Hex 13x13 Openings

I compiled a database of 13x13 hex games on little golem, and I used it to calculate the most popular opening moves. Each move got a score. The score increases with the number of times the move was played. Moves played by highly-rated players were given higher score than those played by low rated players (each 200 ELO difference was worth a factor of 2 in the contribution to score). Also, more recent games were weighted higher than older games when determining the score.

Here are the top few most popular moves:

move: m11 score: 1472 swap score: 19
move: m8 score: 434 swap score: 67
move: k2 score: 373 swap score: 38
move: m10 score: 363 swap score: 66
move: m6 score: 337 swap score: 63
move: l12 score: 325 swap score: 34
move: m7 score: 290 swap score: 59
move: m4 score: 234 swap score: 62
move: m5 score: 189 swap score: 59
move: m12 score: 156 swap score: 16
move: m9 score: 135 swap score: 67
move: m3 score: 100 swap score: 30
move: m1 score: 67 swap score: 54
move: m2 score: 42 swap score: 49
move: l3 score: 25 swap score: 69
move: e2 score: 22 swap score: 60
move: l11 score: 18 swap score: 91
move: c2 score: 16 swap score: 55
move: l2 score: 14 swap score: 72
move: d2 score: 11 swap score: 44
move: k10 score: 10 swap score: 68

Swap score is similar to score, except it measures how often players swapped that opening move (weighted by the ratings of the second player). A move that is always swapped would get swap score 100, and a move that is never swapped would get 0.

Comments:

1) I was surprised by just how popular m11 (which is a3) is. That move is rarely swapped by top players.

2) I was surprised by how unpopular m2 is (even less popular than m1). Is there a reason for this?

3) I noticed that people seem to slightly prefer not to swap k2 and l12 (which is b2). However, people seem to slightly prefer swapping other 2nd row moves, like e2 and c2 (the sample size is small here though). I always assumed that second row moves were all a bit weak, and that b2 and k2 were the strongest of them. This would explain why b2 and k2 are popular while other second row moves are not. But in that case, why do people swap moves like e2 and c2, while not swapping b2 and k2?

I'm interested in hearing your thoughts.

Thanks for this information!

I would like to do a graphical representation of it, could you give the exact formulas you used for both scores?

What do you thing would be a good colouring? I had the idea to use the popularity score as contrast and use the swap score as a blue to red scale, with moves nearer to 50% considered better moves (red) and both 0 and 100% swapped moves as bad (blue) does that make sense?

Thanks for the info, shalev.

I've compiled my own tables of the strength of opening moves. They are not currently in a format that's easy to paste into this message, but I can post them later if there's interest. For now, I'll just describe the approach.

Step 1 is to assign a rank to each player. The official LG ratings are heavily influenced by games lost due to time-outs and the steady influx of 1500-rated new players, so I don't consider the LG ratings very useful. Instead I compute my own ELO-based ratings. I start with a list of all games which (a) were won by resignation or complete connection (i.e. not time-out), and (b) have at least 10 moves. I then look for a set of ELO-ish ratings which best explain the results. More specifically, I find ratings which minimize

bc. sum over all games of square of (E(R(p1) - R(p2)) - result)

Here R(p1) and R(p2) are the ratings of the two players, E(...) is the ELO-expected result (a number between 0 and 1), and "result" is the actual result (either 0 or 1).

Step 2 is compute the "advantage" of each opening move. Suppose player A plays move M against player B and A eventually wins the game. If A is a much stronger player than B, then this doesn't tell us much about strength of M, since A could probably win even with a poor move early in the game. On the other hand, if B is stronger than A then this result is evidence that M is a strong move. Averaging a quantitative version of this over all games which feature move M one gets a measure of the advantageousness of the board position after M. (I do this for popular 2nd, 3rd etc moves as well as 1st moves.)

In fact I don't use all games in step 2 above. Instead I use games in which both players' ratings are within D of the rating of the top player, for various values of D.

Carroll, the exact contribution per game is
2^((rating - 1400)/200 - (2013 - year)/3)
and this is summed over all games in the database (which doesn't include games where both players have low ratings). Again, note that a each 200 ELO gives twice as much contribution, as does each 3 year time difference (also, the year of each game is only a rough estimate, based on the tournament that game is part of). Actually, I think I also divided the final score by 10 and rounded it, to make the numbers nicer.

