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Openings on 19 x 19

Strategic advice from KataHex

I started learning Hex mid-July of 2021, about 2 weeks before I created my LittleGolem account.

katahex/general strategy part 2 (rough draft)

This is not very organized, and some bullet points won't make sense without further context. I also need to decide which ideas are worth keeping. (I have outlines and rough drafts of even more stuff, but I'm worried about making the article too long and diluting it with only rarely-applicable advice.)

• If you can't find a good move, avoid a bad one by playing "skew" relative to existing stones.
  • That is, play a move that interacts with existing stones as little as possible (at least distance 3 from all other stones).
  • The difference between a good move and a blunder is >5x the difference between a great move and a good move.

• Stone "arithmetic": not an exact science, use your intuition; play the move that maximizes the sum of local and global efficiency. I am not actually calculating values when playing, just making mental adjustments.
  • The best move locally is rarely the best move globally. Hex is a game of concessions and tradeoffs. There is no "free lunch," unless you count concepts like inferiority/domination.
  • Conversely, if it looks like your opponent is beating you on one side of the board, there are two possibilities:
    • You played suboptimal moves in that region.
    • Your opponent played too strong in that area, which necessarily means they made concessions elsewhere (usually the opposite side of the board).
  • If the first option is true, you are probably losing anyways, so assume (or hope) the second is true. That means you should tend to tenuki and play the opposite side of the board, if you think your opponent overplayed one side. Indeed, if you continue playing on the same side, your opponent might be able to get out of their mistake by minimaxing, such that they go from being overconnected to being merely connected.

• For brevity, some notation for up-weighting and down-weighting particular moves:
  • chess notation: ??/?/?!/!?/!
  • one way to think about it: you are a machine who has to assign a score (or a prior policy probability) to each move. !? means slightly upweight, ! means upweight, ?? means strongly downweight, and so on.
  • asymmetric scale, one blunder can take multiple "great" moves to recover from; also, brilliancies (!!) are rare and don't come from easily described rules like the ones in this article

•  !? bridge moves, either from your own stone or your opponent's
  • an additional !? if it's part of a bridge ladder that you'll win (you're playing 5-4 acute corner), ?! if your opponent will win
•  ? moves adjacent to your own stone, unless you are minimaxing and know that it's the best move. Be careful of bad minimaxing, where you connect too weakly to one side, allowing your opponent to intrude for territory. More often than not, the concession you make here isn't worth the extra strength.

a3 opening

Nowadays, strong players rarely play a3 on the first move because it's similar to c2 (which is already on the losing side on most board sizes), but a bit weaker. There are some subtle differences between a3 and c2, however, and most strategy advice on the a3 opening is outdated.

If Red plays a3, some older strategy guides recommend that Blue respond d6 or e7 if she wants to play in the acute corner. However, KataHex considers this an inaccuracy; even though d6 and e7 are attacking stones in the sense that Blue would win a bridge ladder towards the acute corner, they don't do a good job blocking Red's edge the way e4 does in an empty corner.

If you want to play the acute corner, 5-5, or 4-4 on smaller boards, is a good choice. (There are other options further away from the corner on larger boards.) KataHex and other bots consistently prefer these moves:

• https://www.littlegolem.net/jsp/game/game.jsp?gid=2175191&nmove=4
• https://www.littlegolem.net/jsp/game/game.jsp?gid=2033243&nmove=2
• https://www.littlegolem.net/jsp/game/game.jsp?gid=2018841&nmove=8
• https://www.littlegolem.net/jsp/game/game.jsp?gid=1950209&nmove=20

One reason may be the efficiency of templates IV2e and V2d, described in more detail below.

