Hearthpwn

I've been doing some simple math and started to realize that the prospects of Arena is not as promising as most players think, especially for the average player.

Imagine 100 players in one closed Arena. After the first game, 50 players will be 1-0, while the other 50 will be 0-1. Then in the second game, the 50 1-0 players get matched up, and 25 of them will be 2-0, while the other 25 will be 1-1. For the 50 0-1 players, 25 will be 1-1, while 25 will be 0-2. So at the end of Game 2, we have 25 2-0, 50 1-1, and 25 0-2. In Game 3, the 25 2-0 and one 1-1 gets matched up, so 12 of them would be 3-0, 14 would be 2-1. The 49 1-1 and one 0-2 would be matched up and will result in 24 players going 2-1, 26 players going 1-2. The remaining 24 0-2 players would go against each other and 12 players would be 1-2, while 12 would be 0-3 and that would end their arena. So, end of Game 3 we have 12 players 3-0, 38 players 2-1, 38 players 1-2, and the rest are out.

Game 4, the 12 players who are 3-0 gets matched, and 6 of them go 4-0, while other 6 goes 3-1. 38 2-1 players gets matched and 19 go 3-1 while 19 go 2-2. 38 1-2 players go head to head and result in 19 players 2-2, and 19 players 1-3 (end of arena). So at the end of game 4, 6 players 4-0, 25 players 3-1, 38 players 2-2, and the rest are out with 1-3.

At this point we have 31% of players out of arena with only 1 or 0 wins.

5th game: The six 4-0 players paired up and 3 will go 5-0, 3 will go 4-1. 25 3-1 players and one 2-2 gets matched, resulting in 12 players 4-1, 14 players 3-2. 37 2-2 players gets matched up. So, 19 gets 3-2 while 18 gets 2-3 and are out.

At this point we have 49% of players out of arena with 0 to 2 wins. 3% of players going 5-0, 15% going 4-1, 33% of players going 3-2.

Game 6: 2 players 6-0, 8 players 5-1, 25 players 4-2, and 16 players 3-3.

At this point, players who are out with 0 to 3 wins are 65%, 2% 6-0, 8% 5-1, 25% 4-2.

I will not proceed further but 4 wins is the number you need to cover 150 gold entry cost (1 pack plus 50 gold reward), and 65% of the players won't make it.

So, maybe it is time to reflect on the ability of an average player and ask ourselves, why are we still recommending that new players go for arena for "better" rewards?

 

It's interesting to see the math on Arena laid out like this. And in a vacuum, I think you're likely right that Arena isn't worth it. But the way this looks at Arena assumes no skill, only luck. And while it's true that 65% of players won't get to 4 wins and make their money back, there are things you can do to greatly increase your chances of being in that winning 35%. There's a reason that streamers (Trump, Kripp, Guardsman Bob, etc) average about 6 wins in Arena. Basically, if you have a fair amount of skill at playing the game, and you have a good idea of which cards are best to pick for value and for curve, you can consistently make your money back in Arena. But you're absolutely right that we shouldn't keep recommending new players to Arena. It takes some work to get good in the Arena. I can remember my first few runs, which all ended something like 2-3 or worse. Now I average closer to 4 or 5 wins. Which, as this math proves, is a result of an increase in ability in both playing and drafting. Really cool post though, I like seeing the percentages clearly presented.

Quote from Urukai_ »

It's interesting to see the math on Arena laid out like this. And in a vacuum, I think you're likely right that Arena isn't worth it. But the way this looks at Arena assumes no skill, only luck. And while it's true that 65% of players won't get to 4 wins and make their money back, there are things you can do to greatly increase your chances of being in that winning 35%. There's a reason that streamers (Trump, Kripp, Guardsman Bob, etc) average about 6 wins in Arena. Basically, if you have a fair amount of skill at playing the game, and you have a good idea of which cards are best to pick for value and for curve, you can consistently make your money back in Arena. But you're absolutely right that we shouldn't keep recommending new players to Arena. It takes some work to get good in the Arena. I can remember my first few runs, which all ended something like 2-3 or worse. Now I average closer to 4 or 5 wins. Which, as this math proves, is a result of an increase in ability in both playing and drafting. Really cool post though, I like seeing the percentages clearly presented.

