Saturday, May 14, 2016

Gamma World: How to Determine Probabilities for Chart A

I've been doing work on a post-apocalyptic or far-future supplement to Seven Voyages of Zylarthen. The idea is sort of to do for Zylarthen what Mutant Future did for Labyrinth Lord--not a clone of Mutant Future (since that would be pointless) nor a precise clone of Gamma World (since that would be pointless and probably a violation of copyright) but a variation on Zylarthen, incorporating much of the Gamma World vibe.

Among other things, I've been looking at the 1st edition Gamma World technology charts--the mechanism by which the game simulates the process by which player-characters figure out the function and use of ancient technological artifacts.

There are three sorts of increasingly complex flow charts representing different sorts of artifacts--from laser pistols (Chart A, the simplest chart) to, say, permanent cybernetic installations (Chart C, the most complicated chart). The idea is that you roll a ten-sided die to negotiate through, with possible bonuses or penalties based on intelligence and how many others are available to help. The longer you take (the more die rolls you make) the greater the chance you'll figure out the artifact out but also the greater the chance you'll end up breaking it or even harming yourself or your party. Chart A looks like this:

So, after discovering an artifact--"a sort of jumble of sticks or tubes containing many grooves and colored knobs"--you start at S and want to get to F. In the process, you want to avoid getting to the skull and crossbones (henceforth A for accident). Depending on the artifact and the kindness or lack thereof of the referee, an "A" result could mean anything from "you break the trigger rendering it permanently inoperable" to "you shoot yourself in the face with a laser at point-blank range--roll 10d6 damage."

Now, I think the consensus is that the "chart system" is original and in theory very cool but it is usually a disappointment in practice. At worst it takes something that should be quite exciting--learning the purpose of some wondrous and powerful device--and turns it into something boring--rolling a die over and over again to track an abstract mechanic with no opportunity for player choice or referee creativity.

What can be done about that is a question for another time. And of course since Gamma World was published, all sorts of tweaks have been offered, in places ranging from the early issues of Dragon Magazine to the latest blog posts.

What I want to do first, however, is to look at the charts (or rather Chart A) from the point of view of probabilities. Given x number of rolls, what is the probability that you will get to F, or get to A or simply get nowhere at all?

For low numbers of rolls, the probabilities are easily calculated. Then it gets tougher:

From S to F (the number of rolls is the number on the left):

1: 0%
2: 0%
3: 10.5%
4: Complicated

Why 10.5%? Because there is only one path that will get you there in three rolls. Using a d10, you need to roll a 1-7, then a 1-5, then a 1-3. The chance of doing that is 70% * 50% * 30% or 10.5%.

From S to A:

1: 0%
2: 0%
3: 0%
4: 0%
5: 0.63%
6: Complicated

Why 0.63%? Because there is only one path that will get you there in five rolls. Using a d10, you need to roll a 1-7, then a 1-5, then a 8-10, then a 9-10 and then finally an 8-10. The chance of doing that is 70% * 50% * 30% * 20% * 30%, or 0.63%.

Though it might be complicated to precisely compute the odds at higher number of dice rolls for getting to F or A, we can roughly calculate them by performing simulations in Microsoft Excel. I imagine some of you mathematics wonks have done this or something like it in the past, or even have a better way of doing it using Excel or some other program. But I thought a few of you might be interested to see how I did it.

First you assign a number to each of the nine squares, circles and diamonds. I assigned them numbers from 101 to 109 so as not to get them confused with the pips 1 to 10 on a d10. I started at the top, going row by row down, and then left to right in each row where there was more than one shape. Thus, the top circle is 101, S is 102, F is 105, the diamond is 108 and A is 109.

We can then render the flow chart into this nested IF/THEN formula, where A1 is where you are and B1 is a random number from 1 to 10:  
Once you get to 105 or 109, you stop, or rather, you "move" nowhere.

So, to simulate one set of rolls, you would create these cells from left to right:
  • A1: "102" (that's because you always start at S)
  • B1: "=Randbetween(1,10)" (that gives you a random integer from 1 to 10)
  • C1:"=IF(A1=102,IF(B1<=7,103,102),IF(A1=103,IF(B1<=5,104,IF(B1<=7,103,101)),IF(A1=101,IF(B1<=2,104,101),IF(A1=104,IF(B1<=3,105,IF(B1<=7,104,107)),IF(A1=106,IF(B1<=2,103,IF(B1<=4,102,106)),IF(A1=108,IF(B1<=3,106,IF(B1<=7,102,109)),IF(A1=107,IF(B1<=1,105,IF(B1<=5,106,IF(B1<=8,107,108))),IF(A1=105,105,IF(A1=109,109,0)))))))))"
That gives you one roll. To simulate the next roll, add two more cells:
  • D1: "Randbetween (1,10)" (or take B1 and copy it to D1)
  • E1: Take C1 and copy it to E1.
And so on, for how many rolls you want.

