Weird thing about the chance one roll has to beat another:

A while ago, EastwoodDC posted a spreadsheet for calculating the chance of rolling any given result on a number of whatever sized dice. I started looking at the chances one roll had to beat another and I noticed something weird.

When rolling 1d6 against 2d6, the chance that they'll tie is equal to the chance of rolling exactly 7 on 3d6, and the chance that 1d6 will beat 2d6 is equal to the chance of rolling 6 or less on 3d6.

When rolling 2d6 against 3d6, the chance that they'll tie is equal to the chance of rolling exactly 14 on 5d6, and the chance that 2d6 will beat 3d6 is equal to the chance of rolling 13 or less on 5d6.

When rolling 3d6 against 4d6, the chance that they'll tie is equal to the chance of rolling exactly 21 on 7d6, and the chance that 3d6 will beat 4d6 is equal to the chance of rolling 20 or less on 7d6.

So I'm pretty sure that, anytime you roll Nd6 against (N+1)d6, the chance they'll tie is equal to the chance of rolling N*7 on (N+N+1)d6 and the chance that Nd6 will win is equal to the chance of rolling (N*7)-1 or less on (N+N+1)d6.

Messed around a little more, and it looks like 1d6 ties 3d6 as often as 4d6 will roll exactly 7; 1d6 beats 3d6 as often as 4d6 rolls 6 or less; 2d6 ties 4d6 as often as 6d6 will roll exactly 14; and 2d6 beats 4d6 as often as 6d6 rolls 13 or less. So I expect that, when rolling Nd6 against (N+2)d6, they'll tie as often as (N+N+2)d6 rolls exactly N*7, and that Nd6 will beat (N+2) as often as (N+N+2) rolls (N*7)-1 or less.

Wait. Those look the same. Can I just follow this pattern for any number of d6?

Nd6 ties (N+M)d6 as often as (N+N+M)d6 rolls exactly (7*N).
Nd6 beats (N+M)d6 as often as (N+N+M)d6 rolls (7*N)-1 or less.

And then, since the "7" there is a result of the average outcome of a single d6, can I extend it to other sizes of dice?

NdD ties (N+M)dD as often as (N+N+M)dD rolls exactly ((D+1)*N).
NdD beats (N+M)dD as often as (N+N+M)dD rolls ((D+1)*N)-1 or less.

Seriously just guessing at this point (could probably figure it out by hand with more spreadsheets, or by pure math from Pascal's Triangle) but I think this is going to work.


EDIT: VanVelding has tackled the math for this here and here.
EDIT ²: and here.

Final thoughts on TRO:3063

I was going to write more about TROs, but I don't have my notes together yet, so... TRO:3063!

It was initially released back around August 25th with the final version posted on September 9th. You can download it from The BattleTech Reader, OurBattleTech, BattleTechUniverse, Lords of the Battlefield, or the BattleTech Engineer. If you are considering a similar project, I highly recommend you look at the advice in Steve's 'Lessons Learned' document. Much of it applies to non-BattleTech, non-book projects as well.

There's already been some discussion of the TRO and the reception has been overwhelmingly positive. It matches, and in some ways exceeds, the quality of the current official TROs -- if I knew nothing about BattleTech, I don't know if I could tell the difference between it and Catalyst's books. (Granted, it was produced as a leisure activity over five years; Catalyst does not have that freedom.)

Looking at it now in retrospect:

Many readers seem curious about why the book covers the units it does or why they're built the way they are. (Power level, prevalence of SPLs & AMS, lack of infantry/spaceships/Clans, etc.) The reasons have been touched on in a few places, but for a book like this, it would be helpful to have a foreword explain more completely what the writers hoped to achieve and some of the thinking behind their decisions.

TR:3058, TR:3060 and TR:3067 show an overwhelming trend towards putting fusion and XL engines on new and upgraded vehicles. (The Tokugawa and Schiltron, for example.) I don't know that it was intentional, but TRO 3063 fits that trend quite well.

The manufacturing dates published by Catalyst (TechManual, Tactical Operations and so on) haven't been accurate enough for me to take them as Gospel, so me and Steve talked about how far R&D might've gotten by 3063. I suppose in-universe editorial comments [set off in square brackets, the way TR:2750 does] could have covered those weapons from a 3070s point of view.

