Choosing not to Parachain

This is a repost about the game Warlocks, and the spell paralysis in general, which I've posted about before.

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I was talking to a friend of mine recently about Warlocks and he asked me an interesting question: Do I think that the fact that I won the annual championship in spite of my decision to restrict my own use of paralysis is evidence that it's not a broken spell?

I think a lot of warlocks are aware of my feelings on paralysis, and some probably noticed that on my profile I declare my "personal rules", which include the fact that I won't gesture F more than 5 times consecutively (meaning 3 paras). Not only does this prevent me from abusing paralysis, I know for a fact that some of my opponents are aware of the restriction, and play on the assumption I won't parachain, so it also limits my ability to threaten with F's.

So am I providing evidence that parachains aren't needed to compete at a high level? Yes. Am I providing evidence that it's not an extremely imbalanced spell? I don't think so.

In order to explain why, I'll start with why I self-imposed this rule in the first place. When I started out, around the time I first approached elo 1600, I started to realize how powerful paralysis was, and though I never thought of myself as a paramancer, there was a time when I found myself using paralysis extensively. What would invariably happen is that in the middle of a match I would cast paralysis at some point, and then think "alright, I'll break the chain and do something else now" just before noticing that my opponent had some kind of spell he was threatening in two turns, and that if I kept up the para just a little longer, I could stop it. After I stopped that threat, I'd say "oh, but now if I para just one more turn, I can also respond to a second threat", and this would continue ad infinitum. I would end up using massive parachains even though by this point I had already begun to feel that parachains made games less dynamic and interesting. I remember Succat (a self-declared "paraholic") once saying that he pleased to see that I had a nice "para trigger", or something to that effect.

However, I felt like paralysis wasn't one of many useful strategies at my disposal, it felt like a crutch. I realized that if I left it up to how I happened to feel during a match I would always find an excuse to continue parachaining, so the only way to force myself to be more creative was to make a rule out of it, and by making it public I would feel pressure to keep at it. I was also hoping, I think, that other people would follow suite and I would see fewer long parachains coming at me, but that part never happened.

After I stopped parachaining (and charming hands to "-", and opening "D/P"), I found I really did explore a lot of different options, and was pleased with this decision for a time. Then when I got to around elo 1800 for the first time, over a year ago, I started to realize that I was going to be playing more often against the very best players in the game, and that they all used parachains and parafod extensively (some of them more than others, but every single one of them used parafod to kill me at one time or another). I felt like my own refusal to use parachains might be an insurmountable barrier that would keep me from being able to compete with them, and I honestly got to a point where I started playing very few friendly and ladder matches, and mostly just played in the League because the monthly rules were one surefire source of continued variation in play, which is what I wanted.

At some point I noticed that the players in the league, even ones who normally used parafod often, were less likely to use paralysis when a monthly rule was in play even if the rule didn't have to do with paralysis. In other words, the fact that we were playing in an unranked space where new strategies were encouraged made people drop their paraholic ways and try new things, and the results were sometimes brilliant. However, when they returned to ranked games, most players would stop thinking outside the box and play safe.

This encouraged me to try more seriously to take on the good players head on with my personal rules in play, and I could almost tangibly feel myself gradually getting the hang of beating paralysis. It's also worth noting, when I say something like "beating paralysis", that I actually used paralysis a lot myself. In fact, when I went on the warlocks archives recently, I was surprised to find that I apparently cast paralysis, per game, more often than average. The difference is that I don't cast long chains - I'll cast paralysis once or twice (or maybe thrice) and then break off into something else, and then come back and cast paralysis again a little later. It's a good spell that I like using, but forbidding myself from relying on it made my weaves more dynamic and forced me to be more creative, so I felt like I was getting the best of both worlds. In fact, in my final match against Xade in the tournament, I opened with paralysis on my left hand.

I really wasn't expecting to do as well as I did in the tournament, but if it's a sign of anything, I think it's a sign of how much I gained specifically by avoiding using paralysis as a crutch and trying to be just a little strange (Bio went so far as to call one of my moves in the final match "insane", which made me smile). That said, being the only player who doesn't use parachains has still been a serious restriction, and I've had a number of games where I believe I would have definitely won had I allowed myself to use a parachain. One of the other noteworthy aspects of the tournament was the appearance of a number of new players around the same time, some of whom experienced players found "shockingly good." The shock, I think, stems from the fact that these players haven't yet learned what the popular weaves are, and so make plays that are unexpected and hard to predict - experienced players would do well to try and learn a little from their games with these new players.