For the swap score, a similar score is calculated for both swapped games and non-swapped games, and the "swap score" is just
100*(swapped weight)/(total weight).

Ignatius, I'd be interested to see your results. I'd also be interested in seeing who the top 10 players are according to your calculations, and what their ratings are.

Your approach for calculating the best first move sounds interesting. Do you find that it is sensitive to noise? I suspect such and approach might require a large database to be accurate.

On the other hand, if you do have a large database then your approach should be superior to mine - I'm only calculating which moves are most popular amongst top players, and I'm trusting the top players to pick the best move. Your analysis, on the other hand, has the possibility of discovering that top players' opening moves are worse than they think.

Hexagons in grayscale according to swap score percentage:

!http://minortriad.com/swapscores.png(swap_scores)!

Thanks to shalev for this information...!

Shalev, here are answers to some of your questions.

I think the data is sensitive to noise, or more specifically suffers from small sample size. And because players are aware of certain popular sequences of moves, it is not the case that each game is an independent test of the opening sequence.

I was definitely interested in discovering what the strength of various moves was, independent of players' opinions of the moves.

One thing I'm uncertain of: Suppose N strong players play games with opening move X, resulting in a certain percentage of wins for the first player (no swap). And suppose also that M less-strong players play games with opening move X, resulting in a certain percentage of wins for the first player. Because there are more weaker players than strong players, M will be significantly larger than N. On the other hand, in the games of weaker players one expects the play to be further from ideal, and so the opening move will have less influence on the outcome. I'm not sure what relative weighting to give few-games-better-play versus many-games-weaker-play.

I'm traveling now, so I won't be able to format the opening move data for a few days. The ratings are easier -- I'll post them in the next message.

Here are the top ??? players who have played at least 30 qualifying games, according to my ratings. (I originally tried for the top 90, but the message was too long. Perhaps I should drop the links from the table.)

In order for a game to be "qualifying", it has to be (1) won by resignation of complete connection (i.e. not out-of-time), (2) have at least 10 moves, and (3) not be against a +-infinity player (see next paragraph).

Suppose a player has won (or lost) all their games. Then their optimal ELO rating is +infinity (or -infinity), and they can be dropped from the rating calculation. Call these "+infinity/0" players (or "-infinity/0" players). Similarly and inductively, if a player has won all their games except those against +infinity/n players, then their optimal ELO rating is +infinity and we call them +infinity/n+1 players. (And similarly for -infinity/n.) I drop from the rating calculation all +-infinity/n players for all n. (The highest n that actually occurs is 3.)

Also, since the ratings are arbitrary up to an overall additive constant, I normalize the ratings so that the rating of the strongest player is zero.

..... It looks like the table I was going to include here is too large to be accepted by the server, so I'll try breaking it into separate messages.

|_. id |_. name |_. wins |_. games |_. rating |
| 6398 | Maciej Celuch | 430 | 495 | 0 |
| 13355 | Daniel Sepczuk | 515 | 583 | -120 |
| 17442 | n/a | 103 | 105 | -137 |
| 16475 | cr0ssfir3 | 72 | 79 | -157 |
| 6992 | psychoanal | 66 | 80 | -200 |
| 21658 | Naked Face | 164 | 205 | -217 |
| 6583 | Hideki Motosuwa | 48 | 56 | -222 |
| 15078 | wojtek | 511 | 579 | -228 |
| 2832 | Leoni | 169 | 201 | -230 |
| 22531 | asdef | 76 | 105 | -249 |
| 10331 | Radek Brozek(cienias_1234) | 79 | 96 | -254 |
| 7861 | wojtex | 151 | 187 | -270 |
| 23321 | Yussuf | 95 | 105 | -274 |
| 19135 | Pawcio | 129 | 148 | -287 |
| 13431 | Pedro Jorge | 54 | 61 | -289 |
| 16617 | Rtv | 84 | 96 | -291 |
| 4189 | Tim Shih | 206 | 262 | -298 |
| 8138 | koontz | 150 | 230 | -308 |
| 15279 | Piotr | 40 | 44 | -310 |
| 8452 | hextreme (lucky) | 86 | 100 | -312 |

