Saturday, April 7, 2018

Easing curves for game balance

To my casual readers, you may skip to the below section on Final Thoughts if the math here seems a bit intimidating.

For Unity you can download my Easings.cs script that borrows from tween.js on GitHub. A delegate for exact choice of easing function can be passed into your custom scripts.

Lerp and easing functions

I have been using easing functions extensively for many years, usually for animation. Easing functions are just curved functions for values $t$ from 0 to 1.

Linear is boring, but sometimes necessary to interpolation. Meaning, a value changes gradually - neither speeding up, nor slowing down as time $t$ goes from 0 to 1. Linear interpolation $lerp$ is a convenient function to compute this linear value:

$lerp(x_0, x_1, t) = x_0+(x_1-x_0)*t$.

Where $x_0$ is the initial value and $x_1$ is the final value as $t$ goes from 0 to 1. These values can be colors, vector positions, numbers, etc.

Easing functions, from

To understand these functions, just choose one and slide your finger from the left and gradually to the right. If the point on the line is closer to the bottom horizontal line, then it is closer to the initial value. If it is closer to the top horizontal line then it is closer to the final value. Some functions can even go beyond the bounds of the initial and final values.

If we have a fireball in a game, then perhaps we might want to have its movement towards its target begin slowly, and then pick up speed until it explodes. In the second row of the figure above, are cubic functions. Notice how $easeInCubic$ would be very suitable for the fireball's movement. On the other hand, $easeOutCubic$, may seem a bit strange, where the fireball shoots quickly and then seemingly slows down before it explodes.

Game Balance

I use these easing functions extensively for game balance as well. In my game, weapons can be upgraded from ranks 1 to 10. At rank 1 it is the weakest, and at rank 10 it is the strongest and does the most damage. The easiest solution would be to just write the damage values manually.

Morning Star

For example, the damage of a morning star for each of the 10 ranks could be manually coded like this:
\[[1, 3, 5, 18, 13, 23, 50, 83, 88, 92]\]
But this gets confusing! It is too many numbers to work with. Plus, it is hard to immediately see the difference between each rank. Notice how the damage can go from 50 to 83, which is a 63% increase. That is massive and throws off the balance of the game. Also the damage at rank 4 is greater than rank 5.. oops! These errors can occur easily if we write the damage values manually.

A generated graph of the damage of the morning star quickly illustrates the flaws.

We need a better solution...

In my game, weapons have an initial damage of $d_0$ at rank 1. In addition, a multiplier that determines the final value $dmul$, where for example a value of 4 would mean the damage at maximum rank would be $d_1 = d_0*dmul$, four times the initial starting value. Easy! The damage for the morning star now can be specified by just two values!

So, let's give it an initially large damage $d_0$ value of 42, but a tiny $dmul$ of 1.25, so fully upgrading it only provides an extra 25% damage increase.

The linear damage of the morning star for each rank would be $lerp(d_0, d_1, t)$ where $t$ is:

$t = (rank - 1)/(maxrank - 1).$

The linear damage increase is a bit unfair because a weapon casually upgraded to rank 2, would completely outshine a new weapon at rank 1. The game balance would feel off. In addition, upgrading from ranks 9 to 10 would provide the same damage increase as upgrading from ranks 1 to 2.

$easeInCubic$ looks like a good choice. So, the revised damage of the morning star would be:

$lerp(d_0, d_1, easeInCubic(t))$.

Similar to how damage is calculated, the cost of upgrading needs to be considered as well. Suppose we use the same curve, $easeInCubic$. This would not be balanced because it would be too easy to upgrade anything to around rank 7!

I think $easeInOutCubic$ would be an interesting choice for the cost of upgrades:

The cost for sequential upgrades always increases, but ranks 4 to 6 would have the largest increases in cost. This would be suitable because most weapons could casually be upgraded to around rank 4 until the costs get awfully pricey. Ranks 7 to 10 are where the costs are very high, but would be very rewarding due to the drastic increase in damage; there is an incentive to max it out.

Final thoughts

Balancing a game is a continued struggle. As written above, I have chosen an approach to help adjust the difficulty of my current game in order to feel balanced and rewarding to play. In-game currency is used to upgrade weapons from ranks 1 to 10. Here is the ideal mindset a player should have while upgrading these weapons:
  • Ranks 1 to 4: "Oh, these were easy upgrades. It feels like I am rich. I upgraded everything to rank 4! I like the slight increase in damage and I am not certain which weapons are my favorites. I just found some new weapons that still look powerful even at rank 1, so I will try those out too."
  • Ranks 4 to 7: "Woh, these upgrades are starting to get expensive! I must be more careful in how I invest my money. There does not seem to be a noticeable improvement on damage by upgrading these, but the enemies are starting to get really tough too."
  • Ranks 7 to 9: "All these upgrades are so pricey, but the damage is going way up now! It looks like the damage at rank 10 is twice as good as rank 7 so it is worth it!
  • Rank 9 to 10: "Well, it was a struggle to upgrade it to rank 9, but looks like it is totally worth maxing it out because it is the largest increase in damage of all the upgrades! Now I can choose something else to max out."

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