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Subject: Average Card Value as a Metric for Money-heavy Strategies rss

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Scott Fasnacht
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I recently did some testing and posted results in this thread: http://www.boardgamegeek.com/thread/421600

From those results, it occurred to me that a money-heavy strategy really cares about one thing: how much money a given card can make for the deck when it's drawn. Cards that make money directly are self-explanatory in that regard. Other cards, however, should be able to be measured in a similar way. For instance, a Smithy will draw you 3 cards if you have an action to play it. That makes the value of the smithy equal to:

(Odds you have an action left to play it) x (value of 3 cards you will draw on average)

While these may be difficult for a person to keep track of during play, they should not be too difficult to work out with a computer.

One that you CAN work out is the first Smithy you buy. Let's say you get a 4/3, and will buy Smithy/Silver. Once you've done so, how much is that Smithy worth? Average value of a card in the deck is then 9/12 = 0.75. However the Smithy can never draw itself, so really the cards it can draw are valued at 9/11. We know the Smithy draws 3 cards, so its value must then be 3(9/11) = 2.45. In other words, with the deck in this configuration, drawing a Smithy is worth 2.45 coin on average. We know this because we can always play it when it's in hand as we currently have no other action cards. Thus with the Smithy included in the deck, the overall average card is valued about (9+2.45)/12 = 11.45/12. A second Smithy, however, could render either one useless, as they would show up on the same turn sometimes, and that would negatively impact their value in the deck. I won't get into such calculations, but I believe I can (given time) figure out how to implement them.

The main point, though, is that if you can determine the expected value of any given card when you draw it, you can use that data for two very nice things:

1. To determine the card to purchase that will improve your deck the most
2. To determine which card(s) to play this turn to achieve the maximum average value from the hand you've drawn.

These calculations are going to get too complex to use in a real game, but if I can get them in use in a program, then in theory the resulting algorithm can play a big money strategy at an optimal or near-optimal level using any given set of cards... And THAT could shed a lot of light on good gameplay principles for cash heavy strategies, in addition to which cards are most important to the strategy, potential synergies, and a myriad of other useful things.

What I'm looking for right now then is feedback, thoughts, whether you think it could work, ideas for how best to approach this... any kind of input at all would be good. So... what do you think?
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Dan Schaeffer
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So how long is it until some genius decides to offer college courses in Dominion math?

What do I think? I think maybe I'd rather just play the game than analyze it.
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Edward
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I'd be surprised if most of the best Dominion players don't do this already. Its limitations -- it doesn't tell you when to switch to VP-buying, it's difficult to calculate its effect on Villages that let you play Smithies, and it definitely does not handle cards like Cellar well.
 
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Scott Fasnacht
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Golux13 wrote:
So how long is it until some genius decides to offer college courses in Dominion math?

What do I think? I think maybe I'd rather just play the game than analyze it.

Really? What an odd thing to say. You're on the dominion strategy forum but apparently don't care about things that might help your dominion strategy.

Further, why would you read a thread titled "Average Card Value as a Metric for Money-heavy Strategies" if you don't want analysis? Does that title not clearly indicate that analysis is coming??? I thought I was doing a favor by overtly announcing what the thread would be about so you could avoid it if you wouldn't be interested. I guess what I was really doing was giving you a chance to read something you'd hate so you could complain about it.

theory wrote:
I'd be surprised if most of the best Dominion players don't do this already. Its limitations -- it doesn't tell you when to switch to VP-buying, it's difficult to calculate its effect on Villages that let you play Smithies, and it definitely does not handle cards like Cellar well.

I totally agree, many players probably do this already, at least to a degree. Certainly keeping track of the average card value of your deck in terms of coins/card is a useful thing to do. I'm looking at ways to extend the usefulness of that potentially to all cards (though when to buy VP cards does still present a problem). Even the odd card like cellar may be able to be evaluated in this way.

