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Dominion: Intrigue» Forums » Strategy

Subject: Duke / Duchy versus Province Analysis rss

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Geoffrey Ulman
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One reasonable way to evaluate the effectiveness of a victory card purchase in dominion is to look at the victory point gained versus coins spent (a sort of mpg for dominion, if you will). You want to get the most bang (victory points) for the buck (coins spent).

A province costs 8 coins and provides 6 victory points: .75 vp/coin.

Meanwhile, duchy costs 5 coins and provides 3 victory points: .6 vp/coin.

(Estates are, naturally, the worst deal out of the bunch at .5 vp/coin).

In order for a duchy / duke strategy to be viable, the total vp/coin ratio must be equal to or better than .75 (otherwise you should just buy provinces). Note: I understand that this analysis ignores the fact that duchys and dukes are significantly easier to buy. Thus, one might accept a slightly lower efficiency.

For the duchy / duke strategy, we can calculate the over all efficiency as follows:

total vp = #duchys * #dukes + 3 * #duchys
total cost = 5 * #duchys + 5 * #dukes

efficiency = total vp / total cost

The following chart shows how efficiency changes with 4, 5, and 6 duchys and 0 through 6 dukes.

[IMG]http://chart.apis.google.com/chart?chs=600x400&chd=t:0.6,0.68,0.75,0.8,0.84,0.87,0.9|0.6,0.67,0.71,0.75,0.78,0.8,0.82|0.6,0.64,0.66,0.68,0.7,0.71,0.72&chds=0,1&cht=lc&chxt=x,y,r,x,y&chxl=0:|0|1|2|3|4|5|6|1:|0|0.5|1.0|2:|provinces|3:||Number of Dukes Purchased|4:||vp/coins|&chxp=2,75&chxtc=2,-600&chdl=6 Duchys Purchased|5 Duchys Purchased|4 Duchys Purchased&chco=ff0000,00ff00,0000ff&chtt=Efficiency+of+Duchy+/+Duke+Strategy[/IMG]

Although simply purchasing enough duchys so that dukes are a better deal than duchys (4 duchys makes dukes 4 points each versus 3 for a duchy) makes each individual duke more efficient than a province (.75 vp/coin for a province versus .8 vp/coin for a duke worth 4 points) it's not enough to make up for the 4 inefficient duchys (.6 vp/coin) which had to be purchased.

Even after 4 duchys then 6 dukes have been purchased, the overall efficiency (.72 vp/coin) is still less than simply buying provinces.

An efficiency of .75 vp/coin can be reached by buying 6 duchys and 2 dukes, or 5 duchys and 3 dukes.

The moral of the story is: as a rule of thumb make sure you have at least 5 duchys to make those dukes worthwhile.
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Mat Nowak
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Here's a previous topic that Chris Martin (chrisjwmartin) posted regarding the optimal number of Dukes and Duchies: Too Many Dukes Spoil the Duchy, OR The Optimal Proportion of Duchies and Dukes
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Geoffrey Ulman
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Cool, thanks for pointing out that thread. It looks like he comes to the same conclusion (buy 5 duchys before buying dukes), which is comforting .
 
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wodan wodan
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I agree, and noted as such in the previous thread. My attitude is that you should aim for the following:
1. Get 4 Duchies
2. Get 4 Duchies and 5 Dukes in alternating order.
Accomplish that and you have 64 points, a lot more than what a Province focused strategy will get.

You should be aiming for lots of Silver if possible.
 
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Yaron Racah
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wodan46 wrote:
I agree, and noted as such in the previous thread. My attitude is that you should aim for the following:
1. Get 4 Duchies
2. Get 4 Duchies and 5 Dukes in alternating order.
Accomplish that and you have 64 points, a lot more than what a Province focused strategy will get.

You should be aiming for lots of Silver if possible.


I have to disagree. If you're aiming for, say, 7 Duchies and 4 Dukes (49 points, more than 8 Provinces), the order in which you get them is irrelevant for you final score. However, the order matters greatly for the opponent's ability to mess up your strategy:

If you mess around with Dukes before you have all your Duchies, Mr. Province can make a fast grab for 2-3 Duchies (maybe in one turn), gaining some points for himself, and reducing your score (5 Duchies + 6 Dukes is only 45 points for 11 cards, not 49).

However, if you get the Duchies early, you can now get the Dukes at leisure (you don't need all of them, and they don't score for anyone else).

In 3-4 player games, it's even worse - if several players are playing Duke/Duchy, everyone will need more Duchies than Dukes, so there will be a mad race for Duchies (as with Gardens), with more than enough Dukes to go round. In this case, you should buy Duchies until they run out, then switch to Dukes.

The "get 3/4 Duchies, then alternate" idea is based on a desire to maximize your points at every stage, which is not important: only final scoring matters.

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Geoffrey Ulman
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yaron, you're right: what you really care about is having spent your money most effectively (most victory points per coin spent) when all is said and done.

With that in mind I created the following chart:



This shows the efficiency (victory points / total coins spent) for every combination of dukes and duchys.