It can be useful to realize that a3 is primarily a blocking move (blocking Blue on the left rather than helping Red connect to top), whereas c2 is half blocking and half connecting. To illustrate the point, it's a bad idea for Red to play both c2 and 4-4 obtuse corner on the top edge (in the absence of Blue's blocking stones), because that's too much strength near the top. However, Red could feasibly play both a3 and 4-4 on the top edge.

a3 can combine well with c5. Red's goal is to leave b4/c5 open as options (these are the same hexes used in the a3 escape trick, but of course, a3+c5 can also connect upwards). Blue should also keep an eye on b4 - not to play immediately because it's too close to her own edge - but as the most efficient option for connecting to the left side near the top. Blue would much prefer to connect using b4 (which is a locally efficient move that neutralizes a3's switchback and upwards connecting potential) than b5 (which only slightly weakens a3's potential, forcing Red to play d5 instead of c5), or a 2nd row ladder with b3 (which removes Red's weakness at b1).

a3 is one of my favorite openings to analyze (I only wish it were more fair as an opening move). Unlike c2 where Blue's best move is typically to tenuki, a3 gives Blue the option to fight in the same acute corner. There is essentially a separate theory of a3 corner joseki.

I find a3 lines particularly educational because a3 changes the corner just enough to make different moves more efficient. You can learn a lot about how an existing stone influences neighboring stones.

• d4: https://hexworld.org/board/#14c1,a3d4e4e3
  • Note Red e4 is near his own edge but adjacent to the blue stone. Blue e3 is counterintuitive but it forms template V2d, which has a hole at Red's a3.
  • Red avoids b5 which is a very weak move, because it makes Blue b4 even more efficient: https://hexworld.org/board/#14c1,a3d4b5b4c4d2. This position is practically Blue's dream; Blue makes full use of d4 and the weakness at a3 with template IV2e (which also has a hole).
    • This is similar to Red 5 being a blunder in this opening: https://hexworld.org/board/#14c1,a10d11c12c11b12b11
• e3 first: https://hexworld.org/board/#14c1,a3e3c4b4d2c3d3c5
• d3 is bad: https://hexworld.org/board/#14c1,a3d3b4
• Why is e4 weak? After Red c5!, the normally locally efficient b4 (even though it's globally inefficient, because it's on Blue's 2nd row) is terrible because Red d3 is a bridge cut. Note that a3 primarily is a blocking move, not a connecting one, so d3+a3 is not so much strength near the top that it renders d3 a bad move. https://hexworld.org/board/#14c1,a3e4c5b4d3
• e5: In some ways, these resemble the low intrusion 5-4 joseki, shifted up one row.
  • https://hexworld.org/board/#14c1,a3e5c5c6d6d5 Red 5 is optional but actually an acceptable bridge peep, because intruding on the other side of the bridge would be too much strength near the top and almost never a good idea.
  • Red must still avoid b5 for the same reason as before: https://hexworld.org/board/#14nc1,a3e5c5c4b5b4d4f3d3 is very favorable to Blue. Observe that Blue 8 at f3 is more efficient than at e2, because e2 is followed by Red f3 which is a bridge cut
  • https://hexworld.org/board/#14nc1,a3e5c5c4d4c6d6e6c8d5g6h4 If Red bridge peeps at d6, then Blue e6 may be a stronger reply than connecting directly. Red 11 at f6 instead would be a blunder because of Blue g4.
  • https://hexworld.org/board/#14nc1,a3e5c5d4c4c6f5h3 There are too many variations after Blue d4 to list them all. Blue 6 at d3 leads to a fun minimaxing sequence but is perhaps a slight inaccuracy: https://hexworld.org/board/#14nc1,a3e5c5d4c4d3b6d2c3d7

Second row ladder vs. connecting directly

Pretty contrived, but holding your opponent to a (one-sided) second row ladder instead of letting them connect directly is worth more than getting a second-row escape yourself.

I came across this intuition a few times when studying various sequences on large boards, and it seemed surprisingly self-evident (almost like it "has" to be true). I wasn't sure why, so I thought it was a fun exercise to convince myself that it actually makes sense from first principles. Here is an informal argument:

• Let x = "holding your opponent to a second row ladder instead of letting them connect directly"
• Let y = "getting a second-row escape yourself"
• To show that x > y, decompose x = m - n, where
• m = "your opponent has a second row ladder"
• n = "your opponent connects directly"
• Also, -y is "your opponent has a second-row escape". Then, x > y is equivalent to m + (-y) > n.
• That is, all else equal, you'd rather allow your opponent a second-row ladder with a second-row escape, than allow your opponent a direct connection. Why is this true? In the first situation (m + (-y)), you have the ability to jump as the defender, and gain territory in exchange for allowing your opponent to connect. On the other hand, if the opponent is connected directly without the need for a ladder escape, you have no such option.

