How is the math in loot boxes?

Question

Game development

We are developing a multiplayer game, and we would like to introduce loot boxes there (but not in the form of some kind of rip-off, but with good, fat chances).
Actually, we want to figure out how to make loot boxes predicted by profitability (examples of loot boxes are chests in CSGO / DotA 2)? Are there any mathematical problems on this topic? (I searched - I didn’t find it, the knapsack problem doesn’t fit here very well). If someone has experience in developing something like this, but is not ready to share information for free, they are ready to pay.

How we see it:
We take, for example, a distance of 5000 loot box openings (the cost of one opening is 10 rubles).
We want to make money on this from strength - 10%, that is, 5000R, which means that 5000 pieces should fall out of this loot box. valuables for a total of 45000R.
And here the question arises - what algorithm to choose these very values ​​in a given number and with a total amount?
Since items in the loot box can have different prices (from 1-2 rubles to 3000-10000 thousand), we would like the number of expensive items to be as large as possible in relation to the number of cheap items.

Answer 1

From a marketing standpoint, you will need to tell the consumer (who opens the case) what the percentage chance of a certain item to drop is. In this case, the algorithm is quite simple - divide certain items into "classes" and set the chance of a certain class falling out (for example, 1% -15% -30% -54% - by the cost of the items, respectively). After that, you set the chance to drop items in a certain class (the scheme is the same). Either you will not indicate this chance, but it is still the key task of the algorithm.
The algorithm itself: you take the definition of 1000 (more if the turnover is large) box openings, and distribute them as a percentage to specific IDs for this thousand (that is, you get, for example, 10 people received gold, 150 received silver, 300 received bronze, 540 will get a donut hole) and randomize it all by setting certain rules of the algorithm (for example, not to fall out in a row, or so that 2 gold falls out for every 200 holes) and make prices.
After 1000 openings of the box, the counter is reset and starts to spin in a new way.
And it's up to you to decide what kind of drop configuration of the same type or different types of items to use. Depends on turnover, marketing, etc. In such matters, it is better not to trust the guys from the forums, but to hire a really good specialist if you yourself cannot think over the config.
It is better to take the counter of the algorithm with a margin, because the smaller the counter, the more "pale" the work of the algorithm is. When your drop counter goes under 10,000, then it all really looks like "Random". For in this case, the number of "combinations" of drop-down items increases for every 10 discoveries (conditionally)

Answer 2

Mathematics is simple, you just need to use the concept of mathematical expectation .
Each item has a certain value.either in rubles, or in game currency, or in time, or it is some kind of analogue of power / strength - it doesn’t matter. Ask your game designer (balancer) what's the price. By attaching a chance to an item, you sort of reduce its cost. That is, if the Sword of a Thousand Truths costs 500 rubles, then with a 1% chance it is checkmate. The expectation will be 5 rubles, obviously. Then just add up all the mat. waiting in the chest - it will be checkmate. waiting for the cost of the chest, so take it as a basis. You can then increase or decrease this chest value by as much as you like, as long as it's "fair" enough for you. As for me, the honesty here is questionable. Many players also understand this math, at least intuitively (when the odds are visible). What is needed here is not honesty, but balance . So it's better to delegate all that mathgame designer , or whoever is responsible for monetization .
Further, in game development they often use just the same, the so-called dishonest random. He may be dishonest in different ways, but this does not change the essence. For example, a more or less fair option: if the Sword of a Thousand Truths (with a 1% chance) does not drop out of the 200th loot box, then the game should cram it in so that the player is ultimately happy. And if this Sword of a Thousand Truths fell out to a player, then so that it would no longer fall out for the next 50 loot boxes, otherwise it was too fat. In general, the chance remains 1%, so there is no deception as such. But, of course, if the players know these details, they will begin to abuse your system in a black-and-white manner, which will eventually lead to a distortion of the odds. As they say, think for yourself, decide for yourself whether you need or do not need the declared good fat chances . Although within the framework of marketing , of course, you can use any words, but it is better to delegate this to the marketer. And here in your question, these words are slightly inappropriate. By the way, loot boxes are banned
in some countries , so keep that in mind. It is understandable. The developer can declare anything, and the real chances can be greatly underestimated or a very dishonest random can be screwed in. By the way, each mechanic is introduced for some purpose . Keep this goal in mind when shaping your loot box composition and other features. That is, the task is to achieve the goal, and not just introduce a feature for show. Monkeying is bad. You need to understand why loot boxes exist in CSGO / DotA and how they make the game better. Maybe you don't need it at all in your game.