by Hayl_Storm (@TL_Hayl)

We've had a bit of a hiatus from these posts but here is the third installment! We're taking a step back from actual Hearthstone game play to look at expected value and some broader concepts in game theory as a whole. One note to get out of the way first, however, is that your timer can become a problem. Trying to think through all of your lines of play can take awhile but you will get faster with time; that said, once the fuse starts ticking it will be time to flex your APM.

Expected Value is the central tenet of essentially all game theory. The main idea behind the concept is to make the decision which will give the best outcome the most often: for example, you’d bet 3:1 odds on heads every time over 2:1 on tails because you’ll win more in the long run. This relates to Hearthstone in the sense that playing one spell may win 40% of games is objectively worse than another one which wins 70% in the same scenario.

One of the crucial elements of consciously trying to maximise EV is understanding how to properly weigh your risks and understanding your potential outcomes. In Hearthstone, the outcome of every play will either increase or decrease your chance to win a game: the correct decision will always be that which increases your chance to win the most.

In terms of how it functionally works when you’re in a game, there are two aspects which you need to consider: chance within a given play, and potential outcomes of a game plan.

Since Hearthstone is a card game, chance is brought to the forefront with every card you draw. Chance is very important to the calculation of EV because to find the best play you must consider not only each possible outcome, but the likeness of each. To use another simple example, if you -- when flipping a fair coin -- get $1 per tails and $0 per heads, you can expect to make $0.50 on every toss because you have a 50% chance of making $1. In Hearthstone terms, you need to be considering the +/- on your chance to win the game rather than dollars earned.

The first type of risk is one that was already mentioned: the risk involved in using answers aggressively. One of the ways this can manifest occurs from the perspective of the aggro player trying to beatdown as quickly as possible. Since Hearthstone is a game of incomplete information -- see poker vs. chess -- it is impossible to definitively calculate your lines: you will have to make assumptions. One simple example, as the beatdown player, would be as follows:

- It is turn seven;

- You have a Polymorph in hand;

- If you cast Polymorph now, you will win the game in three turns;

- Your opponent has no cards in hand and few relevant draws;

- You will lose, however, if your opponent draws xxxxxx legendary and you can’t Polymorph it;

- Your opponent has 15 cards in their deck and no card draw potential.

Your lines, therefore, are either to Polymorph aggressively or play around the legendary. If you choose to take the aggressive line, you will win in 80% -- Ragnaros must be in the top three of fifteen cards -- of games where your opponent doesn’t hit the Ragnaros. The more conservative line, won’t make you lose outright, however, but has more uncertainty. Most of the time decisions will have more variables than this, but you should approach big decisions in this manner: what do you lose to, and how much time do you have?

The easier type of variance to approach occurs in cards like Lightning Storm which have variance directly built into them. The EV calculation here is more straightforward but follows a similar path: what are the odds of X and what is the result of each outcome. Using the Lightning Storm example, we can look at what happens if it does two versus three damage to a minion. Lightning Storm has a 50% of doing two damage and 50% chance to do three. We need to, therefore, look at the outcome of two damage versus three on each minion. This is most relevant when there are one or two minions which greatly swing the result depending on how much damage is dealt. The way to calculate EV, therefore, is to figure out how good/bad the outcome of said key minion dying: if you are marginally ahead if it works but way behind if it doesn’t, see if there is another line you can take.

The dreaded Tinkmaster Overspark -- as he currently exists -- is a perfect example of evaluating potential outcomes. Targeting a generic 2/2 with Tinkmaster would be considered a bad play: 50% of the time, you will give the minion -1/-1; the other 50% of the time, the target will get +3/+3. In the long term, your EV on this play is to give the minion +1/+1 and will be a negative EV in terms of your percent chance to win the game. Tinkmastering a 3/3 also reiterates the point of playing safe from before. Targetting a 3/3 is a pure 1:1 coin flip as the odds are equivalent and the average case is +0/+0. This is still not an advisable play, however, because with in a game losing is the worst possible option. So, therefore, you should hedge on the safe side if EV calculations are close or high variance.

We've had a bit of a hiatus from these posts but here is the third installment! We're taking a step back from actual Hearthstone game play to look at expected value and some broader concepts in game theory as a whole. One note to get out of the way first, however, is that your timer can become a problem. Trying to think through all of your lines of play can take awhile but you will get faster with time; that said, once the fuse starts ticking it will be time to flex your APM.