Some potential applications of this rule (it's a stretch to say that these are direct implications of the rule, but the rule can help with building intuition):

• Helps explain why Blue 4 is surprisingly strong here: https://hexworld.org/board/#19nc1,e15e17d17b18:pc19d18
• Blue is not afraid of the ladder in the 4-4 3-2 obtuse corner: https://hexworld.org/board/#19nc1,c2d16b17e17c15b18d17c19
• Red is just fine in this line of the obtuse corner opening: https://hexworld.org/board/#19nc1,a19:pd15c17c14e16b17b18a18c15a16b15

Insights and tidbits from KataHex (hzy's bot)

• katahex_model_20220618.bin.gz (I'll call this the "strong" net) appears significantly stronger than the "default" net.
• Swap map for 19×19 generated with the strong net, with around 15k visits for most moves. For the red hexes, the number corresponds to Blue's Elo advantage if she swaps Red's move; for the blue hexes, the number corresponds to Blue's Elo advantage if she does not swap Red's move. Smaller numbers correspond to fairer openings. Includes all fair openings as well as a few selected unfair openings, including the strongest move without swap (e10). I used more visits for the fairest moves: 1000k for e3 (49.5% win rate), 500k for n3 (49.2%), 100k for a15 (52.5%).
  • Key takeaways: The swap map at Swap rule#Size 19 uses data that is almost certainly from the "weak" net. Compared to the weak net, the strong net notably thinks a19, n3—p3, and k4—l4 are stronger. I personally trust the strong net's evaluations more; I think it's dubious that the weak net thought l4 was a very fair opening. The nets disagree on whether e3 is winning or losing, though it's so close to 50% that the difference isn't meaningful.

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:

• Is the obtuse corner always winning on larger board sizes? What about the b4 opening? Let P(n) be the statement that "the obtuse corner is a winning opening in n×n Hex without swap." There are a few possible cases; an interesting exercise is to come up with subjective probabilities of each case being true.
  • A. P(n) is always true. If so, can we prove this?
  • B. P(n) is true for infinitely many n, with finitely many counterexamples. If so, what's the smallest counterexample?
  • C. P(n) is true for infinitely many n, with infinitely many counterexamples. If so, does P(n) hold "almost always," "almost never," or somewhere in between?
  • D. P(n) is true for finitely many n. If so, what's the largest such n?
• 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.)

hexanna:

• Thank you, this is amazing! From the Google Translate, the bot is an adaptation of KataGo trained on 13×13 and smaller, using transfer learning to train larger nets on top of the 13×13 net for a short period of time. I may edit the swap rule article later with some insights.
  • The results for up to 15×15 look very reliable to me. This is because many of the subtle patterns suggested by other bots, like leela_bot, appear in these swap maps. For example, on 13×13:
    • a1–c1 are stronger than d1; a2–c2 ≥ d2 ≥ e2 in strength; and a similar relation holds for moves on the third row. See Openings on 11 x 11#d2.
    • b4 is weaker than all of its neighbors, because Blue can fit the ziggurat in the corner.
    • j3 is surprisingly weak and i3 is surprisingly strong. Many people were surprised about this when leela_bot's swap map came out, but the result may be more than just random noise.
    • a10 is the weakest of a4–a10, while a5 is the strongest.
    • b10 is stronger than all of its neighbors, because Blue cannot fit the ziggurat in the obtuse corner.
  • That this bot picked up on all these subtleties, and assigns a win percentage close to 100% for most moves on 13×13, suggest to me that it is probably stronger than leela_bot and gzero_bot. I can't know for sure, though.
  • On the other hand, and the author seems to agree, the 37×37 map looks very unreliable. I see percentages as low as 37% but only as high as 54% (for a move like f1, which should almost certainly be a losing move).
  • The 27×27 map looks more reliable. I'm personally very skeptical that moves on Red's 6th row are among the most balanced moves, but there are some interesting (if somewhat noisy) insights to be had still.