I miss the Heavy PPCs. Not for the PPC itself, but for what having a heavy main gun forces you to do with the rest of your armament. It's more elegant. Less jumbled to read and fewer weapons to roll during play. (On the whole, the designs in this book are probably a little too optimized for my taste, but not moreso than what I remember of Catalyst's trends for the period.)

It's so hard to tell what's changed since I did my editing pass because I reviewed most entries only once, looked mostly for just the biggest issues, and tried to preserve/highlight (instead of replace) the original intent. (If you see a spot with too many three-clause sentences, or where paragraphs are all weirdly equal lengths, that's probably me.) And I really need to thank the other editor(s?) for going through entries after/before I'd gotten to 'em. But man, I'm scrolling through the book, and I know Steve's changed stuff, but it's so hard to pick out. I've only got two so far: looks like he overruled my cuts to the Feng-Niao, and I think he took a couple SPLs off the Sabra.

Mini Campaign System

Some of these rules will look specific to BattleTech (because I'm trying to manipulate 'Mech stats into matching the fluff better), but the basic structure could probably be adapted to other game universes.

Before choosing forces you'll need to calculate three or four values for every 'Mech that will be available to the players:

Production Cost = [(cruising speed)*(armor points)+(jump MP)*(armor mass)/(mass of individual jet)+(heat dissipation)*(maximum possible heat)+(weapon and equipment BV)]/10
Ready Maintenance = 3*(number of weapons, equipment, jump jets and excess actuators) + 2*(maximum possible heat)
Active Maintenance = 1.33*(armor mass)^2/(engine mass)
Repair Cost = (C-Bill price)/100,000

These values are permanent traits of a given design and do not change during play. They're scaled so that a typical 2866-3050AD 'Mech battalion costs 3600 production points and is able to ready 90% of its force, deploy 67% an average distance, repair 22%, and replace 3% every turn.

At the beginning of every turn the players generate 3 objectives. Each objective has a random value and sits a random equidistance from both players. Roll 2d6 for each objective to determine distance, and roll 2d6*20 to determine their values.

There are two ways to play this:

Bean Counting Way: At the beginning of each turn, players add 3600 points (assuming that's how many they bought forces with initially) to their unused points from the previous turn. They must pay the "ready" cost for every unit they wish to use this turn. They must also pay an amount equal to the "distance" roll x "active" cost for every unit assigned to a scout force or a main force. Units which were inactive all turn can be repaired at the end of the turn. New units can be purchased for 8x their "Production" cost.

Fast Way: Before play begins, subtract the sum of each force's "active" costs from half the sum of its "production" cost; record the difference on a token. The force receives this many bonus points each turn (or, if negative, must pay that many points each turn). Use the 2d6*20 roll to determine the equidistance to each objective: if it comes up doubles, then players must pay 2x "active" cost to assign a unit to a scout force or main force; otherwise, pay 1x "active" cost. Deactivating an undeployed unit generates points equal to its "ready" cost. Units that were neither deactivated nor deployed get repaired for free at the end of turn. "Objective" points must be spent to purchase new units (at 1x "production" cost) the same turn they are won.

Now, roll 1d6 to determine the type of objective:

1-2 Stand Up Fight (whoever destroys the most units (powered down units count half) in 10 turns)
3-4 Territory (whoever has the most hexes closer to his ground units than to opposing ground units at the end of turn 10)
5-6 Grab (divide points into 4 equal lots on the middle of the board: any 'mech can carry points (on its back) equal to its tonnage, but if it doesn't have hands another 'mech would have to put them there; weapons fire destroys 1 point per point of damage) 

Once the objectives have been generated, both players secretly assign their units to any of three roles (scout, main body, rear guard) at any of the three objectives. The objectives are then resolved in any order the players choose. Resolve one site entirely before moving on to the next.

Any unit that exits off a neutral map edge is considered to have powered down or fled (as appropriate). 

Scouts

Determine maps randomly, then lay them single file, narrow end to narrow end. Both players reveal their scout detachment. The force with higher average speed gets to choose its home edge. Disregard any units still on the board after 10 rounds.

