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Dominion: Intrigue» Forums » Strategy

Subject: Harem: the Province Test rss

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Jason Woolever
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Akin to the Silver Test commonly used to judge 3-cost cards, I propose a similar test, the Province Test, for Harems. In many ways a Harem is like 2/3 of a Gold plus 1/3 of a Province. Put another way Harem+Harem+Harem =~ Gold+Gold+Province. Obviously this in the absence of cards which affect Treasure cards which tip the balance (more on that later). Many players accept that with an early draw of 8 coins, it is often better to take a Gold over a Province. Yet it surprises me when the same players get star-struck at the sight of Harems, and reach for them with 6 or 7 (or possibly 8) coins, often earlier than a decision between buying Gold versus Province would come up.

So, with Harem on the table and 6+ coins in hand, when contemplating your buy imagine you have 8 coins and Harem was not available. Would you buy a Gold or a Province? If the latter, have at the Harem. If it's the Gold, take the Gold!

Other factors favoring Gold
Adventurer is in your deck - Adventurer favors big treasure
Bureaucrat is in play - Gold is immune
Cellar is in your deck - increasing deck variance favors Cellar
Chancellor is in your deck - increasing deck variance favors Chancellor
Courtyard is in your deck - increasing deck variance may help
Militia is in play - increasing deck variance weakens Militia
Minion is in your deck - increasing deck variance strengthens Minion
Remodel is in your deck - Gold->Province is stronger than Harem->Province
Secret Chamber is in your deck - increasing deck variance favors Secret Chamber
Thief is in play - late-game theft of Harem hurts, late-game theft of Gold is a mere nuisance
Torturer is in play - increasing deck variance weakens Torturer
Tribute is in play - watch your neighbor grin ear to ear if a Harem is revealed

Other factors favoring Harem
Scout is in your deck - Scout can accelerate Harems
Spy is in play - reducing deck variance weakens Spy

References to cards being in your deck refers not only to current holdings, but also future buys.

More often than not Gold wins unless you are firmly in Province acquisition mode. Favor Harem starting mid-game when you fall short of 8 coins.
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Yaron Racah
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Excellent analysis!

One comment: Spy is a pro-variance card, if you're using it yourself. It's anti-variance when used against you by opponents.
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Mason Louie
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How does your wonderful analysis change if instead of ending by Provinces, you wish to end by 3 piles-- especially considering that Harems gets a lot of attention usually?
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Jason Woolever
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Good point on Spy- there are forces on both sides.

Regarding game-end condition, it does have some relevance. In most cases players I suspect players won't know whether they would favor shortening or lengthening the game until mid-game when Provinces are firmly in play, past the transition point of Gold to Harems. If the decision is close, though, it could be the swing factor.
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Jonathan Pickles
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One other ponderable is that gold is usually so good as you are seeking to get 8 for a province. If you are only trying to get the 6 for a Harem then the 2 gold you get from another Harem is more likely to be adequate. So do you get sucked into a Harem spiral?

Once you have tasted polygamy one dozen wives is never enough.
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Tony Chen
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I am not sure I understand your logic. Are you saying that because Gold is better than a Province, then by the same logic Gold should also be better than a Harem? I am not sure it's as simple as that.
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Yaron Racah
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drunkenKOALA wrote:
I am not sure I understand your logic. Are you saying that because Gold is better than a Province, then by the same logic Gold should also be better than a Harem? I am not sure it's as simple as that.


I think that's exactly what he's saying:

If, at a certain point in the game, you're not willing to trade 3 coins for 6 vp (i.e., take a Province over Gold), you probably shouldn't be willing to trade 1 coin for 2 vp (i.e., take a Harem over Gold) - it's the same 2 vp for 1 coin rate.


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Tony Chen
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But there might be a diminishing return for each coin that you add on. Not all coins are worth the same. The first coin might be worth 3VP, the second 2VP, the third 1.5VP.

If we wrote return in VPs as a function of the ratio of coins versus VPs in a card, it might not be monotonically increasing. As in, it could be a curve instead of a straight line. If it is a dome-shaped curve, the peak would be somewhere in between pure VP and pure coin. If it were a straight increasing line, then yes the peak would be at the rightmost end where it is pure coins.

Now that may very well be the case (that the function actually is monotonically increasing). But we can't just automatically assume that that is the case. It might not be.

Tommy wanted to pay $3 for an ice cream. We can't just automatically assume that he will want to pay another $3 for another ice cream on the same day, right?

