Finite Go variant: Loose

[luigi]

illluck, it doesn't matter if it's solved or not. A 25 komi for 5x5 Go would be balanced for all dan players, regardless of whether they know the solution or not. They would draw every time, whereas if two complete newbies play 5x5 Go, it's obvious that a 25 komi won't balance the game.

I disagree. A game being balanced doesn't mean a draw every time. A game being balanced means both players should draw with perfect play. 25 komi balances the game even for beginners, it's just unlikely that games would finish in a draw. If I play a 3 stone handicap game against a 10k, the game is balanced against me, but I'll still win every time. Balancing in the sense of komi just means that neither player should have an advantage at the beginning, not that the outcome is a foregone conclusion.

I think we're not understanding each other. Let's define:

a) "Fair komi" as the komi which ensures a draw with perfect play, and
b) "Balanced komi" as the komi which ensures that Black and White have the same chance of winning, provided that both players are equally skilled.

Of course, "fair komi" is always the same, since it implies perfect play. My hypothesis is that "balanced komi" varies with skill. Consider these two extreme cases on a 19x19 game:

a) Black and White always choose their moves at random.
b) Black and White are 9p players.
c) Black and White are perfect players.

In the first case, the advantage of playing first is virtually meaningless. Therefore, without komi, the winning chances for both players must be really close to equal. With a 6.5 komi, the average result will be really close to +6.5 for White (maybe +6, since Black will have made on average 0.5 more moves than White).

In the second case, 6.5 is the "balanced komi", but needn't be the "fair komi".

In the third case, the "balanced komi" is equal to the "fair komi", and it can be arbitrarily low or high. It can be the case that some incredibly sophisticated sequence of moves ensures a 32 point win for Black, for instance. We simply don't know.

[HermanHiddema]

a) Black and White always choose their moves at random.

In the first case, the advantage of playing first is virtually meaningless. Therefore, without komi, the winning chances for both players must be really close to equal. With a 6.5 komi, the average result will be really close to +6.5 for White (maybe +6, since Black will have made on average 0.5 more moves than White).

Have you tested this assertion?

[MarkSteere]

I'm just debating, btw. No ill will intended. I hold Go and Go players in high regard, the bgg dustup notwithstanding. That was just masterful debating. Yes, I goof on Go, but that's in the context of Go being a great game in the world. Don't take it personal.

Go got me started in game design (along with Reversi, a lesser game). Now, coming full circle, I've got a Go variant. Arghhh! A genetically mutated offspring.

I took the line "Redstone makes use of the pie rule" out of the rule sheet. I don't want to play the role of pie police if people are more comfortable with komi. I don't think the choice of balancing mechanism really matters in a game of hundreds of moves.

Pie wouldn't affect Go gameplay one iota other than rendering irrelevant "The Big Book of Opening Plays". If there were an iota, it would be a positive iota.

One thing to consider. Minus komi. You might still need komi to augment the pie rule. If the komi would be 7.5, if not for pie, you might need a komi of, for example, -1.5 with pie.

[jts]

Pie wouldn't affect Go gameplay one iota other than rendering irrelevant "The Big Book of Opening Plays". If there were an iota, it would be a positive iota.

What do you take the "Big Book of Opening Plays" to be like? There are a few players who take their prepared openings very seriously (there's an active thread right started by a player who's sad that they don't work), but it's not really that important for go, and most people who enjoy the game put vastly more effort into other areas.

I could go into more detail, but you mentioned elsewhere that you don't really play go, so it will probably be easier for you if you explain what you think the relevance of the Big Book is, and we take that as a starting point.

[luigi]

a) Black and White always choose their moves at random.

In the first case, the advantage of playing first is virtually meaningless. Therefore, without komi, the winning chances for both players must be really close to equal. With a 6.5 komi, the average result will be really close to +6.5 for White (maybe +6, since Black will have made on average 0.5 more moves than White).

Have you tested this assertion?

