The Coin Exponential System

Let me describe a simple resolution mechanic that has some nice mathematical properties:
Character with stat X wants to overcome a risky situation of difficulty Y (where Y may be a static difficulty determined by the GM or the corresponding stat of an adversary).

If X ≥ Y then toss X-Y+1 coins and the character succeeds if at least one coin shows a head.

If X ≤ Y then toss Y-X+1 coins and the character succeeds if all the coins show heads.
For example, John wants to climb a wall. He has a climbing stat of level 5 and the GM determines that climbing the wall has a difficulty of 3. John tosses (5-3+1 = ) 3 coins and requires a single head to succeed. Albert tries to hit Bob. He has a hitting stat of level 5 and Bob a not-getting-hitted stat of level 6 so Albert tosses (6-5+1 = ) 2 coins and will succeed only if both of them show heads.

Although the description implies 2 separate cases, the system is completely symmetric (making the toss from the Y point of view yields the same success/fail probabilities for both characters). Moreover, these probabilities don't depend on the individual values of X or Y but just on their absolute difference (i.e. 30 vs 33 is equivalent to 26 vs 29 which is equivalent to -104 vs -101) allowing an infinite stat-range in both directions which works very well with logarithmic ability scales like the ones used in DC Heroes, Underground, Blood of Heroes, TORG or ARM (but as far as I know there isn't any game that uses both, a logarithmic scale and this coin-based exponential resolution mechanic).
 Y \ X |     1     |     2     |     3     |     4     |     5     |
-------+-----------+-----------+-----------+-----------+-----------+
1 | 50% \ 50% | 25% \ 75% | 12% \ 87% | 6% \ 94% | 3% \ 97% |
-------+-----------+-----------+-----------+-----------+-----------+
2 | 75% \ 25% | 50% \ 50% | 25% \ 75% | 12% \ 87% | 6% \ 94% |
-------+-----------+-----------+-----------+-----------+-----------+
3 | 87% \ 12% | 75% \ 25% | 50% \ 50% | 25% \ 75% | 12% \ 87% |
-------+-----------+-----------+-----------+-----------+-----------+
4 | 94% \ 6% | 87% \ 12% | 75% \ 25% | 50% \ 50% | 25% \ 75% |
-------+-----------+-----------+-----------+-----------+-----------+
5 | 97% \ 3% | 94% \ 6% | 87% \ 12% | 75% \ 25% | 50% \ 50% |
-------+-----------+-----------+-----------+-----------+-----------+
Apart from the symmetry and the fact that the system relies just on the absolute stat-difference, this system has three additional properties that must be noted:

1) Equally leveled characters have a 50%-50% success probability (which seems quite natural).

2) There are no ties (the character either succeeds or fails).

3) Even though the odds of beating a strong opposition quickly decay as the level-difference increases (more precisely, for each additional level that your opposition gains your probability of success gets halved), there are no impossible (or sure) tasks.

The latter property means that, although a +1 bonus always provides the same "increase" in power (think, again, in logarithmic scales like the one of DC Heroes, where a +1 bonus makes you twice as strong) there is a natural "diminishing returns" effect built in the system which makes irrelevant such bonus when there is a big level difference between characters: A +1 bonus in a 5 vs 5 conflict is huge whereas this same +1 bonus in a 5 vs 15 conflict means practically nothing but, again, this does not depend on your raw ability score (i.e. 5) but on your score difference (i.e. 5-5 vs 5-15) which is quite natural if you think about it.

Another useful property of the system is that all conflicts are resolved using the same system whether they have an active or a passive opposition. This means that both, difficulties and stats are measured in the same scale, which reduces the task of eyeballing the difficulty of a task to answering the following question: which ability level represents a 50% chance of success in this task?

Finally, coins are pretty ubiquitous but they are a pain in the ass to toss in general so you can use other "binary" randomizer instead (like odd-even or high-low in any even-sided dice). Fortunately, the nature of the system requires the minimum number of coins in every situation (a conflict between a character of level 120 and a character of level 127 requires tossing "just" 8 coins) and the quickly decreasing probability of success for really unbalanced conflicts makes it difficult to think about tossing more than 10 coins at any given time.


Now, my questions:

* Do you know about any RPG that uses this resolution mechanic? (So far I've found just one: a Spanish zombie RPG called Carne Fresca)
* Do you like this resolution mechanic? Do you think that it can be used in practice?

