# HearthSim: Divine Shield Modeling — Part 2

By | July 16, 2014
This is part 2 of our Divine Shield modeling.

### Setup

Let’s take our Super Basic Deck, with a generic no-good Hero (with no hero abilities), and pit it against an opponent with the same deck with Scarlet Crusader replaced by Magma Rager. We will call the player with Scarlet Crusader Player0, and the opponent Player1.

We take $$w_{\rm ds} = 0$$ as a base case and will compare the performance of our AI as we tweak $$w_{\rm ds}$$.

As a side note… the Scarlet Crusader is a much better card than the Magma Rager under this circumstance (and probably under any other circumstances). Player0’s base win rate is something like 64.8%, compared to 57.1% if Player0 used the same deck as Player0 (i.e., Magma Rager instead of Scarlet Crusader).

### Results

The result of running the simulation with different divine shield weighting looks like this:

Raw Data

Recall from part 1 that our initial guess was that the optimal weighting will be somewhere between 0 and 1. Well, we were close. At least in this situation, the optimal weighting seems to be 1. Let’s try to understand the results in more detail.

##### Case: $$w_{\rm ds} < 0$$

When the weight is negative, the AI thinks that having a divine shield on a minion is a disadvantage, and it will prioritize removing the DS. Because attacking the enemy hero doesn’t remove the DS, the AI will pretty much always attack another minion with the Scarlet Crusader. This strategy turns out to be ok, and it maintains a >60% win rate, but it’s obvious that it’s not optimal.

As a side note, with the weight less than -1, the AI will never put the Scarlet Crusader onto the board because doing so will result in decreasing the score. So, there is a lower limit on $$w_{\rm st}$$.

##### Case: $$w_{\rm ds} = 1$$

With $$w_{\rm ds}$$ set to 1, the AI seems to perform optimally. This is because it ends up making decent trade decisions. At this weight, the divine shield on a Scarlet Crusader is worth 4, while if it is used to attack an enemy minion but fails to kill it, it does 3 damage. So, in the attack, the AI loses 4 score from DS and gains 3 from the enemy minion health going down — a losing proposition. The AI typically won’t make that attack and will go after something else with better value, such as hitting the hero. The simulation suggests that that is indeed the correct thing to do.

##### Case: $$w_{\rm ds} > 1$$

When $$w_{\rm ds} > 1$$, the AI starts to consider the DS more and more valuable. At $$w_{\rm ds} = 1.25$$, the DS is worth 5; at $$w_{\rm ds} = 1.5$$, it’s 6; and so on. This increase makes the AI less eager to attack the another minion with the Scarlet Crusader, and it ends up pretty much always going for the face. Needless to say, always going for the face is a losing strategy (see this post for a demonstration of that), so the win rate plummets.

##### Miscellaneous Results

As suggested in a previous comment, I thought it would be interesting to look at the distribution of the duration of the game; i.e, how many turns does a typical game last given different strategies employed by the AI. In this setup, let’s compare the cases where $$w_{\rm ds} = 0$$, $$w_{\rm ds} = 1$$, and $$w_{\rm ds} = 2$$. Below is the plot of the fraction of games that end at a given turn.

Games played with $$w_{\rm ds} = 2$$ clearly tend to end earlier. This trend makes sense from the strategy point: the AI is being quite aggressive, and aggro games tend to be shorter. It’s also a hint that the AI at such weight isn’t losing close games; rather, it is usually on the receiving end of a beat down.

### Summary

In conclusion… I think this is a fair divine shield model for the AI. DS certainly seems to help make the deck better, and as long as you use it to smack and kill other minions, it provides good value for its cost. You’d certainly want to pick Scarlet Crusader over Magma Rager, and most likely Argent Squire over Murloc Raider and so on.

As always, leave any comments or suggestions on our on our discussion board!