Now I have. I happen to have a rudimentary Go script handy which I wrote some time ago. I've just used it to run 10000 random 9x9 games with 0.5 komi and other 10000 with 6.5 komi. Here are the results:

0.5 komi, 9x9 board:

0 -> 58 / 114 = 0.50877192982456
1 -> 59 / 137 = 0.43065693430657
2 -> 46 / 131 = 0.35114503816794
3 -> 63 / 125 = 0.504
4 -> 55 / 111 = 0.4954954954955
5 -> 54 / 120 = 0.45
6 -> 57 / 123 = 0.46341463414634
7 -> 62 / 128 = 0.484375
8 -> 50 / 115 = 0.43478260869565
9 -> 61 / 121 = 0.50413223140496
10 -> 55 / 114 = 0.48245614035088
11 -> 55 / 111 = 0.4954954954955
12 -> 55 / 115 = 0.47826086956522
13 -> 64 / 112 = 0.57142857142857
14 -> 55 / 115 = 0.47826086956522
15 -> 54 / 102 = 0.52941176470588
16 -> 52 / 112 = 0.46428571428571
17 -> 56 / 116 = 0.48275862068966
18 -> 65 / 144 = 0.45138888888889
19 -> 61 / 131 = 0.46564885496183
20 -> 56 / 121 = 0.46280991735537
21 -> 61 / 114 = 0.53508771929825
22 -> 79 / 130 = 0.60769230769231
23 -> 64 / 135 = 0.47407407407407
24 -> 81 / 140 = 0.57857142857143
25 -> 65 / 131 = 0.49618320610687
26 -> 70 / 119 = 0.58823529411765
27 -> 61 / 123 = 0.49593495934959
28 -> 65 / 124 = 0.5241935483871
29 -> 56 / 113 = 0.49557522123894
30 -> 60 / 114 = 0.52631578947368
31 -> 62 / 121 = 0.51239669421488
32 -> 74 / 143 = 0.51748251748252
33 -> 60 / 120 = 0.5
34 -> 67 / 114 = 0.58771929824561
35 -> 51 / 105 = 0.48571428571429
36 -> 49 / 110 = 0.44545454545455
37 -> 73 / 136 = 0.53676470588235
38 -> 59 / 128 = 0.4609375
39 -> 71 / 124 = 0.57258064516129
40 -> 68 / 114 = 0.59649122807018
41 -> 63 / 114 = 0.55263157894737
42 -> 64 / 116 = 0.55172413793103
43 -> 72 / 133 = 0.54135338345865
44 -> 68 / 120 = 0.56666666666667
45 -> 69 / 124 = 0.55645161290323
46 -> 63 / 133 = 0.47368421052632
47 -> 67 / 142 = 0.47183098591549
48 -> 71 / 131 = 0.54198473282443
49 -> 66 / 124 = 0.53225806451613
50 -> 68 / 129 = 0.52713178294574
51 -> 60 / 131 = 0.45801526717557
52 -> 57 / 117 = 0.48717948717949
53 -> 64 / 116 = 0.55172413793103
54 -> 62 / 117 = 0.52991452991453
55 -> 57 / 131 = 0.43511450381679
56 -> 60 / 117 = 0.51282051282051
57 -> 67 / 127 = 0.52755905511811
58 -> 69 / 130 = 0.53076923076923
59 -> 74 / 134 = 0.55223880597015
60 -> 69 / 116 = 0.5948275862069
61 -> 86 / 155 = 0.55483870967742
62 -> 68 / 116 = 0.58620689655172
63 -> 61 / 126 = 0.48412698412698
64 -> 52 / 125 = 0.416
65 -> 81 / 143 = 0.56643356643357
66 -> 66 / 129 = 0.51162790697674
67 -> 53 / 111 = 0.47747747747748
68 -> 53 / 119 = 0.4453781512605
69 -> 61 / 102 = 0.59803921568627
70 -> 66 / 134 = 0.49253731343284
71 -> 67 / 115 = 0.58260869565217
72 -> 76 / 141 = 0.53900709219858
73 -> 49 / 112 = 0.4375
74 -> 50 / 110 = 0.45454545454545
75 -> 54 / 120 = 0.45
76 -> 66 / 126 = 0.52380952380952
77 -> 58 / 110 = 0.52727272727273
78 -> 52 / 109 = 0.47706422018349
79 -> 49 / 105 = 0.46666666666667
80 -> 44 / 99 = 0.44444444444444
81 -> 43 / 115 = 0.37391304347826

Total -> 5054 first player wins / 10000 games = 0.5054 winning probability
_____________________________________

6.5 komi, 9x9 board:

