Hello YouTube, jumpy23 here with the follow-up to my last video. Once again, if you'd rather read it than watch it, the script will be in the description, although I've actually got a couple of visual aids toward the middle this time.

Before I hop into the promised content, I have a few points to address in response to the previous video, that came to me through YouTube and reddit comments. A couple people were asking how to apply these concepts to unit tier lists. I'm not discussing units here, but rather "play," or in other words, strategies. Unit tier lists are a second-order effect, in that we rate how we feel the units contribute to our preferred strategies (efficient or otherwise). I have a few thoughts on how best to run unit tier lists for efficient play, and I might make a follow-up video about that, but it's out of scope for now.

The second thing is that I've received a few comments from people telling me that I've missed the point entirely, and that 'efficiency' is just playing casually but with certain "degenerate" strategies banned. In other words, in the last video, where I said "everyone agrees that playing efficiently is a way to display skill"--these people disagree with that. This is a fundamental, impassable disagreement on efficiency. I have no interest in discussing casual play because of the problems with it I mentioned yesterday: the existence of Schrodinger's Casual, and the fact that arguments tend to be entirely clout- and personal experience-based. If I can't claim the word 'efficiency' for something more rigorous, then so be it, I'll find a new word to discuss this context which involves a display of skill. But I would like to at least try to claim 'efficiency' for my side before I cede that ground.

The last point of order before I get into the meat of my video. If these topics have interested you at all, I can't recommend enough checking out the A-Fish-In-Sea Emblem Discord server, owned by Irysa. This is where people interested in efficient (as I've described it) Fire Emblem play gather. It's also the place where LTCers gather, because there's a lot of overlap in those communities, myself included. The link is in the description.

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Several people have told me that they're looking forward to hearing how I can combine turncount and reliability into a single metric. I hope I didn't build the hype up too much, because the fact is that I don't have a new metric to propose. Rather, I'm hoping to bring attention to a little-known old one that I think is pretty good: Expected Turn Count, or ETC.

ETC was first developed by some old serenesforest users about a decade ago to address this exact same question, how to combine reliability with turncounts. The solution is simple and elegant: Count up how many turns you spend on that map *on average*, including resets, by pursuing a particular strategy. Allow me to pause my FE6 Ironman to put the formula for ETC on the screen.

t = conditional mean turn of map clear
k = conditional mean turn of reset
x = chance of success

ETC = t + (k * (1-x)/x)

For example, if your strategy clears the map in 3 turns 50% of the time, and in 4 turns 50% of the time, then the ETC of the strategy is 3.5. If your strategy resets 50% of the time on turn 3, and clears the other 50% of the time on turn 3, then the LTC of the strategy is only 3, but the *ETC* of the strategy is 6. (The math around resets is slightly unintuitive and out of the scope of this video. Irysa has a video dedicated just to explaining the math behind ETC, which I will link in the description. The point of my illustration is just to show how reset chance affects ETC).

There are several strengths to using ETC to measure efficiency.

1) It's conceptually simple. I defined ETC in a dozen words: "How many turns you spend on that map on average, including resets." That's not a simplification--it is literally exactly the number that ETC gives you. The math is also relatively simple, and doable for anyone who has taken algebra in high school.

2) It distills our two factors--turncount and reliability--into a single number. When splitting hairs between strategies where you can't clearly eyeball the superior one, ETC makes it obvious. Which is better, a strategy which 1 turns 20% of the time and resets 80% of the time, or a consistent 4 turn? ETC tells us that the former strategy expects to spend 5 turns playing the map on average, and we have an answer.

3) It *only* cares about our two factors. This is very powerful as it allows us to compare strategies however we want. This evaluation system doesn't have any intrinsic preference for, say, full recruitment. It lets us answer ALL of the following questions: "Is it more efficient to kill Matthis or recruit him," "Which of these two strategies that kill Matthis are more efficient," "Which of these two strategies that recruit Matthis are more efficient."

4) It lines up with--I would even argue it *explains*--our natural intuition. It's better to reset earlier rather than later under ETC. It's better to wait an extra turn, rather than risk a reset at the end of a map.

5) The ETC of any existing strategy can be easily calculated. Want to know the ETC of an old video? Note all the chances of failure in the video description and add them up. I just calculated the ETC of dondon151's FE6 Chapter 2 while making this video (link in the description). Deke has a 51.5% CoD on turn 3. The boss kill on turn 5 has a 64.6% CoS, and if it fails, the clear extends one turn. The ETC of this clear is about 8.60 turns. That took me 3 minutes. Granted, I cheated a bit because I knew dondon is nice and lists chances of failure in his video descriptions--not all LTCers are this diligent, myself included. But the point stands.

Piggybacking off of this example, Valkama has been calculating the ETC of various Shadow Dragon LTC clears today for shits and giggles. I'm going to talk about what he found as an example, because it's a perfect demonstration of what makes ETC such a powerful tool.

Here are five LTCs of Shadow Dragon's Chapter 6. One of them goes for an extremely unreliable 2-turn, and the other 4 all use similar 6-turn strategies. But they have massively different expected turncounts.