|_. id |_. name |_. wins |_. games |_. rating |
| 5411 | Stanley Kozera | 445 | 628 | -316 |
| 22082 | CoColino | 102 | 118 | -320 |
| 1882 | Arne Regier | 65 | 79 | -322 |
| 15549 | keny-li | 42 | 51 | -329 |
| 9091 | iLyN | 173 | 251 | -331 |
| 8299 | _fubbi_ | 75 | 97 | -337 |
| 1607 | David J Bush | 514 | 658 | -340 |
| 8652 | vacation | 94 | 127 | -345 |
| 15465 | no_name | 86 | 102 | -352 |
| 1657 | Frode Lillevold | 182 | 275 | -357 |
| 10983 | Marek Zakrzewski | 42 | 53 | -367 |
| 7496 | visual | 39 | 47 | -370 |
| 9783 | el koyotte | 51 | 59 | -373 |
| 7018 | Jonatan Rydh | 148 | 195 | -377 |
| 12579 | gryf_ | 93 | 122 | -377 |
| 10612 | kimmuriel | 51 | 70 | -379 |
| 13699 | Keizo | 117 | 142 | -380 |
| 22939 | Monica Rainey | 219 | 277 | -380 |
| 31853 | shalev | 88 | 105 | -383 |

|_. id |_. name |_. wins |_. games |_. rating |
| 1927 | Bill LeBoeuf ? | 130 | 182 | -385 |
| 7569 | Player #7569 | 49 | 60 | -392 |
| 21590 | mangajah | 37 | 47 | -396 |
| 8965 | Adam | 203 | 285 | -398 |
| 9244 | FF | 84 | 112 | -405 |
| 13277 | aduthc | 160 | 200 | -407 |
| 5252 | :/ | 374 | 570 | -407 |
| 14030 | Tao Brahman | 85 | 120 | -408 |
| 12113 | Maciej Bednarski | 80 | 101 | -409 |
| 8462 | pogromca | 81 | 116 | -409 |
| 8977 | Manuel Fernandez | 42 | 57 | -410 |
| 9108 | fight_maan | 78 | 109 | -412 |
| 21306 | skip_to_my_lou | 72 | 83 | -414 |
| 13073 | + | 98 | 121 | -419 |
| 7885 | romekdth | 56 | 83 | -421 |
| 3311 | 150 | 84 | 114 | -423 |
| 17295 | Jaroslaw Herba | 171 | 234 | -424 |
| 2847 | Juliusz Cezar | 40 | 54 | -426 |
| 7577 | Alan Poe | 35 | 40 | -427 |

And here's a more diverse sampling of players.

|_. id |_. name |_. wins |_. games |_. rating |
| 6398 | Maciej Celuch | 430 | 495 | 0 |
| 13355 | Daniel Sepczuk | 515 | 583 | -120 |
| 21658 | Naked Face | 164 | 205 | -217 |
| 7861 | wojtex | 151 | 187 | -270 |
| 23321 | Yussuf | 95 | 105 | -274 |
| 4189 | Tim Shih | 206 | 262 | -298 |
| 8138 | koontz | 150 | 230 | -308 |
| 5411 | Stanley Kozera | 445 | 628 | -316 |
| 22082 | CoColino | 102 | 118 | -320 |
| 1607 | David J Bush | 514 | 658 | -340 |
| 1657 | Frode Lillevold | 182 | 275 | -357 |
| 13699 | Keizo | 117 | 142 | -380 |
| 22939 | Monica Rainey | 219 | 277 | -380 |
| 31853 | shalev | 88 | 105 | -383 |
| 1927 | Bill LeBoeuf ? | 130 | 182 | -385 |
| 14030 | Tao Brahman | 85 | 120 | -408 |
| 21306 | skip_to_my_lou | 72 | 83 | -414 |
| 17295 | Jaroslaw Herba | 171 | 234 | -424 |
| 3205 | Zeycus | 75 | 106 | -436 |
| 10581 | pOmek:B | 113 | 170 | -438 |
| 9328 | Art Duval | 158 | 254 | -458 |
| 3333 | Ignatius J Reilly ? | 74 | 107 | -486 |
| 2259 | Marius Halsor ? | 598 | 937 | -509 |
| 2597 | Niall | 238 | 329 | -511 |
| 12627 | Shumacher;] | 163 | 247 | -538 |
| 5438 | zaszczyk | 455 | 662 | -562 |
| 1728 | ypercube | 569 | 885 | -564 |
| 9540 | EdoEF | 181 | 290 | -570 |
| 6699 | Lukasz S. | 121 | 187 | -570 |
| 16152 | Pessoa | 300 | 477 | -579 |
| 1919 | Jose M Grau Ribas | 834 | 1364 | -595 |
| 11474 | Marcin Pindral | 384 | 589 | -614 |
| 19433 | nietsabes | 124 | 203 | -642 |
| 2844 | Jonathan | 199 | 390 | -656 |
| 20150 | klui | 40 | 52 | -699 |
| 14918 | MarleysGhost ? | 295 | 532 | -702 |
| 9105 | FruG0 | 751 | 1274 | -710 |
| 14080 | Alexandre | 172 | 288 | -715 |
| 28037 | German_Rodriguez_Perez | 135 | 332 | -942 |
| 1224 | Nagy Fathy | 228 | 663 | -1075 |
| 12365 | Thoddy | 162 | 631 | -1283 |
| 11040 | Gerald | 1 | 107 | -5741 |
| 1503 | bill | 10 | 671 | -5828 |