It's all about comparing yourself to the rest of the field, and whether you can rise above the rest and be one of the top 35% of players. As the numbers suggests, only 35% of the players can do it. Improvements in performance is largely due to 2 factors: one is improvement in one's skills, and another is the influx of less experienced fodder players in arena. Essentially, everyone who just started playing arena will be fodders and form part of the bottom 65%.

However, it only looks less promising if you look at it from the perspective of getting back that full 150 gold. If you see it as a format where you pay a small amount of gold (maybe around 20-50 gold due to nett loss) to be able to play with rare, epics, and legendary cards that you don't have, it might seem that every run is quite worth it.

IMO, arena is not a place to grind gold and it shouldn't be advertised as such. It is the one and only gold sink in the hearthstone economy, because comparing 100 players spending 300 gold each on packs and 100 players spending 300 on arena runs, the first group of players will always end up with more packs/cards/dust.

You are not factoring the risk-reward. The scaling of the rewards after 4 wins far outpace the cost of below 4 wins.

Example, last time I did a 0-3 and was rewarded 1 pack and 30g. Thats a -20g loss.

I just finished a 7-3 and was rewarded 1 pack and 180g. Thats a 130g profit.

So winning 1 time of 7-3 will cover 4 occasions when I go 0-3.

Even if you average 3 wins, you should still make a profit...Someone do the maths to make sure ;)

If your Math is right, a break even point should be at like 2 wins and profit at 3 wins in arena, my friends don't play hearthstone with me now(or at all) bc earning cards is a long hard grind, and unless you are above average its too slow to keep up to p2w players, imo quest rewards should go up in value and payouts for arenas be better, just so it's not so daunting the task of collecting s bc at this rate with a 2-3 average win rate in arena and doing dailies i open 1 pack every 2-3days, after paying for an arena entry  i pray i get 4 wins lol

There's a list for what % of players make it to each win going from 0-3 to 12 wins. You can find it on google. Also you just need to go 3 wins to break even I get around 7+ wins easily... Just gotta play arena for a bit.

The list is at the bottom of the page http://hearthstone.gamepedia.com/Arena

Quote from 2pac »

There's a list for what % of players make it to each win going from 0-3 to 12 wins. You can find it on google. Also you just need to go 3 wins to break even I get around 7+ wins easily... Just gotta play arena for a bit.

I don't doubt that you get 7+ wins easily. However, looking at the numbers, you need to realize that you and other 7+ win players are approximately 10% of the total arena players. In order for you to have 7+ wins, more than half of the players will go less than 3 wins. The math shows that it is mathematically impossible for the arena to have even 30% of players going 7-3 or better.

3 wins doesn't break even 100% of the time, because of the random reward. Only 4 wins and above guarantees the 1 pack plus >50 gold reward return.

Quote from Apex1771 »

You are not factoring the risk-reward. The scaling of the rewards after 4 wins far outpace the cost of below 4 wins.

Example, last time I did a 0-3 and was rewarded 1 pack and 30g. Thats a -20g loss.

I just finished a 7-3 and was rewarded 1 pack and 180g. Thats a 130g profit.

So winning 1 time of 7-3 will cover 4 occasions when I go 0-3.

Even if you average 3 wins, you should still make a profit...Someone do the maths to make sure ;)

3 wins give u one pack, 25 gold, and a random reward of gold or dust. So i don't consider that as breaking even. Only 4 wins give u a guaranteed 1 pack plus >50 gold return.

