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my opinion? don't.
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I vote for Geo's method too. Seems fairly straight forward and points awarded as you go, seems to best way. |
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I tend to agree with the Primitive One about third and fourth places. However, if players were to agree, an agreement could be reached between them to determine who is third and who is fourth. (The same went true for 1st and 2nd position in the first 2vs2 rated game) And if you assume a team game, the second player, provided she is in the winning team, should deserve some points for the victory.
Let's do the calculations for the aforementioned 2vs2 game with the system you proposed Slick. - Gecko: 4rd, rating 990 - Belisarius: 3rd, rating 1050 - Slynky: 2nd, rating 1104 - Alneyan: 1rd, rating 1000 So, assuming I plugged the right figures in the formula, we have the following: - Gecko:-16-11-13=-40 points - Belisarius:-18-14+13=-19 points - Slynky: -26+14+11=-1 point - Alneyan: +26+18+16=60 points While it would be very nice for me to gain 60 points (what an unbiased opinion!), I don't feel I deserve such a jump in the Ratings. Likewise, Slynky should win a few points and it does seem harsh for the poor Gecko. I will likely plug the numbers in your own system Slynky to see how it would come out, even if the only example we currently have is a bit peculiar. All in all it does look like a complicated matter, and I do not envy you Slynky for having to sort it out. |
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I second Geo's proposal for the reasons given below. (It isn't as if I could add anything)
I would consider a loss as having no more fighting potential. That is, if a player has no planet left but if he has troop transports/colony ships she should be considered as still being (barely) alive. However, lurking a single fighter in a red nebulae does not qualify as having any sort of military power. And likewise, parking a few warships in the same nebulae should not allow a player to remain alive when the game is obviously lost. (Granted, the case of a player being almost crushed but still able to gain a colony or two is a bit extreme and is unlikely to occur, unless you do want to fight to the bitter end) |
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I would hope we don't have people like that but I suppose there is always a surprise.
But before we get too involved in a discussion like this, IF there is a person like that, I suspect he would act that way in ANY of the scoring formulas we have suggested. Geo's makes it less likely. |
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Preposterous! I disagree with Geo's and all similar proposals. You take your lumps or laurels as the numbers dictate. For goodness sake what could be fairer? Surely fudging the numbers to encourage play is not the intent of the rating system. Shoot, why not go whole hog and give no loss in points for losing, just add points when you win. Criminy, this this fudging around is all fudged up. In case I wasn't clear, I say stick with the basic FIDE system and avoid these systems that will create artificial inflation. |
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Emperor ….. I bring both good and bad news.
First the Bad; the Primitve Hordes are moving in on our Homesystem as we speak, conquering our planets, emasculating our men and committing unspeakable acts with our women. Their leader will stop at nothing less than your head on a pike and the total destruction of our race. - And the good news ? - It seems like all other races have already bowed to the Primitive Hordes and we are the Last to fall. - And this is good in what way ? - We will be awarded a huge amount of points for coming second, making us look like a winner http://forum.shrapnelgames.com/images/icons/icon12.gif GP: There is no inflation in Geos system. The Idea is that coming 9th or 10th is really not that different and should be awarded approximately the same amount of (negative) points, while there is a huge difference between coming 1st and 2nd. |
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Grandpakim, I totally disagree with you when you say my suggestion is somehow fudging the results. On the contary I believe my suggestion is much more accuratly indicitive of the difficulty in winning or losing a large game. What you are propsing in fact is the system with the point inflation. (What does FIDE mean btw? http://forum.shrapnelgames.com/images/icons/icon7.gif ) Are you actually trying to say that winning a 20 person game is 19 times harder to do then winning a 2 person game? Then why give the ultimate winner of the 20 person game 19 times the points? On the other hand, why is the first person out of the 20 person game disportionally penalized to the tune of 19 times what they would lose in a 2 person game. Was the larger game easier so they should have been expected to do better? Does the large game exsist in some alternate reality where its simultaneously easier for those that lose and harder for those that win? http://forum.shrapnelgames.com/images/icons/icon12.gif It makes no sense to me to inflate the points artifically like this.