Not to mention that by adding different sources of VPs and viable end condition into the game might alter the whole equation altogether. You need gold precisely for Provinces. Not so much for Gardens, right?
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David desJardins
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drunkenKOALA wrote:
But there might be a diminishing return for each coin that you add on. Not all coins are worth the same. The first coin might be worth 3VP, the second 2VP, the third 1.5VP.


No, it's pretty much linear.
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DaviddesJ wrote:
drunkenKOALA wrote:
But there might be a diminishing return for each coin that you add on. Not all coins are worth the same. The first coin might be worth 3VP, the second 2VP, the third 1.5VP.


No, it's pretty much linear.


Really? So in a set with no extra buys, you would buy a card with 20 gold and 1 VP rather than a card with 8 gold and 2 VPs?
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David desJardins
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Peristarkawan wrote:
Really? So in a set with no extra buys, you would buy a card with 20 gold and 1 VP rather than a card with 8 gold and 2 VPs?


The value of money is pretty much linear in the range of 1 to 3 money per card, in a large deck that gets shuffled so that the increment in expected money per card or in the likelihood of accumulating any particular amount of money is pretty much linear in the value of the card.

No, I wouldn't buy a card with 1 million gold and negative 100,000 VPs on it. That's silly.
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Andrew Hardin
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You mention pro-variance but what you are really getting at is that Dominion tends to favor high variance early (when your expected coin per turn is low) and low variance middle to late (when your expected coin is high), with some strategies having a drop late in the game as the VP cards begin to lower the expected coin.

I for one think the Silver Test is really a false trail, but your proposed Province Test really seems to me to be more of a Gold Test. I think a Gold Test really is an important aspect of Dominion with regard to the 6 cost cards.

The problem with the Harem is that in exchange for 2 VP you give up 1 Coin each time you draw the card. If Dominion was not a game of breakpoints this would often not be an issue. This can cost you 3 or more VP late in the game if you hit 7 instead of 8 Coin (you lose 3 when you buy that Duchy instead of that Province but you also allow your opponent(s) a chance at 6 VP).

Expected coin is not the only way to judge value in Dominion. In my view, Dominion strategy should focus more on the quantiles (i.e. the probability of getting certain Coin values) and less on the center of the distribution.

Harem+Harem+Harem is often not equal to Gold+Gold+Province. They have the same expectation but very different variance and kurtosis. Which is better depends very heavily on if your strategy favors this approach or not. I am a fan of high variance decks, particularly with the base set but I am not as sure with the newer cards.

- Lex

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David desJardins
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LexH wrote:
Expected coin is not the only way to judge value in Dominion. In my view, Dominion strategy should focus more on the quantiles (i.e. the probability of getting certain Coin values) and less on the center of the distribution.


In small decks (e.g., if you use Chapel to shrink your deck to 8 cards), there can be a significant difference. In large decks (i.e., toward the end of a game when you have 20-25 cards) there's no significant difference.
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David desJardins
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LexH wrote:
I for one think the Silver Test is really a false trail


If the "Silver Test" is, "Don't buy a card costing $3 or more if buying Silver would be better," how can that be false? Are you advocating for buying a card costing $3 or more even when Silver would be better?
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Andrew Hardin
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DaviddesJ wrote:
LexH wrote:
I for one think the Silver Test is really a false trail


If the "Silver Test" is, "Don't buy a card costing $3 or more if buying Silver would be better," how can that be false? Are you advocating for buying a card costing $3 or more even when Silver would be better?


No, that isn't exactly what I am advocating.

I basically stopped looking at Silver as the default buy awhile back. There are situations when I will refuse to take a Silver even if available. For example, if I am playing a chaining deck I will at times focus entirely on getting Gold and chaining cards. For that kind of deck Silver is something I am trying to avoid.

I agree that Silver is often worth having, but the value of Silver varies so much that I no longer use it as the baseline. I apply a more general test that says 'Should I even buy a card? If so, what should I buy?'. The answer is sometimes Silver, but I no longer compare a card directly against Silver. (Edit: What I do is determine if the card fits that strategy I am attempting. If not, I don't buy it even if I have the Coin. If so, then I buy the card that best fits the strategy. Sometimes that is Silver, sometimes not).

- Lex
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Andrew Hardin
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DaviddesJ wrote:
LexH wrote:
Expected coin is not the only way to judge value in Dominion. In my view, Dominion strategy should focus more on the quantiles (i.e. the probability of getting certain Coin values) and less on the center of the distribution.


In small decks (e.g., if you use Chapel to shrink your deck to 8 cards), there can be a significant difference. In large decks (i.e., toward the end of a game when you have 20-25 cards) there's no significant difference.


This is where you and I simply disagree.

Even with 30 cards I think quantiles should be emphasized over expected coin in many cases. If you reshuffled after each hand, and your samples were drawn with replacement I would agree.

