A fact about well-orders and well partial orders I hadn't previously realized

I was looking through Diana Schmidt's habilitation thesis last night for some stuff, and I noticed it stated a fact about well partial orders (and also well-founded partial orders and well-orders) which, although trivial to prove, I'd never realized before.

So suppose you have a partial order (or total order) S which can be written as the union of a chain of its subsets, which are well partial orders (or well-founded partial orders, or well-orders in the total order case). Now, this famously does not imply that S is a WPO (or WFPO, or well-order); in the well-order or WFPO case this will work if you can show the subsets are lower sets (aka initial segments), but in the WPO case even that's not good enough.

The thing that I didn't realize until I saw it pointed out in Schmidt's habilitation is that actually, you don't need any of these extra conditions if the chain itself has uncountable cofinality! Even in the WPO case! Because a failure to be a WFPO or WPO can always be detected as a (countable) decreasing chain or bad sequence. So if there were some bad sequence in the union, you could find the individual elements of it, take the subsets they live in, and then there would necessarily be some element of the chain containing all of them, contradiction.

Of course, this is a strong condition! But it does come up sometimes, and when it does it lets you prove well-ordering or well partial ordering without having to do much of the work that you normally would. I do think using it means you don't get upper bounds on the type, though! You still have to use the more usual methods for that. But if you don't care about the type, this is a shortcut you can make use of.

Slay the Spire vs Dicey Dungeons

OK, so, this is the post I wanted to write, like, months ago!

So I played Dicey Dungeons a while back. Dicey Dungeons gets compared a bunch to Slay the Spire, so after seeing this post where Zvi says Dicey Dungeons is "tier 4" while Slay the Spire is "tier 1", I was like, OK, I guess I have to try out Slay the Spire.

My initial reaction to Slay the Spire was disappointment that it isn't Dicey Dungeons. :P But, y'know, now I've played a lot of Slay the Spire (and also played Dicey Dungeons Reunion), and, well, yeah, it's probably better on the whole? But, I wanted to compare the two a bit and make notes of what each game does better.

So, a big thing that Dicey Dungeons does better is symmetry. One of the things that disappointed me in Slay the Spire was that the enemies, well, they're video game enemies; they have patterns that they execute and that's it. If they do more damage one turn and less damage another turn, it's because, IDK, they just decided to use a less-damaging attack the second turn. There's no internal logic to it; they're just executing patterns that you're meant to deal with.

By contrast, in Dicey Dungeons, your opponents are playing by the same rules as you; they have equipment and dice, and each turn they try to use their roll as best as possible. If they do less damage on a given turn, you can see that it's because they rolled worse.

The fact that your opponents have some actual AI to figure out what to do means that rather than just executing patterns, they can actually adapt to a gain or loss of capabilities. If you shock or weaken their equipment, if you burn or lock their dice, they don't just do the same thing they were going to do, they figure out a new plan. If you face the Copycat, who copies your equipment, it figures out how to use it. Now sometimes the AI seemingly just flubs things, especially where countdowns are involved, but still, the symmetry opens possibilities.

It also means it's possible for you to use enemy equipment or (as in the case of Copycat) for them to use yours. The whole Thief character has using enemy equipment as his central gimmick, and the Finders Keepers episode where you get to *permanently* steal equipment from *different* enemies is great. In the finale episode you get to actually play *as* the enemies! That wouldn't make any sense in Slay the Spire.

Tying into this, another thing I like in Dicey Dungeons is your ability to interfere with your opponents' ability to act. In Slay the Spire, you can't really do this; if an enemy is going to hit you for lots of damage, well, you can put up a lot of block, and you can weaken them, and occasionally you can gain intangible, but other than that last one you can't really directly interfere with their action. Because while you're playing with an actual deck of cards and energy points, your enemies aren't, they're just executing patterns. So while they might junk up your deck, or put status effects on you that reduce your ability to act (although there isn't as much of this as in Dicey Dungeons), you can't do the same to them.

