The January 17 update on *MTG* *Arena* introduced the solution to the “5th card problem.” Now, if you open a 5th rare or 5th mythic in an 8-card pack, it will be seamlessly replaced with a different card of the same rarity from that set. If you already have a full collection, you get 20-40 gems instead. For Limited, if you pick a 5th copy of a rare or mythic, it will also turn into 20-40 gems after the event. Commons and uncommons still go towards the vault, for now.

If you hadn’t seen the announcement video yet, check it out.

Duplicate protection is an excellent change, but at the same time the update nerfs the individual card rewards (ICRs) for Constructed events. While you will still receive three ICRs at any Standard event as before, the upgrade rates were slaughtered. You’ll get fewer rares and fewer mythics, and instead you’ll get more uncommons. In this article, I run the numbers to analyze these changes.

Given the amount of rares you need for competitive mana bases, I will value rares and mythics equally, in accordance with my previous Arena EV article. Although it won’t affect my analysis, the rate at which rare ICRs upgrade to mythics is also reducing considerably across the board, from 1:3 to 1:8.

# Best-of-One Standard Event

For Bo1 Standard events, the following table gives the expected total number of rares or mythics in both the old and new system, based on the old and new upgrade rates. The number of non-upgraded uncommons is equal to three minus the number of rares/mythics.

Game Wins |
E[rares/mythics] OLD |
E[rares/mythics] NEW |

0 |
0.45 | 0.07 |

1 |
0.45 | 0.07 |

2 |
0.45 | 0.07 |

3 |
0.45 | 0.07 |

4 |
1.30 | 0.07 |

5 |
1.30 | 1.06 |

6 |
2.15 | 2.05 |

7 |
2.15 | 2.05 |

As you see, there is now a huge difference between 4 and 5 wins. Given the probabilities of reaching certain records, a player with a 50% win rate (i.e., a 50% probability to win an arbitrary game) has only a 22.7% chance to reach the minimum of 5 wins required to guarantee a rare. In the 11.7% of events where such a player would go 4-3, they now won’t be guaranteed a rare anymore. In terms of expected value, this 50% win rate player will win 0.44 rares/mythics per BO1 Constructed event in the new system, whereas this used to be 0.87 (including upgrades).

# Traditional Standard Event

Match Wins |
E[rares/mythics] OLD |
E[rares/mythics] NEW |

0 |
0.45 | 0.15 |

1 |
1.30 | 0.25 |

2 |
1.30 | 0.25 |

3 |
2.15 | 1.20 |

4 |
2.15 | 1.20 |

5 |
3.00 | 2.15 |

For best-of-3 Standard events, the number of rare ICRs is slashed throughout. In terms of expected value, a 50% win rate player will win 0.63 rares/mythics per Traditional Constructed event in the new system, whereas this used to be 1.45.

# Specialty Constructed Event (such as Momir, Pauper, or Singleton)

Game Wins |
E[rares/mythics] OLD |
E[rares/mythics] NEW |

0 |
1.15 | 0.16 |

1 |
1.15 | 0.25 |

2 |
1.15 | 0.35 |

3 |
1.15 | 0.50 |

4 |
1.15 | 1.15 |

5 |
1.15 | 1.15 |

Whereas you were guaranteed a rare just by entering a Specialty Constructed event, now you will have to win at least 4 games. A player with a 50% win rate has only an 18.8% chance to reach 4 or 5 wins in this event. In terms of expected value, this 50% win rate player will win 0.45 rares/mythics per Specialty Constructed event in the new system, whereas this used to be 1.15.

At the same time, whereas entry costs and gold rewards were unchanged for the Standard events, both were reduced by half for Specialty events.

# How Much Are ICRs Worth?

The value of ICRs will differ from player to player, but under the assumptions from my previous Arena EV article where a rare/mythic wild card is valued at 6 times the value of a random rare/mythic, and random commons and uncommons are valued according to the progress points they give for the vault) a rare/mythic ICR is worth 412 gold and an uncommon ICR is worth 25 gold.

I believe that the assumptions underlying this valuation remain reasonable under the new system for players who are building up their collections. Once your collection grows large, Constructed events become less appealing since you may receive 5th copies of rares/mythics as ICRs. These would convert to gems: 20 gems for rares and 40 gems for mythics. This occurs regardless of whether you have collected all cards of that rarity from that set or not. At least it’s better than the measly vault progress points they used to give, which I used to value at 8 gems for rares and 16 gems for mythics. In any case, the majority of players for whom my analysis is interesting won’t run into duplicate ICRs all that often.