So my general point is this - once you've played warlocks for a long time it can become very comfortable to fall into regular patterns - go for charms and ogres as often as possible, and then when you have enough initiative advantage go for parafod or bolt an opponent to death. However, any time that a strategy becomes common wisdom, it starts to get stale, and forcing yourself to play in ways that are less comfortable is the best possible way to continue learning new things. To return to my original question, paralysis is an extremely powerful spell, one that I think is imbalanced, which is precisely why it, more than anything else, prevents players from developing creative strategies. If paralysis weren't so strong, my decision not to use it wouldn't have had as big an effect on my playing style.

In short, choosing not to parachain can be good for you.

Elo and Trickle-Up Economies

Preface: I realized after writing it that this post is pretty long and could use an abstract. It's an overview of the two scoring systems used in Warlocks viewed as economies. The objective is simply to note what kind of properties will emerge from having an economy with these particular constraints. What does emerge is that the resources in both systems are only acquired from other players, not independently generated, which results in a pyramid of sorts, where a large number of players lower down in the hierarchy provide a source of points for the higher ranked players. Larger economies will result in a higher peak, but otherwise do not effect the average player in these systems.

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As I've mentioned before, I'm an avid player of the game Warlocks, based on Waving Hands. This game uses two different ranking systems for competitive players, and they have some interesting differences between them.

The first, and simpler ranking system is ladder points, and they work as follows: every time a player wins a ladder match, they gain one ladder point; every time a player loses a ladder match, they lose one ladder point; every time a player dies during a ladder match, their ladder points reduce to zero (note that in this game, most matches end with one player surrendering, not dying). Every player starts with a ladder score of zero, and you cannot have negative ladder points. This means that every time a player with no ladder points loses a match, a ladder point is created from the ether, and every time a player with ladder points loses a match, their point is effectively transferred to the winner. There's one more feature of ladder matches that's worth mentioning, and that is that you cannot challenge a player to a ladder match if your relative ladder scores are more than 5 points apart.

The result of these features is that ladder scores rarely get very high. Since your ladder score will get reduced to zero by a single death it takes a lot of skill (or luck) in order to continually grow your score. Moreover, since you cannot challenge an opponent who is more than 5 points apart from you, the high possible ladder score for any player is 7 points higher than the second-highest score (assuming they began 5 points apart and that the higher-ranked player won). This means that in order for me to have a ladder score higher than 20, there need to be other players with a ladder score of at least 15 I can challenge. This means that the upper limit of ladder scores depends on the presence of a population of successful ladder players who collaboratively create ladder point (by playing those with 0 ladder points) and then transferring them up the ladder to the best players.

We'll see a similar dynamic with the second ranking system: elo. Like in chess rankings, elo is a system in which the change in a player's score is weighted depending on their expected likelihood of winning (which is, in turn, based on the competing players' relative elo scores). Each player who registers begins with an elo score of 1500, which defines that score as the expected skill level of an average new player. Each match results in one player gaining a number of points and the other losing an equal number of points - in other words, once again, matches effectively cause a "transfer" of points from one player to another. If a player with a lower score beats a player with a higher score, they earn more points from the win, and if a player with a higher score wins, they earn fewer points. The difference in points earned corresponds to a player's expected likelihood of winning - meaning that if I'm expected to have a 75% chance of defeating an opponent, I will earn 1/3 as many point for winning as he will if he wins, so that over the course of many games, elo scores will stabilize if players tend to win as often as they are expected to given their elo score. Since all starting players start with 1500 points, they begin ranked as equals even though some may be stronger players than others. However, the differences in skill level will fairly rapidly be reflected in their score once they begin playing ranked games.

Let's look at an example. I register a new account and start with 1500 elo. If I play and beat another new player, I will gain 12 points, to have a score of 1512, and their score will go down to 1488. Now the difference in our scores is 24, so if I play that same player again and win, I will gain slightly fewer points than I did the first time. Once the elo difference is over 100 points, I will gain 8 points from a win and my opponent will gain 16 points if he wins - as long as I win approximately twice as often as I lose, the elo difference will remain stable, but if I win more often, it will continue to go up, and if I lose more often, it will go down.

