ゲーム理論の無線リソース制御への応用 / IT SITA 201812

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1. だがわཧ࿦〣ແઢ゙ぬがと੍ޚ〭〣Ԡ༻ ࢁຊ ߴࢸ ژ౎େֶେֶӃ৘ใֶݚڀՊ 2018-12-18 ৘ใཧ࿦ݚڀձ
2. ࣗݾ঺հ — ୅දతݚڀɾڞಉݚڀऀ — ぎじぶゎひぜد〿〣〷〣 — ▶ ゙ぬがと੍ޚ〣֬཰زԿղੳ Prof. Seong-Lyun Kim (Yonsei Univ.), 2016ʙ ▶ ػցֶशɾڧԽֶशɽಛ〠じゐ゘ը૾〠〽぀ゎ゙೾੍ޚ ੢ඌཧࢤઌੜ, 2013ʙ ▶ ゞぐん゛とిྗ఻ૹ, ࣰݪਅؽઌੜ, 2011ʙ ▶ IoT, IEEE 802.11ah, क૔ਖ਼തઌੜ, 2011ʙ ▶ や゚ぶゔゆ゛ひぜと Ӌాউ೭ઌੜ, Dr. Taneli Riihonen, Mr. Sathya N. Venkatasubramanian (Aalto Univ.), Prof. Mikko Valkama, Dr. Ville Syrj¨ al¨ a, Dr. Dani Korpi (Tampere Univ. Tech.) 2010ʙ ▶ ぢそぺふくゅແઢɼແઢ゙ぬがと੍ޚɼれふアてを゚だがわ Prof. Jens Zander (KTH), 2008ʙ ▶ ろ゚ばりひゆな゚゘ɼろ゚ばりひゆɾぎへりひぜɼだがわཧ࿦ ٢ాਐઌੜ, ଜాӳҰઌੜ, 2001ʙ ׯবରࡦ〣؍఺〜〤Ұ؏ 2 / 47
3. まぐゅ゙ひへや゚ぶゔゆ゛ひぜと〣ఏҊ〝࠷ద゙ぬがとׂ౰ தܧہ R ⿿ಉҰप೾਺ଳ〜ಉ࣌〠ૹड৴『぀〝ɼେ　〟ճ〿ࠐ〴ׯব ˠまがやぶゔゆ゛ひぜと (Phase 1+3) ⿿Ұൠత ճ〿ࠐ〴ׯবΛぎアふべઃܭɼぎべ゜そɾぶくでの゚〜ずをアな゚ ˠや゚ぶゔゆ゛ひぜと (Phase 2) 〝⿶⿸৽〔〟ྲྀぁ 〞〖〾⿾〽⿶ํ〜〤〟。ɼ⿶⿶〝〈औ〿〣とたでゔが゙アそΛఏҊ Phase 1 τ1 Phase 2 τ2 Phase 3 τ3 S R D S R D S R D CSR2 CRD2 CSR1 CRD3 S to R R to D time time τ1 τ2 τ3 CSR1 CSR2 CRD2 CRD3 maximize (τ1,τ2,τ3) t subject to t ≤ τ1 CSR1 + τ2 CSR2 , t ≤ τ2 CRD2 + τ3 CRD3 , τ1 ≥ 0, τ2 ≥ 0, τ3 ≥ 0, τ1 + τ2 + τ3 ≤ 1 K. Yamamoto, K. Haneda, H. Murata, and S. Yoshida, “Optimal transmission scheduling for a hybrid of full- and half-duplex relaying,” IEEE Commun. Lett., vol. 15, no. 3, Mar. 2011 3 / 47
4. ୯Ұ〣௨৴ SNR ෺ཧ૚ (OSI ࢀরゑぶ゚ୈ 1 ૚) ৴߸ॲཧ ぎアふべɾ఻ൖ ৘ใɾූ߸ཧ࿦ てをぽア〣௨৴࿏༰ྔ (प೾਺ར༻ޮ཰) ln(1 + SNR) nat/s/Hz ෳ਺〣ಉҰप೾਺ଳಉ࣌௨৴ SI NR ぶがの゙アぜ૚ (OSI ࢀরゑぶ゚ୈ 2 ૚) ແઢϦιʔε੍ޚ ゐぶくぎぎぜなと੍ޚ ࠷దԽཧ࿦ʁ ήʔϜཧ࿦ʁ 4 / 47
5. େ༰ྔ (ແઢ) ௨৴Λ࣮ݱ『぀〔〶〠〤 てをぽア〣௨৴࿏༰ྔ (৘ใཧ࿦) W (1) ln(1 + SNR (2) ) nat/s /௨৴࿏ (3) (1) ଳҬ෯ W Λ֦େ てアる゚゛がぷ૿େ ろ゚ばずを゙ぎ఻ૹ OFDM ゎ゙೾ प೾਺ڞ༻ ˠׯব੍ޚඞཁ खஈɿϦιʔε੍ޚ (2) SNR Λ૿େ దԠมௐɾଟ஋มௐ ߴརಘぎアふべ もがわやさがゎアそ ఻ൖଛ〣খ《⿶ばを ぼ゚ தܧ〠〽぀ෆײ஍ଳ ରࡦ (3) ௨৴࿏〣਺Λ૿େ MIMO ௨৴࿏ (Massive MIMO) ᜚ີج஍ہઃஔ densification ۭؒతप೾਺࠶ར༻ ˠׯব੍ޚඞཁ खஈɿϦιʔε੍ޚ W log2 (1 + SNR) bit/s 〝ಉ」 5 / 47