This is too bad, because I really like being able to interfere with my opponents' ability to do things and to try to lock them out of acting. In Slay the Spire, you can't go for the lockout; you can weaken, but beyond that you just have to be able to block.

Although, by contrast, in an advantage for Slay the Spire, blocking is actually useful in that game. In Dicey Dungeons, blocking is generally too weak to be useful, except on the rare occasion you can get so much block that you can achieve a lockout that way. In Slay the Spire blocking is useful at levels below lockout! Still, I do kind of like how in Dicey Dungeons, inflicting status effects to interfere with action is often more effective than blocking, because, as I said, I think interfering with action is fun. :)

Another thing that bothered me at first when trying Slay the Spire is that it's more of a big-numbers game, where Dicey Dungeons is more of a small-numbers game. I like small-numbers games, where every unit counts. In big-numbers games, you eventually stop worrying about small differences, which I don't like. That said, on playing Dicey Dungeons Reunion just recently, this difference was less than I expected; I imagine it makes more of a difference when the game is harder, and Reunion wasn't that hard for me. I guess it made more difference when I wasn't as good, or was playing on hard mode.

Dicey Dungeons also has more interesting variation between a character's episodes. Slay the Spire doesn't really have any equivalent to this; there's Ascension, but that just makes things harder in a uniform way, and doesn't really introduce any new interesting things (at least until you get to Ascension 17-19, I guess, but I haven't gotten their yet; I've only gotten to Ascension 10, and only with Defect). But Dicey Dungeons has a bunch of interesting variation in its episodes. Some of these are applied uniformly to each character, while some are character-specific, but like... an entirely different set of status effects! That's something that can be applied uniformly to each character while still changing things up significantly.

On the downside, I'd say Dicey Dungeons actually has a bit too much variation in some of its episodes. Ideally, each episode would stand out from the basic format in one way. Often, however, episodes will change up your character's limit break for no apparent reason, despite that not being listed in the episode description? I get that this might be a case of adjusting the limit break to fit the episode, but at least it should be listed if you're going to do that. As it is it's a bit confusing.

And on that note, another thing that Slay the Spire does better is difficulty -- once you get good at it, you can start turning the difficulty way up with Ascension. Dicey Dungeons has hard mode (which is way easier than it used to be :P ), but nothing beyond that. If you want more and more challenge, you're going to have to look elsewhere or, like, mod the game, I guess. (When playing Reunion just recently, I was like, huh, this is pretty easy now, isn't it? Like I only died playing as the Robot, and even then only a few times...)

Although, speaking of difficulty -- one confusing thing about Slay the Spire is how it treats the ending and the heart. Like, OK, let me tell you how I played Slay the Spire.

First, I played as each character until I beat the game as that character. After beating the game as the Defect and unlocking the keys, as Watcher I figured, well, I have to go for the keys, now don't I? It took me a while but eventually I got all three keys, beat the third boss, got utterly wrecked by the hearth, and... didn't get the achievement for beating the game as the Watcher. Nope, you have to either beat the heart or not get the keys. Argh, I hate it when games do this! (I mean, you do unlock Ascension 1 regardless, but I wanted the achievement!) So I went back and played again as Watcher until I won again, but this time not getting the keys. Then I went back to Ironclad and played as each character until I'd beaten the heart as each character (this took a long time). After that I started playing Ascension; in order to avoid getting stuck (beating the heart as Defect, and especially Watcher, each took me a *long* time), I now changed from "switch character on win" to "switch character on loss". I still went for the keys each time, but I counted it as a win so long as I beat the third act; I didn't actually focus on beating the heart. (Although I did manage to beat the heart as Defect on Ascension 3, the only time I've managed to beat the heart with Ascension on.)

