Stars per hour

  (Edited: )

In the laddering bonanza last month, Day9 used the phrase "stars per hour". And I wondered: does playing faster really mean you get more stars on the hour? Even if that means selecting a slightly poorer deck?

Stars per minute is all about speed (play time) and score (stars won minus stars lost). Playing really fast and making mistakes that lose you the game means you can lose a lot of stars really fast. Overthinking your every move means you will gain stars really slowly. This looks like an optimization challenge.

SCORE

When you win a higher percentage of games, it also means you are losing less (uncontroversially so). Which not only means you will be losing less stars to losing, but it also raises your chance to land winning streaks, adding bonus stars to the tally. That is, until you get to rank 5, at which point winning streaks no longer provide bonus stars or a chance of finding me there, so I didn't study that condition.

But is the deck which wins most often also the one that will give you the most stars per hour? 

SPEED

Priest decks are known to generate long matches, regaining life all the time, and hitting hard late game. Rogues, the opposite. When you have equal win percentages between your slow priest and quick rogue, the rogue will be able to rake in much more stars for every hour you play. But what if the slow priest is also a little bit more successful at winning, and hence lose less and streak more?

It was time for some numbers.

NUMBERS

I set up 10,000 simulated matches for 7 contestants. Their play style would go from very fast to very slow, and I predicted their win percentages by pulling numbers out of regions that shall remain seated. This allowed me to skip the matches and go straight to the numbers, which saved a lot of time.

A list of 10,000 random integers between 1 and 100 would do. A player with a win chance of 48 would win only the "matches" smaller than 49. And he would get bonus stars for every third victory and on (fourth, fifth etc).

So I made a table and you can find it in the appendix. The lengths of matches are indicative: I do not mean to suggest that playing slower gives you a higher chance of winning. This is nonsense. But it does give an interesting graph. If you achieve a 47 percent win chance with a deck that plays three times as fast as a deck with 53% winnings, the latter is much preferred. If playing a minute slower than average means you win one game extra every 100 games : take that minute! And playing with a Hmmm... attitude is best!*

*Don't. Always play with a YARRRG attitude.

Enough of this. Lets's see that table.

APPENDIX

  Fold early Fast gambler

I must choose!

Average Hmmm... I must consider... Tortoise
Length of match 5 7 8.5 9.5 10.5 13 16
Win chance 47 48 49 50 51 52 53
Wins 4782 4885 4984 5077 5193 5287 5388
Losses 5218 5115 5016 4923 4807 4713 4612
Bonus 1094 1172 1237 1309 1390 1471 1550
Total stars 658 942 1205 1463 1776 2045 2326
Session length 833.33 1166.67 1416.67 1583.33 1750 2166.67 2666.67
Stars/hr 0.790 0.807 0.851 0.924 1.015 0.944

0.872

Edit:

A table in which the star/hour rate is a constant 1, dependent on the Win Chance and the losses and wins as above.

Win chance 47 48 49 50 51 52 53
Avg match duration for 1st/hr (in minutes) 3.94 5.65 7.23 8.78 10.66 12.27 13.96