Thanks for the ratings!

A lot of the top players on Little Golem moved all over the place in your ratings. Very interesting. I'm surprised by how high I place on your rating compared to people like Art Duval and Marius Halsor. On the other hand, I'm also surprised that so many names I've never heard of are above me.

By the way, do you weigh games less if they are older? I suspect a lot of players improve over time. I've certainly improved a lot over the past year...

There are several reasons for the significant differences between the above ratings and LG ratings. I think the biggest one is forfeited games (ignored by me but not by LG). Look at the LG rating graph for wojtek, for example. A very strong player who periodically forfeits all his games. This affects not only his rating but also the ratings of anyone who loses to him when his rating is artificially low.

Another reason is that I weight old games and new games equally. I would like to experiment with weighting new games more heavily, but this raises complicated issues since players often play games at very irregular rates.

A third reason, I think, is that LG ratings have not had enough time to stabilize. Put another way, the LG ratings are too compressed (difference between top players and bottom players is too small).

A fourth reason, as I mentioned above, is that LG ratings are constantly injecting new, 1500-rated players into the mix. Other sites (e.g. the old PlaySite and some Go sites) handle this problem by making new player ratings more volatile (or assigning them lower inertia, to use a different metaphor). LG treats the rating of a new player the same as more established ratings.

Thanks to shalev, Ignatius, and Tom Ace for presenting this data!

IMO, first moves M1 and M2 are too strong and should be swapped, although I'm more certain about M1 than I am about M2. Conclusions reached on an 11x11 board do not necessarily apply to 13x13. I used to play 1.A3 a lot because it could be regarded as a "trap opening," although not necessarily a sound one. These days I agree with the general consensus that it is probably too weak to swap.

There are two people who know a lot about Hex openings. One is lazyplayer, which might seem surprising since he has played only 38 13x13 games here as of this post, and his rating is 1835. He is better known as a real time player on an 11x11 grid on Game Center. He is really very strong, and has studied 11x11 openings in great detail. The other player is of course Maciej, who usually beats lazy. I don't doubt there are several other players here who know Hex very well, but I suspect these two know more about opening theory than anyone else.

Thanks for your input David!

Could you explain why A3 can be regarded as a trap opening?

Let's name the player with an additional stone on the board when it is opponent turn to play the first player.

I think all openings tends to be easier to play for first player than for second player, because the additional move on the board makes easier for him to guess the right plan. But i've no proof of this, obviously! :)

About A3, i think it is trap in the sense that this difference in difficulty of playing against it is even larger than other moves with same hope of being winning under perfect play...

Basically, it looks unconvincing as a winning move, but in practice you can win with it! :)

David, by now Arek Kulczycki is more or less as strong as me on 11.

We've both experimented a lot on 11, but at least for me i didn't learn much about hex in general. I've learned mainly new tricks for specific positions.

