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[trivialknot]

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 too.  This 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

Rank	Card	Impact		Rank	Card	Impact
1	Rebuild	2.97		110	Throne Room	1.03
2	Mountebank	2.55		111	Cellar	1.02
3	Goons	2.46		112	Market	1.01
4	Cultist	2.43		113	Stonemason	1.01
5	Ill-Gotten Gains	2.38		114	Ghost Ship	1.00
6	Ambassador	2.36		115	Remodel	1.00
7	Governor	2.23		116	Oasis	0.99
8	University	2.19		117	Vagrant	0.99
9	Scrying Pool	2.17		118	Haven	0.98
10	Tournament	2.16		119	Great Hall	0.98
11	Familiar	2.01		120	Watchtower	0.98
12	Minion	2.00		121	Trading Post	0.96
13	Swindler	2.00		122	Doctor	0.96
14	JackOfAllTrades	1.99		123	Moneylender	0.95
15	Chapel	1.96		124	Lookout	0.95
16	Masquerade	1.94		125	Silk Road	0.95
17	Colony	1.94		126	Inn	0.92
18	Platinum	1.94		127	Smugglers	0.92
19	Soothsayer	1.87		128	Pawn	0.91
20	Fishing Village	1.84		129	Forge	0.91
21	Fool's Gold	1.81		130	Rabble	0.89
22	Wharf	1.80		131	Envoy	0.88
23	Marauder	1.80		132	Sage	0.88
24	Witch	1.80		133	Feodum	0.86
25	Sea Hag	1.75		134	Horn of Plenty	0.86
26	Torturer	1.75		135	Death Cart	0.85
27	King's Court	1.74		136	Vault	0.85
28	Black Market	1.68		137	Trade Route	0.84
29	Border Village	1.63		138	Smithy	0.84
30	Tunnel	1.60		139	Farmland	0.84
31	Steward	1.57		140	Expand	0.82
32	Forager	1.57		141	Wishing Well	0.81
33	Market Square	1.55		142	Moat	0.80
34	Plaza	1.52		143	Mint	0.80
35	Remake	1.51		144	Council Room	0.79
36	Urchin	1.51		145	Harem	0.79
37	Gardens	1.50		146	Storeroom	0.78
38	City	1.49		147	Advisor	0.78
39	Highway	1.48		148	Pearl Diver	0.78
40	Wandering Minstrel	1.45		149	Baron	0.78
41	Hunting Party	1.44		150	Hunting Grounds	0.77
42	Alchemist	1.43		151	Taxman	0.77
43	Worker's Village	1.41		152	Scavenger	0.77
44	Lighthouse	1.41		153	Catacombs	0.76
45	Conspirator	1.41		154	Journeyman	0.76
46	Grand Market	1.40		155	Potion	0.76
47	Ruins	1.40		156	Cartographer	0.75
48	Baker	1.40		157	Nomad Camp	0.75
49	Peddler	1.39		158	Treasury	0.75
50	Hamlet	1.38		159	Procession	0.74
51	Squire	1.36		160	Develop	0.74
52	Young Witch	1.34		161	Feast	0.74
53	Ironmonger	1.33		162	Loan	0.73
54	Herald	1.31		163	Band of Misfits	0.71
55	Vineyard	1.31		164	Mystic	0.70
56	Upgrade	1.31		165	Beggar	0.70
57	Prince	1.31		166	Library	0.69
58	Bridge	1.31		167	Noble Brigand	0.68
59	Nobles	1.30		168	Rats	0.68
60	Fortress	1.30		169	Workshop	0.68
61	Jester	1.30		170	Bank	0.68
62	Junk Dealer	1.29		171	Spy	0.67
63	Festival	1.28		172	Armory	0.67
64	Count	1.27		173	Venture	0.67
65	Duke	1.27		174	Duchess	0.67
66	Fairgrounds	1.25		175	Poor House	0.67
67	Knights	1.25		176	Cutpurse	0.66
68	Candlestick Maker	1.24		177	Oracle	0.65
69	Butcher	1.24		178	Rogue	0.65
70	Hermit	1.24		179	Mine	0.64
71	Bishop	1.23		180	Pillage	0.63
72	Bandit Camp	1.23		181	Stash	0.63
73	Margrave	1.21		182	Woodcutter	0.61
74	Walled Village	1.21		183	Talisman	0.60
75	Caravan	1.20		184	Masterpiece	0.59
76	Apprentice	1.20		185	Outpost	0.59
77	Apothecary	1.19		186	Secret Chamber	0.58
78	Hoard	1.18		187	Merchant Ship	0.58
79	Bazaar	1.18		188	Graverobber	0.58
80	Village	1.17		189	Fortune Teller	0.58
81	Haggler	1.17		190	Saboteur	0.57
82	Warehouse	1.16		191	Tribute	0.57
83	Counterfeit	1.15		192	Herbalist	0.55
84	Salvager	1.15		193	Bureaucrat	0.55
85	Laboratory	1.14		194	Treasure Map	0.54
86	Embassy	1.14		195	Cache	0.54
87	Merchant Guild	1.14		196	Philosopher's Stone	0.53
88	Menagerie	1.14		197	Mandarin	0.52
89	Crossroads	1.12		198	Explorer	0.52
90	Stables	1.11		199	Navigator	0.52
91	Ironworks	1.11		200	Thief	0.51
92	Mining Village	1.11		201	Pirate Ship	0.50
93	Spice Merchant	1.11		202	Adventurer	0.50
94	Possession	1.11		203	Transmute	0.49
95	Quarry	1.10		204	Royal Seal	0.48
96	Farming Village	1.09		205	Chancellor	0.47
97	Golem	1.08		206	Harvest	0.45
98	Shanty Town	1.08		207	Scout	0.45
99	Tactician	1.07		208	Contraband	0.42
100	Trader	1.07		209	Coppersmith	0.39
101	Monument	1.05		210	Counting House	0.38
102	Scheme	1.05		211	Copper	0.00
103	Courtyard	1.05		212	Duchy	0.00
104	Embargo	1.05		213	Silver	0.00
105	Island	1.04		214	Gold	0.00
106	Native Village	1.04		215	Curse	0.00
107	Militia	1.04		216	Province	0.00
108	Altar	1.03		217	Estate	0.00
109	Horse Traders	1.03

