Ku ami su, Hi o o; Suzuki, Kaiyu; Ma suzawa, Tomo umi
A icle
N- uple ne wo k sea ch in O hello using gene ic
algo i hms
Games
P o ided in Coope a ion wi h:
MDPI – Mul idisciplina y Digi al Publishing Ins i u e, Basel
Sugges ed Ci a ion: Ku ami su, Hi o o; Suzuki, Kaiyu; Ma suzawa, Tomo umi (2025) : N- uple ne wo k
sea ch in O hello using gene ic algo i hms, Games, ISSN 2073-4336, MDPI, Basel, Vol. 16, Iss. 1, pp.
1-11,
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Ci a ion: Ku ami su, H., Suzuki, K., &
Ma suzawa, T. (2025). N-Tuple
Ne wo k Sea ch in O hello Using
Gene ic Algo i hms. Games,16(1), 5.
h ps://doi.o g/10.3390/g16010005
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A icle
N-Tuple Ne wo k Sea ch in O hello Using Gene ic Algo i hms
Hi o o Ku ami su * , Kaiyu Suzuki and Tomo umi Ma suzawa
Depa men o In o ma ion Sciences, Tokyo Uni e si y o Science, Yamazaki, Chiba 278-8510, Japan;
[email p o ec ed] (K.S.); [email p o ec ed] (T.M.)
*Co espondence: hi [email p o ec ed]
Abs ac : As one o he s onges O hello agen s, Edax employs an n- uple ne wo k o
e alua e he boa d, wi h poin s o in e es ep esen ed as uples. Howe e , his ne wo k
main ains a cons an shape h oughou he game, whe eas he poin s o in e es in O hello
a y wi h espec o game’s p og ess. The p esen s udy was conduc ed o op imize he
shape o he n- uple ne wo k using a gene ic algo i hm o maximize inal sco e p edic ion
accu acy o a ce ain numbe o mo es. We selec ed shapes o 18-, 22-, 26-, 30-, 34-, 38-,
42-, and 46-mo e con igu a ions, and cons uc ed an agen ha app op ia ely shapes an
n- uple ne wo k depending on he p og ess o he game. Consequen ly, agen s using he
n- uple ne wo k de eloped in his s udy exhibi ed a winning a e o 75%. This me hod
is independen o game cha ac e is ics and can op imize he shape o la ge (o smalle )
N- uple ne wo ks.
Keywo ds: O hello; gene ic algo i hm; n- uple ne wo k
1. In oduc ion
The i s ime a compu e O hello agen achie ed a ic o y o e a op- anking human
playe was in 1997, and his was achie ed by Logis ello (Bu o,1997b).
O hello is a wo-playe abs ac s a egy boa d game in which playe s al e na e be ween
placing black o whi e disks on an 8
×
8 boa d. The objec i e is o ha e he majo i y o
one’s colo disks on he boa d when he game ends. A disk mus be placed on a squa e so
ha i sandwiches one o mo e o he opponen ’s disks be ween he newly placed disk and
ano he disk o playe ’s colo , in a ho izon al, e ical, o diagonal line. Any opponen ’s
disks ha become sandwiched as a esul o his mo e a e hen lipped o become he placing
playe ’s colo . When a playe canno make a alid mo e, he u n is passed o he opponen .
The game ends when nei he playe can make a mo e, ypically when he boa d is ull.
Logis ello, he s onges O hello agen , was de eloped o employ he n- uple ne wo k
as an e alua ion unc ion (Bu o,2003). The n- uple ne wo k ep esen s poin s o in e es
on he O hello boa d as uples. Logis ello uses an N-Tuple ne wo k shape ha was hand-
c a ed by an expe . The e ec i eness o he N-Tuple Ne wo k is highly dependen on i s
shape (K awiec & Szube ,2011). Fu he mo e, in Edax, he ne wo k main ains a cons an
shape h oughou he game. Howe e , he poin s o in e es on an O hello boa d change
wi h he p og ess o he game. Fo example, he co ne and edge a eas a e impo an in
O hello, because co ne disks a e ne e lipped and edge disks a e less likely o be lipped.