Also, why should everyone in the 20 person game get full credit for a win each time someone gets knocked out. Why should I get the same amount of points in a large game simply for outLasting Primitve, when we never met and I had no involvment in beating him at all, then I would for beating him head to head in a 1 on 1 game. No artifical point inflation, please. Just give me the points I have earned thank you. Don't give me extra points just because I chose to play in a large game. |
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1 game completed (notations made beside 4 players to indicate a pending adjustment to score when we can come to a reasonable agreement on how to rate multi-player games) |
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On the other hand why should the guy who finished second, beating out 18! count 'em, 18 other players get so much less than the winner? The power struggle at the end may well have been a close thing. The final result may have been decided by luck. Unfortunatley we don't know all the factors. Was it a no-contest and the winner really should get all the points while all the others lose points, or was it anybody's game right to the end and even the first man out should have received some points for his outstanding effort? The fact is we just don't know and, after all, the rating system is all about winning and losing not about how hard it was. Now apology time. I was incorrect in saying there is inflation in Geoschmo's system. I have no excuse for missing this fact and apologize, especially to Geoschmo as well as any others I may have offended. FIDE: Federation Internationale de Echecs (any of our French speaking contributors feel free to correct my spelling) Or in english, The International Federation of Chess. FIDE uses the rating system upon which our own SEIV rating system is based. This system is so good you can accurately compare yourself to someone long dead... as long as you both have (had) a rating. Ratings range from about 2900 for a world champion to about 500 for a tyro who continually loses. The average is somewhere around 1500. A typical Master would be around 2200 points, a Grandmaster 2500. Some years ago, if memory serves, the whole system (or was it just Canada?) had an adjustment because of deflation! This happens because people stop playing but while playing they got better and gained rating points. When they quit or died, those points were lost to the general pool ergo the deflation. Now, back to Geo's system. Not inflation, not deflation, but stagflation. Geo's idea tends to keep Ratings bunched close together. (I for one am disappointed how closely bunched the Ratings still are. I fully expected someone to have achieved 1500 points by now.) It also makes playing rated multiplayer less attractive. No matter how well or poorly you do, there won't be much change. Why bother? After playing for a year and winning, you find you gained 18 points! Well whoopdedoo! Think I'll stick with KOTH! I still can't see any good reason to fiddle with the Ratings in multiplayer. |
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Uh, huh? I don't understand the point you are trying to make here. It's probably me though. http://forum.shrapnelgames.com/images/icons/icon7.gif The point I was trying to make though was in a 1v1 game it's easy to figure out the points. He lost, you were 100% responsible for him losing, so you get 100% of the spoils. If you have 20 players and one loses it's not so easy to say who is the most responsible. We could try to come up with some system to cover that, but it would be hopelessly complicated and very subjective. So the fairest thing in my mind is just divide the points up. Over the course of the game, the better player should Last longer and end up with more points anyway. Quote:
</font><hr /></blockquote><font size="2" face="sans-serif, arial, verdana">Even if you never met him? Even if you never met anyone? http://forum.shrapnelgames.com/images/icons/icon7.gif You could be stuck in a corner and not meet anyone before 2 or three players get knocked out of the game. That's fine. It's the luck of the draw and all. But you haven't exactly earned anything. Certainly not three victories. Quote:
No offense was taken, no appology is neccesary. Were simply having a lively debate on a subject we disagree on. I am not totally sure the Chess system correlates directly to SEIV. You don't have 20 player games in chess after all. If the Ratings system were limited to 1 on 1 games, and we were talking about a 20 player round robin SE4 tourney then I would agree 100% with using that system. But in my mind there is a significant difference between a 20 player tournament of 1 v 1 games and a single 20 player game of SE4. I can see your point about the lack of movement in points with my suggestion. And I did consider one factor that makes it not so bad. Since 20 player games take so long to complete, and players get knocked out one at a time over the course of the game, the winner isn't going to get a one-time 300+ point jump in their standings. It will happen gradually over the course of the game 16 or so points at a time. So in this regard it's not so bad. But frankly human nature is what it is. You can't deny it or try to fight it. No matter how ideal we'd like to think people act, there will be a tendancy for players once they get near the top of the Ratings to "do the math" and have second thoughts about whether joining a 20 player game is worth the risk of getting ganged up on early and taking a 300+ point drop in their Ratings. Do you honestly see no problem at all with this? I just can't think of an acceptable solution to what I see as two separate problems. It's not like we can give the winner credit for 19 victories and only give the loser credit for one loss, can we? Wait, can we? http://forum.shrapnelgames.com/images/icons/icon7.gif It sounds kind of top heavy, but what if we do that. As each person loses they get a loss equal to the average of the points of the players remaining in the game, and each player gets a win against them. So the ultimate winner gets 19 wins. The 2nd place guy gets 18 wins and one loss. 3rd place gets 16 wins, and still just one loss. 4th place, 15 wins, still one loss. et cetera. What effect would such a top heavy system have on the Ratings. Would it cause serious point inflation? Geoschmo [ March 12, 2004, 13:25: Message edited by: geoschmo ] |