If scouts advance off the opposing edge, sum their Cruising MP (plus the range of any Active Probes, plus 1 for every .25 tons of "sensors"). The opposing player then matches that total by revealing Main Body units and summing their Cruising MP (plus the range of any ECMs on the revealed units). The "EW Suite" on a revealed RVN-1X counts as 4 points of Scout probe (even when deployed with the Main Body) and 6 points of Main Body ECM (even when deployed as a Scout). If the opposing player runs out of Main Body units, begin revealing Rear Guard. If the opposing player has no Main Body or Rear Guard, then the Scouts win the Objective.

For each 2.5 tons of communication gear that advances off the opposing edge, the player may ask one yes/no question about the opponent's Main Body or Rear Guard.

Any scouts that retreat off a friendly edge join the Main Body action, but do not count as "revealed" units or contribute to the Main Body's average speed.

Main Body

After scouts finish revealing units, both players secretly decide which of their unrevealed units they want to commit to the Main Body, which they want to send after the enemy's Rear Guard and which they want to send back to their own Rear Guard (units assigned to Rear Guard do not have to pay the "active" maintenance cost). Both players then reveal all Main Body units and any Rear Guard units (like artillery or Mobile HQs) that affect the Main Body engagement.

For each 2.5 tons of communication gear a player has in the Scout force or Rear Guard, the player may choose a single mapsheet. Determine the other mapsheets randomly. The Main Body with higher average speed gets to choose its home edge. Resolve the Objective as described earlier.

If a player sent Main Body units after the opponent's Rear Guard, then lay out a second map for that battle. The Rear Guard player chooses all mapsheets, reveals all forces, and deploys on the map. The attacking player chooses home edge. Play this battle simultaneous with the Main Body battle but do not end until one side or the other has surrendered or withdrawn from the Rear Guard maps.

If a player wins the Objective or advances units off the opposing map edge, then that player may choose to pursue any defeated or retreating opposing forces.

Rear Guard

Lay the maps single file, narrow end to narrow end. If the fleeing player already laid out a second map for the Main Body battle, then use those maps; the victorious survivors of that secondary battle chooses home edge and deploys their surviving forces opposite the pursuing player's home edge. If the defending Rear Guard has not yet been attacked, then that player chooses all maps and home edges and deploys the Rear Guard on the board (unrevealed units originally assigned to Rear Guard may deploy as Hidden Units). Otherwise, determine mapsheets randomly, and the force with higher average speed chooses home edges.

Fleeing units enter from the pursuing player's home edge. Pursuing units enter from the same edge on the following turn. The battle ends after ten turns (if not earlier).

Base Defense

To attack your opponent's resources directly, assign units to Scout, Main Body and Rear Guard like a normal objective. Then:

Bean Counting Way: travel the highest distance rolled for this turn x2 to destroy/steal points or inactive units worth 5x the total Cruising MP of your Main Body.
Fast Way: travel the highest distance rolled for this turn +2 to destroy/steal inactive units worth 5x the total Cruising MP of your Main Body.

Your opponent can assign "Base Defense" forces to Scout, Main Body and Rear Guard as well; however, the defender can reinforce the "Base Defense" Main Body by calling back Rear Guards assigned to other objectives. To do this, the Rear Guard unit cannot have been involved in a battle, and must pay the "distance * active maintenance" cost for whichever objective they were originally assigned to. (The Defender can also pay to "ready" otherwise unassigned units.)

VARIATIONS

Obviously my equations at the top there are aimed at 2866-3050 play, but higher tech shouldn't be too hard. Do maybe 6000 point battalions for the century before and after; count CASE as the equivalent mass of standard armor; make Gauss weapons add their damage to both heat and heat dissipation counts; I'm not sure how to account for gyro or cockpit variants. 

Could add forward bases to shorten travel distances and to avoid keeping your eggs (points) in one basket, or have your resource points travel with the units themselves. Another thing I considered was that any resource points that didn't get used by the end of a turn would move themselves 1d6 distance towards the opposing player, and would require their own set of "Base Defense."

It'd be easy enough to reflect the quality of the battalion's technicians, or their relationship with House quartermasters, by giving the player more or less resource points than were used for purchasing the initial force.

Raising the value of the Objectives, or lowering the price of replacement 'Mechs, would encourage more decisive battles (and fewer retreats).