I would recommend people here to give a try to M1. The corresponding one in 11x11 leads to interesting positions, and on 13 i think that there should more chances for second player. Unlike moves like A3 or A4, its strength (which may or may not be enough to win) is not easy to "see"...

About M2, according to me (and also accoriding to Maciej if i remember well) it is stronger than M1...

Last note. I also think that moves like E2, F2, etc, are weaker than B2. You don't find this in the data because you are not getting rid of games played by weaker players, i guess...

IMO,
first winning MX moves are:
M1,M2,M4,M5,M6,M7,M8,M9
first loosing MX moves are:
M3,M10,M11,M12,M13 (M13 the worst)

M2 is probably the strongest, just a little bit stronger than M1.

Strenght of M1 is based on the fact that you can connect top using the line below short diagonal. It also makes sth like "13x12 board near right edge" for opponent. I mean the opponent can't connect right edge using the short diagonal.

Strenght of M2 is based on the fact that you can "almost" connect top walking one or two lines below short diagonal. The only one way for opponent to block M2 from connecting top is to play M1. There's a lot of opportunities to take top walking horizontally on the 2nd row. That explain the "almost" word. M2 also makes sth like "13x11 board near right edge" for opponent. I mean the opponent can't connect right edge using the line below short diagonal.

The very next move after M3 is L2. L2 is one of the short diagonal move and prevents lot's of M3 connections, so M3 is almost useless while L2 is such a strong move. That's why M3 is bad.

I am surprised that M1 and M2 have about 50 swap score while M4,M5,...,M9 are harder to play with and they have over 60 swap score.

M10 is the most complicated one for me to swap. It's also a "trap opening" but more dangerous than M11/A3. I think M10 is losing and that is my problem:)

What about F3, G3, H3 first moves?

Wow, thanks for your insight Maciej! Do you have any comments on B2 or K2 or any other 2nd row move?

As for F3, G3, and H3, my database shows that they are very rarely played by top players. This means I don't have a good sample size. However, when they are played, their swap score is over 70%, meaning people like to swap them. I don't think this means anything, because my conclusion is that most players simply swap unfamiliar moves. Do you believe they are decent opening moves?

By the way, what do you guys think of L3? EdoEF has been playing this recently. Is it good? And are there any other moves on the L column that can be played? (My guess is no, other than L12.)

Shalev, L5 is also worth some attention. But if you can't beat moves like M4 or M5, it doesn't make much sense to start fighting against L5 or L3.

If you want to try something unusual, 1 F3 2 E9, and 1 M1 2 I10 or 1 M1 2 H9 are my suggestions... or come to IGG and play 11x11 with me... :)

For Maciej i think his evaluations are biased toward favoring first player, and i think this is because he evaluates them according to his experience... as i've said earlier, in practice it is easier to play first player, but this doesn't mean that it can't force a win.

It's like Go, where komi tends to be under-estimated, because nobody is able to squeeze every point from an initial position...

4 years ago i tried to use 1 H3 against Daniel... and failed miserably... but now i think that i could have played much better. :)

http://littlegolem.net/jsp/game/game.jsp?gid=1119735&nmove=10

In particular, now i would play first move one step closer to top side, i would play 7 J12 instead of 7 J10, and something else instead of D5...

What about the games of EdoEF with L3? Can you show us some examples?

Here's one example: http://www.littlegolem.net/jsp/game/game.jsp?gid=1512990

There are also a few such games in EdoEF's table of the current championship:
http://www.littlegolem.net/jsp/tournament/tournament.jsp?trnid=hex.ch.31.2.1
(Note that most of these games are not completed yet, so don't comment.)

If you want more, here are the games starting with L3 in my database (this includes all games except very recent ones, and except games where both players have low ratings):

894534 429774 32937 719015 1210651 1184058 1258385 17045 1565290 1565288 1565286 1565284 1565282 1516951 1515034 1515033 1512996 1512994 1512992 1512990 1498907 1402736 69139 1129309 863792 827365 1358958 16032 1150176 333046 1500107 126991 1180731 877186 852534 845501 1343803 1074330 1181681 1513113 1486090 890460 870353 861057 1353031 1044263 713183 593743 1241613 1229199 1214706 1095123 391858 262622 206371 1513148 1343816 1062091 1082327 952755 919301 896476 838647 733154 733152 733145 733132 730491 846904 17048 1270755 283841 57013 1295259 1147062 1145003 1140128 1140126 1140124 1140116 1139788 1139784 1138911 1138906 1137044 1137041 1135224 1129307 1129305 1129300 1129236 1129231 1125482 1121943 1271746 1258391 1082323 881750 835657 131909 69141 55995 14512 14097 1077369 900580.