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

[Awaclus]

This data predates Adventures, so cards only go up to Guilds.

More importantly than cards only going up to Guilds, the data was collected back when everyone sucked at the game.

[markus]

I don't think that this measure is useful. If I understand it correctly, it is high when you're more likely to gain all other cards in its presence.
Wouldn't that at first glance mean that the card is weak, because I'm more likely to want other cards?
Junkers make people gain Curses, Ruinss, etc, such that they end up high on the list, but not because people want to gain those cards.
Or it really favours engines, such that I want to gain a lot of different cards.

In the end, I can maybe explain why a card has a high or low impact factor, but I wouldn't know the reason, without knowing the card.

I think a more useful version would define categories for cards (Village, Smithy,...) and calculate the impact on those, e.g. am I more likely to gain Villages in the presence of Rebuild or not.

[Titandrake]

I don't think that this measure is useful. If I understand it correctly, it is high when you're more likely to gain all other cards in its presence.
Wouldn't that at first glance mean that the card is weak, because I'm more likely to want other cards?
Junkers make people gain Curses, Ruinss, etc, such that they end up high on the list, but not because people want to gain those cards.
Or it really favours engines, such that I want to gain a lot of different cards.

In the end, I can maybe explain why a card has a high or low impact factor, but I wouldn't know the reason, without knowing the card. The theory is that this is a proxy for measuring how much a card changes what you'd be planning to do.

I think a more useful version would define categories for cards (Village, Smithy,...) and calculate the impact on those, e.g. am I more likely to gain Villages in the presence of Rebuild or not.

A card is high on the list if it is likely to change what cards you gain. So the impact factor goes up both if a card's presence makes you want to gain more of a card, or less of a card.

Things that gain other cards (Cursers, Ruins givers) may be over-represented. That's probably why Tunnel is so high, for example.

idk, I think there are some interesting things in there. Colony and Platinum being so high is some interesting commentary on how strategy changes in Colony games.

[markus]

I missed that the absolute amount of change is added up.

[terminalCopper]

Colony and Platinum being so high is some interesting commentary on how strategy changes in Colony games.

I don‘t think the presence of Colonies has a huge impact on the chosen strategy. If I understand the maths correctly, I assume that Colony games are merely longer, and therefore more cards are bought.

[markusin]

Colony and Platinum being so high is some interesting commentary on how strategy changes in Colony games.

I don‘t think the presence of Colonies has a huge impact on the chosen strategy. If I understand the maths correctly, I assume that Colony games are merely longer, and therefore more cards are bought.

I don't know if Colony games are all that much longer by a number of turns metric. I think it's mostly that Colony games increase the amount you want to build, so you likely have more cards being gained overall.

[markusin]

This data predates Adventures, so cards only go up to Guilds.

More importantly than cards only going up to Guilds, the data was collected back when everyone sucked at the game.

Man, forget Loan, look at Harem's placement here. Edit: okay maybe Harem wasn't so bad back before Adventures came out.

It feels like this metric is good at finding the cards that warp what you do on the board, but not so good at highlighting the cards where you naturally get one at convenience just because it's usually better than not getting it regardless of what you're doing, such as with Junk Dealer, Spice Merchant, Upgrade, Militia, and Ghost Ship.

[trivialknot]

A card is high on the list if it is likely to change what cards you gain. So the impact factor goes up both if a card's presence makes you want to gain more of a card, or less of a card.

Things that gain other cards (Cursers, Ruins givers) may be over-represented. That's probably why Tunnel is so high, for example.

Correct, it looks at the absolute change in gain percentages.  And I agree that it over-represents cards that make you gain other cards, especially Coppers, Curses, Ruins, Estates--things you wouldn't otherwise gain.

More importantly than cards only going up to Guilds, the data was collected back when everyone sucked at the game.

Also correct.

I don‘t think the presence of Colonies has a huge impact on the chosen strategy. If I understand the maths correctly, I assume that Colony games are merely longer, and therefore more cards are bought.

I don't look at the total number cards gained from each pile, just the percentage chance that at least one card was gained from each pile.  I also tried analysis where I looked at the total number of cards gained.  The number one ranked card was Ill-Gotten Gains, and well you can see which method I decided to stick with.

I think Colony is high primarily because it increases the gain percentage of 2 supply piles: Colony and Platinum.

