Introduction
The topic of enhancement is a hot topic of debate within the Tree of Savior community as it plays a huge role in deciding what content a player is able to participate in. Due to the randomness involved in the process, a huge gap in power can arise between the players, leading to a lot of strong feelings about what is reasonable and doable. This very same randomness leads to a lot of misunderstanding about what is likely versus unlikely. Since the typical player doesn’t have a whole lot of resources, it is difficult for them to gain a proper understanding of the randomness behind enhancement. It doesn’t help that the typical human being is already ill-equipped to understand randomness.
Since I’m a nerd I decided to try to actually figure out the chances and costs behind enhancement. All of this is under the assumption that there are no tricks to improve your chances; there are no special times or places or ways to enhance. Every event is independent and you can’t influence the outcome. With this in mind I set up some Monte Carlo simulations in Matlab. This sort of approach basically involves using a whole lot of random samples to get at a solution. In this case, it is to answer the questions of how likely it is to achieve a certain enhancement and how much that would cost. Simulation by large numbers allows us to address the problem of not having enough resources to do this in game. As long as the probability table that I am using is correct, the results should accurately reflect what would happen in game.
If you are interested in the code that I used, you can find a paste of it here. I won’t really go into the technical details here unless someone asks me, but just note that enhancement costs were taken for a masinios bow. The costs would scale proportionately if you used a different piece of equipment. I guess the other thing worth pointing out is that I went in under the assumption that I had 8 potential ‘to work with’ since I would want to reserve 5 of the available 13 potential on a masinios bow for slots. This number would change if I were willing to go golden sockets or if the equip had a different amount of potential to work with.
The Case for Remaking
One approach for obtaining a high enhancement is to just try many times on many weapons using normal anvils. This is the typical approach for players that sell enhanced equipment since using diamond anvils is too cost-prohibitive to turn a profit. Surprisingly, this method won’t cost you as much as you might expect. In these simulations I set it so that it quits when it hits a goal. So if the goal were +11, then the trial would stop when going from +10 to +11, and thus the results would exclude a case where it went from +12 to +11 and then gave up.
The figure here shows the results from my simulations. The lines represent the mean and the error bars represent +/- one standard deviation, so a range covering 68% of the results. One must be a bit wary reading too much into the standard deviation, though, because the distribution is actually heavily skewed and not actually a symmetric normal distribution. I still show it just to get a sense of the variability in results.
Let’s look at specifically the case of making a +16 so we can understand how to read this plot. Here, the cost of enhancement has a mean of 33m, with a lower and upper bound to the error bars at 9m and 57m, respectively. The number of weapons made has a mean of 5.2 with a lower and upper bound of 1 and 9.4, respectively. What this means is that on average you would go through 5.2 weapons and spend a total of 33m across all the weapons trying to enhance them to make a single +16. Be wary of randomness though! This isn’t a guarantee that all you need is 5 weapons and 33m; it’s just an average.
To get a sense of the true total cost you’d need to factor in the price of each weapon. That isn’t too hard since you’d just multiply the cost of one weapon with the total number of weapons required and then add that to the rest of the silver spent. So if it costs you 5m per weapon, then on average it’d cost 26m in weapons and 33m in enhancements for a total of 59m silver.
One takeaway from this result is that players selling +16 weapons for over 100m are actually making a huge profit doing this. Another is that you need a lot of money to sustain this. Even though on average you might need 33m to enhance a +16, you need to be rich enough to take a hit of 60m (or maybe even more) and still feel like it’s okay to keep going.
The Case for Golden Anvils
I always wondered if golden anvils were useful for pushing past +11. The effect of resetting back to +11 if you fail while above +11 reeks of ‘trap,’ but I couldn’t help but wonder what it takes to push your luck far enough. In these simulations, the weapon starts at +6 and tries until the target enhancement was achieved.
In this figure we have the amount of golden anvils consumed in blue and the cost of using them in orange. The number of anvils consumed is the same regardless of whether they’re shining or not, but if they’re shining then the cost of enhancement is zero. If you’re using shining golden anvils, you’d then have to factor in the cost of buying all of them.
This plot suggests that if you got screwed over and ended up with a +6 weapon that you wanted to fix and get back to +11 then it’d take you about 19 golden anvils on average to pull that off. If you were ambitious and wanted to make it to +16, then on average you might expect to need something like 300 golden anvils. If you were forced to buy those golden anvils off the market at 3m each that’d cost you 900m! If I were making a masinios weapon at 100m per attempt, I’d be better off just remaking the weapon since that’d take 533m on average. If the weapon somehow cost 200m to make, I might consider buying shining golden anvils and attempting until I make it to +16.
The Case for Diamond Anvils
Again I wondered what it’d be like if I had all the money in the world. In this case, I wondered what it’d be like if I could use as many diamond anvils as I liked. So I set up the simulation to be the same as the normal anvil case (where it’d bring in a new weapon if a target enhancement wasn’t reached) but it’d instead switch over to diamond anvils after +5. Here’s the modified code.
These two figures show how many attempts would need to be made to make a certain enhancement level. For example, if you wanted to make a +16 weapon, on average you’d go through about 1.6 weapons, spend 28.5m in enhancement silver, and use 19.5 diamond anvils. Surprisingly, even with the triple price of diamond anvils, you’d spend less on enhancing than you would with normal anvils. Unfortunately, diamond anvils cost a lot. If you’re going by Klaipeda’s prices this would cost you about 1 billion silver to buy all the diamond anvils you’d need. Right now this seems to be the least cost-efficient route towards acquiring a +16.
Diamond anvils are probably the best way to get past +16, but seeing how cost-prohibitive it is, I’ll just leave this cumulative distribution plot describing how hard it actually is getting to the highest enhancement levels we see in the game today.
A quick read of this cumulative distribution plot suggests that there is a 14% chance of not making it to +11 while using all diamond anvils, a 50% of making it to +15 and a 2% of getting past +24. Oof. You can still get really punished while using only diamond anvils, and the people with ultra-high enhancement levels are both extremely wealthy and extremely lucky.
