Expected turncount as a(n almost) self-consistent efficiency metric

[Redwall]

For a long time, I favored expected turncount as a useful way of simultaneously valuing high reliability and low turncounts. However, I never thought very hard about accounting for resets. I propose a simple solution: we add in the turns spent in those unsuccessful attempts.

Let 0 <= x <= 1 be the chance of a clear without a reset.

Let 0 <= 1-x <= 1 be the chance of a reset occurring.

Let t be the conditional mean turncount of a clear, given no reset occurs (probability x).

Let 1 <= k be the conditional mean turn on which a reset occurs (probability 1-x).

Note that x, t, and k will not in general be integers.

Assume the player will play until achieving a successful clear that doesn't end in a reset. How many turns will he or she be expected to play if we add in the turns spent in any attempts ending in resets?

The probability that the first attempt doesn't end in a reset is x.

The probability that the first attempt ends in a reset is 1-x. As a result of this, the player will then try again.

The joint probability that the first attempt ends in a reset AND the second attempt does not is (1-x)*x.

The joint probability that both the first AND second attempt end in resets is (1-x)^2.

The joint probability that the first two attempts end in resets AND the third does not is (1-x)^2 * x.

I hope you see where this is going.

<TC> = xt + (1-x)k +(1-x)*[xt + (1-x)k] + (1-x)^2 * [xt + (1-x)k] + ...

= [xt + (1-x)k] +(1-x)*[xt + (1-x)k] + (1-x)^2 * [xt + (1-x)k] + ...

= [xt + (1-x)k] * {1 + (1-x) + (1-x)^2 + ...}

= [xt + (1-x)k]/x

= t + k(1-x)/x

Let us look at some intuitive cases. If x = 0, then the series diverges; a player who plays such a stage until he or she clears it will spend an infinite number of turns trying to win.

If x = 1, then there is no chance of a reset, and the player's expected turncount is simply t.

If x = 0.5, the expected turncount is t + k. Because the odds of an attempt ending in a reset are equal to the odds of clearing without a reset, asking the player to play until a no-reset attempt occurs necessarily yields the same results as asking him or her to play until a reset occurs.

We can spot several trends: for fixed t and k, decreasing x from 1 down to 0 will increase <TC>, appropriately penalizing the player for adopting risky strategies. Additionally, for fixed x and t, increasing k will increase <TC>; we favor strategies that fail sooner rather than later because we will have spent more turns (not to mention human time) on those strategies that fail later.

What did I mean by "almost" in the topic title? Note that I have been vague about the conditions underlying a reset; for example, if I get Sumia, who is responsible for saving many turns, killed in C3 of Awakening, I can technically choose to go on with or without her. The present framework allows the player to reset and incur the resulting C3 turncount penalty; however, the player can also continue without Sumia, in which case he or she must recompute the expected turncount without the reset while accounting for the turns lost in Sumia's absence. In an Ironman playthrough, the latter way of doing things is obviously preferable, but in other contexts, it is up to the player; I suspect that most players will restart upon character deaths and nothing else, so for those players, those will be the conditions underlying a reset. For other players, they may reset if they do not get the lowest turncount possible; that is something also treated by the current metric. Unless a Lord dies, it is up to the player when to reset, and it is for this reason that the metric cannot be completely self-consistent.

While I cannot define for you what is warm and what is cold, I can objectively define terms like temperature that can inform your understanding of warmth and cold; for example, temperature is the partial derivative of energy with respect to entropy at constant particle number and volume. The idea behind this thread is the same: I'm not telling you what you should consider efficient or inefficient, but I am telling you that using expected turncount as a measure can rigorously answer questions like why we believe spending five extra turns to boost reliability by 5% is generally not worthwhile, and why we believe resetting earlier is preferable to resetting later. None of this stuff is all that fancy, and has probably been figured out before by some other members of the forum; however, my hope is that this becomes a more popular way of thinking about decision-making in FE games.

Edited June 9, 2013 by Redwall

[XeKr]

I think the subset of players that value this metric is quite small.

Nice work, though. I think your solution is quite simple and intuitive, just no one (well, I didn’t. >_>) really bothered to sit down and do the math. Tiering is srs business.

It may be helpful to define the conditional mean turncounts, for reference.