# What is the New Net Expected Profit of Playing These Events?

I define the expected net profit of an event as the expected gold rewards plus the gold value of the ICRs (where any rare/mythic is converted into 412 gold and any uncommon is converted into 25 gold) minus the event’s entry fee in gold. For the three types of Constructed events, the following graphs depict the expected net profit under both the old and new system for varying win rates.

Under the old system, a BO1 Standard event had a positive net profit for players whose game win rate was as low as 30%. Under the new system, you need a game win rate of 43% or higher for the event to be profitable. If your game win rate is lower than 43% and you only care about building up your collection, then you would now be better off buying packs than playing a BO1 Standard event. With a 50% win rate, your expected net profit is now 154 gold.

By the way, since gold rewards and the entry fee remain unchanged, you still need a 57% game win rate to go infinite per my previous analysis. That is, if your game win rate is 57% or higher, then your expected gold prizes (excluding the value of ICRs) cover the event’s entry fee of 500 gold.

To enable easy comparison with BO1 Standard, I kept the same scale and presented the net profit as a function of the game win rate. Note that a 60% game win rate corresponds to a 64.8% match win rate.

Under the old system, a Traditional Standard event had a positive net profit for players whose game win rate was as low as 40%. Under the new system, you need a game win rate of at least 47% for the event to be profitable. With a 50% win rate, your expected net profit is 179 gold.

To go infinite, you still need a 55% game win rate per my previous analysis.

Although expected payouts don’t depend heavily on your win rate, these Specialty Constructed events bring a positive net profit for practically everyone, even in the new system—you need a game win rate of only 32%.

Since gold payouts aren’t top-heavy, going infinite is practically impossible in these events. It requires an absurd game win rate of at least 79%.

# What’s the Expected Number of Games per Event?

Best-of-1 events take less time to play than best-of-3 events. Since time is valuable, let’s take this into account.

The expected number of games played in a best-of-1 event is easy to determine for any win rate. Just multiply the probability of each win-loss record by the corresponding number of games played, and then sum up all of these values. This calculation reveals that a player with a 40% game win rate will spend 4.91 games per Bo1 Standard event. It’s 5.73 games for a player with a 50% game win rate and 6.58 games for a player with a 60% game win rate.

For best-of-3 events, the bookkeeping is more involved because the number of games per match depends on your win rate, and it’s also different for a loss or a win. For a player with game win rate W, a match they won consists of

games on average, and a match they lost consists of

games on average. By combining these numbers with the probability of each win-loss match record, we can then determine the expected number of games per event. This doesn’t consider the time spent sideboarding, but that usually doesn’t take long. It also doesn’t take into account the likelihood of seeing longer post-board games, but that’s difficult to account for.

According to this calculation, a player with a 40% game win rate will spend 7.55 games per Traditional Standard event. It’s 9.30 games for a player with a 50% game win rate (this was inaccurate in my previous article) and 11.06 games for a player with a 60% game win rate.

# What Event Type Now Gives the Best Net Expected Profit per Game Played?

Combining the expected number of games per event with the expected net profit, we get the following comparison for the new system.

So, if you are interested in building up your collection in the most time-efficient way possible and can get behind my conversion of ICRs into gold, then you should choose as follows:

- If Specialty Constructed events are available, play Specialty Constructed events if your game win rate is 52% or lower and play BO1 Standard events if your game win rate is between 52% and 56%.
- If Specialty Constructed events are unavailable, play Bo1 Standard events if your game win rate is between 43% and 56%. If your game win rate is smaller than 43%, then just open packs.
- If your game win rate is 56% or higher, then play Traditional Standard events.

# Conclusion

For new *Arena* players who were trying to build up their collection by grinding Constructed events, which includes me, the January 17th update is unfavorable. Players with low win rates will especially have a harder time farming rare/mythic ICRs. Yet even though the value of these events is not as good as before, they are still decent after the nerf. Under my valuation, at win rates as low as 43%, BO1 Standard events remain profitable in terms of the total expected value of both gold prizes and card rewards.

In exchange, we get a solid solution to the 5th card problem, which created massive feel-bad moments before the update, especially for players with large collections. If a reduction of rare/mythic ICRs is the cost we have to pay for that, then I can live with that. My biggest gripe is that I would have liked to have a higher expected number of rares/mythic ICRs for going 4-3 in the Bo1 Standard event, if only to smooth out the rewards a little bit. But I can understand the general concept of slaughtering ICR upgrade rates.

The developers had previously stated that players using the Constructed events as an easy way to build their collection “was never our intent.” From their perspective, the changes are a good way to enact the duplicate protection while meeting their goals for the game’s economy. In my evaluation, I believe that the changes represent an improvement to the game as a whole.

## Discussion