遊戲演算法在撲克牌遊戲大老二的應用與比較

୯ҥᆵ᡼εᏢႝᐒၗૻᏢଣၗૻπำᏢس

୯ҥᆵ᡼εᏢ

ၗૻπำᏢس

ᅺγፕЎ

Department of Computer Science and Information Engineering

College of Electrical Engineering & Computer Science

National Taiwan University

ᅺγፕЎ

ၯᔍᄽᆉݤӧኔլจၯᔍ

. εԴΒ΢ޑᔈҔᆶКၨ

໳ྍᡒ ኗ

97 2

master thesis

ၯᔍᄽᆉݤӧኔլจၯᔍ-εԴΒ΢ޑᔈҔᆶКၨ

The game theory’s application and comparison on pokergame Bigtwo

໳ྍᡒ

Yuan-Lin Huang

ࡰᏤ௲௤ȅֈػၰ ௲௤

Dr. Yuh-Dauh Lyuu

ύ๮҇୯ 98 ԃ 6 Д June 2009

i

୯ҥᆵ᡼εᏢᅺγᏢՏፕЎ α၂ہ঩཮ቩۓਜ 

ፕЎύЎᚒҞ ፕЎमЎᚒҞ

ҁፕЎ߯ɄɄɄ։ȐɄᏢဦɄȑӧ୯ҥᆵ᡼εᏢၗૻπำᏢسֹԋ ϐᅺγᏢՏፕЎǴܭ҇୯ɄɄԃɄɄДɄɄВ܍ΠӈԵ၂ہ঩ቩࢗ೯ၸ Ϸα၂Ϸ਱Ǵ੝Ԝ᛾ܴ

α၂ہ঩Ǻ

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! _________

ȐࡰᏤ௲௤ȑ

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

س Ь Һ

! ! ! ! ! ! ! ! ! ! ! ________

ii

ठᖴ 

གᖴ৪ᢌܹԴৣ໒ޑፐȨႝတჹֽ౛ፕȩ௴วගٮΑךፕЎޑБӛ གᖴࡰᏤԴৣֈػၰޑӕཀᆶЍ࡭ᡣךёаࣴز೭ঁᚒҞ

གᖴךޑৎΓޑᜅշόฅךό཮ٰۺࣴز܌

གᖴךޑܻ϶ǵӕᏢ๏ϒךႴᓰکЍ࡭ᡣךૈ୼໩ճֹԋ೭ፕЎ

iii

ύЎᄔा 

ၯᔍᄽᆉݤӧႝတΓπඵችύ΋ޔתᄽ๱ख़ाޑفՅϐ΋Ǵᙖҗ჋၂ᡣႝ

တ׳ԖඵችޑވၯᔍǴ௖૸ՖᒏΓᜪޑඵችǶၯᔍᄽᆉݤޑࣴزவԐයޑᚈΓ ၯᔍǵֹӄၗૻၯᔍǴۯ՜ډӭΓၯᔍǵόֹӄၗૻၯᔍǴԖӚԄӚኬޑᄽᆉ ݤǶҁЎ٬Ҕځύ΋٤ᄽᆉݤǴ঺Ҕӧኔլจၯᔍ-εԴΒǴКၨځᓬӍ٠уа

૸ፕǶ

ᜢᗖӷǺ

ၯᔍᄽᆉݤǴΓπඵችǴӭΓၯᔍǴmaxⁿᄽᆉݤǴኔլจၯᔍ-εԴΒ

iv

Abstract

Game algorithms play an important role of artificial intelligence, we try to find human intelligence by making computer play games more wisely. Researches of game algorithms from earlier two-player games and perfect information games extend to multi-player games and imperfect information games. There are many kinds of game algorithms. In this thesis, we apply some of these algorithms to the pokergame Bigtwo. We also analyze the various algorithms.

Key words:

Game algorithm, artificial intelligence, multi-player game, maxⁿ algorithm, pokergame Bigtwo

!

Ҟ!!ᒵ!

α၂ہ঩཮ቩۓਜ /////////////////////////////////////////////////// j!

ठᖴ ////////////////////////////////////////////////////////////// jj!

ύЎᄔा ///////////////////////////////////////////////////////// jjj!

Bctusbdu ////////////////////////////////////////////////////////// jw!

ಃ΋ക! ᙁϟ /////////////////////////////////////////////////////// 2!

ಃΒക! ၯᔍᄽᆉݤޑᐕў /////////////////////////////////////////// 3!

3/2!നԐයޑၯᔍࣴز /////////////////////////////////////////// 3!

3/3!ᚈΓֹӄၗૻၯᔍޑࣴز////////////////////////////////////// 5!

3/4!ӭΓၯᔍޑࣴز ///////////////////////////////////////////// 8!

ಃΟക! εԴΒޑވݤᆶೕ߾ //////////////////////////////////////// 22!

ಃѤക! ჴᡍ೛࿼ᆶኧᏵ //////////////////////////////////////////// 25!

ಃϖക! ่ፕ ////////////////////////////////////////////////////// 27!

ୖԵЎ᝘ ////////////////////////////////////////////////////////// 28!

!

!

კҞᒵ

კ 2!BMQIB!QSVOJOHǶ /////////////////////////////////////////////////// 6!

კ 3!CFUB!QSVOJOHǶ //////////////////////////////////////////////////// 7!

კ 4!NBY^OǶ ////////////////////////////////////////////////////////// 9!

