The statistics of arena farming

[Fayceless]

While thinking about a discussion taking place elsewhere on these forums, it occurred to me that the common ways of deciphering a farming deck's chances at getting a rare spin are flawed.  Since arena matches do not occur in nice, tidy sets of 2 to 5 games, a simple formula to determine a deck's potential success does not work.***   So I set about trying to figure out the real odds of getting those rare spins.

***Wizy posted a nice formula below that can be applied to decks of any win-rate.  You'll probably need a calculator unless you can do X^-1 in your head.

I wrote a short program to simulate the results of 100,000 games, over and over again.  Nothing complex, just a long list of wins/losses based on a win percentage.  The program then counts how many spin-winning runs the imaginary decks would get.

Here's what I got.  The table below shows estimates for how many rare spins a deck with X win % gets in 100 games played, plus electrum earned. (based off recent arena trends)  Numbers are rounded.

Key: S = total number of rare spins won  E = total electrum earned.

BRONZE          SILVER           GOLD          PLATINUM         S        E      S        E      S         E     S         E 100.00% 20      800     25      5900    33      17800   50      16800 90.00%  14      570     19      5150    27      15900   43      15000 80.00%  10      340     14      4400    21      13900   36      13150 70.00%  6       110     9.5     3700    15      12050   29      11300 60.00%  3       -120    6       3000    11      10100   22.5    9450 50.00%  2       -350    3       2200    7       8150    17      7700 40.00%  1       -578    1.5     1450    4       6200    11      5800 30.00%  0       -810    1       700     2       4300    7       4000 20.00%  0       -1040   0       0       1       2350    3       2200 10.00%  0       -1270   0       -750    0       450     1       300 0.00%   0       -1500   0       -1500   0       -1500   0       -1500

The next table is another (more accurate) representation of the data I got.  It shows how many games, on average, a deck will take to win a rare spin.

Bronze  Silver  Gold    Platinum 100.00% 5       4       3       2 90.00%  6.94    5.24    3.72    2.35 80.00%  10.26   7.21    4.77    2.81 70.00%  16.5    10.55   6.38    3.47 60.00%  29.65   16.79   9.07    4.44 50.00%  62      30      14      6 40.00%  161.1   63.44   24.38   8.75 30.00%  586.5   174.94  51.48   14.44 20.00%  3000+   780     155     30 10.00%  N/A     N/A     1000+   110 0.00%   N/A     N/A     N/A     N/A

Thanks to Wizy for more accurate numbers!

As you can see, Bronze loses efficiency very, very quickly, while even a 50% platinum farm deck can net a lot of upped cards - it takes, on average, just under 6 games per spin.  It's no wonder Platinum is more popular than gold - even 40% gets a card in 9 games.  And if you can manage to get to 75-80% in platinum, you've hit the spot where no deck in any league can out-perform yours.

Of course, none of this takes into account TTW or game length.  You'll still have to do that on a deck-by-deck basis, but if you really want to get into it, you can use these numbers, with stats from individual decks, to figure out which deck is a more effective farm deck.

Enjoy the numbers.  Feel free to ask anything or point out any glaring mistakes I've made.

[YawnChainHow]

Wow, nice numbers. Could you calculate at what win% you break even in electrum earned in Gold and Plat?

Also, beneath the second table, you seem to use 'games' and 'turns' interchangeably. I hope it's the former rather than the latter in both cases, or I'm really missing out.

[Wizy]

I've noticed some of your numbers in the second table are slightly off, e.g. 50% in platinum: 5.95 and 50% in bronze: 60.6
Here is a slightly more accurate table of average games played per spin:

Bronze  Silver  Gold    Platinum 100.00% 5       4       3       2 90.00%  6.94    5.24    3.72    2.35 80.00%  10.26   7.21    4.77    2.81 70.00%  16.5    10.55   6.38    3.47 60.00%  29.65   16.79   9.07    4.44 50.00%  62      30      14      6 40.00%  161.1   63.44   24.38   8.75 30.00%  586.5   174.94  51.48   14.44 20.00%  3000+   780     155     30 10.00%  N/A     N/A     1000+   110 0.00%   N/A     N/A     N/A     N/A

[Fayceless]

Wow, nice numbers. Could you calculate at what win% you break even in electrum earned in Gold and Plat?

Also, beneath the second table, you seem to use 'games' and 'turns' interchangeably. I hope it's the former rather than the latter in both cases, or I'm really missing out.

