Article: 121942 of rec.games.frp.dnd Newsgroups: rec.games.frp.dnd Path: news.tuwien.ac.at!newsfeed.ACO.net!Austria.EU.net!EU.net!howland.reston.ans.net!gatech!newsfeed.internetmci.com!news.kei.com!nntp.coast.net!torn!nott!cunews!freenet.carleton.ca!FreeNet.Carleton.CA!an803 From: an803@FreeNet.Carleton.CA (Eric Paquette) Subject: Population distribution per level Message-ID: Sender: an803@freenet3.carleton.ca (Eric Paquette) Reply-To: an803@FreeNet.Carleton.CA (Eric Paquette) Organization: The National Capital FreeNet Date: Thu, 15 Feb 1996 13:56:24 GMT Lines: 26 Bonjour, I'm searching for a mathematical formula describing the population distribution of the various levels. I tried 0.9/10^x where x is the level. Unfortunately, you need a population of around 1.1*10^20 to have at least one 20th level character. I tried 0.9/2^x where x is the level. Unfortunately, the sum of all those persentages totals more that 100%. The formula: 0.5/2^x where x is the level works well but I find it hard to beleive that 50% of all inhabitants are 0th level. Anyone has a reasonable formula for population distribution? With a population of 1000000, I would like at least one 20th level character. Also, the 0th level characters must be a reasonable percentage. It doesn't have to be a formula in the form of A/B^x where A is the initial percentage, B is a constant and x is the level. Any form of formula is acceptable. Thanks in advance. ALP, Eric -- Ramachil sur IRC; adresse électronique/E-mail: an803@freenet.carleton.ca, s968313@aix2.uottawa.ca (}-8 oO()Oo (}-8 Eric M. Paquette (}-8 oO()Oo (}-8 "A mouse is nothing but a sophisticated paperweight!" (}-8 Article: 121944 of rec.games.frp.dnd Path: news.tuwien.ac.at!newsfeed.ACO.net!Austria.EU.net!EU.net!newsfeed.internetmci.com!uwm.edu!math.ohio-state.edu!magnus.acs.ohio-state.edu!drusmith From: drusmith@magnus.acs.ohio-state.edu (Dru A Smith) Newsgroups: rec.games.frp.dnd Subject: Re: Population distribution per level Date: 15 Feb 1996 14:36:22 GMT Organization: The Ohio State University Lines: 42 Message-ID: <4fvgd6$sjj@charm.magnus.acs.ohio-state.edu> References: NNTP-Posting-Host: bottom.magnus.acs.ohio-state.edu In article , Eric Paquette wrote: >Bonjour, > > I'm searching for a mathematical formula describing the population >distribution of the various levels. > >I tried 0.9/10^x where x is the level. Unfortunately, you need a >population of around 1.1*10^20 to have at least one 20th level character. >I tried 0.9/2^x where x is the level. Unfortunately, the sum of all those >persentages totals more that 100%. >The formula: 0.5/2^x where x is the level works well but I find it hard to >beleive that 50% of all inhabitants are 0th level. >Anyone has a reasonable formula for population distribution? With a >population of 1000000, I would like at least one 20th level character. >Also, the 0th level characters must be a reasonable percentage. >It doesn't have to be a formula in the form of A/B^x where A is the >initial percentage, B is a constant and x is the level. Any form of >formula is acceptable. >Thanks in advance. I did this many many years ago, and may still have the stats. I'll have to check. But I did it with more than just levels in mind. I also tried to distribute the population of "adventurers" with the statistics given in the DMG for "class of random NPC". That way, the total # of 1st level people out there are somthing like 45% fighter-types, 30% thief types, 15% cleric types, 8% magic-user types, and 2% others ( psionicists, monks, death masters, bards, etc). Can't quite remember the total # of adventurers compared to the population as a whole. I'll check. Dru Smith ---------------------------------------------------------------- *** starting 10/1/95, new e-mail: dru@charybdis.ngs.noaa.gov *** "...all life is only a set of pictures in the brain, among which there is no difference betwixt those born of real things and those born of inward dreamings, and no cause to value the one above the other" -- H.P. Lovecraft, The Silver Key Article: 121960 of