Thursday, April 14, 2016

A Hearthstone Brain-Bender (and why I enjoy the puzzle that is Aggro Shaman)

One of the strengths of Hearthstone and other top card games is the depth of strategy one can find in different major archtypes of constructed decks.  Although I personally enjoy making a combo deck work or using control decks, there's also beauty in some of the aggressive decks in the current "Wild" format.

Plus, it just feels terribly powerful to always be dictating the pace by making the questions and threats which all need to be answered for a slower deck to survive and possibly defeat you. Zoo is a shining example of this phenomenon performing well on the Hearthstone ladder, and aggressive Paladin strategies also do well.

But the deck I find to be really fun to pilot is aggressive Shaman, thanks in large part to the puzzle-like quality that comes from playing cards with the overload mechanic. Unlike many turns with a deck like Face Hunter (which also has decisions, don't get me wrong), a good Shaman player must look ahead and plan for the unknown when dealing with the high power and big drawback of overload spells and minions.

I finally came across a perfect example of the fun complexities at the end of a thrilling game I played today with aggro Shaman against a Paladin.

Fair warning, this is a bit of a brain bender with math involved, but I think we all can learn from analyzing such situations. We all must yearn to play better if we want to be successful, after all, and that takes hard math and metrics sometimes.


The situation plays out like this: the opposing Paladin uses his board of minions to clear three minions from my board on his 8th turn and then swings with a Truesilver to go to 14 life, and he clearly has lethal damage for his next turn. None of these minions have taunt.

So the goal will be to deal 14 damage.

I only have a Doomhammer with some charges left in play. My hand going into the 8th turn is Crackle, Lava Burst, Earth Shock (which would've been handy if a taunt was played). I have 6 total mana available because two overload applies from the previous turn. I start the turn by drawing a second Crackle.

What do you do?

Seriously, look over that situation and figure out how you would spend the six mana available. There's certainly going to be some RNG randomness determining whether you can win from this position, but the second Crackle opens up a series of potential lines of play, each having a different Estimated Value (EV) of winning. Can you, in 70 seconds, evaluate what maximizes the EV here and then execute the play before time runs out?

It's precisely the type of challenge, whether in a single turn or over multiple turns, that I love about aggro Shaman and other more puzzle-style decks like Patron Warrior. Because we have the benefit of more than 70 seconds now, let's figure out what the correct line of play would have been.

We can simplify the circumstance a bit by swinging for 4 damage with two hits of the Doomhammer, leaving the Paladin at 10 life. With 6 mana, the likely options with that hand boil down to:
  • Play Lava Burst, Crackle (this would've been the only play if Earth Shock were needed to clear a path for the Doomhammer hits, but I digress)
  • Play double Crackle with a potential Shaman Totem hero power summon as well
So what do you do first? Play Lava Burst, a first Crackle, or do the hero power? Each ends up with a different potential EV of winning!

Option 1 - Lead with Lava Burst

Let's start with the simple. If you lead with Lava Burst, it will bring the Paladin to 5 life. But the 3 mana left will allow for only one normal Crackle. With a 25% chance on each outcome (3/4/5/6 damage), that means the chances of winning are exactly 50% (EV=50%).

Not bad, but can we do better?

Option 2 - Lead with Hero Power

Let's instead say you lead with the hero power, which will leave you with 4 mana and commit you to the double Crackle plan. You might be very tempted to do this because the swing in potential win percentage is obviously dramatic when a spellpower totem is rolled from the hero power.

More specifically, the math plays out like this. There are 16 potential results from playing two normal Crackles (3/4/5/6 and 3/4/5/6, paired together have 16 total outcomes ranging from 6 damage to 12 damage). Of those 16, only 6 deal 10+ damage. That means a 37.5% win chance if the two Crackles are not powered up. This will happen 75% of the time using the Shaman hero power, but let's come back to that number in a moment.

However, in the 25% chance a spellpower boosting totem is the result of the Shaman hero power, the expected chances of winning jump all the way up to an astounding 81.25%! That is because when firing off the 16 possible outcomes of double powered-up Crackle (4/5/6/7 plus 4/5/6/7), 13 of the 16 potential outcomes will lead to 10+ damage. Winning over 80% of the time sounds like a great bet, if the fates favor you in the hero power.