Consider that a cellar, when best used, will redraw every card in your hand that is underweight - that is, any that are worth (or will generate from being played) less coin than the average card in the deck. So then its value is (value of draws) - (value of discards). This could be roughly extended to all situations across the deck in order to figure out the value of buying one for the deck:

(avg value of draws) - (avg value of discards)

Averaging it across the deck...

(4)(percent of underweight cards in deck)(avg amt an underweight card is underweight by)

That approximates the number of cards in hand with cellar that are underweight (4 x % underweight cards) and finds the avg amt that redrawing them will upgrade a hand. In many cases a huge percent of a deck is underweight in this game - say you bought gold and silver until the deck is worth 1.2 coin per card. Make it 7 copper, 3 estate, 4 silver, 1 gold = 18/15 = 1.2. 10 of 18 cards are underweight with an avg value of 7/10. This would give (4)(10/18)(1.2-0.7) = 1.11 which is the expected amount on avg that the cellar gains your hand in this deck. Buying a cellar in this case will actually detract from the average value of the deck since its average benefit is less than that of an avg card in the deck - you will draw more coin later by buying nothing this turn instead of a cellar, and the same is true of a copper valued at 1.0. I fear this may often be the case at least until mid-game with cellar, and it likewise mirrors how I have felt when I have drawn cellars in big money strategies - they just don't seem to pull their weight most times.

To make best use of cellar a deck needs a high percentage of underweight cards with a large disparity between those cards and the average value of a card in the deck. Since these two factors seem to typically oppose each other, my guess is that cellar can never become as valuable as a silver piece. Maybe it could in a combo deck where the combo pieces themselves become more valuable than anything in the deck due to the huge combo they enable.

Villages, and action management cards in general, are the real problem with what I've proposed. Still not quite sure how to tackle that one...
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Dave G
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The_Foz_4 wrote:
Golux13 wrote:
So how long is it until some genius decides to offer college courses in Dominion math?

What do I think? I think maybe I'd rather just play the game than analyze it.

Really? What an odd thing to say. You're on the dominion strategy forum but apparently don't care about things that might help your dominion strategy.

Further, why would you read a thread titled "Average Card Value as a Metric for Money-heavy Strategies" if you don't want analysis? Does that title not clearly indicate that analysis is coming??? I thought I was doing a favor by overtly announcing what the thread would be about so you could avoid it if you wouldn't be interested. I guess what I was really doing was giving you a chance to read something you'd hate so you could complain about it.


Because many of us subscribe to all the threads for a given game. Also, because we like to complain. More importantly, though, how much good is this kind of analysis if you can't do the math in your head? Do we really need a statistical formula to figure out that a silver is better than a smithy if I don't have any money in my deck?

I'm pretty good at Dominion, and I can barely do arithmetic at a third grade level. This seems like overkill. Fair enough if you want to do it and write about it, but I see no need to go jumping on people who think it's a little odd.
 
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Dan Schaeffer
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The_Foz_4 wrote:
Golux13 wrote:
So how long is it until some genius decides to offer college courses in Dominion math?

What do I think? I think maybe I'd rather just play the game than analyze it.

Really? What an odd thing to say. You're on the dominion strategy forum but apparently don't care about things that might help your dominion strategy.


I read the strategy forums to learn about interesting card and card-combination strategies. Also, to make fun of people who claim that Chapel breaks the game or that Chancellor is a totally useless card.

As Dave noted, I am subscribed to the game, so I see all the threads. Some of them I read in depth. Others of them make my eyes glaze over. This one was closer to the latter, in part because it is the eighteen-umptillionth thread attempting to "solve" Dominion mathematically. Your analysis might be spot-on, or it might be a load of horse crap; one thing I'm certain of, though, is that it won't "help my Dominion strategy," because I don't do complex mathematical analysis in my head at the Dominion gaming table.

Finally, I wasn't complaining about it. You asked for readers' thoughts at the end, so I gave you mine. If you wanted only in-depth analytical responses, you should have submitted it to a peer-reviewed journal. I hear "The Journal of the American Dominionological Association" is always looking for submissions.
 