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[ March 12, 2004, 23:04: Message edited by: narf poit chez BOOM ] |
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[ March 13, 2004, 01:34: Message edited by: narf poit chez BOOM ] |
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There is precedent though for your idea. In business it is common to add a fancy title and a slew of new duties but only a 10% raise in pay (or sometimes none). Not fair? Of course not, but it happens all the time. Quote:
Warning: The following is not intended as a slight or insult but only to help me make a point. The system you describe penalizes the good players while rewarding the poor players. To me this sounds like communism or a welfare state. You've probably guessed, I'm not a big fan of either. Quote:
The very nature of SEIV makes this inequity inevitable and has led yourself and others to find ways to eliminate the element of luck-- a project I heartily endorse! But make no mistake, you will never eliminate luck without changing the essential character of the game. The chess rating system works perfectly for any solo (as opposed to team) competition that is scored by win/draw/loss. It would work great for tennis but not so well for the 100 meter dash (the time clock is best here) and very poorly for hockey (where personnel changes taint the result). It works better for chess then SEIV because of the luck factor. In chess there is no luck. In SEIV luck tends to keep Ratings close by allowing a very poor player to win more games than he should. Quote:
Finally, your latest suggestion would be very inflationary. New players joining the system would have an impossibly steep hill to climb. Geo, I went into such detail regarding the FIDE system to emphasize the large point spread. Currently SEIV has less than 200 points between best and worst. What that means is our best player against our worst in a 16 game match would have a 10/6 result. Hardly remarkable. The system is young, the spread will grow. A personal, 300 point drop does not scare me. Does it really scare any of you? |
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I see both points here and I must say, I fall inside them. That is to say, somewhere in the middle of what Grandpa Kim and Geo are discussing. Alas, I'm not sure of a solution.
On one hand, a multi-player game is like a tournement of several games yet it is unlike it. Of course, due to luck, SE4 is unlike a chess game. It becomes a bit more unlike a chess match in a multi-player game. More luck and more events to "upset" a good player. And, in multi-player games, it's the only place to add the dimension of diplomacy (not found in 1 x 1 games) and this is yet another side of a skilled player. (that AND team coordination). If we made scoring like a series of 1 x 1 games for each to be rated, I fear there would be no one joining in them. After all, even the best diplomat would be at a disadantage if starting in the corner where a lack of empires to "chat" with would be a bit of an unfair start. (but, that is the "luck" discussion we have had several times...yet, who would want to chance the loss of a 100 points?) In contrast, 5 (for example) different games would "balance" out bad starting positions...assuming a person wasn't unlucky for 5 games in a row http://forum.shrapnelgames.com/images/icons/icon10.gif . To make it a personal example, I'd much rather play (and get the rating from) 5 1 x 1 games than take a chance on being Last in a game with 5 rating players. While I recognize I might possible end in first place, I wouldn't want to wager a bad starting position on it. Without looking back at the formula Geo has suggested, I would propose a "final adjustment" to it that goes like this: after all the points have been computed, we add (and subtract) 1 point for each person beaten (and lost to). A variation of this would be, instead, to add (and subtract) the number of points equal to the number of rated players in the game TIMES the number of players beaten (or lost to). So, in a 5-player game, the loser would get an adjustment of -4 or (considering the second suggestion) -20. Just some food for thought. |
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Quoting myself:
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When it all comes down to it, fitting square pegs in round holes ( http://forum.shrapnelgames.com/images/icons/icon12.gif ) like we are trying to do when we adapt a chess formula to a different sort of game, it's ONE game, not a Swiss System pairing of 7 rounds. Having said that, I'd be willing to go for a 50-point gain (or loss). I MIGHT feel differently IF all multi-player games were played on Geo's Balance mod with an uninterested party positioning all the players in fair starting positions (etc.) but that is probably not going to happen on the majority of multi-player games. Nature of the "beast" and all. Another "addition" to rated games might be to add a rule that says: Before game start, if a player has changed his mind about having himself rated in any particular game, he may withdraw his request to be rated. This would give a person the chance to change his mind if the number of players requesting to be rated surpassed the limit of what he was willing to gain/lose. Again, I just don't know...tossing out ideas. |
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Scared, no. But I'd be a little ticked off about it probably. http://forum.shrapnelgames.com/images/icons/icon7.gif
I am not sure how we got off on the issue of fairness of starts. I guess that is a factor for some, but not so much for me. I think all that evens out in the long run. |
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Well, that's my thought, anyway... http://forum.shrapnelgames.com/images/icons/icon7.gif . |
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Since my Last post I've come to realize there is quite a difference between one on one and multi-player.