[pacovf]

Have you considered other normalizations for kingdom piles? How does this list change if you give them the same weight as for basic supply piles?

[trivialknot]

Have you considered other normalizations for kingdom piles? How does this list change if you give them the same weight as for basic supply piles?

Yes this is really easy to do, and I'm willing to post results for alternative analysis.  The hard part is actually just thinking out a mathematical formula that makes sense.

The current formula is:

Impact(supply pile X) = Sum over supply pile Y abs( P(gain Y|X and Y are in supply) * P(Y is in supply|X is in supply) - P(gain Y|Y is in supply)*P(Y is in supply) )

I was thinking that I could try an alternate formula like this:

Impact(supply pile X) = Sum over supply pile Y abs( P(gain Y|X and Y are in supply) - P(gain Y|Y is in supply) )

This would eliminate the weight given to base cards (or e.g. Potions in the case of potion cards), but also it would eliminate the contribution of each card to its own impact factor.  Is this calculation something people would be interested in seeing?

Another thing I wanted to do was take the impact vectors, and apply Principal Component Analysis.  This would identify a few dimensions that can be used to differentiate cards.  But I'm waffling on how I should define the impact vector.

[Commodore Chuckles]

This data predates Adventures, so cards only go up to Guilds.

More importantly than cards only going up to Guilds, the data was collected back when everyone sucked at the game.

And the proof is that there are cards lower than Scout.

[markus]

I don't like the own-effect of the impact factor, because the presence of the card in the kingdom shouldn't affect the average probability to gain it.

But shouldn't it be smaller for cards that show up more frequently (Platinum), as it boils down to:
P(gain X|X is in supply) * (1 - P(X is in supply)),
and P(X is in supply) is higher for Platinum? (It can be offset by a higher probability of gaining Platinum.)

The other thing that I noticed with the current formulat is that a card Y that is gained independently of all other cards, still contributes to their impact factors:
P(gain Y|Y is in supply) * (P(Y is in supply|X is in supply) - P(Y is in supply) )

In that sense, I like the alternate formula more than the current one.

Wishlist:
1) Also show a version that excludes Copper, Curse, Ruins, Estate (am I missing something?)
2) For each card, show the top 3 cards that contribute the most in a positive or in a negative way to the impact factor.
3) I (still) would like to see the impact factor (and number of cards gained from that pile) for categories of cards.

4) Analysis with games from Shuffle iT.

[trivialknot]

I don't like the own-effect of the impact factor, because the presence of the card in the kingdom shouldn't affect the average probability to gain it.

It made sense to me because the idea was to see how much the card changes your gains, and surely the presence of a card changes your ability to gain itself.  But I can calculate it the other way too.

But shouldn't it be smaller for cards that show up more frequently (Platinum), as it boils down to:
P(gain X|X is in supply) * (1 - P(X is in supply)),
and P(X is in supply) is higher for Platinum? (It can be offset by a higher probability of gaining Platinum.)

Not quite.  The own-effect for Duke is about %Duke*0.95, and the own-effect for Platinum/Colony is (%Platinum+%Colony)*0.875.

The other thing that I noticed with the current formulat is that a card Y that is gained independently of all other cards, still contributes to their impact factors:
P(gain Y|Y is in supply) * (P(Y is in supply|X is in supply) - P(Y is in supply) )

That is technically true, but what I didn't tell you is that P(Y is in supply|X is in supply) is just a number I hard-coded in.  I didn't bother taking into account the fact that if X takes up a slot in the kingdom, it decreases the probability of Y.  But that's a really good point, if my formula includes such an effect, does the formula even make sense?

Wishlist:
1) Also show a version that excludes Copper, Curse, Ruins, Estate (am I missing something?)
2) For each card, show the top 3 cards that contribute the most in a positive or in a negative way to the impact factor.
3) I (still) would like to see the impact factor (and number of cards gained from that pile) for categories of cards.

4) Analysis with games from Shuffle iT.

I can do 1) and 2) fairly easily, although I'm thinking that I should also remove the other base cards and Potion.  I feel like it doesn't make sense to put synergy with Duchy on the same level as synergy with Count, because Duchy is in every game.  I could keep in Platinum and Colony because those are only 2.5x as likely to show up.

For 3) the difficulty is whose card categories do I use?  I hope that the PCA analysis, when I do it, will provide some insight into card categories.

AFAIK Shuffle IT data doesn't exist.

[trivialknot]

Introducing the synergy factor!

The synergy factor of a supply pile is the sum of the absolute changes in the gain percentages of other kingdom piles, due to this supply pile's presence.  That means I don't count the contribution from base cards, Potions, Ruins, Platinum/Colonies, or from the card itself.  The math also ends up excluding the contribution of potion cards to Potion, and the contribution of Looters to Ruins.  Finally, I left out Prince and Walled Village because there's less data for them and it was impacting statistics in a bad way. 

Furthermore, for each card I show the top 2 synergies, and top anti-synergy.  While these are interesting, keep in mind the error margins can be larger than the differences between the top few synergies.

