only if they disagree do you actually play the game. This way you could get through a much larger number of games (albeit relying on the players' estimation for most).
2) If you're evaluating a single card, use a bidding for VP mechanism to restrict access to that card. This way, even if you only play the six games, you still get more information from the VP that different players bid. We did something similar with the Bidding for Position Crew Battle on the Discord a few weeks ago.

[ipofanes]

... a scenario in which nobody gains the card, but it makes an impact by merely being present in the kingdom (e.g. Salt the Earth).

That's my experience with Salt the Earth too. Nobody ever gains it, and if one tries to, I slap his hand.

[weesh]

1. How often (top?) players gain the card

Some cards are powerful, like chapel, but you get full benefit out of a single copy.
It would be silly to say village is more powerful than chapel just because people buy more villages over a game.

---

Also, "power" and "strong" are not really interchangeable, for the purposes of a nitpicky f.ds thread.

Imagine two baseball players. 
1) .350 batting, few home runs
2) .200 batting, lots of home runs

I'd call the second player more powerful, but I'd rather have the first player on my team.

---

That said, raw power of a dominion card could be determined by it's peak use.  What is the most absurd thing that it can do? 
Page thus ranks very highly for me, because complete immunity plus infinite actions is quite absurd compared to many other cards. 
The fact that it takes a lot of time to get rolling doesn't factor into my definition.

Granted, this isn't a useful definition of "power"

[trivialknot]

My intuitive definition of "powerful" is something that makes you change a lot of decisions.  Pearl Diver is not very powerful because even if you gain one in almost every game, it only really changes a few decisions--the choice to get Pearl Diver over nothing.

If we wanted to make this definition into something that can be objectively and practically measured, I propose the following procedure:
1. Take a large number of games between top players.  Calculate the average number of copies of each card gained.
2. Now look at just the games where X is in the kingdom.  Again, calculate average gains for each card.
3. Calculate the "distance" between the percentages calculated in #1 and #2.  The larger the distance, the stronger the card.

A few problems with this procedure:
a) We're trusting top players to make good decisions.
b) You'd need a really large data set.  Possibly impractically large.
c) This ignores decisions that don't involve gaining cards, or gaining the same cards at different times.  Special mention to events.
d) Many cards (e.g. junkers, Cache) force you to gain cards without it necessarily being a "decision", and this might cause their power to be inflated.
e) This would still underestimate the power of e.g. walled village, but I'm okay with this.

[Dingan]

Not that card cost equals card strength but .. I wonder how playtesters come up with the costs of cards. Do they just use intuition? Do they use some formula(s)? As far as I know, the only "formula" they follow is don't price card X less than or equal to card Y if card X is strictly better than card Y (using your favorite definition of strictly better).

[faust]

Also, gaining the card is probably always allowed for the purpose of ending the game on piles

This is problematic as for some cards (Messenger, Stonemason), the ability to quickly end the game is part of what makes up their strength.

[markusin]

Geez, there are actually a bunch of factors to consider here.

The way I would define card strength "Council Room stat" style would be to consider the change in win percentage per extra copy of the card you gain over your opponent(s), but giving more weight when closer to the average number of gains for that card across all games. So for Chapel, you care most about the win % per gain over your opponent when you have ~1 Chapel in deck. For Village, you might care most about the win % per extra gain over your opponent(s) when you have 5 Villages. This way you can better get a sense of how good a card like Peddler, Governor, or Magpie is when you win the split. By centering around the average number of gains for the card in question, you can kind of take into account how easy it tends to be to gain the card, as well as how often the card is gained in general, and to put more weight on win percentage changes occurring when say Village is split 6/4 than when it's split 9/1. In weird cases like Stonemason pileouts, this method still considers cards to be stronger as the likelihood of the card being used as part of a pileout increases.

I don't know, what I described above is not exact or anything. I'm just trying to spitball ideas here.

[markusin]

Geez, there are actually a bunch of factors to consider here.

