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  1. ❮ ✒ ù ✤ ✤ ✒ þ á ◗ ❈ ù ❋ ✻ ❃ ✿ ✽ ✸ ■ ✜ þ ❆ ❮ ❈ ❃ ❆ ❅ ❄ ❃ ◗ ✽ ✒ ✼ ✻ ✺ á � á ✒ þ P ✽ ✾ ❆ ✺ ✺ ✽ ✾ ✾ ✶ ❃ ❅ ❈ ❄ ❃ ✵ ✁ ✽ ✼ ✻ ✺ ❅ ✵ ❃ ❏ ❖ ❅ ◆ ▼ ▲ ❑ ❑ ■ ✽ ❍ ✻ ● ❆ ø ❆ ✽ ❃ ✾ ✽ ✏ ❑ ● ✻ Û ❍ ■ ❏ ❑ ▲ ❆ Ð ▼ ◆ ❅ ❖ ❃ ✽ ❆ ✽ P ✾ ✼ ✽ ❃ ❄ ❅ ❆ ❃ ✾ ❃ ✽ ✺ ✺ ❅ ❈ ✽ ß ④ ❆ ■ ✺ ❆ ❑ ❑ ❏ ■ ❍ ✻ ● ✻ ▼ ❆ ✽ ❃ ✽ ❋ ❈ ❅ ✺ ▲ ◆ ✸ ❃ ✽ ✿ ◗ ❃ ❈ ❋ ✻ ✿ ❅ ✽ ✸ ■ P ❆ ✽ ❃ ❖ æ ô ✺ ❃ q ■ ◗ ✸ ❈ ❋ ✻ ✿ ❆ ✽ ✸ ■ P ❆ ✽ ❃ P ✽ ❅ ❄ ✽ ø ✾ ▼ ❃ ❆ ❅ ❃ ❃ ◆ ❅ ✽ ✼ ✻ ✺ ❖ ❖ ◆ ✺ ❃ ✾ ✾ ❃ ❃ ❆ ❅ ❄ ✻ ✺ ❋ ✽ ✼ ✻ ✺ ❈ ◗ ✽ ✺ ▼ ● ▲ ❑ ❑ ❏ ■ ❍ ✻ ❆ ❅ ✽ ❆ ✽ ❃ ✽ ✿ ❈ ✺ ✾ ❅ ❅ ✾ ✾ ✽ q ✺ ✺ q q ❆ ➙ ❈ ✽ ❃ ✽ ❆ ❆ ● ❃ ❅ ❍ ✒ ✏ ➤ ô ø ù ✒ ø ù ø ❄ ù ➤ ✺ ✻ ✼ ✽ ❃ ➤ ✻ ■ ❈ ❍ ❅ ◆ ▼ ▲ ❑ ❑ ❏ ■ ✻ ❃ ● ❆ ❑ ❆ ✽ ❃ ✽ ▲ ❖ ✻ ❏ ✸ ◗ ❑ ❈ ❋ ✻ ❃ ✿ ✽ ■ P ❆ ✽ ❙❯❚❲❱✩❳❀❨❬❩✔❭ ❪❴❫✜❵✹❛ ❜✬❵❞❝❢❡✔❣✐❤ ❥❢❦ ❪❴❧★❵❞♠❬♥✜♦✐❛✵❝❞❵♣❤ rsq ❡✷♠ �✂✁☎✄✝✆✟✞✡✠☛✁✌☞✎✍☛✏✎✑ ❪✉t☎✈①✇❘❣②❛☛♦✐③①q ③⑤④ ❪⑦⑥✘❥✐r❲❵❞♠⑧♥②❜⑨❝✹⑩✹❥✐③✹❝❞❵ ✒✔✓✖✕✘✗✌✙✖✚✜✛✣✢✖✤✎✥✦✚★✧✩✙☎✪✬✫✘✭✘✮✰✯✲✱✖✢ ✳✵✴ ✶✷✳✹✸ ✾❀✿ ❁❂✸ ❇✵❄ ✼✷❉ ❊✵❋ ✾✵❏ P❘❏ ✳✵✴ ✶✷✳✹✸ ✾❀✿ ❁❂✸ ❇✵❄ ✼✷❉ ❊✵❋ ✾✵❏ P❯❶ ❷❹❸❻❺❼❭✎❽❯❾❿❽☎➀➁❽✌❭➂❸❻➃✩➄✦➅➇➆✔➃✩➈➊➉s❳❬❭✎❺➋❽ ➬➱➮ ➆➁❭✎❽✝➈➊✃✰❐❻❸❻❺❼❭✎❽ ③✐❣②❛✵❵❞➎✌q ❝❞❡✷❥✐➏➐❤ ❵✲➑➒♥✜❣✐❣②♥✜③⑤❵②③✲❡s➓➔♠→♥✜❤ ♦✲❡❀q ♥✌③➣q ♠⑧❥↔❝❢♥✜③⑤❡❀q ③⑤④✐❵✹③✹❝❂❦↕❣✐❤ ❥✐③ ➌❀➍ deterministic chance r❲❵♣❤ rsq ❡➛♠s➓➜♦✐③✐❤ ➝❢❵✹❤ ❦➒❡✷♥↔➞☎③✹➎➒④①♥✐❥✐❤ ➟☎r➁♦⑤♠❬❡➠❥✐❣✐❣②❛ ♥⑤➡✐q r❲❥⑤❡❬❵ perfect information chess, checkers, backgammon go, othello monopoly ❫✖❤ ❥✐③↔♥②❜✡❥⑤❡➛❡❬❥✹❝✹➝✐➢ imperfect information bridge, poker, scrabble ❥✐❤ ④②♥➐❛✬q ❡❂⑩①r➥❜➦♥➐❛⑨❣①❵✹❛ ❜✬❵❞❝❢❡s❣✐❤ ❥❢❦✰➧✬➨✎♥✜③✰➩➂❵✹♦✐r❲❥✐③✐③✌➟➂➫✲➭②➯②➯✌➲ nuclear war ➞☎③②q ❡❬❵➣⑩✲♥➐❛✬q ➳❢♥✜③✌➟✘❥✐❣①❣②❛ ♥⑤➡✐q r❲❥⑤❡❬❵♣❵❢➵②❥✐❤ ♦✹❥⑤❡❀q ♥✜③✟➧✷➸✘♦⑤♠❀❵①➟✡➫✲➭✹➯✐➺✩➻✖➼✦⑩⑤❥✐③✐③⑤♥✜③✌➟✡➫✲➭①➺①➽✩➻ ➼✌❥➐r➁♦⑤❵②❤ ➟✖➫✲➭①➺①➾✹✈➐➺②➚➐➲ ❣②❛☛♦✐③②q ③⑤④➪❡✷♥↕❛✵❵❞➎✜♦✹❝❞❵➶❝❢♥①♠→❡➛♠s➧❀➹↔❝⑤➘✖❥②❛ ❡❂⑩✲❦⑤➟✖➫✲➭①➺①➴➐➲ ✳✵✴ ✶✷✳✹✸ ✾❀✿ ❁❂✸ ❇✵❄ ✼✷❉ ❊✵❋ ✾✵❏ P✝➷ ✳✵✴ ✶✷✳✹✸ ✾❀✿ ❁❂✸ ❇✵❄ ✼✷❉ ❊✵❋ ✾✵❏ P✝❒ ❨❬❩s❨→❺❼❸✎❰ ❨→❩s❨→❺à❸✡❰➇❸➊❳✷❐➊➈➊➃✦❨➛❱✩➅✔❺ ❫✜❵✹❛ ❜✬❵❞❝❢❡✔❣✐❤ ❥❢❦➶❜✬♥➐❛✘➎✐❵❢❡❬❵✹❛☛rsq ③①q ♠→❡→q ❝①➟✖❣①❵✹❛ ❜✬❵❞❝❞❡☛Ï→q ③✲❜✬♥➐❛✬r✔❥⑤❡❀q ♥✜③♣④✐❥✐r❲❵❢♠ â❞ã❢ä➛å☛æ ç❂ã❻è✎é ê❂é ë✐ì❂í❞î✵ï✩ð→ñ→é ò✵é ó❞ê✲ô õ✷ö✬÷☎ø ù②ú✬û➛å☛â❞ú☛ã❢ü✖ö☛ý✎þ ÿ❀ø ö ✂✁ þ ✄� ç→ú✖û ✆☎ →ä ✂✝ ➠þ ÿ✖æ ã ✟✞✡✠ →ð ☞☛ ❀ì ☞✌ ❂ó ✍☛ ❢ò ✂✎ õ✷ö☛÷☎ø ✏✍✑ ❞ç ➎✐❵❞❥✌➢✡❝②⑩✲♥➐♥①♠❀❵♣r❿♥✹➵①❵s❡➛♥✰❣②♥①♠❂q ❡❀q ♥✜③♣Ñ⑧q ❡❂⑩➒⑩①q ④✌⑩✹❵❢♠→❡➁Ò➶Ó➦Ô✜Ó✵Ò➒Õ✲Ö↕×②Õ②Ø Ù✩Ú ì ✍✓✆✔ ❢ð ✕✎ ÿ ✆✏✗✖ ➶è✡é ê❢é ë✐ì❂í✲î ì ✍✓✆✔ ❢ð→ô ✛ ②ô õ✷ö☛÷☎ø õ✷ö☛÷✦ø ✘✙✠✚✠☞✓ ÿ ✢✜ ➏①❵❢♠→❡➠❥✹❝✹⑩①q ❵❂➵②❥✐➏✐❤ ❵➪❣✹❥❢❦⑤♥✹ÜÝ❥⑤④✐❥①q ③⑤♠→❡✔➏①❵❢♠→❡❲❣✐❤ ❥❢❦ û→ã ✣✑ ú✬û✷å☛â❢ú☛ã ✥✤ ✦☞✧ ☎þ ì ✍✓✆✔ ❢ð ✕✎ ÿ ✩★✫✪ ✦✬✤ ✦☞✧✭✦☞✪ ✮✗✦☞✧✰✯ ÿ ✂✏ Þ✦ß ➟✜➾✹Ï❀❣✐❤ ❦↔④✐❥✐r❲❵✐➢ â❞ã❢ä➛å☛æ ç❂ã❻è✎é ê❂é ë✐ì❂í❞î ì ✍✓✆✔ ❢ð→ô ✳ ✵õ✷ö☛÷☎ø ù①ú✬û✷å☛â❢ú☛ã❢ü✖ö ✙✴☞✁ ✷✬✸ ☛ö ✄✶ ✴ ❀ø 3 ✱✲✁ ö ✂✁ MAX á ✺✹ ✜ð ✚☛ ❂ë✐é ê❂ì ✻✓ ☛î ✹ ✌ð→ò ✲✌✍✎ õ✷ö☛÷✦ø ù⑤å ✗✝ ❞û❬ã ✱✰✁ ö ✂✁ A 1 A 2 A 3 ú✬û➛å☛â❢ú☛ã ✟✼✽✌ ❀é ✓ ➛é õ✷ö☛÷✦ø ✌✕✛✺✎ ✱✲✁ ö ✂✁ ü➦û✦æ á❲ë✐ì❂í ✿✪ ø✹å ✗✝ ❢û❬ã û ☞✾ ✯✺✤ ❀✡❁✬❀✰❂✆✧❃✪ ❄❅✱✲✁ ö ✗✁ ú✬û➛å☛â❢ú☛ã ✥✤ ✤ ➐è✎é ê❂é ë✐ì❂í❞î ì ✍✓✆✔ ❢ð ✡❀✗❆❈❇✣✔ ❢ñ❂ñ❀ð→ò➦ò➦ó ✍☛ ❂ò✷ô 3 2 2 ✦✚✧✽✦✚✪ ✮✂✦☞✧✰✯ ✱✲✁ ö ✂✁ MIN ü➦û û ☞✾ ú✬û➛å☛â❢ú☛ã ✥✤ ✤ ➐è✎é ê❂é ë✐ì❂í❞î ì ✍✓✆✔ ❢ð ✙❀✗❆✺❇✣✔ ❢ñ❂ñ❀ð→ò➦ò➦ó ✍☛ ❂ò✷ô ✦✚✧✽❉ ❀✰★❈✧✄✯ ✱✲✁ ö ✂✁ A 13 A 21 A 22 A 23 A 31 A 32 A 33 A 11 A 12 3 12 8 2 4 6 14 5 2 ✳✵✴ ✶✷✳✹✸ ✾❀✿ ❁❂✸ ❇✵❄ ✼✷❉ ❊✵❋ ✾✵❏ P❯P ✳✵✴ ✶✷✳✹✸ ✾❀✿ ❁❂✸ ❇✵❄ ✼✷❉ ❊✵❋ ✾✵❏ P ❋❊

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