Print

Please login or register.
1 Hour 1 Day 1 Week 1 Month Forever
Login with username, password and session length

[WanderingWinder]

Any baseball fans?  WPA per gain (or buy, or trash, or whatever) seems applicable.

Okay, so given no one seemed to want to derail the Mountebank argument, I'll answer my own question.

http://www.fangraphs.com/blogs/get-to-know-wpa/

WPA ...

I've actually twice mentioned something similar in the past. Interestingly, rrenaud's reply suggests something very similar to the OP of this thread.

Anyway, I think something along these lines is a great idea if someone has the know-how and expertise to implement it. Baseball is surprisingly similar to Dominion in the way it's analyzed, and I'm sure we could borrow some great ideas from sabermetrics.

WAR and WPA are very different. WPA is basically incalculable for Dominion, because it requires a way to evaluate current probability of winning a game, which if we could do, would make the statistic pretty pointless (the game boring as well)

Which is why baseball has failed as a sport?

Look, just because you can calculate win probability doesn't make the game meaningless (or boring).  Isn't one of the greatest Dominion moments when you come back from what seems like an insurmountable lead to win?  Besides the fact that it would be difficult to run WPA calculations in real time as you are playing in your head, unless you are a stats savant.

The point to using WPA is in being able to make the comparison that purchasing one card improved your odds of winning X amount more than purchasing another card.  It doesn't mean you actually buy it because one had a WPA that was two hundredths of a percentage point higher.  But it's a way of comparison that actually takes into account kingdom and game state.  And Win Probability Added is a way to value cards other than words like "strength" or "power" that don't mean anything.

You don't buy Card X because "ooh it is strong."  You buy it because it directly increasing your chance to win the game.  That's what WPA could measure.

Well, baseball still works because even though you know what it is you are trying to get done, it's still difficult to do it. The intrigue of the sport is in the physical execution much more so than the evaluation. *I* can evaluate baseball reasonably, but I sure as heck can't play it very well.

Dominion is a game all about making decisions. If you have a chart which tells you what the mathematically best decision is, it takes all the skill out of the game.

Okay, you are right, if it is something which is available but not accessible by players (not at all clear since we aren't at all near to doing this, and it seems more likely to me that this would be the result of revolutionary simplification of the problem rather than brute force over every possible gamestate....), then it can still be interesting to play, though less interesting for post-game analysis.

Regardless, my larger point is that WPA isn't at all calculable for Dominion.

[Kirian]

All the talk of using baseball-style analysis makes me think two things:

(1) I wonder if Nate Silver plays Dominion?

(2) Can we have a Fantasy Dominion league?  Like, everyone drafts a bunch of players, then how those players do over the course of the week is somehow... OK, probably not.

[ashersky]

(2) Can we have a Fantasy Dominion league?  Like, everyone drafts a bunch of players, then how those players do over the course of the week is somehow... OK, probably not.

It only works if the best Dominion players can't also be managers.  And you'd need them all to play a specific number of games against each other.  It would be more akin to Fantasy Football, I think.

[SCSN]

This is very interesting, but I see a couple issues. The first is that this can't produce a unique ranking list. What we actually get are two rankings: one based on % usefulness, one on impact factor. You have to arbitrarily choose how to weight the two components of the ranking. When A > B on both lists we can conclusively say A is better than B. But when the two components disagree, you can get A > B or B > A just by changing how you combine % gained and impact. You might say there's no weighting, we just use the numbers as they come out, but this is an arbitrary weighting because the two scales have no relationship to one another, it isn't clear how much they each contribute to overall strength.

Good stuff. The fact that you can't combine the numbers as it is, is because I tried to solve one of the problems that is adequately solved by the impact factor (namely, how to avoid that spammable cards end up really high just because you often spam them) by making % usefulness a binary measure: either you buy the card at least once or you don't buy it at all.

