Maximise success using Upgrade Rarity

This guide explains the optimal way to use the Upgrade Rarity table, so that you can maximise your chance of success and save precious Arkforge. This is not an opinion piece, but mathematically proven results.

Motivation

In August of 2015 the Supreme rarity (gold) was introduced. At the same time, the Salvage Matrix was changed to use a new Upgrade Rarity process. This new screen allows you to gamble on updating gear.

I have read lots of incorrect statements about the chances of rolling up your weapon. Given a success rate of 5%, quite a few people have said that twenty rolls will give you a 100% chance of an upgrade. Yes, it's true that 5 times 20 equals 100, but that's not at all how probability works.

A simple example will make this clear. We know that when we roll a die we have a 1 in 6 chance of getting a 5. Rolling it six times in no way guarantees we get a 5. In fact, you could roll that die 20 times and still not get the 5 you want. Or, you could just as easily get several fives. Anyone who has played dice games will know this well!

It's the same with Upgrade Rarity. Rolling a 5% chance twenty times or even fifty times will not guarantee anything (though your odds increase the more you roll). People are sinking all sorts of Arkforge into weapons this way, and then complaining that they feel ripped off.

To help people save cash, I thought I would calculate the odds correctly. And furthermore, I will also figure out the optimal strategy for the Upgrade Rarity process.

Four probability rules

1. A probability is commonly given in percent, but we need to express it as a decimal to perform any calculations. That 5% I keep mentioning is the same as 0.05. (Percent simply means "divide by 100".)

2. Every "roll" has only two states: it can succeed or fail. The chance of success plus the chance of failure always add up to 1. So, if our success rate is .05, our failure rate is 1-.05 = .95.

3. If an event depends on several conditions, each with individual probabilities, we need to multiply these together to get the total probability for that event. For example, the chance of rolling a 5 on a die is 1/6. The chance of rolling two fives in a row is 1/6 times 1/6 or 1/36.

4. We will want to know the cumulative chance, after a certain number of rolls, of getting a given result. For a cumulative sum, we simply keep a running total of individual probabilities for each round.

These four rules are key to everything that follows.

The gambling system

Trion has given us all sorts of options in the Upgrade Rarity process. We can try our luck at 5%, or 10%, or 15%, etc., all the way up, in 5% increments. Or we can slap down 3200 Arkforge and have a perfect 100% chance of an upgrade -- no gambling involved. Which approach makes the most sense?

To answer this, two things are needed. First, you need to decide on the risk you are willing to take. Second, you need the probability tables.

For example, say you are happy with a 50% chance of upgrading your gun. How many times would you need to spend 100 Arkforge at a 5% success rate to reach this cumulative probability? How much will you spend in total?

That is the problem I will solve, and not just one way but two. Some of you will prefer one explanation to the other, but they both give the same numbers, since they are two ways of looking at the binomial distribution.

Explanation 1

Let's work out the chance of success for the first few rounds, and look for the pattern. I will take the first table (5% chance) as an example.

After one round our probability of success is 0.05. If we succeed, we stop rolling, because there is no reason to continue. We won! Thus, the chance of success after one round is that very same 0.05 or 5%.

Consider the more likely outcome: we fail in round 1 so we try again. Our chance of failing round one is 0.95. Our chance of success in round two is 0.05. Then we apply Rule 3 above for a combined probability of 0.95 times 0.05, or 0.0475. Add this to the round one total, to get the cumulative amount (Rule 4). 0.05 + 0.0475 = 0.0975 = 9.75%. Our chance has almost, but not quite doubled.

If we get to round three, we must have failed twice before (in rounds one and two). Our probability of being successful is hence 0.95 x 0.95 x 0.05 = 0.045125. The incremental chance is smaller than before. We add this to the previous result, to get the running total. Chance of success after three rounds is 0.0975 + 0.045125 = 0.142625 = 14.26%.

The pattern should now be obvious. For each round we multiply the chance of success in that round by the chances of failure in all previous rounds.

Explanation 2

Consider the chance of success at a given round, say round five. The chance of success is 1 minus the chance for failure (Rule 2 above). So, let's instead calculate the chance for failure, since it works out to be a simpler concept.

We fail on round five if all five rolls fail. So that's a probability of 0.95 for the first roll, 0.95 for the second roll, same for the third, fourth, and fifth. We multiply all these together to get the result (Rule 3). That's 0.95 multiplied by itself 5 times, otherwise known as 0.95 to the power 5. We use a caret to represent this exponential.