6. ۭؒతप೾਺࠶ར༻ Spatial Channel Reuse ࠶ར༻〟「 प೾਺ɾ࣌ؒΛ෼ׂ「〛ׂ౰ p1 G11 p1 G11 G22 W 2 ln ( 1 + G11 p1 σ1 2 ) + W 2 ln ( 1 + G22 p2 σ2 2 ) ଳҬ⿿൒෼〠 ࣌෼ׂ〣৔߹〣ࣜ ࠶ར༻⿴〿 ಉҰप೾਺〜ಉ࣌〠௨৴ p1 p2 G11 p1 G12 p2 G11 G22 G12 W ln ( 1 + G11 p1 G12 p2 + σ1 2 ) +W ln ( 1 + G22 p2 G21 p1 + σ2 2 ) ׯবൃੜ ׯব〣෼෍ΛࡶԻ〣〷〣〝ಉ」〝 ݟ〟『৔߹〣ࣜ 6 / 47
7. ࠷దૹ৴ిྗ੍ޚ໰୊〣ఆࣜԽ p1 p2 G11p1 G12p2 G11 G22 G12 ௨৴࿏༰ྔ〣૯࿨〣࠷େԽ໰୊ sum rate maximization problem max (p1,p2) [ ln ( 1 + G11p1 G12p2 + σ1 2 ) + ln ( 1 + G22p2 G21p1 + σ2 2 )] nat/s/Hz ໨తؔ਺〤ඇԜؔ਺ (ූ߸൓స「〛࠷খԽ〠「〔〝　〠ඇತؔ਺) ▶ ཧ༝ɿょひなߦྻ ̸= ਖ਼ఆ஋ ▶ ݁࿦ɿ(ඇઢܗܭը๏〣⿸〖ར༻「〹『⿶) ತܭը๏͸ద༻ෆՄ 7 / 47
8. ばをぼ゚ׂ౰〷ؚ〶〔࠷దແઢ゙ぬがと੍ޚ໰୊ maximize (pi,ci)i∈N ∑ i∈N ln ( 1 + Giipi ∑ j∈N\{i} Gijpj (cj = ci) ಉҰばをぼ゚ׯব + σi 2 ) (1) (1) 〜ばをぼ゚ (ci) Λݻఆ「〔࠷దૹ৴ిྗ੍ޚ໰୊ (࿈ଓ࠷దԽ) 〣 ໨తؔ਺〤ඇԜ (1) 〜ૹ৴ిྗ (pi) Λݻఆ「〔࠷దばをぼ゚ׂ౰໰୊ (཭ࢄ࠷దԽɾ ૊߹【࠷దԽ) 〤 NP ࠔ೉ [Raniwala+2004; Hayashi+2009] ⿶』ぁ〠【〽ɼ࠷దແઢ゙ぬがと੍ޚ໰୊〣 ࠷దղ͸ղੳతʹ͸ٻ·Βͳ͍ Ұൠత〠〤ɹ ܦ ݧ ଇ めゔが゙とふくぜと ɹ [Raniwala+2004] A. Raniwala, K. Gopalan, and T.-C. Chiueh, “Centralized channel assignment and routing algorithms for multi-channel wireless mesh networks,” ACM SIGMOBILE Mobile Comp. Commun. Rev., vol. 8, no. 2, Apr. 2004 [Hayashi+2009] S. Hayashi and Z.-Q. Luo, “Spectrum management for interference-limited multiuser communication systems,” IEEE Trans. Inf. Theory, vol. 55, no. 3, Mar. 2009 8 / 47
9. ࠜຊతݪҼ〣೺Ѳ〝ɼࢁຊ〣ぎゆ゜がば ໰୊Λ೉「。「〛⿶぀〣〤ɼׯব〜⿴぀ ࠜڌ (1) 〜ׯব߲Λແ「〠『ぁ〥 ∑ i∈N ln  1 + Giipi (((((((((((( ( ∑ j∈N\{i} Gijpj (cj = ci) + σi 2   Ԝؔ਺〝〟぀ɽૹ৴ిྗ (pi) 〠ؔ「〛୯ௐ૿Ճ〜⿴〿 ࠷େૹ৴ిྗ〠ઃఆ『ぁ〥〽⿶ɽಛஈ໾〠〤ཱ〔〟⿶⿿... ࢁຊ〣ぎゆ゜がば 1. だがわཧ࿦ɾڧԽֶश ׯবΛ૬ޓ࡞༻〝ଊ⿺ɼ૬ޓ࡞༻ 〣औ〿ѻ⿶ΛՄೳ〝『぀だがわཧ ࿦ɾڧԽֶशΛద༻ 2. ֬཰زԿ ゘アはわ〠ଘࡏ『぀ૹ৴఺⿾〾 〣ׯব〣෼෍Λղੳత〠ѻ⿸〈〝 〣〜　぀֬཰زԿΛద༻ 9 / 47