1 win of 7-3 covers 4 occassions of 0-3. For most players, it is >4 times more likely that they finish 0-3 than 7-3. In fact, most will go more than 10 arena runs below 4 wins before they hit their first 7 win.

Quote from Livesage »

Good effort, but this doesn't reveal anything that we don't already know. It is a statistical fact that the median win rate of arena has to be 3 wins. So it is to be expected that 49% of the players will have 0-2 wins. Then there is a middle tier of players with average 3-3 runs (which, in your simulation, comprises between players in the 50-65% range). It is harder to get above average win rates (only 35% of players make it) so the rewards also scale exponentially as a result.

What I disagree with is that the break even point is 4 wins. 4 wins is actually the number of wins where you are guaranteed to break even and will almost always make a profit as well. But at 3 wins you more often than not also break even if your third reward comes in the form of gold or dust (assuming 1 dust is worth roughly the same as 1 gold). And sometimes even at two wins you can break even if you are slightly lucky.

The reason why people recommend newbies to play Arena is because Arena is an investment. You have to accept some losses initially to get better at it; there is no other way to learn other than by playing. It's like why people go to college - it is a losing proposition initially because you have to pay tuition fees and forgo any income you would have earned from working for 4 years, but in return you (generally) draw a higher salary once you graduate.  

Some of us might already know the whole math. However, judging by advices such as "don't buy packs now, save up for arena it will be more worth it", it makes me think that we are misleading the newbies into thinking that by investing 150 gold into an arena run they are automatically making the more "worth it" short term investment. When in truth, they will definitely lose a lot of gold (compared to just buying packs with every 100 gold) in arena before they even start to break even. And after that, they need to push their way to the top 20% of players who average more than 5 or 6 wins to get enough profit to cover their initial investment. And finally be one of the top 10% who can go infinity.

It's a very steep success curve where the success ultimately depends on how many bottom-of-the-barrel fodders are available. The truth is, we can't all be winners in the arena, it doesn't work. In order for one to go 5-3, he is probably sending 10 players out with -ve rewards. The entire system sustains only a minority of successful players, so why are we suggesting for newbies to try their luck to see if they are top 10% arena material when it is 90% likely that they are not?

If i'm a seasoned arena veteran, the only good reason for me to introduce newbies to this format would be to get more fodders. After all, it is more likely that they will aid me in my 7-win quest than cause me to go 6-3 instead of 7-2.

The fact remains that if a player buys packs with every 100 Gold, it is 65% likely that he will have a larger card collection than if he went for arena, for the forseeable future.

The reason is simple; fresh meat for the grinder. ;)

Another reason to recommend arena, is because it's a unique experience that gives you (temporary) access to every card in the game.

With the daily rewards averaging 45-50g (depending upon how efficiently you utilise your rerolls), casuals can still experience arena once every ~3 days regardless of how well they perform. That's fine for most casuals.

 

1. I see you are not factoring in dust and card rewards. If a player's purpose is to build a cardpool those are just as important as gold. With that in mind an arena run breaks even at 3 wins with a small chance of achieving that at one or two wins as well, while being certainly profitable above that number.
2. One other reward you gain by playing in the arena, the most valuable for a newer player, is experience. You get to play with cards you do not know, in a much larger variety of situations than what you would encounter in constructed play.
   For the type of players who learn from their games, the arena can provide much more experience in a much faster pace than constructed can give you, therefore it is better even for newer players and even if they average out one or two wins.

The only type of player i would not suggest arena to, is the ones who get easily frustrated, or want to be absolutely casual from the beginning, without being willing to go through a harsh learning period. For those players, playing vs AI would be the best thing to do before moving on to casual play, since they would not be willing to go through a string of defeats before managing to get into the game,

Edit: The average arena rewards are actually worth about 170 gold, based on calculations using these stats, so achieving a balance of 150 gold worth of rewards requires an average of about 2.65 wins.