rec.games.frp.dnd Path: news.tuwien.ac.at!newsfeed.ACO.net!Austria.EU.net!EU.net!howland.reston.ans.net!newsfeed.internetmci.com!uwm.edu!math.ohio-state.edu!pacific.mps.ohio-state.edu!freenet.columbus.oh.us!magnus.acs.ohio-state.edu!news From: "Thane R. Hecox" Newsgroups: rec.games.frp.dnd Subject: Re: Population distribution per level Date: 15 Feb 1996 16:06:08 GMT Organization: The Ohio State University Lines: 14 Message-ID: <4fvllg$sqa@charm.magnus.acs.ohio-state.edu> References: NNTP-Posting-Host: hh3mac1.acs.ohio-state.edu Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Mailer: Mozilla 1.22 (Windows; I; 16bit) I've always found the old A*(1/2)^x to work out fairly well, where A is the total population and x is the level This gives (If I remember correctly about one 19th level char per million. A rough figure that seems to work well in alot of campaigns. Having said that I would strongly suggest NOT having fixed numbers like that in a game world...it's just not very condusive to good game mechanics. What if you want a group of evil high level baddies to have banded, they would have to represent a HUGE number of 'regular' people. My game world has a much higher distribution of regular joes, but then again the 'leveled' npc's seem to be the influential personas in the world so my pc's have no problem encountering them. Good-luck.. Thane Article: 121968 of rec.games.frp.dnd Newsgroups: rec.games.frp.dnd Path: news.tuwien.ac.at!newsfeed.ACO.net!swidir.switch.ch!nntp.coast.net!torn!news!server.uwindsor.ca!latimer From: Curtis Latimer Subject: Re: Population distribution per level In-Reply-To: X-Nntp-Posting-Host: server.uwindsor.ca Content-Type: TEXT/PLAIN; charset=US-ASCII Message-ID: Sender: news@news.uwindsor.ca (Usenet) Organization: University of Windsor, Ontario, Canada References: Mime-Version: 1.0 Date: Thu, 15 Feb 1996 23:43:59 GMT Lines: 50 > I'm searching for a mathematical formula describing the population > distribution of the various levels. > > I tried 0.9/10^x where x is the level. Unfortunately, you need a > population of around 1.1*10^20 to have at least one 20th level character. > I tried 0.9/2^x where x is the level. Unfortunately, the sum of all those > persentages totals more that 100%. > The formula: 0.5/2^x where x is the level works well but I find it hard to > beleive that 50% of all inhabitants are 0th level. > Anyone has a reasonable formula for population distribution? With a > population of 1000000, I would like at least one 20th level character. > Also, the 0th level characters must be a reasonable percentage. > It doesn't have to be a formula in the form of A/B^x where A is the > initial percentage, B is a constant and x is the level. Any form of > formula is acceptable. > Thanks in advance. Hi, You might want to try P(x) = 625000 * e^-x/1.5 This gives a formula that allows for 1, 20th level character for every 1,284,465 people in a population. Unfortunately it also shows that 0th level characters make up 48.7% of the population. I thiink that roughly half of the people in a population being 0th level is a pretty desent estimate. Not everyone is meant to be an adventurer. And not every farmer needs to be 1st level or more. Most surfs in a kingdom would be 0th level anyways. To have level means that the character is special in some way. Maybe of good breeding stock or so on.., I tried to get the percentage lower and even higher but that led to either having 1 20th level character for trillions of people in a popultion, or hundreds of 20th level characters per million in the population. So I would suggest to you to use this formula, or don't use expontial formula's. Maybe try a ratio system. ie 3 19th characters per 20th level character, 4 18th level characters per 19th level character... Curtis Latimer Article: 122014 of rec.games.frp.dnd Newsgroups: rec.games.frp.dnd Path: news.tuwien.ac.at!newsfeed.ACO.net!paladin.american.edu!hookup!lll-winken.llnl.gov!news.larc.nasa.gov!lerc.nasa.gov!magnus.acs.ohio-state.edu!math.ohio-state.edu!howland.reston.ans.net!surfnet.nl!news.nic.surfnet.nl!rug.nl!root