კ 5!NBY^O!QSVOJOHǶ //////////////////////////////////////////////////// :!

კ 6!NBY^O!FYDFQUJPOǶ/////////////////////////////////////////////////// :!

!

!

!

!

߄Ҟᒵ

߄਱!2 ѤঁUZQF!BႝတޑКᖻ่݀Ƕ /////////////////////////////////// 25!

߄਱!3 ΟঁUZQF!Bک΋ঁUZQF!C)ཛྷ൨ుࡋ 7*ႝတޑКᖻ่݀Ƕ///////////// 26!

!

1

ಃ΋കʳ ᙁϟ!

Γπඵችӧႝတࣴزޑሦୱ΢΋ޔࢂ዗ߐޑࣴزϐ΋ǴᏢޣॺ೿ठΚܭ פ൨ӵՖᡣႝတՉ୏ள׳ຠ߈ΓᜪޑՉࣁޑБݤǴ׆ఈૈᡣႝတ׳Ԗඵችޑ ᔅշΓᜪǴԶၯᔍႝတ AI ޑࣴز൩ࢂځύ΋ঁࣴزޑБӛǶᙖҗӵՖᡣႝ

တވၯᔍވள቉্ǴکΓᜪ࣬КΞࢂӵՖǴٰ௖૸ΓᜪޑඵችᆶᐒᏔޑඵች ޑ౦ӕǶ

ၯᔍᄽᆉݤவനԐය brute-force પኪΚཛྷ൨Ǵډࡕٰޑ alpha-beta ׯؼ ཛྷ൨ݤаϷ΋٤ೕ߾ϯޑۓကǴ೿٬ளႝတӧވၯᔍ΢ຫٰຫமǴࣗԿঐᎯ ܭΓᜪϐ΢Ƕ1997 ԃՋࢩ්ᐒᏔΓ-ుᙔ(Deep Blue)Ѻ௳ΑШ්ࣚЦ׳ёᇥ

ࢂฦ΢Α΋ਔޑଯঢ়Ƕ

Զ೚ӭԐයၯᔍᄽᆉݤЬाࢂࡌҥӧᚈΓֹӄၗૻޑၯᔍ΢(two-player perfect information game)ǴՠΨԖຫٰຫӭۯ՜ޑᄽᆉݤ໒ۈ๱౳ܭӭΓၯᔍ (multi-player game) ǴаϷߚֹӄၗૻၯᔍ(imperfect information game)Ƕ

ҁЎ჋၂ஒᄒԿҞ߻ࣁЗр౜ޑ΋٤ᄽᆉݤǴ঺ҔӧӭΓޑኔլจၯᔍ- εԴΒ΢Ǵ٠КၨځᓬӍǴයఈૈפрׯ຾ᄽᆉݤޑБӛǶ

ӧಃΒക္Ǵ཮ϟಏ΋٤வа߻ډ౜ӧ܌р౜ၯᔍᄽᆉݤޑ࣬ᜢࣴزǶ

ಃΟകᇥܴኔլจၯᔍ-εԴΒޑၯᔍೕ߾ϷވݤǶಃѤകࢂჴᡍޑ೛࿼Ϸኧ ᏵǶಃϖകࢂ૸ፕϷ่ፕǶ

2

!

ಃΒകʳ !ၯᔍᄽᆉݤޑᐕў!

!

3/2!നԐයޑၯᔍࣴز!

΋૓ၯᔍᄽᆉݤޑᏢޣᇡࣁǴനԐ໒ۈബҥԖس಍ޑၯᔍᄽᆉݤޑۈ઒

ࢂহၭܭ 2:5: ԃ܌ว߄ޑЎക!\9^!Ƕځࣁ΋ጇࣴزӵՖ٬ҔำԄΠՋࢩ්

ޑЎകǴԜጇЎകό໻໻௖૸ΑӵՖΠՋࢩ්Ǵ׳ᗺрΑ܌Ԗၯᔍᄽᆉݤ܌

ᔈ၀ԵໆޑӚ໨ाનǶ!

२ӃǴ΋ঁ්ֽёа೏ಒϩࣁ೚ӭၗૻٰж߄౜ӧޑዬय़Ǵٯӵ්ηޑ Տ࿼ᆶঁኧǵ፺ډ໵η܈қηǵᄒԿҞ߻ዬय़ࣁЗޑ්ηوݤǾ฻฻Ƕჹܭ ٿঁވৎٰᇥǴ೭٤ၗૻࢂֹӄϦ໒ޑ)qfsgfdu!jogpsnbujpo*Ƕ!

ࡽฅၗ਑ࢂֹӄϦ໒ޑǴٗሶךॺ൩ૈଞჹჹЋΠ΋؁ޑ؂΋ᅿёૈ܄

଺ཛྷ൨Ƕऩࢂдو೭΋ЋǴ፺ډךॺਔךॺԖٗ٤؁ኧёаᒧ᏷Ǵௗ๱дΞ ёаӵՖوΠ΋؁Ǵനࡕפр΋ۓჹԾρԖճޑ؁ኧǶ౛ፕ΢೭ࢂᒤளډޑǴ ՠ٣ჴ΢ाीᆉр܌Ԗޑ؁ݤǴךॺ཮ሡाीᆉεऊԖ 21²³¹ᅿᡂϯǴջ٬؂

΋؁ךॺѝ޸΋డࣾ)21^.7t*Ǵךॺϝሡा 21^:1ԃωૈीᆉֹΠ΋؁၀ӵՖوǶ ќ΋ᅿБݤࢂᆉр܌Ԗёૈޑዬय़ǴᙖԜפрӵՖ຾Չډ܌ᒏȨയճȩޑዬ य़ǴԶ܌Ԗޑዬय़ᕴӅԖ 75"0!43")9"*³)3"*⁷!ᅿǴஒ߈ 21⁵⁴ᅿǴ܌а೭Ψࢂ

3

όϪჴሞޑǶ!

