by Lucas Karahadian
In Blizzard's digital card game Hearthstone, Arena mode is a gametype that tests players deck construction skills with a random card set. Players are to construct a 30-card deck by choosing one card at a time from a randomized set of three.
After making a deck, players enter a play format with the following rules:
- Players will only play against other players with the same win-loss ratio for the current Arena run (A run is considered to be the entire process from deck creation to elimination and prize claiming).
- Players are eliminated when they reach 3 losses.
- Players are victorious when they reach 12 wins.
With these rules in mind, I set out to determine the distribution of wins in completed Arena runs. My approach was to propagate percentages through a matrix representing win-loss rates. Luckily, this process can be simulated without the use of any random number generation and without considering win-rates of the individual players. At each win-loss bracket, regardless of who is playing, half of the players win and half of the players lose. The results are below and can be recreated using option 1 in ArenaSim.java
Using these results we can calculate the percentage of runs ending at different wins by looking at column 3 and row 12.
The most interesting thing to notice about this data is at the extremes. The percent of 0-win runs is less than 1-win runs, and the percent of 11-win runs is less than 12-win runs. My interpretation of this occurrance is that since the general movement of the players is towards the 12-win ceiling, there are many chances to hop off of the 0-win track and no influx of players into that track since it's the floor, and likewise at the 12-win ceiling there are three different win-loss results that end in 12 wins.