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Thread: TROPHY OFFERS: Formulas discovered!!

  1. #1
    Forum Master zachUVA's Avatar
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    TROPHY OFFERS: Formulas discovered!!

    NEW: There is now a convenient table (located just below the graphs) providing the link between cup offers (win offer for the attacker) and the cup difference between the two players!



    Ok,
    technically speaking they are models (not exact formulas), but after taking several hundred observations of cup differences and the win/loss offers I saw, I have found equations that accurately model how cup offers are calculated.

    Win Offer Equation
    Y=-0.63599+(60.07066)/(1+0.991798e^0.00576x)


    • Where x is your cups minus your opponents cups and y is the number of cups you can win from 3-starring the opponent on offense.
    • For those who do not know, e is a mathematical constant that is approximately 2.71828
    • This is also the equation for calculating how much you can LOSE on defense, but you need to enter your opponents cups minus your cups to use as x, since they are they are the ones on offense. This does not work below 1,000 trophies, since losses are smaller than gains there.
    • This equation is the same for matchmaking offers and for revenge offers.
    • The equation appears to be accurate for any difference in cups between opponents, however the minimum is 0 and the maximum is 59, and the answer given by the equation will not be accurate if it is outside that range.
    • This equation appears to hold at ALL cup levels, whether it be 400 or 4000. That being said, the test range for win offers did not include sub-200 or any cup counts above 3400, but I see no evidence that it wouldn't be accurate everywhere.
    • The answer the equation gives is never off by more than 1 trophy (actually it's never off by that much, but it can be off by a few tenths of a cup, meaning the rounding can sometimes give a result that's off by 1).



    Loss Offer Equation


    Y=39.0907-(39.0619)/(1+0.993e^0.00595x)


    • Where x is your cups minus your opponents cups and y is the number of cups you can lose from failing to earn a star on offense.
    • For those who do not know, e is a mathematical constant that is approximately 2.71828
    • This is also the equation for calculating how much you can WIN on defense, but you need to enter your opponents cups minus your cups to use as x, since they are they are the ones on offense.
    • This equation is the same for matchmaking offers and for revenge offers.
    • The equation appears to be accurate for any difference in cups between opponents, however the minimum is 0 and the maximum is 39, and the answer given by the equation will not be accurate if it is outside that range.
    • IMPORTANT NOTE: THIS EQUATION IS ONLY VALID FOR MATCHES ABOVE 1,000 TROPHIES. Any matches under 1,000 trophies have a reduced loss offer on offense (I suspect that defensive wins award the cups predicted by the equation though, but I'm not sure). I do not currently have an equation for loss offers below 1,000 trophies, since it becomes more complicated there.
    • Above 1,000 trophies the equation appears to work everywhere. Again, testing was only done up to 3400 trophies, but there is no evidence that this doesn't hold all throughout champions league.
    • As with the win equation, the answer is never off by more than one trophy.


    For BOTH formulas, round normally. If the result is 23.4, expect 23. If it's 23.7, expect 24.


    Visual Representations

    I understand that these equations can be hard to picture, so I'll post graphs of the win and loss offers plotted against the cup difference between opponents (again, I used my cups minus opponents cups for these calculations).



    As you may notice, these are not even remotely linear. They are S-shaped, which is why I had to use logistic regression to come up with the above equations.

    Table

    Win Offer Cup Difference (attacker's cup count minus defender's)*
    0 >720(?)
    1 575-719(?)
    2 505-574
    3 454-504
    4 414-453
    5 379-413
    6 350-378
    7 324-349
    8 300-323
    9 279-299
    10 259-278
    11 241-258
    12 224-240
    13 207-223
    14 191-206
    15 176-190
    16 162-175
    17 148-161
    18 134-147
    19 121-133
    20 108-120
    21 95-107
    22 83-94
    23 71-82
    24 59-70
    25 47-58
    26 35-46
    27 24-34
    28 12-23
    29 1-11
    30 -11 to 0
    31 -23 to -12
    32 -34 to -24
    33 -46 to -35
    34 -58 to -47
    35 -70 to -59
    36 -82 to -71
    37 -94 to -83
    38 -107 to -95
    39 -120 to -108
    40 -133 to -121
    41 -147 to -134
    42 -161 to -148
    43 -175 to -162
    44 -190 to -176
    45 -206 to -191
    46 -223 to -207
    47 -240 to -224
    48 -258 to -241
    49 -278 to -259
    50 -299 to -279
    51 -323 to -300
    52 -349 to -324
    53 -378 to -350
    54 -413 to -379
    55 -453 to -414
    56 -504 to -454
    57 -574 to -505
    58 -719(?) to -575
    59 <-720(?)