ࡽฅόёૈӈр܌ԖޑёૈǴٗሶךॺޑ฼ౣࢂǴᅰёૈޑவҞ߻ޑ௃

༈ύפрӳޑΠ΋؁ǶӧԜдΨගрΑ΋໨ߚதख़ाޑ೛ۓǴٗ൩ࢂቩֽڄ ኧ)fwbmvbujpo!gvodujpo*ǶቩֽڄኧޑфҔǴӧܭղᘐҞ߻ዬय़ٿБ༈Κޑ ᓬӍǴٯӵ්ηഭຫӭޑ΋Бϩኧᔈ၀Кၨଯǵ࡭Ԗό೏ߔᏲޑࣤӟ)܈ࠤ ൕǵЬ௲*ᔈ၀Ԗၨଯޑϩኧ฻฻Ƕӧ฼ౣ؁ݤཛྷ൨္ǴቩֽڄኧתᄽޑفՅ ൩ࢂǴ྽ךॺ،ۓو೭΋؁ډΠ΋ঁዬय़ਔǴךॺӵՖޕၰ೭΋؁ჹךॺٰ

ᇥࢂӳޑᗋࢂᚯޑǻԜਔ൩ा᎞ቩֽڄኧٰղᘐΑǶ!

ԖΑቩֽڄኧϐࡕǴ฼ౣޑཛྷ൨൩ᡂԋ΋ᅿ nby.njo ޑཛྷ൨Ƕ྽፺ډך ॺՉ୏ਔǴךॺा൨פӵՖന٫ϯךॺޑϩኧޑ؁ݤ)nby*Ǵϸϐ྽፺ډჹЋ Չ୏ਔǴд཮൨פӵՖന٫ϯдޑϩኧޑ؁ݤǴ࣬ჹٰᇥջࢂ٬ךॺϩኧന եޑ؁ݤ)njo*Ƕ!

ჹ܌Ԗёૈޑ؁ݤ଺ཛྷ൨ޑБԄǴךॺᆀϐࣁ Uzqf.BǶ٣ჴ΢೭ኬޑᄽ ᆉݤࢂߚதؒԖਏ౗Ǵ୺Չрٰޑႝတமࡋࢂߚதޑ১ޑǴӢࣁӧԖज़ޑਔ ໔ϣдࣁΑཛྷ൨܌Ԗёૈޑ؁ݤǴ٬ளд܌ૈ௖઩ޑుࡋࢂߚதభޑǴдѝ

ૈղᘐௗΠٰޑٿΟ؁ዬय़ֽ༈ޑᡂϯǶ!

΋ঁӳޑΓᜪވৎΨό཮ჹ܌Ԗёૈ؁ݤ଺ۯ՜ԵໆǴдѝ཮ࡘԵځύ

൳ᅿ؁ݤǶՠࣗሶࢂӳޑ؁ݤǴ΋ঁКၨᙁൂޑղձٯη൩ࢂǺёаӞ௞ჹ Б්ηޑ؁ݤᔈ၀ࢂӳޑǶଞჹ೭٤ӳޑ؁ݤ଺ཛྷ൨Ǵךॺᆀϐࣁ Uzqf.CǶ

4

ࡷ΋٤ٰ࣮Кၨख़ाޑ؁وΠѐǴёа࿯࣪ࡐӭਔ໔Ǵቚуਏ౗Ƕ!

നࡕǴךॺᔈ၀ᗉխ؂ԛӧӕኬޑዬय़ΠǴᡣႝတ଺ֹӄ࣬ӕޑ،ۓǶ Ӣࣁ൩നࡕٰ่݀ᇥǴ೭΋؁҂Ѹࢂന҅ዴޑᒧ᏷ǴࣗԿёૈࢂᏤठѨ௳ޑ চӢǶךॺёаቚу΋٤ᒿᐒᡂኧǴ٬ႝတӧ଺،ۓਔό཮؂ԛཛྷ൨ޑ໩ׇ

کᒧ᏷ޑБԄ೿࣬ӕǶ!

!

3/3!ᚈΓֹӄၗૻၯᔍޑࣴز!

ԖΑ୷ҁޑس಍ࢎᄬϐࡕǴࡕٰࣴزၯᔍޑᏢޣॺǴ೿ӧགྷᒤݤ٬ำԄ ޑਏૈගଯǶεठ΢ޑࣴزБӛࢂǺ٬Ҕ׳ᆒዴޑቩֽڄኧǴ٬Ҕ׳זೲཛྷ ൨ޑᄽᆉݤǶځύԖঁനݱҔǴ൳ЯѸഢޑ΋ঁᄽᆉݤ൩ࢂ bmqib.cfub!

qsvojoh!\6^Ƕ!