Howe e , s ones a e a ely placed in hese a eas ea ly in he game. The e o e, i is no un il
he middle o he game ha he co ne a eas ep esen poin s o in e es .
To sa is y his dynamic p ope y, we op imized he shape o he n- uple ne wo k o
maximize pe o mance o ce ain numbe s o mo es. We expec o imp o e he ne wo k’s
pe o mance by app op ia ely using he op imal shapes depending on he game’s p og ess.
Games 2025,16, 5 h ps://doi.o g/10.3390/g16010005
Games 2025,16, 5 2 o 11
2. N-Tuple Ne wo k
The n- uple ne wo k was p oposed o cha ac e ecogni ion by Bledsoe e al (Bledsoe
& B owning,1959). Subsequen ly, he n- uple ne wo k was applied o classi ica ion (Rohwe
& Mo ciniec,1996) and eg ession (Kolcz & Allinson,1996) asks. In O hello, he n- uple
ne wo k was i s used by Bu o in Logis ello (Bu o,1997a), and hen popula ized by Lu-
cas (Lucas,2008). I has been used ecen ly in games such as 2048 and Connec Fou (Konen
& Baghe i,2020;Thill e al.,2012).
An n- uple ne wo k comp ises
m
uples, each o which ep esen s a poin o in e es
on he boa d. The o e all ne wo k is ope a ed by sampling he inpu boa d a each uple,
whe e he i- h uple consis s o a sub egion o n squa es on boa d
ai⊂ {
1,
. . .
, 64
}
,
|ai|=ni
and weigh
wi
. Th oughou his pape , he e m “ uple shape” e e s o his sub egion
ai
,
and “ne wo k shape” e e s o i s se
a={a1
,
. . .
,
am}
. The weigh s
wi
ha e a one- o-one
co espondence wi h he possible s a es o he egion
ai
. Because he e a e h ee possible
s a es o a squa e in O hello—black s one, whi e s one, o emp y— he numbe o possible
s a es o a egion aiis 3ni; hus, wihas 3nielemen s.
Each uple ou pu s a weigh co esponding o he inpu boa d
x
, and he sum o he
ou pu s o each uple is he o al ne wo k ou pu (Equa ion (1)).
(x) =
m
∑
i=1
wi
ni
∑
j=1
3j−1x(aij)(1)
whe e
wi[k]
deno es he
k
- h elemen o
wi
, and
x(i)
deno es he s a e o he i- h squa e o
x
. The alue o
x(i)
is 0, 1, o 2 i he squa e has no s ones, a whi e s one, o a black s one,
espec i ely.
The ollowing is an example o he compu a ion o Equa ion (1) on he boa d
x
ep esen ed in Figu e 1. Fo simplici y, he ne wo k consis s o only one uple, whose
shape is
a1={
56, 57, 58, 59, 60, 61, 62, 63
}
, highligh ed in blue in Figu e 1. In his case,
x(a11)
,
x(a12)
,
. . .
,
x(a17)
a e 2, 1, 1, 2, 2, 1, 0, and he e na y no a ion o
∑ni
j=1
3
j−1x(a1j)
is
2112210, which is he conca ena ion o hese numbe s. The e o e, (x) = w1[2112210(3)]
Figu e 1. Example o indexing.
When symme ic sampling is used, he ne wo k ou pu is he sum o he eigh applica-
ions o (Equa ion (1)) o he alid e lec ions and o a ions o he boa d (Equa ion (2)).
F(x) =
4
∑
i=1
( o i(x)) +
4
∑
i=1
( o i(mi o (x))) (2)
whe e
o i(x)
is a unc ion ha o a es he boa d
x
by 90
i
deg ees, and
mi o (x)
is a
unc ion ha lips he boa d
x
ho izon ally. These ope a ions can be used o gene a e a o al
o eigh su aces. Figu e 2illus a es an example o his ope a ion.
Games 2025,16, 5 3 o 11
Figu e 2. Example o symme ic sampling.
Th oughou his pape , he e m “ne wo k” deno es an n- uple ne wo k ha inco po-
a es symme ic sampling.