Instead of using that % usefulness stuff it's better to just multiply at each node the probability of that node's occurrence (e.g. there's an 8.3% chance that "having 5 to spend at T1" occurres for you in a non-baker game) with the impact factor of your purchase, and then for each card sum over all nodes where that card gets bought. This way you get a single number that represents the card's power within that strategy. Do the same for each other Nash-Equilibrium in your set of NEs, add the numbers per card and then divide them by the total number of NEs, and then do exactly the same for each kingdom, add the card's power ratings per kingdom and divide by the total number of kingdoms that card has appeared in ($1-$3 cost cards appear a bit more often b.c. of being a YW-bane).

The other issue might not be an issue at all, I can't decide. It seems like your measure of impact will overly punish cards which continue to be useful throughout the game, but become less important. A card like Wharf can be extremely important to buy in the early turns, and then sort of scale down in importance (but have a lot of points in the game where it might be an optimal purchase, but the alternative is not so bad). Since you propose taking an average of all decision points Wharf's early game impact will be diluted by the later turns. Your measure of impact is going to heavily favor early game cards which suck after the first few turns, I suspect Chapel will lead the list, but the other trashers (even some mediocre ones like Trading Post) should be high too, and cheap card gainers also. These cards impacts won't be diluted in any way because they aren't purchased at all after a number of turns. Maybe this is okay, and perfectly in line with a natural concept of "impact", it just seemed like it might not be great (especially in the Wharf case).

I think this too is solved by the new definition: you only buy a Chapel once, which will have a very high impact, but you'll likely buy multiple Wharfs, and the impact of each Wharf purchase adds up to the point that it ends up higher than Chapel (if indeed it's better than it, which in this particular example may or may not be the case).

[markusin]

I've been thinking about this question for a bit. I think SCSN is really on to something.

The complex part of all this is that the behaviour of the marginal benefit that the cards give when gained varies. Cards like Mountebank and Chapel have an overall huge positive effect on win%  on the first gain, less so on the second gain, and then perhaps negative marginal benefit after the third gain. However, Fool's Gold will have negative marginal effect on win% on the first gain, but then increase win% by large amounts after the 7th gain or something. Spammable cards like Hamlet and Ironmonger probably have steady marginal win% increases per gain.

So I was thinking that you could combine the total effect on win% at each quantity level (1 copy, 2 copies, 3 copies, etc.) where the total effect on win% is positive (because, in general, no one forces you to get 5 Chapels or 7 Mountebanks or only 1 Treasure Map). But then you look at FG and see that it's very unlikely that you get all 10 of them. So a rating system I would look into would consider the stats across the different number of copies you could have of a specific card, but then have a weighted average with the weight of the values decreasing as the number of copies of the card increases. So the win% with 10 FG might be very high, but it will have less weight for the average than the value with 7 copies. Even just getting 1 or 2 FG has the effect of potentially blocking your opponent's FG and can be turned into a gold later.

What I'm saying above probably isn't very clear, but ultimately I'm suggesting that the impact of a card gain should be weighted against the probability that you can actually make that purchase. I think this would also make the card comparisons more fair when considering cards of different costs. So Cellar wouldn't be as bad compared to Warehouse than it would be otherwise, and Moneylender wouldn't be as bad as it would otherwise be when compared to Counterfeit.

[Witherweaver]

I'm looking forward to seeing Moneycard when it comes out.

[timchen]

One thing I find intriguing is that strictly speaking the ranking (a) is useless. You never want to compare two cards directly when they are not both in a game. But just because of its closeness to (c) that is what HMM suggested, which is an ordered list, it becomes more useful than (b).

I guess after all it's too hard for human mind to put something of O(N^2) instead of O(N) into perspective...

[ghostofmars]

I would define strength of a card in the following fashion:

Consider a kingdom of any 9 cards. Which card has the most impact on the strategy I would play?
Of course one would have to average over all possible kingdoms to get the true rating, but a much smaller subset should already give a reasonable guess.

Example:
In the following kingdoms, what has more impact X = Duke or Mountebank?
Crossroads, Lighthouse, Herbalist, Urchin, Horse Traders, Ironworks, Jack of All Trades, Count, Harem, +X
Fools Gold, Throne Room, Rats, Stables, Tactician, Jester, Explorer, Venture, Grand Market, +X
Poor House, Crossroads, Remake, Feast, Militia, Venture, Outpost, Explorer, Nobels, +X

[AJD]

I would define strength of a card in the following fashion:

Consider a kingdom of any 9 cards. Which card has the most impact on the strategy I would play?

