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Subject: How sophisticated are the strategic opportunities in the First Game set of Kingdom cards? rss

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Mike Shapiro
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My gaming buddies and I have logged a few dozen plays of Dominion, and so far we have stuck almost exclusively to the 10 Kingdom cards Donald Vaccarino recommends for the first game:
* Cellar,
* Market,
* Militia,
* Mine,
* Moat,
* Remodel,
* Smithy,
* Village,
* Woodcutter, and
* Workshop.

We've clung to these cards not out of timidity but out of a kind of purist obsession: even the ten simplest cards in the box offer a wealth of strategic opportunities, and we are eager to understand the underlying dynamics of these cards before we start futzing around with Gardens and Festivals.

If the core critique of Dominion is correct, there will be one strategy for this set whereby one player can engineer a Province-buying machine more quickly and more reliably than through any other strategy. According to this critique, once this strategy is uncovered the players will then rush to execute it, with the player who goes earliest and draws most luckily necessarily winning the game.

At present, I am wrestling with whether this critique is persuasive.

After a sequence of refinements, my buddies have developed what they call the Village Eater strategy. If you've played with this set at any length you've likely seen this in action: the Eater buys and Workshops as many Villages as he can early in the game, although this is a cardinal sin according to at least one strategist on this board; s/he then buys Smithies, then 1 or 2 Mines and a pile of Markets. By the 11th or 12th hand, the Eater is reliably drawing all or nearly all of her or his deck; by the 13th or 14th hard, the Eater buys the first Province; by the 16th hand, the Eater is likely buying 2 Provinces per turn.

Is this strategy unanswerable?

The Village Eater does not scale well: in a three-player game, two Eaters will neutralize each other in the race for Villages while a third player can cozily Mine her or his way to a deckful of Gold. The strategy is also vulnerable to nearly every other card in the box: as soon as the Witch comes into play, for example, the Eater is drawing -- and making -- Curses instead of the Village/Smithy combo that s/he has to achieve early in the hand. If the Chapel is swapped in, the Eater has less than a 50% chance of winning.

But what about the core question of this post: is the Village Eater strategy unanswerable in a two-player game using the ten cards of the First Game set?

A Militia-heavy strategy tends to be ineffective: an Eater can have a half-dozen Moats in the deck that will couple well with Villages. A Gold-heavy or Gold/Remodel strategy tends to be too slow: the Eater is getting up to speed by the 14th or 15th hand, after all.

Although there are delicate combinations of Militia, Mine, and Remodel (with appropriate numbers of Villages and Smithies) that can come close to the Eater for speed, I have not yet found the combo that will get me there by hand 15 or 16.

What do you think? Is this evidence that the game is strategically flat or simply evidence that the group needs to spend more time finding an answer for this strategy?
 
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Everett Scheer
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village-smithy is arguably the best strategy with the basic set. The tactics here is when to move to buying cash and ramp to provinces, and whether mine or market is better for your deck.
 
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Edward
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Considering a strategy of buying nothing but treasures can get to four provinces on average by turn 16-17, I'd say that strategy needs some work.

Two optimization ideas:

* Militia. Playing an early game militia limits you to buying something up to 3, or if you're lucky 4.
* Remodel. Late game allows you to convert your golds into provinces automatically; instead of needing 16 money for 2 provinces, you only need 11.

The default set is "good" in the sense that there's no clear-cut strategy to it (unlike, for example, Chapel/Lab/Market/Feast/Cellar + useless other cards); but I'd suspect any winning strategy would revolve around a Village/Smithy/Cellar chain + Remodel + Militia.
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Felix Rodriguez
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If you JUST do two smithies and ignore the villages you will, on average, get four provinces by turn 15. (See http://www.boardgamegeek.com/thread/400371).

So if your opponent buys his first province by turn 13, he has likely lost. For this reason, I'm not a fan of the village strategy for the starting hand at all.
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Alexander Flurie
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theory wrote:

* Remodel. Late game allows you to convert your golds into provinces automatically; instead of needing 16 money for 2 provinces, you only need 11.


Do note that this requires 11 money and an action versus 16 money and two buys.
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Heiko Hartmann
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mshapiro wrote:
If the core critique of Dominion is correct, there will be one strategy for this set whereby one player can engineer a Province-buying machine more quickly and more reliably than through any other strategy. According to this critique, once this strategy is uncovered the players will then rush to execute it, with the player who goes earliest and draws most luckily necessarily winning the game.

At present, I am wrestling with whether this critique is persuasive.