10. ϙΞιϯ఺աఔ [Haenggi2012, §2.4] Ґஔ⿿ಠཱ〟ɼີ౓ λ 〣఺Λ〒⿸ݺ〫 ྖҬ B ⊂ Rd 〠ؚ〳ぁ぀఺〣਺ Φ(B) 〤 ظ଴஋ λ|B| 〣れぎぬア෼෍〠ै⿸ P ( Φ(B) = n ) = e−λ|B|(λ|B|)n n! ▶ d = 1 〟〾れぎぬア౸ண ▶ B 〣த〠఺⿿〟⿶ P ( Φ(B) = 0 ) = e−λ|B| ▶ B 〣த〠఺⿿⿴぀ P ( Φ(B) ̸= 0 ) = 1 − e−λ|B| [ࢁຊ 2017] ࢁຊ ߴࢸ, “な゚゘ぼひぷゞがぜ〣֬཰زԿղੳ〣جૅ,” Japanese, in ৴ֶٕใ, CQ2017-57, Aug. 2017 http://bit.ly/RCS201705 10 / 47
11. ֬཰زԿ⿿޿。࢖いぁ぀ܖػ〝〟〘〔 [Andrews+2011] 〣ॏཁ〟݁Ռ な゚゘ج஍ہ〣ҐஔΛϙΞιϯ఺աఔ ゕがづ〤࠷ۙ๣ج஍ہ〠઀ଓ (Poisson-Voronoi cells) ड৴ిྗมಈ〤ࢦ਺෼෍〠ै⿸ (゛ぐ゙がやこがでアそ) ڑ཭ݮਰఆ਺ α > 2 Ҏ্〣Ծఆ〜ɼSIR 〣໘త෼෍ (cdf, ccdf) ⿿௚઀ٻ〳〿 α = 4 〜ดܗࣜ (〔〕「ɼ[ࢁຊ 2017] શ෦〣ཧղඞཁ) -20 -10 0 10 20 30 w/o fading P(SIR > θ) = · · · = 1 1 + 2θ α−2 2 F1 ( 1, 1 − 2 α ; 2 − 2 α ; −θ ) = √ θ arctan √ θ (α=4) 『〟い〖ɼ[Andrews+2011] 〠〽〿 な゚゘ぼひぷゞがぜ〠 ֬཰زԿղੳ⿿ద༻Մೳ〝ࣔ《ぁ〔 0 1 −20 −10 0 10 20 30 P(SIR ≤ θ) θ (dB) −20 −10 0 10 20 30 with fading [Andrews+2011] J. G. Andrews, F. Baccelli, and R. K. Ganti, “A tractable approach to coverage and rate in cellular networks,” IEEE Trans. Commun., vol. 59, no. 11, Nov. 2011 11 / 47
12. ແઢ LAN ஗⿶〟⿴... 12 / 47
13. प೾਺ばをぼ゚ ड৴ిྗ
14. ੨〝੺〣 2 ばをぼ゚ɾ3 ௨৴૊ ड৴ہ〜〣ಉҰばをぼ゚ׯবిྗ⿿࠷খ〣ばをぼ゚Λબ୒『぀〝〞⿸ 〟぀⿾ 14 / 47
15. interference serious interference 15 / 47
16. serious in- terference interference ▶ ผ〣ड৴ہ⿿ਂࠁ〟ׯবΛड々぀〽⿸〠〟〘〔〕々 ▶ ゚がゆ⿿ى〈぀ ▶ 〞〣〽⿸〠ఆࣜԽ「〛ٞ࿦『ぁ〥ྑ⿶⿾ʁ 16 / 47
17. ہ 1: max c1∈{blue,red} u1 (c1 , c2 , c3 ) ہ 2: max c2∈{blue,red} u2 (c1 , c2 , c3 ) ہ 3: max c3∈{blue,red} u3 (c1 , c2 , c3 ) ▶ I := {1, 2, 3}ɿ ૹड৴ہ૊〣ఴࣈू߹ ▶ ci ɿૹड৴ہ i ∈ I 〣ばをぼ゚ ▶ ಉҰばをぼ゚ׯবిྗ Ii (c) := ∑ j∈I\{i} ಉҰばをぼ゚〜⿴ぁ〥 ड৴ہ i ⿿ૹ৴ہ j 〽〿ड々぀ిྗ Gij P (cj = ci ) ▶ ui (c) := −Ii (c) ූ߸൓స〜࠷େԽ໰୊〠 ▶ Gij ɿૹ৴ہ jɾड৴ہ i ؒ〣 ిྗརಘ ▶ Pɿૹ৴ిྗ ▶ (cond) := { 1 if cond is true, 0 otherwise. 17 / 47
18. ہ 1: max c1∈{blue,red} u1 (c1 , c2 , c3 ) ہ 2: max c2∈{blue,red} u2 (c1 , c2 , c3 ) ہ 3: max c3∈{blue,red} u3 (c1 , c2 , c3 ) 1. ہ 1 〤ɼɹ ㈃㈥㈏㈭ ม਺ ɹ c1 Λ੨〝੺⿾〾બΜ〜ɹ −ड৴ׯবిྗ ໨తؔ਺ ɹ u1 Λେ　。「〔⿶ 2. ہ 1 〣໨తؔ਺ u1 〤ɼࣗہ 1 〣ɹ ㈃㈥㈏㈭ ม਺ ɹ c1 〕々〜〟。 ଞہ 2,3 〣ɹ ㈃㈥㈏㈭ ม਺ ɹ c2 , c3 〠〷ґଘ『぀ؔ਺ ( ::: ׯব〣Өڹ) 3. ֤ہ〤ɼଞہ〣ɹ ㈃㈥㈏㈭ ม਺ ɹ〤มߋෆՄ 18 / 47