Rank	Card	Synergy Factor	Top synergy		2nd synergy		Top anti-synergy
1	Rebuild	25.8	Feodum	0.25	Feast	0.24	Laboratory	-0.34
2	Chapel	16.9	Treasure Map	0.29	Market Square	0.14	Lookout	-0.42
3	JackOfAllTrades	16.9	Feodum	0.21	Duke	0.13	Sea Hag	-0.30
4	Governor	16.8	Rats	0.19	Militia	0.18	Bandit Camp	-0.34
5	Cultist	16.6	Death Cart	0.18	Trader	0.16	Rabble	-0.27
6	Masquerade	16.6	Possession	0.15	Bandit Camp	0.12	Sea Hag	-0.39
7	Wharf	16.6	Bank	0.18	Quarry	0.17	Catacombs	-0.33
8	University	16.1	Library	0.30	Journeyman	0.26	Quarry	-0.13
9	Ill-Gotten Gains	15.8	Trader	0.20	Gardens	0.17	Festival	-0.22
10	Minion	15.2	Lighthouse	0.18	Horse Traders	0.17	Ghost Ship	-0.31
11	Goons	14.3	Quarry	0.22	Native Village	0.22	Hoard	-0.26
12	Fishing Village	13.8	Envoy	0.28	Smithy	0.26	Village	-0.33
13	Fool's Gold	13.4	Storeroom	0.17	Nomad Camp	0.16	Peddler	-0.19
14	Fairgrounds	13.1	Death Cart	0.19	Spy	0.18	Chapel	-0.02
15	Ambassador	13.1	Lighthouse	0.17	Native Village	0.13	Sea Hag	-0.37
16	Mountebank	12.6	Trader	0.23	Watchtower	0.22	Witch	-0.30
17	Tournament	12.2	Band of Misfits	0.16	Tactician	0.10	Bridge	-0.21
18	Prince	12.0	Oracle	0.30	Quarry	0.24	Jester	-0.21
19	Steward	12.0	Bandit Camp	0.10	Bazaar	0.10	Lookout	-0.26
20	Forager	11.9	Poor House	0.18	Fool's Gold	0.16	Trade Route	-0.37
21	Sea Hag	11.9	Moat	0.17	Lighthouse	0.17	Familiar	-0.32
22	Border Village	11.8	Rabble	0.26	Journeyman	0.24	Nobles	-0.13
23	Torturer	11.7	Lighthouse	0.17	University	0.14	Rabble	-0.27
24	Remake	11.5	Poor House	0.32	Fortress	0.13	Moneylender	-0.28
25	King's Court	11.5	Saboteur	0.27	Woodcutter	0.21	Bank	-0.17
26	Scrying Pool	11.4	Storeroom	0.21	Woodcutter	0.21	JackOfAllTrades	-0.17
27	Urchin	11.4	Fortress	0.16	Lighthouse	0.10	Lookout	-0.26
28	Wandering Minstrel	11.2	Workshop	0.21	Armory	0.19	Farming Village	-0.30
29	Walled Village	11.2	Hunting Grounds	0.24	Council Room	0.23	Advisor	-0.19
30	Hunting Party	11.2	Altar	0.15	Horn of Plenty	0.15	Laboratory	-0.26
31	Plaza	11.0	Library	0.22	Watchtower	0.18	Mining Village	-0.31
32	Gardens	10.9	Beggar	0.39	Workshop	0.26	Apprentice	-0.13
33	Swindler	10.7	Masterpiece	0.30	Chancellor	0.28	Fool's Gold	-0.16
34	Tunnel	10.5	Storeroom	0.28	Embassy	0.22	Militia	-0.19
35	Fortress	10.4	Procession	0.20	Develop	0.19	Worker's Village	-0.15
36	Black Market	10.3	Fairgrounds	0.17	Quarry	0.13	Smugglers	-0.18
37	City	10.2	University	0.18	Bridge	0.16	Cartographer	-0.18
38	Worker's Village	10.2	Catacombs	0.21	Torturer	0.21	Mining Village	-0.21
39	Witch	10.2	Lighthouse	0.15	Trader	0.14	Torturer	-0.26
40	Highway	10.1	Talisman	0.21	Ironworks	0.18	Mystic	-0.19
41	Familiar	10.0	Transmute	0.25	Golem	0.20	Witch	-0.13
42	Vineyard	9.9	Herbalist	0.24	Rats	0.23	Cultist	-0.18
43	Baker	9.9	Altar	0.11	Count	0.09	Treasury	-0.18
44	Festival	9.6	Hunting Grounds	0.22	Watchtower	0.21	Market	-0.17
45	Ironmonger	9.6	Workshop	0.22	Ironworks	0.17	Farming Village	-0.23
46	Bazaar	9.4	Smugglers	0.18	Smithy	0.16	Treasury	-0.14
47	Village	9.4	Rabble	0.18	Smithy	0.18	Shanty Town	-0.11
48	Soothsayer	9.4	Lighthouse	0.15	Watchtower	0.13	Witch	-0.24
49	Marauder	9.3	Lighthouse	0.13	Death Cart	0.12	Militia	-0.16
50	Bridge	9.3	Native Village	0.20	Inn	0.11	Nomad Camp	-0.25
51	Squire	9.1	Torturer	0.20	Rabble	0.19	Native Village	-0.14
