People liked my article, “Chess involved luck, and other propositions“, so I’d like to add a bit more discussion on a related topic.
In turn-based games like Chess, there’s a slight asymmetry between players, in that one of the players moves before the other one does. And moving first seems to be an advantage. This has been demonstrated through statistical analysis of various chess tournaments and databases. Depending on which data are used, the first mover wins anywhere from 52% to 55% of the time.
First mover advantage can be considered as third factor, independent of either luck or skill. If you flip a coin to decide who goes first, then first mover advantage is one component of luck. But it’s the sort of luck that you can eliminate by say, choosing a tournament structure where players alternate white and black.
There’s apparently a lot of historical discussion of first mover advantage in chess, but at this point I may as well drop the pretension that I know anything about chess. The game that I’m a lot more interested in, is competitive Dominion. Dominion is a turn-based game, and also has a first player advantage. The community compiles a ton of statistics from games online, and the statistics show that in two player games among top players, the first player wins about 58.8% +/- 0.2% of the time (excluding ties).
Given how large the first player advantage is in Dominion compared to chess, some people have complained that this is a design flaw. I’ve seen many proposals for how to “fix” it. One possibility is to prevent the game from ending at the end of the first player’s turn. Unfortunately this would be a different game–at high-level play, the ability to end the game on your own turn is an important aspect of strategy. Another possibility is to have players bid for the right to be first player. Unfortunately, bidding strategy is so difficult that much of the game ends up hinging on that early decision. My favorite proposal is to play rando-Dominion. It’s a normal game of dominion, but if the first player wins, you roll a die. If you roll a 1, then the second player wins instead.
It’s debatable whether first player advantage is actually a detriment to the game. The larger the first player advantage is, the less the game rewards skill. Generally, skilled players want a game that rewards skill more, because they want to win more. On the other hand, skilled players winning more means that new players win less, which becomes a barrier to entry.
There’s also something to be said about the value of a game that makes you work harder to win more. People like challenges, right? A game with first player advantage challenges you to win consistently even when you’re put at a disadvantage some of the time.
Anyway, that’s most of what I wanted to say. The rest is a mathematical discussion that the reader may skip.
A first player advantage of 59% means that first player advantage explains about 18% of the outcomes. What about the other 82%? That largely depends on the source of the data. This data is compiled from ladder games online among top players. The ladder system tries to match players of nearly equal skill. If all the games were between players of precisely equal skill level, then we would have to conclude that the other 82% is explained by chance. Suddenly, a large first player advantage doesn’t look so bad when what it’s being measured against is chance.
Okay, but ladder games obviously can’t match players of precisely equal skill level. So let me get another statistic… In these games, the more skilled player wins about 67% of the time.
This implies (through algebra) that the better player wins 76% of the time when they’re first, and 58% of the time when they’re second. I want to make clear that this is not a measure of how much skill matters compared to first player advantage. It’s more a measure of how successful the algorithm is at matching people of nearly equal skill level.
Let’s apply the model I discussed in my previous article, the Elo rating system. I’m going to fudge it a bit by declaring that the standard deviation in a single player’s performance is equal to 1 (instead of 223). I find that the better player has a mean performance of about 0.57 higher than the worse player. The first player receives a performance boost of about 0.32. These calculations aren’t very exact, by the way, I’m just plugging them into
Win% = 1/(1 + exp( -M*pi/sqrt(6) )
where M is the difference in mean performance between the players. A more sophisticated calculation would consider the distribution of skill among players, but I’m not bothering with that.