A Statistical Analysis of Strategy in Wizards Of The Grimoire

Hey all!

My name is Joe and I co-created Wizards of the Grimoire (WotG). What is equally relevant to this post, however, is that I also coded the Wizards of the Grimoire Health Counter App!

WotG is a two-player game where wizards battle to the death to determine who will become the next Keeper of the Grimoire. Both players draft spells from a common spell pool. At the beginning of the game, one player will be chosen to draft the first spell into their engine, while the other player will be given the opportunity to attack first. The player who attacks first has only 2 cards on their first turn while the player who picks a spell first will have 3 spells on their first turn. Each turn thereafter the players are able to add a new spell to their repertoire until both players have 6 in total at which point they can't accumulate any more spells.

Now that the game has gotten into people's hands, lots of games are being played. Fortunately for me, many of these games are being played using the app as a health tracker. And since the app uploads button press data to a server, I am able to track the changing of health totals across a game.

I've written some scripts to analyze this data and I've found some really interesting stuff! My goal is to track a few key statistics so that, if any future expansions come along, we can modify the game in order to ensure balance. In addition, these statistics might inform us as to what "optimal" play looks like.


What kind of data does the app collect?

The app doesn't know anything about what cards are being used. The only data that is being collected that is useful for analyzing gameplay is data regarding health totals. Based on this information, however, I can determine: which player went first, how much damage was dealt each turn, and who won.

In total, there are now over 300 games in the database. In practice, however, I have had to remove many of them. For any game where the players didn't input health totals below 0, I can't determine which player won. In addition, there are many blocks of data where users aren't actually playing the game but are actually just fiddling with the app.

Having scrubbed through the dataset, I've found over 80 games of WotG that we can feel confident in.

So what can we learn from these games?



Who wins more, the player who picks the first spell, or the player who attacks first?

In my dataset of 80 games, the player who picks the first spell won 36 times, and the player who attacked first won 44 times. In other words, the player who picks first has a 45% win rate, while the player who attacks first has a 55% win rate.

- I believe that, as the player base becomes more familiar with the game, this will even out more. Attacking first is a clear advantage that will always be fruitful, regardless of skill level. Picking first, however, will only be advantageous when the correct card is picked. For this reason, I think skilled players will be able to win more consistently while picking first while unskilled players will be disadvantaged when picking first.


How Close Are Games?

A game where it is difficult to determine who will win until the final turns is a healthy game.

In my dataset of 80 games, the median health left over for the winning player at the end of the game is 9.5 (SD = 7.3). This means that 70% of games end with the winning player having between 2.2 and 16.8 health. This seems to indicate that, in the majority of games, it isn't clear who will win until the final turns.


Are Games This Close When The Player Who Picks First Wins? What About When The Player Who Attacks First Wins?

When the player who picks first wins, the median health left over is 9.5 (SD = 7.4). When the player who attacks first wins, the median health left is also 9.5 (SD = 7.2).

- This is interesting and not what I expected. Given that the player who picks first has a 45% chance of winning, I thought that they would also be winning their games by a smaller margin. In other words, it would stand to reason that the player who picks first (and is therefore disadvantaged), when they happen to win, would be just squeaking their wins by.


How Many Turns In The Average Game?

The mean number of turns per player in a given game of WotG is 8. 04 (SD = 1.03). That is quite a tight margin. This means that 70% of games end with each player having taken an average number of turns between 7 and 9.

The mean number of turns taken by the player who picked first when they won is 7.56 (SD = 1.16). The mean number of turns taken by the player who attacked first when they won is 8.4318 (SD = 1.30). The distributions of game wins based on the turn on which they occurred can be seen in figure 1.


So, as one might expect, the average number of turns required to win as the player who picks first is 1 less than the average number of turns required to win as the player who attacks first.

Average Damage Per Turn

For each player's first 4 turns of the game, there is a disparity in the number of spells available to each player. The player who picks the first spell has 1 extra spell available to them on every turn. This should lead to differences in average damage per turn. Indeed, what we find is that the player who picks first does more damage every turn until both players have 6 spells, as can be seen in figure 2.

 On the 5th turn, when both players have 6 spells. the trend of the first pick dealing more damage per turn disappears.

How large is this advantage? The average disparity in damage per turn between the two players is 0.65 damage per turn. In other words, the player who picks first will, generally, deal an extra 2.5 damage over the first 4 turns from their card advantage.

Personally, I think this is a really healthy disparity. We have to keep in mind that the player who picks first is necessarily disadvantaged because their turns are always taken after the player who attacks first.



What insights can we glean from this data in order to win more often at WotG? Here are some hypotheses I've derived from the data that I think are interesting.

- Firstly, Figure 1 illustrates how important turns 7 and 8 are in a game of WotG.- Based on that chart, the first point in a standard game of WotG in which a player has an opportunity to win is their 7th turn. It is quite common for the player who picks first to win on their 7th turn, however, it is uncommon for the player who attacks first to win on their 7th turn. This makes sense, because the player who picks first deals, on average, an extra 2.5 damage over their first 4 turns due to card advantage. So, in general, it seems that the player who picks first should aim to finish the game on their 7th turn, and the player who attacks first, rather than trying to finish their opponent on their 7th turn, should aim to survive 1 extra turn, and then try to end the game.

- A second insight that can be made by this graph is that, when the game goes beyond each player's 8th turn, the player who attacks first almost always wins. This, again, makes sense. If each player has taken 8 turns and is still yet to end the game, both players are probably very low at this point. Since both players are low, whoever attacks first will win the game. It is of critical importance to the player who picks first to either end the game on turn 7 or 8.

- Given that games that go past each player's 8th turn are much more often won by the player who attacks first, it seems to me that a reasonable strategy for the player who attacks first is to build a deck that slows down their opponent. The player can rely on the incremental damage from their basic attacks + a 1 or 2 damage spells to get them in range of a kill. Then, using their utility delay spells, they can stop their opponent from winning on turn 7 or 8. Finally, they can finish their weak opponent off on turn 9.

- Finally, I think that another important insight into this data has to do with a mechanic in the game called Basic Attacking. At the end of the every turn, a player has the oppotunity to discard a single mana from their hand and deal damage equal to its power; on a standard turn I feel confident in estimating that this does ~3 damage. Given that the average damage done on a turn when 6 spells are available to the player is around 7 - 8 damage (as can be seen in the final bars of figure 2), I think this data shows the importance of basic attacking. If a basic attack does ~ 3 damage on average and the average damage per turn is ~7-8 then basic attacking comprises ~40% of your damage per turn. Oftentimes I watch new players play the game and they don't basic attack. It is critical to do so.



How Long Does The Average Game Take?

We've described WotG as taking between 20 and 40 minutes on our website and on the game box. How long does it take in reality?

When averaged across the 80 games in the dataset, the median game length was 35 minutes (SD = 13 minutes). This means that most games take between 22 and 48 minutes. Seems like our description is reasonably accurate. I expect that, as people play the game more, the total time necessary to play the game will go down. First time playthroughs will be especially slow, and our dataset is currently proportionally overloaded with first-time playthroughs.


Thanks so much for reading through this!

If you actually read through this whole post then I laud your enthusiasm and consider you to be a truly worthy protégé. Let me know your thoughts on my analysis! Are there any further queries you can think to investigate? Do you disagree with my takeaways? Any confusions?

Best of luck with your games,