If you haven’t already, you might want to check out the articles of Matthew Watkins at PureMTGO.com. Matthew datamines a huge number of MTGO matches, analyzes them and showcases his results for different formats. I’m sure that he puts a lot of time and effort into his articles and I personally find his approach very interesting. However, in his last article Ars Arcanum: Magic 2013 Limited Overview, I stumbled over something. In his statistical analysis of the impact of the decision to play or draw, he states the following:

After over 600 games, I saw that playing won the game 49% of the time, while drawing won the game 51% of the time. This is essentially a statistical tie.

Note that 600 is the total number of observed games. Here are his statistics:

Players chose to play in 93.2%, to draw in 6.8%.

Players choosing to play first win 47%.

Players choosing to draw first win 16%.

Instead of focusing on which decisions the players are making, I will just look at the player having the choice to play or draw. The game win average from the perspective of that player is

93.2% * 47% + 6.8% * 16% = 43.8%

As this is less than 50%, it means that the player with the decision to play or draw is at a significant disadvantage. In Matthew’s words:

This data seems to suggest that most Magic players are making exactly the wrong decision!

This would mean that the average player is so bad at making his or her play/draw decision that it would be advantageous for him to flip a coin instead. Even better than asking a coin is to do the exact opposite of what he believes is right. However, from a game theoretical point of view, it is always better to make this choice rather than your opponent. So, the question remains: Why do the players with the decision to play or draw lose more games?

The answer might be that Matthew’s statistics are biased towards players with sub-50% win percentages. In game 1, the right of choice is decided by the flip of a coin. In the subsequent game(s), the loser of the previous game gets to decide. This means that, on average, the disadvantaged player (in terms of his game win percentage) gets to decide whether to play or draw more often than his opponent. If Matthew did in fact combine the results of the first and subsequent games of a match, this introduces a bias explaining the 43.8% result.

It is interesting to note that this bias is larger the more heterogeneous the skills of the competing players is, even if we assume perfect play/draw decisions. On the other hand, if and only if all players are at exactly the same skill level, we would in fact get the 50% result one intuitively expected.

Matthew, if you happen to read this, I would be quite interested to take a look at your raw data :). Keep it up!

-Simon