Arena Doesn't Pay For Itself: The Math

Arena Discussion
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It doesn't.

This isn't about how there isn't enough gold, or that arena isn't fun, or whatever else. This is the simple fact that arena does not and never will pay for itself. If two people exist such that each player goes 3-3, they both lose. If one goes 4-3 while another goes 2-3, they both lose. If one goes 5-3 while another goes 1-3, they both lose. If one goes 6-3 while another goes 0-3, they both lose. If three people exist; one who went 7-3, one went 0-3, and one who went 0-1, one loses, one breaks even, and one is losing. If they go 8-3, 0-3, and 0-2, one loses, one is greatly losing, and one is slightly profiting. If four exist, and they go 9-0, 0-3, 0-3, and 0-3, three of them lose greatly while one wins moderately.

There is a bit more complication to this (those 3 losses came from somewhere else, so someone else is slightly winning, etc), but the fact that 3-3 vs 3-3 is a double-loss proves the case on the simplest level.

I've seen a lot of people on the forums claiming "7+" wins averages out the arena entry, so if you work really hard you can become good enough to pay for your arena forever!!!!!!!!!

There are two ways this can be.

1. Assume Arena uses an MMR system.

Only if you tank your MMR before entering, or MMR doesn't exist, or if you are the best player in the world will you pay for your arena forever.

I've seen a lot of explanation to these people about why that doesn't quite work out that way. Usually they say something like "Dude, the statistics have been worked out or, like, something, man. It's, like, mathematically proven! It's totally proofed that only 10% of the time arenas break even..!"

Of course, those who say that don't fully understand it, so they make math sound like a foreign concept, as if they are just reporting what the scientists out on the field are putting in their reports. The typical response is "Yeah well just make your deck really well and bam you can always get high wins! Practice hard!" or whatever.

The one really basic idea that a LOT of people seem not to be understanding is as follows:

If you go 9-0 in an arena, that means three other people went 0-3. This should be obvious. Your 9 wins translated to 9 losses somewhere. Sure, it wasn't 3 people losing 3 each, but you get the picture. That's how it averages out to be. There's no way of changing that. Even if your 9 wins caused 9 other dudes to go 9-1, that then means we have 81 losses elsewhere unaccounted for. Building it up like that and believing that these losses don't catch up to you later is the foundation of a pyramid scheme: the fact is that you eventually run out of people to soak the losses. The bottom line? All those people going 0-3.

That means, in order for you to make bank off of your 9-0 arena record, it wasn't just YOUR arena entry that paid for it; it was yours AND THEIRS. To go 9-0 in an arena means that FOUR people entered the arena, paying a total of 600 gold. Out of those four people, ONE of you reaped the big bonuses while THREE of you got screwed with losses.

If the MMR system worked effectively, then you could reasonably expect that -- for anyone who has already reached their set MMR (ie, not a new player) -- going 3-3 will be average. Sometimes 4-3 will happen, as will 2-3. Every time you go 4-3, someone else went 2-3. Every time you go 5-3, someone else went 1-3. With a good MMR system, you'll probably almost never go more than 4-3, no matter how good you are.

The only case where you would consistently go 9-0 would be the case of being the #1 player in the world. Guess what that means, then? The guys who are numbers 2 through 4 would be losing to you consistently, forcing them down and creating new 2's through 4's to lose to you, who would also go down and cycle through. What's that prove? That being #2 in the world doesn't mean you go 9-0, or 7-3. It means you go... 3-3, on average.

So enough with this "just practice!" business. Unless you become by far the best in the world, you won't be 9-0'ing the competition consistently. Even second in the world would 3-3, and, in reality, unless you're BY FAR 1st in the world, you'd probably only be going something like 4-3 or 5-3 anyway.

And before anyone says "Oh I've gone 9-0 6 times already I must be best in the world lulz." No. I supposed all of this with a functional MMR system that has had time to settle. Once this game is live, THAT is what you will have. Right now, it's a choppy MMR system with new people being introduced in massive (beta invite) waves.

Ie, you are going 9-0 because three noobs are going 0-3 and the MMR system is pairing you up for some reason.

