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SettingFraming

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Opening Probabilities: A Study
« on: January 25, 2017, 05:37:18 pm »
+36

Feel free to skip around this article. I know it's long with lots of text, and it's perfectly fine if you just go to whatever section(s) you find interesting.

An Introduction

Dominion is, at it's core, a game of probabilities. This is something we know and love, especially because the probabilities of Dominion are often so vast that they get in touch with our pure thought-stuff, and reckon with the limits of our reasoning.
   
But there is also much that we can know about Dominion, and that is what this article is about. Specifically, this article is going to be dealing with the first few turns of Dominion in very concise, exact ways. While there is much more to Dominion than the opening turns, they are often the most important ones and largely set the pace of the game. Also, those turns are rather simple, and you should nearly exactly know what chances you are giving yourself. Hereís what you need to know when considering openings, from both hitting numbers and trashing cards standpoints.

The Economy is Thriving
   
Weíre a salty bunch, Dominion players. We often get super mad if we do something like donít hit 5. But how unlucky are we? Iíll start this article gently with some basic openings, and compare the differences. The first table is the probability of hitting a number at least once on turns 3-4, while the second table shows you the probability you have of hitting at least a number on both turns 3 and 4.

Openings and $$4$5$6$7
Poacher/Silver100%91.8%37.8%6.3%
Silver/Silver100%91.2%42.5%8.8%

Openings and $$3/3$4/4$5/5
Poacher/Silver94.7%62.6%8.2%
Silver/Silver94.7%64.6%14.9%


The difference between a Peddler Variant (in this case Poacher) and Silver in the opening may surprise you, mostly because thereís not much of one. We all know that opening a cantrip card is great for cycling, and that cycling is great, but also youíre really not at all harming your chances of hitting a number by doing so. By opening Tournament/Silver your odds of hitting 5 are almost exactly the same, while your odds of hitting 6 or 7 are very slightly lower. Maybe if your plan is to do something like Hireling-Big Money then Silver/Silver is defensible since you have a 5% greater chance to hit $6, but other than that youíre always going to want the Peddler variant.

Playing or Praying the Chapel


Alright, letís move on to some less obvious stuff. One of the angriest moments of Dominion is if your Chapel misses the shuffle. So, whatís the difference between opening Cantrip/Chapel, and Silver/Chapel?

Openings and Cards Trashed034
Chapel/Cantrip9%0%91%
Chapel/Silver16.6%30.3%53.1%

It turns out thereís a pretty significant difference between average number of cards trashed with Chapel when opening with a Silver (or other stop card) and a cantrip. Namely, Chapel/Silver trashes on average 3.03 cards while Chapel/Cantrip trashes 3.64.

The reason for this is two-fold, the first being that having a Silver in your deck instead of a cantrip increases the chance that Chapel misses the shuffle by more than 7%, and the second being that the Silver will be sitting (nearly) uselessly in a collision with your Chapel 30.3% of the time. With strong trashing, the importance of opening a cantrip may be less about the presence of the effect of the cantrip, and more about just getting the heck out of the way of whatever else youíre doing.

Going Big or Going Home with Double Terminals

Another question often faced in the opening is pretty straightforward: Should I open double terminal? Well in order to answer that, you need to know how good the terminals are (i.e. is it worth risking collision), but also the chances of actually getting to play them. The table below shows %íes of the time that you get to play both cards or if you just get to play one of them, either because they collided or one missed the shuffle.

Terminal Plays012
%1.5%60.6%37.8%

A couple main points from this table: Both your cards will miss the shuffle 1.5% of the time. This is definitely something to get salty about, and often means you just lose. You only get to play both cards on turns 3/4 37.8% of the time, though of course it can be really, really good if you get to do so. Finally, you get to play only one of the cards 60.6% of the time. So you shouldnít be expecting to be able to play both cards if you open double terminal, though it can be a very real possibility.