This list includes games which are not finished yet.

Also, here are the G3 games:
514360 1110371 1110084 1091705 1091142 57070 284624 829110 829108 965209 1014310 1256600 1076779
And F3 games:
218962 218066 965213 1038572 1261099 579344
And H3 games:
1119735 47173 1332990 1561253 1209040 262589 1074402 829449 1402461

Here's a link to the first move stats I mentioned above. Comments and questions are welcome.

http://canyon23.net/hex_first_moves/

Ignatius, very interesting...

I wonder whether there is a good way, in theory, to measure the relative strength of first moves (or any other moves in the game).

One proposal I've heard is to consider how many moves it takes to win (assuming the losing player is playing optimally with a goal of making the game last as many moves as possible). I think someone (U of Alberta?) worked this out for a small board (6x6 or 7x7).

Another idea is to look at the proportion of winning and losing responses further down in the tree. If I play a winning move, then of course 100% of my opponent's responses are losing. But for some of those responses it might be that only a small percentage of my possible replies are winning. So we could say that the strength of the initial move #1 is the minimum, over all possible replies #2 by my opponent, of the percentage of winning moves #3 for me. Or this criterion could be applied recursively through the entire game tree.

It would be interesting to see some of these measures graphed for small boards to see whether there's any sort of convergence.

Ignatius, i think that as board size increases, the game really changes. And there are more and more positions that are very balanced in the sense that winning player has very few winning moves...

Thanks for the data, Ignatius. It according to your data, 2nd row moves are surprisingly weak - even standard ones like B2 and K2. I'm surprised by this, and I'm not sure if I agree.

Also, it looks like every opening stone on the M file is at least a slight advantage, except for M11 (which is A3), M12 (which is A2), M3, and M4 (the data is conflicted on M4). This roughly agrees with what Maciej said, once you take into account lazyplayer's observation that it's easier to play if you have the first stone.

(Note that I'm not even looking at moves like M13 or L2, which are obviously not playable).

Another weird observation is that B2 is rated similarly to A2 (or even slightly worse). This is weird because B2 is actually STRICTLY BETTER than A2 (every path that connects through A2 would also connect if B2 were played instead).

Also, about the question of measuring the relative strength of first moves in theory, I think looking at the proportion of winning and losing moves is a bad way to go. What we really care about is the proportion of winning and losing "natural moves" - moves that a good player will analyze and consider playing. The definition of "natural moves" changes with the strength and even play style of the players - but this makes sense, because the strength of different opening moves also varies with the strength of the players.

Even programs that use Monte-Carlo type searches generally weigh some moves higher than others when randomly placing stones. I think what'd be really nice is to take a good Hex program, make it play against itself using different opening moves, and measure the proportions of wins. The result would obviously depend on the program, but I would guess they would be fairly accurate if the program is sufficiently good and plays in the same style as humans.

Does anyone feel like trying this with MoHex? I don't have a copy of it. Also, does anyone know how good MoHex is on a 13x13 board? Can a 1900 player beat it half the time?

shlev, Ignatus page is very good also because it tells us for which moves there is interesting data... in the case of players rated above 2000, it seems there is interesting data only for first column and b2, c2, and k2. Indeed B2 is rated better than A2 if you ignore weaker players... :)

BTW, i agree that data is more or less as expected by me... even if i'm somewhat surprised by wild differences between A column moves...

Mainly difference between A6 and A7, and between A9 and A10...

A11 not being easy to beat may be surprising for some (but not for me because i was already believing that the obvious answer can't be immediately played).

shalev, I don't personally have any opinions about how well the statistics predict actual winning moves. I don't think my intuition at the beginning of the game is very good. In fact, I did all this opening move statistical analysis in order to try to learn about the opening from other players.