Bmqib.cfub!qsvojoh! ࢂ΋ঁ୷ܭ njo.nby ཛྷ൨ۯ՜рٰуזཛྷ൨ਏ౗

ޑ΋ঁᄽᆉݤǴځཷाࢂᅰໆ෧Ͽሡाཛྷ൨ޑ࿯ᗺኧҞǴჹܭό཮ቹៜډཛྷ ൨่݀ޑ࿯ᗺ൩Ӄ୊௞όѐ଺ཛྷ൨ǶѬЬाϩԋٿᅿ୊ຊݤ;!bmqib ୊ຊ!'!

cfub!୊ຊ!Ƕ!

Bmqib ୊ຊǺౢғܭ nby!opefǴଷ೛ӧ nby!opef ޑಃ΋ঁϩЍཛྷ൨ֹࡕ ளډΑ΋ঁ wbmvfǴௗ๱ᇆ൨ಃΒঁϩЍǴ߾ಃ΋ঁϩЍޑ wbmvf ёа׎ԋ΋

ঁ଺ࣁղձޑ cpvoeǶ྽ϩЍ 3/2 ໺ӣ΋ঁॶɦcpvoeǴٗሶௗΠٰϩЍ 3/3 ϷϐࡕޑϩЍ൩ؒѸा଺ཛྷ൨ΑǴӢࣁϩЍ 3 ޑॶԿϿ཮ɦcpvoe!)njo!

5

opef*ǶჹϩЍ 2 ٰᇥǴѬޑॶԿϿ཮ɧcpvoeǴ܌а 3/3 аࡕཛྷ൨рٰޑॶ

೿ࢂؒԖཀကޑǶ၁ـკ΋Ƕ)қՅ࿯ᗺж߄ nby!opefǴ໵Յ࿯ᗺж߄ njo!

opef Ƕ*!

კ 2!bmqib!qsvojohǶ!

Cfub ୊ຊǺౢғܭ njo!opefǴଷ೛ӧ njo!opef ޑಃ΋ঁϩЍཛྷ൨ֹࡕ ளډΑ΋ঁ wbmvfǴௗ๱ᇆ൨ಃΒঁϩЍǴ߾ಃ΋ঁϩЍޑ wbmvf ёа׎ԋ΋

ঁ଺ࣁղձޑ cpvoeǶ྽ϩЍ 2/3/2 ໺ӣ΋ঁॶɧcpvoeǴٗሶௗΠٰϩЍ 2/3/3 ϷϐࡕޑϩЍ൩ؒѸा଺ཛྷ൨ΑǴӢࣁϩЍ 2/3 ޑॶԿϿ཮ɧcpvoe!

)nby!opef*ǶჹϩЍ 2 ٰᇥǴѬޑॶԿϿ཮ɦcpvoeǴ܌а 2/3/3 аࡕཛྷ൨р

ٰޑॶ೿ࢂؒԖཀကޑǶ၁ـკΒǶ!

6

კ 3!cfub!qsvojohǶ!

!

ӧᏢ཮ bmqib.cfub!qsvojoh ࡕǴךॺ໒ۈࣴزӵՖᡣ΋ස hbnf!usff ܌ ሡཛྷ൨ޑ࿯ᗺኧ෧ډനϿǴΨ൩ࢂૈ୼଺നӭޑ୊ຊǶਥᏵࣴزךॺว౜Ǵ ऩࢂ΋සᐋѬޑ௨ӈ୼ӳޑ၉ǴѬ൩ёа଺നӭޑ୊ຊǶࣗሶࢂ΋ঁӳޑ௨ ӈǻ൩ࢂᅰໆᡣ཮ԖКၨӳޑ่݀ޑ࿯ᗺ଺ࣁϩЍύޑಃ΋ঁ࿯ᗺǶךॺவ

΢य़ޑۓကύёа࣮рǴऩࢂӧಃ΋ঁ࿯ᗺύךॺ൩פډ΋ঁࡐӳޑॶǴٗ

ӧཛྷ൨ځࡕޑϩЍਔǴךॺதத൩ёа୊௞΋٤όѸाཛྷ൨ޑ࿯ᗺǶܭࢂǴ ӵՖ଺΋ঁӳޑ௨ӈޑཛྷ൨ᐋ)hppe!npwf!psefsjoh*Ǵ൩ԋࣁ۳ࡕࣴزޑख़ ᗺϐ΋Ƕ!

ځჴ೭ၟךॺѳதΠ්܈ѺจਔޑࡘԵኳԄࢂߚதᜪ՟ޑǴ೯தךॺӧ ࡘԵΠ΋؁၀ӵՖوਔǴ΋ۓ཮Ӄࡷ΋٤࣮ଆٰКၨԖճёკޑ؁ݤѐགྷΠ

΋؁ϐࡕޑֽ༈ǶٯӵΠΑ೭΋؁ёа࠶ુჹБޑ ljohǴ൩཮ᓬӃԵቾǶԶ

7

Кၨόख़ाޑ؁ݤǴ۳۳ᘍӧനࡕωѐࡘԵځёૈ܄ǶٯӵوΑ΋؁ჹ්ֽ

คᜢᆙाޑγծǶ!