The Central Limit Theorem doesn't apply in this case, since each hand is not independent. If it did I would agree that with even 5 cards the distributions are similar enough. But since much of the game you are dealing with 20 or less cards this isn't the case. Without a nice bell curve for the distributions you get longer tails that create many hands that vary heavily from the expected mean. (It certainly doesn't hurt to improve your expected coin, but the quantiles would reflect that too).

More importantly, the presence of cards such as the Cellar and Chancellor allow for selective removal of unwanted cards that bias the results in favor of Gold over Silver in many cases. Building decks towards high variance can create systematic advantages if the cards are right.

I simply don't have the time right now to prove or disprove this I tend to rely on empirical evidence from direct play only. I may be completely wrong on this.

- Lex
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Tony Chen
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DaviddesJ wrote:
drunkenKOALA wrote:
But there might be a diminishing return for each coin that you add on. Not all coins are worth the same. The first coin might be worth 3VP, the second 2VP, the third 1.5VP.


No, it's pretty much linear.
It is linear if you are talking about expected return on coin value per hand. In Dominion, VPs aren't scored by adding up your total coins every turn. But you knew that.

It's not as simple as Gold is better than Province, therefore Gold must be better than Harem also because Harem is in between the two.

So instead of 3 Gold, you get 3 Harems. During the game, that's effectively getting 3 Silvers instead of 3 Golds. How many Provinces do you think it will cost you? Might be more than 1 to make up for the 6 VPs, but the reasoning isn't as simple as: always trade in 2VP for 1 coin.

Consider 3 Harems, 1 Gold, and 1 Province, versus 3 Gold and 2 Provinces. Same number of VP, same expected return of coins. But the variance is different. I agree with Lex that the different variances may have very significant effects on how frequently one will hit a certain mark. In mid game, when your average income is slightly higher than 8, I'd argue that it is better to have a lower variance. Depends on the type of deck you are building too. Again like Lex said, if your deck is all about chaining together action cards so that you draw your entire deck, then average coins is irrelevant, you just need 2 Gold and a Silver. You wouldn't want the extra Silvers that may decrease your chances of chaining together the action cards.
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Yaron Racah
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LexH wrote:
Expected coin is not the only way to judge value in Dominion. In my view, Dominion strategy should focus more on the quantiles (i.e. the probability of getting certain Coin values) and less on the center of the distribution.

Harem+Harem+Harem is often not equal to Gold+Gold+Province. They have the same expectation but very different variance and kurtosis. Which is better depends very heavily on if your strategy favors this approach or not. I am a fan of high variance decks, particularly with the base set but I am not as sure with the newer cards.


Higher order cumulants like skew and kurtosis (or quantiles, which are another way of looking at the same thing) are essential for analysing heavily skewed distributions. For example, they are used when analysing income distributions, which can have some people making 10 or 100 times as much as the average, and the average can be double the median (as it is in Israel, regretfully).

The distribution of card values in a Dominion deck isn't anything like that. Your best card (typically Gold) is usually worth about twice as much as your average card. Laboratories (another great card) is worth double your average card, almost by definition. Sometimes you'll have cards exceeding that, but not by much. So your best card deviates from the average about as much as your worst card (Estate/Province). This means that the skew, while non-zero, is rather small, and we can safely neglect it.

[if you find the time to calculate the kurtosis of a typical Dominion deck, and more specifically how it is affected by the Gold/Harem decision, and how that affects the Province acquisition rate, that could be illuminating.]

Negligble higher order cumulants, however, don't mean that everything is linear. You are correct in pointing out that the 2nd cumulant - variance - plays an important role. Specifically, as we are mainly concerned with the 8-coin threshold, we want higher variance when our expected value is below that threshold, and lower when it is above.

However, except for the early game (in which we all agree Gold is the way to go), your expected value should usually around 8, or slightly lower. This is because of the self correcting nature of the game: if you can score Provinces often, you will, lowering your expected value. If you can't, you'll get Gold and other useful cards, raising your expected value.

Given that the expected value of coins produced is not very far from 8, variance doesn't play a big role, either. And this means that the OP's linear approximation is rather a good one.

[You are, however, correct in pointing out that card-selection does make variance a good thing. You mention Cellar and Chancellor, to which I would add Militia. In the presence of these cards, I would be less likely to buy Harems. But see my next post for a qualification of that statement.]
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Yaron Racah
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drunkenKOALA wrote:
So instead of 3 Gold, you get 3 Harems. During the game, that's effectively getting 3 Silvers instead of 3 Golds. How many Provinces do you think it will cost you? Might be more than 1 to make up for the 6 VPs, but the reasoning isn't as simple as: always trade in 2VP for 1 coin.