Quote from Tahladnas »

Quote from Livesage »

Good effort, but this doesn't reveal anything that we don't already know. It is a statistical fact that the median win rate of arena has to be 3 wins. So it is to be expected that 49% of the players will have 0-2 wins. Then there is a middle tier of players with average 3-3 runs (which, in your simulation, comprises between players in the 50-65% range). It is harder to get above average win rates (only 35% of players make it) so the rewards also scale exponentially as a result.

What I disagree with is that the break even point is 4 wins. 4 wins is actually the number of wins where you are guaranteed to break even and will almost always make a profit as well. But at 3 wins you more often than not also break even if your third reward comes in the form of gold or dust (assuming 1 dust is worth roughly the same as 1 gold). And sometimes even at two wins you can break even if you are slightly lucky.

The reason why people recommend newbies to play Arena is because Arena is an investment. You have to accept some losses initially to get better at it; there is no other way to learn other than by playing. It's like why people go to college - it is a losing proposition initially because you have to pay tuition fees and forgo any income you would have earned from working for 4 years, but in return you (generally) draw a higher salary once you graduate.  

Some of us might already know the whole math. However, judging by advices such as "don't buy packs now, save up for arena it will be more worth it", it makes me think that we are misleading the newbies into thinking that by investing 150 gold into an arena run they are automatically making the more "worth it" short term investment. When in truth, they will definitely lose a lot of gold (compared to just buying packs with every 100 gold) in arena before they even start to break even. And after that, they need to push their way to the top 20% of players who average more than 5 or 6 wins to get enough profit to cover their initial investment. And finally be one of the top 10% who can go infinity.

It's a very steep success curve where the success ultimately depends on how many bottom-of-the-barrel fodders are available. The truth is, we can't all be winners in the arena, it doesn't work. In order for one to go 5-3, he is probably sending 10 players out with -ve rewards. The entire system sustains only a minority of successful players, so why are we suggesting for newbies to try their luck to see if they are top 10% arena material when it is 90% likely that they are not?

If i'm a seasoned arena veteran, the only good reason for me to introduce newbies to this format would be to get more fodders. After all, it is more likely that they will aid me in my 7-win quest than cause me to go 6-3 instead of 7-2.

The fact remains that if a player buys packs with every 100 Gold, it is 65% likely that he will have a larger card collection than if he went for arena, for the forseeable future.

It's a cycle. New players start as fodder in order to learn and get better. Then they become good and start profiting off the newer players who are now the fodder. You can choose to avoid arena and not be fodder, but then you will also never become better and reap the potential rewards as you grind your 10 gold per 3 wins.

Let's say someone plays 2 hours a day and games take 10 min each. So in one day, they play 12 games. Let's say they have a 50% win rate so they win 6 games and gain 20 gold per day. In one month they gain 600 gold. In 6 months they gain 3600 gold. Not including any of the daily quest gold (which would be gotten by both arena and constructed players), a constructed player would be able to buy 36 extra packs in 6 months.

Now let's say they spend the same amount of time playing arena. Say in the first month, they do poorly and go 0-3 wins. They do 2 such runs a day and get 2 packs (worth 200g) but no extra gold rewards. So they lose 100g per day. After 1 month they lose ~3000 gold.

In the next 2 months, they improve and average 3-4 wins which let's them break even. Then in the next 3 months, they get much better and start to average 6-7 wins which is a 100g profit per run. If they do 1 run a day, that's 9000 gold in 3 months.

After the same 6 months, they would be 6000 gold up which would let them buy 60 extra packs compared to 36. And since they are now averaging 6-7 wins, they can get 3000g profit per month (if they play 1 arena a day) compared to the static 600g per month playing constructed.

Just about anyone can hit that 6-7 win average after 3-6 months of experience. It's not an issue of whether they can be a successful arena player, it's an issue of whether they are willing to make the initial sacrifice because as you can see, losing a lot of gold in the first month for someone who doesn't have much gold to begin with is hard to bear. But once you reach the point of getting 6-7 wins, your profit vs playing constructed is way higher.

