First player advantage

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.


  1. says

    It’s strange that there are no board games with simultaneous moves. Only sports move simultaneously, or a game like Twister.

    Whether chess, checkers or any other, both players could decide their move in secret and move at once. If they move to the same square, both pieces could be eliminated or the pieces reset and players select another move.

  2. says

    There are certainly board games with simultaneous moves. Like Seven Wonders. Or uh, Escape: The Curse of the Temple. Those just aren’t the board games under discussion.

  3. khms says

    What many or even all of you might already know, but I’ll add it because it hasn’t been mentioned: I don’t know the first thing about dominion, but chess is a member of a class of games (no explicit chance component, turn-based, finite number of options every turn, clear win/lose definitions) where, if you had perfect knowledge (see previous article on that), there’s always one player – first for some, second for others – who can force a win; that is, if they do everything right, there is no possible way the other player can win. From which is´t is obvious that in these games, if the had perfect knowledge, the (first/second depending) player should win 100% of the time.

    So you might argue that the actual percentage one player wins better than the other, compared to that 100%-0%, tells us something about how well we actually understand the game in question.

  4. says

    One piece of information that’s missing, is the draw rate of chess. I think the 52-55% statistic excludes draws, because apparently at the highest levels of chess, the draw rate is around 55%. Given this information, many people think that under perfect play, chess likely ends in a draw (and “white wins” is the runner up hypothesis).

  5. consciousness razor says

    Don’t forget about draws. Here is the wiki article on Zermelo’s theorem (with my emphasis):

    in any finite two-person game of perfect information in which the players move alternately and in which chance does not affect the decision making process, if the game cannot end in a draw, then one of the two players must have a winning strategy (i.e. force a win). An alternate statement is that for a game meeting all of these conditions except the condition that a draw is not possible, then either the first-player can force a win, or the second-player can force a win, or both players can force a draw.

    You can ask about the first position, before either player has moved. Can both players can force a draw at the beginning of the game?
    If not, since white is thought to have an advantage, you may be inclined to think white is the player who can force a win (with perfect play). But chess isn’t actually “solved” for us — our best chess engines aren’t that good — so it’s not proven that white’s advantage means that they can force a win. This means that, as far as anyone knows right now, the game could be a draw (with perfect play on both sides), which would be consistent with the theorem above. I suspect that chess is a draw, because white’s advantage isn’t always great enough to force a win. That is, it seems fairly plausible to me that black’s best choices will suffice to lead them out of whatever trouble white’s advantage creates, so black can at least force a draw in all cases. But a forced win for black may (for all we know) require that white doesn’t play perfectly.

  6. says

    One of my favorite simultaneous-movement games was Diplomacy. Unfortunately, I have not had a chance to play it in decades.

    It’s possible in Diplomacy to be in a situation where you must guess correctly what the other person will do in order to win. In simplest terms, piece A can do x or y. If A does X, and you do R, you win. If A does Y, and you do S, you win. Sometimes the X/S or the Y/R combinations will result in no change in the game state, sometimes is altering which player is favored or making a draw inevitable. In the first case, perfect play may result in a draw even when the first player should eventually win an infinite game.

  7. dangerousbeans says

    @khms or we can just argue that because chess is in that category it’s boring 😛

    the only games i can think of where i can’t see a ‘first player advantage’ type effect are combat sports, time trials in cycling and motor racing, and golf. in the later cases there isn’t a direct competition between players, it’s player vs the course. combat sports are simultaneous (obviously) and have very simple environments.
    poll positions, starting with ball possession, starting as offence/defence in a payload race, shooting first or second in archery and similar, betting later in poker. these situations all affect your knowledge of the environment, and whether you get to control or have to react to things

  8. says

    This is found in Magic: The Gathering as well (while some decks prefer to go second, most prefer first). They try to mitigate it by having the first player skip their first draw step, but providing they hit their land drops (variance is a huge factor too and going first doesn’t matter if you don’t find land or only find land) they will always be in a better position to get aggressive as well as react to their opponent.

    The card draw advantage goes away in multi-player formats like Commander because the first player gets to draw as well. I don’t know what percentage difference it makes between multi-player and two-player, but multi-player has been studied and going first is definitely and advantage.

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