ନΑӵՖ෧Ͽ࿯ᗺቚуཛྷ൨ਏ౗аѦǴΨԖ٤Γ஑ݙܭόӕБӛޑࣴز

΢ǴႽࢂჹЋኳಔ)pqqpofou!npefm*\3^Ƕځ௖૸ޑࢂӵՖਥᏵჹЋόӕǴࡌ ҥ׳Ԗਏޑ฼ౣཛྷ൨ǶӳКᇥԖ٤ΓΠຝ්ਔࡐ཮ҔଭǴٗሶΨ೚ӵՖӞ௞

дޑଭ཮ࢂКၨख़ाޑ฼ౣǶԖᜢ pqqpofou!npefm ޑࡌҥΨԖ೚ӭϩЍǴம ϯᏢಞݤ)sfjogpsdfnfou!mfbsojoh*\21^൩ࢂځύ΋ᅿǴЬा೿ࢂਥᏵόଶ ޑ usbjojoh ٰፓ୏ϩኧǴ٬฼ౣന٫ϯǶ!

߈ٰᗋԖ΋ঁΜϩ჻ཥޑᄽᆉݤǴћ଺ npouf.dbsmp! \2^ǶԜᄽᆉݤନ ΑёаᔈҔӧၯᔍ฼ౣཛྷ൨΢ǴΨёаቶݱӦၮҔӧځдٯӵߎᑼीᆉ΢

य़ǴځЬᡏεठ΢ࢂճҔεໆኳᔕኧᏵ଺಍ीǴᙖҗεኧ౛ፕǴפډ΋ঁᖿ ߈ܭന٫ှޑှǶ঺Ҕӧၯᔍᐋ΢ǴѬԖ΋ঁֹӄόӕܭহၭޑӦБǴ൩ࢂ

ѬόሡाቩֽڄኧǶѬᐋ΢࿯ᗺޑሽॶࢂ࿶җኳᔕډಖᗺਔǴ่࣮݀ࢂគ܈

ᒡ)܈΋ঁॶ*Ǵฅࡕӆஒ่݀۳ӣ໺ډ؂΋ঁ࿶ၸޑ࿯ᗺ଺ಕᑈǶ!

Զᒿ๱ႝတೱ่ೲࡋᡂזǴаϷӭਡЈᅌᅌදϷǴѳՉᄽᆉݤΨ໒ۈ೏

ᔈҔ่ӝډځдᄽᆉݤ္\4^Ǵ೿ࢂࣁΑૈ׳ගϲำԄीᆉޑೲࡋǶ!

!

3/4!ӭΓၯᔍޑࣴز!

ନΑᚈΓၯᔍѦǴΨຫٰຫӭΓ໒ۈ׫ΕࣴزӭΓၯᔍޑሦୱǴӭΓၯ

8

ᔍԖ೚ӭόӕܭᚈΓၯᔍޑࡷᏯ\:^Ǵคݤֹӄ঺ҔᚈΓၯᔍޑ࿶ᡍӧځ΢Ƕ!

२ӃӀࢂnby.njo!tfbsdiǴ൩όࢂёаޔௗ঺ҔޑᄽᆉݤǴӢࣁԜਔވ ৎς࿶ᡂԋ 4 Γ 5 ΓǶӢԜԖΓගрΑnby^o!bmhpsjuin\7^Ǵځཷۺၟnby.njo

࣬྽ᜪ՟Ƕ!

२ӃڗжΑচҁ࿯ᗺ΢ѝԖൂ΋ޑॶǴ؂ঁ࿯ᗺ΢౜ӧׯԋԖ΋ঁ

O.uvqmf ॶޑତӈǴಃ j.uvqmf ޑॶ൩ж߄๱ވৎ j ޑϩኧǶਥᏵ nby ࿯ᗺޑ চ߾Ǵӧވৎ 2 ޑ࿯ᗺ΢Ǵჹځη࿯ᗺ଺ཛྷ൨ਔǴ൩ࡷᒧ٬ވৎ˞Ԗനଯϩ ኧޑ࿯ᗺ)uvqmf.2 ϩኧനଯޑ*Ƕځд࿯ᗺ߾٩Ԝᜪ௢ǶӵკΟǶ!

კ 4!nby^oǶ!

Nby^o!ΨԖ܌ᒏޑbmqib.cfub୊ຊݤǴచҹࢂ୷ܭuvqmfॶޑᕴک໪ڰۓ )΢ज़*аϷځॶԿϿाεܭ 1)Πज़*Ƕٗሶଷ೛࿯ᗺbޑuvqmf.2!ޑॶу΢࿯

ᗺgޑuvqmf.3 ޑॶς࿶εܭᕴکǴٗ࿯ᗺgޑځᎩη࿯ᗺࢂёа۹ౣό଺ཛྷ൨ ޑǶ၁ـკѤǶ!

9

კ 5!nby^o!qsvojohǶ!

ՠࢂ bmqib.cfub ډΑӭΓၯᔍΨࢂԖٯѦޑǴځচӢԖΒǴ΋ঁࢂջ٬

೏୊ຊ௞ޑ࿯ᗺϝฅԖёૈቹៜډ᏾සၯᔍᐋޑॶǹځΒӧܭѳᑽᗺڗ௭ޑ όӕǴ཮೷ԋ᏾ঁၯᔍᐋޑόӕǶځ΋Ǵ၁ـკϖǴऩਥᏵ bmqib.cfub ୊ຊǴ g ᔈ၀ࢂा೏୊௞ޑǴՠࢂ٩Ᏽ g ޑॶޑόӕǴ᏾සᐋޑ่݀཮όӕǶ!

კ 6!nby^o!fydfqujpoǶ!