I think it's correct to assume that, for a given set of 10 cards, there is one type of strategy which will have the best chances to win. Of course this 'one strategy' still has some decisions and considerations which must be done during play to tweak the deck any further.
I don't think that this is really a downside of this game though. Finding this 'best strategy' (as I said, there may be a lot of minor variants of this strategy) is part of the game. As soon as you just exchange one of the 10 cards for another, the strategy may change (either a totally different strategy becomes viable or you have to slightly adjust your old strategy) and the 'search' for a good strategy is open again.

Did you try a heavy 'Remodel' deck? Put in some villages (this will also lower the amount of villages available for the other players - but you need a few of them as well), workshops (to buy Remodels) and (if you can get it early on) some Mines. Buy a lot of Remodels and try to convert 4-cost cards (including the Remodels themselves) to Gold and Gold to Provinces. In the late game you may even use Remodel on village/workshop for some Duchys. You need Gold to convert to Provinces, so buying Silver is also often a good option when you have a Mine.
Hints: If you think that you have the lead you may even remodel a Province into a Province to speed things up. Of course you only do this if you have nothing else to remodel. In the beginning it could be an option to remodel Estates into Remodel to speed your deck up.

Overall I think you have a tough time stopping the village+smithy combos. This strategy is relatively immune to Militia (if they are good players they will ignore this card and don't buy a Moat. But even if they do, the Moat combines well with the village...) and your only options are to deplete their resources (e.g. also grab villages) or to build a deck which is faster.
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Mac Mcleod
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workshop+remodel on the open, gives you a good shot at 2 villages on your 3rd and 4th turns. (workshop->village, buy village) (remodel estate->village, buy village). More likely to end up with 3 villages. Unlikely to get stuck with 2 villages.

I have gotten six villages by turn 5 this way.

Then when you sense you are in the last trip through your deck, a third to half of your villages can become duchies.


---
with regard to
I think it's correct to assume that, for a given set of 10 cards, there is one type of strategy which will have the best chances to win. Of course this 'one strategy' still has some decisions and considerations which must be done during play to tweak the deck any further.

I agree in some cases this is true.
For example lab+chap is just gross and throne+workroom+garden is gross and both are hard to counter and if two folks go for them, then one of them will usually wein.

On the other hand, others reacting to your deck modify it's chances of winning. In some cases, it shifts the balance between different strategies depending on how many people go for it. Gardens and Markets both being a good example-- 3 markets is "eh", but 6 markets is pretty potent. 8 gardens is potent, 4 gardens is "eh".

Two people going for the ideal strat can cancel each other out.
One person going for the ideal strat can be canceled by a counter strategy.

This would be more true in a 4 player game than a 2 player game.



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Andrew Hardin
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If I open 5/2 I am probably going to try to counter Village/Smithy with Market/Cellar.

But for a 4/3 split I would either go Village/Smithy or Silver/Militia.

The opening deck splits into what I see are 3 viable strategies:

+Action/+Card (Village/Smithy)
Deck Cycling with Treasure (Market/Cellar)
Big Money with Attack (Silver/Militia)

The Remodel is worth having as a Gold -> Province or Silver/Village -> Duchy play.

These are a pretty bland set of 10, lacking Festival, Lab, Chapel and containing two of the weaker cards (Mine/Woodcutter) and the Workshop is not a card I like to use much without Gardens. I am going to have to take a look at the Village/Workshop strategy mixed with the Remodel. I don't usually like to add a lot of cards to my deck.

- Lex
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Mike Shapiro
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LexH wrote:
...and containing two of the weaker cards (Mine/Woodcutter)...


I'm surprised to hear the Mine characterized thus. Doesn't it help tighten a deck by trashing Coppers?
 
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Andrew Hardin
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mshapiro wrote:
LexH wrote:
...and containing two of the weaker cards (Mine/Woodcutter)...


I'm surprised to hear the Mine characterized thus. Doesn't it help tighten a deck by trashing Coppers?


Some people value the Mine higher than I do. But I don't care for the card since buying it eats up one of your critical (and limited) 5 cost card buys.

The problem is timing. Unless you buy it as part of a 5/2 split the Mine is coming up at the earliest at Turn 3. It will cycle back in no earlier than Turn 5. It will probably come up 4-7 times in the entire game that you can use.

It has no +Action so it can interfere with another play.

The 5-cost cards as a group are the best cards in the game (except the underpriced Chapel which is really a 5-cost card pretending to be cost 2 and the distinctly unusual Adventurer). The penalty you pay for these cards is they take critical time to acquire. The Mine compares poorly to the other 5-cost cards I would rather buy in most situations. Ironically, the card would be underpriced at 4.