[Edit, August 24: I've changed the "Ready Maintenance" equation so that LRM-5 boating no longer breaks even.]

Campaign for Characterful Efficiency

Some people say that writing flawed and inefficient designs is more fluffy, colorful, interesting or fun (I occasionally see it expressed as "canonical designs are required to suck"), but that's not automatically true. Machines can make good use of their weight while still having a lot of character.

efficient and characterful

Regardless of how flawed or how perfect a vehicle is, much of its character comes from things like actuator placement, armor distribution, armament, weapon placement, firing patterns and game balance. These things establish a consistent and non-arbitrary aesthetic which prevents designs from seeming arbitrary and soulless.

Now, many fans define "good" design as making efficient use of tonnage. And it makes sense to judge designs by their mass - mass is central to construction and customization, Technical Readouts are organized by mass, campaign supplies can be limited by mass, and RPG characters have some control over the mass of their assigned vehicles. Other fans define "good" design as making efficient use of C-Bills. And that makes sense too - C-Bill expenses are central to mercenary campaigning.

But for game design and world design none of that makes sense.

To balance a game you balance the actual gameplay stats - the original designers knew what they were doing when they balanced the old classic 'Mechs against each other instead of against their tonnage. They even wrote this attitude into the in-universe worldview. In most places it's subtle but, notably, TR:3025's writeups divide 'Mechs mostly by armor class instead of by weight class: fast scouts 3-4.5 tons; light support 5-6.5 tons; mediums 7-9.5 tons; heavies 10-12 tons; assaults 13+ tons. Or consider the CDA-2A Cicada, JM6-S JagerMech, CGR-1A1 Charger, BNC-3E Banshee and CP-10-Z Cyclops. They all cost a lot of tonnage and C-Bills for what they do. Yet TR:3025 condemns only the Charger and the Banshee, and it only condemns them for being under-armed. What sets those two apart from the others? They're underarmed relative to their armor. Not just their mass.

The BattleTech universe was designed to make vehicle mass and sale price essentially irrelevant at the faction scale - the Great Houses are very limited in their ability to change the raw materials, industry and expertise available to any given facility. So every facility builds whatever it can, as much as it can, and the House is going to install all that tonnage *somewhere*; and how much effort the Houses put into producing an item, and how much their manufacturers profit from building it, appears to be many times greater than the item's sale price. So yeah, there is a cost and difficulty to producing any given design, which can make that design look more or less plausible as part of a faction's army, but tonnage and C-Bills are a very small and indirect part of that.

Don't get me wrong. I do think it would have been nice if the construction rules had been balanced a little more carefully. (Doubling the mass required for heatsinks, armor and gauss weapons would be a good first step.)

The real issue, as I see it, is that standard play will never (and was never meant to) reflect the long-term benefits of certain design decisions. So I was thinking about the "Operational Game" from the old Tactical Handbook, and about how cards in the CCG have variable Resource costs, and I think I can create a mini-campaign system which gives "wasteful" designs the advantages that they're supposed to have.

Next time: The Campaign system.

Editing TR:3063

Technical Readout:3063 is an outstanding not-for-profit fanbook being put together by Steve Satak (Centurion13) and Bill Burt (Sounguru). Their goal was to create a set of vehicles which are not only compatible with the canonical universe (the book is meant to fill the seven year gap between TR:3060 and TR:3067) but also pull their weight better in-game than more "characterfully" flawed official designs do. 

The book doesn't skimp on art, either

In June of '09 I helped revise the chapter introductions, for which Steve was kind enough to give me a writing credit. Now, I don't edit for a living and I don't know how my work compares to (say) editors at Penguin or McGraw-Hill, but I did try to hold the book to a professional standard. Steve appreciated this and so, from August of 2010 to October of 2011, I gave each TRO entry a good once-over.

This meant shuffling content around to get it under the correct headers, cutting redundant or vacuous text, slimming "Notable Crews" down to size, replacing slang and colloquialisms, fact checking, massaging the text for rhythm and flow, and finally returning it to Steve and his co-writer, Geoff, to insert new material as needed.

EDIT: We were using the "compare" and "merge" features of MS Word (Steve) and Open Office (me) to integrate changes, and it now looks like a good portion of my work did not actually carry forward into the final "merged" documents. 

C'est la vie.

