# The math behind Hearthstone’s new Heroic Brawl: Is it worth your gold?

Last Updated January 23rd, 2022

Blizzard has recently announced the new Heroic Brawl mode, an expensive mode with a 1000 gold entry fee that offers the promise of massive prizes if you can build up a good win-streak. But that’s a gigantic, capitalized, IF. The prize chart for the mode was recently revealed, and to the dismay of many players it turns out you can lose about as big as you can win.

So what are your chances to actually come home in the money? Is it worth playing Heroic Brawl, or are you better off saving your gold for Arena runs and expansion packs? I did some research and enlisted the help of Robert Bauer, a local mathematician, teacher, and tutor, to find out.

## To The Winner Go The Spoils

Let’s start by examining the rewards for Heroic Brawl. First, let’s assume that the entry fee gives you one run attempt. If you win 0 times in your run, you get one lonely pack, a 100 gold value, losing 900 gold in the process. If you win 12 times, you get 50 packs, around 1100 dust, around 1100 gold, and 3 golden legendary cards. That’s approximately a 12,600 gold value, a MASSIVE win compared to the 900 gold loss at 0 wins. If you were to stop your examination there, Heroic Brawl would seem like an obvious value.

Except, there will be far more people who have 0 wins than 12 wins. To accurately figure out what your chance of winning is, you have to simulate the game itself. Let’s assume that Heroic Brawl is set up the same way Arena is, in that players play other players with similar records. Everyone starts at 0-0. Everyone plays everyone else, and by definition half of the pack wins and half loses, leaving half of players at 1-0 and the other half at 0-1. Then the pack plays each other again and you get a quarter at 2-0, a half at 1-1, and a quarter at 0-2. The process repeats itself over and over again, further thinning the pack until we have at least one player in every possible distribution of wins and losses.

Luckily, Gamepedia already did all the work figuring that out for us.

That’s a lot of numbers, but now we know how likely it is for any given player to hit any combination of wins and losses. If we then figure out how many wins it takes to profit, we know how likely any given player is to end a run “in the money.”

## Doing The Math

Let’s look at the prize chart again. We need to be looking for a prize that is roughly equivalent in value to the entry fee of 1,000 gold. According to the chart, that happens at about five wins.  That will get you six packs (a 600 gold value), 220 gold, and 220 dust (approximately a 200 gold value). All of this is approximately a 1,000 gold value.

Looking at the charts above, we can see that only 22.66% of people playing Heroic Brawl will break even, and only 14.45% will profit! The vast majority of Heroic Brawl players are just going to lose gold by investing in it. The odds, as it were, are stacked against you.

Let’s compare this to Arena, which has a similar payout structure. In Arena, 3 wins will get you a card pack (a 100 gold value), 25-35 gold, and a random reward of 20-25 Gold, 20-25 Dust, a common, a rare, or even a card pack. On average, this will be at least a 150 gold value, the entrance fee for arena.  If you look at the charts above, 50% of people will break even when they play arena. Half the players “win” and half “lose.” It’s a fair game. The only thing keeping you from profit is your own skill.

So, hypothetically, the top half of Arena players will always win, while the bottom half loses. Meanwhile, only the top quarter of Heroic Brawl players will win while everyone else loses. Arena is looking like a much better value.

But what about that jackpot at 12 wins? Isn’t it possible that the reward is so high it’s worth it to play just for a chance at hitting it? Well, you are right. There is a possibility that high enough rewards at higher payouts might make the brawl mathematically worthwhile, even if most lose. These higher rewards would, effectively, have to make up for all the gold you would lose attempting to get there.

Our friendly neighborhood mathematician helped us find the answer to this question by figuring out the expected value of any given run.

The expected value of any random variable is the average of all possible values that variable can have. This is useful because it allows you to figure out, approximately, what value you are getting from any random action.

Let’s assume you roll a six-sided die over and over again. Statistically you are likely to see each of its faces equally in your distribution. So average up the faces ((1+2+3+4+5+6)/6) and you’ll find that you get 3.5, the expected value of the die. If, instead of a six, the die had a face reading 100, the expected value of the die would go up dramatically, in this case to 19.5. This is roughly synonymous to a Heroic Brawl prize distribution heavily weighted toward the top payouts.

So to find the expected value of a run, we take the entire distribution, every single possible outcome, appearing as often as it is expected to appear, and average it together just like we did with the die.

The results, unfortunately, support our theory that Heroic Brawl simply isn’t worth it. A Heroic Brawl run has an expected value of 711 gold. That means, given enough runs, an average player will expect to lose 289 gold for each attempted Heroic Brawl run. No matter how many times you play, you are going to eventually end up losing money.

Arena, however, is much more generous. Averaging together all payouts gives us an expected value of 222 for an expected profit of +72 gold each Arena run. The numbers don’t lie. Playing Arena is a good value and playing Heroic Brawl isn’t. The odds are just stacked against you.

Why is this? Let’s talk economics.

## The Price of Doing Business

Hearthstone isn’t just a game, it’s a business. Blizzard makes money when people purchase packs, Arena runs, or Heroic Brawl runs with cold hard cash. But people don’t have to purchase cards with cash. They can purchase them with in-game gold. It’s in Blizzard’s best interest to get players to spend their cash instead of gold. However, it’s also in Blizzard’s best interest to, simply put, not be a dick to its fanbase and squeeze them for money.

We are on the verge of a new expansion and players have begun saving gold. I myself have already saved up 5,000 gold, enough to buy fifty packs of whatever new expansion comes out. Blizzard, on the other hand, charges \$40 dollars for a 50 pack pre-order, and they would really like me to buy those packs with money, not gold.

Enter Heroic Brawl, a new and fun way to play Hearthstone with high-risk and high-reward. You only profit in Heroic Brawl if you can routinely end up in the top 14% of players. The top 14% of players already have decent collections.

These are the players that you watch on stream blow tens of thousands of gold and cash, opening up literally thousands of packs all at once. Blizzard was not making money on these players anyway, because they already have the gold to buy what they need from the new expansion. It’s THESE players that stand to profit from Heroic Brawl. They get huge rewards of cards, gold, and dust, which they don’t need.

The players who need the rewards from Heroic Brawl are intermediate-to-new players who are still building their collections. They are the only players that stand to really benefit from the massive rewards the mode promises. They also just so happen to be the people least likely to see a profit when making a Heroic Brawl run.

Remember, SOMEBODY has to lose to allow Heroic Brawl to function. We can’t all be winners. So all of those losers either A) give Blizzard 10 bucks or B) remove their hard earned 1,000 gold from the in-game economy. Removing gold from the economy makes it more likely that players will spend real-world money on cards they need out of necessity. Either way, Blizzard profits.