After simulating the desired number of rolls, add three cells to record whether you ended up at F, ended up at A or failed to arrive anywhere. After, say, ten rolls, your ending point will be in U1. You will thus put a 1 in V1 if U1 is 105 (you ended up at F), a 1 in W1 if U1 is 109 (you ended up at A) and a 1 in X1 if U1 is anything else (you failed to arrive at F or A). You can do this automatically with more IF/THEN formulas:
  • V1: "=If(U1=105,1,0)"
  • W1: "=If(U1=109,1,0)"
  • X1: "=if(V1+W1=0,1,0)"
You now have one row in Excel simulating ten (or how ever many) rolls through Chart A. This of course doesn't give you the odds of anything. It simply tells you what happened one time.

Now, pull that row down, say 1000 times. You may have to pull A1 down separately to get 102 in each cell. Otherwise B1 may become 103, C1 may become 104 and so on. Obviously, you don't want that.

Now you have 1000 rolls. And it only took you a few seconds.

Finally, add three SUM formulas at the top or the bottom (or anywhere else, really). It's probably better to add them at the top, thus instead of your sets of rolls going from rows 1 to 1000, you can make them go from rows 5 to 1004 or whatever. You want the sum of F's, the sum of A's and the sum of no results. Assuming you've pulled the whole set down so as to start at row 5, you then have:
  • V1: "=SUM(V5:V1004)"
  • W1: "=SUM(W5:W1004)"
  • X1: "=SUM(X5:X1004)"
Let's try it for the first set of 1000 rolls. I get:

F (success): 503
A (accident): 32
No Result: 465

What about a second set of 1000 rolls? As you probably know, just by changing what is contained in one cell--say a dummy cell somewhere--every random result (all 10,000 of them) is automatically re-rolled by Excel.  So, now I get:

F (success): 558
A (accident): 33
No Result: 409

Here's a third set:

F (success): 528
A (accident): 28
No Result: 444

We have discovered a few things:
  1. After ten rolls, while we do not know precisely what the chance of success is, it looks like it's 50% to 55%.
  2. The chance of an accident is much much lower--only around 3%.
  3. Roughly 40% to 45% of the time, you get nowhere.
I find that interesting. It's certainly not obvious from the chart. Among other things I would have thought the chance of an accident would be higher.

Of course it presumably would be higher if the character had a low intelligence. On the other hand it would probably drop do almost negligible if the character had a high intelligence.

I hope some of you find this useful and helpful. Though again, I imagine a few of you have already done this on your own. Despite its seeming complexity, it actually doesn't take very long to set up, at least if you're familiar with some of the simpler Excel formulas. Of course, writing the formula for Charts B or C would take somewhat longer.

If you can believe it, I did write the formula for Chart C and then I lost it (or maybe it's some anonymous "Workbook1" in my documents). But I did save the results. After ten rolls on Chart C the numbers are:

F (success): 90
A (accident): 78
No Result: 832

Yeah, there's an almost 10% chance you'll figure out how to operate that permanent cybernetic installation after only a few hours. On the other hand, there's also a close to 10% chance you'll blow yourself up (or whatever) in the same amount of time. 

Is that how you thought it would turn out?

Please let me know if you have any questions or if I made any mistakes on the above. I think it's all correct, but it's not very difficult to make mistakes with so many formulas. 

NEXT POST: An oddity that these results reveal about Chart A...    


  1. Excellent game. I liked Metamorphosis alpha better as it was simpler but more fun.

  2. My god, do you store an extra brain in a jar or something for this kind of thing? Great post. Very interesting.
    And no just viewing the charts at face value I thought there was a higher than 10% chance of blowing yourself up.

    1. Re: brain in jar. Thanks! And the brine is mid-price gin.

      You can use the spreadsheets to tweak the chances. For example, you can see what adding +1 to the rolls does. Then the question becomes what SHOULD the chances be?

      I'm not sure I know. What might be interesting is incorporating that into the decision making process. For example, after clicking a button, the "artifact" starts to emit a high pitched whine that steadily gets louder. Do you throw it away and run, or do you press the next flashing button and point the device at a tree?