Hahah, thanks for catching that.  As for break-even, I'll look into it.  It doesn't take very many wins.

@Wizy: interesting, where did you get those numbers?

[teffy]

Very interesting. I´m surprised nobody ever had this simple idea (or I don´t know that).

I have also tested it. Your numbers seem to be right.(Haven´t tested electrum)
And it´s a really short program.

[Wizy]

@Wizy: interesting, where did you get those numbers?

I've used this math:
Example for 50% in platinum:
In order to win a special spin you need to light ring A and ring B
you have to play 2 games to light ring B
in order to play 2 games for ring B you have to light ring A twice
you have to play 4 games for ring A to light it twice
2+4=6

50% in bronze:
rings A,B,C,D,E
2 for E, 4 for D, 8 for C, 16 for B, 32 for A
2+4+8+16+32=62
i.e. X+X^2+X^3+X^4+X^5 ; X=sucess rate^-1 in this case: 0.5^-1=2
i.e. 2+2^2+2^3+2^4+2^5=62

Another example totally different, 80% in silver:
1.25+1.25^2+1.25^3+1.25^4=7.207=7.21~

Then used my favorite program: Excel, to easily calculate the rest ;p

[Fayceless]

Nice.  I haven't taken a math class in years.     There's bound to be some uncertainty in the numbers I posted, as it was based off a quick simulation.  Even at 100,000 games, there's some variation between tests.  Maybe I'll ramp it up to a few million and see if it turns out closer to the numbers you posted.

Edit: Yes, simulating at 5,000,000 games verifies that the numbers you posted are indeed correct.  Not that I doubted you!  I just wanted to be 100% sure.  Seems 100k wasn't a big enough sample size.  Updating OP with your #s.

[fsk]

It's hard to calculate break-even electrum because:

• You can EM.
• Award depends on HPs left.
• You can sell the cards you win.
• If you get a rare you don't need from the bonus spin, you can sell it.

[Wizy]

I got double ninjaed but I will submit the post anyway, I think it is still relevant unless you have already corrected the spins.Regarding the spins in the first table, while your results are close to the real ones, some are not exact. The problem is that no matter how many times you run such a program, the results are always flawed.
Example: if the program "plays" 100,000 games with a winning rate of 10%, it is supposed to win exactly 10,000 games to give exact results, however it will rarely ever win exactly 10,000 therefor the results are flawed.

With maths, not only the results are more accurate, it is also easier to do.
Imagine there is a town in wich you know there are 10 houses and 3 people in each house. If you want to know how many people are in the town, woud you visit every house and count each person one by one? It is easier if you simply do 10*3=30 (31 if you enter the town, doh)

tl;dr -Here are the wrong average spins (per 100 games) corrected:
Silver 70%=9
Platinum 60%=23

[teffy]

[...]Yes, simulating at 5,000,000 games verifies that the numbers you posted are indeed correct.  [...]

I tried 100.000.000. PC is still fast enough. (10 sec per entry for me, Java)

Advantage of math solution:
- often more exact
- (often) easier to calculate
- more elegant

Disadvantage:
- Thought mistakes can happen, especially with probabilities and statistics and users. However, Wizy did a good job (and posted a good explaination).
- Requires a special way of thinking, and sometimes you don´t know, how complex your problem is. It´s sometimes easier to write a program.

If there´s a nice math solution, I often prefer (and like to see) the math solution. However, it´s good to confirm that with a program.

[Fayceless]

tl;dr -Here are the wrong average spins (per 100 games) corrected:
Silver 70%=9
Platinum 60%=23

Both of those come out really, really close to .5, it would seem.  I'll just put .5 because, especially in silver, I think it gives a better idea of how many spins a deck can expect to get.

[Wizy]

You are right 9.49 and 22.5 I did just let Excel round them
One thing I realized, there is no way to know the win rate of a deck with precision, thus my obsession for exactitude was unnecessary, since either way the final result will never be exact. o.o

Anyway, things I find interesting:

Be it for rares or electrum, bronze is not worth a zanz compared to other leagues, with whatever deck.

50%vs plat gives more rares than 70%vs gold, IMO Getting 50% win rate vs platinum is much easier than getting 70% vs gold.

Most plat farmer decks have a win rate of about 60%, the only way a gold farmer could get more rares than that would be with about 80% win rate, wich is close to impossible vs gold. Whoever says that his deck can reach 80% vs gold lies or didn't test enough games.
Even so, gold looks like the right choice for score and electrum.