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The pricing of items and a method to evaluate Hero power level

In card games like Artifact, playing a card comes with a cost: the card itself and the resource called mana. Additionally, in Artifact, there is a third type of resource: gold. Gold is different in that it allows you to acquire a special kind of card called an item, which can then be played for no additional cost. We are going to see how this resource relates to the building blocks of cards, the basic stats.

Analyzing the cost of the item cards can provide us with insight into the gold cost of each mechanic in the game. We are going to be taking a look at the cost of the most basic mechanic that an item is capable of: adding basic stats to a hero (attack, armor or health). Luckily for us, we already know plenty of items like this and they all follow a simple pricing formula.

Let A be the attack value given by any item. Then

G_{A} = 2A − 1

Where G_{A} is the gold cost of the item. Similarly, let R be the armor value. Then

G_{R} = 4R − 1

Where G_{R} is the gold cost of the item. Let H be the health value. Then

G_{H} = H − 1

Where G_{H} is the gold cost of the item.

G_{A} G_{R} and G_{H} can be easily confirmed for every item revealed with the text: "Equipped hero
has +X basic stat." With these, we should be able to gauge the gold cost of other mechanics
present in items such as Blade of the Vigil (Fig. 1) which costs 7 gold and adds 2 attack, which
G_{A} puts at 3 gold. Therefore it puts +2 cleave at 7 − 3 = 4 gold, and the same goes for Blink Dagger's effect (Fig. 1): "Move equipped hero to another lane." It is important to note that these
gold values represent the cost of adding the respective text to an existing item and not the cost of
an item with that text which is something we don't know yet.

Another possible use of this is to evaluate the power level of each item. If you're not familiar with this term power level can be understood as the relation between the actual value of a card and its cost — a card being under-costed suggests that its power level is high. Take Stonehall Cloak and Stonehall Pike (Fig. 1) for example: both items are engines which give initial stats worth 3 gold and both generate stats worth 1 gold after each combat phase. According to our calculations they should cost the same, but Stonehall Cloak costs 1 less gold, suggesting that its power level is high.

Fig 1: Sample of items in the game

We now know a little bit more about how items are priced. What does that tell us about other cards? Let us entertain the question: if heroes were items, how much would they cost? Adding items of the same type poses a problem: the sum of their costs isn't equal to the cost of a single item with the sum of their stats. This is due to the constant −1 and when evaluating heroes we would like to be able to say how much stats we need to add to make their gold values equal to each other. To do that we need instead to look at the difference D in the cost between 2 sets of items:

D = (G_{A} + G_{R} + G_{H}) − (G_{A'} + G_{R'} + G_{H'} ) = 2A + 4R + H − 2A' - 4R' - H'

The constant is thus eliminated from the equation and we are left with a number which is still meaningful: it tells us the difference in the gold cost between two heroes if these were treated like items equipped on a hero card with all its stats set to zero. Now let us define E such that

E (A, R, H) = 2A + 4R + H

Notice that D = E(A, R, H) − E(A', R', H'). From now on let us call the quantity defined by E effective stats. This number is useful in estimating the power level of heroes because each time we add to the effective stats of a hero we are looking at a number which can be used to calculate the gold cost of a new hero card with new values and not the same hero card equipped with a new item.

We may now proceed to calculate the effective stats of every hero that has been revealed.