Consider 2 scenarios: </font>
In the first, there will always be one player out first effectively losing to all 19 opponents and one winner effectively beating all 19. In the round robin the chances of one player winning (or losing) all 19 of his games is absurdly small. No-one in KOTH has put 19 victories together back to back, and I don't think it will ever happen. This makes it both easier to win and easier to lose in multiplayer. To accomodate this fact, I suggest computing the new Ratings normally then dividing by 2. This still leaves a substantial change in Ratings but recognizes that the same feat is much harder in one on one. Slynky, remember that while you are playing that rated multi-player game you could easily play 20 one on one's. This should help smooth out the element of luck. Also you could be playing several, rated multi-player games and as Geo points out, things tend to even out. Some will give you a good start, some bad. By playing many games, the element of luck diminishes. It is players like I, who play a small number of games who should be worried about luck. |
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Heh ya,
I really dont mind what system we end up with. But, it needs to be decided with the number of multi-player games going at the moment. I would prefer a system that penalises the 1st loser in a 20 player game at around 3 times the loss of a one on one - so there is some extra penalty but not enough to stop people rating big games. P.S, Slynky, I won my KOTH v Warlord Ragnarok, can I have some points so I have more to give you when you get a chance http://forum.shrapnelgames.com/images/icons/icon12.gif |
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Still needing to come to an agreement on multi-scoring before we "cast the die". |
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In the insterest of compromise though, I think Gpa's "Round Robin/2" suggestion is a step in the right direction. I also think Joachim has a decent point that losing a 20 player game should cost more then just losing one game, as the "fractional" system allows for.
The problem is how do you come up with a modifier that works for all number of players in a game? Dividing by 2 might give you a number you like for a 20 player game, but what about a 5 player game, or a three player game? Here's a suggestion. Let's call it the "2/N" system. Take a 20 player game. The first player eliminated, you calcualte the points lost to each player as you would in the round robin system. But then you multiply each of those by a modifier which is equal to 2/number of players. In a 20 player game the modifier would be .1. So assuming equal rankings the player would lose 1.9x points. Each of the 19 winners would get .1x points. This is twice the amount gained in the fractional system. Plus keep in mind each player isn't getting an equal share here. x in this system would be a different number for each player, unless they all have the same ranking. In this system assuming equal rankings the points would range fomr -1.9x for the loser, to +2.2x for the winner. Less on the high end but more on the low end then the straight fractional system. The advantage of this system over a straight x/2 modifier is that the formula works for any number of players. 2/2 (1x1 game) = 1. This system also has no point inflation since 100% of the points awarded to the winners comes from the losers. It has a little multiplication, but that's what we are wanting, to give a winner of a big game a little more then a 1x1 game. |
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Reading down only Gramps seems to be against the principle of the “fractional” system, all the other reservations seems to be on the number of points on offer.
The easy way to increase the point-spread of the "fractional" system is by multiplying the results with a fixed number based on the game size. This can be just an easy formula like (1+ gamesize/20) or we could make a fixed table. Or we could just go totally non-mathematical and make a table for different gamesizes. A little work to set up, but very easy for Slynky to use later http://forum.shrapnelgames.com/images/icons/icon12.gif |
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Games needing rating computation (in this order):
4 player multi-player (Alneyan, Belisarius, Gecko, Slynky) Alneyan vs Primitive Joachim vs Warlord Ragnarok I can compute the Last game (it is dependent on nothing else) but if I compute the second game without doing the first one, Prmitive gets less points (possibly...and probably only 1 point less). So, should I wait for what appears to be an agreement on rating multi-player and then do the second game or go ahead and go ahead and compute the second game (which, I suspect, is mostly up to Primitive). |
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Point inflation is a problem with the "unequal" system I suggested most recently. Point inflation is bad because you are giving ome players more points then you are taking away from other players. This makes it harder on new players to climb the rankings.
