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Video Games and Mathematics

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In this post I'm sharing some of my thoughts about this subject after having made thinking about mathematical problems that come up in the context of video games a hobby of mine.

I have seen some interesting things. A situation from Borderlands 3 about a gun where each bullet has a chance to cause a damage over time effect was very similar to a situation about a caster in World of Warcraft where each cast had a probability to speed up casting for a set duration, and where both of these effects can be refreshed while active. What would the average casts per second be? Or the average damage per second in the case of Borderlands 3?

There's also a problem in Fire Emblem about the probability of winning a back and forth exchange between your unit and an enemy unit which shares in common with a problem about the chances of victory in a Monster Hunter World hunt. What's the probability in either case? The Monster Hunter World problem requires some major assumptions initially, while the other three have exact solutions right away.

Markov chains seem to be a powerhouse for modeling many video game scenarios. That's the common link between all four problems above, and countless others. I've seen some other interesting kinds of problems come up while thinking about video game phenomena, including topics like tier lists, pathing, strategy optimization, geometrical configurations, and onwards.

A finite markov chain consists of a finite number of possible states, where each state has some probability to move to each other state (and itself) that does not change over time. I can answer any questions about how this would be used to model any of the four problems above.

With a finite markov chain you can answer questions about how often states are visited, what the probability of going from one state to another after a given number of steps is, and how long on average it will take for one state to reach another.

This seems to me to be one of the core concepts for understanding video game phenomena and solving related problems. Player behavior is another concept perhaps. Learning algorithms could be useful sometimes. Some statistical techniques are useful. A hill climbing algorithm came up when I tried to solve a problem about Hearthstone and Magic the Gathering. There are undoubtedly many others.

tl;dr

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( Mathematics is often used to solve practical gameplay problems, but it can also be used as a tool for curiosity in its own right. I give an example of one concept that comes up a lot and can be used to solve problems that seem complicated. )

Discussion

How likely is it for a practical game problem to be solved by some mathematical technique at a level meaningfully better than a solution from the combined wisdom and intuition of players?

What is a mathematical concept that is useful for solving game problems?

What is a game phenomena that inspires an interesting mathematical concept?

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So I have been doing some math and programming related to this.

More Mathematical Details With an Example (pdf)

https://github.com/grassyseas/Markov-Chains/blob/master/AGameRelatedMarkovChain.pdf

Algorithms Written in Python for Basic Markov Chain Calculations

https://github.com/grassyseas/Markov-Chains/blob/master/MarkovCode.py

If this happens to interest you.

An Application for Monster Hunter World

This was made before I realized the connection to Markov Chains, and since I've ended up solving the same problem in four different ways and am glad to say they all give the same outputs. The idea was to assess how effective a defensive skill was in an action game by creating a model based on a real player's gameplay and then running lots of simulations with a pRNG to get an averaged out result of victories vs. losses.

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