19. ɹ ㆛Μ㇊㇃ㆎㆋ㆟ ઓ ུ ܗ ɹ 2 ਓだがわ (two-player strategic form game)    max x1 f1(x1, x2) max x2 f2(x1, x2) ඇڠྗだがわཧ࿦〜ѻいぁ぀࠷〷てアゆ゚⿾〙جૅత〟໰୊ (ܦࡁֶ〜〣) ڭՊॻ〜〤ɼ〈⿸໌ࣔత〠ॻ⿶〛⿶〟⿶〈〝⿿ଟ⿶ ࠷దԽཧ࿦〣දه〠׳ぁ〛⿶぀ํ〠〤ɼ〈〣දه〜આ໌「〔⿶ 19 / 47
20. ઓུܗ 2 ਓだがわ〣ଊ⿺ํɾॏཁੑ ɹ ҙ ࢥ ܾ ఆ ओ ମ ゆ゛がん ɹ 1 〠〝〘〛 max x1 f1(x1, x2) ▶ ゆ゛がん 1 〤ࣗ෼〣རಘؔ਺ f1 〣஋⿿࠷େ〝〟぀ બ୒ (ઓུ) x1 Λબ〫 ▶ རಘؔ਺ f1 〣஋ɾऔ぀〮　બ୒ x1 〤 ଞ〣ゆ゛がん 2 〣બ୒݁Ռ x2 〠ґଘ max x2 f2(x1, x2) ▶ ゆ゛がん 1 〣બ୒݁Ռ x1 ⿿ ゆ゛がん 2 〣໨తؔ਺ f2 〠ӨڹΛٴ〱『 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ઓུܗ 2 ਓだがわ〤〈〣ҙຯ〜ҙࢥܾఆओମؒ〣૬ޓ࡞༻〣දݱ〠੒ޭ 20 / 47
21. (୯໨త) ࠷దԽ໰୊ max (x1,x2) f(x1, x2) ▶ ిྗ〝ばをぼ゚Λ⿸〳。ઃఆ「〛 と゚がゆひぷΛ্〆〔⿶ ଟ໨త࠷దԽ໰୊ max (x1,x2) ( f1(x1, x2), f2(x1, x2) ) ▶ と゚がゆひぷ〝஗Ԇ྆ํΛ ྑ。「〔⿶ ▶ ゕがづ 1 〝ゕがづ 2 〣 と゚がゆひぷΛ ूத੍ޚత〠ྑ。「〔⿶ ઓུܗだがわ    max x1 f1(x1, x2) max x2 f2(x1, x2) ▶ ֤ओମ 1, 2 ⿿ࣗ෼〣ग़དྷ぀ൣғ〜 と゚がゆひぷΛ্〆〽⿸〝 「〛⿶぀ 21 / 47
22. ʮだがわཧ࿦〤໾〠ཱ〙〣ʁʯ〝⿶⿸യવ〝「〔࣭໰Λड々぀ ઐ໳༻ޠΛ࢖⿺〥ɼ⿼〒〾。〈⿸⿶⿸〈〝Λฉ　〔⿶〣〜〤〟⿶⿾ Q1. (୯໨తɾଟ໨త) ࠷దԽ໰୊〣࠷దղΛٻ〶぀〔〶〠 だがわཧ࿦〤࢖⿺぀⿾ʁ Q2. (୯໨తɾଟ໨త) ࠷దԽ໰୊〣࠷దղΛ :::::::: ෼ࢄత〠ٻ〶぀〔〶 〠だがわཧ࿦〤࢖⿺぀⿾ʁ Q3. だがわཧ࿦〤ແઢ௨৴てとふわ〣՝୊Λղܾ〜　぀〣⿾ʁ 22 / 47
23. Q3 〤ஔ⿶〛⿼⿶〛ɼࣅ〔〽⿸〟ҎԼ〣࣭໰〠౴⿺〾ぁ〳『⿾ʁ Q4. ࠷దԽཧ࿦〤ແઢ௨৴てとふわ〣՝୊Λղܾ〜　぀〣⿾ʁ 〷〘〝۩ମత〠『぀〝 Q5. ແઢ௨৴てとふわ〣՝୊〤࠷దԽ໰୊〝「〛ఆࣜԽ〜　぀〣⿾ʁ 〒⿸⿶⿺〥࠷ۙ〈Μ〟ٙ໰Λ〽。ฉ　〳『〢 Q6. AI/ػցֶश〤໾〠ཱ〙〣⿾ 23 / 47
24. ՝୊ ఆࣜԽՄೳ ղ々぀ 24 / 47
25. だがわཧ࿦ಛ༗〣༻ޠ 〈〈〳〜〣આ໌〜ࠞ】〛࢖〘〔 ࠷దԽཧ࿦〝だがわཧ࿦〣༻ޠΛ੔ཧ ࠷దԽཧ࿦ だがわཧ࿦ ࠷దԽ໰୊ だがわ game — ゆ゛がん player (ܾఆ) ม਺ variable x ઓུ strategy a, s ม਺〣੍໿৚݅ ઓུू߹ strategy set A ໨తؔ਺ f རಘؔ਺ payoff function u ゆ゛がん〣ఴࣈू߹ Iɾઓུू߹ (Ai)i∈I ɾརಘؔ਺ (ui)i∈I 〣 3 〙૊ triplet ( I, (Ai)i∈I, (ui)i∈I ) ΛઓུܗήʔϜ〝ݺ〫 ઌ〰〞〣࠷దԽ໰୊〣૊〝ଊ⿺〛ࠩ「ࢧ⿺〟⿶ triplet Λ࢖⿸〣〤ɼそ゘やΛɼ௖఺ू߹ V 〝ล〣ू߹ E 〣 tuple (V, E) 〝「〛ఆٛ『぀〽⿸〟〷〣 25 / 47
26. だがわཧ࿦ಛ༗〣දݱ ▶ 『〮〛〣ゆ゛がん〣ઓུू߹〣௚ੵ A := ∏ i∈I Ai A Λఆٛ『぀ཧ༝ɿ 『〮〛〣ゆ゛がん〣ઓུ〣よぜぷ゚ a := (a1, . . . , a|I| ) Λ a ∈ A 〝දه「〔⿶ ui : A → R 〝දه「〔⿶ ▶ ઓུू߹〣ゆ゛がん i Ҏ֎〠ؔ『぀௚ੵ A−i := ∏ j∈I\{i} Aj A−i Λఆٛ『぀ཧ༝ɿ ゆ゛がん i Ҏ֎〣ઓུ a−i := (a1, . . . , ai−1, ai+1, . . . a|I| ) Λ a−i ∈ A−i 〝දه「〔⿶ BRi : A−i → Ai 〝දه「〔⿶ a−i Λఆٛ『぀ཧ༝ɿ ui(ai, a−i) 〝⿶⿸දه⿿ศར〕⿾〾 26 / 47