Steward: A Double Terminal Case Study

Double Steward is one of those hot topic debates that keeps us up at night. Okay maybe not so much, but anyways, I want everyone to sleep well. There are a few key questions with Double Steward that you need to ask. How important is being thin? Am I fine with the $2ís? Do I need to hit numbers any time soon? How good are two Stewards in my deck long term?

Sometimes the answers to those questions are marginal, and then itís really important to know what double Steward actually does for you. Here are some tables detailing (a) how many cards Steward actually lets you trash, assuming you always choose to trash, and (b) what is the difference between opening Steward/Steward and Steward/Silver in terms of hitting numbers (again, assuming always trashing).

Openings and Cards Trashed024
Steward/Steward1.5%60.6%37.8%
Steward/Silver16.7%83.3%0%

A couple quick notes on the above table:
  • You should see the usefulness of knowing the double terminal opening odds for Steward/Steward.
  • Steward/Steward trashes 2.72 cards on average, while Steward/Silver trashes only 1.6. Recall that Chapel/Silver trashes on average 3.03 cards, meaning that Steward/Steward is remarkably similar to Chapel/Silver at thinning rates.

And hereís the economy table.

Openings and $$2$3$4$5$6
Steward/Steward100%57.1%31.9%5.3%0%
Steward/Silver100%98.7%76%40.7%8.8%

Openings and $$2/2$3/3$4/4$5/5
Steward/Steward66.1%1.3%0%0%
Steward/Silver98.2%42.8%7.1%0%

In terms of economy, itís not even close. Steward/Steward means that you arenít going to be able to buy high-$ cards for a while, since even your chance of hitting even just 3 just once before the next shuffle is only 57.1%, and if you were planning on picking up a $4 village before shuffling, well good luck with that since you only have a 31.9% chance of hitting $4. Steward/Silver is much, much stronger at nearly everything, providing reasonable chances of hitting most numbers, though of course being weaker than a straight-up double economy opening. Of course, one can choose to use their second Steward for economy, but that is a dubious plan for risking the double terminal. One of the main benefits of opening double Steward may in fact be in reducing the chance that your trasher misses the shuffle, giving yourself a 98.5% chance to trash at least once instead of only an 83.3% chance.

An Outroduction

Of course there is much more to playing Dominion than just knowing the percentages. You need to build a deck, and you need to know where youíre going. But knowing the percentages can help inform you and help the decisions you make be just a little less in the dark. I plan on doing more follow-ups to this article, some with more specific or in-depth focuses, or even just different concepts such as durations in the opening.

Until then, I hope we can all keep learning, and be just be a little less bad at Dominion.

Addendum: It's been brought to my attention that it would be good to include this, so here's a link to an old Wandering Winder article that shows probabilities of hitting price points for a variety of economy-based openings: https://wanderingwindergames.blogspot.com/2015/05/dominion-opening-theory-money-matters.html
« Last Edit: January 26, 2017, 12:18:06 pm by SettingFraming »
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shark_bait

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Re: Opening Probabilities: A Study
« Reply #1 on: January 26, 2017, 11:59:46 am »
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Here are some of my suggestions.

Bonfire - Maybe something like average number of cards in deck at a given turn compared to the Chapel and Steward variations.

Baker - In the opening, how much does having a single coin token smooth the probability distribution regarding 5/2 and 4/3?  If you can save the token for turns 3-4, how does this impact the percentages for hitting certain price points or combinations? 

Borrow - Similar to Baker for how this impacts the opening.  What kind of trade off is expected regarding money for the Coin/Card trade?  Is there a quantifiable difference that can justify the loss in tempo?


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Dingan

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Re: Opening Probabilities: A Study
« Reply #2 on: January 26, 2017, 03:41:02 pm »
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By any chance can we get one of those cool probability table thingies for Masquerade (Masq/Silver vs. Masq/Masq vs Masq/cantrip vs Masq/nothing etc.)?
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SettingFraming

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Re: Opening Probabilities: A Study
« Reply #3 on: January 26, 2017, 04:48:38 pm »
+1

By any chance can we get one of those cool probability table thingies for Masquerade (Masq/Silver vs. Masq/Masq vs Masq/cantrip vs Masq/nothing etc.)?