From: A de Kreij Subject: Re: Population distribution per level Content-Type: text/plain; charset=iso-8859-1 Message-ID: Sender: root@rug.nl (Operator) Nntp-Posting-Host: molgen29.biol.rug.nl Content-Transfer-Encoding: 8bit Organization: Rekencentrum der Rijksuniversiteit Groningen References: Mime-Version: 1.0 Date: Thu, 15 Feb 1996 15:19:31 GMT X-Mailer: Mozilla 1.1IS (X11; I; IRIX 5.3 IP22) X-Url: news:DMtLE0.Cxz@freenet.carleton.ca Lines: 38 an803@FreeNet.Carleton.CA (Eric Paquette) wrote: >Bonjour, > > I'm searching for a mathematical formula describing the population >distribution of the various levels. > >I tried 0.9/10^x where x is the level. Unfortunately, you need a >population of around 1.1*10^20 to have at least one 20th level character. >I tried 0.9/2^x where x is the level. Unfortunately, the sum of all those >persentages totals more that 100%. >The formula: 0.5/2^x where x is the level works well but I find it hard to >beleive that 50% of all inhabitants are 0th level. snip. >ALP, >Eric >-- >Ramachil sur IRC; adresse électronique/E-mail: an803@freenet.carleton.ca, > s968313@aix2.uottawa.ca OK, what you need is some kind of statistical funtion. I don't know them by heart but you can find a statistics book in any reasonable school/college library. You could use normal stochastic distribution (bell shaped) in which case 10th lvl would be average. Probably better is something like Student's distribution or one of the other non equal sided distributions. Play around with the coefficients a bit to obtain the peak at the level you want and you're all set. Alternatively if you want lvl 1 to be most prominent use an exponential function # = A / exp(lvl) The disadvantiage of this aproach is that the integral is not 100% unless you do some extensive tricks (how much calculus do you want to perform?) so i would prefer the statistical aproach (all the distributons equal 100%). have fun! Arno kreijade@biol.rug.nl Article: 122017 of rec.games.frp.dnd Path: news.tuwien.ac.at!newsfeed.ACO.net!Austria.EU.net!EU.net!newsfeed.internetmci.com!in2.uu.net!news.u.washington.edu!root From: aaron@iphc.washington.edu (Aaron Ranta) Newsgroups: rec.games.frp.dnd Subject: Re: Population distribution per level Date: Thu, 15 Feb 96 09:05:25 Organization: International Pacific Halibut Commision Lines: 23 Message-ID: <4fvp49$fdg@nntp4.u.washington.edu> References: NNTP-Posting-Host: 128.95.250.37 Mime-Version: 1.0 X-Newsreader: WinVN 0.93.6 In article , an803@FreeNet.Carleton.CA says... > >Bonjour, > > I'm searching for a mathematical formula describing the population >distribution of the various levels. > >I tried 0.9/10^x where x is the level. Unfortunately, you need a >population of around 1.1*10^20 to have at least one 20th level character. >I tried 0.9/2^x where x is the level. Unfortunately, the sum of all those >persentages totals more that 100%. >The formula: 0.5/2^x where x is the level works well but I find it hard to >beleive that 50% of all inhabitants are 0th level. Yes, it is hard to believe. Far more then 50% should be 0th level. I'd guess upwards of around 80% or even more. Article: 122039 of rec.games.frp.dnd Path: news.tuwien.ac.at!newsfeed.ACO.net!Austria.EU.net!EU.net!howland.reston.ans.net!vixen.cso.uiuc.edu!ux6.cso.uiuc.edu!jayv From: jayv@ux6.cso.uiuc.edu (verkuilen john v) Newsgroups: rec.games.frp.dnd Subject: Re: Population distribution per level Date: 15 Feb 1996 20:26:05 GMT Organization: University of Illinois at Urbana Lines: 53 Message-ID: <4g04st$gms@vixen.cso.uiuc.edu> References: NNTP-Posting-Host: ux6.cso.uiuc.edu I don't have the time to work out the math right now, but something along the following lines seems reasonable (hopefully I didn't make a boo-boo, as this is basically off the top of my head--I'm sure Travis Hall will correct me if I did :). Assume that there is a probability P1(n) of dying between level n-1 and level n and a probability P2(n) of not advancing between level n-1 and level n. Starting at the initial point of 1, the fraction