Saturday, June 4, 2011

First Mover Advantage

First Mover Advantage (FMA) is relevant to many games, including two of the oldest games: Go and Chess.  In many cases, this can be a 'king maker' as long as the player who goes first does not make a mistake.  In other games, it greatly impacts the strategy and tactical doctrine of game development.  There are many ways to attempt to resolve FMA, but not all of them function well, and not all games can be 'fixed'.

Definition of FMA
First Mover Advantage is the inherent advantage the player who moves first in a game has over the other players and should win the game.

Severity of FMA
A game's FMA severity will differ from light to severe.  In severe games, a lesser player can beat a better player if the lesser player goes first, even if the lesser player makes some mistakes.  Light FMA games provide a slight advantage to the first player which, if the player makes a mistake, may cost them the game. Such games are more balanced, but can become tricky to find

Example of Severe FMA
Consider the following simplistic game:
Objective: First person to place a unit in the square marked with 'X' wins
Rules: 1) Each unit may move one space at a time.
          2) Only one unit may occupy a space at a time
          3) Units may not capture other units

And now the board:
Horribly Imbalanced First Mover Advantage
 Obviously, whomever goes first will win.  There are many games with First Mover Advantage, some statistically proven, and others where it is obvious from playing the game.  In other games, the rules may be so complex as to hide the FMA issue and it may not be seen for some time.  The challenge is to identify these games quickly.
Symmetrical Games/Maps
Many times games begin with a symmetry, or close enough to symmetry, which supposedly makes things 'balanced'.  However, the symmetry may provide an illusion of 'balance' which hides the first mover advantage.

Detecting FMA in Symmetrical Games
Test 1: Symmetrical Move Method
A basic way to detect First Mover Advantage in Symmetrical games is to perform a 'Balance Test'.  In this type of test, the first player makes a legal move and the other player makes the same (or equivalent) move. This continues until one player cannot, due to a rules violation, make the same move.  Properly testing FMA using this methodology requires testing all, or very many, possible move variations.

Consider Chess as an example of testing FMA this way.  The game begins with symmetry:
Starting Positions
White, the first player, then moves the King's Pawn forward 2 spaces.  Black responds with the same, returning the symmetry.  White then moves  the Queen Pawn forward 2, and Black responds by moving the Black Queen Pawn forward 2.  Finally, white moves the King's Bishop to Queen's Knights 5.  The board with these moves is shown below:
Symmetry Broken
Technically, to restore symmetry, Black must move the Black King's Bishop forward.  However, the rules of the game declare the king cannot be in Check.  Therefore, Black's next move must be to either place a unit between the bishop and the king, or move the king.  Either of these breaks the 'symmetry', and so it can be assumed White has a First Mover Advantage.

In fact, statistics show that chess does suffer from FMA, but not by much.  I believe the statistical advantage of White to Black wins is somewhere around 52% to 56%.  However, this is only among players of equal caliber.  If I were to play Kasparov, I would expect to get trounced, whether I was White or Black.  This test simply shows FMA may exist, not how significance of the advantage.

Test 2: Reactionary Moving
Symmetry testing is great, but not complete.  Often times it leads to situations where either one player or the other makes an "idiotic" move just to test symmetry when a better move exists.  Similarly, many games have non-symmetrical setups, unit strengths or objectives which prevent Symmetrical testing.  In these cases, other tests are required.

'Reactionary Moving' is one such test.  In 'Reactionary Testing', the first player makes a move and the second player then makes the best of all possible moves against the move made.  If the second player's moves are always made 'in reaction' to the start player, First Mover Advantage probably exists.
Best Move?
A major issue with Reactionary Moving is subjectiveness.  First, 'best possible move' must be determined. This may not be seen for several moves if playing a complex game.  Second, determining if a move is made 'in reaction' to the first player is difficult.  Both of these elements require an objective analysis of the game and expertise in determining the outcome.

Test 3: Statistics
Another measure of whether a game has FMA is statistics.  In some games, such as Dominion, the game has so many randomizing factors as to make symmetry testing invalid and 'reactionary moving' too open to interpretation.  Statistics comes as the means of last resort for determining FMA.


William said...

What amazes me is that I read a thread once with some people denying that the first mover advantage exists in Dominion!

Then of course there are some games, such as 2-player Small World, where going first is a disadvantage.

Finally, you can eliminate the FPA entirely by adapting simultaneous play :-)

Harald Korneliussen said...

There is an important test for first mover advantage that should be mentioned: the pass test. If it is legal for the first player to pass, and this would leave him in a situation identical to the second player, then you can conclusively say that there is no second mover advantage at the very least, and either there is first mover advantage (it is not rational to pass) or the game is horribly broken (it is rational to pass).

This only applies to perfect information games, though. If there is hidden information, by definition you do not know what state you leave your opponent in (in case of Dominion, you do not know his gold split).

Chance Folmar said...


Excellent Observation. I will have to add that to the FMA Tests in a follow up blog. Thank you for the comments.