# Mathematical calculation of current hand, minions, spells card worth

In chess or other similar strategy games you know the value of each piece and can easily calculate the current situation or what will be the best move according to those values.

Is there such a system for heartstone?

Example of what I am asking:
Yeti: 4/5 = 9 total value
Giant: 8/8 = 16 total value
Argent squire: 1/1 + special = 2 + divine shield value

Is something like this developed by players or blizzard?

This question is asked to find out if there is a way to maximize the efficiency of our moves on a mathematical basis.

• I'd dispute the 'easily' part. Strategy for winning the game >> point value of the next move. The system in chess exists largely for scoring draws, it's not usually considered when playing as a decision-making aid.
– TZHX
Commented Feb 4, 2014 at 9:34
• It is considered as a decision making aid in AI algorithms. Commented Feb 4, 2014 at 9:48
• but it's not the end-all for strategy, high-end chess AIs also look at positioning and how many squares you control to decide how good a move is Commented Feb 4, 2014 at 11:42
• There probably is some sort of formula along these lines in use to guide the designers on card abilities vs mana cost... but it would be far from foolproof, as evidenced by the fact that some cards are considered better than others. Commented Feb 4, 2014 at 13:24
• If this is a question about why the AI plays the way it does, that's developer intent and off-topic for this site. If this is a question about what players are doing in the meta, that's fine. The question could be improved by clarifying this distinction. I will attempt to answer according to the latter. Commented Feb 4, 2014 at 15:34

# TL;DR: see summary at bottom

Vanilla Test
I have seen some of the "Pro's" on their streams refer to a minions value by asking if it passes the "Vanilla Test". That is, you can get a textless common card, such as a Bloodfen Raptor that is a 3/2 minion for 2 mana. So when you are looking at other 2 cost cards, you compare it to the other available 2 cost cards.

Mana Cost
That said, your example of value on the Yeti does not take into account the mana cost. If you were going to develop some system like this for yourself, you would also NEED to take into account the mana cost of the cards, as a bare minimum.

For example, Yeti is a 4/5 for 4, maybe a formula like this (4+5) / 4 = 2.25 value.

However, cheaper cards may easily break this system, such as a worgen infiltrator, 2/1 for 1 which would look like (1+2) / 1 = 3.00 value.

Now, having 10 cards that are 2/1 for 1 drops at 3 value in your deck is not necessarily better than 10 cards that are 4/5 for 4 drops at 2.25 value because the amount of cards available to you is limited.

Mana Curve
This puts a much higher importance on "Curving Out". By that I mean, having a good mana curve in deck such that there is a high probability of being able to spend 1 mana on turn 1, 2 on turn 2, 3 on turn 3, 4 on turn 4, 5 on turn 5, etc.

So into the above formula for minion evaluation of (Health + Life) / Mana Cost, you probably also need to take into account how the card fits into your mana curve. The other random variables you would need to account for are Card Text (like abilities) and tempo.

Text
While a Venture Co. Mercenary at 7/6 for 5 mana is very good (7+6)/5= 2.6 value... It has an opportunity cost because while it lives your minions will all cost +3 more. It essentially is adding 3 to the denominator of all your other card values while it is in play.

There are too many different text variables to assign a point range for each, but generally I would say -3 to +3 based on the strength of the text. For example: Ysera has a very powerful ability and would probably get a +2 or +3 based on it, whereas the Venture Co. Mercenary would be in the -2 to -3 range.

Tempo
The other thing to take into account is tempo. Having 30 cards with super high value, even taking into account their text and mana curve, if they all cost 4-10, you will be so far behind on turn 4 when you start playing that many fast decks may have such a lead it is impossible to come back. So a 10 cost card with a super high value, while strong, has a huge cost in not being able to be played early.

An example... Mind Control is often considered to be a super powerful card. If your opponent plays a 12/12 deathwing, wipes the board and discards their hand, and you follow that up with Mind Control, you can often hear the sound of your opponent's soul dying(dramatic exaggeration). However, having 10 Mind Controls in your deck would fortell near certain doom for you.

So the way I value tempo is, will this card get in play and be valuable in time to make a difference? 1-3 cost drops get a +1-2 value. 4-5 cost drops get 0-1. 6+ get -2 to 0 based on my analysis of the cards affect on tempo.