I still use the Mine if the game fits the Mine strategy. But it is one of the rarer cards in my deck (the 3 cards I use the least are Mine, Thief and Woodcutter).

- Lex

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Mac Mcleod
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A mine question.
I used to upgrade coppers to silvers first.

But lately, I've been going silver to gold first (so 1c,1g is the result instead of 1s,1s).

Any thoughts on this from the math wizs?

Mines are a little weak but with extra actions and the chance to get one early they often factor in my losses to miners.
 
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Felix Rodriguez
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maxo-texas wrote:
A mine question.
I used to upgrade coppers to silvers first.

But lately, I've been going silver to gold first (so 1c,1g is the result instead of 1s,1s).

Any thoughts on this from the math wizs?

Mines are a little weak but with extra actions and the chance to get one early they often factor in my losses to miners.


1c, 1g is, IMHO preferable. Another way to think about it is the initial breakdown. Do you prefer 5/2 or 4/3? That's essentially the decision you're making. Also, if you have remodels you can turn golds into provinces.
 
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Everett Scheer
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Kaelistus wrote:
maxo-texas wrote:
A mine question.
I used to upgrade coppers to silvers first.

But lately, I've been going silver to gold first (so 1c,1g is the result instead of 1s,1s).

Any thoughts on this from the math wizs?

Mines are a little weak but with extra actions and the chance to get one early they often factor in my losses to miners.


1c, 1g is, IMHO preferable. Another way to think about it is the initial breakdown. Do you prefer 5/2 or 4/3? That's essentially the decision you're making. Also, if you have remodels you can turn golds into provinces.


It depends on the situation. With this set, I'd prefer to go 1s,1s so that i don't have things that don't upgrade later. In sets with chapel/moneylender, I'd prefer the opposite, since I want to trash the copper.

Edit: horrible omition of words that described the opposite of my point
 
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Andrew Hardin
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maxo-texas wrote:
A mine question.
I used to upgrade coppers to silvers first.

But lately, I've been going silver to gold first (so 1c,1g is the result instead of 1s,1s).

Any thoughts on this from the math wizs?

Mines are a little weak but with extra actions and the chance to get one early they often factor in my losses to miners.


Silver to Gold instead of Copper to Silver is useful if you want to increase the variability of your deck. Copper to Silver does the opposite.

Increasing variability is useful if you need to get to 8 Coin quickly but your average Coin draw is somewhat low. Decreasing variability is useful if your deck is already high enough (particularly if you are drawing 8 regularly).

Assuming you have 1 Copper, 1 Silver you end up with either:

1 Copper, 1 Gold, or 2 Silver

Imagine you marked both cards and kept track of your draws. The former case will be either 1 higher or 1 lower whenever you draw them unless you draw both together. You will draw low one time, and high the next, or vice versa.

The 1 Copper, 1 Gold deck has higher variability and stronger negative correlation.

If your average Coin/Treasure ratio is 4 then you want higher variability to get more chances to hit 5 or 6 (you trade that you get more hands with 2 or 3 instead of 4).

If your average Coin/Treasure ratio is right at 7 then you probably want higher variability (pushing 7 up to either 8 or 6), while if your deck is at 8 you want more stability (less 7 and 9 and more 8 and 8). Too much variability will push you to 9 and 5 which is not what you want.

The situation is even more extreme if you do this twice, producing 2 Copper, 2 Gold instead of 4 Silver. The second case is highly variable and in my view usually better. But I like high variability decks mixed with the Chancellor (I do this partly because my opponents don't and I win by hitting more Provinces than they do).

Because of the way I play my Treasure/Ratio is rarely high enough that I don't want the 'high/low' variability that this strategy introduces. Particularly since the low values of 2 and 3 produce lots of Cellar and Silver buys mixed with Gold.

- Lex
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Ian Kelly
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maxo-texas wrote:
A mine question.
I used to upgrade coppers to silvers first.

But lately, I've been going silver to gold first (so 1c,1g is the result instead of 1s,1s).

Any thoughts on this from the math wizs?

Mines are a little weak but with extra actions and the chance to get one early they often factor in my losses to miners.


In simulation (where the only cards bought were mine, silver, gold, and province, and the metric was the number of turns to reach 4 provinces), I have found that preferring to upgrade copper over silver is modestly faster than the reverse -- about 16.37 turns vs. 16.39 turns with a single mine. In the absence of a reason (even a mild one) to prefer the 1c,1g split I would pick 1s,1s.
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Andrew Hardin
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Peristarkawan wrote:
maxo-texas wrote:
A mine question.
I used to upgrade coppers to silvers first.

But lately, I've been going silver to gold first (so 1c,1g is the result instead of 1s,1s).