0 -> 48 / 124 = 0.38709677419355
1 -> 50 / 133 = 0.37593984962406
2 -> 44 / 123 = 0.35772357723577
3 -> 47 / 116 = 0.4051724137931
4 -> 56 / 125 = 0.448
5 -> 49 / 124 = 0.39516129032258
6 -> 49 / 131 = 0.37404580152672
7 -> 51 / 127 = 0.40157480314961
8 -> 43 / 127 = 0.33858267716535
9 -> 46 / 122 = 0.37704918032787
10 -> 39 / 115 = 0.33913043478261
11 -> 38 / 103 = 0.36893203883495
12 -> 48 / 125 = 0.384
13 -> 53 / 130 = 0.40769230769231
14 -> 39 / 112 = 0.34821428571429
15 -> 51 / 137 = 0.37226277372263
16 -> 39 / 112 = 0.34821428571429
17 -> 53 / 108 = 0.49074074074074
18 -> 51 / 119 = 0.42857142857143
19 -> 51 / 129 = 0.3953488372093
20 -> 37 / 92 = 0.40217391304348
21 -> 48 / 113 = 0.42477876106195
22 -> 58 / 122 = 0.47540983606557
23 -> 37 / 109 = 0.3394495412844
24 -> 46 / 132 = 0.34848484848485
25 -> 62 / 128 = 0.484375
26 -> 46 / 131 = 0.35114503816794
27 -> 51 / 141 = 0.36170212765957
28 -> 50 / 117 = 0.42735042735043
29 -> 46 / 129 = 0.35658914728682
30 -> 49 / 121 = 0.40495867768595
31 -> 45 / 122 = 0.36885245901639
32 -> 53 / 122 = 0.4344262295082
33 -> 56 / 119 = 0.47058823529412
34 -> 40 / 95 = 0.42105263157895
35 -> 43 / 109 = 0.39449541284404
36 -> 45 / 118 = 0.38135593220339
37 -> 42 / 116 = 0.36206896551724
38 -> 43 / 122 = 0.35245901639344
39 -> 49 / 129 = 0.37984496124031
40 -> 50 / 117 = 0.42735042735043
41 -> 39 / 113 = 0.34513274336283
42 -> 45 / 121 = 0.37190082644628
43 -> 58 / 122 = 0.47540983606557
44 -> 56 / 138 = 0.40579710144928
45 -> 54 / 115 = 0.4695652173913
46 -> 49 / 142 = 0.34507042253521
47 -> 36 / 100 = 0.36
48 -> 48 / 118 = 0.40677966101695
49 -> 55 / 128 = 0.4296875
50 -> 52 / 125 = 0.416
51 -> 54 / 135 = 0.4
52 -> 63 / 121 = 0.52066115702479
53 -> 53 / 130 = 0.40769230769231
54 -> 47 / 132 = 0.35606060606061
55 -> 42 / 109 = 0.38532110091743
56 -> 51 / 126 = 0.4047619047619
57 -> 42 / 91 = 0.46153846153846
58 -> 48 / 127 = 0.37795275590551
59 -> 46 / 108 = 0.42592592592593
60 -> 60 / 132 = 0.45454545454545
61 -> 49 / 127 = 0.38582677165354
62 -> 52 / 113 = 0.46017699115044
63 -> 46 / 115 = 0.4
64 -> 52 / 132 = 0.39393939393939
65 -> 43 / 120 = 0.35833333333333
66 -> 60 / 124 = 0.48387096774194
67 -> 38 / 116 = 0.32758620689655
68 -> 64 / 145 = 0.44137931034483
69 -> 57 / 119 = 0.47899159663866
70 -> 59 / 136 = 0.43382352941176
71 -> 49 / 137 = 0.35766423357664
72 -> 50 / 118 = 0.42372881355932
73 -> 46 / 131 = 0.35114503816794
74 -> 40 / 123 = 0.32520325203252
75 -> 64 / 140 = 0.45714285714286
76 -> 38 / 91 = 0.41758241758242
77 -> 35 / 124 = 0.28225806451613
78 -> 50 / 132 = 0.37878787878788
79 -> 47 / 141 = 0.33333333333333
80 -> 32 / 104 = 0.30769230769231
81 -> 41 / 133 = 0.30827067669173

Total -> 3951 first player wins / 10000 games = 0.3951 winning probability

The first column represents where the first stone was placed. 1 is a1, 81 is j9 and 0 is a pass. Not surprisingly (for me), 0.5 seems to be the ideal komi for random players, whereas 6.5 komi results in an unbalanced game.

(Rules used: area scoring, no suicide, positional superko.)

[luigi]

I took the line "Redstone makes use of the pie rule" out of the rule sheet. I don't want to play the role of pie police if people are more comfortable with komi.

