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Thursday, October 13, 2011

FMA Argument for Dominion

William, a friend of mine, recently left a comment on the FMA article found here: http://gamesstrategyandtactics.blogspot.com/2011/06/first-mover-advantage.html.  I then received some emails arguing if FMA exists in dominion.  Rather than become embroiled in an discussion, I'm going to answer this with simple example to prove FMA exists.

Dominion Basics
I won't go into too many specifics of Dominion except as it is relevant to this example:


Dominion "estate" card,
completely useless during the
game, but worth 1 VP
at the end

  1. Dominion is a game where player's use cards to purchase other cards (namely: 'Treasure' cards are used to purchase other cards,
  2. Players may do the following on their turn: Play one card for an 'Action', then 'Buy' one card, Discard, and then draw.
  3. Discard phase: All just purchased cards and cards in the player's hand are discarded
  4. Player draws up to 5 cards.
  5. If a player cannot draw up to 5 cards, the discard is then shuffled and cards are drawn until the player has 5 cards in hand.
Yes, there is more to the game than this, but to test for FMA, the above is all we need.

Assumptions
For this example we will make the following assumptions:
  1. It is a 2 player game,
  2. All players will draw the same set of cards,
  3. All player's are attempting to get the same card, 'Militia'
Treasure card: vital for
buying new cards
Assumption 1: Keeps the example small, in reality, more could be added, but the example takes longer.
Assumption 2: I expect the most comment from this example.  We will discuss the 'randomness' later in the article
Assumption 3: Others could say there are better cards out there, but bear with me on this as it is necessary to the example.

First Hand
Each player will have the following cards available in their hand:
1 Estate - effectively useless card for the first hand
4 Coppers - enough to purchase the Militia card.

Per our Assumption above, each player purchases the 'Militia' card.  This card, when played as an action, forces other player's to discard down to 3 cards.  It costs 4 gold to buy, but that is not a problem the first turn.

Second Hand

Militia: Armed and Dangerous? 

The second hand will have 2 Estates and 3 Coppers.  We know this because the game starts with every player having the same 10 cards: 3 Estates and 7 Coppers.  With this card every player purchases some other card, irrelevant to this discussion.

Just an interesting point here about dominion.  Both of the two hands above could have been drawn in ANY ORDER and it would not change the discussion.  In other words, the first player could have drawn 2 Estates and 3 copper for their first hand, but would have then had 4 copper and 1 estate to purchase the Militia.  This is due to the shuffling mechanic: No player reshuffles cards in their deck until after the they must draw a card and cannot.

Given probability and statistics, the two above hands are the 'expected' outcome for most player's on the first turn.  Players may have 5 treasures in their first hand (and 2 in their second), but we can safely assume the above two hands will occur on the first turn with reasonable expectation.

Put simply, we aren't doing something crazy about the above.

Third Turn
At this point every player will have shuffled their deck and drawn five new cards.  Thus, it is the first player's turn.  The first player draws their Militia card and plays it as an Action.  This card gives the first player 2 coins for purchases and every other player must discard 2 cards.

In this case, FMA raises its head.  No player can stop the play of this card (we will get to counters, such as Moat, later).  This first player is the first player to even have this option.  So, if the First player draws an "average" draw he will have:

  1. 1xMiltia
  2. 3xCopper
  3. 1xEstate
Dominion 'Mine' card.
The first player, upon playing the Militia, therefore has 5 coins to purchase a card.  Every other player must discard 2 of the above cards upon the first player playing Militia.  The second player will discard an Estate (as it is useless during the game) and 1 copper.  This leaves the second player with just 4 coins to purchase a card.

FMA Confirmed
From the above example we can see the first player has an option other player's do not.  From this, the first player has a First Mover Advantage.

Does it Detract from the Game?
FMA does not detract from my playing Dominion.  I enjoy Dominion.  It is a great game and I recommend everyone play it.  I know FMA exists in dominion, but I don't believe it is significantly large, especially not in a 2-player game.  However, the majority of statistics I've seen gathered on plays of Dominion indicate the odds of winning decrease as the player order increases (First and Second player win majority of games, then third player, then last player).

Arguments Against FMA
'But wait!' I hear people say.  It seems the more I hear about discussion for FMA in Dominion, the more I hear a "but wait" argument.  So, I'll discuss some basic arguments against my conclusion of FMA and my counter to them.  Many players take Dominion as a game of "skill", and they argue vehemently against anything to the contrary.  However, the below are common arguments.  If you have an argument I don't have covered here, leave it in the comments or send me an email: I welcome the discussion!

Moat: Counter to FMA?
The above only happens less than 50% of the time
True.  The above scenario (when the first player draws the Militia), will only happen 41.66% of the time.  Still, that is 46% of the games in which the start player is guaranteed a 1 coin advantage.  Not much of an advantage to be sure, but still an advantage.

There are better cards to buy than Militia
Absolutely agree with you.  However, Militia is simply an example to show the first player will have an option the other player's do not, and they will have the option to use that card first.  Militia is simply used as an example to amplify the results and show an 'extreme' version of what can happen.

The Moat would Counter Militia
Again, absolutely true.  However, there are two arguments against this as a counter for FMA.

First, there are usually better cards than Moat to purchase.  So what was the decision behind purchasing Moat?  If it was to counter Militia, then the player is making a purchase based on a move by the First player.  This is effectively 'breaking symmetry', a classic definition of FMA.

Second, particularly with games with more than two players, the "earlier player mover advantage" becomes more pronounced.  Consider, in a three player game, if the first player choose Militia and the second player also chooses Militia, both of these players have the option of playing militia before the third player.  Even if the first player does not draw the card, the second player might and will play it, affecting the third player.

Neither a Borrower nor Lender Be
- William Shakespeare
Working out the odds, if Militia is the 'card of attack', there is a slightly better than 65% odds the Third player will be hit by Militia.  In a Four player game, the fourth player has over an 80% odds of seeing the Militia card played against them if the first three player purchase a Militia.  This should clearly display the odds of being attacked by a specific card increase the later one goes in turn order.

There is too much randomness later on
True.  However, good players can turn a small advantage into a winning strategy.  In fact, that is a goal of many players.  I am not a good Dominion player (I don't play enough), but I fairly good at many other games.  I have a knack of turning a seemingly insignificant disparity into a game winning situation.  The same is true of Dominion.

So Skill doesn't matter in Dominion
Absolutely not.  An unskilled player (such as myself) against a skilled player will get slaughtered.  If I were to play chess with Mr. Kasparov, he could probably let me remove a Pawn of my choosing of his from play and he would still wipe the floor with me.  Skill does matter and skilled players will overcome a FMA of a lesser skilled opponent.  But, this argument does not dispute that FMA does not exist.

What other argument are there?
Let me know if you have another arguments against FMA.  Again, I welcome the debate!


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