Comments

  • Not sure I grok the mechanic completely but Prince Valiant and Hollow Earth Expeditions are both coin pool systems.
  • Prince Valiant would be the closest game I can think of.

  • Thank you for the replies but both games set the difficulty by asking for more or less "heads" in your toss rather than modifying the number of coins that you toss:

    CES: Toss |Ability-Difficulty|+1 coins and get a success with 1 head (or all heads, depending on the case).

    Prince Valiant / Ubiquity System: Toss Ability coins and get a success with Difficulty heads.
  • My interpretation and another explanation, correct me if I'm wrong.

    Everything has a rating from -inf to inf. For example, maybe Alice's Archery is 5 and Bob's Archery is 3.

    Your chance of failing when you have an x better rating is 1/2^(x+1).
    Your chance of succeeding when you have an x worse rating is 1/2^(x+1).

    Always test to find out if "the unlikely thing happens". So if your rating is higher, test to see if you lose. You lose if a massive coincidence occurs! For example, if your rating is 3 higher, toss 4 coins, and if they all come up heads, you lose! But if your rating is lower, test to see if you win via massive coincidence.

    Could be implemented by the rule "underdog always rolls" to be simpler at the table, and nicer, because you don't really want to be rolling "to see if you fail".

    Could be generalized to not have quite such a steep drop-off at nearish levels by, for example, saying the "massive coincidence" is rolling all 3-6 on a number of d6s except ties which are just 50/50. That's 50%, 44%, 30%, 20%, ...
  • What's the game where you play homeless people?

    That game had a cool system where you had traits rated in multiples of "penny," "nickel," "dime," or "quarter" and when it was time to resolve things, you took all the attribute stats and put your change in a cup and dumped it on the table. All the heads added up.

    So if you had Strength=2 Dimes and Punching=3 Pennies, and you wanted to hit someone, you'd drop 13 cents into your cup, give it a good shake, and dump it. If the coins showing heads were a dime and two pennies, your score was 10+1+1 = 12.

    Neat, huh?
  • Note that basically all multiple-coin mechanics are a dice pool of d2s.

    Another thing you can do with coins is similar to an exploding die. Basically, flip a single coin, and add 1 every time you get a heads, but stop when you get a tails.

    So the GM can say "you need 2 successes" and you flip a coin. If it's tails, you're done. You lose. If its a heads, you have 1 success and the next must be a heads for you to get 2 successes and win.

    Stats can modify that. A Beekeeping-3 skill could mean that you get three attempts.

    Overall, though, coins have a slower handling time than dice. They're harder to pick up, harder to "roll," and just end up being d2 rolls unless you do something clever like use their coin values. You might as well use cards (red/black) or d6es (1-3/4-6, as Burning Wheel does).
  • My interpretation and another explanation, correct me if I'm wrong.

    Everything has a rating from -inf to inf. For example, maybe Alice's Archery is 5 and Bob's Archery is 3.

    Your chance of failing when you have an x better rating is 1/2^(x+1).
    Your chance of succeeding when you have an x worse rating is 1/2^(x+1).

    Always test to find out if "the unlikely thing happens". So if your rating is higher, test to see if you lose. You lose if a massive coincidence occurs! For example, if your rating is 3 higher, toss 4 coins, and if they all come up heads, you lose! But if your rating is lower, test to see if you win via massive coincidence.
    This is exactly what I meant... but correctly explained. Thank you very much for taking the time to rephrase the whole thing into something more clear and readable!
    Could be implemented by the rule "underdog always rolls" to be simpler at the table, and nicer, because you don't really want to be rolling "to see if you fail".
    Yes, I thought about that version of the rules and it will probably be the best way to explain them but I wanted to emphasize the symmetry of the whole thing in this thread so I end up using the more convolute one...
    Could be generalized to not have quite such a steep drop-off at nearish levels by, for example, saying the "massive coincidence" is rolling all 3-6 on a number of d6s except ties which are just 50/50. That's 50%, 44%, 30%, 20%, ...
    Yes, I can see how the steepness of the success probability can be a problem and I thought about it before (since 0.707 * 0.707≈0.5 you can go as fas as moving the massive coincidence to rolling 1-7 on a number of d10s without breaking the system too much) but, by now, I'm too much in love with the fact that using coins is quite handy and don't require to specify a special case for tied ability scores.