Er... Mark, Redstone is an annihilation game. Komi isn't applicable.

One thing to consider. Minus komi. You might still need komi to augment the pie rule. If the komi would be 7.5, if not for pie, you might need a komi of, for example, -1.5 with pie.

Using the pie rule and a komi at the same time would be twice as inelegant as any one of these rules by itself. If the pie rule needs a komi, why not use just komi alone?

[luigi]

The 5x5 example does not generalize, because with the correct komi, black cannot win. As such, it has no implications for larger sizes.

Lets talk about 5x6 go instead. With perfect play, it is a 4 point win for black. Can you show that a 4 point komi is wrong for weaker players?

Same as before:

4 komi, 5x6 board:

10000 random games generated
3476 Black wins
305 ties
6219 White wins
0.36285 success degree for Black

0.5 komi, 5x6 board:

10000 random games generated
5141 Black wins
0 ties
4859 White wins
0.5141 success degree for Black

[illluck]

OK, thanks a lot for the interesting data. However, the data for players who do know how to play (randomness is a pretty huge assumption :p) is more relevant. The OGS data on that post I linked is somewhat dated, but does suggest that komi works reasonably well among actual players.

[HermanHiddema]

@Luigi

Very nice, can you tell me what your algorithm considers "random"? E.g. is "pass" considered a valid move that may be randomly selected?

EDIT: I see that it is (move 0).

So how does it decide when to count? When both players happen to pass in succession?

[topazg]

Also, this comparison isn't between weak Go players and strong players, it's between random play and perfect play, which is a _huge_ difference.

Even at 30k, learning that a single stone atari'd on the second line is dead comes almost immediately. By 20 minutes of playing, even a rank beginner will have a strategy and a plan, and be aware of enough things that the play is a long way from random. If for these players 0.5 komi is correct, that becomes really interesting in how komi scales with strength, but I suspect even by 20k the komi should be much closer to 6.5 than 0.5 - that's only my intuition speaking, I don't have data, and anecdotally I know most DDK players I've spoken to about komi consider it almost completely irrelevant to the game result.

I'm not sure how much useful information can be taken from random playout stats.

[MarkSteere]

I took the line "Redstone makes use of the pie rule" out of the rule sheet.

Oops, komi can't be used in Redstone because there's no score. Doh!

[Putting pie rule back in rule sheet.]

[luigi]

So how does it decide when to count? When both players happen to pass in succession?

That's right. It's completely random, even at that.

[luigi]

Also, this comparison isn't between weak Go players and strong players, it's between random play and perfect play, which is a _huge_ difference.

Even at 30k, learning that a single stone atari'd on the second line is dead comes almost immediately. By 20 minutes of playing, even a rank beginner will have a strategy and a plan, and be aware of enough things that the play is a long way from random. If for these players 0.5 komi is correct, that becomes really interesting in how komi scales with strength, but I suspect even by 20k the komi should be much closer to 6.5 than 0.5 - that's only my intuition speaking, I don't have data, and anecdotally I know most DDK players I've spoken to about komi consider it almost completely irrelevant to the game result.

I'm not sure how much useful information can be taken from random playout stats.

The only thing that is proved by my random games is that the balanced komi varies with skill. If the balanced komi is 0.5 for some players and 6.5 for some others, there must be a skill level for which the balanced komi is 1, 1.5, 2, 2.5, etc. However, it probably grows very quickly as soon as one leaves random play, and starts to grow very slowly very early in the skill progression, so I agree with you that for any two real players it will always be closer to 6.5 than 0.5.

My intuition is that balanced komi is a function of the logarithm of the player's skill. Maybe something like this:

[jts]

What you may be missing, luigi, is that a skilled player can play conservatively to protect his komi advantage. You can say that each of those skills (skill at using the first stone, skill at conservative play) would grow as a logarithm; since you've made it a monotonic function, that will be an arbitrary consequence of how you scale "skill level". But that doesn't mean that the correct komi grows as a logarithm.

[luigi]

What you may be missing, luigi, is that a skilled player can play conservatively to protect his komi advantage. You can say that each of those skills (skill at using the first stone, skill at conservative play) would grow as a logarithm; since you've made it a monotonic function, that will be an arbitrary consequence of how you scale "skill level". But that doesn't mean that the correct komi grows as a logarithm.

I'm not sure I understand what you're trying to say. Do you mean balanced komi grows even quicker with skill than a logarithmic function because of that? Could you elaborate on that?