The chance to succeed is therefore given by: 1 - (0.95^5). Wow, that was simple!

Implementation

I wrote a short Python program to do the grunt work. Both of the algorithms are dead easy to implement. I decided to use the first five tables (5%, 10%, 15%, 20%, 25%, 30%), but then skip in increments of 10%. I doubt the values in between make much difference. And I figured you don't want to be reading tables all day!

Trion has chosen a confusing way of presenting the costs on the Upgrade Rarity screen. The base cost of 100 gets added to the incremental cost, which depends on the percentage success you have chosen. These two values sum to the total cost for each attempt. (I am not sure why they didn't simply show one number, for the actual cost. This way is unnecessarily confusing.)

For example, each roll at 10% costs 175 Arkforge, each roll at 15% costs 250 Arkforge, etc.

The program iterates through a sequence of rolls. Each run terminates when one of the following conditions is met. Either the chance for success hit 100%, in which case there is no reason to continue. Or, the cost hits 3200, in which case we really should have just bought the upgrade with certainty (this costs 3200).

Result Tables

In each of these tables the first column is how much you spend. The second column is the cumulative chance of success. The first entry makes it clear which odds are being used. "100 5.00" means it costs 100 arkforge to get a 5 percent chance of success. So obviously we are using the 5% table.

Analysis

What can we learn from all this? Let's examine the first table.

First, it's apparent that we get less for our money the more rounds we go. Yes, we are still increasing our chance of success, but by less and less each time. This is a law of diminishing returns that's very common in probability problems. It's also what the house doesn't want us to know. The house always counts on the gambler not being too smart.

We can look at a couple of landmark values. To have a 50% chance of success, we need to spend 1400 Arkforge. Maybe read that again to realise how expensive this process is!

After spending 3200 Arkforge, we still have a one in five chance of failure. Obviously that compares badly with simply paying out the 3200 in the first place.

So, how risk averse are you? Will you take a chance on the 5% table?

Maybe we'd better compare the other options before deciding.

Combining the Data

How to summarise this data in a meaningful way? I decided to list the results by probability threshold. This reflects the key question: What odds are you comfortable with?

Say you want a 60% chance of success overall. Is the .1 probability table better than .4 table? Well, this depends on which has the lowest cost at this probability level.

In this table the rows indicate the probability you want to aim for, from 10% to 100%. (I realise that no-one is going to want only a 10% chance of success... but the row is here for completeness.) The column headers indicate the table that gives us the corresponding cost.

Here's how you use the table. Say you are happy with 30% total chance of success. Read off the numbers from the third row. Which is the lowest value? 575 Arkforge. Look up that column to see which percentage you should be rolling on. Turns out the cheapest options is the 30% table.

Analysis

The same pattern repeats itself almost universally. In most cases, you should just buy once on the table that corresponds to the percentage chance you want. Rolling multiple times on lower probability tables is a worse choice.

There are two minor exceptions.

If you want a 50% chance, then roll twice on the 30% table, to save 25 Arkforge on average (1150 versus 1175). Besides, this actually gives you a 51% chance. You have a slight edge either way.

If you want a 60% chance, then roll six times on the 15% table, saving 50 Arkforge on average (1500 versus 1550). This actually gives you a 62.29% chance. It's the largest deviation from the general rule. But still not compelling.

Psychology

Using these results consistently requires discipline. In reality, how many people are going to roll 50% once and then accept a negative result? Most would roll again, increasing their chance to 75% and their outlay to 2350. But that is counter-productive. It would have been better to roll once on the 80% table to begin with (for 2400 Arkforge).

If you find this happening, it's pretty good evidence that you are actually not happy with only a 50% chance of success. Re-evaluate this number upwards.

Some readers will still ask: Which upgrade option is the best? But there is no definitive answer. It all depends on how risk averse you are. Or how lucky you feel. But at least this article has armed you with the facts.

Trion might have given us fixed percentage results for a fixed price. The only reason the lower percentage options exist at all is to sucker you into using them. They don't make any mathematical sense. But they are psychologically appealing.

Yep, it's all psychology.

tl;dr

Avoid temptation. Save up for the percentage table that equates to the chance of success you want. Then buy it once and stick with the result. Any other approach is suboptimal.

The only question remaining is... Which odds you are happy accepting?