Let's suppose, hypothetically, that we had statistics for a very large number of games between good (but not perfect!) players. Suppose opening moves X, Y and Z have winning percentages of 52%, 62% and 72%. Does this information tell something about human psychology? About objective truths related to the Hex game tree (independent of human psychology)? About both of those? If we ran the experiment with Hex players who were also good Go players (or also good Chess players), would the numbers come out differently?

Hi guys, do you have any new observations?

lazyplayer's idea that a game is easier to play for the first player seems true indeed

First of all, on one hand, my intuition was that it should always be easier for the player who has a winning position. This intuition also has reasonable causes. On iggc we (me and lazyplayer) played a lot of games with multiple undos (often we play over 30 moves and then undo to, say, 5th move and play on again). During these experiments I noticed that "easiness" of play strongly correlates with the winning percentage.

But wait, does the winning percentage of x% indicate that we are x% sure that the position in question is winning? Heh, not really! At the moment my opinion is that a winning position leads to an easiness of play, but easiness does NOT indicate that a position is winning.

It seems to me that easiness comes from the volume of choice - the bigger choice you have the harder the game. Generally moves that are closer to edges give more choice to opponents, because they don't threaten to finish the game quickly. For example this 3rd move:
http://www.trmph.com/hex/board#11,b2g5b8
is less threatening than this:
http://www.trmph.com/hex/board#11,b2g5e7
so the position after b8 is more complex and harder to play for white than after e7, yet I don't know which move is stronger.

Since the opening move is inevitably close to an edge it should be indeed harder for the second player, hence easier for the first player. Another factor is that an opening move that is well-known is likely to be followed by more well-known moves which is important because players have relatively small choice of well-known moves. The latter maybe explains the low win percentage of the most common openings like b2 an a3.

In short it can be summarized exactly like lazyplayer did, it is easier to plan for the first player, but this doesn't indicate that they have a winning position. A practical conclusion can be - swap everything :) But on a very high level it's a mistery.

Well, this comment is probably too long and too obvious, but if you reached this point, note a more interesting observation of mine! From my experience all moves on A line (>A3) are very strong on 11x11, but playable. In recent months I learned that all moves on A line (also >A3) are almost deadly on 13x13. I started examining 19x19 size and noticed that A line opening is again at least decent on rows >3.

So... I pose a crazy theory that A4 opening is of the same strenght on all (playable) board sizes! :)

Arek, "difficulty" is something like number of "reasonable" moves / number of winning ones. But this formula must be applied to the whole tree of possible "reasonable" continuations. And there are alternations between one player winning and the other winning. So it's very hard to define "difficulty"... :)

I may remember wrong, but isn't A4 or A5 proven to be losing on 8x8? Does anyone remember?

A4 is losing on 8x8, but A5 is winning:
http://webdocs.cs.ualberta.ca/~hayward/hex/

If you look at the 7x7, the pattern seems to be that A(N-3) is winning on NxN, which is why A5 wins on 8x8.

It's also interesting to note that E2 is winning on 7x7, but no corresponding piece is winning on 8x8 or 9x9. In general, if you look at small boards, it does seem to be the case that opening moves must become more "central" as the board gets larger. However, I agree with Arek that this doesn't match the experience humans have playing 11x11, 13x13, and 19x19 - most experts notice little "weakening" of moves like A4 as you move to larger board sizes (or even no weakening at all).

By the way, I've recently been playing some games with B2, and it seems quite playable. I'd also like to advocate some other moves on the second row.

In particular, if I start losing too much with B2, I plan to switch to C2, which is just a bit stronger (I can prove C2 is strictly stronger than B2 using a strategy stealing argument, but in practice it doesn't seem easy to get much extra strength out of C2 compared with B2). Note that C2 is winning on 7x7 while B2 is not.

If my opponents start swapping B2 and I start losing too much to that, I plan to switch to either A2 or D2. I think D2 should be somewhat weaker than B2, and it should lead to interesting games. I think A2 is underrated. Possibly A2 is actually stronger than A3 (or at least, players know how to play against A3 by now, but not against A2). Of course, A2 is strictly weaker than B2, so it's only an option if B2 is considered too strong.