In many s udies, n- uple ne wo k e e s o a Random Snake-shaped ne wo k (K awiec
& Szube ,2011;Lucas,2008;Szube e al.,2013;Thill e al.,2012). Howe e , his is no
he case o he ne wo k used in Edax (Figu e 3). The e o e, in his s udy, he shape o he
n- uple ne wo k is no limi ed o Random Snake-shaped.
Figu e 3. Ne wo k shape o Edax.
The ne wo k ou pu s an es ima e o he inal game esul —i.e., he s one di e ence—
o he inpu boa d. The e o e, he sea ch o weigh s is a eg ession p oblem ha can be
sol ed using he g adien descen me hod. In his s udy, he squa ed e o is used as he
loss unc ion.
3. Rela ed Resea ch
Wojciech e al. (Ja´skowski,2014) p oposed a me hod o analyzing he shapes and
weigh s o n- uple ne wo ks using an e olu iona y s a egy. They concluded ha a ne wo k
wi h 32 uples o size 2 pe o med bes .
Howe e , he s udy by Wojciech e al. did no accoun o la ge ne wo k shapes, such
as ha used in Edax, no did hey employ di e en ne wo ks wi h espec o he game’s
p og ess. The objec i e o his s udy was o cons uc a ne wo k ha ou pe o ms Edax.
To achie e his, we op imized he ne wo k’s shape o a size simila o ha o Edax o each
p og ession o he game, and cons uc ed a ne wo k using he esul s o his sea ch.
Games 2025,16, 5 4 o 11
4. Me hods
The p oposed me hod uses a gene ic algo i hm (GA). A GA is an op imiza ion me hod
inspi ed by he heo y o e olu ion, which posi s ha he i es indi iduals a e mo e
likely o su i e and ep oduce. In a GA, mul iple indi iduals, each ep esen ed by a
ch omosome-like da a s uc u e, a e gene a ed. These indi iduals a e e alua ed based on
a i ness unc ion ha quan i ies how well a pa icula solu ion sol es he gi en p oblem.
The algo i hm p oceeds by selec ing indi iduals wi h highe i ness alues, and hen
applying gene ic ope a o s such as c osso e and mu a ion o gene a e new o sp ing. This
p ocess is epea ed i e a i ely un il a sa is ac o y solu ion is ound.
The p oposed me hod uses a dis ibu ed GA (DGA) (Belding,1995) o de e mine
he op imal ne wo k shape o each mo ing boa d. A DGA is an algo i hm ha di ides
he popula ion in o mul iple islands and pe o ms he GA on each island independen ly.
The p oposed me hod uses
N
islands, whe e he
k
h island is op imized a he
k
h mo e.
Unlike in he con en ional DGA, each island is op imized using BRKGA (Gonçal es &
Resende,2011). Th oughou he sea ch, he ne wo k size is assumed o be cons an .
4.1. Gene and Decode
In BRKGA, a gene is ep esen ed by a ec o o eal numbe s a in e als o
[
0.0, 1.0
]
.
BRKGA equi es a de e minis ic algo i hm called a decode ha ans o ms inpu genes
in o solu ions o co esponding p oblems.
In he p oposed me hod, he shape o a uple is decoded om genes o he same size
as he sea ch egion. Each squa e o he sea ch egion is assigned an elemen o he gene
ia one- o-one co espondence and only
ni
elemen s a e selec ed om he squa e wi h
he smalles alue o cons uc
ai
. In his s udy, he sea ch a ea o uples was limi ed o
24 squa es in h ee ows om he edge o he boa d. Figu e 4illus a es an example o his
ope a ion. Because he ope a ion was pe o med o a numbe o uples, each indi idual
has a eal ec o o 24 m elemen s as i s gene.
Figu e 4. Decode Ope a ion (ni=8).
4.2. Fi ness
The i ness quan i ies he deg ee o op imali y o a solu ion in a GA. In his s udy,
i ness is de ined as he p edic ion accu acy o he Logis ello book skele on da ase .