Notionally I'd continue until I'd beaten Ascension 20 with each character, but then it'd have been way too long before I ever wrote this post, so I stopped before then. :P Of course I might go back and do that now... :P

Anyway, the question all this raises is -- where does beating the heart fit into this? The game treats it confusingly. There are achievements for beating the heart with each character, but nothing for doing so with Ascension on (you'd think there'd be an achievement for "beat the heart on Ascension 20 with all characters", but no, there isn't). I don't like when games have unclear multiple goals like this. (Have I ranted here before about how much I dislike specials in pinball? I don't think I have.) Am I supposed to go for the heart or not??

Like each difficulty level is effectively two difficulty levels -- one for without going for the keys, one for with going for the keys. What order am I supposed to play them in?? I went with, 0, 0*, 1, 2, 3, ..., 20. But how do 1*, 2*, etc, fit into this? After I beat Ascension 20 with each character, should I go back and try to beat the heart on Ascension 1? Maybe I should just go straight for the heart on Ascension 20? Or maybe I should go back down to somewhere in the middle? I don't know!

(Oddly, on speedrun.com, there are categories for Ascension 20, but there are absolutely no categories for beating the heart. That's surprising, I wonder why that is?)

That said one thing Slay the Spire does better than Dicey Dungeons is making risk/reward decisions in navigating the dungeon. In Dicey Dungeons, even though you don't have to go for max level, I always do (yes, even as Inventor, where this is the riskiest!). Notionally there's a tradeoff there, but... I feel like the way the game is put together really encourages it (e.g. the way that beating every enemy gets you to *exactly* max level), and I'd be disappointed if I didn't do it. By contrast, in Slay the Spire, there is no notion of 100%ing a run. There's no getting away from the need to make serious risk/reward decisions in climbing the Spire. And yeah obviously the keys are an exception here, because as mentioned I always go for the keys, the same way in Dicey Dungeons I always go for 100%; but in Slay the Spire there's lots of risk/reward decisions you have to make even if you decide you're always going for the keys no matter what.

One more thing I have to knock Slay the Spire for is unlocks. Now I'm not talking about unlocking difficulty levels or characters, or like how you have to unlock characters and episodes in Dicey Dungeons. Rather I mean unlocks that permanently change the game -- unlocking cards and relics. I don't like this. Also due to the order I played things in it took me forever to unlock everything, because by the time I beat the heart as all four characters, I was still missing a bunch of Ironclad and Silent unlocks...

The fact that you unlock the keys is especially bad when you combine it with what I said above about difficulty and the heart. So it's like -- if you're *not* going for the keys, now you have to *avoid* one particular elite? Not going for the keys doesn't interfere with your game in any other way, but now each act will have one particular elite you have to avoid! Annoying! I mean, I always go for all the keys (that bit with the Watcher aside), but if I weren't, I'd be annoyed...

However I just remembered that Dicey Dungeons actually has unlocks too! Although not to the same extent -- it only has one. But it's done in a fairly perverse way; when you unlock the Jester as a character, he stops appearing as an enemy! Oy. Slay the Spire is definitely worse on the whole unlocks thing but I want to say that I am just opposed to this sort of unlocks in general. And *losing* content is just perverse!

I also have to mention game length here. Each run of Dicey Dungeons feels like it ends a bit early, whereas Slay the Spire feels like it ends at the right point. Although in terms of wall-clock time, oof, a run of Slay the Spire can easily take like 2 hours (or longer if you're new). I mean you can save and come back, but c'mon, who wants to do that? :P

Overall I'd probably give the nod to Slay the Spire over Dicey Dungeons these days just because of the ability to turn up the difficulty more; Dicey Dungeons is pretty easy for me these days (at least, when not on hard mode). Also, when I was playing Reunion, I was like, huh, I didn't get many good status-effect inflictors... is that just a Reunion thing? I don't remember the base game being like that... anyway, if you haven't played this sort of game as much, and don't insist on that level of difficulty, there's a lot to like about Dicey Dungeons that Slay the Spire doesn't do!