Shalev, B2 always and everywhere was the old Arek's theory. C2 instead was my favorite many years ago... you walked in already walked in paths... :)

lazyplayer: ah, I see. And what was your conclusion? Is C2 strong on 11x11? Why did "the old Arek" stop playing B2? =P

Shalev, try beating B2 on 7x7 and you'll see what we mean when we say that blue is winning but can have a very hard game. I think on larger board, b2 is even more likely to be objectively losing, but this is almost entirely compensated by the fact that optimal play for blue is harder as the board size goes up... :)

C2 is better as you said and can make a difference. But i don't think it is enough to change the objective result on 13. On 7x7 on the other hand, C2 wins and B2 loses, for example.

Shalev, I abandoned B2 because lazyplayer proved to me that it most likely loses on 11x11. C2 is worth dedicating some time to explore it. Nevertheless on 13x13 I concider B2 pretty strong, but I avoid it just because most people know it very well.

Btw, I'm strongly against comparing facts from small boards () with larger, playable boards. It has already been noticed that the effect of scale is not intuitive. Moreover on small boards the effect of move being "fitting" or not dominates other features and a global view.

Arek, I find B2 is not as well-explored on 13x13 as you might think. If you look at the game I annotated, you'll see I mentioned that my 3rd move was already non-standard, and pretty much never previously played by top players (despite being an interesting, and perhaps natural, move).

On the other hand, I agree that B2 is not as exciting as the A13 move you recently played against me =P

Maybe I mean a different thing. You're right that there are plenty unexplored possibilities to vary the play after B2. However, B2's purposes are very clear and straightforward. Moves around the middle of opponent's edge are much more exciting, because one may use them in many ways. Problem is that these moves seem far too strong ;)

Maybe A11 is a more reasonable option than I've always thought. I'm willing to try it in future games against lower ranked players.

With the recent announcement of the HexWiki being back online (http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=625) I wanted to ask if you guys mind if I use some of the information that you've put together on this thread onto the page http://hexwiki.amecy.com/index.php/Openings_on_13_x_13#The_first_move , specifically to update the play percentages of the first moves (the content on the current page is likely as of 2007) as well as to ask TomAce if I can use the diagram with the swap percentages.

It is interesting to compare Maciej's notes for 13x13 first moves  with smaller board sizes 1x1 through 9x9 which have all been solved by computer.  If X is the last column , X1 wins on all board sizes (1x1 through 9x9) as Maciej asserts for 13x13.  X2 wins on all boards  except 4x4. X3 loses on all boards except 6x6. X4 wins on all boards 6x6 through 9x9.  A1 of course loses on all boards 2x2 and up. A2 loses on all boards except 9x9 (!). A3 loses on all boards 4x4 through 9x9 except 6x6.  So Maciej's assertions for 13x13 have been (with only a few exceptions) proven to be true on board sizes up through 9x9.

I'm going to bump this thread with a picture:

This is from my own database of games. There's about 2600 games in there and I only used games between players with high ratings. The numbers are how many times an opening was played (I've only included openings with more than 20 games), and the percentage is the number of games won with that move. A move is counted as a win if either a) it isn't swapped and black wins, or b) it is swapped and white wins (games where a player times out are not included). In other words, this is how strong the move itself is, independent of swapping. A good opening should be near 50%, moves that have high percentages should be swapped, moves with lower percentages shouldn't.

This mostly agrees with Maciej Celuch's comments, except m10/a4, which appears to be a strong opening. a11 is a very bad opening, interestingly you can see that the equivalent move loses on smaller boards that have been solved by computers (7x7, 8x8, 9x9). a12 is too strong and should be swapped.

It's interesting also that b2 and c2 appear to be weaker than a2, even though I can't see how white could have a stronger position with the former. I think what happens is that b2/c2 appear more threatening, and the player making the next move plays on the same side of the board, while a2 appearing unthreatening results in more players responding with weaker moves in the opposite corner. If that's true then a2 should be considered a weak opening.

It's also odd how few players play openings along black's edges. Based on trends from solved smaller boards, I would expect the first row to all be losing moves (except m1). That means there should be some inbetween moves on the second or maybe third row that are close to being 50% openings which could be usable, but no one seems to play there.

Thoughts?