New Katahex, new swap map for large boards.

We have gone from 24 to 8 "balanced" opening positions on size 19.
Check the Hex discord for details.

E3, F17, N3, O17, S1, A19, A10, S10, S5, S4, A15, A16, P3, O3, D17, E17

are all still below 61% and above 40% win rate.

Not that it matters much for human play. But still, if I can memorize at least a good pie offer, I'd rather play the best one.

Stronger nets typically have win rates further away from 50% because they're better at converting small advantages into a win.

Consider the c2 opening on 19x19. The previous best version of katahex, 20220618, has Black's win rate at around 42.0% when I analyze the move locally. The current version, 20240812, has a win rate of 37.1%.

One way to "normalize" for this difference is to consider, for a given net,

• W = the win rate of the strongest move without swap

• 1-W = the win rate of playing against the strongest move (in other words, the win rate of a "pass" move)

and interpolate between the two, to see what "fraction" of a stone a particular move is, according to that net. It's empirically useful to do this interpolation in (logistic) Elo terms. If you do that with the c2 opening, with the 20220618 net the c2 opening is 42.0% = -56 Elo, the strongest move (e10) is W = 90.0% or +382 Elo, and therefore, 1-W = 10.0% or -382 Elo. Interpolating, we see that c2 is 42.7% of the way between passing and the best move. Intuitively, c2 is "42.7% of a stone"; moves that are worth >50% of a stone should be swapped.

If you do the same exercise with the 20240812 net, c2 is 37.1% = -92 Elo, the strongest move is +539 Elo, so passing is -539 Elo. Interpolating, c2 is 41.5% of the way between passing and the best move. The point is that this 41.5% is very close to the 42.7% we got from a totally different net, with very different win rates.

One could repeat the same calculation for different moves. For a4, the 20220618 net says 61.7% win rate or +83, which is 60.9% of the way between -382 and +382. The 20240812 net says 66.7% win rate or +121, which is 61.2% of the way between -539 and +539. There are moves where the two nets are further apart, but I think this represents a genuine disagreement between the two nets.

For the swap maps on HexWiki, I applied this normalization method and called a move "balanced" if it was between 40% and 60% of a stone. Intuitively, these balanced moves give a swap advantage of <=10% to the second player. By normalizing, the number of "balanced" opening moves shouldn't change too much as nets get stronger, though some variation is normal, especially as the evaluations for neighboring moves aren't completely independent data points. (The map currently on the wiki reflects an even older net, 20220313. This net had genuinely different opinions on some a-column openings. I may soon update it to the newest net.)

There is more that I could go into, like on 15x15, the 20240812 net has a 99.4% win rate for the top move. That's so close to 100% that the error bars in Elo terms are huge. There are ways around this: you could evaluate the win rate of the 2-2 obtuse corner (b14) on 20220618, find that it's worth close to 75% of a stone, and work backwards using the win rate of b14 on 20240812 (which is around 92.8%, with more reasonable error bars in Elo terms) to calculate another estimate of the win rate of a "full" stone.

Could you please provide a link for this Hex discord?

The Hex discord is https://discord.gg/QXu9UqDdAC