2. What if the Arena doesn't use an MMR system?

This is where you get your "10%" figure, or so. Essentially what the figure checks is how likely you are to win each match, assuming a 50% chance on each match.

In that scenario case, these are your probabilities to win each game:

Going 0-3 (0.5^3 * 1 permutation): 12.5%
1 permutation: L-L-L

Going 1-3 (0.5^4 * 3 permutations): 18.75%
3 permutations: W-L-L-L, L-W-L-L, and L-L-W-L

Going 2-3 (50% ^ 5 * 6): 18.75%
6 permutations: W-W-L-L-L, W-L-W-L-L, W-L-L-W-L, L-W-W-L-L, L-W-L-W-L, L-L-W-W-L

Rather than continuing to write the permutations, I will just tell you how many there are. Suffice it to say, though, it's basic probability involving "choose" in math. Since the last factor is always "LOSE" (which is how your arena streak ended), the first scenario (0-3) is really just "2 Choose 0" which is 1. The second scenario (1-3) is "3 Choose 1" which is 3. The third scenario (2-3) is "4 Choose 2" which is 6, etc.

Going 3-3 (0.5 ^ 6 * P): 15.625%
P: 5 Choose 3 = 10

Going 4-3 (0.5^7 * P): ~11.7%
P: 6 Choose 4 = 15

Going 5-3 (0.5^8 * P): ~8.2%
P: 7 Choose 5 = 21

Going 6-3 (0.5^9 * P): ~5.5%
P: 8 Choose 6 = 28

Going 7-3 (0.5^10 * P): ~3.5%
P: 9 Choose 7 = 36

Going 8-3 (0.5^11 * P): ~2.2%
P: 10 Choose 8 = 45

Now it's a bit different. If you hit 9, that's means you ended on a Win. The last is still guaranteed (to be a win), so we are still choosing out of our (TotalGames-1). However, since we know it's a win, we checking for permutations that involve 1 less win than actually happened (because the ending game was a win!)

Going 9-2 (0.5^11 * P): ~2.2%
P: 10 Choose 8 = 45

Going 9-1 (0.5^10 * P): 0.88%
P: 9 Choose 8 = 9

Going 9-0 (0.5^9 * P): 0.2%
P: 9 Choose 9 = 1

Adding these up, we get the following:

Chance of winning 6 or fewer games: 12.5% (0 wins) + 18.75% (1 win) + 18.75% (2 wins) + 15.625% (3 wins) + 11.7% (4 wins) + 8.2% (5 wins) + 5.5% (6 wins) = ~91%

Thus, 9% chance of winning 7 or more. Just to see this in math: 3.5% (7 wins) + 2.2% (8 wins) + 2.2% (9-2) + 0.88% (9-1) + 0.2% (9-0) = ~8.98%. Seems about right for estimations. Now you know how people got the "Less than 10% of the time!" business.

But what does this mean to you, as a player? I'm sure you're sitting there going "I'm great. I win way more than 50% of the time! So this obviously doesn't apply to me!"

That's true. If there were no MMR system and you got randomly assigned players, then you reasonably could go 9-0 more frequently than 0.22% of the time. (Meanwhile, THOSE bad players would end up going 0-3 more than 12.5% of the time!)

In reality, we don't actually have a 50% chance to win going into any game. We aren't playing mirror matches against perfect clones. The truth isn't that we have a 50% chance to win. No, the truth is that 50% of games are won. This should be obvious: every game has exactly 1 winner and 1 loser. As we all know, a draw in the arena does not affect your score!

So what does this mean? This means that if YOU are a good enough player, YOU won't go 9-0 0.22% of the time. It means that 0.22% of ALL PLAYERS will go 9-0. Roughly 9% of ALL PLAYERS will go 7-3 or better.

In this scenario, players reasonably COULD become so good that they can pay for their own arena admission: you just need to be in the top 9% of the playerbase.

Does that fact mean I am wrong, though? No. I simply said that arena does not pay for itself. It doesn't -- even if you were in the top 9% of the playerbase, it's not simply YOUR arena fee that is paying your future entries, it is the other 91%'s fees that are paying for it!