ځΒǴ܌ᒏޑѳᑽᗺࢂࡰ྽ࢌ΋࿯ᗺӛΠཛྷ൨ځη࿯ᗺਔǴၶډٿঁ)܈

а΢*η࿯ᗺӕϩޑ௃ݩǶӧᚈΓၯᔍਔǴӢࣁᕴک࣬ӕΞѝԖٿঁވৎǴ܌

10

аӧځύ΋ঁވৎޑϩኧ࣬ӕޑ௃ݩΠǴќ΋ঁވৎޑϩኧΨ཮࣬ӕǶՠډ ΑӭΓၯᔍǴ΋ঁވৎϩኧ࣬ӕǴځᎩޑވৎϩኧࠅό΋ۓ࣬ӕǶଷ೛ވৎ 4 ޑ࿯ᗺ b ޑη࿯ᗺځॶϩձࢂ)4-2-7*ک)5-1-7*Ǵჹܭ၀࿯ᗺǴٿঁޑॶ೿

ӕኬ٬ވৎ 4 Ԗ 7 ϩǴՠჹܭވৎ 2 کވৎ 3 ٰᇥ൩ԖৡΑǶ܌аӧѳᑽᗺ ӵՖࡷᒧ࿯ᗺΨࢂߚதሡाࡘԵޑǶ!

ନΑཛྷ൨ᄽᆉݤޑόӕǴӭΓၯᔍᗋԖ΋ঁதр౜ԶѸ໪ԵቾޑୢᚒǴ ൩ࢂόֹӄၗૻ)jnqfsgfdu!jogpsnbujpo*Ƕ΋૓ӭΓၯᔍ္Ǵވৎѝૈޕၰ Ծρ܌ᏱԖޑจǴаϷᘚय़΢рၸޑจǴՠࢂჹЋᏱԖࣗሶจࢂคݤளޕޑǴ

ٗሶ܌ᒏޑཛྷ൨ᄽᆉݤǴ༈ѸΞѸ໪଺΋٤࣬ჹᔈޑ׳ׯǶ٬Ҕ Npouf!Dbsmp

଺ኳᔕۯ՜௢ෳჹЋจಔࢂঁᗋόᒱޑБݤ\5^Ƕ!

11

ಃΟകʳ εԴΒޑވݤᆶೕ߾!

ҁകஒϟಏӭΓኔլจၯᔍ.εԴΒޑވݤǶ!

२Ӄވৎ೯தࢂ 3.5 ΓǴஒ΋ୋኔլจ 63 ஭จѳ֡ϩଛ๏؂ঁΓǴջё

ྗഢ໒ۈၯᔍǶ!

ၯᔍޑވݤ;!

२Ӄ໒ֽҗ࡭Ԗఘ޸ 4 ޑވৎॄೢрจǴдёаவЋύѺрൂ஭܈ҺՖ จಔǴՠ΋ۓளх֖ఘ޸ 4ǶӧдϐࡕޑވৎǴѸ໪ၟ๱рӕኬจࠠ٠ЪКд ޑจεޑจಔǶऩࢂдคݤрКҞ߻рจεޑจಔǴ܈ࢂόགྷрจǴ߾ qbtt ඤΠ΋ঁވৎǶऩࢂ΋ঁΓрจࡕǴځдΟৎ)ଷ೛ࢂѤΓ*ऩࢂ೿ qbttǴ߾

дёа،ۓௗ๱ाрՖᅿᜪࠠޑจಔǴவдЋύѺр၀จಔǴԶځдΓѸ໪

ၟ๱рӕኬจࠠޑจǶӵԜ٩ׇ፺ࢬрจǴޔډԖΓஒЋύޑจӄ೽Ѻрѐ ջࣁയճǶ!

จޑελ;!!

޸Յࣁ໵ਲɧआЈɧБ༧ɧఘ޸!

ᗺኧ!3ɧBɧLɧRɧǾɧ4!!!

ၯᔍޑจࠠ;!

ϩࣁൂ஭ǵ΋ჹǵϖଫޑ੝ਸจࠠ!

12

x ൂ஭ȐTjohmfȑǺҺՖ΋ଫЋύޑจǴрจਔѸ໪εܭ΢ৎрޑ΋ଫจǶ ٯӵǴB КBǴB ΞεܭLǵR ฻ǴаԜᜪ௢Ƕ!!

x ΋ჹȐQbjsȑǺҺՖٿଫӕኧӷ܈ӷ҆ޑจǴрจਔၟൂଫ΋ኬǴѸ໪

εܭ΢ৎрޑจǶٯӵBB ࢂεܭLLǶ྽ᗺኧ࣬ӕਔǴКၨၨεޑ޸ՅǶ!!

