Monday, 30 September 2013

Two-Player Dominion and Bidding

A colleague of mine just finished acquiring all the Dominion expansions and is planning a tournament.  The format will be two-player round-robin - though as with most round robins, one assumes that not all the games will be played unless there is a very small turnout.

Some time ago I was very big on Dominion.  I used to play a lot of solitaire three player games where I would have two decks go pure silver-gold-province and see how many points I could get with the third deck.  But everything I know about two player Dominion comes from simulation.

Two player Dominion is a very skewed game.  If you play a silver-gold-province strategy then the second player takes about 70% of the wins.  Note that this is very different than winning 70% of the time, nearly 30% of the games are ties, as you might expect.  The problem is that the tie breaker gives the win to whoever has had fewer turns in the game.  That means that if the second player is the second one to buy a fourth province then the game is a draw, but if the first player is the second one to buy a fourth province then the second player wins.  This means that if the second player acquires a fourth province before the first player, then the first player needs to buy a different point card.

By toying with different variants on silver-gold-province I concluded that the best I could come up with was for the first player to break estate or duchy parity when given the option and for the second player to match it.  That is, if you are the first player and you have five you should buy a duchy instead of a silver if you currently have the same number of duchies.  The second player should buy that duchy if the first player has more duchies.  The same goes with having two and buying estates.  There may be further refinement possible, but at the time I was looking for something I could easily program a computer to run ten thousand games of and the way I had built the code limited how complex the decision tree could be.

Anyway, by using that the first player/second player break/restore duchy/estate parity strategy, the second player only takes home about 53% of the wins and there are far fewer ties.

When one player has an advantage in a game, the World Boardgaming Championships will use a bidding system to equalize the advantage.  A player can bid a number of points and then the person with the higher bid picks their seat.  If the first player always wins a game, then the theory is that there should be some number of points that we could handicap the first player by to make the game fair or close to fair.

Of course the obvious problem with bidding systems is that there probably isn't a bid that makes the game 50/50 between players of equal skill.  Dominion is a simple example of this.  Let's add a bidding system to two-player Dominion.  We know second seat is better, but how much should you bid for it?  It turns out that answer is that if you bid any amount then second seat is downright awful.  Since second seat's advantage comes solely from the tie-breaker system, a bid of any size negates that advantage and gives the game back to the first player.  The first player starts taking 60% or more of wins and there are no strategies to mitigate the difference.  No matter what the second player does the first player can just ride their extra turn to take games.

53% is not really all that skewed the grand scheme of things.  Who wins the game with identical strategies is random to begin with, so adding another random element to the beginning is probably not breaking the game.  The point of bidding is that there should still be a point to playing after the coin is flipped, and if you are conceding with a 47% chance to win then you deserve to lose anyway.

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