# You are not playing enough constructed gauntlet

Whenever I can, I am spamming the constructed gauntlet queue.

I am writing this blogpost on a Friday night; today, I have entered the queue 8 times. I have gone 5-x seven times, 4-3 one time, but that is because I am running on the high end of variance.

Even if I were to turn a few of those 5-x records into 0-3 records, I would still be way ahead.

My secret deck that is rocking the queues? Your friendly neighborhood Majramp (Titania’s Majesty, Ruby/Wild) deck.

I am almost entirely playing against other majesty decks; I may have 1 or 2 out of 6 matches be a non-majramp deck, but not many.

The tricks I have in my deck are about the same that everyone else is running at this point, I like to think that I have some things that push the margins in my favor, but not drastically more than any other deck at this point.

Essentially, I am flipping coins. Also, I am flipping coins for a profit.

I also know that you are not in the queue because I am able to write this article in between matches.

### Gauntlet is Positive Expected Value

I am sharing this with the rest of the community at some cost to myself, since this if more of you are playing gauntlet, that inevitably devalues the profits for me of playing in the constructed gauntlet queue, However, I will be sacrificing the overall value of packs (and set 3 cards) for faster gauntlet queues, which will benefit everyone (as we will play more games per hour). Naturally, if you find Titania’s Majesty to be boring or annoying, this is not for you. If, however, you do not mind doing the gauntlet grind, or if you have a killer anti-majramp deck, I am here to convince you that gauntlet is for you.

Gauntlet is high variance due to the best-of-1 match structure. That does not matter for us and, if anything, is something that helps quite a bit. Best-of-1 reduces the time it takes us to go through a gauntlet tournament and, the faster we can run gauntlet, the more packs we make per hour.

Right now, given the prize distribution of gauntlet, if you enter with the same deck that everyone else is playing, you are engaging in a positive-EV gamble. That is, going into gauntlet with a deck that merely flips coins with other decks is a favorable activity. You will make a profit from gauntlet. To show this, let us turn to a little math. (Everything is available on this Google spreadsheet, but I wanted to share the math a bit as well.)

Let’s assume a world where everyone is equally skilled and are running identical decks. Your chance of winning is 50%. Some games you lose at the coinflip, some you do not, but your chances are roughly even as everyone else. For the sake of argument, let’s also assume that you can net 170p for your packs and 1235p for your primals (both after the AH tax).

To calculate the expected value of any given run of gauntlet, we need to figure our likelihood of winning 0, 1, 2, 3, 4, and 5 games. Calculating zero games is the easiest as the only way to win zero games is to lose three times in a row. We do this .5*.5*.5 (.5^{3} = .125) or 12.5% of the time. So, in 87.5% of cases, we are walking away with one pack, not quite making up for our 500p investment, but recouping something at the very least. Next, we need to figure out one win. To get one win, we need three losses and one win, which is effectively .5^{4}. Now, the trick here is that the order does not matter for our outcome, but it does matter in calculating probability as there are multiple ways to get three losses and one win. So, .5^{4} is simply 6.25%, but there are three ways in which we can achieve this. We can go WLLL, LWLL, and LLWL. Of note, we cannot go LLLW because our third loss would knock us out of gauntlet. consequently, we multiply our result by 3 and get 18.75%. For two wins, we have our probabilty to the fifth power, but we have six ways to achieve two wins and three losses, so we also have a 18.75% of achieving that. For three wins and three losses, we have .5^{6} with 10 different combinations of eligible wins and losses. For four wins, we have 15 combinations. There is one way we can achieve 5 wins, WWWWW, and there are five ways we can achieve five wins and a loss, and 15 ways in which we can achieve five wins and two losses. You can find the breakdown of all possible ways to finish gauntlet here. Our probability breakdown is:

Games Won | 0 | 1 | 2 | 3 | 4 | 5 |

Probability of Outcome | 0.125 | 0.1875 | 0.1875 | 0.15625 | 0.1171875 | 0.226563 |

So, a couple of important notes. First, the probabilities all add up to 1 (or 100%). This is important; if it were any number other than one, we would know for sure we made a mistake. Second, the plurality of finishes are in the five-win category. Now, the majority are less than five wins, but the plurality of cases, just over one in five finishes, will be five wins even with a 50% record.