The way I would define card strength "Council Room stat" style would be to consider the change in win percentage per extra copy of the card you gain over your opponent(s), but giving more weight when closer to the average number of gains for that card across all games. So for Chapel, you care most about the win % per gain over your opponent when you have ~1 Chapel in deck. For Village, you might care most about the win % per extra gain over your opponent(s) when you have 5 Villages. This way you can better get a sense of how good a card like Peddler, Governor, or Magpie is when you win the split. By centering around the average number of gains for the card in question, you can kind of take into account how easy it tends to be to gain the card, as well as how often the card is gained in general, and to put more weight on win percentage changes occurring when say Village is split 6/4 than when it's split 9/1. In weird cases like Stonemason pileouts, this method still considers cards to be stronger as the likelihood of the card being used as part of a pileout increases.

I don't know, what I described above is not exact or anything. I'm just trying to spitball ideas here.

Okay, actually this needs to consider the average on the largest number of copies of the card in question a player in the game gains. So if a card is usually split 6/4, you give the most weight on the win % per extra gain for the player with 6 copies of the card, I guess accommodating for the fact that having +2 copies over another player is the minimum in that case. When the card is commonly split 7/3, you take into account that +4 copies over another player is the minimum when you have 7 copies of the card.

[Titandrake]

My intuition is that a card is powerful if strategy discussion tends to warp around that card. It's hard to not think about Wharf / Cursers / multi-card trashers / non-terminal trashing when thinking about how to approach a board. It's pretty easy to pretend Raider doesn't exist. You usually have to think about Village because it's not only a source of +Actions, it's one of the best sources of +Actions in the game (the +1 Card on Village is actually ridiculously important and getting it for $3 is a bargain.)

Mandarin is usually ignorable. Sure, it becomes very important to consider if Capital in the kingdom, but otherwise it's not a big deal. In contrast, if you drop Capital into a random board, it's usually worth thinking about. So I'd say Capital is stronger than Mandarin.

If neither player buys Familiar, it doesn't mean it's a bad card, it probably means they have an idea for how to counter it in a way that justifies not spending the buys on Potion / Familiar. There was a game I played against Sicomatic a while back, where I considered a HT/Duke slog against an Ambassador deck. Given my shuffles, I would have gone for it if Familiar wasn't on the board. But Familiar was on the board, and the threat of -10 VP from Curses was too big, so I decided to mirror Ambassador, and then neither of us ended up buying a Familiar because it didn't make sense to.

Metrics based on number of gains, win rate with and without, etc. have the issues that Awaclus mentioned. I feel the right approach is to compute those metrics and then treat them as an interesting source of data. If you try to treat them as some ground truth number indicating power level, you're going to run into issues.

I think the funniest one from the CouncilRoom days was that buying Curse was one of the most winning plays you could make, because people only buy Curses when they're winning and have the 3-pile. I suspect most metrics have a failure case like that somewhere.

[markusin]

Even funnier, I think according to Council Room Curse/Curse was not the worst opening.

[trivialknot]

After thinking about it, I think my proposal above might be statistically feasible.

In ben_king's analysis of councilroom data, there were about 8000 games with each card (just looking at 2P pro rated games with the top 100 players).  The cards only go up to Guilds, so there are 200 kingdom cards.  And that means there are about 400 games with any given pair of cards.  So you could calculate, e.g., gain percentages with a 5% margin of error.  Now you have two vectors: gain percentages in the average game, vs gain percentages in the average game that has card X.  When you calculate the distance between these two vectors, the margin of error is about 0.05*sqrt(200), or about 0.7.  I think this is small enough that you could detect differences between cards.

Another interesting thing you could do is take all the vectors, and perform principal component analysis.  I'd expect the first component to be engine/BM favorability, but the second component, who knows?

If I had the data in the appropriate format, I'd just try the analysis and see what happens.  Probably there are some pathologies in this method, but you never know until you try it.  As it is, I don't know how to access the data, and I imagine there'd be a lot of data prep work.

[ben_king]

If I had the data in the appropriate format, I'd just try the analysis and see what happens.  Probably there are some pathologies in this method, but you never know until you try it.  As it is, I don't know how to access the data, and I imagine there'd be a lot of data prep work.

If you'd like, I could send you the data that I have.  It's all just plain text logs though.

[Chase Adolphson]

Even funnier, I think according to Council Room Curse/Curse was not the worst opening.

That literally does not make any sense. Maybe if I knew the kingdom it might make more sense.

[pacovf]

Even funnier, I think according to Council Room Curse/Curse was not the worst opening.

That literally does not make any sense. Maybe if I knew the kingdom it might make more sense.