Arena does not pay for itself on the whole. It is a gold sink. A fun, addictive gold sink. Think of slot machines: even though someone occasionally hits the jackpot and gets WAY MORE than they invested, MOST people just lose money. That's how the casino makes it!

Thus, the average player has absolutely no hope whatsoever of ever earning back their continuous entry into an arena. So please, stop using this "the arena can pay for itself if you practice!" excuse to justify or explain away other things. If every single player in this game practiced their hearts out, only 9% of us can do it. If someone became good enough to do it, that is only because they dethroned one of us who were previously doing it.

This isn't an issue you can practice away.
It's not supposed to...?
This is the perfect stats that I was simply too lazy to posts to the ignorants claiming this game is not a money sink. Watch them revert back to how it's a TCG(false) or that card are worth anything(also false) and that your rate of gold is fine(lie). I have no problem with Blizzard making money, but it doesn't need to start a casino of gaming that's going to cost more money than WoW
If you go 9-0 in an arena, that means three other people went 0-3. This should be obvious. Your 9 wins translated to 9 losses somewhere. Sure, it wasn't 3 people losing 3 each, but you get the picture.


We don't get the picture because its WAY more likely that its 9 people going 0-1.

Your math is very off. If you are a decent player and decide to buy into the arena it IS worth it.
09/02/2013 09:14 AMPosted by Mand
It's not supposed to...?

Didn't say it was supposed to. Said it doesn't, and so people should stop claiming it does.

We don't get the picture because its WAY more likely that its 9 people going 0-1.

Your math is very off. If you are a decent player and decide to buy into the arena it IS worth it.

I like how you countered math with vague guesswork. I assume that's how we put men on the moon, right?

I already addressed the fact that going 9-0 doesn't really mean 3 people are going 0-3. Like I said, 0-9 could mean that 9 people go 9-1, and those 81 losses could be explained by 81 people going 9-1, but eventually, there IS a bottom line: people going 0-3, 1-3, 2-3, etc.

What do you think happens when you go 9-0 and it makes 9 people go 0-1? Think all 9 of them go 9-1? No. For every three people going 9-0, there are nine people going 0-3. Sure, you could once again claim to me that that's not true: three people going 9-0 is probably 27 people going 0-1.

The world is made of a finite number of people. You cannot simply keep adding 0-1's forever. Eventually, some of them become 0-2, and then 0-3.

When you make someone lose, they lose. If they win, someone else had to lose. The fact is that when you win 9 and lose 0, you are upsetting the balance: you just took 9 potential wins out of the system, the losses need to come through SOMEWHERE.

If you truly believe that everybody can win, then you are the reason why pyramid schemes work.

http://en.wikipedia.org/wiki/Pyramid_scheme

Take a good look. You will see a lot of similarities between the reasoning why people might fall for a pyramid scheme (and the math for why it can't work), and you should be able to see how it ties into this conversation.

We can't all be winners. Those losses need to go somewhere.
While I agree with certain aspects of the post - it's a mistake to say taht with a proper MMR system that you'd almost always win 50% of your games in arena - while *on average* you would win 50% of your games, when you draft good decks you will still win >50% of the time and with bad drafts you will win <50% of the time. This will make it significantly more likely to get either lots of wins or lots of losses, even with a properly functioning MMR.

Conceivably this could be dealt with by the MMR if they had both an arena MMR and only ever matched you with people of equal MMR and an equal number of wins/losses in arena at the same time (neither of the 2 on their own will change this) - but I highly doubt there will be enough players for a system like that to work all the time even at release.

One other interesting thing to note, is that even if everyone had a 50% chance of winning every game, you will on average win slightly less than 3 games per arena run. The math is a bit counter-intuitive, but because the arena ends at 9 wins even though you haven't gotten to 3 losses yet, the average number of losses in an arena run will be slightly less than 3 and thus the average number of wins will also be slightly less than 3.
I've always thought it strange that they set the bar at 7 wins of max 9 to earn your fee back. Considering the oft quoted figure of 50/50 win/loss, then you'd expect 3 wins would be reasonably achievable for most people eventually, which leads me to think that 5 wins would be a more acceptable number to earn your fee back with.