52	Margrave	9.1	Village	0.15	University	0.15	Ghost Ship	-0.16
53	Hamlet	9.0	Library	0.21	Smithy	0.19	Native Village	-0.25
54	Young Witch	8.9	Tunnel	0.15	Band of Misfits	0.13	Expand	-0.23
55	Herald	8.9	Armory	0.26	Workshop	0.25	Mining Village	-0.13
56	Grand Market	8.9	Quarry	0.26	Vault	0.21	Hoard	-0.18
57	Embassy	8.8	Tunnel	0.11	Farmland	0.07	Catacombs	-0.18
58	Bandit Camp	8.8	Envoy	0.19	Hunting Grounds	0.16	Festival	-0.18
59	Farming Village	8.7	Torturer	0.20	Hunting Grounds	0.16	Mining Village	-0.20
60	Upgrade	8.5	Rats	0.31	Poor House	0.28	Apprentice	-0.18
61	Colony	8.5	Mine	0.13	Taxman	0.12	Duke	-0.24
62	Platinum	8.5	Mine	0.13	Taxman	0.12	Duke	-0.24
63	Duke	8.4	Beggar	0.21	Bureaucrat	0.15	Possession	-0.11
64	Conspirator	8.3	Sage	0.14	Ironworks	0.13	Bank	-0.11
65	Junk Dealer	8.3	Market Square	0.11	Quarry	0.09	Trading Post	-0.26
66	Butcher	8.3	Rats	0.23	Fortress	0.12	Remodel	-0.18
67	Caravan	8.2	Workshop	0.26	Ironworks	0.23	Advisor	-0.22
68	Market Square	8.2	Apprentice	0.17	Vineyard	0.11	Worker's Village	-0.15
69	Mining Village	8.2	Torturer	0.18	Rabble	0.16	Fortress	-0.16
70	Lighthouse	8.2	Mint	0.11	Horn of Plenty	0.08	Moat	-0.20
71	Bishop	8.1	Fortress	0.19	Rats	0.18	Gardens	-0.18
72	Laboratory	8.0	Altar	0.15	University	0.15	Mystic	-0.15
73	Salvager	7.9	Rats	0.19	Fortress	0.12	Remodel	-0.20
74	Native Village	7.8	Bridge	0.19	Torturer	0.15	Pearl Diver	-0.16
75	Knights	7.8	Moat	0.16	Rogue	0.15	Duke	-0.17
76	Apprentice	7.8	Rats	0.30	Fortress	0.16	Gardens	-0.13
77	Envoy	7.7	Inn	0.14	Worker's Village	0.09	Nomad Camp	-0.14
78	Horn of Plenty	7.7	Spy	0.16	Journeyman	0.15	Workshop	-0.05
79	Stash	7.7	Envoy	0.15	Scavenger	0.13	Market	-0.11
80	Hermit	7.7	Market Square	0.12	Wishing Well	0.11	Sea Hag	-0.17
81	Stables	7.6	University	0.11	Altar	0.09	Mystic	-0.16
82	Trader	7.6	Feodum	0.38	Duke	0.11	Sea Hag	-0.18
83	Quarry	7.5	Market	0.16	Forge	0.15	Trader	-0.09
84	Crossroads	7.5	Torturer	0.17	Great Hall	0.14	Native Village	-0.14
85	Alchemist	7.4	Possession	0.22	Transmute	0.17	Smithy	-0.12
86	Tactician	7.4	Storeroom	0.18	Secret Chamber	0.16	Soothsayer	-0.11
87	Peddler	7.4	Worker's Village	0.13	Salvager	0.12	Fool's Gold	-0.09
88	Monument	7.4	Golem	0.12	Band of Misfits	0.10	Scavenger	-0.16
89	Menagerie	7.3	Ironworks	0.14	Outpost	0.13	Envoy	-0.12
90	Ruins	7.3	Fairgrounds	0.09	Watchtower	0.09	Baron	-0.12
91	Haggler	7.1	Mining Village	0.14	Spy	0.13	Rogue	-0.11
92	Count	7.0	Duke	0.18	Grand Market	0.05	Trading Post	-0.16
93	Candlestick Maker	7.0	Vineyard	0.16	Peddler	0.12	Pawn	-0.21
94	Ironworks	7.0	Silk Road	0.23	Gardens	0.21	Armory	-0.19
95	Merchant Guild	6.9	Quarry	0.10	Golem	0.09	Nomad Camp	-0.12
96	Embargo	6.9	Stash	0.11	Venture	0.10	Pawn	-0.11
97	Golem	6.9	Apothecary	0.17	Saboteur	0.12	Throne Room	-0.07
98	Throne Room	6.8	Armory	0.13	Band of Misfits	0.13	Mining Village	-0.08
99	Procession	6.8	Adventurer	0.17	Catacombs	0.15	Horn of Plenty	-0.06
100	Apothecary	6.7	Possession	0.20	Golem	0.19	Loan	-0.11
101	Altar	6.7	Stash	0.16	Mystic	0.16	Expand	-0.10
102	Spice Merchant	6.7	Fool's Gold	0.11	Market Square	0.09	Moneylender	-0.24
103	Militia	6.6	Band of Misfits	0.12	Lighthouse	0.11	Cutpurse	-0.18
104	Counterfeit	6.6	Fool's Gold	0.10	Envoy	0.10	Loan	-0.14