Efficiency
The last and most important part is that you have to meassure card efficiency. A Leper Gnome for 1 mana is a 2/1 and deals an additional 2 damage on dying in most cases. However, while it is great for early tempo, drawing one of these bad boys after turn 6 makes me cry because it has a very small value and takes up an entire card slot in hand and an enitre draw, but most likely it will not even trade 1 for 1 on the board.

So the way I value efficiency is, how much effect will the card have when on the board? Following is a very general guide for how much value a card adds to hand based on cost in my estimation: Cards that cost 1 mana automatically get a +3 for efficiency. Cards that cost 2-3 get a +2 or +1. Cards that cost 4-5 get +1 or +0. Cards that cost 6+ get a -1 to -3.

EDIT:
The above guide should not be true for every case however. Cards like Harvest Golem, Cairn the Bloodhoof, and Sylvanas Windrunner are almost like 2 cards in 1 and in my estimation, that should be represented in the denominator. I know that those are all Deathrattle abilities that should be represented in the "Text" section, but in certain cases the ability is so powerful that I think it needs to be part of the denominator.

New & Ammended examples below are bolded. I tried to add a few examples with efficiency values resulted in non-4 denominators.

# SUMMARY

This is not a perfect or comprehensive system and would need more work, but if I were to put together a value formula for minions, it would be something like this:

(D+H+T+M+P) / (C+E)
D = Damage
H = Health
T = Text modifier
M = Mana curve modifier
P = Tempo modifier
C = Cost
E = Efficiency

Examples:
Wisp = (1+1+0+0+1)/(0+3) = 3/3 or 1.00
Leper Gnome - (2+1+1+1+2)/(1+3) = 8/4 or 2.00
Argent Squire - (1+1+2+1+1)/(1+3) = 7/4 or 1.75
Bloodfen Raptor - (3+2+0+2+2)/(2+2) = 9/4 or 2.25
Knife Juggler - (3+2+2+2+2)/(2+2) = 11/4 or 2.75
Questing Adventurer - (2+2+1+1+1)/(3+1) = 7/4 or 1.75
Dalaran Mage - (1+4+1+0+1)/(3+1) = 7/4 or 1.75
Harvest Golem - (2+3+2+2+1)/(3+0) = 10/3 or 3.33
Chillwind Yeti - (4+5+0+1+1)/4 = 11/4 or 2.75
Ysera - (4+12+3-2+0)/(9-4) = 17/5 or 3.6
Deathwing - (12+12+2+-2+2)/(10-3) = 26/7 or 3.71

• Your method of calculating efficiency seems to end up turning the denominator into 4 or 5 for all cards. That means it doesn't really have any effect and you're ending up just adding damage+health and a few modifiers. Commented Feb 5, 2014 at 17:45
• M = Mana curve modifier, what is this? Commented Feb 5, 2014 at 18:01
• @bwarner The efficiency variable is subjective to interpretation, but I thought it was the best way to evaluate value because there is a trade-off in value to cost. If you were to remove it, the rest of the formula would not work out well. That was the best way I could come up with to create a comprehensive theoretical numerical value, if you have a suggestion for how to improve it I will gladly try to take that into account. Commented Feb 5, 2014 at 20:07
• @Esq Mana Curve Modifier was a value I assigned to a card based on how well it helps you fill out a strong mana curve (typically centered around 2-4). This value is not static based on the mana cost however, but is rather my analysis of how much impact a card has on the board if played on the turn it first becomes available. As an example, Harvest Golem is a super strong turn 3 drop and fills out the mana curve very well, whereas a Dalaran Mage is a very weak drop in most cases on turn 3. I would give them a +3 and +0 respectively to represent that value. Commented Feb 5, 2014 at 20:12
• @DiabloMonkey: I think that bwarner's point was that if you took C+E out of your equation entirely then it would make little difference. You say the formula wouldn't work out well but it would be exactly the same in your examples because C+E is always 4. It may just be your examples and if so you might want to include some cards whose C+E is at the higher or lower end of the scale to demonstrate how your scoring works for these cards. Commented Feb 5, 2014 at 23:17

I haven't seen any numerical systems yet, and I don't think they would be very useful. The value of a minion depends on much more than the sum of the attack and health points.