I'm happy with the max number, it forces people to try new decks and keeps it fresh (yes, I know, also forces people to pay for entry more often too). I just think that 7 wins is too high with only 3 losses until you're out. Make it 5 wins to earn back your fee and either drop the cap to 7 wins total or add extra prizes to compensate and balance it.

Out of curiosity, what's the point where you equal out the cost of a deck vs. the cost of arena entry? Taking into account the difference in gold cost (considering standard single deck gold cost of 100g), at what number of wins do the additional prizes add up to the 50g extra fee for the deck?
The arena most definetely pays for itself and more.
Even if you suck completely and go 0-3 u still get a pack of cards worth of 100g.
at 3-4 wins you get a prize worth of the 150g you payed to enter.
plus its hella fun to play arena.
So if you are competent arena is worth it, if you suck don't touch it and go buy booster packs.
09/02/2013 09:19 AMPosted by cloudberry
If you go 9-0 in an arena, that means three other people went 0-3. This should be obvious. Your 9 wins translated to 9 losses somewhere. Sure, it wasn't 3 people losing 3 each, but you get the picture.


We don't get the picture because its WAY more likely that its 9 people going 0-1.

Your math is very off. If you are a decent player and decide to buy into the arena it IS worth it.


Actually it's not 9 people going 0-1.

It's one person going 0-1.
It's another person going 1-1.
It's another person going 2-1.
It's another person going 3-1.
It's another person going 4-1.
It's another person going 5-1.
It's another person going 6-1.
It's another person going 7-1.
It's another person going 8-1.

At least, that's how it will work once there is a sufficient Arena population for the matchmaking system to function properly. Right now, its tolerance on who matches up against whom is far looser, and it's possible for two people to play each other in back-to-back games.

But what I describe above is what they said during the Arena livestream about how Arena matchmaking is going to work.
Huge gaps in your math, but the general point you make is true. This is of course what everyone already knows. To make more gold in the arena than you spend, you have to be very good. That's really the bottom line. Are you suggesting that they give high rewards to those who do not do so well?

That said, it is really, really easy to make it so spending your gold in the arena is at least better than spending it on packs. When you add up the dust and gold you do get (on top of the pack), even for a 3 or 4 win, it is almost always better than the 50 extra gold you spent to enter.
Well, I agree with you. Arena does not pay for itself for everyone. That's fact, and that's math.

Now, how about being the one that wins ? :)
I have 4 win, 1 loss so far
obviously only 15ish percent will earn gold,

but arena cant get MMR, it would punish good players, and give 9wins to ppl who purposefully lose in ranked
OP, I commend you on going to the effort here, but unfortunately, nobody cares.

The people who understand and agree already know.
The people who don't already know are choosing not to.

They'd rather say
"I watch so and so on a stream and he only paid for one arena entry and now it pays for itself"
or
"If you're good enough you can win, it's all about skill"

Somehow, the fact that you can get absolutely horrible cards to start with, let alone bad draws, let alone any number of factors that go into winning or losing, they're convinced that if you try hard enough, you'll always win.

But in the end, it's their loss.
Casinos make millions out of stupid people.
Arena is just a casino disguised as a free to play game.
If the people who are bad or average simply stop playing arena guess what happens?

The above average / good people start losing money.
Excellent thread OP, I had similar suspicions about the arenas but seeing the actual math really puts things in a different light.
09/02/2013 09:31 AMPosted by Mand


We don't get the picture because its WAY more likely that its 9 people going 0-1.

Your math is very off. If you are a decent player and decide to buy into the arena it IS worth it.


Actually it's not 9 people going 0-1.

It's one person going 0-1.
It's another person going 1-1.
It's another person going 2-1.
It's another person going 3-1.
It's another person going 4-1.
It's another person going 5-1.
It's another person going 6-1.
It's another person going 7-1.
It's another person going 8-1.

