Imagine that you know, without a doubt, that you are going to lose a multiplayer game of Shadowverse or Hearthstone—you should concede immediately. Assuming your win rate exceeds 50%, continuing only takes away from your total play time, and slows your climb up the ranks. OK cool—this is just common sense, albeit slightly against human nature.
But what about a more realistic scenario: after 5 minutes of play you believe you have a 10% chance of winning. Now is it better to (a) concede and move on to the next match, or (b) keep playing with a low chance of victory?
1. When to Concede
Based on my statistical analysis, some Shadowverse and Hearthstone players should concede when there is low probability of winning. This decision is most relevant for aggro players with high win ratios.
For these successful, aggressive players, it is generally more efficient to concede when your probability of winning falls below 7%. For strong control players, it is still more efficient to concede when your win probability drops below 5%.
Read through the following section to understand how I reached these results. I also provide an equation to determine exactly when you should concede!
2. The Math Behind Conceding
Let r be your win rate, let t_quit be the duration of the current game, and let t_avg be the average duration of your ladder games. Data on the average duration of Hearthstone games can be found here. Although the decks listed in that article are outdated and Hearthstone specific, it still seems reasonable to say that aggro decks have a t_avg of about 5 or 6 minutes, while control decks have a t_avg of more like 9 to 10 minutes. Finally, let t_saved be an estimate of how much longer you think the current game will last—do you think the game will take another minute, 3 minutes, 5 minutes, etc.? Given these definitions, we can use the likelihood ratio test—
In this formula, p(w|s) should be read as the probability of winning given the current game state. So the variables r, R, t_quit, and t_avg are known, and you just need to estimate how much time you think you will save, t_saved, as well as your current probability of winning, p(w|s).
In order to visualize this formula, we can check out the following plots. On the horizontal axes I vary your win rate, r, and your average game duration, t_avg. Hence, the bottom corner corresponds to bad players using aggro decks, the left corner represents good aggro players, the top corner indicates good control players, and the right corner shows bad control players. The vertical axis gives the critical value for p(w|s), and each plot considers a different amount of time saved (1, 3, or 5 minutes).
For example, let’s say you are an excellent control player (top corner). Then if you think conceding will save you 1 minute, it is only efficient to concede when your probability of winning is less than 2%. If you think conceding will save you 3 minutes, it’s correct to concede when your probability of winning is below 5%. Finally, if you think conceding will save you 5 minutes, then you should concede if your probability of winning is around 7.5%.
Quitting a game which you would have won has a steep penalty. Not only have you wasted the time spent playing that game, but you also must win two more games to make up for your loss. Hence, we usually don’t concede until we are sure we have lost! For experienced Shadowverse and Hearthstone players, however, I have shown that it can actually be correct to concede before the “final” turn. As a rule of thumb, conceding is worth the risk if you have a high win rate and your probability of losing the current game is above 95%.