Consider 3 Harems, 1 Gold, and 1 Province, versus 3 Gold and 2 Provinces. Same number of VP, same expected return of coins. But the variance is different. I agree with Lex that the different variances may have very significant effects on how frequently one will hit a certain mark. In mid game, when your average income is slightly higher than 8, I'd argue that it is better to have a lower variance. Depends on the type of deck you are building too. Again like Lex said, if your deck is all about chaining together action cards so that you draw your entire deck, then average coins is irrelevant, you just need 2 Gold and a Silver. You wouldn't want the extra Silvers that may decrease your chances of chaining together the action cards.


I think you're attacking a straw man here. No one suggested getting Harems before getting any Gold, or after only getting 1 Gold. Actually, if we spell out the results of the OP's logic, I think you'll find them quite reasonable.

The OP is saying: get Harem over Gold whenever you would get Province over Gold. So the question is: when would you get Province over Gold?

1. As is almost cliche by now, a pure-money strategy gets to 4 Provinces in ~17 turns. In that time, it makes about 6 passes through the deck. Effective variants of that strategy will get there a little earlier, but might churn a little faster (due to Smithy, Cellar, etc.), so we can take 6 passes as a number-of-thumb (I'll qualify that later for different sorts of decks).

2. What are the benefits of getting Gold over a Province? Each time you draw that Gold later in the game, it might get you a Province. However, if that is the case, it means you have 5 coins in your hand other than that Gold: if you don't have the Gold (due to having bought a Province over it), you can always get a Duchy (or something better). So the Gold isn't giving you more than 3 points every time you draw it. That means you have to draw it at least 3 times to justify taking it over a Province.

Taken together, these two points mean that I would take Gold on the 2nd and 3rd passes through the deck, switching to Province after your 3rd reshuffle. According to the OP's logic, this is also the point I should start taking Harem over Gold (if I can't have the Province). So I'm only taking Harems during the 4th and 5th passes (on the 6th pass, of course, I take Duchy over both of them).

How many times am I going to get 6 or 7 coins during those two passes? Two? Three? So we're only talking about 2-3 Harems, and even that, late in the game, after getting some Gold. It seems a very sound progression to me.

Now, in an engine deck (especially a well-trimmed one), I go through most of my deck every turn. That means that up until the very end, I expect to see each acquisition several times. Naturally, that inclines me to take Gold over Province for quite a while - perhaps getting to the point where I get all my Provinces in 2-3 final killer turns. Until those turns, I would not be inclined to take Harems, but that also follows from the OP's logic: I would not be inclined to take Provinces, either (and in those last turns, I'm taking Provinces and Duchies anyway, so the question is moot).

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Tony Chen
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Quote:
The OP is saying: get Harem over Gold whenever you would get Province over Gold. So the question is: when would you get Province over Gold?

I knew that.
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Tony Chen
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Quote:
1. As is almost cliche by now, a pure-money strategy gets to 4 Provinces in ~17 turns. In that time, it makes about 6 passes through the deck. Effective variants of that strategy will get there a little earlier, but might churn a little faster (due to Smithy, Cellar, etc.), so we can take 6 passes as a number-of-thumb (I'll qualify that later for different sorts of decks).

2. What are the benefits of getting Gold over a Province? Each time you draw that Gold later in the game, it might get you a Province. However, if that is the case, it means you have 5 coins in your hand other than that Gold: if you don't have the Gold (due to having bought a Province over it), you can always get a Duchy (or something better). So the Gold isn't giving you more than 3 points every time you draw it. That means you have to draw it at least 3 times to justify taking it over a Province.

Taken together, these two points mean that I would take Gold on the 2nd and 3rd passes through the deck, switching to Province after your 3rd reshuffle. According to the OP's logic, this is also the point I should start taking Harem over Gold (if I can't have the Province). So I'm only taking Harems during the 4th and 5th passes (on the 6th pass, of course, I take Duchy over both of them).

That shows that Province is better than Gold starting on the 4th pass.

And a Gold may still be better than a Harem even on the 4th and 5th pass. Because it could still theoretically get you 2 or 1 more Provinces. But according to the OP, Harem > Gold whenever Province > Gold. But that is clearly not always the case. When Province > Gold starting on the 4th pass, Gold may still be better than Harem.

And a pass 1 Harem over Gold. That could cost you 5 Provinces, or none.

All you have shown is that a pass 4 Province over Gold could cost you at most 2 Provinces, or 6 points relative to Duchies. And it is worth 6 points at least.