At least, that's how it will work once there is a sufficient Arena population for the matchmaking system to function properly. Right now, its tolerance on who matches up against whom is far looser, and it's possible for two people to play each other in back-to-back games.

But what I describe above is what they said during the Arena livestream about how Arena matchmaking is going to work.


Where did the other 27 losses go? Those players didnt get their wins against bots, someone must have lost them. This is what would actually happen:

if you had 1024 players, after the first round of matches:
512 1-0, 512 0-1
Should be obvious enough.

After the 2nd round of matches (assuming everyone plays against people of the same win/loss amounts)
256 2-0, 512 1-1, 256 0-2

256 lose from the 1-0 bracket, and 256 win from the 0-1 bracket for a total of 512 going 1-1.

3rd round:
128 3-0, 384 2-1, 384 1-2, 128 0-3 (removed from arena)

4th round:
64 4-0, 256 3-1, 384 2-2, 192 1-3 (removed from arena)

Note that it stops being symmetrical since the 0-3 bracket stops playing.

5th round:
32 5-0, 160 4-1, 320 3-2, 192 2-3

6th round:
16 6-0, 96 5-1, 240 4-2, 160 3-3

7th round:
8 7-0, 56 6-1, 168 5-2, 120 4-3

8th round:
4 8-0, 32 7-1, 112 6-2, 84 5-3

9th round:
2 9-0 (arena finished), 18 8-1, 72 7-2, 56 6-3

10th round:
9 9-1, 45 8-2, 36 7-3

11th round:
22.5 9-2, 22.5 8-3

At this point I realized I should've done the math with 2048 players instead of 1024 players to avoid fractions, but oh well, you could just multiply all the results by 2.

The end results for this then, if you used 2048 players instead of 1024 players, and assuming everyone is *always* matched up with someone of the same win/loss record will be:

256/2048 people go 0-3
384/2048 1-3
384/2048 2-3
320/2048 3-3
240/2048 4-3
168/2048 5-3
112/2048 6-3
72/2048 7-3
45/2048 8-3

4/2048 9-0
18/2048 9-1
45/2048 9-2
stop treating this !@#$ like statistics class.

if you played lebron james 1 on 1 basketball, 1st to 20, 10 games...

your win % isnt going to be 50%. if you're wanting to win out in arena, maybe you have to play better than opponents, have a better understanding of deck types, know about pick orders etc etc etc.

do things that will increase your win% to above 50%, its pretty simple.
OP, I commend you on going to the effort here, but unfortunately, nobody cares.

The people who understand and agree already know.
The people who don't already know are choosing not to.

They'd rather say
"I watch so and so on a stream and he only paid for one arena entry and now it pays for itself"
or
"If you're good enough you can win, it's all about skill"

Somehow, the fact that you can get absolutely horrible cards to start with, let alone bad draws, let alone any number of factors that go into winning or losing, they're convinced that if you try hard enough, you'll always win.

But in the end, it's their loss.
Casinos make millions out of stupid people.
Arena is just a casino disguised as a free to play game.


This post couldn't be any more true, the people who don't recognize this are just ashamed of how much money they've spent or simply like being taken advantage of and feel like they actually matter to Blizzard and are not another penny in the bank
stop treating this !@#$ like statistics class.

if you played lebron james 1 on 1 basketball, 1st to 20, 10 games...

your win % isnt going to be 50%. if you're wanting to win out in arena, maybe you have to play better than opponents, have a better understanding of deck types, know about pick orders etc etc etc.

do things that will increase your win% to above 50%, its pretty simple.


Not it's not that simple actually, infact my last arena I played against a guy who right off the bat put down two blood imps, I have ZERO choices of AOE cards so that meant infact he had +2 health on every unit for the rest of the game for the cost of 2 mana on turn 1 because of course he got the coin. This goes to show how by simply trying hard doesn't get you wins, a lot of it is luck and even luck plays into people getting 8-0. By your stupid logic I bet you're 100-0 in this game right? No you're not, stop acting like you know anything because you clearly don't and i'm sorry you failed out of stats class

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