Any thoughts on this from the math wizs?

Mines are a little weak but with extra actions and the chance to get one early they often factor in my losses to miners.


In simulation (where the only cards bought were mine, silver, gold, and province, and the metric was the number of turns to reach 4 provinces), I have found that preferring to upgrade copper over silver is modestly faster than the reverse -- about 16.37 turns vs. 16.39 turns with a single mine. In the absence of a reason (even a mild one) to prefer the 1c,1g split I would pick 1s,1s.


That makes sense given the strategy is basically Big Money. Big Money relies upon high Treasure/Card ratio. With a high Treasure/Card ratio you should be trying to avoid large swings. You will also notice it is rather slow compared to better approaches. I am not sure this result would translate well in a regular game.

As an aside, do the confidence intervals on the simulation estimates overlap (at 0.02 I suspect they do)? What you probably have is no statistically significant difference but I am curious. Even if they are statistically significant is there any practical significance to this difference?

Oh, and how fast is Big Money in general (is buying the Mine even a positive play?)

- Lex
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Todd McCorkle
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My boring default strategy for the suggested starting set is:

silver/smithy (assuming a 3/4 split).

For the rest of the game, I ignore the action cards and simply...
-buy a province with 8+ money
-buy a gold with 6-7 money
-buy a silver with 3-5 money

(I'm sure this looks familiar to those who frequent these boards).

My rationale... I absolutely hate it when the only action card in my hand is a smithy and I draw nothing but action cards when I play it. It didn't matter how many villages to smithys I had, I would still draw the smithys by themselves. Best way around that seem to be to have no other action cards than the one smithy.

I also noticed that I tended to do better the fewer action cards I had. I don't know if that's because of my play style, lousy luck with card distributions, or *shrug*.

I was a little disappointed with the game when I discovered how well this works. Fortunately, it doesn't work for all setups (especially if a witch is in play), but that's a small saving grace. The Intrigue spoilers have me excited about the game again. I hope it mixes things up enough.
 
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Ian Kelly
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LexH wrote:
As an aside, do the confidence intervals on the simulation estimates overlap (at 0.02 I suspect they do)? What you probably have is no statistically significant difference but I am curious. Even if they are statistically significant is there any practical significance to this difference?


I didn't test it before, so I ran it again with 500001 trials each way.

Average 1: 16.377223
SD 1: 1.629428
Average 2: 16.398319
SD 2: 1.659916
Pooled Standard Deviation: 1.6447
Pooled DF: 1000000
95% Confidence Interval for the Difference ( -0.0276 , -0.0146 )
T-Value -6.4133
Sample 1 ≠ Sample 2: P-Value = < .00001
Sample 1 > Sample 2: P-Value = >.99999
Sample 1 < Sample 2: P-Value = < .00001

Quote:
Oh, and how fast is Big Money in general (is buying the Mine even a positive play?)


Pure Big Money, i.e. buying no kingdom cards at all, takes 16.8 turns, so yes, a single Mine is a net improvement. But you're right in suspecting that buying a single Mine is not optimal. The fastest single-card buys that I've found (not accounting for any effect the card may have on opponents) are:

Council Room at 14.56
Envoy at 14.59 (the difference is significant, and not surprising)
Chapel at 14.8
Smithy at 14.9

For comparison, ChapLab clocks in at about 14 with 1 Chapel and 2 Laboratories.
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Les Marshall
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mshapiro wrote:
After a sequence of refinements, my buddies have developed what they call the Village Eater strategy. If you've played with this set at any length you've likely seen this in action: the Eater buys and Workshops as many Villages as he can early in the game, although this is a cardinal sin according to at least one strategist on this board; s/he then buys Smithies, then 1 or 2 Mines and a pile of Markets. By the 11th or 12th hand, the Eater is reliably drawing all or nearly all of her or his deck; by the 13th or 14th hard, the Eater buys the first Province; by the 16th hand, the Eater is likely buying 2 Provinces per turn.


And some people have the audacity to claim this game is weak on theme..Sheesh! One look at the box shows how a simple abacus can win you a kingdom. cool
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I should also add that I've been playing around a lot with simulating Village-Smithy strategies, and I have yet to find one that beats 15 turns on average. But it may be that I just haven't developed a sufficiently sophisticated strategy yet.
 
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Andrew Hardin
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Peristarkawan wrote:

I didn't test it before, so I ran it again with 500001 trials each way.