The "round-robin" style on the other hand has point multiplication. It's not inflation in the sense that you aren't generating points out of ether, but the massive point swings generated by such a system are an unfair and potentially damaging factor. If you play two games, lose one, and win one, you should basically be back close to where you started. Not exactly of course, becasue it will be factored by the the relative abilities of the players you played. But if you play a 20 player game and lose, you will have to play and win 20 1x1 games to get back close to where you started. That's just not fair IMHO. Gpa's recent suggestion, call it "Round-Robin/2" has the same problem, only to a lesser degree because of the artifical manipulation of the results. My original "fractional" suggestion doesn't have any point inflation or point multiplication. Gpa's point that it doesn't allow for enough rank movement isn't really accurate. The reason there isn't much rank movement isn't because of the point system being used, it's because SE4 games take a long time to finish. It's inherantly slow. Much slower then chess of course. I am not wedded to the "fractional" system at all. I would welcome an alternative suggestion. I just haven't seen one yet that isn't what I wouldn't call seriously flawed. |
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How about this
Rs = standard rating change Rm = modified rating change for multiplayer n = number of players Rm=Rs(1.5 root [n-1]/[n-1]) For three players 1.59/2 or .794 of a one on one game For 10 players 4.33/9 or .481 For 20 players 7.12/19 or .375 Just to simplify things I will consider that a victory over a single player will give a victor a 16 point jump in the Ratings. In a three player game the victor would currently get 32 points. With this system he gets 25. In a ten player, instead of 144, he gets 69. In a twenty player instead of 304, he gets 114. I could live with this. I tried it using square root but the final numbers were just too low for my taste. I don't know if anyone likes this or if Slynky is willing to use it, but it seems like a good compromise to me. Is a hundred point loss to the first man out in a twenty player game too much? I think it's okay. What does everyone else say? Edit: Note this works for one on one games too since any root of 1 is 1. I don't know how your calculating program is set up but you could possibly just insert this into the formula. [ March 17, 2004, 04:13: Message edited by: Grandpa Kim ] |
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Not only Primitive will lose points (several, depending on the formula), but I will gain more Slynky. If you compute my score for the 2vs2 game with 985 (after my defeat with Primitive) instead of my current 1000, I will win several points in the bargain.
And obviously, you do not want to compute Primitive's game first, lest he takes over your nice seat at the head of the Ratings. http://forum.shrapnelgames.com/images/icons/tongue.gif (And with Asmala coming, you ought to be wary Slynky, as this seat might become slippery. *Chuckles*) |
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Too get the ball rolling, maybe we could put this to a vote:
A: Linear system: Equal point spread between the positions. B: Some sort of Logarithmic system: Significant difference between 1st and 2nd, Small difference between Last and second to Last. C: Something completely different. I’m all for B http://forum.shrapnelgames.com/images/icons/icon12.gif |
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B
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Once Geo & Kim started talking Mathematics, they lost me.
