A Little More Swing Die Theory
by Ryan Keane
Here’s a general model for determining the “optimal” swing die each game. This model assumes that each character being used has 1 swing die and no Option, Poison, Null, Turbo Swing, Mood Swing, or Winslow dice, as these affect one’s hand size for scoring purposes. Also note that the model is based on a simplified game ending where the larger player is left with 1 die; it does not predict the outcome of games where the larger player is left with 2 or more dice (I’ve left this out for simplification) or where there is a “stalemate” (both players still having dice at the end).
Player A is the character with the larger initial hand size (i.e. hand total w/o swing die). Note that Auxiliary and Reserve dice, once added to the hand, should be included in the initial hand size total.
Player B is the character with the smaller initial hand size.
Player A’s formula for swing die choice:
Ax<=Bx-Di+3/2*(An)-1/2
Player B’s formula for swing die choice:
Bx<=Ax+Di-3/2*(An)-1/2
Where
- Ax: Player A’s swing die size
- Amax: A’s maximum swing size
- Bx: Player B’s swing die size
- Di: Player A’s initial hand size minus Player B’s initial hand size (greater than -1 by definition).
- An: The size of one of Player A’s dice which must be kept for him to win. This value can vary from Player A’s smallest to largest die, including his swing die. Optimally, Player A wants An to equal his smallest die, while Player B wants An to equal Player A’s largest die. However, in practice one must vary An to find the optimal swing die that fits in one’s swing range (eg. 4-20).
Let’s do an example:
You, playing Avis, just lost against Hammer(4) (i.e. with a d4 swing die).
Avis: 4,4,10,12,X Hammer: 6,12,20,20,X
What swing die should you use next round?
I’m player B because my initial hand size is 30, whereas Hammer’s is 58; so we use the second formula above. Ax=4 and Di=28, so…
For An=4:
Bx<=4+28-3/2(4)-1/2=25.5
With a Swing d20 (since Avis can’t use a d24 swing), Player A will lose with just a d4 left.
For An=6
Bx<=22.5
With a Swing d20 (since Avis can’t use a d21 swing), Player A will lose with just a d6 left.
For An=12
Bx<=13.5 With a Swing d13, Player A will lose with just a d12 left.
For An=20
Bx<=1.5
Since Avis can’t use a d1 swing, no possible swing will result in Player A losing with just a d20.
So now it comes down to a choice between a d20 (Hammer(4) can win with 3 out of his 5 dice) and a d13 (can win with 2 of his dice). I’m not sure what the optimal choice is – you’ll need another model :).
Another example:
You’re Hammer and just lost to Avis(20).
In addition to the An values included in Hammer’s initial hand size, we’ll also include the case where you win with just your swing die left
(An=Ax=2(Di-Bx)+1).
For An=6
Ax<=20-28+3/2(6)-1/2=0.5
You can’t win with just a d6, no matter what swing you use.
For An=12
Ax<=9.5
With a Swing d9, you can win with a d12 (3 of your 5 dice).
For An=17
Ax=17
With a Swing d17, you can win with a d17 (3 of your 5 dice).
For An=20
Ax<=21.5
With a Swing d20, you can win with a d20 (3 of your 5 dice).
So a d20 gives you the most bang without giving up anything in terms of hand scoring.
How about against Avis(13):
For An=6
Ax<=13-28+3/2(6)-1/2=-6.5
You can’t win with just a d6, no matter what swing you use.
For An=12
Ax<=2.5
You can’t win with just a d12, no matter what swing you use.
For An=20
Ax<=14.5
With a Swing d14, you can win with a d20 (2 of your 5 dice).
So d14 is your best swing.
On a side note, it would be interesting to know which pair-offs reach swing equilibrium, where both players are using their optimal swing dice and do not change them when they lose, and which pair-offs oscillate, each player changing their swing die each time they lose to counter their opponent’s swing choice. If equilibrating pair-offs exist, then for those pair-offs you would always know the optimal swing die to start with. Maybe I’ll work on that next…