User:Hexanna

11×11 swap map

(This could go in the "Handicap" article, but much of it is personal speculation and in my opinion not necessarily article-worthy.)

The numbers indicate my guess of what "percent of a stone" each opening move is for Red, where 100 is the best possible move, 0 is equivalent to passing, and 50 is the border between a winning and losing opening. Even though every move is technically 0 or 100 under perfect play, some losing moves are harder to refute than others.

One proposal for how to quantify the strength of stones more rigorously is through three-move equalization. Let's say you place 2 red and 1 blue stones and they are spread out enough to not interact with each other — placing the red stones at opposite corners or edges, for example. Then, the other side should generally swap if the following holds:

(sum of values of the two red stones) > (value of blue stone) + 50

Here, "value" means "percent of a stone" based on the above swap map, where you flip blue stones over the long diagonal first. This rule is a natural generalization of the swap rule, where the other side should swap if (value of red stone) > 50. For example, here is a three-move opening that I think should be quite fair. The red stones have value 65 + 65 = 130, and the blue stone has value 80 since it's equivalent to a red stone at i4:

Theoretically, if 11×11 were strongly solved, you could take all such three-move equalization openings and whether they are winning for Red or Blue, and figure out which stone values allow the formula to make the fewest mistakes in classifying each position as a win or loss. The values obtained this way should be a good indication of how strong a stone is under imperfect play, such as in handicap games. It would be interesting to see the result of such an exercise on a computationally tractable board like 8×8, though on such a small board the stones clearly "interact" with each other significantly.

Random unsolved questions

Most of these are very difficult to answer, and I would be happy if even a few were answered in the next few years:

• Hex on large boards
  • If you trained a strong neural net AI on a 19×19 or larger board, what would its swap map look like?
    • Is the obtuse corner always winning on larger board sizes?
    • What about a move in the middle of Red's third row, like j3 on 19×19?
  • Is the 4-4 or 5-5 obtuse corner still a good move in the early opening, or is it better to play closer to the center?
    • For instance, imagine a board with an obtuse corner and sides extending to infinity. 4-4 is likely quite locally efficient with respect to this obtuse corner, for the same reason bots think it is optimal in 13×13. But 4-4 might not be locally optimal, and some other move (say, the 7-7 or 8-8 corner move, or something even further from the corner) could be ever-so-slightly more efficient on the infinite board, for deep tactical reasons that require far more space than on the 13×13 board.
  • If top humans or bots played 37×37 without the swap rule, how much of an advantage (in Elo terms) does the first player have, in practice?
• Solving 10×10 and 11×11 Hex
  • Which opening moves are winning/losing on 10×10 and 11×11?
  • Which move is the "fairest", or informally the hardest to prove as winning/losing (analogous to a6 on 9×9)?
  • With three-move equalization, what is the "fairest" 3-move opening on 10×10 or 11×11?
• Kriegspiel Hex (Dark Hex), a variant with incomplete information
  • Under optimal mixed strategies, what is Red's win probability on 4×4?
  • For larger boards (say, 19×19), is Red's win probability close to 50%?
    • If so, a swap rule might not be needed for Kriegspiel Hex, which would be neat.
    • If not, imagine a variant where Red's first move is publicly announced to both players, and Blue has the option to swap it. Which initial moves are the fairest now?

replies by Demer:

• https://zhuanlan.zhihu.com/p/476464087 has percentages, although it doesn't translate these into a guessed swap map and I don't know anything about the bot they came from.
  • ​ It suggests that [on 13x13, g3 is the most balanced opening] and [on 14x14, g3 should not be swapped].
  • On 27x27 without swap, it likes the 4-4 obtuse corner opening slightly more than anything else nearby.
• As far as I'm aware, even 3×4 Dark Hex has not been solved. ​ (https://content.iospress.com/articles/icga-journal/icg180057 apparently gives "some preliminary results" for that size.)

Article ideas

• Motifs — very loosely related to joseki; small local patterns that occur in the middle of the board, usually representing optimal play from at least one side but not necessarily both sides
  • Motifs have some notion of "local efficiency" (not to be confused with efficiency) — some motifs are, on average, good or bad for a particular player. Strong players anecdotally try to play locally efficient moves on large boards where calculating everything is impractical. It would be useful to have some of these rules of thumb written down. Can be thought of as a generalization of dead/captured cells, where LE(dead cell) = 0, and LE(X) ≤ LE(Y) if Y capture-dominates X.
  • Here are some examples. In the first motif, Red 1 is often a weak move. Blue's best response is usually at a, or sometimes at b or c as part of a minimaxing play. But d is rarely (possibly never) the best move, because Red can respond with a, and Blue's central stone is now a dead stone. So, for any reasonable working definition of "local efficiency" LE, we have LE(d) < LE(a), and LE(b) = LE(c) due to symmetry, though it is unclear whether Blue a or b is more likely to be better, assuming there are no other nearby stones.

The motif below seems quite common on large boards, and in my experience it is usually good for Red, who allows Blue to connect 2 and 4 in exchange for territory.

The following motif is quite clearly good for Blue, who captures the two hexes marked (*):

Sometimes, a player will attempt to minimax by placing two stones adjacent to each other, like the unmarked blue stones below. Red has several options, such as the adjacent block 1, though a far block is often possible too. It would be enlightening to know, absent other considerations, which block is the most "efficient" for Red, so that on a large board, Red could play this block without thinking too hard. Of course, in general the best move depends on the other stones on the board, and there's no move that strictly dominates another. The best move may even plausibly be to "play elsewhere."

In my experience, it's usually better locally for Red to play in x in the following cases to create a trapezoid or crescent, rather than y.