You betcha. It will be in the next article.

Here are some of my suggestions.

Bonfire - Maybe something like average number of cards in deck at a given turn compared to the Chapel and Steward variations.

Baker - In the opening, how much does having a single coin token smooth the probability distribution regarding 5/2 and 4/3?  If you can save the token for turns 3-4, how does this impact the percentages for hitting certain price points or combinations? 

Borrow - Similar to Baker for how this impacts the opening.  What kind of trade off is expected regarding money for the Coin/Card trade?  Is there a quantifiable difference that can justify the loss in tempo?

Bonfire: You betcha. Baker/Borrow I'll do, but I'll save them for a separate article that's more about decision making in those scenarios (and maybe definitely I'll also include Save!).

I'm going to be releasing a follow-up to this with a big batch of common openings shortly, which means probably tomorrow. After that one, I have three to four more follow-ups planned, all of which I hope to come up with more clever names.
  • Durations in the Open
  • Everything about Potion
  • Handling Flexibility
  • Weak Trashing
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UmbrageOfSnow

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Re: Opening Probabilities: A Study
« Reply #4 on: January 26, 2017, 09:32:04 pm »
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Why are turn 3/4 terminal collision and 1 terminal missing the shuffle while you play the other on either turn 3 or 4 grouped together in the same percentage?  Those seem like potentially very different cases.
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markusin

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Re: Opening Probabilities: A Study
« Reply #5 on: January 26, 2017, 10:19:45 pm »
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Why are turn 3/4 terminal collision and 1 terminal missing the shuffle while you play the other on either turn 3 or 4 grouped together in the same percentage?  Those seem like potentially very different cases.

I found that odd too, but figured the relation between them was that you only managed to play one terminal in turns 3/4. If that terminal is your Steward, that means you only trashed two cards before the second reshuffle instead of four or zero.

The way the two cases affect your buying power is different, but that is already covered in the money group.
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luser

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Re: Opening Probabilities: A Study
« Reply #6 on: January 27, 2017, 01:08:43 am »
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Steward stuff there is nonsense. Double steward is great because it gives you more options to trash. Economywise only way silver is better is when it collides with steward then you could get 3 cost instead nothing for steward collision. It is more likely to hit 5 and trash with double steward than with silver as you could use t3 steward as silver if it hits five and trash estates on t4 or trash estates at t3 and use second one as silver on turn 4.
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allanfieldhouse

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Re: Opening Probabilities: A Study
« Reply #7 on: January 28, 2017, 11:44:59 am »
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Steward stuff there is nonsense. Double steward is great because it gives you more options to trash. Economywise only way silver is better is when it collides with steward then you could get 3 cost instead nothing for steward collision. It is more likely to hit 5 and trash with double steward than with silver as you could use t3 steward as silver if it hits five and trash estates on t4 or trash estates at t3 and use second one as silver on turn 4.

I mean...he gave the percentages of hitting 5 with each opening. Are you saying his percentages are wrong?

After reading both posts again, I think the OP is assuming you're always using Steward for trashing and never for the $. And you're saying that sometimes you should trash with one and get money with the other? If you're planning on doing that, shouldn't you just get the Silver in the first place?
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Titandrake

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Re: Opening Probabilities: A Study
« Reply #8 on: January 28, 2017, 10:06:18 pm »
+1

Steward stuff there is nonsense. Double steward is great because it gives you more options to trash. Economywise only way silver is better is when it collides with steward then you could get 3 cost instead nothing for steward collision. It is more likely to hit 5 and trash with double steward than with silver as you could use t3 steward as silver if it hits five and trash estates on t4 or trash estates at t3 and use second one as silver on turn 4.

I mean...he gave the percentages of hitting 5 with each opening. Are you saying his percentages are wrong?

After reading both posts again, I think the OP is assuming you're always using Steward for trashing and never for the $. And you're saying that sometimes you should trash with one and get money with the other? If you're planning on doing that, shouldn't you just get the Silver in the first place?