Notably, if the winning player is ahead by enough elo, they effectively gain no points from victory, so many high-ranked players will simply refuse to play ranked matches with much lower-ranked players (since they have nothing to gain and much to lose if they make a mistake). In practice, the maximum effective difference between players who can fairly compete in ranked matches is a little over 200 points. Any more of a difference and fluke wins by inexperienced players will unduly throw off the scores of high-ranked players.

All of this together suggests some interesting features of the elo economy - since a winning player gains as much as their opponent loses from a match, the sum elo score of the player population cannot grow except by the addition of new players, and that the existence of players with more than 1500 elo requires the existence of players with less than 1500 elo. Moreover, a player can only effectively grow their elo by playing opponents with an elo score within 200 points of their own, which suggests that growing your elo depends on a population of players with elos near your own, so the highest possible elo in the system depends on the number of successful players, which is in turn limited by the number of total players. That is, a population of new players is needed in order to support the elo growth of players with elos between 1500-1700, and a population of players with elos of at least 1700 is needed to support the elo growth of players with elos between 1700-1900.

As of this writing, there are 1577 players who have registered to play Warlocks, about 200 of which never played a ranked duel. Of the players who have played ranked games, 281 have an elo higher than 1500, and 419 players have an elo lower than 1500. The lowest elo in the system is 1298 (202 points lower than the average) and the highest elo is 2106 (606 points higher than the average). This suggests that in practice, a large population of weak players is needed to support the heightened elo scores of a relative few. There are two reasons for this: first, players who repeatedly lose will likely stop competing at some point, and players who repeatedly lose will have their elos fall to the point where they no longer effectively feed the elo growth of stronger players.

Since the value of a win is weighted by the likelihood of the win, players who perform as well as expected will have stable elos - if you are about twice as good as the average new player (meaning twice as likely to win), your elo should stabilize around elo 1600. However, once a player enters the higher echelons of play, the relative dirth of other high-ranked players makes it harder to play enough balanced games to maintain a representative elo. In a population of players with elos from 1400 to 1600, it is unlikely for me to grow my elo above 1800, no matter how good I become at the game.

So the grand result is this: The total size of the elo economy of the game is determined by the number of players in the system, and the larger the total elo economy is, the higher the elo ratings of the best players can be, but that for the vast majority of players, the size of the elo economy will have no impact on their personal elo scores. That is, as a resource, the total quantity of elo in the system will only effect the players at the top.

Now there are obvious disanalogies between the elo economy in Warlocks and market economies in the real world, but it nonetheless serves as an interesting model of a competition driven economy. This is also not meant in any way to be some kind of moral statement about how "just" the elo system is - the numbers simply represent the fact that some players win more often than others, and it is the explicit goal of the elo system to represent this. I simply believed that the unintended emergent features of the system are noteworthy, since they result from the interactions of thousands of players.

Waving Hands: Paralysis and "Broken" Mechanics

I recently had a request for another post on Waving Hands, and since we're in the midsts of the Warlocks 2010 championships (in which I'm well placed to make the finals right now, and on which Waving Hands is based), I decided to oblige.

A lot of discussion about the game by avid players revolves around a single spell, paralysis, which is considered by many, including myself, to be unbalanced, and possibly even broken. Its abusability has even led to the formation of a guild, the Paramancers, who specialize in this one spell. One of the interesting questions to arise from this discussion, however, is what exactly it means for a spell to be "broken."

When we talk about game balance, we generally assume that what we have in mind is a series of equally viable options. If one strategy is disproportionately represented or effective, then it is unbalanced. This, at first, seems like a fair criterion for a balanced game - if, in Starcraft, almost no one ever played Zerg, but 90% of players played Protoss, that would be a clera sign that something was amuck in the balance of the game. However, if we push the idea a little further, it starts to become murkier. Isn't the very idea of "strategy" supposed to be that some moves ARE better than others, and that finding the good moves is what makes the game fun? Rock, paper, scissors is a perfectly balanced game, because no move is better than any other, but for that reason it's impossible to strategize about anything other than player psychology, so the game is shallow.