That's an interesting definition because it allows for a card to be strong even in a kingdom in which you don't want to actually gain it. For instance, there's lots of games in which the presence of Embargo in the supply has a huge impact on what a good strategy is, but if all players play a good strategy none of them has any reason to gain Embargo.

[WanderingWinder]

I thought about this earlier today, and come home to find markusin has posted something similar. But I will expound my slightly differently-nuanced view:

So, the problem is that we are trying to evaluate card X and card Y, etc. But the first copy of card X will not have the same value as the 8th copy. So the 1st Sea Hag is better than the first Monument, the 4th Monument is definitely better than the 4th Sea Hag.

So this would lead us to consider sets of cards, {3 Sea Hag, 1 Monument}>{4 Sea Hag}, I am not going to list out so many combinations. But this fails, too, for a few reasons. First, accessibility of the different sets; this is very dependent on cost, of course, but also what the cards do - it is harder to get to 3 hag+monument than 1 hag+3 monument (in a vacuum). Second, your opponent is doing stuff, too; there's attacks here, there's losing splits here, and there's denial here, i.e. it's not so much about what gives you the best set, it's about what makes your set most better than the opponent's set. Third, it doesn't actually give us a way to evaluate which card is better than the other - if the optimal configuration is 2x monument + 1 sea hag, which of these cards is more important? You can try figuring out taking which one out most hurts you, but because of the non-conservative way these things add to value, it's not clear that such an approach would actually answer the question.

One thing I find intriguing is that strictly speaking the ranking (a) is useless. You never want to compare two cards directly when they are not both in a game. But just because of its closeness to (c) that is what HMM suggested, which is an ordered list, it becomes more useful than (b).

I guess after all it's too hard for human mind to put something of O(N^2) instead of O(N) into perspective...

Strictly speaking, this post, the entire discussion, these forums, the game, all of it is useless. Oh wait, usefulness is a value judgement...

Clearly, some people do 'want to compare two cards directly when they are not both in a game'.

I am not sure that I understand what your last point is supposed to mean; surely you can't generalize this at all and would really rather just say that IN THIS CASE, 'the human mind' (also an over-generalization; clearly some human minds exist which this isn't true for) much more lends itself to the 'linear' problem in question rather than the 'quadratic' problem in question.

I mean, if we are just using O(n), the size of n as well as any coefficients, lower order effects, etc. are surely going to come into play when talking about human minds - right?

[GeoLib]

One thing I find intriguing is that strictly speaking the ranking (a) is useless. You never want to compare two cards directly when they are not both in a game. But just because of its closeness to (c) that is what HMM suggested, which is an ordered list, it becomes more useful than (b).

I guess after all it's too hard for human mind to put something of O(N^2) instead of O(N) into perspective...

Strictly speaking, this post, the entire discussion, these forums, the game, all of it is useless. Oh wait, usefulness is a value judgement...

Clearly, some people do 'want to compare two cards directly when they are not both in a game'.

I am not sure that I understand what your last point is supposed to mean; surely you can't generalize this at all and would really rather just say that IN THIS CASE, 'the human mind' (also an over-generalization; clearly some human minds exist which this isn't true for) much more lends itself to the 'linear' problem in question rather than the 'quadratic' problem in question.

I mean, if we are just using O(n), the size of n as well as any coefficients, lower order effects, etc. are surely going to come into play when talking about human minds - right?

I believe timchen means that in no game setting is it relevant whether one card is better than another unless both cards are in the game. The fact that I would prefer a Masquerade over an Ambassador on this board doesn't matter at all if Masquerade isn't in the kingdom. However, it is much easier for us (generally) to look at a single ordered list of all the cards O(N) rather than remembering all of the pairwise comparisons of all the cards when both are present in the kingdom O(N^2)

[WanderingWinder]

One thing I find intriguing is that strictly speaking the ranking (a) is useless. You never want to compare two cards directly when they are not both in a game. But just because of its closeness to (c) that is what HMM suggested, which is an ordered list, it becomes more useful than (b).