Sometimes they differ, even if you always play 1 Steward as +$2. On turn 3 you draw Steward + 2 Copper + 2 Estate. On turn 4 you draw Steward + 3 Copper + 1 Estate. Now imagine the first Steward was a Silver instead.

I don't think the Steward stuff is nonsense. It's worth thinking about. But in my experience the main factor between Steward/Steward and Steward/Silver is how much you care about the trashing, and how worried you are about having the extra terminal so early in the game, then making a judgment call based on the tradeoffs.
« Last Edit: January 28, 2017, 10:07:56 pm by Titandrake »
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weety4

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Re: Opening Probabilities: A Study
« Reply #9 on: February 01, 2017, 02:55:04 am »
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But in my experience the main factor between Steward/Steward and Steward/Silver is how much you care about the trashing, and how worried you are about having the extra terminal so early in the game, then making a judgment call based on the tradeoffs.
Totally agree. It is mainly a risk decision for the first shuffle and beyond that village availability is a factor that influences your decision.
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timchen

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Re: Opening Probabilities: A Study
« Reply #10 on: February 02, 2017, 10:54:03 am »
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Yeah, opening double steward and hope to use one of them to spike $5 or $6 is definitely something that should be considered. It is an optionality that cannot be worth less than nothing. And comparing this to opening silver/steward one can see that it wins against steward/silver when silver is in the worse hand and loses when the two stewards collide (not including the trashing benefit).

If the goal is to spike a $5 (along with trashing ofc) then I think steward/steward is the way to go. If spiking a $6 is desired then steward/silver is probably better. Much of the probability of $6 is from silver colliding with steward.
 
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SettingFraming

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Re: Opening Probabilities: A Study
« Reply #11 on: February 09, 2017, 10:27:32 am »
+6

I know I said I was going to do more articles on this, and while I still plan on it I've had a lot of other stuff going so my priorities are a bit jumbled. Instead of delaying until I get an entire article written however, I thought the Masquerade numbers were interesting enough (and the fact that they were requested) to go ahead and post here.

Attending the Masquerade

Openings and $$4$5$6$7
Masquerade/Silver100%93.6%35.0%0%
Masquerade/Masquerade97.2%35%6.1%0%
Masquerade/Poacher99.5%75.2%28.7%2.9%
Masquerade/Warehouse96.3%45.9%4.4%0%
Masquerade/Nothing100%59%6.7%0%

Openings and $$3/3$4/4$5/5
Masquerade/Silver94.7%61.9%6.9%
Masquerade/Masquerade90.8%37.5%8.4%
Masquerade/Poacher94.7%59.2%11.9%
Masquerade/Warehouse94.9%59.5%5.8%
Masquerade/Nothing94.9%59.5%5.8%


Cards Trashed012Average
Masquerade/Silver16.6%83.4%0.0%0.83
Masquerade/Masquerade1.5%60.7%37.8%1.36
Masquerade/Poacher9.1%85.5%5.4%0.96
Masquerade/Warehouse1.5%79.7%18.8%1.17
Masquerade/Nothing9.1%84.4%6.5%0.97

Well, those are a lot of numbers. The conclusions are really for you guys, but here are some quick take-aways:
  • Double Masquerade is a huge downgrade in economy, and only offers a relatively modest uptick in trashing. It can still be correct, but you have to be prepared for the large economic downside.
  • Masq/Silver offers significantly better economy than Masq/Poacher, and the trashing benefit of Masq/Poacher is minimal. So don't just auto-buy the Peddler variant.
  • Masq/Warehouse seems like a reasonable opening, as it not only really cycles you, but it also offers better economy than Masq/Masq.

Some trivia on the above:
  • The fact that you're passing from a 7 card hand and receiving from a 5 card hand makes a noticeable, though small difference in Masq's economy. You're a few percentage points more likely to pass an estate and receive a copper than vice versa.
  • When discarding (such as with Warehouse), Estates are discarded first, but one Estate is left in hand if Masquerade is about to be played. The above numbers aren't smart enough to discard all of the Estates in hand if an Estate is likely to be drawn in the next two cards.
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