    *The ranges were obtained using the data I gathered AND by using the predictions of the formula to fill in the gaps. It is quite possible that some of these ranges are off by a cup or 2. However, I have very little data at the extremes, so the ranges for 0-10 and 50+ cup offers are approximate at best.



    Additional Commentary


    • SC’s consistent claims that matchmaking and trophy offers are purely based on the cup difference between the two players is fully supported by the data I gathered. TH level, experience level, base strength, and everything else had no impact on cup offers. There were also no inconsistencies in the data; every data point fit the general trend.




    • Why are there no formulas to find out the cup difference given a win or loss offer (like in normal matchmaking)? Well, I used a logistic regression model to fit the data, but to do this it has to use decimals and rounding to get to the whole numbers that are shown ingame. What this means it ingame we are basically presented with a step function, and if you plot the cup difference as a “function” of the offers, you can have 10, 20, 200+ y values for a given x value. This is not even close to being a function. That being said, anyone can use the equations above to find estimated ranges of cup differences by taking all x values that give a result that rounds to 25 (for example) and noting the range of cup differences that produce that result. If demand is high, I can do this later.




    • Another interesting observation I made while collecting this data is that matchmaking's normal range is heavily biased to show you opponents who are lower than your own cup level. If someone has exactly the same number of cups as you, they will be worth 30 for a 3 star and 20 for a loss on offense. The normal matchmaking range gives players win offers ranging from 13 to 34, with 30 being relatively high in that range. From my observations I found no opponent more than 58 cups above me, and no opponent more than 223 cups lower than me. From this, I can estimate that matchmaking's range is roughly 300 (75 above to 225 below perhaps?) in most ranges. Of course, this does not apply in champions or anywhere where matchmaking struggles to find opponents.




    Well, I guess that's enough for now, but I'll be happy to provide any more info that I can based on any questions. I plan on making a "simplified" version soon that uses a linear approximation to estimate offers within normal matchmaking (useless for revenge offers that have high win or high loss offers), for those who want something easier to work with at the expense of accuracy.

    Please ask questions, provide any further insight/analysis, and discuss the results.



    PS: All data was gathered by using a second device to record cup counts while the offers were shown on the main device to ensure accuracy. Also, I just want to point out that Supercell had no involvement with this project of mine. No outside help was provided, nothing was reviewed by anyone before posting, etc., so there could very well be flaws with this, which I hope people will point out if they see any.

    -Zach
    Last edited by zachUVA; July 8th, 2015 at 07:04 PM. Reason: added table

    |Trophy Offer Formulas Discovered!|
    ^Now with a handy table linking offers and cup differences^

    RIP Smash Land

  2. #2
    Forum Master zachUVA's Avatar
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    Simplified version

    Admittedly, the above post is fairly complicated at times, so let's make this one a little easier to work with.

    Win formula:

    Y=-0.0794x+29.35838


    • Y is the number of cups you can win, and x is your cups minus your opponent's cups.
    • This is only useful for matches found in normal matchmaking ranges (13-34 cups).



    Loss formula:

    Y=0.0531x+19.60453


    • Y is the number of cups you can lose, and x is still your cups minus your opponent's cups.
    • This is only useful for matches found in normal matchmaking ranges (13-34 cups), and this formula is not accurate below 1,000 cups.



    Both formulas become less accurate the further you go from the stated ranges!


    Visual Representations:

    These graphs show the predicted values the formulas give and plots them against the real values I observed.





    Hopefully that makes it at least a little easier to understand, but I wasn't really sure what to include and leave out. Please let me know if anything is unclear or could be explained better.
    Last edited by zachUVA; July 29th, 2014 at 08:20 PM.

  3. #3
    Forum Superstar firelizard's Avatar
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    1st! Interesting thread zach

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    Forum Elder KlarkSmith's Avatar
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    Please consider taking the time to not read this.
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    Impressive work. *bow*

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    Senior Member Epiqtime's Avatar
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    The loss minimum may actually be 1 since I gained 0 cups on defense once but the attacker lost 1 cup. Minimal trophy deflation
    TH9 l 188 Skulls l Lvl 98



    BK 10 l 300M GG l AQ 8





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    Unbelievable. Awseome job.

    Currently Maxing out TH9.
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    "I think you are fishing for drama."


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    Cool...numbers...
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    Cool

    Sounds interesting.
    But I'm going to postpone the reading.

  9. #9
    Centennial Club obeybanana's Avatar
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    Very interesting.
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  10. #10
    Millennial Club UnknownShadows's Avatar
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    Very nice work zack!
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