I'll abstain from the Vote, but am if forced will lean towards B. Just no-one ask me to do any calculations. |
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Yeah, you can always tell when the Baron is playing with his socks off: he has 20 ship fleets. http://forum.shrapnelgames.com/images/icons/icon10.gif
Primitive, this part doesn't matter to me; it's the truncation I'm interested in. (Look that up in your Funk & Wagnall! http://forum.shrapnelgames.com/images/icons/icon12.gif ) Edit: And possibly more important. How easy will it be for Slynky to implement? [ March 18, 2004, 02:55: Message edited by: Grandpa Kim ] |
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Games needing rating computation (in this order):
4 player multi-player (Alneyan, Belisarius, Gecko, Slynky) Alneyan vs Primitive Joachim vs Warlord Ragnarok Primitive vs RexTorres (reading all comments on a group rating system...still looking for something that doesn't "skew" the Ratings too much and doesn't result in 20 minutes of computations...I think, if I understand Geo's Last suggestion, that it would result in a lot of manual computation/adjustment. Let's not give up yet.) |
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Multiplayer Ratings
I've been messing with a spreadsheet and some formulas and are half way to a system with a table. The value from this table should be multiplied with the score from the standard calculation, using loosing players ranking vs the average of the remaining players rankings (or if Slynky is lazy, the average of all players). There is a lot of factors to play with, but I would like some input before I spend more time (And yes, I have noticed there are less points lost for coming Last in the 20 player game than in the 10 player game http://forum.shrapnelgames.com/images/icons/icon12.gif ) The table so far gives these values for: 3 players: 1,51 / -0,08 / -1,44 4 players: 2,04 / 0,29 / -0,74 / -1,59 10 players: 3,73 / 1,58 / 0,71 / 0,17 / -0,25 / -0,60 / -0,91 / -1,20 / -1,48 / 1,75 20 players: 5,26 / 2,56 / 1,59 / 1,05 / 0,68 / 0,40 / 0,18 / -0,02 / -0,20 / -0,36 / -0,52 / -0,66 / -0,80 / -0,93 / -1,06 / -1,19 / -1,31 / -1,44 / -1,56 / -1,68 Too high ? Too low ? Stupid idea to start with ? And if anybody has the time for a crash course in displaying tables, it would be highly apreciated http://forum.shrapnelgames.com/images/icons/icon12.gif |
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Slynky,
Please make the Koth game Electrum v Primative a rated game. Thanks |
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Primitive, that looks okay to me, though I think the winner in a three player should get a bit more. I understand what you are trying to do, but...
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1 game completed (Baron Grazic vs Fire) 1 game added (Electrum vs Primitive) (If I have missed a posting other than what is listed below, please let me know) Games needing rating computation (in this order): 4 player multi-player (Alneyan, Belisarius, Gecko, Slynky) Alneyan vs Primitive Joachim vs Warlord Ragnarok Primitive vs RexTorres |
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Just bumping my questions from 3 Posts down.
Anybody else than Grandpa care at all ? Slynky ? |
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As to my two cents, again I dont mind what system as long as it is not too much to completely discourage large rating games. |
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Personally, I'm just waiting for you to decide, Slynky. After all, you're the one who has to deal with it. If I can't live with your decision, that's my problem. http://forum.shrapnelgames.com/images/icons/icon7.gif
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I think I am in favor of Geo's formula with a twist. Let me see if I can explain it easily.
His original formula (a few pages down) was (to summarize) to average the score of the people who beat the first person out (and so on) and apply points. This resulted, as some people disagreed with, a result that simulated a single game. So, my twist would be to adjust this "straight" score to vary the spread so that losses and wins would spread out more. And this is what people have been trying to do with other suggestions. So, the best I can figure and the easiest on me would be to figure it the way Geo said and then multiply the points gained (or lost) by the square root of the number of players in the game being rated. So, (duh!), if 4 players were in the game, the square root modifier would be 2. This is probably a common number of people for a multiplayer game. But, if 9 (to make it easy) were in a game, the multiplier would be 3. One has to see 16 people in a game before the multiplier became 4. This, in my opinion, would lesson the effect (both on winners and losers) in a big game...while giving a pretty fair representation for the most common game. In other words, the bigger the game, the less chance that a loser would suffer a huge loss from a bad position (and less the chance a winner would gain a huge amount of points based upon his lucky starting position). I think I will go with this, which I think is a fair representation of the suggestions we have read and attribute any disparity in such a rating to a "true chess" rating formula to luck, and a lack of true comparison to any chess tourney/game. |
Re: SE4 Rating System
Thanks, Grandpa Kim. http://forum.shrapnelgames.com/images/icons/icon7.gif
I am convinced of a few things: (1) Nothing we have talked about will be as precise as is measured in a chess game/tourney. (2) Luck will even itself out over a number of games but it is more apparent that it will equal out over 1 x 1 games more as, was pointed out, than multi-player games as they Last for so long. (3) No one would like being rated in a game of multiple players that would result in a possible loss of 100+ points even though there is a possible gain of 100+ points. (4) Generally, I think people believe a multi-player game should result in more points lost or gained than a single game but not as much as equal to a single game times the number of players in a multi-player game. (I hope that makes sense) (5) We want a way to compute scores as people are eliminated from a multi-player game. |
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