The da ase is a eco d o ma ches gene a ed by Logis ello’s sel -play, ep esen ing ap-
p oxima ely 120,000 games. In his s udy, 80% o he da ase was used o measu e he le el
o i ness, wi h he emaining 20% used as es da a. When measu ing he le el o i ness
in a ce ain gene a ion, we andomly sampled 80% o he da a o i ness measu emen s
as aining da a, wi h he emaining 20% ese ed as alida ion da a. Subsequen ly, each
indi idual on he
k
h island was ained wi h he aining da a o each game co esponding
o he
k
h mo e a e he weigh s we e ini ialized o ze o. The loss was hen measu ed
Games 2025,16, 5 5 o 11
using alida ion da a co esponding o mo e
k
o each game. The loss was de ined as he
i ness o each indi idual.
4.3. Measu e o Pe o mance
To e alua e he p oposed me hod’s gene alizabili y a he h mo e, we measu ed i s
pe o mance agains he es da a. Fo a gi en ne wo k, upon ini ializing all weigh s o
ze o, we ained weigh s using he da a o he
h mo e o each game in he i ness es da a.
Subsequen ly, we measu ed he loss using he da a o he
h mo e o each game in he es
da a, e e ed o as he es sco e a mo e .
4.4. Mig a ion
Mig a ion is an ope a ion ha akes place a ixed gene a ion in e als in he DGA.
Du ing mig a ion, indi idual in o ma ion is exchanged be ween islands. In his s udy,
mig a ion was pe o med ollowing he gene a ion– ansi ion ope a ion.
Suppose ha islands a e a anged in an ascending o descending o de o
k
. When an
island mig a es, i eplaces copies o indi iduals om neighbo ing islands in a sequence
wi h he same numbe o indi iduals om i s own island. In his case, indi iduals om
neighbo ing islands a e selec ed in o de o i ness, wi h he indi iduals o be eplaced
andomly selec ed om he mu an popula ion in BRKGA. The popula ion size on each
island does no change be o e o a e his ope a ion. The numbe o indi iduals subjec ed
o his ope a ion is deno ed by a pa ame e known as he mig a ion size.
Because each edge island has only one neighbo , he numbe o indi iduals selec ed
o mig a ion om his island is equi alen o he mig a ion size. In con as , because each
non-edge island co esponds o wo neighbo s, he numbe o indi iduals om each o
hose neighbo s is hal he mig a ion size.
5. Expe imen s
Th ee expe imen s we e conduc ed in his s udy. In he i s expe imen , we op imized
he ne wo k shape o each ep esen a i e mo e using he p oposed me hod. To e i y he
e ec i eness o he op imized ne wo k shape, we e alua ed he p edic ion p ecision in he
da ase in he second expe imen , and he pe o mance in he game in he hi d expe imen .
5.1. Main Expe imen
This expe imen was conduc ed o de e mine he shape o he n- uple ne wo k using
he p oposed me hod. The size o he ne wo k is equi alen o ha used in Edax. The
numbe o islands was se o eigh , and he numbe o mo es o be op imized on each
island al e na ed acco ding o
k=
14
+
4
k
o
k=
1,
. . .
, 8, wi h each island e e ed o as
he island a
k
mo es.The se ings o DGA and BRKGA a e lis ed in Table 1, whe e he
BRKGA pa ame e s a e deno ed as (BRKGA).
Table 1. DGA and BRKGA se ings.
Pa ame e Name Value
Mig a ion In e al 25 Gen
Mig a ion Size 10
Numbe o Island 8
Island Size 100
pe,pm(BRKGA) 20, 20
ρe(BRKGA) 0.7
The ollowing expe imen al esul s we e ob ained: Figu es 5and 6depic he change
a es o he i ness and es sco es o he bes indi idual on each island in each gene a ion
Games 2025,16, 5 6 o 11
(mo ing a e age a an in e al o 250). The a e o change was based on he a e age o
0–50 gene a ions o each island. Fo
k
islands, he es sco e was measu ed a he
k
h mo e.
Fo islands wi h 18 and 22 mo es, he a e o change in bo h he i ness and es sco es
emained close o 1. In con as , es sco es o islands a 34–46 mo es con inued o dec ease
un il app oxima ely 2000 gene a ions; he es sco e o he island a 42 mo es con inued o
dec ease e en a e 4000 gene a ions.