In the case of Warlocks, some spells are clearly more popular than others, but this creates a self-balancing factor: the more commonly a strategy is used, the more predictable it is, and in a game that relies on predicting your opponent's moves as much as Warlocks does, that can be a fatal weakness. This means that if you use less popular spells, you can take your opponent by surprise, and in doing so may be able to make up for the features of the spell that make it less popular.

So in the case of paralysis, the fact that it is used much more often than most other spells is, in some sense, unbalanced, but that doesn't necessarily make the game worse. This is where we can draw a distinction between an unbalanced and broken spell, because a broken spell will interfere with the overall playability of the game. The problem, in this case, is that paralysis DOES interfere with the overall playability of the game. The criteria that can be used for a broken strategy might include:

1. Whether or not predicting the strategy makes it possible to "punish" the player who is predictable.

2. Whether or not there exists effective counters to the strategy.

3. Whether or not there is a good motivation for a player to use a different strategy.

As I mentioned before, there exists a guild called the Paramancers, who specialize in paralysis. In other words, just by being in this guild, they are effectively announcing to their opponents before the duel begins what strategy they will use, and yet they are still successful players. This simply wouldn't be the case with any other strategy - if I announced that every game I play, whenever possible, my right hand will constantly be casting antispell, I would lose every game I played (even though, in a given game, my right hand might end up making those gestures anyway).

Discussing counters will require a more detailed discussion of the spell itself, which requires some background knowledge of Warlocks. So if you've been reading up to now simply because you liked the idea of distinguishing "unbalanced" and "broken" mechanics, here's your chance to escape.

Paralysis - FFF

In Warlocks, you submit gestures on your left and right hands that add up, over several turns, to spells, that take effect when the gestures of that spell are completed. The gestures needed to cast the spell paralysis are FFF, and it causes one of your opponent's hands to be "paralyzed" into the same gesture on the next turn (well, except that W is paralyzed into P and S into D). The important thing is that since the gestures of paralysis are so symmetrical, gesturing another "F" on the next turn allows you to cast it again immediately (because now your last three gestures were, once again, "FFF"), resulting in "parachains" where one hand gestures "FFFFFFF..." ad infinitum. There is one restriction on parachains built into the basic rules, however - on consecutive turns, you can only paralyze the hand you already were paralyzing.

At first, this doesn't seem terribly abusive. If I use an endless parachain, I can keep one of my opponent's hands tied up, and effectively make us both play one-handed. This makes the spell useful on its own, in case you have some other advantage you want to hold onto, or force your opponent to respond to you and a summoned monster with only one hand available. It also grants a small initiative advantage, because as soon as the paralyzer decides to end his parachain, he can immediately begin casting a new spell, whereas the paralyzed player must suffer the effect of the final turn of paralysis before he can move his hand freely again, leaving him one turn behind on one hand. This is a significant advantage by itself.

However, the real problem with paralysis occurs when you start changing targets. Since it counts as a mind-affecting enchantment, it cancels with other mind-affecting enchantments, which means that you can cast paralysis on yourself to counter an opponent's interrupt, and then go back to paralyzing them on the next turn. What's more, when you stop paralyzing your opponent for one turn, and then start again, you can switch which hand you're affecting, allowing you to alternate and restrict your opponent's use of both his hands. Finally, paralysis cast on a monster stops the monster from attacking that turn, which means that a parachain can be used, on any given turn, to disrupt either of your opponent's hands, hold his monster at bay, or counter one of his interrupts on your (including his own paralysis). This versatility in a spell that can be cast every single turn is incredibly powerful.

There already exists, however, a standard variant that helps to make paralysis easier to disrupt, called "parafc". It means that when paralysis is used on an F gestures, the F is paralyzed into a C gesture. The significance of this is that it makes paralysis targeted at yourself (to counter another enchantment) into a risky move, because if your opponent was bluffing, and doesn't complete their own enchantment, you've paralyzed yourself, and either cannot continue your parachain (because now you've gestured FFFc), or must disrupt your other hand. The effect of this rule is that many situations in which a parachain could not be countered are now situations in which your opponent can generate "50/50" opportunities to disrupt you.

The problem with this solution seems, to me, fairly straightforward. While it's better to have a 50% chance to disrupt a parachain than a 0% chance, if I really know, before my opponent has made a single move, what he's going to do, I should have a 100% chance of countering him. Paralysis can still be cast every turn, can restrict both of your opponent's hands, and can be used defensively - now it's simply that you have run a 50% risk of disrupting yourself when using it defensively (which is the case with every other mind-affecting enchantment, anyway). The fact remains that even playing parafc, Paramancers can play effectively even when their opponents know what strategy they are using, and this simply shouldn't be the case.