Recursive swap

Not really a serious suggestion, just for fun. One advantage of "recursive swap" over multi-stone swap is that opening preparation plays a smaller role, because both players have control over the first n stones.

RECURSIVE_SWAP'[k, depth, color]:
  if depth = 0:
    [color] continues playing as normal.
  else:
    [color] plays a move. [~color] can either
      swap[k], or
      RECURSIVE_SWAP'[k+1, depth-1, ~color]

RECURSIVE_SWAP[n]:
  RECURSIVE_SWAP'[1, n, Red]

RECURSIVE_SWAP[0]:

Red continues playing as normal.

RECURSIVE_SWAP[1]:

Red plays a move. Blue can either

• swap, or
• continue playing as normal.

RECURSIVE_SWAP[2]:

Red plays a move. Blue can either

• swap, or
• play a move, after which Red can either
  • swap2, or
  • continue playing as normal.

RECURSIVE_SWAP[3]:

Red plays a move. Blue can either

• swap, or
• play a move, after which Red can either
  • swap2, or
  • play a move, after which Blue can either
    • swap3, or
    • continue playing as normal.

Analysis

RECURSIVE_SWAP[0] is the same as playing with no swap.

RECURSIVE_SWAP[1] is the same as playing with the swap rule.

RECURSIVE_SWAP[2]:

• Red shouldn't play a move that's too strong or it'll be swapped.
• If Red plays a weak stone, Blue should try to play a move just strong enough that Red will be indifferent to swap2. (If a "fair" move is half a stone, and Red plays a weak move worth x < 0.5 stones, Blue should play a move worth x + 0.5 stones.)
• Red should try to play a weak move that's also hard for Blue to equalize (so that Red gets a sizable advantage when deciding whether to swap2 or not).

RECURSIVE_SWAP[3]:

• If Red plays a move worth x > 0.5 stones, Blue should swap.
• If Red plays a weak stone worth x < 0.5, Blue should play a move worth less than x + 0.5 (or else Red will swap2). If Blue's move is worth y, then Red should play a move as close to (y - x) + 0.5 as possible, so that Blue's swap3 decision is difficult.
• Red should play a weak move that's hard for Blue to find a tricky reply to (where a "tricky" reply is one that makes it hard for Red to equalize, such that Blue has an easy time deciding whether to swap3 or not).

Miscellaneous:

• Infinite recursive swap is (under perfect play) strategically equivalent to Reverse Hex, because each player must try to play a losing move as long as possible, or else their opponent will swap and win.
• On any board size, assuming the opponent has the swap option after the board is completely filled, I believe RECURSIVE_SWAP[n] is a win for Red if n is even, and a win for Blue if n is odd.
  • I'm bad at proving things rigorously, but I think this follows from the fact that Reverse Hex is barely a win for the winning side (in that the losing side can delay the loss until the whole board is filled).

Fjan2ej57w's question 7, "how much space of an empty board would be filled if both sides play optimally":

Stack Overflow answer for reference. My conjecture is that Hex without swap asymptotically requires Θ(n^2) cells, and more generally, a Demer handicap of Θ(f(n)) stones requires Θ(n^2/f(n)) cells, for all f(n) between Θ(1) and Θ(n). My intuition is that on 1000000×1000000 Hex, the first-player advantage is minuscule, and even a handicap of n^(1/2) = 1000 stones, say spaced out evenly across the short diagonal, would require on the order of "1000 columns and 1000000 rows", n^(3/2), to convert to a final connection. Another interesting question is to find a constructive winning strategy with an o(n) (sub-linear) handicap.

reply by Demer: ​ ​ ​ ​ ​ ​ ​ Even one with ​ n/6 - ω(1) ​ handicap would be interesting. ​ ​ ​ (improving on ​ https://webdocs.cs.ualberta.ca/~hayward/papers/handicap.pdf )