Actually this math isnt work. You are facing opponents with same win quantity, not with same win/lose rate.

Okay, here's a blanket to reply to the couple of comments comparing Arena to University or the work force:

It is mathematically possible for everyone from a class to have solid jobs upon graduation. It's even possible for the entire university to produce graduates who are able to land decent jobs. Nothing is preventing that from happening. It all depends on the student himself.

However, in the zero sum game that is the arena, it is mathematically impossible to have EVERYONE getting 4 wins.

That's the difference. The society's workforce doesn't need a person to be jobless in order to sustain a doctor. Being a successful engineer doesn't mean that you are causing 5 people out of their jobs. However, in Arena, if you are going 7-0, you did cause 3 players to crash out with less than 3 wins.

Quote from Tahladnas »

Okay, here's a blanket to reply to the couple of comments comparing Arena to University or the work force:

It is mathematically possible for everyone from a class to have solid jobs upon graduation. It's even possible for the entire university to produce graduates who are able to land decent jobs. Nothing is preventing that from happening. It all depends on the student himself.

However, in the zero sum game that is the arena, it is mathematically impossible to have EVERYONE getting 4 wins.

That's the difference. The society's workforce doesn't need a person to be jobless in order to sustain a doctor. Being a successful engineer doesn't mean that you are causing 5 people out of their jobs. However, in Arena, if you are going 7-0, you did cause 3 players to crash out with less than 3 wins.

That's not even true. There are a finite amount of positions for any job in society. If your town has 3000 people and can employ 10 doctors (doctors being 12 wins), there are 2990 other people who are not able to become doctors. Of course it's not zero sum like arena but a similar concept applies. The 10 people who are becoming doctors aren't going to be 10 random people, they would be people who invested their time studying in university.

And if the town has employment opportunities for only 2500 people, then 500 people are going to miss out. Whether you are one of the 500 unemployed (0 wins), one of the 10 doctors (12 wins) or someone in between depends on how much time you invested.

Okay, so I couldn't resist doing some math.

https://docs.google.com/spreadsheets/d/16XHsBO1UiTL5J3cYXrl9Xl_xUdNVkY9pnE-52ODdVvQ/edit?usp=sharing

This is a spreadsheet I made concerning Arena results. Data about rewards are taken from ArenaMastery.

There are two tabs in the spreadsheet.

How to read (General):

Perfectly Random Matchmaking:

 How to use "Perfectly Random Matchmaking":

 Perfectly Selective Matchmaking (Very Maths-y, can skip):

  How to use "Perfectly Selective Matchmaking":

 Results (for people too lazy to read the spreadsheet):

From the spreadsheet, we see that a person must be at least Tier 6 to break even in gold;

a person has to be somewhere around Tier 9 to go 1 Arena a day, counting 60 quest gold/ranked gold per day;

and a person has to be at least somewhere between Tier 11 and Tier 12, let's say Tier 11.5, to go infinite in Arena.

Now, Tier 6 people average less than 3 wins per Arena (in fact 2.9 wins per Arena) but they still break even. Their winrate in Arena is actually 42.8%, far below average. This shows that 3 wins per Arena is indeed enough to break even, and you do not need a 50% winrate to break even. This is because you face worse players at lower win/loss ratios in Arena, and going upwards of 3 wins in the occasional Arena yields big benefits.

Tier 9 people average 4.62 wins per run. If your average is 4.62 wins or above, you should be able to do 1 arena a day.

Tier 11.5 people, taking an average over Tier 11 and Tier 12, average around 6.5 wins per run. So, contrary to popular opinion, you don't have to get 7 wins every run to go infinite. The occasional 12 wins (around once every 12 runs) will more than compensate for a couple of subpar Arenas.

Let me know if you find any mistakes! You can check my calculations by clicking on each cell, but beware: some sums are over 4 lines and a couple hundred characters long!