Cells are colored by the total number of dukes + duchys. Thus, 3 dukes 1 duchy, 2 dukes 2 duchys, and 1 duke 3 duchys, are all the same shade of gray because 4 total cards have been bought.

For each grey shaded diagonal, the cell in red is the optimal ratio of dukes to duchys given you've purchased that many total cards.

Looking at the pattern or red (optimal) cells, we come to the following conclusion:

Always aim for ending with 2 less dukes than your total number of duchys.


In the absence of any competition for these cards, you should follow this rule all the way up. This ensures that if the game ends unexpectedly you'll have an optimal mix. Of course, if one set is in danger of running out, you may want to grab as many of those as you can .
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Geoffrey Ulman
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wodan46 wrote:
I agree, and noted as such in the previous thread. My attitude is that you should aim for the following:
1. Get 4 Duchies
2. Get 4 Duchies and 5 Dukes in alternating order.


yaron wrote:
The "get 3/4 Duchies, then alternate" idea is based on a desire to maximize your points at every stage, which is not important: only final scoring matters.


I should note that, not surprisingly, the rules of thumb quoted above do always land you in a red/optimal cell on the chart. So following them will result in the highest possible duke/duchy score for your number of cards.
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Jeff Goris
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One thing that seems to get overlooked in the Duke/Duchy vs Province victory cards is how the number of pure victory cards in your deck slows down your deck. Sure, duchies and dukes can be more efficient in terms of VP per Coin, but they are not more efficient in terms of VP per buy or VP per Victory Card.

Eg, 6 duchies and 3 dukes is worth 36 points costing you 45 coins, whilst 6 provinces worth 36 points costs 48 coins. Yes, the Dukes/Duchies are more efficient per coin. However with Dukes/Duchies you required 9 buys vs just 6 for Provinces. Similarly, with Dukes/Duchies you have 9 extra useless Victory cards slowing down your deck vs. 6 for Provinces.

I'm not saying that one is better or worse than the other. It's just that there's a lot more to consider than VP/coin when deciding which strategy to go for at the beginning of a game.
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Geoffrey Ulman
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Hmmm good point... I hadn't considered that. Of course, with good support cards like scout, etc... you could mitigate the problem -- so like any other card valuation it comes down to what other cards are in the mix. Which is of course what keeps the game interesting .
 
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Yaron Racah
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ulmangt wrote:
Looking at the pattern or red (optimal) cells, we come to the following conclusion:

Always aim for ending with 2 less dukes than your total number of duchys.


Actually, looking at your table, the optimal number of Dukes is 3 less than the number of Duchies. Of course, if the total number of Dukes+Duchies you get is even, then a difference of 3 is impossible, and then 2 and 4 are equally good.

This is also common sense: An extra Duke gives you 1 point per Duchy. An extra Duchy gives you 3 points, plus 1 point per Duke. So they are equal when [Duchies = Dukes + 3].

ulmangt wrote:
In the absence of any competition for these cards, you should follow this rule all the way up. This ensures that if the game ends unexpectedly you'll have an optimal mix. Of course, if one set is in danger of running out, you may want to grab as many of those as you can .


This is correct, except that you should always count on competition (even if no one is playing Dukes/Duchies, your Province-buying opponents might always make a late game grab (to ruin your strategy, or just for the points. Therefore:

ulmangt wrote:
I should note that, not surprisingly, the rules of thumb quoted above do always land you in a red/optimal cell on the chart. So following them will result in the highest possible duke/duchy score for your number of cards.


is incorrect:
If you follow the rule of thumb quoted, you will most likely not end the game with the optimal number of Dukes/Duchies, because Duchies will be taken while you mess around with Dukes. You will likely have 5 Duchies and 6 Dukes, or something like that.

If you really want to get the optimal proportion (say, 7 Duchies and 4 Dukes in a 2 player game), buy 7 Duchies first, then 4 Dukes.
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Greg Payne
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yaron wrote:
Actually, looking at your table, the optimal number of Dukes is 3 less than the number of Duchies.

Wasn't this mentioned in the original thread about the Duchy? Namely: Get 3 Duchys, then either alternate Duchy and Duke, or get to $10 with 2 buys and get one of each.
 
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Yaron Racah
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Lardarse wrote:
yaron wrote:
Actually, looking at your table, the optimal number of Dukes is 3 less than the number of Duchies.

Wasn't this mentioned in the original thread about the Duchy? Namely: Get 3 Duchys, then either alternate Duchy and Duke, or get to $10 with 2 buys and get one of each.


Yes, I was just noting that his table supports this (admittedly trivial) conclusion as well.

But I still insist that you don't want to actually alternate them. The "3 difference rule" is good for planning your final tally, but once you have, get all Duchies, than Dukes.
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Tony Chen
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You can't look at it like that (VP/coin). What if you could get 30VP for 30 coins on one card, at a rate of 1VP/coin? Would you try to build up for that? Or how about 2 Estates, versus a 4 cost 2VP card? They have the same VP/coin, but they are very different. Even if the Estates were free, people still wouldn't buy them at an infinite rate of return VP/coin-wise. In fact, people go through the trouble of trashing them away.
 
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