This gets us closer to our expected utility, but we know need to figure out our prizes. Generally speaking, you get one pack per prize unless you hit five wins then you get six packs. To calculate our expected value for each outcome, we take the probability of an outcome and multiply it by the value of the outcome. I have substituted market prices for packs and included the possibility of your packs turning into primals assuming a 2% probability. Our expected value for each outcome is:

Games Won | 0 | 1 | 2 | 3 | 4 | 5 |

Expected Value | 0 | 36.50625 | 72.919875 | 91.03561 | 90.92289 | 263.0356 |

To calculate the final expected utility, then, we sum all possible outcome states and minus our costs, so summing the above table and subtracting the costs, we get an outcome of 54.42p. Thus, even if you go in there with no edge whatsoever, this is a positive endeavor.

*What if I have an edge?*

Naturally, we may go in with better decks, teched-out majesty decks, decks that outright beat 90% of the field, or we may think we are better skilled than other players. Likewise, we may be worse, so it is useful to us to find out how our payout changes if our headsup matchup rate is different than even.

As you can see, you can go into gauntlet as an underdog and still do better than even if you are trending above 46.907%. If you find a strong anti-meta deck and trend at 60%, you are making over 240p a gauntlet run and gaining back over 48% as a return on investment.

*Great, Zubrin, now everyone is going to do this and the price of packs are going to fall, surely this is not profitable anymore.*

So, we can calculate that as well. For simplicity, we will keep the winrate at 50% and vary the price of packs. Also, we will assume that the primal value is roughly 7.65 that of a normal booster to match the values we had before. Then, we can just vary the value of the boosters to see when we stop making money and when we are okay.

For a 50% player, our breakpoint is 154p (post-tax) return on boosters (160p pre-tax). If our winrate is higher or lower, our breakpoint changes as to when gauntlet is or is not a positive investment.

Why aren’t you queuing for gauntlet? It is feeling lonely in this queue right now with the other players, such as JadiimJedi, Fridged, Shinshire, Janome, BossHoss, and a few other known, competitive players pushing that EV despite claims that gauntlet is too random.

I get paid a lot more at work and constructed gauntlet isn’t much more enjoyable.

I am finishing my master thesis today and after tomorrow you can expect to frequently see me and my anti Titania deck in the gauntlet again 😉

The EV for comp. constructed is better – Gauntlet is instant gratification. Also you need to play hundreds of gauntlets to have the above calculation to matter and chances are you have a disadvantage for the first many runs untill you learn the format.

Unfortunately, Gauntlet has taken away the playerbase needed for constructed which I am kind of sad about.

The EV certainly can be much better elsewhere, but it is really only an opportunity cost if the the other options are real options. VIP is certainly better.

yeah, but the constructed community is rather small and doesn’t look like it can support competing formats – so everybody is playing the most accesible with the lowest EV – even though it would be better for most to play in regular tournaments.

But the article was interesting from a value-farming perspective. I still think it is too risky for a 50% player, though.

I have ground up ~5000 plat in two days time.

Compare it to arena ev and it’s actually very close. I did the math with a fast deck can clear in about 30 mins get 6600 gold plus random loot at least wort 4 k usually. So in and hour make on the low side 20k in an hour with the insane exchange rates currently going on this is equivalent ev with far less variance.

I think a LOT of people cant get anywhere near 30 minute clear times, even with the best decks. Probably due to computer specs.

Apologies for the poor graph viewability on mobile—an artifact of them being generated on the fly and our mobile theme not being able to force them to a maximum width. For now, please view the article in landscape to see the entire graphs. Will look into a better solution for future mathematical articles.

Hey Zubrin, I think in order for you to sell this experience, it needs to be benchmarked against it’s competitor – Sealed Gauntlet. The fact remains that – a) people don’t feel confident they have the cards, b) people don’t feel confident they can beat other skilled players, c) there is no guarantee the plat put in will generate any packs (mainly why these players will try sealed instead). I was planning to do a write-up on why I thought Sealed Gauntlet was bad for the game using some calculations based on time-commitment and output into the economy… Maybe another day.