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Edward
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Quote:
Maybe it could in a combo deck where the combo pieces themselves become more valuable than anything in the deck due to the huge combo ithey enable.


Exactly. They are invaluable in Village/Smithy chains.

There's another factor that a blind ACV calculation ignores -- the relative importance of "getting to" certain values. For example, in a Gardens game, it is critically important to consistently get to 4 coin; in a Duke/Duchy race, it is critically important to get to 5. It's not just a linear progression. Accordingly, in Gardens games, Workshops are way better than Silvers even though Silvers probably help your ACV more.
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Edward
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Golux13 wrote:
The_Foz_4 wrote:
Golux13 wrote:
So how long is it until some genius decides to offer college courses in Dominion math?

What do I think? I think maybe I'd rather just play the game than analyze it.

Really? What an odd thing to say. You're on the dominion strategy forum but apparently don't care about things that might help your dominion strategy.


I read the strategy forums to learn about interesting card and card-combination strategies. Also, to make fun of people who claim that Chapel breaks the game or that Chancellor is a totally useless card.

As Dave noted, I am subscribed to the game, so I see all the threads. Some of them I read in depth. Others of them make my eyes glaze over. This one was closer to the latter, in part because it is the eighteen-umptillionth thread attempting to "solve" Dominion mathematically. Your analysis might be spot-on, or it might be a load of horse crap; one thing I'm certain of, though, is that it won't "help my Dominion strategy," because I don't do complex mathematical analysis in my head at the Dominion gaming table.

Finally, I wasn't complaining about it. You asked for readers' thoughts at the end, so I gave you mine. If you wanted only in-depth analytical responses, you should have submitted it to a peer-reviewed journal. I hear "The Journal of the American Dominionological Association" is always looking for submissions.

Excuse me, I wasn't aware all topics in this forum had to cater to what you wanted to read. I happen to be interested in the OP's sort of analysis; the fact that there are "eighteen-umptillion" such topics suggests that many others on this forum share my sentiment. The fact that you personally find this "complex mathematical analysis" too much for you doesn't mean there's anything wrong with these kind of discussions on a geeky forum specifically designed to discuss Dominion strategy, in a topic specifically labeled "Average Card Value as a Metric for Money-heavy Strategies".

These kinds of dismissive posts discourage others from pursuing their own analysis and contribute nothing towards the discussion.
 
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Dan Schaeffer
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theory wrote:
Golux13 wrote:
The_Foz_4 wrote:
Golux13 wrote:
So how long is it until some genius decides to offer college courses in Dominion math?

What do I think? I think maybe I'd rather just play the game than analyze it.

Really? What an odd thing to say. You're on the dominion strategy forum but apparently don't care about things that might help your dominion strategy.


I read the strategy forums to learn about interesting card and card-combination strategies. Also, to make fun of people who claim that Chapel breaks the game or that Chancellor is a totally useless card.

As Dave noted, I am subscribed to the game, so I see all the threads. Some of them I read in depth. Others of them make my eyes glaze over. This one was closer to the latter, in part because it is the eighteen-umptillionth thread attempting to "solve" Dominion mathematically. Your analysis might be spot-on, or it might be a load of horse crap; one thing I'm certain of, though, is that it won't "help my Dominion strategy," because I don't do complex mathematical analysis in my head at the Dominion gaming table.

Finally, I wasn't complaining about it. You asked for readers' thoughts at the end, so I gave you mine. If you wanted only in-depth analytical responses, you should have submitted it to a peer-reviewed journal. I hear "The Journal of the American Dominionological Association" is always looking for submissions.

Excuse me, I wasn't aware all topics in this forum had to cater to what you wanted to read.


And I never suggested that they should. Do I think the OP has wasted valuable time and brain-power on the analysis he's made? Yes. Do I think he should be prohibited from doing so, or from posting about it? No. Pretty simple.

Quote:
I happen to be interested in the OP's sort of analysis; the fact that there are "eighteen-umptillion" such topics suggests that many others on this forum share my sentiment.


Mazel tov to all of you.

