This is my first post in what will be a continuing series on math in games – board games, video games, tabletop games… mind games?

I’m very excited about a board game called Gloomhaven. It’s awesome! Buy it. Play it with friends. Seriously.

While Gloomhaven is about your **unique** **story**, as your group makes choices and encounters foes of all sorts, the combat (which I also love) that drives the game is based on math. So, we can analyze character options and tricky decisions with numbers! Yay, numbers! I’m a physics person, but I can love math too.

The first major decision you might make is when you either

- level up, or
- gain three precious checkmarks

## Which perk do I choose?

Ah, so many options! These perks impact your attack modifier deck, which you draw from for each attack. Do you reduce the number of possible bad outcomes? Add one great outcome? What about rolling modifiers?

If you suffer from “analysis paralysis” on regular turns, then this may be a nightmare. Luckily, a little number-crunching can go a long way here.

## Math time

Let’s assume you start with the standard deck of of 20 cards (because the rules say you do), which looks like this:

Then it’s simple (maybe) to determine the odds of each result and the average damage of your “Attack 3” action.

Here’s the distribution of the standard attack modifier deck:

This shows that your “Attack 3” action has a 7.14% chance of missing, the same chance of getting “x2” and doing 6 damage, but is usually (76.19%) going to fall within the 2-4 damage range.

*Whoa, wait. *There are 20 cards, and one of those is a miss card. Why isn’t the miss chance 1/20 = 5%… ??? Good question, thoughtful reader! I will leave the mystery of this to be explained in my next post. 🙂

### How perks change your damage distribution

Now that we know our initial damage “profile”, we should investigate which perk gives the largest bump up from that starting point.

First, we’ll agree to ignore a few things beyond the scope of our study:

- Elemental infusions
- Status effects (e.g. Immobilize, Muddle)
- Forced movement (e.g. Push)
- Multiple targets (for now – may look at this in another post)

These can all improve your damage, experience, or survivability. However, it’s hard to quantify and compare these benefits. We will only look at cards that directly and immediately impact the damage of an attack.

**Remove four “+0” cards**

You can see above that removing a bunch of “+0” cards does what you expect. It doesn’t change the average, but it now becomes way less likely that you will simply do your standard damage.

People may choose this if they simply like variance and find “+0” cards boring. Or, later in the game, you may choose this perk to increase the odds of landing those other awesome cards you have been adding.

**Replace a “-1” card with a “+1” card**

Here’s a more significant change. The previous distributions all centered around 3 damage, but this one increases your average damage, as expected. The new average damage is 3.095. It basically does what you think it would.

**Add a “-2” card and two “+2” cards**

This also does what you think it would, but what is interesting is that the average damage is 3.083, slightly (very slightly) less than the previous perk. Of course, this deck would have better odds to punch through heavy shielding.

**Remove negative item effects**

To evaluate this perk, let’s assume you are wearing a fairly standard armor available at the beginning of the game that could add two “-1” cards to your deck. . How much would that hurt you compared to the standard deck?

The average damage in this case is 2.913, so on average you’re losing about 0.087 to gain that armor. Taking this perk would return you to the standard with average damage of 3.

Actually, two other perks would get you the same average damage if you are wearing this armor, so you just have to decide which distribution you like better. Here they are:

Same average damage, but still pretty different decks!

This is just one example of how decks should not be compared by simple averages. After all, a deck of all “+0” cards has the same average damage as one an equal number of only “+2” and “-2” cards, but they would play *very* differently!

### Summary

Which perk you should choose will depend on which enemies you expect to fight, what goals you have in your attacks, etc. However, some simple math can give you an insight into which deck might suit your play-style best.

This post is just a summary of simple stuff that shouldn’t be too surprising, but may be a good reference. Remember that the results could change if your attack value is less than 3, or if you are engaging shielded enemies, or have advantage… yeah, lots of stuff can change. Hopefully this is a good basic starting point.

Next post I’ll dive into more interesting math, how reshuffling on Miss and x2 cards can impact decks, and discuss the interaction of advantage with our decks in Gloomhaven.

This is a wonderful analysis, I love that you’re working in terms of probability distributions instead of simple point values. I see this is part of a larger series that I’m delighted to dive into and I hope you share all the gritty details of how you calculated these. Thanks!

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