surviving to level 1 is F(1) = (1 - P1(1)) (1 - P2(1)) (Note that I'm assuming independence here. See below.) Now if you just wanted to keep this probability of death or no advance the same, taking F(n) = [(1 - P1) (1 - P2)]^n would give you the fraction at level n. However it is unreasonable to assume that P1 and P2 are constant over n. Rather P1 should be a monotonically decreasing function of n, reasoning that there is a certain amount of natural selection making survival to higher levels more likely given that you've survived to the one you're at now. Contrarily, P2 should be a monotonically increasing function of n, reasoning that aging, retirement, lack of opportunity, etc., will remove some eligible survivors from the pool of advancers. Hence, the formula would be n F(n) = F(n-1) * (1 - P1(n)) (1 - P2(n)) = Prod[(1 - P1(i)) (1 - P2(i))] i=1 Basically I'm trying to take in account the dependencies across levels by altering P1(n) and P2(n) rather than writing complicated interdependency formulas. This is a nice quick and dirty assumption that will get you a decent approximation. As a first cut try P1(n) = P1(n-1)^2 and P2(n) = Sqrt(P2(n-1)), for several P1 and P2 to see how things turn out. (Note that for x in [0, 1], sqrt(x) >= x and x^2 <= x, which is kind of counter-intuitive.) Give it a whirl on your favorite spreadsheet and see how it comes out. I've done similar calculations to obtain a stationary population from a given 1 year mortality curve from an initial value and the calculation took only a few minutes to set up. -- J. Verkuilen jayv@uiuc.edu A conversation overheard in the wierd world of Mattel(tm) talking dolls: Talking Barbie(tm): "Math is tough." Talking Ken(tm): "Group theory is intuitively obvious." Article: 122098 of rec.games.frp.dnd Path: news.tuwien.ac.at!newsfeed.ACO.net!Austria.EU.net!EU.net!howland.reston.ans.net!ix.netcom.com!netnews From: harkerii@ix.netcom.com (John Edwards) Newsgroups: rec.games.frp.dnd Subject: Re: Population distribution per level Date: Fri, 16 Feb 1996 05:15:01 GMT Organization: Netcom Lines: 30 Message-ID: <4g13bg$7o3@ixnews2.ix.netcom.com> References: Reply-To: harkerii@ix.netcom.com NNTP-Posting-Host: ix-den16-21.ix.netcom.com X-NETCOM-Date: Thu Feb 15 9:05:52 PM PST 1996 X-Newsreader: Forte Free Agent 1.0.82 The ever-so-wise an803@FreeNet.Carleton.CA (Eric Paquette) once said: ->Bonjour, -> I'm searching for a mathematical formula describing the population ->distribution of the various levels. Well, first of all, do you really need this? I ask this because, for myself (who is basically lazy) I don't see any need. Remember that levels (and classes, for that matter) are sort of a special thing. Not everybody gains levels. Only those who actually work at it (ie: Adventureres, Guardsmen, etc) will actually advance. The general populace of a city/town/country will, for the most part be between 0 and 2nd level. The only time I will worry about stats for NPC's is if I have a character who likes to break into homes/businesses and steal things, or if they decide to run rampant and kill everybody, and even then, the majority of the populace will run away. The easiest way to deal with general populations is to simply give them a single roll of a d4 and leave it at that. If you feel a need, roll lots of stats for guardsmen. Even more, if the players are prone to do bad things in town. (I even included a gold dragon masquerading as a shopkeeper for the thief to rob. Boy was he suppriesed...) John Fritz get up for god's sake. Get up! They've killed Fritz. They've killed Fritz. Those lousy stinking yellow fairies, they've killed Fritz! Those horrible atrocity-filled vermin. Those despicable animal warmongers. They've Killed Fritz Max - Wizards Article: 122133 of rec.games.frp.dnd Path: news.tuwien.ac.at!newsfeed.ACO.net!paladin.american.edu!gatech!swrinde!howland.reston.ans.net!math.ohio-state.edu!magnus.acs.ohio-state.edu!news From: Thane Newsgroups: rec.games.frp.dnd Subject: Re: Population distribution per level Date: 16 Feb 1996 14:37:17 GMT