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What distribution or function models Face-to-Face rolls?

Discussion in 'Access Guide to the Human Sphere' started by GingerGiant, Dec 18, 2018.

  1. GingerGiant

    GingerGiant Well-Known Member

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    I'm trying to get a sense of the odds of winning a given Face-to-Face roll. Given we know the target number, the number of dice to be rolled for each side, and the distribution of faces on a d20, can we mathematically model the Face-to-Face roll (instead of brute-force through massive amounts of dice rolls)? Is there a distrubtion or function that can calculate the odds of at least one success? Multiple successes?
     
  2. ijw

    ijw Ian Wood aka the Wargaming Trader. Rules & Wiki
    Infinity Rules Staff Warcor

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  3. Robock

    Robock Well-Known Member

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  4. Robock

    Robock Well-Known Member

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    I know someone made nice charts, sets of charts, for various BS and Burst. But I can't find it, It might have been lost long time ago.
     
  5. GingerGiant

    GingerGiant Well-Known Member

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    The scariest thing in the analysis is probably the binomial coefficient. Everything else is fairly straightforward, but I think the most valuable thing is calculating things with matrices. I'd really like to find a quick way to estimate odds, without plugging in a calculator. While the odds of a normal roll are fairly simple, Face-to-face is a very complicated mechanic Maybe the best thing to do would be to make up a cheat sheet with various combinations of bursts and attributes (in steps of +/- 3).
     
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  6. RogueJello

    RogueJello Well-Known Member

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    Having asked this question once before, partially because I wanted to be able to do the math in my head during a game, the answer was No there is no known way to do this simply. Might be some mathematical genius who plays Infinity will enlighten us in the future, but for now it's not possible.
     
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  7. GingerGiant

    GingerGiant Well-Known Member

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    Okay, I mathed up a formula for approximating YOUR odds of winning a Face-to-face roll with Burst 1. The formula is "[My Skill x (20 - Their Skill / 2)] / 400". It has a tendency to overestimate the success of the lower skill when there is a big difference in skill, but not by more than 10%. For example, 9 vs 15, the actual likelihood of the 9 winning is 21%, but the estimate is close to 28%. The actual odds of the 15 winning is 57.5%, and the estimate is 58%. I still don't have a good trick for quickly estimating the odds when one side has more burst, though.
     
  8. Section9

    Section9 Well-Known Member

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    I will be honest, I don't like any approximation that over-estimates your odds that much.

    What I've done is gotten the local group to agree to make a decision and then break out the Dice Calc to see which option we should have gone with.

    I can ask my Dad (math PhD) if there is a simple-ish way to crunch those probabilities.
     
  9. GingerGiant

    GingerGiant Well-Known Member

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    I did an experiment to see how playing "Low Roll Wins F2F" compares to the normal rules, and the thing that makes F2F rolls complicated (even with just Burst 1 vs. Burst 1) is the way the outcomes intersect each other in the successful range.

    This table represents a Blue Troop with BS 15 trading fire with an Orange Troop with BS 8 (probably due to modifiers), each with Burst 1: https://drive.google.com/file/d/1_IJprif1XIuksTgwdCFvnu0Gzb6Jl9BB/view?usp=sharing
    The negative numbers (highlighted Orange) correspond with the outcomes that count as net successes for the Orange Troop. Vice versa for blue. Two's are crits.
    Higher attribute dominates the chart because the additional successful outcomes granted just by having higher skill are "more successful" than any of the values rolled in the the lower attribute's area of success, and it's this factor that rewards manipulating modifiers. For comparison, I did an experiment where the low value beats the high values (within the range of success), and this removes the higher skill's ability to eat into the lower skill's success region. https://drive.google.com/open?id=1J-Q5vXVVofG1mV1VOUXItfxU5DncjjG0
    It's "fairer," in that the roll is more likely to result in success for either Troop involved, but by that same token, much more random. It's probably not the game you want to play, but the picture illustrates the fact that the outcomes space is much more straightforward. The reason for this is that the extra region of successes granted by having a higher attribute is less valuable than the successes in the region where the results overlap.
     
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  10. toadchild

    toadchild Premeasure

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    Always interesting to see different approaches; I see you’re coming at this from a very different angle than I was! I can see the appeal of a quick estimate you can calculate in your head.

    I spent a bit of time trying to come up with a closed form solution to FtF odds but never got anything I was happy with. So I stuck with what I know and here we are today.

    Feel free to PM me if there are any statistical tables I can help you generate.
     
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  11. meikyoushisui

    meikyoushisui Competitor for Most Ignored User

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    It's easiest to block the math into four groups: 1) both players fail, 2) both players succeed, 3) player A fails, 4) player B fails. Here, "Succeed" means "rolls under or equal to their target value", not "wins the f2f roll." This is a naive model, and we'll ignore crits for now. Keep in mind this is also just chance to win the f2f roll, not to wound or tank a wound.

    This is really easy to model in a B1 vs B1 roll. Let's say you are B1 at 16 and your opponent is B1 at 12.
    There is a 20% * 40% chance that both players fail (8%). (You roll 17-20, opponent rolls 13-20)
    There's an 80% * 40% chance that you succeed (32%). (You roll 16 or below, opponent rolls 13+)
    There's a 20% * 60% chance that only your opponent succeeds (12%) (You roll 17+, opponent rolls 12 or below.)
    There's an 80% * 60% chance that you both succeed. (48%) (Both players roll under target value.)

    Now we subdivide the last value, to represent the odds of success for each player in the case where both beat their target values.
    If you roll a 13-16, you win (25% of your possible rolls in this case (remember we're subdividing the 48% now) we're looking at, so 12% of the grand total.)
    The remaining odds will be split evenly between the players (so 18% each.)

