## Monday, April 9, 2012

### Janowski model extended to match play

When I was looking at match equity tables I wondered whether you could extend Janowski's model (using a "cube life index" to interpolate between dead and live cube equities as a proxy for the jumpiness of game state) to match play. I'm pretty sure this is what GNUbg does based on their docs.

Turns out it's pretty straightforward if you assume the same match equity table as you calculate with Tom Keith's algorithm, which is a live cube model - it assumes game-winning probability changes diffusively, and that W and L (the conditional points won on win or lost on loss) are independent of the game-winning probability. That's the same as Janowski's live cube limit, except W and L are calculated from entries in the match equity table instead of the usual money scores for wins, gammons, and backgammons.

The Keith algorithm mentioned before gives you the cubeful match equity in the live cube limit. The dead cube limit has cubeful equity that's linear in P, running from -L at P=0 to +W at P=1. The model cubeful equity is just the weighted sum of the live and dead cube cubeful equities.

I implemented this to see how it compares to using a Janowski money model for doubling decisions in tournament play. That's of course inefficient - it doesn't account for match equity - but it's an interesting benchmark to show how much a match-aware doubling strategy matters.

Checker play was Player 3.4 optimizing on cubeful equity, and I ran 40k matches (with variance reduction) for a range of match lengths and cube life indexes. For a given cube life index, both the match and money doubling strategies share the same cube life index, which seems a fair comparison. Remember, for money play, a cube life index of x=0.70 was optimal.

The entries in the table are average points per match in the 40k matches. All values have a standard error of +/- 0.005ppm.

Match
Length
x=0x=0.25x=0.50x=0.75x=1
3+0.056+0.047+0.051+0.066+0.054
5+0.073+0.080+0.091+0.105+0.067
7+0.115+0.138+0.137+0.142+0.061
9+0.115+0.148+0.150+0.154+0.081

The main conclusion here is that using a proper match-aware doubling strategy makes a huge difference in outcome. The impact is larger for longer matches because any per-game edge gets magnified over a match. In principle for very long matches the match strategy should become the money strategy, but for matches up to length 9, anyways, that is not apparent in the statistics.

This assumes the same match equity table as before, which is based on the live cube model. Really I should recalculate the match equity table assuming the same Janowski model for cube decisions, which I think will change it a bit. But I'll leave that for another day.

Or I could extend my jump model to match play - it should be a relatively simple extension, since like with Janowski it's just about changing W and L to be based off match equities.