27. ૒ߦྻだがわ (ఆٛ) ڭՊॻ〠〈Μ〟ද⿿〽。ग़〛。぀ a1 \ a2 X Y Z A 7, 6 1, 7 2, 3 B 4, 2 3, 5 6, 2 C 2, 1 1, 3 2, 1 ▶ ゆ゛がん਺⿿ 2 e.g. I = {1, 2} ▶ ઓུू߹⿿཭ࢄత〟৔߹ e.g. A1 = {A, B, C}, A2 = {X, Y, Z} ▶ རಘؔ਺ u1 (a1 , a2 ), u2 (a1 , a2 ) 〣 දݱํ๏ Լਤ〣〽⿸〠ଊ⿺぀〝ղ〿〹『⿶〝ࢥ⿸ A B C X Y Z 2 4 6 8 u1 (a1 , a2 ) u2 (a1 , a2 ) a1 a2 27 / 47
28. ఆٛɿ࠷దԠ౴ (ରԠ) Best response (correspondence) A B C X Y Z 2 4 6 8 u1 (a1 , a2 ) u2 (a1 , a2 ) a1 a2 ఆٛɿa1 〠ର『぀ゆ゛がん 2 〣࠷దԠ౴ (ରԠ) BR2 (a1 ) := { a⋆ 2 ∈ A2 u2 (a⋆ 2 , a1 ) = max a2 u2 (a1 , a2 ) } ଊ⿺ํɿa1 Λݻఆ「〔〝　〠ɼu2 Λ࠷େԽ《【぀〠〤 a2 Λ〞⿸『〮　⿾ ରԠ correspondenceɿଟՁؔ਺ set-valued function 〣〈〝 28 / 47
29. (७ઓུ) べひてゔۉߧ (pure-strategy) Nash equilibrium A B C X Y Z 2 4 6 8 u1 (a1 , a2 ) u2 (a1 , a2 ) a1 a2 ࣍〣ࣜΛຬ〔『 (a⋆ 1 , a⋆ 2 ) { u1 (a⋆ 1 , a⋆ 2 ) ≥ u1 (a1 , a⋆ 2 ) u2 (a⋆ 1 , a⋆ 2 ) ≥ u2 (a⋆ 1 , a2 ) ޓ͍ʹ࠷దԠ౴ʹͳ͍ͬͯΔઓུͷ૊ (a⋆ 1 , a⋆ 2 ) べひてゔۉߧ〠⿼⿶〛〤 (ࣗ෼͚͕ͩ) ઓུΛมߋͯ͠΋ɼརಘؔ਺͕޲্͠ͳ͍ ࠞ߹ઓུべひてゔۉߧ〤লུ 29 / 47
30. ແઢ௨৴てとふわ〠⿼々぀ʮઓུʯ ゕがづؒ〠ぷ゛がへざや〣⿴぀ࣗ༝౓ てとふわ ࢖༻『぀つがもと (LTE/ແઢ LAN)ɼずを゙ぎɼՁ֨ ܾఆɼなずゔ゙ふく ぼひぷゞがぜ ઀ଓઌج஍ہɼゃぢな゚ɾやこわぷな゚ɼ TCP ᫔᫓੍ޚぎ゚っ゙どわ ଟݩ઀ଓ प೾਺ばをぼ゚ɼのぐわと゜ひぷɼ CSMA/CA (଴〖࣌ؒ〣࠷খ஋ CWmin) ෺ཧ૚ ૹ৴ిྗ (ૹ৴〣༗ແ)ɼ(ෳૉ) ૹ৴৴߸ɼぎアふべࢦ ޲ੑ 30 / 47
31. だがわཧ࿦〤 ઓུܗだがわΛ〤」〶〝『぀《〳》〳〟だがわ〝 ֤だがわ〠ؔ『぀ۉߧ֓೦〟〞〣ू߹ମ جૅ〝〟぀ઓུܗだがわ〝べひてゔۉߧΛ〳』ཧղ「〛⿼。ඞཁ 〈〣ޙ〤ɼ⿶。〙⿾〣ڵຯਂ⿶ぷゃひぜΛ঺հ 31 / 47
32. 1 ઌ〠આ໌「〔ばをぼ゚બ୒だがわ〜ऩଋੑΛอূ『぀〠〤〞⿸『ぁ 〥〽⿶⿾ʁ 2 れふアてを゚だがわ〜〤࠷దԠ౴はぐべゎぜと〜ऩଋ『぀⿿ɼむ゛が ぷ࠷ద〜〟⿶べひてゔۉߧ〠ؕ぀〈〝〷⿴぀ɽԿ〝⿾〟〾〟⿶⿾ʁ 32 / 47
33. ఆٛɿBest-response dynamics ࠷దԠ౴はぐべゎぜと ֤࣌఺ k 〠⿼⿶〛ɼ୯Ұ〣ゆ゛がん (i 〝『぀) 〣〴⿿ɼઓུΛ ai[k] ⿾〾 ࠷దԠ౴ BRi(a−i[k]) 〠มߋɽ『〟い〖 ai[k + 1] ∈ BRi(a−i[k]) A B C X Y Z 2 4 6 8 u1 (a1 , a2 ) u2 (a1 , a2 ) a1 a2 k i a1[k] a2[k] 0 2 A X 1 A 2 〒〣ଞ〣ゆ゛がん〤ઓུΛมߋ「〟⿶ɽ『〟い〖 a−i[k + 1] = a−i[k] 33 / 47
