More Exploding Dice: T&T Saving Throws

I received a request from Christian Lindke, author of Cinerati, about the probability distribution of a different sort of "exploding" dice: Saving throws in Tunnels and Trolls. I have previously written about Exploding D10, and this is a closely related problem. I'll let Christian explain it ...

I am getting ready to do a few posts on Tunnels and Trolls.  Specifically, I will be proposing an alternate combat resolution system -- one based on an existing system within the game.  I want to play around with the concept of using T&T "Saving Throws" as the basis for all mechanical resolutions in the game.

In T&T the saving throws are resolved by comparing a statistic to a difficulty # and using the result as the basis for a 2d6 roll.  The actual equation is [15 + (level of challenge x 5)] - Statistic = Target Number.  So a character with a Luck of 12 attempting a 1st level challenge would need a result of (15+5-12=8) eight or better to succeed on the attempt.  Figuring out the probabilities on a basic 2d6 roll is a simple affair, but in T&T a player who rolls doubles keeps the result, re-rolls and adds the result to the prior sum until the player no longer rolls doubles.  It's an open ended doubles system.  What would be basic equation be to determine probabilities in this case.  Logically, given that the chance of doubles is 1/6, it seems that the probabilities would be the same as a normal open ended d6 roll for each die (which I believe produces an average of 4.3 per die), but I'd like an equation I can use to determine the game balance as characters advance etc.  I'd like to use the base probability of an "average" character of a given level attempting a task as the basis for my new system.

If you could be of assistance, it would be greatly appreciated.

A friend of mine had noted that asking me questions like this is like teasing a small child with a shiny toy, holding it just above my reach; You just know I'm going to jump up and try to get it - and I did. ;-)

As I said, this is very similar to how the probability for Exploding D10 work in the MechWarrior 3rd edition RPG. With D10X you count the roll if is in the range 1-9, and if it is a "10" you count the 10 and roll again (that the explosion). Rinse and repeat until done. In T&T we have a 2d6 roll that explodes if a tie is rolled on the two dice instead of the highest value on either one.

As with D10X, this breaks down into two three parts - which are a geometric series, the value of rolling ties tie, and the value of rolling no-ties.

The probability of rolling a ties on 2d6 is 1/6, so the geometric series starts of like this:
0 ties with probability = 5/6 = 0.8333
1 tie with probability = (1/6)*(5/6) = 5/36 = 0.1389
2 ties with probability = (1/6)*(1/6)*(5/6) = 5/216 =0.02315
3 ties with probability = (1/6)*(1/6)*(1/6)*(5/6) = 5/1296 = 0.003858

... and so on out to infinity. The average number of ties in a series is (1/6)*1/(5/6) = 1/5 = 0.2, which is nice it you only want to know the average 2d6X roll (which is 8.4, btw), but we need the entire probability distribution.

Now there are six different ways to roll a ties, and they have values of 2,4,6,8,10,12, all with 1/6 probability, which is just the same as 2 times the value of a 1d6 roll. I will note this as 2*1d6. Combining this with the geometric series, we get this:

a "0" with probability = 5/6 = 0.8333

a 2*1d6 roll with probability = (1/6)*(5/6) = 5/36 =0.1389
a 2*2d6 roll with probability = (1/6)*(1/6)*(5/6) = 5/216 = 0.02315
a 2*3d6 roll with probability = (1/6)*(1/6)*(1/6)*(5/6) = 5/1296 = 0.003858
... and so on out to infinity.

NOTE: The probability of zero ties is really 0.8333 (the first bar); I chopped off the graph to show the what ties actually add in, which really isn't very much. I calculated the probabilities out to 6 consecutive ties, which is only a little bit of overkill. It could go on forever, but unless you are infinitely lucky (and have a lot of spare time on your hands) your streak of ties will eventually come to an end, where you add the sum of the the final not-tied dice to your previous total. That probability is a modification of the usual discrete triangular 2d6 distribution, and it looks like this (2d6-ties):

The final step to pull this all together is a bit messy, so pardon my hand-waving over the details (if you really want the nitty-gritty, look in the accompanying spreadsheet). You multiply each probability in the geometric series with each the probabilities in the 2d6-ties distribution and sum up  the corresponding values of the roll, then tally up the total probability for all combinations with the same sum. It looks like this:

This looks a lot like the 2d6 distribution with a long "tail" tacked onto the right hand side. Here's the cumulative probability:

Finally, a table of the rolls and cumulative probabilities, though if you really want to numbers it's probably easier to grab them from the spreadsheet. Again, the table theoretically should go to infinity, but in practice you will only rarely see any rolls greater than 20.

I'm curious to see what Christian does with this, and I'll link to his post(s) on the topic when he has them up.

Have fun rolling the dice!


[Edit: Typo corrected, added decimal probabilities.]
[Edit2: Added link to spreadsheet, which I swear I did once already.]

