## Supermechs Optimal Box Opening Simulation

## Preface

Players often ask which type of box is best to buy. While the expected Token cost per mythical item is lowest with Gold boxes, there is a possibility of receiving no mythical items. In comparison, the appeal of Mythical and Ultra Mythical boxes is the certainty of mythical items and with a increased chance of receiving a mythical item which the player does not yet own. In considering the power of mythical items, an optimal player may wonder at which point it is more beneficial to buy mythical boxes?

In other words, what is the smallest number of tokens that a player should expect to spend to acquire the most number of mythical items?

##1) Introduction

There are four types of Token boxes (1) Silver boxes, (2) Gold boxes, (3) Mythical boxes, (4) Ultra Mythical boxes. I assume that box opening takes place during a special event (75 percent discounted price and double the chance of a mythical item). Then, each of the five cards in a Silver (Gold) box has a 14 percent (30 percent) chance of being a mythical item. Since the price a discounted Silver box is 68T and the price of a discounted Gold box is 120T, clearly **it is never optimal to buy Silver boxes**. Similarly, if one is to buy mythical boxes, *it is strictly better to buy Ultra Mythical boxes rather than Mythical boxes*. Thus, we can restrict the type of boxes to Gold and Ultra Mythical boxes.

###1.1) Box Opening Strategy

With zero mythical items owned, clearly, choosing a Gold box is the optimal strategy. On the other hand, if the player has all but one mythical item, buying a Ultra Mythical box is the optimal strategy. So, the question can be framed as:

What is the optimal number of Gold boxes one should open before opening Ultra Mythical boxes to minimize costs?

## Assumptions

Unless I counted wrong, there are currently 145 mythical items available. I assume that the all mythical items are drawn with equal probability. While this may not be true, it is a necessary assumption.

##2) Simulation Approach

Let the number of Gold boxes opened be N_{gb}. The simulation experiment is simple. Fix a value of N_{gb}

- Open N
_{gb}Gold boxes - Open Ultra Mythical Boxes until the player has at the target number of unique mythical items. Let the number of Ultra Mythical boxes opened be N
_{um} - Then, the Token cost as a function of N
_{gb}(and N_{um}which is a function of N_{gb}) is T(N_{gb}) = 120*N_{gb}+ 450*N_{um}.

Steps (1)-(3) are repeated 200 times to approximate average Token cost given a fixed value N_{gb}.

From the argument given in Section 1.1, T() is convex in N_{gb} so a minimum N_{gb}* exists.

###2.1) Gold Boxes VS Ultra Mythical Boxes

Recall that each Ultra Mythical box guarantees three mythical items where each mythical item has a 30 percent chance to not be from your inventory. The number of unique mythical items (unique meaning not already owned) follows a hypergeometric distribution with parameters N=145 (number of possible mythical items), n=3 (number of draws), and K (number of mythical items not yet owned). The expected number of unique mythical items from any individual Ultra Mythical box is given by n*(K/N) which depends on k. This quantity decreases in the number of unowned mythical items.

For Gold boxes, while the probability that any item is mythical is 30 percent. If we we have exactly 70 percent of the possible mythical items, then a single item from a Gold Box has approximately 30 percent chance of being an unowned mythical item. The expected benefit of Gold vs Ultra Mythical boxes in the objective of owning every mythical item while minimizing Token cost is then characterized by the proportion of unowned mythical items remaining *before* opening Ultra Mythical boxes. Some considerations

- The number of mythical items follows a binomial distribution with parameters p=.3 and n=5 (number of draws), the expected number of mythical items is then given by n*p = 1.5
- Token per mythical item differs between boxes

If we consider again the scenario where the player owns exactly 70 percent of the possible mythical items, then the expected price of a single unowned mythical item is lower when buying a Gold box, so the critical value of unowned mythical items K* must be less than 145*0.3. The exact value of K* (and thereby proportion of unowned/unowned mythical items) depends on points 1) and 2) above and may or may not be hard to find, however, I was unable to find an exact solution, however a reasonable conjecture would be that the optimal proportion of unique mythical items owned before opening Ultra Mythical boxes is a multiple of 0.7. Simulation suggests that the multiple is 1.16. *So it is more profitable to open Ultra Mythical boxes when one owns at least 81 percent of the possible mythical items.*

## Simulation Results

First lets look at Token cost following the procedure in Section 2. The X-axis is a fixed value of N_{gb} which admits a value of N_{um}(N_{gb}) or, the number of Ultra Mythical boxes required to open before owning at least one of each mythical item. Given the values N_{gb} and N_{um}, we can compute the total Token cost. Again, prices are assumed to be the regular discounted prices (120T,450T). The dots represent individual simulations and the solid line represents the average over simulations. Following the rule for N_{gb}* = 145*1.16, the minimum average one can expect to spent just under 35,000T in pursuit of owning every mythical item.

The associated number of Ultra Mythical boxes given N_{gb} is given by:

So that’s a lot of Tokens right? The prudent Supermech pilot may then ask: what is the relationship between opening boxes and the number of different mythical items? The next figure explores this relationship. The figure plots the proportion of mythical items owned against the number of boxes opened where boxes opened before the N_{gb}*-th (or 168th) box are Gold boxes and the ones opened after are Ultra Mythical boxes. Individually, the segments before and after N _{gb}* are decreasing since each new mythical item owned means the pool of unowned mythical items is smaller. After the strategy transitions to opening Ultra Mythical boxes, there is a dramatic increase in the marginal proportion ownership. This instantaneous increase is followed by a decreasing trend for the same reasons as with the Gold boxes.

The last graph also reveals the average performance of a dedicated but not necesarrily dollar-paying-player. If we count up all sources of equivalent gold boxes (daily missions, campaign Tokens, campaign mythicals, achievements, and free Gold boxes from special events), the player reasoanble has access to 20 to 40 percent of the possible mythical items.

That means that, following the assumptions about equiprobability draws, then the expected number of unique mythical items by category is

##3) TL;DR

- Only open Gold and Mythical boxes.
- Open Gold boxes until the proportion of unique mythical items you own exceeds 0.7*1.16, then open Ultra Mythical boxes.
- Since this works out to over 150 Gold Boxes, an easier to remember rule would be to never open Ultra Mythical boxes.

and most importantly

- This is a guideline which hinges on lots of assumptions. At the end of the day, this is a game, so do what makes you happy. I admit I’ve opened Silver boxes (due to impatience and belief that This Time Will be Lucky! and Time is Running out on the Sale!) and I’ve also opened an Ultra Mythical box when I owned next to no mythical items because its
*ULTRA*.

##4) FAQ

Q: Y u do dis?

A: Because I’m a jelly scrub who has a grand total of zero God Modes, zero Lava Sprays, zero of any of the God Mode legs, and zero of many more notable mythical items and wanted to rationalize my failures in the Arena.

Q: You’re wrong!

A: I might be! I did this in a night and I make mistakes (just like how I make mistakes in the Arena - hah) so feel free to point them out!

Q: Is the code available?

A: Yes, it is written in Matlab. If you’re interested, hit me up and Ill send it to you.

Q: Why is this in Fanart?

A: What is art?