Organization: The Ohio State University Lines: 4 Message-ID: <4g24qt$aqi@charm.magnus.acs.ohio-state.edu> References: <4g23cv$aqi@charm.magnus.acs.ohio-state.edu> NNTP-Posting-Host: oh13pc08.acs.ohio-state.edu Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Mailer: Mozilla 1.22 (Windows; I; 16bit) Ack...disregard...and read the next post...I'm still getting used to being more than a lurker. Article: 122247 of rec.games.frp.dnd Newsgroups: rec.games.frp.dnd Path: news.tuwien.ac.at!newsfeed.ACO.net!Austria.EU.net!EU.net!newsfeed.internetmci.com!vixen.cso.uiuc.edu!howland.reston.ans.net!ix.netcom.com!netcom.com!wjr From: wjr@netcom.com (Wayne J. Rasmussen) Subject: Re: Population distribution per level Message-ID: Organization: NETCOM On-line Communication Services (408 261-4700 guest) X-Newsreader: TIN [version 1.2 PL1] References: Date: Sat, 17 Feb 1996 04:30:23 GMT Lines: 41 Sender: wjr@netcom5.netcom.com Eric Paquette (an803@FreeNet.Carleton.CA) wrote: : Bonjour, : I'm searching for a mathematical formula describing the population : distribution of the various levels. : I tried 0.9/10^x where x is the level. Unfortunately, you need a : population of around 1.1*10^20 to have at least one 20th level character. : I tried 0.9/2^x where x is the level. Unfortunately, the sum of all those : persentages totals more that 100%. : The formula: 0.5/2^x where x is the level works well but I find it hard to : beleive that 50% of all inhabitants are 0th level. : Anyone has a reasonable formula for population distribution? With a : population of 1000000, I would like at least one 20th level character. : Also, the 0th level characters must be a reasonable percentage. : It doesn't have to be a formula in the form of A/B^x where A is the : initial percentage, B is a constant and x is the level. Any form of : formula is acceptable. : Thanks in advance. I use to believe in formula for things like this. Especially having the bias toward lower levels as far as distribution goes. At some point, I began to not like this. It may have started when I was running a higher level campaign. Several years ago, I noticed something interesting. If you look at the Professional Go organizations in Japan, you will see a graph where the peaks are at both ends of the scale and bottoms in the lower-middle to middle. For those who don't know, Go is a game which is almost 3500 years old. To become a pro in Japan is very tough, I equate this to those who are capable of level advancement in dnd. The Ranks for Pro Go players are shodan, one dan, to nine dan. It isn't easy to make to 9 dan, but, over a long time you can make it. A major fault with this comparision is that go players don't die when they lose a game. So who is to say that the distribution can't be the same in a dnd world? In the end, the GM has to decided how he wants a distrubution for his world. sorry to prattle, :( wayne Article: 122255 of rec.games.frp.dnd Path: news.tuwien.ac.at!newsfeed.ACO.net!swidir.switch.ch!in2p3.fr!univ-lyon1.fr!jussieu.fr!oleane!tank.news.pipex.net!pipex!newsfeed.internetmci.com!uwm.edu!math.ohio-state.edu!magnus.acs.ohio-state.edu!news From: Thane Newsgroups: rec.games.frp.dnd Subject: Re: Population distribution per level Date: 16 Feb 1996 14:12:47 GMT Organization: The Ohio State University Lines: 7 Message-ID: <4g23cv$aqi@charm.magnus.acs.ohio-state.edu> References: NNTP-Posting-Host: oh13pc08.acs.ohio-state.edu Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Mailer: Mozilla 1.22 (Windows; I; 16bit) Ok sorry, I guess the inferrence I made wasn't clear enough. In my previous post I mentioned A*(1/2)^x works out to about 1 19th level per million what I didn't mention is that you can scale that to have almost any % of 0 level people you want. For example: 50%--in my opinion (and obviously others) way too low Article: 122264 of rec.games.frp.dnd Path: news.tuwien.ac.at!newsfeed.ACO.net!swidir.switch.ch!in2p3.fr!univ-lyon1.fr!jussieu.fr!oleane!tank.news.pipex.net!pipex!newsfeed.internetmci.com!news.kei.com!nntp.coast.net!col.hp.com!news.dtc.hp.com!lf.hp.com!apollo.hp.com!hpwin055.uksr!news!keary