105	Ghost Ship	6.6	Farming Village	0.14	Lighthouse	0.13	Militia	-0.11
106	Feodum	6.6	Masterpiece	0.41	Trader	0.30	Throne Room	-0.10
107	Market	6.5	Quarry	0.19	Conspirator	0.11	Band of Misfits	-0.10
108	Horse Traders	6.4	Tunnel	0.11	Gardens	0.10	Nomad Camp	-0.15
109	Warehouse	6.4	Tunnel	0.17	Treasure Map	0.12	Oasis	-0.18
110	Rabble	6.3	University	0.12	Village	0.12	Library	-0.10
111	Shanty Town	6.3	Rabble	0.14	Council Room	0.13	Wishing Well	-0.12
112	Doctor	6.3	Venture	0.09	Merchant Ship	0.08	Lookout	-0.20
113	Jester	6.3	Spy	0.09	Native Village	0.08	Rogue	-0.12
114	Watchtower	6.2	Festival	0.13	Rats	0.11	Oracle	-0.11
115	Nobles	6.2	Quarry	0.20	Trade Route	0.13	Harem	-0.14
116	Courtyard	6.2	Worker's Village	0.08	Bandit Camp	0.07	Pearl Diver	-0.17
117	Moneylender	6.2	Venture	0.09	Stash	0.09	Cutpurse	-0.14
118	Inn	6.1	Torturer	0.13	Hunting Grounds	0.13	Cartographer	-0.10
119	Vault	6.1	Grand Market	0.10	Tunnel	0.09	Envoy	-0.12
120	Lookout	6.1	Mystic	0.09	Treasure Map	0.09	Loan	-0.18
121	Silk Road	6.0	Workshop	0.16	Baron	0.14	Laboratory	-0.08
122	Hoard	5.9	Harem	0.13	Silk Road	0.12	Conspirator	-0.09
123	Smugglers	5.9	Apprentice	0.10	Duke	0.09	Workshop	-0.07
124	Vagrant	5.9	Vineyard	0.09	Harem	0.08	Pearl Diver	-0.24
125	Trading Post	5.9	Feodum	0.12	Grand Market	0.10	Altar	-0.11
126	Possession	5.9	Apothecary	0.20	Alchemist	0.17	Conspirator	-0.09
127	Island	5.8	Ironworks	0.20	Hoard	0.10	Remodel	-0.11
128	Haven	5.7	Envoy	0.10	Treasure Map	0.08	Pearl Diver	-0.21
129	Forge	5.7	Rats	0.12	Catacombs	0.10	Gardens	-0.11
130	Beggar	5.7	Gardens	0.25	Counting House	0.13	Fortress	-0.08
131	Journeyman	5.7	Horn of Plenty	0.07	Bank	0.07	Council Room	-0.12
132	Masterpiece	5.6	Feodum	0.32	Gardens	0.15	Cellar	-0.08
133	Remodel	5.6	Rats	0.14	Fool's Gold	0.13	Horn of Plenty	-0.08
134	Stonemason	5.5	Catacombs	0.10	Feodum	0.09	Altar	-0.07
135	Cellar	5.5	Tunnel	0.12	Bureaucrat	0.08	Pearl Diver	-0.15
136	Smithy	5.5	Festival	0.09	Village	0.09	Envoy	-0.18
137	Workshop	5.4	Gardens	0.15	Silk Road	0.13	Stonemason	-0.08
138	Feast	5.4	Stash	0.09	Witch	0.07	Fortress	-0.07
139	Storeroom	5.2	Tunnel	0.12	Tactician	0.10	Woodcutter	-0.11
140	Pawn	5.2	Vineyard	0.11	Stash	0.09	Pearl Diver	-0.19
141	Catacombs	5.2	University	0.10	Bank	0.08	Journeyman	-0.13
142	Scheme	5.2	Conspirator	0.12	Armory	0.09	Wishing Well	-0.16
143	Nomad Camp	5.2	Golem	0.09	Highway	0.07	Woodcutter	-0.10
144	Rats	5.2	Apprentice	0.14	Butcher	0.14	Spice Merchant	-0.07
145	Expand	5.2	Rats	0.09	Marauder	0.08	Bank	-0.09
146	Library	5.2	Festival	0.09	University	0.09	Envoy	-0.10
147	Armory	5.2	Advisor	0.12	Vineyard	0.12	Workshop	-0.12
148	Venture	5.1	Altar	0.09	Bank	0.08	Graverobber	-0.10
149	Trade Route	5.1	Band of Misfits	0.08	Scrying Pool	0.07	Develop	-0.11
150	Baron	5.0	Silk Road	0.10	Cartographer	0.07	Nomad Camp	-0.11
151	Treasury	5.0	University	0.09	Expand	0.08	Mystic	-0.10
152	Hunting Grounds	5.0	Native Village	0.10	Worker's Village	0.09	Council Room	-0.09
153	Sage	5.0	Trader	0.07	Golem	0.07	Wishing Well	-0.11
154	Advisor	4.9	Armory	0.19	Ironworks	0.14	Spy	-0.09
155	Scavenger	4.9	Stash	0.20	Lookout	0.07	Navigator	-0.12
156	Council Room	4.9	Bank	0.11	Village	0.09	Library	-0.07
157	Wishing Well	4.9	Mystic	0.08	Horn of Plenty	0.07	Sage	-0.10