A | R | H | E | |
---|---|---|---|---|

Axe | 7 | 2 | 11 | 33 |

Beastmaster | 5 | 0 | 12 | 22 |

Bristleback | 8 | 0 | 12 | 28 |

Centaur Warrunner | 4 | 0 | 14 | 22 |

Keefe the Bold | 6 | 1 | 11 | 27 |

Legion Commander | 6 | 1 | 8 | 24 |

Mazzie | 6 | 3 | 6 | 30 |

Pugna | 6 | 0 | 9 | 21 |

Sven | 5 | 0 | 11 | 21 |

Tidehunter | 2 | 1 | 18 | 26 |

Timbersaw | 4 | 0 | 11 | 19 |

Ursa | 7 | 0 | 10 | 24 |

A | R | H | E | |
---|---|---|---|---|

Bloodseeker | 7 | 0 | 7 | 21 |

Bounty Hunter | 7 | 0 | 7 | 21 |

Lich | 5 | 0 | 9 | 19 |

Lion | 6 | 0 | 5 | 17 |

Necrophos | 5 | 0 | 6 | 16 |

Phantom Assassin | 6 | 0 | 8 | 20 |

Debbi the Cunning | 7 | 0 | 5 | 19 |

Sniper | 5 | 0 | 6 | 16 |

Sorla Khan | 8 | 0 | 6 | 22 |

Storm Spirit | 4 | 0 | 6 | 14 |

Winter Wyvern | 6 | 0 | 6 | 18 |

A | R | H | E | |
---|---|---|---|---|

Crystal Maiden | 2 | 0 | 5 | 9 |

Earthshaker | 2 | 0 | 7 | 11 |

J'Muy the Wise | 3 | 0 | 8 | 14 |

Kanna | 2 | 0 | 12 | 16 |

Luna | 3 | 0 | 8 | 14 |

Meepo | 4 | 0 | 5 | 13 |

Outworld Devourer | 4 | 0 | 6 | 14 |

Prellex | 3 | 0 | 5 | 11 |

Skywrath Mage | 3 | 0 | 6 | 12 |

Venomancer | 2 | 0 | 6 | 10 |

Zeus | 3 | 0 | 7 | 13 |

A | R | H | E | |
---|---|---|---|---|

Abaddon | 4 | 0 | 9 | 17 |

Chen | 4 | 0 | 9 | 17 |

Drow Ranger | 4 | 0 | 7 | 15 |

Enchantress | 4 | 0 | 8 | 16 |

Fahrvhan the Dreamer | 4 | 0 | 11 | 19 |

Lycan | 4 | 0 | 10 | 18 |

Magnus | 4 | 1 | 9 | 21 |

Omniknight | 5 | 0 | 12 | 22 |

Rix | 3 | 0 | 7 | 13 |

Treant Protector | 4 | 0 | 10 | 18 |

Viper | 4 | 0 | 10 | 18 |

Axe having the highest effective stats and no ability is no coincidence. Crystal Maiden on the other hand is the hero with the lowest effective stats but her ability is very powerful, it affects every lane and allows you to play more mana than your opponent which is known to be a very powerful effect in other card games.

Notice that the different colors have a very different range of effective stats. The gap between blue and red heroes is particularly big and we can't expect a stronger hero ability to compensate for it in its entirety. In addition to this, stronger heroes are also better at generating gold. The answer to these differences in power lies elsewhere, in cards with incredibly high power level such as Aghanim's Sanctum (Fig. 2).

For the aforementioned reasons, when analyzing heroes it is important to compare them to heroes of the same color. It is also important to take their signature card into account. Legion Commander^{1} for instance has 24 effective stats and her ability's effective stats is given by 6−3 = 3
(in Barbed Mail)^{2}. This puts her at 27 effective stats which is 6 less than Axe. However, her
signature card is Duel (Fig. 2) which is 4 mana cheaper than Berserker's Call and stronger at
picking of single heroes (it doesn't have the condition of the hero being an enemy neighbor). Duel is available in the first round and scales very well into the late game, shutting down a hero while allowing for a bigger tempo swing the more mana you have to spend after you play it. While Berserker's Call can be stronger depending on the situation, its safe to say that its rarely worth the opportunity cost (its available starting on round 4), the extra 4 mana and the extra condition.

Fig 2: Aghanim's Sanctum and Sig Cards of Red Heroes

To arrive at this method we assumed that heroes are balanced in a similar way to items. It is reasonable to do this because these items exist to improve hero cards and so their effect on the game depends largely on the same things. If it turns out to not be the case then our method falls apart, but so far its applications have yielded expected results.