The sea ch esul s o each island a e shown in Figu es A1–A8 in he Appendix A,
showing he bes - i indi idual in he las gene a ion o each island.
012302345ÿ00748ÿ9
72227201313423024
3!3"3 2
2
020
Figu e 5. Fi ness o each island.
012302345ÿ00780ÿ9
72227201313423024
83 3!382
2
020
Figu e 6. Tes Sco e o each island.
5.2. E alua ion o Tes Sco es
This expe imen was conduc ed o e alua e he esul s o Sec ion 5.1. We measu ed
he es sco es a each
k
o he sea ch esul s co esponding o each island in Sec ion 5.1
using he p oposed ne wo k and Edax ne wo ks (Table 2).
Table 2. Tes sco es o each ne wo k.
Mo es Ne wo k Name
18 22 26 30 34 38 42 46 Edax
18 28.11 28.36 28.08 28.05 28.02 28.17 28.06 28.24 28.81
22 26.06 25.88 25.65 25.65 25.81 26.05 26.21 26.54 27.31
26 24.49 23.82 23.54 23.50 23.55 24.04 24.56 25.01 26.07
30 22.67 22.02 21.33 21.02 21.15 21.64 22.24 22.91 24.88
34 21.29 20.41 19.45 19.09 18.87 19.00 19.75 20.24 23.77
38 21.99 20.96 20.22 19.90 19.16 18.63 19.03 19.79 23.17
42 22.36 22.27 21.80 21.40 20.64 19.18 18.55 19.11 22.41
46 23.06 23.18 23.29 23.10 22.48 20.49 19.04 19.01 22.06
Games 2025,16, 5 7 o 11
Fo 30–46 mo es, he ne wo k ha was op imized o ha numbe o mo es exhibi ed
he bes pe o mance. Howe e , o 18–26 mo es, he ne wo ks op imized a 30 o 34 mo es
exhibi ed he bes pe o mance.
5.3. E alua ion o Ma ch Pe o mance
Whe eas, in Sec ion 5.2, we e alua ed he pe o mance o agen s on he da ase , in his
expe imen , we e alua ed he pe o mance o he agen s in a game. Fi s , wo agen s we e
p epa ed, deno ed Agen s 1 and 2.
Agen 1 used he ne wo k shape ob ained om Sec ion 5.1, whe eas Agen 2 used
he same ne wo k shape as Edax (Table 3). In his able, “island a
n
mo e” e e s o he
esul s o he sea ch o ha island in Sec ion 5.1. Bo h agen s used independen weigh s
o each mo e.
Table 3. Ne wo k shape o each agen .
Mo es Ne wo k Shape
Agen 1 Agen 2
be o e 32 Island a 30 mo es
Same as Edax
33–36 Island a 34 mo es
37–40 Island a 38 mo es
41–44 Island a 42 mo es
a e 45 Island a 46 mo es
A e ini ializing he weigh s o he wo agen s o ze o, he ollowing p ocedu e was
ca ied ou :
1. A o al o 64 games we e played and he winning pe cen age was measu ed.
2. The weigh s o bo h agen s–ne wo ks we e ained using da a om he games in (1).
3. I he a ge numbe o games was no eached, (1) was epea ed.
O hello agen s such as Edax and Logis ello mo e acco ding o hei book opening a
ea ly s ages o he game. In o he wo ds, he e alua ion unc ion is a ely used ea ly on in
games, and i s pe o mance becomes impo an only mid-game. Acco dingly, we e alua ed
he pe o mance o he e alua ion unc ion a e he middle o he game. Speci ically,
he game was played om he ini ial s a e o he boa d a he 24 h mo e o a game andomly
selec ed om he da ase . The pa ame e s in he minimax sea ch o he game ee we e se
o 7 and 20 o he mid-game and end-game dep hs o sea ch, espec i ely.
The esul s o 300,000 games a e shown in Figu e 7, wi h he win a e o Agen 1
exhibi ing s abili y a app oxima ely 0.75.
012302345ÿ00748ÿ9
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3!3"3 2
2
020
Figu e 7. Winning Ra e.