158	Moat	4.9	Harem	0.07	Festival	0.07	Pawn	-0.10
159	Develop	4.9	Rats	0.15	Poor House	0.13	Envoy	-0.10
160	Oasis	4.8	Tunnel	0.13	Soothsayer	0.07	Wishing Well	-0.12
161	Woodcutter	4.8	Ill-Gotten Gains	0.08	Gardens	0.08	Baron	-0.06
162	Pirate Ship	4.8	Oasis	0.10	Merchant Ship	0.08	Envoy	-0.12
163	Adventurer	4.8	Stash	0.08	Venture	0.08	Peddler	-0.06
164	Cutpurse	4.7	Hoard	0.08	Band of Misfits	0.07	Navigator	-0.09
165	Great Hall	4.7	Ironworks	0.19	Silk Road	0.13	Sage	-0.10
166	Talisman	4.7	Feodum	0.11	Highway	0.09	Smugglers	-0.07
167	Taxman	4.7	Harem	0.09	Soothsayer	0.08	Moneylender	-0.08
168	Bureaucrat	4.6	Feodum	0.11	Duke	0.09	Vineyard	-0.06
169	Secret Chamber	4.6	Stash	0.09	Tactician	0.08	Poor House	-0.07
170	Herbalist	4.6	Philosopher's Stone	0.14	Stash	0.09	Moat	-0.07
171	Cache	4.6	Monument	0.07	Young Witch	0.07	Conspirator	-0.07
172	Loan	4.6	Venture	0.08	Baker	0.07	Oracle	-0.07
173	Mystic	4.6	Altar	0.08	Rebuild	0.07	Tribute	-0.10
174	Band of Misfits	4.6	Duke	0.07	Margrave	0.06	Bishop	-0.10
175	Mint	4.5	Fool's Gold	0.08	Harem	0.06	Trading Post	-0.08
176	Transmute	4.5	Trader	0.08	Apothecary	0.08	Stash	-0.06
177	Thief	4.5	Venture	0.10	Familiar	0.08	Stonemason	-0.07
178	Harem	4.5	Silk Road	0.13	Hoard	0.09	Ironworks	-0.07
179	Farmland	4.5	Trader	0.09	Silk Road	0.07	Band of Misfits	-0.06
180	Philosopher's Stone	4.4	Herbalist	0.14	Duke	0.09	Throne Room	-0.11
181	Mine	4.4	Harem	0.09	Catacombs	0.08	Band of Misfits	-0.05
182	Bank	4.4	Venture	0.12	Tactician	0.07	Vineyard	-0.07
183	Harvest	4.4	Hoard	0.07	Duke	0.07	Highway	-0.05
184	Chancellor	4.4	Envoy	0.10	Stash	0.10	Quarry	-0.04
185	Cartographer	4.4	Tunnel	0.12	Moneylender	0.06	Bazaar	-0.09
186	Oracle	4.4	Ill-Gotten Gains	0.09	Venture	0.07	Fortune Teller	-0.09
187	Rogue	4.3	Soothsayer	0.08	Doctor	0.07	Candlestick Maker	-0.07
188	Pearl Diver	4.3	Cartographer	0.07	Jester	0.07	Lighthouse	-0.09
189	Noble Brigand	4.3	Beggar	0.09	Venture	0.07	Ironworks	-0.05
190	Duchess	4.3	Stash	0.12	Venture	0.07	Workshop	-0.08
191	Contraband	4.3	Expand	0.09	Envoy	0.08	Festival	-0.06
192	Spy	4.3	Doctor	0.08	Golem	0.07	Quarry	-0.07
193	Outpost	4.3	Trading Post	0.07	Navigator	0.07	Horn of Plenty	-0.05
194	Saboteur	4.3	Embargo	0.07	Sage	0.07	Armory	-0.07
195	Royal Seal	4.2	Cartographer	0.08	Venture	0.07	Stash	-0.07
196	Navigator	4.2	Tunnel	0.08	Farmland	0.08	Oasis	-0.07
197	Tribute	4.2	Ghost Ship	0.09	Familiar	0.07	Pillage	-0.05
198	Treasure Map	4.2	Chapel	0.07	Sea Hag	0.07	Throne Room	-0.08
199	Death Cart	4.1	Rats	0.11	Band of Misfits	0.07	Nomad Camp	-0.08
200	Scout	4.1	Harem	0.08	Band of Misfits	0.07	Quarry	-0.06
201	Counting House	4.1	Beggar	0.10	Treasury	0.08	Tactician	-0.05
202	Poor House	4.1	Vault	0.09	Island	0.06	Remake	-0.09
203	Mandarin	4.0	Mint	0.07	Familiar	0.07	Envoy	-0.07
204	Fortune Teller	4.0	Ill-Gotten Gains	0.07	Mystic	0.06	Poor House	-0.05
205	Merchant Ship	4.0	Marauder	0.06	Inn	0.06	Harvest	-0.06
206	Coppersmith	3.9	Apothecary	0.07	Council Room	0.06	Quarry	-0.07
207	Pillage	3.9	Alchemist	0.08	Hoard	0.06	Saboteur	-0.06
208	Graverobber	3.9	Rats	0.08	Marauder	0.07	Possession	-0.06
209	Explorer	3.9	Duke	0.07	Silk Road	0.06	Envoy	-0.08
210	Potion	3.2	Herbalist	0.08	Woodcutter	0.07	Quarry	-0.04