Quote:
The fact that you personally find this "complex mathematical analysis" too much for you doesn't mean there's anything wrong with these kind of discussions on a geeky forum specifically designed to discuss Dominion strategy, in a topic specifically labeled "Average Card Value as a Metric for Money-heavy Strategies".


I never said there was anything wrong with it. The OP solicited thoughts, I provided them. Then, when questioned, I responded directly to the points made (e.g., about whether this kind of post would help my Dominion game).

Quote:
These kinds of dismissive posts discourage others from pursuing their own analysis and contribute nothing towards the discussion.


Given the proliferation of these hyper-analytic threads, I very much doubt that's the case.
 
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Edward
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Golux13 wrote:
Quote:
Excuse me, I wasn't aware all topics in this forum had to cater to what you wanted to read.


And I never suggested that they should. Do I think the OP has wasted valuable time and brain-power on the analysis he's made? Yes. Do I think he should be prohibited from doing so, or from posting about it? No. Pretty simple.

Quote:
The fact that you personally find this "complex mathematical analysis" too much for you doesn't mean there's anything wrong with these kind of discussions on a geeky forum specifically designed to discuss Dominion strategy, in a topic specifically labeled "Average Card Value as a Metric for Money-heavy Strategies".


I never said there was anything wrong with it. The OP solicited thoughts, I provided them. Then, when questioned, I responded directly to the points made (e.g., about whether this kind of post would help my Dominion game).

I suppose, then, if I didn't like Dominion as a game at all, then I should come in every topic that (explicitly or implicitly) solicits others' opinions, and proceed offer my opinion on how everyone in this forum is wasting their time playing a game that I am not interested in.
 
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Scott Fasnacht
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Golux13 wrote:
Your analysis might be spot-on, or it might be a load of horse crap; one thing I'm certain of, though, is that it won't "help my Dominion strategy," because I don't do complex mathematical analysis in my head at the Dominion gaming table.

djgutierrez77 wrote:
More importantly, though, how much good is this kind of analysis if you can't do the math in your head? Do we really need a statistical formula to figure out that a silver is better than a smithy if I don't have any money in my deck?


Some of what I discuss will be applicable directly in-game. The rest I actually plan to use to run simulations. The idea is, if I can figure out how to make a computer calculate the monetary strength of each card, I can make it play the game applying that to make decisions using a big money strategy (which is too difficult for humans to do in game due to the intense calculations). Then we can glean general guidelines for play from the results of many games that the computer can play out quickly (Simulations I ran for another thread were playing 100k games in seconds, so we can simulate data that would take lifetimes of dominion play).

We might determine that a cellar is never worth owning if the computer chooses to never purchase it when given the option across thousands of games. Or we might find out that it's only worth purchasing at a certain point in the game, or if certain cards exist in the set for that game. The trick is figuring out the math first so the computer can successfully apply it at a high level of play. Without that we will never have results that can be boiled down into more simple guidelines for actual use by humans in real games. Some of those guidelines you can gain intuitively from playing the game a lot, but others may be non-intuitive and thus can only be discovered by this sort of analysis.

As an example of this, how many players knew that if your deck was in the configuration I gave earlier with 1.2 ACV, that you would rather buy nothing than a Cellar? Or that you might EVER want to opt not to buy something if you could afford a card, for that matter? Most instinctively reach for a cellar if it's the only 2-cost card and they have a 2-coin turn. It's easy to think that buying a card must be better than not buying anything at all, but as I demonstrated this is not necessarily the case. It's precisely this sort of counter-intuitive knowledge that I hope to ultimately uncover by unraveling all this nasty math.
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theory wrote:
I suppose, then, if I didn't like Dominion as a game at all, then I should come in every topic that (explicitly or implicitly) solicits others' opinions, and proceed offer my opinion on how everyone in this forum is wasting their time playing a game that I am not interested in.


You should? I don't see how that logically follows from anything I've said. I guess if you really hated Dominion, you could post something like that. I think you'd probably be better off posting a negative review of your own, but I suppose if someone solicited feedback on their post calling Dominion the Best Game Ever, you could respond there.