Edited June 9, 2013 by XeKr

[dondon151]

has already been proposed before (by me, no less)

[SRC]

While not technically wrong, I think a method like this is going to be too grindy to be very helpful except on turn 1-2 clears. (Which are not terribly hard to calculate, anyways.)

I think you're better off just skimming most variables and making sure you're not pushing your luck on the "whoppers". (Relying on crits, lots of dodges, lots of sub 90% or so true accuracy hits) Just by those alone, it shouldn't be too difficult to tell if a strategy is generally reliable, unreliable, or requires absurd luck/RNG abuse.

[Chiki]

My issue with this is that it doesn't account for some resets (an enemy not having a certain amount of HP to clear the chapter in one less turn, for example). In fact, it's impossible to account for it because we don't know the likelihood of an enemy having a certain amount of HP.

It also becomes very messy when we try to calculate the probability of a unit being able to kill a boss with stats that were gained in previous chapters. How would be able to account for that with this system?

Ex: Avatar will need to gain certain levels in Prologue and Chapter 1 to be able to kill a boss in Chapter 2.

So it isn't enough just to account for resets in Chapter 2, but you also have to account for resets in Prologue and Chapter 1. And that's.. I think that would ruin the idea of the expected turncount of one chapter.

But, yes, with some adjustments I think this is definitely a gigantic improvement, even if this has been suggested before, because you've attempted to explain this in precise language.

Edited June 9, 2013 by Chiki

[Redwall]

It may be helpful to define the conditional mean turncounts, for reference.

Noted, thanks.

has already been proposed before (by me, no less)

I did some Googling just now and saw a whole discussion in the FE9 thread. Looks like I'm late to the party ._. Nonetheless, I think it's worthwhile to bring up this idea again when people have ostensibly efficient Awakening playlogs that differ in their turncounts by nearly a factor of 2 (I've seen a playlog of Awakening w/ 186 turns by C19's end and another with 106 turns by that time).

My issue with this is that it doesn't account for some resets (an enemy not having a certain amount of HP to clear the chapter in one less turn, for example). In fact, it's impossible to account for it because we don't know the likelihood of an enemy having a certain amount of HP.

That is true, but it's not so much a problem with the criteria as it is with the fact that at this point, Awakening stat and skill distribution is sort of a black box. If we knew their stat distribution, it wouldn't be a problem.

It also becomes very messy when we try to calculate the probability of a unit being able to kill a boss with stats that were gained in previous chapters. How would be able to account for that with this system?

Ex: Avatar will need to gain certain levels in Prologue and Chapter 1 to be able to kill a boss in Chapter 2.

Messy, but doable. Remember the calculation I did in the HM tier list the other day? x, t, and k as they are defined can, as you point out, depend on your stats, but we can compute the likelihood of getting those stats using the binomial distribution.

So it isn't enough just to account for resets in Chapter 2, but you also have to account for resets in Prologue and Chapter 1. And that's.. I think that would ruin the idea of the expected turncount of one chapter.

Yes, the Sumia example in an Ironman context would force you to consider the long run.

Edited June 9, 2013 by Redwall

[Chiki]

That is true, but it's not so much a problem with the criteria as it is with the fact that at this point, Awakening stat and skill distribution is sort of a black box. If we knew their stat distribution, it wouldn't be a problem.

No one has any idea how stat distribution works in the other games either, so I doubt we'll be able to figure it out.

There's a few other things that are impossible to calculate. For example, an enemy unit in Awakening can randomly have certain skills (this doesn't apply just for Lunatic+). For example, having the HP+5 skill in Chapter 2.

[CT075]

No one has any idea how stat distribution works in the other games either, so I doubt we'll be able to figure it out.

i actually dissected the autolevel routines in the GBA games a long time ago but i lost my notes (for the record this isn't me resetting 800 bazillion times to check the stat distribution this is just me opening the code to find out how the stat rolls are coded)

i can't give you the full rundown right now (although once i get around to redoing my work i will) but your assertion that "no one has any idea" is false

the basic idea is that the levels are rolled pretty much the same way player levels are rolled with some exceptions and mods here and there that i don't remember

Edited June 10, 2013 by CT075

[Chiki]

Even so, it's still true for the non-GBA games.