# Defining and Calculating EV

Expected Value is the central tenet of essentially all game theory. The main idea behind the concept is to make the decision which will give the best outcome the most often: for example, you’d bet 3:1 odds on heads every time over 2:1 on tails because you’ll win more in the long run. This relates to Hearthstone in the sense that playing one spell may win 40% of games is objectively worse than another one which wins 70% in the same scenario.

One of the crucial elements of consciously trying to maximise EV is understanding how to properly weigh your risks and understanding your potential outcomes. In Hearthstone, the outcome of every play will either increase or decrease your chance to win a game: the correct decision will always be that which increases your chance to win the most.

In terms of how it functionally works when you’re in a game, there are two aspects which you need to consider: chance within a given play, and potential outcomes of a game plan.

# Play safe / Chance of Risk

Since Hearthstone is a card game, chance is brought to the forefront with every card you draw. Chance is very important to the calculation of EV because to find the best play you must consider not only each possible outcome, but the likeness of each. To use another simple example, if you -- when flipping a fair coin -- get $1 per tails and $0 per heads, you can expect to make $0.50 on every toss because you have a 50% chance of making $1. In Hearthstone terms, you need to be considering the +/- on your chance to win the game rather than dollars earned.

Aside: for the most part, losing the game is considered to be more negative than winning is positive; therefore, in a truly 50-50 scenario the general advice is to hedge towards the conservative. This line is based on the fact that humans just have a tendency to weight loses more than wins: especially when you feel like you got robbed by chance.

The first type of risk is one that was already mentioned: the risk involved in using answers aggressively. One of the ways this can manifest occurs from the perspective of the aggro player trying to beatdown as quickly as possible. Since Hearthstone is a game of incomplete information -- see poker vs. chess -- it is impossible to definitively calculate your lines: you will have to make assumptions. One simple example, as the beatdown player, would be as follows:

- It is turn seven;

- You have a Polymorph in hand;

- If you cast Polymorph now, you will win the game in three turns;

- Your opponent has no cards in hand and few relevant draws;

- You will lose, however, if your opponent draws xxxxxx legendary and you can’t Polymorph it;

- Your opponent has 15 cards in their deck and no card draw potential.

Your lines, therefore, are either to Polymorph aggressively or play around the legendary. If you choose to take the aggressive line, you will win in 80% -- Ragnaros must be in the top three of fifteen cards -- of games where your opponent doesn’t hit the Ragnaros. The more conservative line, won’t make you lose outright, however, but has more uncertainty. Most of the time decisions will have more variables than this, but you should approach big decisions in this manner: what do you lose to, and how much time do you have?

This thinking applies to the control player in reverse: what do I need to swing this game and how much time do I need to stall?

# Potential Outcomes

The easier type of variance to approach occurs in cards like Lightning Storm which have variance directly built into them. The EV calculation here is more straightforward but follows a similar path: what are the odds of X and what is the result of each outcome. Using the Lightning Storm example, we can look at what happens if it does two versus three damage to a minion. Lightning Storm has a 50% of doing two damage and 50% chance to do three. We need to, therefore, look at the outcome of two damage versus three on each minion. This is most relevant when there are one or two minions which greatly swing the result depending on how much damage is dealt. The way to calculate EV, therefore, is to figure out how good/bad the outcome of said key minion dying: if you are marginally ahead if it works but way behind if it doesn’t, see if there is another line you can take.

**Here is a case where your EV to kill the Damaged Golem 100%, the Yeti 0%, and the Harvest Golem 50%.**The dreaded Tinkmaster Overspark -- as he currently exists -- is a perfect example of evaluating potential outcomes. Targeting a generic 2/2 with Tinkmaster would be considered a bad play: 50% of the time, you will give the minion -1/-1; the other 50% of the time, the target will get +3/+3. In the long term, your EV on this play is to give the minion +1/+1 and will be a negative EV in terms of your percent chance to win the game. Tinkmastering a 3/3 also reiterates the point of playing safe from before. Targetting a 3/3 is a pure 1:1 coin flip as the odds are equivalent and the average case is +0/+0. This is still not an advisable play, however, because with in a game losing is the worst possible option. So, therefore, you should hedge on the safe side if EV calculations are close or high variance.