    In fact, I'm currently more inclined to use an equally steeped logarithmic scale to measure abilities, such as the one of DC Heroes, where adding 1 to a stat makes it twice as powerful (which is something that works well with the fact that you halve the probability of failing by doing precisely this).
  • What's the game where you play homeless people?
    Kingdom of Nothing?
    Note that basically all multiple-coin mechanics are a dice pool of d2s.

    Another thing you can do with coins is similar to an exploding die. Basically, flip a single coin, and add 1 every time you get a heads, but stop when you get a tails.

    So the GM can say "you need 2 successes" and you flip a coin. If it's tails, you're done. You lose. If its a heads, you have 1 success and the next must be a heads for you to get 2 successes and win.

    Stats can modify that. A Beekeeping-3 skill could mean that you get three attempts.

    Overall, though, coins have a slower handling time than dice. They're harder to pick up, harder to "roll," and just end up being d2 rolls unless you do something clever like use their coin values. You might as well use cards (red/black) or d6es (1-3/4-6, as Burning Wheel does).
    Yes, I understand, there are many variants on this kind of mechanics (Bean!, Nickel & Dime, Coin's Hard Edge, Clink!, etc...). I was simply looking for games that implement the specific kind of d2s pools that I described above since it has some nice theoretical properties that I can't believe no one has noticed before.
  • I think there is either a typographical error or a math/logic problem here. I'm a bit of a math dunce, but at the same time, strangely, a language as code/system nerd. So, I notice something peculiar about these two statements:

    If X ≥ Y then toss X-Y+1 coins and the character succeeds if at least one coin shows a head.

    If X ≤ Y then toss Y-X+1 coins and the character succeeds if all the coins show heads.
    What exactly happens if X = Y ? It's ambiguous because the second inequity statement breaks the first. Both if-then statements can be true at the same time. That's going to crash the system pretty fast.

  • I think there is either a typographical error or a math/logic problem here. I'm a bit of a math dunce, but at the same time, strangely, a language as code/system nerd. So, I notice something peculiar about these two statements:
    If X ≥ Y then toss X-Y+1 coins and the character succeeds if at least one coin shows a head.

    If X ≤ Y then toss Y-X+1 coins and the character succeeds if all the coins show heads.
    What exactly happens if X = Y ? It's ambiguous because the second inequity statement breaks the first. Both if-then statements can be true at the same time. That's going to crash the system pretty fast.
    When X=Y both cases are equal (toss a coin and your character succeeds if it shows a head) so you can put the equality in either one or both.
  • I love a lot of things about this, like the symmetry, measuring everything on the same scale, and so forth.

    In terms of presentation, I would say, "When you're up against something or someone that's beyond you, flip coins. You always get one coin to flip. Add to it the difference between your score and theirs (one coin per point). If they all come up heads, you win!"

    Things I don't like:

    1. Flipping coins is not easy or fun. (Draw cards or roll binary dice instead; no problem.)

    2. The drop-off is so steep that it would be hard to get any kind of reasonable spread or fine distinction between characters/obstacles. That could be fine for some games, but very limiting for other designs. You only have an effective range of a couple of points to play with. (For example, if you want the possible range between characters/obstacles to be -4 to +4, everything needs to be rated as -2 to +2, a very tight scale. One point makes a huge difference, cutting your odds in half or doubling them.)


    Could be generalized to not have quite such a steep drop-off at nearish levels by, for example, saying the "massive coincidence" is rolling all 3-6 on a number of d6s except ties which are just 50/50. That's 50%, 44%, 30%, 20%, ...
    This sounds promising. Can you explain the math there?

  • When X=Y both cases are equal (toss a coin and your character succeeds if it shows a head) so you can put the equality in either one or both.
    Ah, I wasn't paying close enough attention to the "then" part. Of course, now it's obvious, you just flip one coin in either case.
  • If we use fudge/fate dice instead of coins, can we get a range of 3 outcomes out of this (a la AW)? If you're outmatched, keep the worst result - if you outmatch your opposition, keep the the best one.


  • Could be generalized to not have quite such a steep drop-off at nearish levels by, for example, saying the "massive coincidence" is rolling all 3-6 on a number of d6s except ties which are just 50/50. That's 50%, 44%, 30%, 20%, ...
    This sounds promising. Can you explain the math there?
    "Massive coincidence" is code for a bunch of dice all doing the same thing, because that's pretty easy to execute and check. The original mechanic was a bunch of dice all hitting one side of the split, like 4-6 on a d6. Or heads on a d2. Here, let's say no one's the underdog - flip a coin, however you like. Let's say you're the underdog by 1 - then roll 2d6, looking for (3-6) and (3-6), happens (2/3)*(2/3)=4/9 of the time, or 44.4%. If you're the underdog by 2, roll 3d6, looking for (3-6) to happen all three times, or (2/3)*(2/3)*(2/3)=(2/3)^3=8/27=29.6%.
    (2/3)^4 ~= 20%
    (2/3)^5 ~= 13%
    etc.
  • Thanks, Guy! That's what I thought you meant, but the rounding threw me. Excellent!