I think there are two ways to balance out paralysis as it stands, and they each essentially involve removing one of the spell's advantages - either it shouldn't be cast every turn, or it shouldn't be able to target either of an opponent's hands. I, in fact, already play with the former restriction self-imposed: I do not allow myself to cast paralysis more than three times consecutively, and this is a restriction that is public, and which my opponents know about and take advantage of. The reason I play with this restriction is simple - because playing without extended parachains makes the game more dynamic and interesting, and if I didn't have an explicit rule, I would end up using them simply because they are so advantageous. However, a fiat "do not cast paralysis more than three times" rule is not a very elegant solution, so this restriction, were it generally enforced, would best take the form of a chance in the gestures of paralysis such that it was impossible to cast every turn.

And restricting which hand can be paralyzed would also balance out the spell effectively. My preferred way of handling this would be to say that whichever hand I cast paralysis with is the hand that gets paralyzed (if I gesture "FFF" with my right hand, I can paralyze your right hand). This is significant for a reason that goes beyond the ability of a parachain to alternate hands every other turn - even though only one hand can be paralyzed, as long as the caster gets to choose which hand it is that is affected, and the choice is made after the spell is successfully cast, the target of paralysis has to restrict which gestures he makes on BOTH of his hands, in order to avoid having a gesture on either hand that will become particularly unusable once paralysis takes effect. In fact, it might be a downside of the parafc variant that any spell featuring an F can be so disastrously disrupted by paralysis (given how few spells use the C gesture). The result of this is that on a turn in which I expect to be targeted by paralysis, I will try to gesture either a W (paralyzed into WPP - counterspell) or PS (paralyzed into PSDD - charm monster) on each hand to prevent my opponent from being able to totally disrupt my spellflow. However, since my opponent can alternate the paralyzed hand every other turn, that means that in theory I have to be prepared to make one of those restricted gestures on each hand every other turn. If I knew ahead of time which hand would be paralyzed, this would not be an issue.

There's plenty more to say on the topic, but this is plenty for one post. I like to hope that some of my idealistic self-imposed restrictions gain traction in the community at large, or that at some point a further evolved version of the game will address some of these issues, but in the meantime paralysis is a legitimately broken aspect of an otherwise unbelievably well-crafted game that we must live with.

Waving Hands 2: Finite Resources

So, given how fan-ish I was in my previous post on Waving Hands (aka Spellcaster, aka Warlocks), it shouldn't surprise anyone that I've continued playing the game in the interim (though a little less frequently now that my boss has figured out what the strings of random letters on my monitor are), and I think it's time to return to what makes the strategy feel particularly deep.

In particular, it's occured to me that several of the elements I discussed before are enabled by a single feature - finite resources. In this case, your primary resource is your HP, which you "spend" for better position and the chance to complete major spells. The key here is that your HP starts off at a set number (15), and generally only decreases. I mean, there are healing spells, but they're less efficient than their analagous damage spells, and in any case your HP can't go above 16. The healing spells have their uses, but in general, the game is about gradually moving toward 0 HP.

Why does that matter? First of all, it keeps the game from dragging out - every turn that passes brings you closer to the conclusion because resources are always being spent faster than they're generated. More importantly, perhaps, it prevents either player from establishing an unbeatable advantage, like you can achieve in many real-time strategy games. In Age of Empires III, if I played the Dutch and survived to the Imperial Age, I got an economy going that was so absurdly efficient that it was virtually impossible to defeat me (because I could replace units as fast as they died without putting any significant dent in my resources). In warlocks, you can't build up your resources and then steamroll your opponent - you only have the resources you started with, and even if you have a clear HP advantage, you can always be defeated by a single Finger of Death or permanent enchantment, so you're never safe.