There are rankings by several well-known players for cards in the Arena. But this is for deciding on which cards to draft, and is not really useful to calculate a specific situation in the game.

Minions with an identical sum of attack and health can be valued quite differently. A low health compared to the attack (and cost) is often considered a severe disadvantage. A Core Hound with 9/5 for 7 mana is problematic because it can be taken out by much cheaper minions.

Other considerations are certain health thresholds, e.g. minions with more than 4 health will survive a Flamestrike.

• And some card values differ on the current situation - take e.g. Arcane explosion against 1 minion versus 5 minions. Similar situation with Darkscale Healer. Commented Feb 4, 2014 at 16:47

It sounds like you are trying to evaluate game positions, rather than the comparative value of cards during deck-building. Assigning a point value to each card doesn't really work for this though. A better simple evaluation is to say "If nothing changed from here on, which player would be able to force a win, and how long would it take them?" This is often referred to as your "clock". It doesn't take into account what you have in your hand, but it can help you evaluate the effect of playing a certain card. For example, playing this card would take me from 4 turns to win down to 2 turns to win. Note that sometimes neither player can force a win from the current position, which suggests that it is fairly even, though there can still be differences in terms of available resources and such.

Players use the concepts of mana curve and card ranking in deck construction (not during play).

Due to the limited number of plays on a turn, it's fairly easy for a human to work out the branches of each decision. A scoring system for during-play decisions would not be helpful and therefore does not exist.

I've attempted to do this on my own. Some problems occur if you compare 2 cards like Chillwind Yeti and Oasis Snapjaw. The majority of Arena experts will agree that Yeti is better, despite each card totaling to 9 for the same cost of 4 and having no text. (Snapjaw is a Beast, but that is mostly irrelevant unless you're a Hunter).

This could mean a few different things. a) More power is favorable than more toughness at 4 mana (4 power Yeti versus 2 on Snapjaw). b) Closer proximity of power/toughness (balance) is more favorable than a wider spread of power/toughness (4/5 is better than 2/7) at 4 mana. c) Oasis Snapjaw is underrated, and equally as good as Chillwind Yeti (the experts are wrong).

I originally had a suspicion that c) was a real possibility. Snapjaw can potentially trade with more smaller creatures than Yeti can. From personal experience however, Yeti has been a much better play on turn 4 than Snapjaw, for some combination of reasons a) and b) depending on the circusmtances. I can reduce their life total faster than Snapjaw would, or I could defeat a 3 or 4 toughness minion in a spot where Snapjaw could not. So c) is highly unlikely given functionality and robustness of the Yeti to impact the game.

This means that there must be some way to account for power relative to cost, toughness relative to cost, and ratio of power/toughness. We need to quantify the reasons why Yeti is better than Snapjaw. It should either get bonus points for having increased power, and/or bonus points for having a closer ratio of power to toughness (4/5) than Snapjaw (2/7).

As a thought experiment, how many points of toughness would we have to add to consider Snapjaw better than Yeti with an eye test? 1, 2, 3, 4 or more? I'd probably take a 2/12 over a 4/5 if both cost 4. This would mean that Snapjaw's power/toughness ratio would become worse, yet the card would become stronger! So, power/toughness ratio would need to be kept in check by actual toughness.

At this point I am at a loss to create a formula that expresses these opinions. Perhaps someone else can use these ideas to help mathematically quantify the differences. Instead of saying each point of power is worth the same, we could say that the 4th point of power is worth some amount more than the 3rd point of power on Yeti. We could also say that power/toughness balance has a minor yet still significant impact on card value, but should be less determinant than actual points of power and toughness.

Once these attributes are accounted for, we could start to compare cards at different mana costs and add that to formulaic expression of card value.

Bonus answer: If you REALLY want to compare cards and know their real values, you can track win percentages based on cards being drawn and played. This is results based and would require a large sample, but would ultimately give you quantifiable material. It'd be nice to have a formula without needing to use brute force.

Take a look at this links. They links are talking about the old Yu-Gi-Oh game, but the theory is pretty much the same and can be easily adapted to be used with HearthStone.