[AJD]

I don't like the own-effect of the impact factor, because the presence of the card in the kingdom shouldn't affect the average probability to gain it.

It made sense to me because the idea was to see how much the card changes your gains, and surely the presence of a card changes your ability to gain itself.

Yeah, to put it another way, if, in a game with Counting House, you're not much more likely to gain Counting House than you are in a game without Counting House, that's evidence that Counting House is a fairly low-impact card whose presence on the board doesn't usually matter.

[ben_king]

I love that Swindler's top synergies are terrible cards that you get Swindled into.

[markus]

I don't like the own-effect of the impact factor, because the presence of the card in the kingdom shouldn't affect the average probability to gain it.

It made sense to me because the idea was to see how much the card changes your gains, and surely the presence of a card changes your ability to gain itself.

Yeah, to put it another way, if, in a game with Counting House, you're not much more likely to gain Counting House than you are in a game without Counting House, that's evidence that Counting House is a fairly low-impact card whose presence on the board doesn't usually matter.

But that runs against the idea of "how much does the presence of the card change the gain probabilities of cards?". By that definition, the presence of Counting House doesn't affect the probability of gaining Counting House conditional on it being present.

I wouldn't call it "synergy factor", if it sums up the absolute values of synergy and anti-synergy.

[markusin]

Remake's top synergy card: Poor House 

Looking at the stats, it's a higher synergy pair than everything except Gardens/Beggar, Trader/Feodum, and Masterpiece/Feodum.

[markus]

For 3) the difficulty is whose card categories do I use?  I hope that the PCA analysis, when I do it, will provide some insight into card categories.

I know, that people can argue what defines a Village. I would suggest that you pick whatever definition you like. Given that most villages are uncontroversial, it shouldn't matter too much for the results.

I'm skeptical that PCA will deliver anything useful, but I'd be happy to be wrong about that.

[trivialknot]

Now that I have both these lists, here are my thoughts on interpretation.

The "Impact factor", while not perfect, is the best objective measure of "card strength" that I have seen so far.  In the gain percentage list, the top three cards were Tournament, Governor, and Border Village; here they are Rebuild, Mountebank, Goons.  And that seems to me a little more on point.  I bet if you compared this to the 2014 Qvist rankings you would find that they are a closer match.

But there is an obvious bias towards cards that either force or incentivize you to gain copper/curse/estate/ruins/potion.  I think dumping 10 curses into players decks should rightfully be considered impactful, but maybe not quite this much.

The synergies and anti-synergies are very interesting, and you can think up a reason for why almost every card on that list is there.*  I'm not sure the sum absolute value of synergies is especially meaningful though.

*ok, I have no idea why Duchess promotes Stash.

The synergy factor looks like the most promising data for PCA.  My prediction is that the first principal component will be "terminals vs villages".  The most villagey cards will be strong villages that encourage even weak terminal draw and weak terminal payload, while discouraging weak villages.  The least villagey cards will be strong terminal draw that encourage even weak villages, while discouraging weak terminal draw.

[Donald X.]

For 3) the difficulty is whose card categories do I use?  I hope that the PCA analysis, when I do it, will provide some insight into card categories.

Hey you can use your math to figure out categories.

1. Each card determines a category. You won't actually do each card, because we care about e.g. villages rather than e.g. Poor Houses.
2. So, pick a card, e.g. Village.
3. Iterate through all other cards.
4. For each card, calculate the sum of the absolute value of the differences between gain percentages for Village and the other card, for all other cards. For example, for Poor House, first you say, okay how much does Village change the gain percentage for Ironworks, how much does Poor House change the gain percentage for Ironworks, take the absolute value of the difference; then, Forager, how much does Village change the gain percentage for Forager, how much does Poor House change the gain percentage for Forager, and so on. In the end you have this number that says, how similar in changing gain percentages is Poor House to Village.
5. Sort from lowest to highest; the ones at the top are the most similar to the chosen card, e.g. Village.
6. Draw the line where it looks like it goes.

This of course includes power level; a card we recognize as a village might be so weak that it's beaten by cards we are sure aren't villages. Still, could be fun.

[samath]

For 3) the difficulty is whose card categories do I use?  I hope that the PCA analysis, when I do it, will provide some insight into card categories.

Hey you can use your math to figure out categories.

In other words, you have a matrix of interaction strengths. Treat each column (or row, or some combination to account for interactions in both directions) as a vector, and run a clustering algorithm on the results. (Donald suggests measuring distance with the 1-norm, but you could probably use any reasonable norm. You might need to lower the dimensionality by random projection first.)

[Awaclus]

*ok, I have no idea why Duchess promotes Stash.

I do! When you gain a bunch of terminals for free, you don't have so much terminal space left so you'll buy Stash or Venture instead of a terminal $5. I don't think this effect would be at all visible today, but I'm like 90-93% sure that was really the reason in 2014.

[markusin]

*ok, I have no idea why Duchess promotes Stash.

I do! When you gain a bunch of terminals for free, you don't have so much terminal space left so you'll buy Stash or Venture instead of a terminal $5. I don't think this effect would be at all visible today, but I'm like 90-93% sure that was really the reason in 2014.

Or like, on board where you go for a Stash deck and you buy Duchy, Duchess is a "might as well" addition. Or am I misinterpreting the statistic.