I guess after all it's too hard for human mind to put something of O(N^2) instead of O(N) into perspective...

Strictly speaking, this post, the entire discussion, these forums, the game, all of it is useless. Oh wait, usefulness is a value judgement...

Clearly, some people do 'want to compare two cards directly when they are not both in a game'.

I am not sure that I understand what your last point is supposed to mean; surely you can't generalize this at all and would really rather just say that IN THIS CASE, 'the human mind' (also an over-generalization; clearly some human minds exist which this isn't true for) much more lends itself to the 'linear' problem in question rather than the 'quadratic' problem in question.

I mean, if we are just using O(n), the size of n as well as any coefficients, lower order effects, etc. are surely going to come into play when talking about human minds - right?

I believe timchen means that in no game setting is it relevant whether one card is better than another unless both cards are in the game. The fact that I would prefer a Masquerade over an Ambassador on this board doesn't matter at all if Masquerade isn't in the kingdom. However, it is much easier for us (generally) to look at a single ordered list of all the cards O(N) rather than remembering all of the pairwise comparisons of all the cards when both are present in the kingdom O(N^2)

Well, okay, but that doesn't generalize. Also, why is it in big-O notation?
Finally, the pairwise comparisons themselves are irrelevant, shouldn't we be remembering something that is O(N^10) (okay, actually more because of young witch, Black Market, colonies, shelters, basic cards even, etc.)?

[GeoLib]

One thing I find intriguing is that strictly speaking the ranking (a) is useless. You never want to compare two cards directly when they are not both in a game. But just because of its closeness to (c) that is what HMM suggested, which is an ordered list, it becomes more useful than (b).

I guess after all it's too hard for human mind to put something of O(N^2) instead of O(N) into perspective...

Strictly speaking, this post, the entire discussion, these forums, the game, all of it is useless. Oh wait, usefulness is a value judgement...

Clearly, some people do 'want to compare two cards directly when they are not both in a game'.

I am not sure that I understand what your last point is supposed to mean; surely you can't generalize this at all and would really rather just say that IN THIS CASE, 'the human mind' (also an over-generalization; clearly some human minds exist which this isn't true for) much more lends itself to the 'linear' problem in question rather than the 'quadratic' problem in question.

I mean, if we are just using O(n), the size of n as well as any coefficients, lower order effects, etc. are surely going to come into play when talking about human minds - right?

I believe timchen means that in no game setting is it relevant whether one card is better than another unless both cards are in the game. The fact that I would prefer a Masquerade over an Ambassador on this board doesn't matter at all if Masquerade isn't in the kingdom. However, it is much easier for us (generally) to look at a single ordered list of all the cards O(N) rather than remembering all of the pairwise comparisons of all the cards when both are present in the kingdom O(N^2)

Well, okay, but that doesn't generalize. Also, why is it in big-O notation?
Finally, the pairwise comparisons themselves are irrelevant, shouldn't we be remembering something that is O(N^10) (okay, actually more because of young witch, Black Market, colonies, shelters, basic cards even, etc.)?

I don't know. I just followed his original format. And yes, generally analyzing a kingdom is a much more complicated proposition than just comparing two cards. I was just trying to clarify what timchen was saying. Obviously a list of cards (or even all of the pairwise interactions) cannot be a complete guide on what to buy, otherwise Dominion would be solved!

[timchen]

Yeah, basically what GeoLib said.

I don't particular mean that a pairwise comparison is very useful if we can remember it, it is just that the way ehunt starts this thread is talking about a pairwise comparison, and strictly in the sense of thinking about a pairwise comparison method (a) actually is not useful.

I just realized that the reason (a) is actually more useful than (b) is because that we are NOT looking at pairwise comparison any more but instead use it to form an ordered list. This list will supposedly give us a baseline how important a card is in average which is helpful for us to judge a board. And it is just like an anecdote saying in this sense card A is better than card B.

As to the question whether a pairwise comparison data (if it exists) is useful, I think it is. At least for now we do use this data on functionally similar cards. Do I get a witch or mountebank, etc, etc.