ϖ஭จಔӝ-!ભձҗեԿଯϩձࣁǺ!

x ໩ηȐTusbjhiuȑǺҺՖϖଫจೱុ໩ׇՠόӕ޸Ǵࢂϖᅿύനեભձ ޑǶԖӭᅿीᆉελޑБݤǴѠ᡼தـޑԖǺB.3.4.5.6 ྽բനλǴനεޑ߾

ࢂ 3.4.5.6.7Ƕόёр K.R.L.B.3ǵR.L.B.3.4ǵL.B.3.4.5Ƕ!!

x မᝳȐGvmm!IpvtfȑǺΟచу΋ჹǶаΟచ௨ελǴȨ΋ჹȩჹܭ฻ભ

ࢂคቹៜޑǴٯӵ L.L.L.3.3 ک L.L.L.4.4 ӧ฻ભ΢ࢂ๊ჹ΋ኬޑǴόၸԜ௃

ݩόёૈр౜ȐӢࣁจύό཮Ԗ 7 ஭࣬ӕޑȑǶ!!

x ៓ЍȐGpvs!pg!b!LjoeȑǺѤଫኧӷ΋ኬޑӆуҺՖ΋ଫจǶၟမᝳ΋

ኬǴಃϖଫจჹ฻ભࢂడคբҔޑǶඤѡ၉ᇥǴ5.5.5.5.4 ࢂεܭ 4.4.4.4.L ޑǶ!

x ӕ޸໩ȐTusbjhiu!GmvtiȑǺϖᅿύനமεޑಔӝǶѬ่ӝΑӕ޸ک໩

ޑ੝܄ǴΨ൩ࢂᇥϖଫೱុޑจǴԶϖଫ೿ࢂӕ΋ᅿ޸Ƕനமޑӕ޸໩ࢂ!

3.4.5.6.7Ƕ!!

13

x ΋చᓪ!Ψ൩ࢂЋύ 24 ஭จֹӄؒԖख़ፄǶόၸԜᅿಔӝᐒ౗ߚதեǶ!!

рจਔǴѸ໪٩ྣ΢ৎѺрޑจࠠрӕኬᜪࠠޑจǴൂ஭൩ൂ஭Ǵ΋ჹ൩

΋ჹǴ໩η൩໩ηǴမᝳ൩မᝳǶ୤៓Ѝǵӕ޸໩ǵ΋చᓪᆀϐȨ܁ނȩǴӧ ҺՖਔংࣣёрจǶऩࢂ΢ৎрΑ៓Ѝǵӕ޸໩Ǵ߾գѝૈрКдεޑ៓Ѝ܈

ӕ޸໩)qt/ӕ޸໩ёаᓸ៓Ѝ*Ǵ΋చᓪ߾၀ވৎς࿶рֹจޔௗയճǶ!

!

ीϩБԄ;! !

྽ԖΓрֹจਔǴ߾ځᎩΟৎ؂Γ؂Ԗ΋஭Ћจᆉ 2 ϩǴᕴکуډគޑވ ৎޑᑈϩ္Ƕٯ;ވৎ΋рֹจǴވৎΒᗋԖ 4 ஭จǵވৎΟᗋԖ 7 ஭จǵވ ৎѤᗋԖ 3 ஭จǴ߾ވৎ΋ள 22 ϩǵވৎΒ! .4 ϩǵވৎΟ! .7 ϩǵވৎѤ .3 ϩǶ!

14

ಃѤകʳ ჴᡍ೛࿼ᆶኧᏵ!

ௗ๱ᇥܴჴᡍޑ೛࿼ᆶኧᏵ!

ᜢܭႝတޑΓπඵችǴӧԜ೛࿼ΑٿᅿႝတǴϩࣁ Uzqf!B ک Uzqf!CǶ!

Uzqf!BǴࢂᙁൂޑೕ߾ϯႝတǴځрจޑೕ߾൩ࢂԖจૈр൩рǴόԵቾ ߥ੮Ћύޑεจ܈จࠠǶᕴϐ൩ࢂό؃គจǴѝ؃עจޑኧໆ෧ԿനϿǶԜᅿ

ႝတஒ଺ࣁکќ΋ᅿႝတ࣬Кޑ୷ྗǶ!

Uzqf!CǴࢂ٬Ҕnby^oᄽᆉݤޑႝတǶӧԜךॺଷ೛ךॺޕၰ܌ԖჹЋޑจǴ ӵԜ΋ٰךॺωૈ٬Ҕཛྷ൨ղᘐ҂ٰޑֽ༈وӛԶόࢂ᎞ଷ೛Ƕࡽฅࢂཛྷ൨ٗ

྽ฅΨሡाቩֽڄኧٰղᘐ྽߻ֽ༈ޑϩኧǴεठ΢ਥᏵ܌ഭЋจኧҞکεจ ኧҞک੝ਸจٰࠠीᆉᕴϩǶԜѦሡा΋ঁനεుࡋޑୖኧǴځཀကࢂࡰ۳Π ཛྷ൨ډಃoԛވৎрจ)qbttΨᆉрจ*Ǵ൩ଶЗཛྷ൨ӣ໺ϩኧǶ!

จֽ٬Ҕ΋ί൑࣬ӕ໶ኧౢғޑ࣬ӕจֽǶ!

߄΋ࢂѤঁႝတࣣ٬Ҕ Uzqf!B ޑКᖻޑ่݀!

߄਱!2 Ѥঁ Uzqf!B ႝတޑКᖻ่݀Ƕ!

! ႝတ΋! ႝတΒ! ႝတΟ! ႝတѤ!

ϩኧ! .455! 33! 357! .235!

യ൑! 352! 361! 371! 35:!

!

15

җ߄΋ύёа࣮рѤঁႝတ۶Ԝയ౗ࢂৡόӭޑǴ߄Ңയॄᔈ၀ֹӄࢂ᎞

ЋจޑӳᚯԶۓǶ!