A pass 3 Harem over Gold: it is worth 2 points itself. How many Provinces is it going to cost you? Anywhere from 0-3, or 0-9 points. Can you say for certain how many?

A pass 3 Province over Gold: it is worth 6 points itself. How many Provinces is it going to cost you? Anywhere from 0-3, or 0-9 points. The chances of missing Provinces are a lot higher now though: everytime you get 5-7 coins instead of 7 with everything else.

What if the chances of hitting 5-7 is 6 times as likely as the chances of hitting exactly 7? Say the chances of hitting exactly 7 is 1/6 per pass, so on the final 3 passes it will cost you 3/2 or 1.5 points. Then the chances of hitting 5-7 is 1 (or 99.99%) per pass, so 9 points on the final 3 passes. So with Harem, you are getting 2 points at the cost of 1.5 points, and with Province, getting 6 at the cost of 9.

Of course, the chances of hitting 7 versus 5-7 might not be as lopsided as 1:6. But do you know what the exact ratio is? Gold over Harem may very well be worth it, (since I need to spell this out: when one would also favor Gold over Province), but it's not because of a linear relationship.
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David desJardins
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drunkenKOALA wrote:
I agree with Lex that the different variances may have very significant effects on how frequently one will hit a certain mark.


Well, it's not a matter of "may". Either it does or it doesn't. I'm saying that in general it doesn't. There are mathematical reasons for that, which the next poster describes pretty well.
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Tony Chen
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When I don't know the answer to something, I don't pretend that I do.

And in my experience, Moneylender > Militia > Bureaucrat > Moneylender. I ascribe that precisely to the effects of variance. In fact, the OP himself believes in the effects of the variance.
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Andrew Hardin
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yaron wrote:

Higher order cumulants like skew and kurtosis (or quantiles, which are another way of looking at the same thing) are essential for analysing heavily skewed distributions. For example, they are used when analysing income distributions, which can have some people making 10 or 100 times as much as the average, and the average can be double the median (as it is in Israel, regretfully).

The distribution of card values in a Dominion deck isn't anything like that. Your best card (typically Gold) is usually worth about twice as much as your average card. Laboratories (another great card) is worth double your average card, almost by definition. Sometimes you'll have cards exceeding that, but not by much. So your best card deviates from the average about as much as your worst card (Estate/Province). This means that the skew, while non-zero, is rather small, and we can safely neglect it.


Not often I hear anybody use the word cumulants instead of moments.

The thing that I have observed in Dominion is that focusing on the expected value of the card ignores that the hand is what you are working with. The expected coin is based upon a sum, not an average, and is therefore exposed to increasing variability as hand size increases. A gold may only be worth 3 times as much as a Copper but the hand consists of however many Gold you can get.

I don't normally focus that much on the kurtosis but it is of interest when dealing with the behavior of some of the Chapel deck strategies. I don't exactly have nice little formulas to calculate them in practice I admit. In general I focus on the variance (I tend to ignore the skew). The variance is affected by a finite population correction due to the limited deck sizes.

In speed games I often go with decks that have average expected coin closer to 6 or 7 rather than 8. The time it takes to get expected coin to 8 is sometimes not something I focus on. Some of my Chancellor deck strategies (particularly Chapel/Chancellor) may have no more than 8 Coin in the entire deck, with the expected coin sitting more around 7.

- Lex
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Andrew Hardin
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DaviddesJ wrote:
drunkenKOALA wrote:
I agree with Lex that the different variances may have very significant effects on how frequently one will hit a certain mark.


Well, it's not a matter of "may". Either it does or it doesn't. I'm saying that in general it doesn't. There are mathematical reasons for that, which the next poster describes pretty well.


My own analysis of the game mostly agrees with what you are saying. In the early part of the game the fact that your hands are not independent (and in fact are highly negatively correlated) also combines with the benefits of getting early Gold and specific cards to make an early high variance strategy work. Later in the game this effect is dampened to the point where I don't focus on it that much (though if I have a high variance style I am willing to go Treasure thin).

I often play a strategy based upon ramping up very quickly, snatching up Provinces and winning by Province exhaustion with 4-5 Provinces, while my deck is sputtering terribly and is reduced to buying Duchy and Estate cards late in the game. I note that you play with an infinite Province variant that removes some of the benefits of this strategy. Your variant gives later players a better chance of catching up (I don't know how often you play this but it would affect my strategy). I have won a portion of my games knowing that if my opponents had just one more turn I would lose.

Because I focus so heavily on the opening of the game, when negative correlation is important and the expected coin is far less than 8 this is a very important consideration in the type of game I play. By the time my deck has an expected coin of 8 I have either won or am likely not going to win.

- Lex
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