Now, in order for this to work, it doesn't require that there is actually no resource generation at all, just that resource generation is limited. In addition to healing, I would count monster summoning as "resource generation," in which case it might seem possible to build resources (by summoning endless hordes of monsters). The fact of the matter, though, is that getting out multiple summoned monsters is hard. Not only can your opponent disrupt or dispel you, but once you have a monster summoned, you have to pay attention to them, or they might be charmed by your opponent, and become their ogre or troll. Monsters are a good investment, but they have to be maintained, and can be expensive to cast in the first place, so the decision to bring out a monster is itself an important strategic decision. While summon ogre is a popular opening spell, you can win a match without ever summoning anything.

And that's what really matters here - that every decision is meaningful. In most RTS games, resource gathering is so central, that it single-handedly determines the outcomes of the game, and decisions like army makeup and where and when to deploy units are secondary. While I immensely enjoyed AoE3, over time the games start becoming identical - the game was about perfecting my rapid development strategy. Games of Waving Hands, on the other hand, are wildly different and unpredictable, and that's what keeps me fascinated with it.

Speaking of strategy games, I just got my hands on a copy of Valkyria Chronicles, so now I have something other than Valentine's Day dinner on my to-do list this weekend.

Waving Hands

I was recently exposed to Spellcaster, or "Waving Hands," and was immediately excited to try it out. The logistics prevented me from trying it for a while, until I found a site that hosts automated sessions between players at RavenBlack Games (I'm Ellipsis on there, so if anyone wants to create an account and send me a challenge, feel free). Now that I've gotten to actually play the game (a dozen or so times) I can say that my excitement was justified.

Spellcaster is a game about dueling wizards. Nothing terribly innovative there, seeing as how an entire gaming medium (the collectible card game) was spawned by a game with the same premise. The innovative part is how you cast spells - by making gestures with your two hands (usually abbreviated as letters you can write down in sequence), combinations of which create spells. Now, this is conceptually very cool, but that's not what makes Spellcaster a good game. It's the fact that the gameplay is very simple (just consists of picking a gesture to make with each hand each turn), but very strategically deep. We can be a bit more specific, though, in talking about what makes the game good:

1) Micro-goals with feedback: Each individual spell is made up of several coordinated actions and could be counterspelled or otherwise interrupted by an opponent. As a result, every individual spell that you complete gives you a sense of accomplishment (proportional to the difficulty of completing the spell).

2) Engagement Level: The game allows you to control how much time you invest in it. There is no time limit between turns, so you can take as long as you want to make a decision. You can check on the progress of the game every once in a while (or just get an email reminder when it's your turn) and play at work. Alternatively, you can spend time strategizing, trying to figure out your opponent's plans, etc, and this time pays off. The ability of the player to control how they play can be very valuable.

3) Balance: Sure, the big spells are flashy and do a lot of damage, but one interruption from your opponent spoils the effort you put into preparing that spell. Meanwhile, the minor enchantments only affect your opponent for one turn, but they're easy to pull off and can disrupt his rhythm. What particularly pleases me about the balance is that even though there are different play styles, every spell is useful to every player - sometimes, a situation just calls for "remove enchantment" or for "anti-spell" and this is just as true for a defensive player as for a fan of the big, flashy spells.

4) Dynamic: You will never succeed at this game by picking a "dominant" strategy and going through the motions. The game relies very strongly on predicting what your opponent is going to do (so you can counter their attacks and make sure they're unable to defend themselves from yours), and the result is that play is very fluid and often unpredictable (your opponent is working very hard to make sure it's upredictable!).

5) Come Backs: The game does not provide any means for the winning player to secure and easily hold onto his lead. If one player has 5 monsters out and is invisible and immune to fire, cold, and physical attacks, all it takes it one dispel magic to level the playing field. Similarly, even if you hit your opponent with a strong spell, they might have something up their sleeve that they've been planning for a while that will take away your advantage. The point is that I've had many games where I was clearly ahead and ended up losing, and many in which I was clearly behind and still pulled out a victory.

This post, of course, isn't just an extended advertisement for the game - these basic elements that make the game compelling are elements that should be included in any game that strives to be strategic, because they are the elements that create a rich play experience. To balance out the post a bit, I can quickly point out the major problem with the game: steep learning curve. The only people who play this game are those who committed themselves to learning how to play it, and who are willing to repeatedly look over the 40+ entry spell-list trying to figure out what kind of options they and their opponents have, and what the advantages and disadvantages of each are. If a game had this kind of strategic depth and was able to lead players in and teach them the skills they needed in a more incremental way, it would be the stuff of greatness.

-Silent Ellipsis