34. ఆٛɿれふアてを゚だがわ [Monderer+1996] 『〮〛〣ゆ゛がんɾઓུ〠ؔ「〛ɼ࣍ࣜ⿿੒〿ཱ〙 (れふアてを゚) ؔ ਺ ϕ : A → R ⿿ଘࡏ『぀ઓུܗだがわΛɼϙςϯγϟϧήʔϜ〝ݺ〫 ui(ai, a−i) − ui(a′ i , a−i) = ϕ(ai, a−i) − ϕ(a′ i , a−i) (2) 『〟い〖ɼઓུ〣มߋ (a′ i ⿾〾 ai) 〠〽぀ རಘؔ਺ ui 〝れふアてを゚ؔ਺ ϕ 〣มԽྔ⿿ಉ」 ʲ෺ཧֶʳ ∂ui ∂ai = ∂ϕ ∂ai ⿿『〮〛〣 1 ≤ i ≤ n 〠〙⿶〛੒〿ཱ〙〝　 ϕ Λ (u1, . . . , un) 〣れふアてを゚ؔ਺〝ݺ〫 [Monderer+1996] D. Monderer and L. S. Shapley, “Potential games,” Games Econ. Behav., vol. 14, no. 1, May 1996 a−i 〤 ai Ҏ֎〣ઓུ〣૊ 34 / 47
35. れふアてを゚だがわ〜࠷దԠ౴はぐべゎぜと⿿ऩଋ『぀ূ໌ ࣌఺ k 〜୯Ұ〣ゆ゛がん (i 〝『぀) ⿿རಘ⿿্⿿぀〽⿸〠ઓུΛ ai[k] ⿾〾 ai[k + 1] 〠ߋ৽ (࠷దԠ౴はぐべゎぜと) ui(ai[k], a−i[k]) < ui(ai[k + 1], a−i[k]) 『぀〝ɼれふアてを゚だがわ〣ఆٛ (2) 〠〽〿 ϕ 〷૿Ճ『぀ ϕ(ai[k], a−i[k]) < ϕ(ai[k + 1], a−i[k]) 〈〣〽⿸〟ઓུ〣ߋ৽Λ a[0], a[1], a[2], . . . 〝දه『぀〝 ࣍〣ෆ౳ࣜ⿿੒ཱ ϕ(a[0]) < ϕ(a[1]) < ϕ(a[2]) < · · · ઓུू߹ A(a 〣औ〿⿸぀ൣғ) ⿿༗ݶ〜⿴ぁ〥 〒ぁҎ্ ϕ ⿿૿Ճ「〟⿶఺ (=ۉߧ఺) 〠༗ݶճ਺〜౸ୡ れふアてを゚だがわ〠⿼⿶〛〤ɼ֤ゆ゛がん⿿རಘؔ਺〠ج〚　 (རݾత〠ɾݸਓ߹ཧత〠) ઓུΛมߋ「〛〷ɼϕ 〤୯ௐ૿Ճ「ɼऩଋ (2) ui(ai, a−i) − ui(a′ i , a−i) = ϕ(ai, a−i) − ϕ(a′ i , a−i) 35 / 47
36. ઌ〠ɼばをぼ゚બ୒⿿゚がゆ『぀〝⿶⿸ྫ〜 ঺հ「〔ɼಉҰばをぼ゚ׯবిྗ〣࿨ (Λූ ߸൓స「〔〷〣) Λརಘؔ਺ ui(ai, a−i) = − ∑ j∈I\{i} GijP (ai = aj) 〝ઃఆ『぀だがわ〤ɼれふアてを゚だがわ〜 〤〟⿶〈〝⿿ূ໌〜　぀ (൓ྫΛ࢖〘〔؆୯〟) ূ໌ (७ઓུ) べひてゔۉߧ⿿ແ⿶〈〝⿿⿴぀⿾〾 36 / 47
37. れふアてを゚だがわ〣ඞཁे෼৚݅ [Ui2000; ӉҪ 2007] རಘؔ਺⿿ ui(a) = ∑ S⊆I:i∈S ΦS(aS) 〣ܗ〠ॻ　׵⿺〾ぁ぀ ؔ਺ ΦS : ∏ i∈S Ai → R (૬ޓ࡞༻れふアてを゚) ⿿ଘࡏ『぀〈〝 〈〣৔߹ɼれふアてを゚ؔ਺〤 ϕ(a) = ∑ S⊆I ΦS(aS) ྫɿI := {1, 2, 3} u1(a1, a2, a3) = Φ{1} (a1) + Φ{1,2} (a1, a2) + Φ{1,3} (a1, a3) + Φ{1,2,3} (a1, a2, a3) 〝⿶⿸ܗ〠෼ղ「〔〝　ɼΦ{1,2} (a1, a2) 〝 Φ{1,2,3} (a1, a2, a3) ⿿ u2 〜 〷ڞ௨〜⿴ぁ〥れふアてを゚だがわ [Ui2000] T. Ui, “A Shapley value representation of potential games,” Games Econ. Behav., vol. 31, no. 1, Apr. 2000 37 / 47
38. p. 36 〣だがわ⿿れふアてを゚だがわ〜〟⿶ผূ [ࢁຊ 2016] ui(ai, a−i) = − ∑ j∈I\{i} GijP (ai = aj) ⿿૬ޓ࡞༻れふアてを゚ ΦS(aS) Λ࣋〙〝『ぁ〥ɼS = {i, j} 〜ɼ Φ{i,j} ( a{i,j} ) = −GijPj (ai = aj) Ұํɼuj ⿾〾ٻ〶〔 −GjiPi (ai = aj) 〝Ұக「〟⿶ =⇒ ΦS(aS) 〤ଘࡏ「〟⿶ =⇒ れふアてを゚だがわ〜〟⿶ 38 / 47