Ruadath: 2 Turn ETC 209.85 (LTC No Restrictions)
Espinosa: 6 Turn ETC 49.6045 (No Forge No Warp)
Dondon: 6 Turn ETC 29.588 (0% Full Recruit)
Reploids: 6 Turn ETC 21.7989 (0% Warpless)
Valkama: 6 Turn ETC 9.8599 (Full Recruit)

The formula makes it very clear where these turns are being bled. First of all, you can see how harshly it penalizes Ruadath's strategy, which has a less than 1% chance of success. But even among the similar strategies, ETC heavily discourages going for 10-20% crits and dodges, which Espinosa did, or relying on 50% enemy movements, which dondon151 did. The difference between Reploids's and Valkama's ETCs is even more interesting: it almost entirely comes down to Valkama leaning on a hit-forged Javelin, and the +20 hit per attack accumulating in a big way over time. If you were to "eye test" these 5 LTCs, you would easily see that Ruadath's was the least reliable, but among the other 4, I'm not sure if you would notice a difference. But with math, we're able to clearly see how careful improvements in reliability shake out.

I won't argue that ETC is perfect. In fact, I'm going to spend the next segment of the video discussing some of the problems with ETC, and how I suggest mitigating them. What I contend is that ETC *exists*, it's *better than what we're doing currently*, and it's *worth the minimal cost to switch*.

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People who are already in the know tend to have two main complaints about ETC.


The first category of complaint can be described as a list of all the ways that ETC breaks down around the edges. For example, ETC doesn't say anything about rigging in prep menus, such as Tellius Bonus EXP level ups or enemy stat rigging. Another area where ETC falls down is with things that don't require resets at all, just absurd time investment, like grinding silver marks in SoV.

Frankly, these are water off a duck's back to me. These are not insurmountable problems, just little blemishes that need patches taped on. We already address these kind of balancing concerns when deciding on contexts currently. In fact, I would go so far as to turn this on its head and say that properly tuned, ETC's ability to handle reliability is a strength when it comes to off-map behavior as well. For example, the Engage tier list I'm currently running bans S rank Bond Rings entirely. But if we were to treat resetting in Somniel as a turn 1 reset, then with ETC, we could mathematically decide whether or not it's worth it to reset for Dire Thunder, instead of just needing an arbitrary ruling that "feels about right."


The other category of complaint strikes at the core of ETC. This is when people say that ETC is too hard. The math of ETC is simple, but it somewhat buries the lede. "Find all the chances of failure, and decide what to do (backup strat or reset) if they occur." This can lead to branching strategies, which also branch, and branch infinitely.

There are two ways that this complaint manifests. The first is in calculating the ETC of particularly complex maps with lots of branches. Think big routs, long winding corridors, etc. A good compromise for this is not to bother with backup strats, at least at first. Simply treat any chance of failure as a reset, and you have a "ceiling" for the strat's ETC. Then working backward, you can prune and develop the most impactful branches to optimize the ETC downward, if you care to. Another good compromise was suggested by Irysa, which was: just don't do the math and someone else will probably do it for you, lmao. This wraps back to what I said was one of the advantages of using ETC, which is that anyone can calculate it given a video (or even written description) of a clear. This reduces the barrier of entry for new players to as low as it could possibly go, and from my experience with the efficient play community, it is more than happy to do this work to welcome new players.

The other complexity problem is when discussing what I'll call "full-game ETC." I've discussed ETC as a way to evaluate a strategy on an individual chapter, because that's what it's really good at. Now where ETC struggles is when people treat it like a different flavor of LTC and try to post complete "low ETC runs." Even more than just the fact that you're trying to optimize the bejeezus out of 20 to 25 chapters, the way that Fire Emblem decisions compound on each other is what makes this such a problem. An ETC run that wants a unit at a certain level by a certain chapter, and tries to optimize their XP grind for maximum reliability... well it's hard, to say the least. If I know I want a 12/5 Cormag in Chapter 15, what is the ETC of getting a 12/5 Cormag by that point? Some people in the efficiency community see this problem as a reason to discard ETC entirely, but I disagree. ETC is a great tool at telling us the best way to finish a map. And even without perfect optimal investment, if we can keep finishing maps optimally, we'll be doing a damn sight better than we have in the past.

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So there you have it, my comprehensive discussion of Expected Turn Count. It's an incredible tool for describing the efficiency of a strategy on a map with given preconditions--again, using my definition of efficiency as "low turns and high reliability." It is theoretically possible to use it to describe the reliability of setting up these preconditions, such as unit stats but also things like low-use weapons for weapon breaks, but I would recommend not doing so.

I want to point out at the end that I am not married to ETC. Any metric that combines reliability with turn count is probably fine. What I think about ETC is that it's good, and that's good enough for me. I do NOT want to let perfect be the enemy of good here. Let's adopt ETC now, and if someone comes along with something better later, we can move to that, having already gotten our foot in the door of numerical analysis.

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Wrapping this back into the first video, we have my complete definition of efficient Fire Emblem play. "Efficiency in Fire Emblem is minimizing turncount while maximizing reliability. Given two strategies, the strategy with the lower Expected Turn Count is more efficient." Hopefully a definition like this can help move Fire Emblem discussion into a more evidence-based future, where clout and bandwagon size aren't the most important pieces of an argument. Thanks for watching.