    This gives us an eyeball value of 62% to win this f2f, 30% for the opponent, and 8% for neither.
    If I'm eyeballing, I usually just add my odds of success due to opponent failure to my odds of success if we both succeed.

    You can apply the same set of math to higher burst exchanges, but it can break down a bit because as B and BS values get higher, the odds of any given player failing get lower.
    ex: you w/ B5 at 17 vs opponent w/ B2 at 15
    There is an almost zero percent chance both players fail (.15^5 * .25^2)
    There is a ~6% chance that you win and opponent fails (1/16).
    There is an almost zero chance you fail and opponent succeeds.
    Of the remaining 94%, there is a 1-(15/17^5) chance you roll 16 or 17 (~45%)
    Then of the remaining case (where both players roll 15 or below on all dice, so roughly 50% of what's left), you split 5/7 and 2/7, so approx 35% vs 15%.
    So this is approx 85% for you and 15% for opponent.

    Crits will break this a little bit, but crits always favor the player with higher B. The other thing this doesn't take into account is the odds of both players rolling the same highest value (a few percent.) Remember that your odds of wounding will almost never be higher than 70% per hit for normal ammo (higher burst situations where eyeballing this math is harder).

    Comparing to the dice calculator, it looks like this system is fairly accurate. For example, in the last case, with using custom units with ARM0 and DAM15 weapons, it looks like Player 1 had a 73% chance to wound, Player 2 has a 14% chance to wound, and there's a 13% chance nothing happens. Though @toadchild, I would love if the custom damage value could go up to 20, if that's not an issue, it would help a lot crunching numbers based solely on winning f2f rolls.

    And for anyone who hasn't checked out the drop-down box on the dice calculator that shows the raw stats, I highly recommend it .
     
    #11 meikyoushisui, Dec 19, 2018
    Last edited: Dec 19, 2018
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  12. GingerGiant

    GingerGiant Well-Known Member

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    You can hit “show the math” (or whatever it is) to get he details and see the hit and wound chances
     
  13. meikyoushisui

    meikyoushisui Competitor for Most Ignored User

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    Yeah, I'm just looking at that now, the last couple sentences were an edit. It looks like my method works roughly but tends to undershoot the likelihood of no successes, due to ties (which appears to scale up with the dice count.)
     
  14. GingerGiant

    GingerGiant Well-Known Member

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    Okay, I approached the problem from a new angle that allows me to easily compare 1 burst against variable burst, in the case where I don't concern myself with anything other than "success". Assuming the Reactive Troop is Burst 1 in the face-to-face, I can compute the probability of Success, Failure, or Draw for each face of the Reactive Unit's d20, in light of a single d20 roll by the Active Toop. Using the Binomial Distribution, I can easily calculate the likelihood that the Active Troop attack is completely unsuccessful, and subtract that from 1 to determine the likelihood that the Active Troop achieves any degree of success.

    Here's a table showing the results of a given pair of attributes in F2F, for various Burst values (for the Active Troop):

    Code:
    A vs  R     Burst 1   Burst 2   Burst 3   Burst 4   Burst 5
    10    10    36.25%    56.44%    68.08%    75.08%    79.51%
    15    9     62.25%    82.01%    89.11%    92.03%    93.39%
    15    12    54.75%    74.49%    82.96%    87.25%    89.74%
    16    6     72.25%    89.01%    93.24%    94.42%    94.79%
    12    11    43.25%    63.99%    74.65%    80.61%    84.23%
    18    12    69.00%    84.43%    89.79%    92.21%    93.43%
    5     12    12.50%    22.13%    29.56%    35.33%    39.83%
    4     9     12.50%    22.63%    30.84%    37.51%    42.94%
    EDIT: I haven't yet told the table how to handle circumstances where the Active Troop has lower BS than the Reactive Troop. Figured it out.

    We can actually extend this in both direction (haven't gotten a table configured yet), by allowing the Reactive Troop's dice to have variable weights for each result, based on the number of dice in burst.
     
    #14 GingerGiant, Dec 19, 2018
    Last edited: Dec 20, 2018
  15. atomicfryingpan

    atomicfryingpan Well-Known Member

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    I use mix of blind faith, optimism, and feeling like I deserve to win the rolls.
     
  16. FatherKnowsBest

    FatherKnowsBest Red Knight of Curmudgeon

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    Luck is one of my skills
     
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  17. RogueJello

    RogueJello Well-Known Member

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    I think it's sad that the game rewards this skill. I'd like to suggest that we move away from this by using a pre-rolled system, where people are just allowed to assume they got the rolls they needed. :)
     
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  18. FatherKnowsBest

    FatherKnowsBest Red Knight of Curmudgeon

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    hahaha! Reminds me that there was once a diceless RPG called "Amber". @Section9 probably played it/heard of it I'd imagine.
     
  19. RogueJello

    RogueJello Well-Known Member

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    I've heard about it, essentially used the character's from Roger Zelanzy's Amber, which is a court intrigue fantasy series. IIRC, all characters had their characteristics in ranks against the other characters, and higher ranks trumped lower ones. Would not work in most games.
     
  20. Section9

    Section9 Well-Known Member

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    Heard of Amber, read through it once. Game basically relied on a players ability to tell an awesome story, off the cuff. That's nowhere near my skillset.

    =========

    I think I saw my Dad's eyes glaze over when I asked him about how to model the probabilities we work with. I admitted that I could do it if it was one die versus one die, but I was lost as soon as it got to multiple dice.

    So sadly, looks like Brute Force is the only way to do it right now.
     
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