Probably an artifact of low statistics.

[singletee]

Even funnier, I think according to Council Room Curse/Curse was not the worst opening.

That literally does not make any sense. Maybe if I knew the kingdom it might make more sense.

Someone who opens Curse/Curse is probably a skilled player doing so as a joke. On the other hand, someone who opens, for example, Festival/Estate is probably a player who hasn't learned that villages aren't great to open with, or that sometimes it's best to buy nothing.

[crj]

Someone who opens Curse/Curse is probably a skilled player doing so as a joke.

There's certainly something unusual going on! Maybe they're deliberately handicapping themself in a game with a much weaker player?

I'm struggling to think of even an edge case in which a Curse/Curse opening could be the strongest play, though. Um. Defiled Shrine in a six-player game?

[markusin]

Someone who opens Curse/Curse is probably a skilled player doing so as a joke.

There's certainly something unusual going on! Maybe they're deliberately handicapping themself in a game with a much weaker player?

I'm struggling to think of even an edge case in which a Curse/Curse opening could be the strongest play, though. Um. Defiled Shrine in a six-player game?

I'd be satisfied if there were legitimate cases where it's not the absolute worst opening, assuming the player proceeds to play competently after the opening.

[sudgy]

Someone who opens Curse/Curse is probably a skilled player doing so as a joke.

There's certainly something unusual going on! Maybe they're deliberately handicapping themself in a game with a much weaker player?

I'm struggling to think of even an edge case in which a Curse/Curse opening could be the strongest play, though. Um. Defiled Shrine in a six-player game?

I'd be satisfied if there were legitimate cases where it's not the absolute worst opening, assuming the player proceeds to play competently after the opening.

As a mind game to make the opponent think they're worse?  I once underestimated an opponent because they opened with copper and I didn't think about Duke.

[terminalCopper]

Someone who opens Curse/Curse is probably a skilled player doing so as a joke.

There's certainly something unusual going on! Maybe they're deliberately handicapping themself in a game with a much weaker player?

I'm struggling to think of even an edge case in which a Curse/Curse opening could be the strongest play, though. Um. Defiled Shrine in a six-player game?

I'd be satisfied if there were legitimate cases where it's not the absolute worst opening, assuming the player proceeds to play competently after the opening.

In a game with Mountebank, curse/curse will likely be a better opening than estate/estate.

[Dingan]

Someone who opens Curse/Curse is probably a skilled player doing so as a joke.

There's certainly something unusual going on! Maybe they're deliberately handicapping themself in a game with a much weaker player?

I'm struggling to think of even an edge case in which a Curse/Curse opening could be the strongest play, though. Um. Defiled Shrine in a six-player game?

I'd be satisfied if there were legitimate cases where it's not the absolute worst opening, assuming the player proceeds to play competently after the opening.

In a game with Mountebank, curse/curse will likely be a better opening than estate/estate.

Would Curse/Curse be better than nothing/nothing? Nothing/Curse?

[crj]

I'd be satisfied if there were legitimate cases where it's not the absolute worst opening, assuming the player proceeds to play competently after the opening.

OK. By that weaker standard, what about buying Curse, Curse, Ambassador, Ambassador on T1-4? There are going to be kingdoms where that will beat people who for some reason ignore Ambassador entirely.

[KingPeter]

I'd be satisfied if there were legitimate cases where it's not the absolute worst opening, assuming the player proceeds to play competently after the opening.

OK. By that weaker standard, what about buying Curse, Curse, Ambassador, Ambassador on T1-4? There are going to be kingdoms where that will beat people who for some reason ignore Ambassador entirely.

In a discussion about "strong cards", I think we should assume mid to high level play.  Curse is not a strong card; Ambassador is.

[Cave-o-sapien]

1. How often (top?) players gain the card

I suspect if we took the data from the past year of rated games on Shuffle iT and for each card counted the number of games in which a top-100 player gained it on their turn, we'd have a pretty decent estimate of card strength.

Of the concerns raised about this approach, the Pearl Diver one seems the most problematic, but I don't think it's a fatal flaw. You might just have to deal with that amount of noise. You could try adjusting the counts with "opportunity cost weights", but those might be tricky to tune/assign.

In any case, I'd love to be able to try this and I'd be very curious to see, in particular, the difference between Chapel and Pearl Diver.