Average 1: 16.377223
SD 1: 1.629428
Average 2: 16.398319
SD 2: 1.659916
Pooled Standard Deviation: 1.6447
Pooled DF: 1000000
95% Confidence Interval for the Difference ( -0.0276 , -0.0146 )
T-Value -6.4133
Sample 1 ≠ Sample 2: P-Value = < .00001
Sample 1 > Sample 2: P-Value = >.99999
Sample 1 < Sample 2: P-Value = < .00001


Interesting, so there is a very slight shift in favor of going with Copper -> Silver instead of Silver -> Gold. That really doesn't surprise me with Big Money.

1 million d.f. I don't see that often. With that many simulations your results are basically certain. The question to me is can this be extrapolated to games other than Big Money+Mine. Then again, this has hardly any practical significance. You could go with either and end up just about the same.

Your results about the +Card and Chapel strategies being so fast solo doesn't really surprise me. The interesting thing I would almost certainly rather have the Envoy than the Council Room since the Council Room would almost certainly accelerate my opponents greater than the 0.03 difference.

What I would love to see are the full distributions. The mean does have meaning but I am curious about the other aspects (skew, variance, kurtosis, .05 and .95 quantiles and such).

- Lex
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Cameron McKenzie
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LexH wrote:
(except the underpriced Chapel which is really a 5-cost card pretending to be cost 2 and the distinctly unusual Adventurer).


Adventurer gets such a bad rap, and I think it's undeserved. I think a lot of people miss one of the major advantages of the adventurer. They simply think of him as an action that gives you +coins with the number varying based on the treasure distribution of your deck and a little bit of luck.

What is often overlooked is the fact that in his search for treasure, the adventurer discards every card he comes across, which means you don't have to draw it and your deck cycles faster. He's kind of a mini-chancellor in that regard. Sure, you may discard some decent actions you don't want, but you shouldn't be playing an action heavy deck with adventurer anyway. More importantly, he discards all the victory cards he finds, which equates to +cards in future turns as you will be drawing something other than the victory cards you skipped.
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Mike Shapiro
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Thank you, Lex and Ian, for this glorious statistical work! Though it is good for me to remember that these are solo numbers, and that the game's depth comes from thinking through the interplay of these combinations against the unaccounted-for opponent's deck.

This might get me thrown off the nearest cliff, but seeing these numbers begins to put me in the mind of understanding Dominion as a modified poker: played at levels higher than I can fathom, both games combine advanced statistics with psychology and game theory. Seeing the numbers doesn't flatten the game so much as reveal its possibilities. I love this stuff.
 
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Andrew Hardin
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MasterDinadan wrote:
LexH wrote:
(except the underpriced Chapel which is really a 5-cost card pretending to be cost 2 and the distinctly unusual Adventurer).


Adventurer gets such a bad rap, and I think it's undeserved. I think a lot of people miss one of the major advantages of the adventurer. They simply think of him as an action that gives you +coins with the number varying based on the treasure distribution of your deck and a little bit of luck.

What is often overlooked is the fact that in his search for treasure, the adventurer discards every card he comes across, which means you don't have to draw it and your deck cycles faster. He's kind of a mini-chancellor in that regard. Sure, you may discard some decent actions you don't want, but you shouldn't be playing an action heavy deck with adventurer anyway. More importantly, he discards all the victory cards he finds, which equates to +cards in future turns as you will be drawing something other than the victory cards you skipped.


I rather the like the Adventurer, it is just a distinctly unusual card in the 25, costing the same as Gold. However, since I have so few Action Cards in my deck more often than not a Laboratory fits my needs better.

But, in a Gardens or Witch game the Adventurer can be my best friend.

- Lex
 
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Andrew Hardin
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mshapiro wrote:
Thank you, Lex and Ian, for this glorious statistical work! Though it is good for me to remember that these are solo numbers, and that the game's depth comes from thinking through the interplay of these combinations against the unaccounted-for opponent's deck.


All credit goes to Ian for the work he has done. I wish I had the time to really do this kind of work myself.

I will say that some care should be taken to recognize that what works for Big Money does not always work for strategies that combo. But I still firmly believe that all Dominion strategy should be based upon at least knowing you can beat Big Money.

The one thing Ian is really showing me is that a good strategy should be looking to seal a win by around Turn 14. This just points out how fast Dominion games are and why some of the slower strategies are a mess.

Quote:

This might get me thrown off the nearest cliff, but seeing these numbers begins to put me in the mind of understanding Dominion as a modified poker: played at levels higher than I can fathom, both games combine advanced statistics with psychology and game theory. Seeing the numbers doesn't flatten the game so much as reveal its possibilities. I love this stuff.


You have hit what I most enjoy about this game. And unlike Poker you don't have to sit around and fold 85% of the time to get to the good parts.

- Lex
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