Difference between revisions of "User:Jdavis/Forging"

Revision as of 02:04, 2 August 2018

This article explains why Spiral Knights players should consider forging 5-star items on medium (70%-80%) chances rather than 100% chances. It describes theory and experiment that you can interpret to make the best decision for your level of comfort with risk.

Assumptions

We ignore forge bonuses such as prize boxes (which are rare) and double level-ups (which do not appear to be affected by forging chances). That is, we assume that they do not happen.

We assume that the tries are independent of one another and that the chances are as stated. Data sets collected by players do not offer any statistically valid evidence that the chances are otherwise.

We also assume that the knight never gives up, no matter how bad her luck is.

Theory

When a knight uses 100% forging chances, heating one 5-star item costs a total of 453 Radiant Fire Crystals.

When a knight uses medium chances, the total cost of heating a 5-star item varies. The minimum cost, achieved when all forgings succeed, is a mere 302 Radiants. But usually the cost is greater than that.

We can use basic probability theory to compute the expected cost, which one can think of as the cost incurred by a knight experiencing average luck.

The key fact from probability is that we expect to need 1 / p trials to complete a task, if each one has probability p of success. For example, forging from heat level 1 to heat level 2 has p = 0.80 and so we expect 1 / 0.80 = 1.2 forgings to succeed, for an expected cost of 1.2 * 6 = 7.2 Radiants.

If we add up the expected costs for all nine forgings from heat level 1 to heat level 10, then we find that the expected total cost is around 419 Radiants. That is, a knight experiencing average luck will spend 419 Radiants.

Some knights will experience better-than-average luck and spend fewer than 419 Radiants, down to the minimum of 302. Some knights will experience worse-than-average luck and spend more than 419 Radiants. In fact, there is no theoretical maximum. A very unlucky knight might spend more than 1,000 Radiants.

Experiment

To get a more detailed picture of the possible costs, we perform a computer simulation. Working in Python 3.x, we use a random number generator to compute 1,000,000 simulations of knights forging 5-star items from heat level 1 to heat level 10. For each of the 1,000,000 simulations, we record how many Radiants were required.

In accordance with the theory, the minimum number of Radiants used was 302 and the average was 419.447. The maximum was 1,258.

In the plot, the horizontal axis shows the number of Radiants used, from 302 to 1,258. The vertical axis shows how many of the 1,000,000 simulations used that number of Radiants.

557,324 of the simulations used fewer than 419 Radiants. 139,708 of the simulations used between 419 and 453 Radiants. 302,968 of the simulations used more than 453 Radiants.

The median number of Radiants used was 400. The average was greater than the median, because a few simulated knights incurred massive costs that pulled up the average. These simulations fill the "long tail" on the right side of the plot. The 95th percentile was 597 Radiants used.

In other words, about 70% of the simulations used fewer than 453 Radiants, and 30% used more than 453 Radiants. Even after taking the unlucky latter group into account, forging on medium chances saved an average of about 34 Radiants over forging on 100% chances.

Analysis

When forging on medium chances, the only guarantee is that you will spend at least 302 Radiants. You risk spending more than that. But probability helps you manage that risk.

Some knights are highly risk-averse, so they wonder, "How bad could the costs be?" And the answer is, "There is no limit to how bad they can be." But this attitude is overly pessimistic. A person who is unwilling to accept any risk cannot ride in an automobile, cross a street, walk in a crowd, or do much of anything in life.

A more reasonable kind of pessimist asks, "How high would the cost be, if my luck were worse than 95% of other knights?" And the answer is around 597. So that kind of pessimistic knight should forge at 100% chances and accept the 453-Radiant cost.

A milder pessimist might ask, "How high would the cost be, if my luck were worse than 70% of other knights?" And the answer is around 453. So it makes little difference whether that knight forges at 100% chances or medium chances.

However, the most reasonable assumption is arguably that you will experience average luck. Then forging on medium chances costs you somewhere around 419 (average) or 400 (median) Radiants. So you should forge on medium chances and save 34-53 Radiants per item.

The assumption of average luck is especially appropriate for knights who heat many items over a long time period. These knights might start off lucky or unlucky, but their cumulative luck tends to approach the average as they heat more and more. This phenomenon is called regression toward the mean. Knights who are on Cradle for a short time and heat only a few 5-star items are more probable (but still not highly probable) to experience extremes of good or bad luck.

Advice

I have heated many items over several years. So I take a long view of things. I forge on medium chances. I am able to ignore the pain of a failed forging, because I know that I'm saving precious Radiants overall.

I have also given this advice to many knights on the Spiral Knights forums. However, the consensus seems to be in favor of forging at 100% chances, because the pain of a failed forging is unacceptably unpleasant. These knights are risk-averse --- maybe not at the 95% level, but somewhere beyond the 70% level.

Ultimately, each knight should pick the level of risk that she can tolerate. This guide exists, just to help you understand the compromises that you're making. In particular, if you tend to forge at 100% chances, then please know that accepting a bit more risk could lower your Radiant costs by approximately 10%.