If you did, I suspect you'd get people asking you why you bothered. Then you could explain yourself, and maybe you'd get someone persistently griping about your original comment and your explanations until you got bored with replying to them. And who knows how long that might take?

But I bet if you did that in every Dominion thread, (a) your explanation, no matter how reasonable, would start to pall; and (b) you'd get banned as a spammer. So no, I guess you shouldn't do that after all.
 
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Andrew Hardin
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I have a couple of problems with using your ACV calculation:

1. Dominion is about sums, not averages. Averages have misleading properties when viewed in isolation. The value of a card is not how much Coin it gives you in isolation but how much Coin it gives you in aggregate.

2. The first issue brings up the problem of the second. The value of a Smithy depends on the value of the cards the Smithy draws. The total Expected Coin is NOT the sum of the Expected Coin for each Card. (Edit: More generally, cards often work together or in opposition and the effect is not always linear).

3. Expected Coin is a measure of centrality. The P(X >= 8) where X is the Coin in your hand is only partly described by this one measure.

Those are the caveats. The real problem is that a mathematical model of this type does not describe the behavior of a real Dominion game with much accuracy. It does describe the behavior of Big Money or Chapel fairly well but it won't do much for describing the behavior of a Chancellor/Laboratory/Cellar strategy.

- Lex
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Scott Fasnacht
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LexH wrote:
I have a couple of problems with using your ACV calculation:

1. Dominion is about sums, not averages. Averages have misleading properties when viewed in isolation. The value of a card is not how much Coin it gives you in isolation but how much Coin it gives you in aggregate.

2. The first issue brings up the problem of the second. The value of a Smithy depends on the value of the cards the Smithy draws. The total Expected Coin is NOT the sum of the Expected Coin for each Card. (Edit: More generally, cards often work together or in opposition and the effect is not always linear).

The combination of these issues is something to which I may have found a solution. Instead of trying to solve the underlying math including huge amounts of dependencies, I believe it will be effective to run numerous sampling tests instead. For instance, to choose a card to buy, you can run a separate test for each of the possibilities. You add a copy of the card to the deck, then have the computer draw a bunch of hands from the new deck. Play those hands, record the coin totals. In this way instead of trying to mathematically solve the ACV of cards, which as you pointed out is quite prone to pitfalls, you can instead estimate the effectiveness of the card by sampling the results it yields. With that data you can calculate not only ACV, but could also find other metrics like P(Coin >= 8). With direct comparisons, you should be able to determine which card pulls its weight best at the current moment.

Such a setup is, however, extremely shortsighted. It may miss the usefulness of a Village->Smithy chain if neither card alone is effective for the deck right now. The way around this is branching. If instead of one card, the computer simulates combinations of cards, say to a depth of 3, then the usefulness of such combinations becomes apparent. The concern is that of complexity, however. If there are 10 possible cards to buy this turn, then if you include 2 subsequent buys there may be 1000 combinations of 3 buys, all of which require thousands of hand draws to evaluate the effectiveness of the resulting deck

For reference, my previous testing program was completing 100k games in a few seconds. Still, there will be some level of complexity that is too much. I plan to start off only considering the effect of the current buying round. If it runs smoothly, I will layer on extra complexity as it seems possible. I'd rather end up with a program that does too little initially than one that is too zealous and never runs to completion.

Quote:
The real problem is that a mathematical model of this type does not describe the behavior of a real Dominion game with much accuracy. It does describe the behavior of Big Money or Chapel fairly well but it won't do much for describing the behavior of a Chancellor/Laboratory/Cellar strategy.


Yes, I've already said that I'm trying to teach the computer how to run a big money strategy. Concerning what you said about Big Money and Chapel, I'm especially curious if the inclusion of Chapel in a set gets the program to make a chapel deck. It may do so if it perceives Chapel to be helpful to its ACV. I'm guessing this will typically be the case since big money quickly makes the whole starting deck underweight.
 
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