I want to first thank Alex Botts for inspiring this article and for giving me some ideas about how to improve some of my articles with new stats to look at. His ideas will really let me dig a little deeper in my analysis. If I’m ever on Double Dodge for any reason I will have only nice things to say to him during the rapid fire questions.
Today I want to talk about how I get the values for the expected damage table. I use an expected value calculation to find the expected damage, and even the percent chance of special playbook results. So lets look at what expected value calculations are first.
Calculating expected values is fairly straight forward, in theory. For every possibility you take the value of that possibility and multiply it by the probability of getting that possibility, you then add up all those values to get the expected value.
Let’s take a simple example: you have a friend that will pay you $1 if you flip a heads on a quarter. So in this set up there are 2 possibilities, heads or tails. If you flip heads you get $1, if you flip tails you get $0. The chance of getting heads or tails is 1/2 (1 in 2, or 50%). So, the calculation: probability of getting heads times the value of getting heads plus the probability of getting tails times the value of getting tails, or .5*1+.5*0=.5 or $0.50. You can expect to get $0.50 from this situation. That is not to say that you will get $0.50, that’s not even possible in this situation. What it is saying is that if you did this same situation, the average of the money you will get is $0.50.
It gets more complicated when you are doing things like rolling multiple dice and picking results from a table. So let’s look at an example that has to do with guild ball. Let’s say you have a one die character play against a model with 5 defense. A straight up probability calculation would say that 2 out of 6 results would be a success, or succeed 1/3 of the time. Because their are only two results (hit or not) it is easy to calculate this way, but lets check it with the expected value. We need to take the possibilities (each die side) and find the probability of each along with the value. I’ll call the die not getting a 5 or greater 0 (a failure) and getting a 5 or greater 1 (a success). From 1 to 6, the calculation goes: 1/6*0+1/6*0+1/6*0+1/6*0+1/6*1+1/6*1 which reduces to 1/6+1/6=2/6=1/3. So the same result. In this example this is not easier than doing a straight up probability calculation, but with how the playbooks work, this is the best way to get an average for damage.
The damage varies as we go up the playbook, and the dice probabilities for each result are harder to calculate. Let’s take getting one hit against a player with 4 defense with a player with 6 TAC and 1 damage on the first column as an example of part of a problem. To get exactly one result, 1 of the 6 dice rolled has to be a 4 of higher. The probability of getting 4 or better is 1/2, but we have to multiply that by the probability of all the rest of the dice getting below 4, 1/2 each for a total of (1/2)^5 or 1/32. Those two numbers multiplied together give us 1/64, but we aren’t even done with the probability yet. We then have to multiply that results by the combinations of getting that one die. In this problem there are 6 different combinations for that one die to be 4 or greater, each of the different dice. So now we are at 1/64*6, but we have yet to multiply it by the value of this, which is the 1 damage we will do. Now you have 1/64*6*1 or 3/32. That is just the 1 die hits results, we would then have to calculate value for 2 dice hitting, 3 dice hitting and so forth up to 6. With different numbers of combinations for that result (for example, 2 dice hitting would have 15 different combinations of dice getting hits). After calculating all that we would then need to add it up to get the final expected value. Then I have to factor in armor when that applies, but that is just an adjustment in what their playbooks look like. In this example I would create a new “column” in front of the playbook with 0 damage before calculating. It makes sure that 1 successful die is always ignored, mathematically.
Lucky for me, technology can help. I use excel to do my calculations. I have created formulas that do the combinations calculation (there is a function for that) and multiplies it by the probability of each damage result and the damage it would do there and adds it all up for me. I even use it to figure out the probability of getting something like a Knock Down from the playbook, or triggering a character play, by assigning 1 to any number of hits that I could pick it and 0 to any results too low. Because I have this set up already I can easily look at different things by changing the values of the playbook. One thing I’ll be doing along with the damage results is looking at the momentum you get from picking damage results. That should help compare people on more than just the level of how much damage they can do.
After figuring out the expected damage for each defense/armor combo in the game I do a weighted average of those values for similar defense/armor groups. Like 2/3,3/2,4/1, and 5/0 all have similar expected damage because 1 defense and 1 armor are close enough in probabilities to simplify this so their are fewer numbers to look at when I make my tables.
Full example: Brisket (poster girl of Guild Ball) attacks an enemy model with 5 defense and no armor. Her playbook damage potential is 1,2,2,2 and she has 4 TAC. I’m going to use the google sheets syntax for the formula. combin(a,b) is a function for finding the combination of b objects out of a total objects (order not mattering for the b objects). So, for this problem, combin(4,2) is the number of different ways we can get 2 success out of 4 dice. Here is the formula:
In each of those terms it is the combination of 4 dice with the number of success (0 in the first term) times the probability of getting a success (1/3) raised to the number of success (0 in the first term) times the probability of getting a failure (2/3) raised to the number of failures (4 in the first term), this all gives the probability -or chance- of getting that number of successes, times the damage with that many successes. In this case the expected damage is 1.21 damage.