Games 2025,16, 5 8 o 11
6. Discussion
Acco ding o he sea ch esul s o Sec ion 5.1, islands wi h a smalle
k
exhibi ed no
uples ocused on he co ne s, whe eas hose wi h a la ge
k
exhibi ed uples ocused on
he co ne s and edges. Howe e , his was no he case o he island in Mo e 18, which
con ained some uples ocused on he co ne s. In he da ase , he e we e e y ew games in
which s ones we e placed a he co ne s in Mo e 18. The e o e, he ne wo k is subop imal.
In addi ion, he es sco es in Sec ions 5.1 and 5.2 show ha islands wi h a smalle
k
a e less likely o p og ess du ing he sea ch and ice e sa. The e a e se e al possible
explana ions o hese esul s. One such explana ion is he sea ch ange o he uples, which
was se o h ee ows a he edge o he boa d. We belie e ha his ange was inapp op ia e
a ea ly s ages o he game, as in he case o Mo e 18. Ano he possible explana ion is
he di e si y o he da ase s. Because he da ase used in his s udy included games wi h
o e lapping de elopmen s du ing ea ly s ages o he game, we belie e ha islands wi h a
small klacked di e si y in he da ase , nega i ely a ec ing he sea ch esul s.
In a compa a i e expe imen be ween he p oposed ne wo k and Edax, he con igu a-
ions in Sec ions 5.2 and 5.3 ou pe o med Edax bo h on he da ase and in an ac ual game.
One o ou u u e asks will be o op imize mo es ea ly in he game, as in he case
o Mo es 18 and 22. This can be achie ed by changing he sea ch a ea o he uples,
using a da ase wi h a wide a ie y o games, o mig a ing o an a ea la ge han he
neighbo ing islands.
In his s udy, we compa ed he pe o mance when he ne wo k weigh s we e ained
om he ini ialized s a e o align wi h he expe imen al condi ions. Howe e , o demon-
s a e ha he p oposed ne wo k uly ou pe o ms he Edax ne wo k, i is necessa y o
compa e i s pe o mance wi h ha o a ained Edax ne wo k.
In his s udy, he ne wo k size was assumed o be cons an h oughou he sea ch.
Howe e , in he u u e, i may be possible o pe o m sea ches wi h a a iable ne wo k size.
An inc eased ne wo k size may imp o e pe o mance a he cos o inc easing he numbe
o weigh s and he compu a ional cos o in e ence. This ade-o mus be conside ed when
conduc ing a sea ch.
The me hod is independen o he game cha ac e is ics o O hello, and can op imize
he shape o la ge (o smalle ) n- uple ne wo ks. The e o e, we expec ha ou me hod
can be applied o games such as Connec -Fou and 2048, whe e n- uple ne wo ks ha e
p o en e ec i e. In such cases, i may be necessa y o edesign he i ness unc ion. This
pape de ined he i ness unc ion as he p edic ion accu acy on a da ase , which allowed
us o design he i ness unc ion o ma ch he pe o mance in ein o cemen lea ning,
as demons a ed in Expe imen Sec ion 5.3. This app oach, al hough compu a ionally mo e
expensi e, elimina es he need o p io knowledge in he o m o a da ase , allowing o i s
b oade applica ion o a wide ange o p oblems. In addi ion, when applied o la ge s a e
spaces, such as Go, he ne wo k size should be la ge . In o de o e i y i s applicabili y
and scalabili y, i needs o be e alua ed in a ious en i onmen s, no jus compa ed o a
single agen , as in his s udy.
7. Conclusions
In his s udy, we employed DGA o op imize he shape o he n- uple ne wo k wi h he
objec i e o maximizing i s pe o mance ega ding ep esen a i e mo es. The shapes o se -
e al ne wo ks ob ained h ough his sea ch we e expe imen ally e alua ed, wi h he esul s
indica ing ha he esul ing ne wo k shape is mo e accu a e han ha o he Edax ne wo k.
By adjus ing he use o ne wo k shapes wi h espec o he game’s p og ess, he p oposed
mechanism also ou pe o med Edax. We expec ha a ne wo k cons uc ed in his manne
can be used o gene a e agen s ha u he ou pe o m Edax. Ou me hod is expec ed o