39. Ұൠ〣 れふアてを゚ ઓུܗだがわ だがわ ゆ゛がんɼઓུɼརಘ Λܾ〶〔〕々 べひてゔۉߧ 0 Ҏ্ 1 Ҏ্ 〣਺ ࠷దԠ౴ ऩଋ「〟⿶ ઓུۭؒ⿿༗ݶ〟〾 はぐべゎぜと Մೳੑ ༗ݶճ਺〜ऩଋ ઓུܗだがわ〣ɼѻ⿶〹『⿶ぜ゘と ࿈ଓ࠷దԽ໰୊〠ɼತܭը໰୊〟〞〣 ѻ⿶〹『⿶ぜ゘と⿿⿴぀〣〝ಉ༷〣Ґஔ〚々 れふアてを゚だがわ =⇒ ゚がゆ「〟⿶ɽٯ〤ݴ⿺〟⿶ 39 / 47
40. れふアてを゚だがわ〣Ԡ༻ ▶ ઌ〣ɼばをぼ゚બ୒⿿゚がゆ「⿸぀ྫ〜ग़〛　〔 ಉҰばをぼ゚ׯবిྗ〣࿨ (Λූ߸൓స「〔〷〣) ui(c) := −Ii(c) = − ∑ j̸=i ඃׯব GijP (cj = ci) Λརಘؔ਺〝『぀だがわ〤れふアてを゚だがわ〜〤〟⿶ ▶ だがわ (3) 〤れふアてを゚だがわ [Nie+2006] u(3) i (c) := − ∑ j̸=i ( ඃׯব GijPj + ༩ׯব GjiPi) (cj = ci) (3) ূ໌ɿΦ{i,j} ( c{i,j} ) = −(GijPj + GjiPi) (cj = ci) ▶ Ҡಈ௨৴ 4G やこわぷな゚〭〣ׯব੍ޚ〟〞〠Ԡ༻ [Mustika+2011] 40 / 47
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    ।ڭतͷࢁ ຊߴࢸࢯ "1ͷνϟϯωϧ࠷దԽΞϧΰϦζϜʹήʔ Ϝཧ࿦ΛԠ༻ɻΦʔόʔϥοϓ໘ੵ͕࠷΋ খ͘͞ͳΔνϟϯωϧΛॱ࣍બ୒͍ͯ͘͠ http://cloud.watch.impress.co.jp/docs/news/1016971.html 41 / 47
42. È ざがみ゘ひゆ໘ੵ⿿࠷〷খ《。〟぀ばをぼ゚Λ֤ AP ⿿ॱ࣍બ୒「〔⿶ AP 〣〴〠〽぀つぐぷつがよぐɼࣗಈత〟ૹ৴ిྗɾばをぼ゚ઃఆ [Kamiya+2016] S. Kamiya, K. Nagashima, K. Yamamoto, T. Nishio, M. Morikura, and T. Sugihara, “Self-organization of coverage of densely deployed wlans considering outermost aps without generating coverage holes,” IEICE Trans. Commun., vol. E99-B, no. 9, 2016 42 / 47
43. ざがみ゘ひゆ໘ੵ〣ධՁख๏〝れふアてを゚だがわ རಘؔ਺ 1: ࣮ࡍ〣ざがみ゘ひゆ໘ੵ〕⿿ɼධՁ⿿໘౗ རಘؔ਺ 2: ࣮ࡍ〣ざがみ゘ひゆ໘ੵ〽〿େ　⿶⿿ɼධՁ⿿༰қ ⿶』ぁΛརಘؔ਺〝「〛〷ɼれふアてを゚だがわ〝〟぀〈〝Λ ૬ޓ࡞༻れふアてを゚Λ༻⿶〛ূ໌ [Kamiya+2017] [Kamiya+2017] S. Kamiya, K. Yamamoto, T. Nishio, M. Morikura, and T. Sugihara, “Spatial co-channel overlap mitigation through channel assignment in dense WLAN: Potential game approach,” IEICE Trans. Commun., vol. E100-B, no. 7, 2017 43 / 47
44. 1 ઌ〠આ໌「〔ばをぼ゚બ୒だがわ〜ऩଋੑΛอূ『぀〠〤〞⿸『ぁ 〥〽⿶⿾ʁ 2 れふアてを゚だがわ〜〤࠷దԠ౴はぐべゎぜと〜ऩଋ『぀⿿ɼむ゛が ぷ࠷ద〜〟⿶べひてゔۉߧ〠ؕ぀〈〝〷⿴぀ɽԿ〝⿾〟〾〟⿶⿾ʁ 44 / 47
