How the Poisson Distribution Can Help Predict Video Game Combat: A Toy Example
The Poisson distribution is a powerful tool that allows us to understand the probabilities associated with X number of events occurring over a given period, provided we know the average number of times the event occurred over the period.
The distribution has been shown to be applicable to a variety of phenomena, including understanding:
- The number of network failures per day.
- The number of hungry patrons expected to visit a fast food restaurant in an hour.
- The number of customers making a complaint about a service in a month.
In this post, we’ll develop a toy example of utilising the Poisson distribution to help us better understand an important element of many video games; combat.
Having played a fair amount of Elder Scrolls Online (ESO) recently, I wanted to explore how the Poisson distribution could be used to understand the probabilities of X number of ‘hits’ occurring during a combat event.
This is simply a toy example to further explore and explain the Poisson distribution — but it still proves useful nonetheless.
Let’s set up some definitions for our example:
- A “combat event» refers to a player fighting an NPC, e.g. a wolf, vampire, guard, with a start and event time (typically resulting in the death of the NPC or player).
- A “hit” refers to a weapon coming into contact with either the NPC or the player, this can be any weapon in-game, so a bow, sword, and so on.
At this point, it’s also important for us to set up the conditions for the Poisson distribution, it is applicable when the following holds:
- An event (or hit, in our instance) can occur any number of times in a period.
- Events occur independently; if one event occurs then it doesn’t impact the probability of an event occurring in the same period (we’ll ignore power hits which can daze an NPC or player, in this instance).
- The rate of occurrence does not change due to time.
- The probability of the event occurring is proportional to the length of time being measured.
The Poisson distribution can be defined as follows:
With λ referring to the average of X and k being the event we want to model.
Therefore, if we assume in our chosen video game, ESO, that the average amount of “hits” in a combat session is 20, and we want to find the probability of achieving 15, this can be calculated as follows:
λ = 20, k = 15
We can calculate probabilities such as the chance of achieving ’5 or less’ hits in a combat session by using the rule of sum for probabilities.
For example, the probability of achieving 5 or less hits in a session would be:
Closing Remarks
Of course, our example from ESO relies on combat sessions being driven by chance (and not ability), or if by ability, that the NPC combat ability scales sufficiently alongside the player, as well as ignoring hits which have the ability to daze another fighter. However, it is certainly a fun thought experiment to explore Poisson distributions through.