Pink Explosion

 I've been working on a pair of starship minis (Panther Class Cruisers from the Galactic Knights line). I hadn't done anything fancy; black primer, layers of metallic medium, black glaze, a little white drybrush, but I wasn't completely happy with it. They looked drab, and I thought it needed something more, some color maybe. I reached into my paint drawer and pulled out a bottle of Vallejo Sunset Red, which might be fairly described as very pink. A tiny bit of this stuff, I thought, should do the trick.I went to squeeze out a little bit into a bottle cap to work with, but the tip was plugged. No problem - I popped the tip off and used a sculpting pick to poke out the opening and stir up the partly cured paint that was blocking the hole. I added a few drops of rubbing alcohol to the bottle, pressed the tip back on, and gave it a good shake. (Rubbing alcohol is a solvent to acrylic paint, and I've has some luck rehabilitating old bottles of paint this way.)
Back to the painting - I went to squeeze out a bit of paint again, and the darn thing was still plugged up. I squeezed it harder and ...

   SPLAT!!!   

 ... the newly loosened tip came off, splattering bright pink paint all over my painting table, my minis, my clothes, my hands, and (I now see) even a bit on my computer keyboard and screen. What a mess! Reasoning that I needed to clean my hands off before I spread the mess, I dropped the minis into the water dish to keep the paint from curing and ran to the sink to clean myself up. Once that was completed, I put the minis under running water and scrubbed the pink off with a lightly stiff nylon brush. This was fairly successful, but the rough handling also took some paint off exposed corners of the minis, which will require some repairs. I am considering re-primering them white and painting them in pink, to commemorate the event [edit: See the results here].

Game theory shows GROUPS follows most successful member

Brian of Scrapyard Armory tipped me to this short article on game theory, which is quite interesting, but the author summarizing the article for Ars Technica really botched one thing. Here is the first paragraph with my insertions/corrections in red.

Game theory shows cultural evolution
follows most successful member

Game theory has become a useful way to evaluate strategies for survival in cultural evolution scenarios. In a new study, scientists set up a model where human players engage with each other and compete for resources, and can change their strategies for doing so in various ways. They found that as more rounds of the game were played, the human players developed a tendency to imitate the best player, causing the players as a group to tend to play the game the same way. This implies that in cultural evolution, as one member of a species group enjoys more and more success, its methods become hard to ignore for the others, which will eventually follow its lead.

The error was confusing biological evolution with cultural evolution. So if Brian and I are playing a game, and Brian consistently wins, I am likely to start playing the game the same way Brian does. That's how groups change their social behavior, and it is a form of cultural evolution. For it to be biological evolution, *I* would have to change my biology to be more like Brian, or at least so that my offspring are more like Brian (and if that happens, I'm going to be VERY suspicious about what Brian has been up to with my wife! ;-) ). People do not change their biology when they adapt new behaviors, but groups can change their culture to adopt new ways of doing things. (See Lamarkian Evolution)

Other than that, I'm OK with the article, and I've even looked up the original publication. Maybe I'll have more to say after I give that a read.

Grav Tanks

Some interesting minis and other gaming materials at Topgun Marketing.

[Hat Tip to Panzerfaust 150s Random Gaming Rambles]

BattleTech Pods Return to OwlCon in Houston, Feb. 19-21, 2010

Passing on the good word from my friends at MechCorps and Virtual World Entertainment:

BattleTech Pods Return to
OwlCon in Houston, Feb. 19-21, 2010

For Immediate Release

MechCorps' Mobile Armor Division will be stationed in the Ley Student Center on the campus of Rice University in Houston, Texas during OwlCon XXIX "The Year We Make CONtact" on February 19-21, 2010.

Some Battletech events at OwlCON too!

The last photo is actually for Mechacon, but is too good not to use again.

Also, I have confirmation that the Pods will be at ORIGINS 2010, which makes me happy, as that's likely my only big CON this years.

Traveller Minis

Here's a welcome and some pictures for everyone visiting from the Star-ranger.com forums.

Some minis I just finished for use with Squadron Strike. These are highlighted with the metallic sliver ink I bought a few days ago. From these pictures it looks like I dipped them in it, but I really just hit a few spots for extra emphasis.










These are from the AdAstra Zhodani Fleet Box 1 set. I have them temporarily mounted on the bases that come with the box. And yes, I was playing with the effects. :-)





More Pics to tell about, but I must get to sleep. Click the final pic to link thru to my Picasa album where you can see all the pics.

Traveller Minis - Zhodani Fleet

- G'nite.

Metallic Inks for Miniatures

I went shopping at Michael's Saturday and found something new; metallic inks from Ranger Inks. These are alcohol based inks (note: not for kids!), they apply very easily with a brush, leave a near perfect metallic sheen, and thin/clean up easily with a bit of rubbing alcohol. I can't begin to tell how impressed I am with this product; I added some silver ink highlights to some spaceship minis that I had already highlighted ... and WOW did it make a difference. The reflectiveness is much greater than I get with metallic paint, and really makes the minis pop.

This is a product created for rubber stamp collectors, but I think it make have great potential with miniatures painting too. Click thru for a demo video.

[Update: added photo and link to photos]