45. ղܾࡦ〣 1 〙ɿlogit / Boltzmann-Gibbs larning Ai ⿿཭ࢄత〟〝　ɼ֤ઓུ ai ∈ Ai Λ Boltzmann-Gibbs ෼෍〠ै〘〛ܾఆ [Lasaulce+2011, §5] exp ( βui(eai , x−i) ) ∑ a′ i ∈Ai exp ( βui(ea′ i , x−i) ) ▶ xi ∈ ∆(Ai), xi(ai): ai Λબ〫֬཰, eai : ai 〣〴Λ 100%〜બ〫෼෍ ▶ x−i ∈ ∏ j̸=i ∆(Aj) ▶ ui : ∏ j ∆(Aj) → R 〝վ〶〛ఆٛ ▶ βɿඇෛఆ਺ɽ౷ܭྗֶ〜〤Թ౓ɽ〈〈〜〤ʮ߹ཧੑ〣౓߹⿶ʯ 〈ぁ〤ɼརಘؔ਺ΛҎԼ〣〽⿸〠ઃఆ「〔৔߹〣࠷దԠ౴〝ಉ」 ∑ ai∈Ai xi(ai) ui(eai , x−i) − ∑ ai∈Ai xi(ai) log xi(ai) ごアぷ゜ゃが (4) 45 / 47
46. だがわཧ࿦〝ڧԽֶश〣ܨ⿿〿 u1,t u1,t+1 a1,t a1,t+2 a2,t−1 a2,t+1 a2,t+3 ઓུܗ 2 ਓだがわ ࠷దԠ౴はぐべゎぜと Rt Rt+1 at at+1 st st+1 st+2 ろ゚ぢやܾఆաఔ ڧԽֶश 46 / 47
47. ࢀߟจݙ だがわཧ࿦Ұൠ ▶ ࢁຊ ߴࢸ, “だがわཧ࿦〣ແઢ௨৴〭〣Ԡ༻,” Japanese, ৴ֶࢽ, vol. 95, no. 12, Dec. 2012 ▶ Ԭా ষ, だがわཧ࿦৽൛, Japanese. ༗൹ֳ, 2011 れふアてを゚だがわ ▶ ӉҪ وࢤ, “れふアてを゚だがわ〣جຊతੑ࣭,” Japanese, Ұڮ だがわཧ࿦ゞがぜてゖひゆ 2007, Mar. 2007 ▶ D. Monderer and L. S. Shapley, “Potential games,” Games Econ. Behav., vol. 14, no. 1, May 1996 れふアてを゚だがわ〣ແઢ௨৴〭〣Ԡ༻ ▶ S. Lasaulce and H. Tembine, Game Theory and Learning for Wireless Networks: Fundamentals and Applications. Academic Pr., 2011 ▶ K. Yamamoto, “A comprehensive survey of potential game approaches to wireless networks,” IEICE Trans. Commun., vol. E98-B, no. 9, Sep. 2015 47 / 47
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50. [Yamamoto+2011] K. Yamamoto, K. Haneda, H. Murata, and S. Yoshida, “Optimal transmission scheduling for a hybrid of full- and half-duplex relaying,” IEEE Commun. Lett., vol. 15, no. 3, Mar. 2011. [Yamamoto2015] K. Yamamoto, “A comprehensive survey of potential game approaches to wireless networks,” IEICE Trans. Commun., vol. E98-B, no. 9, Sep. 2015. [ӉҪ 2007] ӉҪ وࢤ, “れふアてを゚だがわ〣جຊతੑ࣭,” Japanese, Ұ ڮだがわཧ࿦ゞがぜてゖひゆ 2007, Mar. 2007. [ࢁຊ 2012] ࢁຊ ߴࢸ, “だがわཧ࿦〣ແઢ௨৴〭〣Ԡ༻,” Japanese, ৴ֶ ࢽ, vol. 95, no. 12, Dec. 2012. [ࢁຊ 2016] ——,“ແઢ௨৴〝れふアてを゚だがわ,” Japanese, ܭଌ〝੍ ޚ, vol. 55, no. 11, Nov. 2016. [ࢁຊ 2017] ——,“な゚゘ぼひぷゞがぜ〣֬཰زԿղੳ〣جૅ,” Japanese, in ৴ֶٕใ, CQ2017-57, Aug. 2017. [Ԭా 2011] Ԭా ষ, だがわཧ࿦৽൛, Japanese. ༗൹ֳ, 2011. 50 / 47