Since the book is nearing completion, I wanted to share some of my tools & resources:

How to Write Fluff
My old guide (I recall somebody, I believe Welshman or Paul, crediting Cray with supplying a similar outline to Catalyst's writers) and other notes from my 2007 review of TR:3025. That book averages 200 more words per entry than Steve's book does, so I'm pretty sure I reduced the following maximums (though not the minimums) by 25% before applying them to TR:3063.

59-227 overview
126-496 capabilities
109-571 battle history
 17-252 variants
 50-311 notable pilots (34-151 each pilot)

http://thesaurus.com/
My new favorite website. Ever get that single, perfect word stuck on the tip of your tongue? Nine times out of ten, this site helped me track from wrong (though related) words to the one I actually wanted.

http://isatlas.teamspam.net/
The Inner Sphere Atlas is probably the one site I used most (usually to check planetary history or to look for manufacturing leads).

The BattleTech Style Guide 
Doesn't have everything you need to know for a TRO, but it's a solid start.

The Sarna BattleTech Wiki
Though generally accurate, their editors & contributors tend to parse things weirdly, so you should corroborate their assumptions/facts/conclusions against another source whenever possible. Luckily for me, most of TR:3063's references went back to the Field Manuals (which I have) instead of the FCCW SB or related novels (which I don't).

Longwalker's BattleTech Archive
I knew the online community maintained stat summaries (I've got a printout from 1990!) but this (fall 2004) was the first time I'd seen TRO fiction online too. The entries looked strangely slice & diced to me, but I saved html copies anyways, which I've been using to check on word usage and brand-name equipment. Longwalker's archive is now defunct, but someone at OBT has discovered it and other interesting sites inside the Internet Wayback Machine.

http://www.mechground.com/
In 2005 I found Mechground, which was more complete than Longwalker's site but displays each entry as an image instead of as searchable text. I have software (ACDsee) that allows a mouse wheel to page through image files, which I find more convenient than navigating inside a pdf, but my copies seem to be deteriorating (my Quickdraw is mostly unreadable now). I'll probably create my own replacements sooner or later.

Obstacles to Designing This Blog

I have been using xanga for my blogging (I dislike how it displays my archives) and my browser's "open all in tabs" bookmark function to check other blogs for updates -- Blogspot gadgets handle these things much better. So I'm trying to design a blog here, but I've hit some problems, and blogger's help guides are shockingly unhelpful.

1. The sidebars waste so much space with their giant margins. I zeroed out every value that the "edit HTML" page let me zero out and the margin still didn't go away. The sidebar text is so stupidly cramped! I can't sacrifice one column to give the other room because I need both the "Blog List" and "Archive" to be visible without scrolling.
2. Need a color scheme, once I figure out where all the page elements are hidden.
3. Need to find how the "Recent Comments" gadget limits the size of each snippet, and splice that code into the "Blogs List" gadget.
4. Need to find a version of the twitter gadgets that (like the "Blogs List" gadget) compiles tweets from several different twitter feeds, and then figure out where I can put it that has enough room. An extra column running off the right side of the screen, maybe.
5. The mandatory "Attribution" is superfluous. (Hell, it even duplicates a link from the toolbar up top.) Since users can't drag-and-drop rows and columns like they do gadgets, or adjust margins/padding/spacing/shadows on a similar easy diagram (ie, can't create new templates), there simply aren't enough to warrant "Attributions." I've tried moving the gadget somewhere less wasteful, but then a new one spawns and adds extra fields to the foot of the layout.
6. The default "Profile" gadget has two separate links with the exact same URL, neither of which occupies the box's title ("About User")? Assuming I can uncramp the sidebar, definitely sticking my bio inside a simple text box instead.
7. The point of the "Followers" gadget is to help people who like your blog find other people who like your blog? Blogger keeps telling me that it adds to my "Reading list," but half the reason I'm trying Blogspot at all is to track people's updates from my public front page.
8. Why can't I "configure blog posts" to have the hour (or other mobile items) alongside the date (or other immobile items)? How come "Day, Month Date, Year" is listed twice -- shouldn't one entry reverse the order?
9. The editor has options and effects I can't get to or see without posting at least one entry.
   

If I do adopt Blogspot, I am really going to miss the ability to indent text inside a post.