That's Combinatorics

I had some people over after the Serenity movie on Saturday to play board games. I learned two new games and played a third I’d only played once before (Louis XIV, Endeavor, Ra:Dice Game). All three were games I enjoyed and would like to play again. Louis XIV had an interesting ending that sparked a bit of debate due to a hidden scoring mechanic in the game that I thought was a little interesting. I ended up writing a script to simulate the ending which I ran about half a million times to crunch some numbers about how the game ‘should’ end from the game state we got to.

Without going into the specifics of the game itself, know that there’s only three way to score points in the game. The first is accomplishing a task and is worth 5 points. The second is to collect tokens each worth 1 point. (These tokens have pictures on them but you get them blindly so you have no control over the pictures.) The third is to have the most of a given picture of token. Each picture that you have the most of is worth 1 bonus point. The first two sources are easy to calculate, the question becomes how many of the bonus points do you expect to get?

Now, the token pile is split into 6 pictures, and there are 10 of each picture. We played a three player game and ended up with a 9-11-13 split. The debate broke down into how reasonable is it for a given player in that position to score up enough bonus points to win. I was the 9 and had completed an extra task over the course of the game, so I had an extra 5 points. Pounder had the 11 and Duncan had the 13. Since each token is worth 1 and I had 5 extra points I was in the lead going into the bonus phase. I was up by 1 point on Duncan and 3 on Pounder. The bonus points ended up being split 2-2-2 (ties go to no one) so I won, but ‘should’ I have won? How reasonable would it have been for Pounder to come back from down 3 to pass me? (This came about from a debate about who to ‘screw’ if you have the choice.) Assuming no tiebreakers, the approximate odds from my simulation for each result are:

Nick Wins 24.06%
Pounder Wins 0.02%
Duncan Wins 39.50%
Nick-Duncan Tie 35.88%
Nick-Pounder Tie 0.03%
Pounder-Duncan Tie 0.02%
Nick-Pounder-Duncan Tie 0.50%

Turns out being down 3 with the 11 of a 9-11-13 split is a very bad spot to be. Pounder has practically no chance of outright winning and not much better of pulling off a tie. Even if he wins all tiebreakers (in the specifics of the game he won tiebreakers, then I beat Duncan) he doesn’t even win 1 game in 100. I outright win 24% of the time which is pretty reasonable, so despite it being a little favourable to have gotten a 2-2-2 split it’s not like I stole the win. When you consider I had the tiebreaker on Duncan my win chance shoots up to almost 60% which is pretty reasonable. And indicates a very close game took place. If Pounder had some way to give Duncan or I a point during the game it would have had a pretty big impact on the outcome of the game. (All those ties turn into straight wins for Duncan and some number of my wins turn into ties.)

A question then is how many points is a chit worth? It’s worth 1 plus the amount of bonus you earn divided by the number of them it took to get there. In the specific game my chits were worth more because it only took 9 of them to get the same number of bonus points Duncan got with 13. But how much are they worth in general? This is a question I should be able to answer with combinatorics but it’s been a good 10 years since I took an enumeration course and I just can’t work it out right now. (Aidan said he was going to try!) I should probably get a book and refresh myself on it. (And on stats, so I could work out confidence intervals for my simulations instead of just assuming they’re good enough…) But for the specific breakdown of 9-11-13 I can say that the 9 expects to get .86 bonus points, the 11 expects to get 1.4 bonus points and the 13 expects to get 2.1 bonus points. (With 1.6 wasted to ties.) On a per tile basis, my chits were worth 1.09, Pounder’s were worth 1.13, and Duncan’s were worth 1.16. The more you get the more they’re worth, but they still aren’t worth very much extra.

The final question then becomes: “Is this a good mechanic”? (The initial debate on Saturday started with this very question.) My viewpoint hasn’t changed given the numbers but I’m curious if anyone else has an opinion on the matter before I go into details.