Act Bosses - Relative drop chance yields (Mephisto & Andariel)

[drmalawi]

The objective of this post is to investigate how drop chances from Mephisto and Andariel changes with player settings. This post can be seen as a continuation and completion on what I and @Luhkoh have discussed in other places here on the forum.

Introduction
In single player, we often use the \playersx command to emulate more (unpartied) players in the same area. The effect of this command is to decrease the value of the “no drop” parameter value of monsters in their Treasure Class (TC) thus effectively increasing the drop chance of items from almost all monsters (uniques and some superuniques do already have no drop = 0 as base value). Due to how the formula for lowering the no drop parameter, we only see changes for odd number of players setting, i.e. /players3 and /players4 will result in the same no drop value.

Act bosses and drop calculators is a funny story. The popular calculators (Atma, Silospen & German Drop Calculator) are not able to fully reconstruct the peculiarities of act boss drop chances. In particular, act bosses has the picks parameter value equal to 7. This means that game will do 7 picks across their sub-TC’s, if and only if at least one of the first 6 picks resulted in a no drop. The drop calculators also seems to ignore that essences are included in the Act boss TC in Hell difficulty, thus altering the precision of the drop chance.

Here, I have simulated drops from Mephisto and Andariel in order to determine the relative yield in drop chances between different players settings. What the absolute drop chance is for a given item, I will leave for a future project.

I chose to simulate my results (data and code below) so that we do not need to get involved in probability and combinatorial calculations.

This work was done for 1.13-1.14 patches of Diablo 2 Lord of Destruction.


Data

Mephisto Hell, TC weights

Gold = 5
Act 4 (H) Equip A = 52
Act 4 (H) Junk = 5
Act 4 (H) Good = 3
Essence = 1
Sum of item weights = 66

No drop (base, players 1) = 15
No drop (players 3) = 2
No drop (players 5 & 7) = 0

We can from the above see that on players 5 and higher, Mephisto will always drop the maximum of 6 items.

Andariel (Quest) Hell, TC weights

Act 2 (H) Equip A = 19
Act 2 (H) Good = 3
Essence = 1
Sum of item weights = 23

No drop (base, players 1) = 19
No drop (players 3) = 6
No drop (players 5) = 2
No drop (players 7) = 1

Code

(example of Mephisto players 3)

Spoiler

const int kills = 1000; //change for different accuracy

int  res[kills];

    int nodrops = 0;
    int items = 0;
    int roll = 0;

    int nodropparam = 2; //change to relevant value
    int itemsparam = 66;
    int gridsize = nodropparam + itemsparam;

    default_random_engine generator{ static_cast<long unsigned int>(time(NULL)) };
    uniform_int_distribution<int> distribution(1, gridsize);

    for (int n = 0; n < kills; n++)
    {   nodrops = 0;
        items = 0;

        for (int m = 1; m < 7; m++)
        {   roll = distribution(generator);

            if (roll < nodropparam+1)
                nodrops++;
            else
                items++;
        }

        if (nodrops > 0) { // conditional 7th pick
            roll = distribution(generator);
            if (roll > nodropparam)
                items++;
        }

        res[n] = items;
    }

//write the res array to file and analyze data.

Results

I have done some roundings to closest 0.5% unit.
This means that probabilites lower than 0.5% are not included here.

Mephisto Hell

Players 1
Probability(3 items) = 2%
Probability(4 items) = 10%
Probability(5 items) = 26.5%
Probability(6 items) = 61.5%
Average = 5.465 items per kill

Players 3
Probability(5 items) = 1.5%
Probability(6 items) = 98.5%
Average = 5.984 items per kill

Players 5, 7
Probability(6 items) = 100%
Average = 6 items per kill

Relative yields
p3 vs. p1 : 9.5% higher drop chances
p5 vs. p1 : 9.8%
p5 vs. p3 : 3.2%

Andariel (Quest) Hell

Players 1
Probability(1 item) = 3.5%
Probability(2 items) = 13%
Probability(3 items) = 22%
Probability(4 items) = 29%
Probability(5 items) = 22%
Probability(6 items) = 10.5%
Average = 3.819 items per kill

Players 3
Probability(2 items) = 0.5%
Probability(3 items) = 3.5%
Probability(4 items) = 12%
Probability(5 items) = 27%
Probability(6 items) = 57%
Average = 5.359 items per kill

Players 5
Probability(4 items) = 1%
Probability(5 items) = 9%
Probability(6 items) = 90%
Average = 5.879 items per kill

Players 7
Probability(5 items) = 4%
Probability(6 items) = 96%
Average = 5.958 items per kill

Relative yields
p3 vs. p1 : 40% higher drop chances
p5 vs. p1 : 54%
p7 vs. p1 : 56%
p5 vs. p3 : 9.7%
p7 vs. p3 : 11%
p7 vs. p5 : 1.3%

Conclusion

Higher drop chances on higher player settings must be balanced with the fact that the run time usually goes up (since monsters get more hit-points and deals more damage on higher player settings). The numbers presented here can be used to figure out if it is “worth” to increase the player setting to achieve higher chance per unit of time.

As an example, doing Mephisto on players 3 instead of players 1 results in a 9.5% higher drop chance. Let’s say that your Mephisto run times on players 1 are 20 seconds on average. If you can’t do players 3 Mephisto in 22 seconds or faster, the increase in player setting is not “worth” it.

 

[ResTTe]

Interesting calculations as always. Finally we have some reliable data in terms of %gain between different players settings and don't have to rely on the wrong values that drop calculators give us. Good job!

 

[Luhkoh]

Good job! I'm glad this is somewhere where I can link it to show people now. This should really be common knowledge, but the seventh drop mechanic makes it tougher to calc optimal playersettings unless these ratios are calc'd.