Originally published by Sharon Choong Kam Chong on Dec. 25th, 2015.
It’s already time for winter break! I wish you all happy holidays! Here is my last blog for the semester: part three of the Gaming Stats blog. Hearthstone is a card strategy game which is, as Blizzard says, “deceptively simple”.
Part III:
Hearthstone is a card game released in 2014 where players collect monster and spell cards to be played in several game modes. It is free-to-play but players can also choose to pay to obtain card packs or to access content, instead of using the in-game gold.
New players begin with basic cards, and as they win Ranked/Casual games and complete quests, they receive in-game gold to purchase card packs or to pay for access to Arena and Solo Adventures (which have unique card rewards). Tavern Brawls also offer a way to clear quests or get card packs. This blog will talk about the Ranked/Casual Play and Arena modes.
The goal of each game is to defeat the opponent by reducing their health to zero. Each player initially has 30 health points. Before the game starts, players choose a hero to impersonate and each hero has its own “hero power”.
It is a turn-based game: within each turn, one player draws a card from their deck, plays cards from their hand and/or uses their hero power. Each card costs mana to play. Using one’s hero power burns 2 mana. Mana replenishes at the start of every turn. Games always start with 1 mana for each player, which increases by one at the start of the player’s next turn until players each have 10 mana to use. This means that the game strategy is dynamic with the number of turns, as low mana limits gameplay in the early game. Another dimension to keep in mind is that in the event that players run out of cards in their deck, they lose health every turn that their deck is empty; meaning that drawing too many cards or extending the game too long is dangerous.
RNG:
Also known as Random Number Generator, it is not surprising for RNG to be used in a card game model — but it is the main source of frustration of many Hearthstone players. Here are some interesting questions to examine:
In the game, there is RNG in every draw of a card. How do I optimize my deck’s mana curve to always have something to play every turn?
It is difficult to find out the best distribution of mana costs for all situations; after all, a perfect mana curve may not be necessary because sometimes using hero power (for 2 mana) derives more value than playing cards, some minions can change the mana cost of cards, and drawing/discovering extra cards can increase the chance of having something to play every turn. Yet, there have been attempts to estimate the optimal mana curve.
Check out this Mana Curve Analysis Tool in its beta version from Reddit’s chippydip, which is a simple model of a player’s card advantage throughout a game given their deck’s mana curve and a scale to weigh in the importance of hero power in the deck’s strategy. I say simple model, because it simply assumes that players will always favor playing their most mana-costly cards before less costly ones, and always favor using hero power when mana is left after playing cards, which often is not the case in real games.
This Excel sheet from HearthMath shows the probability of at least one copy of a card with x copies in a deck (x-axis) being drawn by turn y (y-axis). For instance, the probability of drawing your only legendary in an Arena deck by turn 7, when at an earlier turn you drew 2 cards in addition to the usual start-of-turn draw, is 30% according to the table. The model is based on the hypergeometric distribution which simulates card draws without replacement. Of course, this disregards probabilities of drawing the card from mulligan, which would complicate the statistics. An empirical optimization model with an actual, large enough sample of games would be even better as it might implicitly take into account all game conditions.
How many card packs do I need to open to be sure to get at least one legendary card?
As of The Grand Tournament expansion, there are 500 craftable cards obtainable in packs, of which 73 are legendary. According to studies, the chance of a legendary card is about 1%. Hence, the probability of getting at least 1 legendary in a card pack is 1-(1-0.01)^5 = 5%. For a probability of 100% (more precisely 99.96%) of getting at least 1 legendary, we would need about 800 cards. In other words, 160 packs need to be opened to be certain to get at least one legendary.
Are decks better off without RNG cards?
RNG cards are those that have additional randomized effects, such as Knife Juggler, Mad Bomber or Ragnaros.