So adding it all together, there's a 75% chance of a 37.5% EV and a 25% chance of increasing that to a 81.25% EV based on the result of the Shaman hero power. Doing the math...the total win percentage chance of using hero power first comes out to (.75*37.5% + .25*81.25%) = 48.4% (EV=48.4%).

So despite the allure of maybe getting lucky on that hero power and having an 80+% chance to claim the victory, the actual EV of such a line of play is LESS than that of just leading with Lava Burst. Surprised?

A quick side note: the math would obviously change if there were already another totem left on the board by the opponent. Indeed, just having one other totem increases the chance of getting the spellpower totem to 33% instead of 25%, and that scenario would raise the EV of this line of play to about the details matter, a lot!

Option 3 - Lead with the Crackle

This turns out to be another math-intensive of the options thanks to the RNG associated with the first Crackle (3/4/5/6). There are really three different outcomes of note after the first Crackle to consider.

First, if the Crackle hits for 5 or 6 (50% chance of this), Shaman will win automatically by playing the Lava Burst to finish off the remaining 4 or 5 health on the Paladin. In other words, in 50% of the Crackle outcome, the Shaman wins 100% of those games.

Second, if the Crackle hits for 3 (25% chance of this), there's going to be 7 life left on the Paladin. There's only one way to have that happen, which is use two of the four remaining mana to hero power, hoping for the spellpower totem, and then the final two mana on the other Crackle (hoping to hit for 7 damage as buffed up). There's a 25% chance of the spellpower totem, multiplied by a 25% chance the second Crackle then maximizes damage at 7 on the (4/5/6/7) choice. In other words, for this 25% of the time, the Shaman will only win (.25*.25) = 6.25% of the time.

Third, if the Crackle hits for 4 (25% chance of this), the Paladin will be at 6 health which is out of range for Lava Burst. The best available option is to then hero power to try and buff up the chances that the second Crackle using the final two mana will provide 6 or more damage. There's a 25% chance the spellpower totem arrives, which would result in a 50% chance of the second Crackle hitting for 6 or 7 to win the game...and in the 75% chance of another totem, the Shaman still might RNG 6 damage from a normal Crackle for a 25% win chance. Totaling this math: (.25*50% + .75*25%) = 31.25% chance of winning in this circumstance, which again happens 25% of the time based on the first Crackle result.

Once again, we have to sum the probabilities. That math for the three scenarios calculated above results in (.50*100% + .25*6.25% + .25*31.25%) = 59.37% chance of winning (EV=59.37%).


Thus, the best option by far is to lead with the first Crackle. Could you figure this out by approximation in the few seconds you have to make a decision? 

For that matter, did I?

That latter answer is NO! I ended up greedy and rolled the hero power first, which actually had the least chance of success (48.4%), a staggering 11% worse than the best line of play. While my luck was poor on the hero power with no spellpower totem, the luck turned around on both Crackles as I rolled a 6 and then a 5 to finish the Paladin off with 15 total damage on the turn (adding the 4 from the Doomhammer).

However, the play was nuanced enough that I wanted to know if I made the right decision, regardless of the positive outcome. And as it turns out, I could not have been more wrong in the line of play.

Lesson to learn: you can always find better lines of play somewhere in a game or deck, especially with complex puzzles like those presented by decks like aggro Shaman.

Furthermore, when faced with a closed world problem like this one, the correct way to think it through quickly would have been to realize that Lava Burst, then Crackle obviously provides 50% chance to win (the simple scenario above), but if you lead with the Crackle you have that same 50% chance plus whatever other small additional chance to win is provided by the second line of play available with 4 mana left (hero power then second Crackle instead of Lava Burst). That inherently is going to be better than 50%, which means you could quickly determine this line of play at least was not the WORST option (AKA, what I ended up on).

I don't know that a regular person could think through the "hero power first" math (which is not as clear on its face) and leave enough time to then implement the plays of the turn before the timer runs out. So while it was enticing to me and could have been better, the normal player facing this scenario for the first time likely should've gone with the sure bet of Crackle first, which as it turns out, is a way better EV to win anyway.

Hopefully you enjoyed this hypothetical brain-bender and even if you dislike playing decks like aggro Shaman, you might be able to respect the complexity and depth of this archtype and the Hearthstone game as a whole in view of this example.

Until next time, keep finding those better lines of play and never just rely on luck.

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