    I used to have a Fudge version of this - it has many of the same features this mechanic describes (and could even be replicated with coins, if you want).

    I called it "*dF".

    Basically, you each have a level. (Numbers or words, it doesn't matter.) Then one side (or both; you could set it up either way) rolls a Fudge die (+, 0 , -). Whatever result comes up, it counts, and you can keep rolling. If your next roll matches it, that one counts, too. Keep going until you get a result which doesn't match.

    For instance, you have a Stat of 3. You roll a Fudge die and get a [+]. You're up to 4! Now you roll again, hoping to match that. Another [+]? Add it to the pile: You're at a 5! Next roll comes up with something else, though - a [-] or a [0], so we ignore it and you're done: You have two {+} dice on the table and a starting Stat of 3, so your outcome is a 5.

    (Oh, there's one feature of the mechanic being described here that it DOESN'T replicate: you can have a tie. That's a nice feature. I suppose, in my Fudge version - which, again, could be done just as easily with coins - you'd have to arrange a tiebreaker, like highest stat wins or best roll wins; I liked ties for the purpose of the game I was using, so it was never an issue.)
  • Things I don't like:

    1. Flipping coins is not easy or fun. (Draw cards or roll binary dice instead; no problem.)

    2. The drop-off is so steep that it would be hard to get any kind of reasonable spread or fine distinction between characters/obstacles. That could be fine for some games, but very limiting for other designs. You only have an effective range of a couple of points to play with. (For example, if you want the possible range between characters/obstacles to be -4 to +4, everything needs to be rated as -2 to +2, a very tight scale. One point makes a huge difference, cutting your odds in half or doubling them.)
    Yes, your second point is also my main concern but I tend to prefer coarse-grained systems where you don't play much with modifiers (or, even better, not at all) and you don't rely on the mechanics to make your character unique.

    Right now I'm working with:
    Your four attributes (Power, Reflexes, Intelligence and Spirit) are measured with the following scale:

    -2 - Terrible
    -1 - Poor
    0 - Human Average
    1 - Good
    2 - Excellent

    You can rate them as you want as long as they end up summing 0.
    Which provides the following range of probabilities:
    Atr-Diff | Success Prob.
    ---------+--------------
    +4 | 96.875 %
    +3 | 93.750 %
    +2 | 87.500 %
    +1 | 75.000 %
    +0 | 50.000 %
    -1 | 25.000 %
    -2 | 12.500 %
    -3 | 6.250 %
    -4 | 3.125 %
  • edited May 2018
    Nice idea. Lincoln Green (by Epidiah Ravachol) also comes close to this, check it out.
  • edited May 2018
    Kingdom of Nothing comes to my mind for coin mechanic. I've used coins for a while but returned to dice. I find dice more fun and practical.

    I'd prefer to i.e. roll a d8:
    roll at least 5 = 50%, 7 = 25%, 8 =12,5%
    roll an 8 and then 5 = 6,25%, etc...
  • Nice idea. Lincoln Green (by Epidiah Ravachol) also comes close to this, check it out.
    Thank you! I took a look. It's work in progress, right? I will check it from time to time for updates.


    Kingdom of Nothing comes to my mind for coin mechanic. I've used coins for a while but returned to dice. I find dice more fun and practical.

    I'd prefer to i.e. roll a d8:
    roll at least 5 = 50%, 7 = 25%, 8 =12,5%
    roll an 8 and then 5 = 6,25%, etc...
    Yes, coins are a pain in the ass to toss. In practice I use a set of special D6s that I found in a educational store: They have 3 faces showing a plus symbol and 3 faces showing a minus symbol. It isn't really difficult now a days to find some kind of "binary dice".

    This being said, the Coin Exponential System is designed so you have to toss the minimum number of coins able to "represent" the difference of skill. This means that it is really rare to toss more than, say, 6 coins at the same time and, even if the skill difference is as huge as 35, you can toss the coins in groups of 4 or 5 and stop as soon as one of them shows a head.
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