From: keary@news.an.hp.com (Teo Keary) Newsgroups: rec.games.frp.dnd Subject: Re: Population distribution per level Date: 16 Feb 1996 14:39:49 GMT Organization: Hewlett-Packard Company Lines: 12 Sender: keary@news.an.hp.com (Matt Keary) Distribution: world Message-ID: <4g24vl$2am@hpwin055.uksr.hp.com> References: NNTP-Posting-Host: hpanq7.an.hp.com In article , an803@FreeNet.Carleton.CA (Eric Paquette) writes: |> The formula: 0.5/2^x where x is the level works well but I find it hard to |> beleive that 50% of all inhabitants are 0th level. Do you find this high or low? I don't find it hard to swallow that half or more of a given world's population is below 1st level. The PC's are supposed to be unusually powerful hero-types (or at least potentially so), even at low levels. (IMHO) - Teo Article: 122296 of rec.games.frp.dnd Path: news.tuwien.ac.at!newsfeed.ACO.net!paladin.american.edu!gatech!newsxfer2.itd.umich.edu!newsfeed.internetmci.com!btnet!zetnet.co.uk!dispatch.news.demon.net!demon!mail2news.demon.co.uk!infernal.demon.co.uk From: lucifer Newsgroups: rec.games.frp.dnd Subject: Re: Population distribution per level Date: Fri, 16 Feb 96 19:54:26 GMT Organization: The Dark Abyss Lines: 30 Distribution: world Message-ID: <9602161954.AA001gi@infernal.demon.co.uk> References: X-NNTP-Posting-Host: infernal.demon.co.uk X-Newsreader: TIN [AMIGA 1.3 950726BETA PL0] X-Mail2News-Path: relay-4.mail.demon.net!post.demon.co.uk!infernal.demon.co.uk Hi.. : I'm searching for a mathematical formula describing the population : distribution of the various levels. : [Snip] : Eric It might be easier to group them into categories, ie: Per 100000000 Grouping Level ------------- -------------- --------- 90000000 Mundane 0th level. 9000000 Neophyte 1st level. 900000 Expert 2-5th 90000 Master 6-10th 9000 Whatever 11-15th 900 Wow 16-20th 90 Wow+ 21-25th 10 Silly people 26-30th. [30th seems tops] : "A mouse is nothing but a sophisticated paperweight!" (}-8 Only if it is dead! >:) w `o' Lucifer : "I don't know the meaning of the word surrender!... |--O-' : I mean I know it, I'm not dumb! Just not in this | / \____^ : context." - The Tick. Article: 122451 of rec.games.frp.dnd Path: news.tuwien.ac.at!newsfeed.ACO.net!Austria.EU.net!EU.net!newsfeed.internetmci.com!info.ucla.edu!library.ucla.edu!ihnp4.ucsd.edu!munnari.OZ.AU!bunyip.cc.uq.oz.au!student.uq.edu.au!s316903 From: s316903@student.uq.edu.au (Travis Hall) Newsgroups: rec.games.frp.dnd Subject: Re: Population distribution per level Date: 18 Feb 1996 23:17:42 GMT Organization: University of Queensland Lines: 25 Message-ID: <4g8c2m$4vh@dingo.cc.uq.oz.au> References: NNTP-Posting-Host: student2.cc.uq.oz.au X-Newsreader: TIN [version 1.2 PL1] Curtis Latimer (latimer@server.uwindsor.ca) wrote: : Most surfs in a kingdom would be 0th level anyways. 0-level surf? Now that's flat! :) (Sorry, couldn't resist.) Okay, okay, I'll contribute something useful. Have you considered using a function not described by a single equation? (Or not by an easily-found equation, anyway.) The equations which have been suggested all suffer from the fact that they asume the relationship between 0-level and 1st level characters will be the same as that between 1st level and 2nd level characters, which I doubt very much is the case. I recently have had to work out the demographics for a mages guild, specifically the largest guild of wizards in the world, which rules a fairly large nation and has about 6000 members. (The nation's population is something like 4000000.) I knew that I really couldn't afford more than 1 in 500 at the most to be wizards, and needed the numbers of wizards at various levels to scale gradually upwards. I think, with the numbers I worked out, there are about 1.7 times as many wizards of a given level as there are wizards of 1 level higher, at the low levels, and more like a ratio of 1.4 at the high levels. I haven't bothered with an equation, though, since the relationship changes towards the higher levels. The Wraith