߄ΒࢂஒಃѤৎႝတ٬ҔUzqf!C!ǴځᎩႝတ٬ҔUzqf!BޑКᖻޑ่݀Ƕ)ࡕ य़ࡴဦࡕޑኧӷ߄Ң٬Ҕnby^o!ਔཛྷ൨ޑനεుࡋǶ*!

!

߄਱!3 Οঁ Uzqf!B ک΋ঁ Uzqf!C)ཛྷ൨ుࡋ 7*ႝတޑКᖻ่݀Ƕ!

! ႝတ΋! ႝတΒ! ႝတΟ! ႝတѤ)7*!

ϩኧ! .:65! .319! .61! 2323!

യ൑! 324! 339! 362! 419!

!

җ߄ύёа࣮рǴӧӕኬޑ΋ί൑ύǴႝတѤޑയ౗ၟϩኧࣣε൯ගϲǴ ёளޕUzqf!Cޑႝတᔈ၀КUzqf!Bޑமࡋଯ೚ӭǶՠऩाӢԜղۓnby^oᄽᆉݤ ൩ࢂКද೯٬Ҕڰۓೕ߾ޑႝတٰளமǴΨقϐၸԐǴӢࣁךॺځჴᗋёаӧ

ೕ߾ϯႝတ΢ӭуΕ೚ӭೕ߾ٰ٬ளѬᡂமǴѝࢂ཮٬ำԄ׳ፄᚇᕷᅥᜤа౛

ှǴӧፓ᏾ׯؼ΢ΨКnby^oٰள֚ᜤ٤Ƕ!

ќѦ଺ࣁჴᡍኧᏵޑКၨǴךॺම࿶עႝတѤ Uzqf!C ޑుࡋፓଯ܈ፓեǴ ՠว౜่݀ؒԖׯᡂǴᔈ၀ࢂቩֽڄኧ೛࿼ਔ໻Եቾډߥ੮εจϷจࠠǴؒԖ Եቾډрจਔऩࢂ೏ځдވৎᇂၸޑࡕ݀Ǵ܌а߄Ңځдވৎޑрจόϼ཮ቹ ៜԾρޑрจǶӵԜ΋ٰǴ՟ЯΨ൩ѨѐΑཛྷ൨ځдވৎрจޑཀကǶ

16

!

ಃϖകʳ !่ፕ!

ҁԛჴᡍύǴךॺว౜ѝाᙁൂޑ೛࿼ቩֽڄኧჹ܌Ԗёарޑจ଺ϩኧ ޑཛྷ൨ٰ،ۓ฼ౣǴځਏ݀ջёКද೯ೕ߾ϯޑႝတޑமࡋӳ೚ӭǶ!

Կܭӵ݀ा଺׳ుቫޑཛྷ൨ǴхࡴղᘐჹЋ཮р٤ࣗሶจǴٰ،ۓԾρा

рࣗሶจǴ߾ቩֽڄኧ൩཮ᡂளΜϩፄᚇǴሡा଺׳ፄᚇޑղᘐǶԜѦ࿯ᗺޑ ᕭ෧Ψஒ׳уख़ाǴӢࣁཛྷ൨ளຫుǴ࿯ᗺޑᘉቚஒ฻КભኧޑቚуǴ཮٬ཛྷ ൨ਔ໔ᛈቚǶ!

҂ٰޑࣴزନΑёа׳ుΕޑ௖૸׳ଯભፄᚇޑೕ߾ϯႝတBJکnby^oᄽ ᆉݤޑႝတBJϐѦǴΨёа၂๱уΕჹЋኳಔǴᙖҗமϯᏢಞ଺р׳ӳޑղ ᘐǶNpouf.Dbsmpޑኳᔕ،฼ཛྷ൨Ψёа၂๱ჴ଺рٰǴЀځӧόֹӄၗૻޑ ၯᔍ္ѬԖѬᐱ੝ޑᓬᗺǶ!

17

ୖԵЎ᝘

[1] B. Bouzy and B. Helmstetter. Monte-Carlo Go Developments. Advances in Computer Games, Many Games, Many Challenges, 10^th International Conference ACG2003 , Graz, Austria, 2003.

[2] David Carmel and Shaul Markovitch. Learning and Using Opponent Models in Adversary Search. 1996.

[3] Guillaume M. J. B. Chaslot, Mark H. M. Winands, and H. Jaap van den Herik.

Parallel Monte-Carlo Tree Search. Computers and Games: 6th International Conference, CG 2008.

[4] Matthew L. Ginsberg. GIB: Imperfect Information in a Computationally Challenging Game. Journal of Artificial Intelligence Research, 14:303–358, 2001.

[5] D. E. Knuth and R. W. Moore. An Analysis of Alpha-Beta Pruning. Artificial Intelligence, 6:293–326, 1975.

[6] Richard E. Korf. Multi-Player Alpha-Beta Pruning. Artificial Intelligence, 49:99–111, 1991.

[7] Valavan Manohararajah. Parallel Alpha-Beta Search on Shared Memory Multiprocessors. Master’s thesis, Graduate Department of Electrical and Computer Engineering, University of Toronto, Canada, 2001.

[8] C. E. Shannon. Programming a Computer for Playing Chess. Philosophical Magazine, 41(314):256–275, 1950.

[9] Nathan Sturtevant. Current Challenges in Multi-Player Game Search.

Computers and Games: 4th International Conference, CG 2004.

[10] Matthew R. Wahab. A Reinforcement Learning Agent for 1-Card Poker. 2004.