Opening Probabilities: A Study

This article was adapted from a forum post, originally by SettingFraming, aka breppert.

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. Here’s what you need to know when considering openings, from both hitting numbers and trashing cards standpoints.

The Economy is Thriving

We’re a salty bunch, Dominion players. We often get super mad if we do something like don’t hit 5. But how unlucky are we? I’ll 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/Silver 100% 91.8% 37.8% 6.3%
Silver/Silver 100% 91.2% 42.5% 8.8%

 

Openings and $ $3/3 $4/4 $5/5
Poacher/Silver 94.7% 62.6% 8.2%
Silver/Silver 94.7% 64.6% 14.9%

The difference between a Peddler Variant (in this case Poacher) and Silver in the opening may surprise you, mostly because there’s not much of one. We all know that opening a cantrip card is great for cycling, and that cycling is great, but also you’re 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 you’re always going to want the Peddler variant.

Playing or Praying the Chapel

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

Openings and Cards Trashed 0 3 4
Chapel/Cantrip 9% 0% 91%
Chapel/Silver 16.6% 30.3% 53.1%

It turns out there’s 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 you’re 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 %’es 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 Plays 0 1 2
% 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 shouldn’t 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 $2’s? 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 it’s 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 Trashed 0 2 4
Steward/Steward 1.5% 60.6% 37.8%
Steward/Silver 16.7% 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 here’s the economy table.

Openings and $
$2 $3 $4 $5 $6
Steward/Steward
100% 57.1% 31.9% 5.3% 0%
Steward/Silver
100% 98.7% 76% 40.7% 8.8%

 

Openings and $
$2/2 $3/3 $4/4 $5/5
Steward/Steward
66.1% 1.3% 0% 0%
Steward/Silver
98.2% 42.8% 7.1% 0%

In terms of economy, it’s not even close. Steward/Steward means that you aren’t 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 you’re 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 a Wandering Winder article that shows probabilities of hitting price points for a variety of economy-based openings.

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