Chapter 4

_�Dõðql

4.1 HÜA!˘‰�Ò

4.1.1 ˘‰�Òql

øOíHÜA!l��_øu¨Ös_¶}, }�u=1qìJ£HÜA�˙� H

ÜA�˙ùÆŠ�²ì_ñíç3WÑ, *QYm7� Ç,m7ígMƒ²µ|h

í¥@, _ñÊç3¬˙ç2øÍí|c¥�·u˘k¥_jÞí{æ, 7=1

qì†u d¸ø_u¦íYjd†, _ñD_ñ5ÈC6u_ñDÍ$5Èí�

�j�·.⧃vYjd†íd¸�

˘‰�Òí=1qì3bu*s_!…¡b (x, X)íqìÇá,1 “x/X”[ý˛

�˘‰í_ñ.â* X_bå52‘²| x_U{, 7;Wú2U{_bíÖ›ø

)ƒ.°íÀç (¦u�»çí/_ªWyÎJ2Ý·b), BbI¡b y[ýú

2U{í_b, Sy¸ Ny†}�[ýú2 y_U{íÝÀ,çJ£2ÝíAb, F

1Ék˘‰�ÒíqlJ£˘‰�Òí|_d_í«n, Walker and Young (2001) ød2�ÝÌ�í�Ü� Chen

and Chie (2003) †u;Wv¹ÇÕql7A�˘‰�Ò� FJ…¹í�Òql3bu•/ Chen and Chie (2003) ød

íqì�

69

70 CHAPTER 4. _�Dõðql

J y = 0, 1, . . . , x�Bk˘‰ÇÝíȽÙb (drawing periods), BbÇÕJ¡

b r̄[ý, FJà‹˘‰�Òør‚nÇÝøŸ, † r̄ = 7�

©ø‚.°×üíÝÀ,ç Syøâ©‚óú@í2ÝAb NyFÌ}, à‹/

øÝáí2ÝAb Ny = 0, †v_ÝáíÝÀ,ç Syøø–Úlƒ-ø‚� Ê

F�íÝá³å, w2|?D�ù˘Iíÿ˘kåÝíÀç Sx7, çÍ, à‹v‚

åÝí2ÝAb*ÿ Nx = 0, v‚íåÝÀç6}ÚlB-ø‚, 7/ÄÑ2Ý

œ0óúüíÉ[, ¦É�åÝíÝÀ,çn�œ}\Gƒ-ø‚, FJøOF

˚íÝÀß� (rollover) ùÆNÌA2åÝ7å˘Úlƒ-ø‚íÛï, 7d.

,6N|, ÝÀß�ÿu¨A˘‰�ÒßÞ˘‰#Ï (lottomania) í3bŸÄ,

c Beenstock, Goldin and Haitovsky (1999)�2 °ší8$6êÞÊÅq, *

Ç4.1 2BbÿªJÀUõƒ�Òí�»ç�DÝÀß�×Ûpéí£óÉ�37d

.,‡ú‡ø‚�ÝÀß�í�»¾ (rollover draw) péòk‡ø‚³�ÝÀ

ß�í�»¾ (regular draw) íÛï¢ÇÕ˚Ñm=^@ (halo effect), ck

Creigh-Tyte and Farrell (1998) ¸ Walker and Young (2001)�

©‚ÝÀ,çí×ü·OÎøìíªW}º5, \¦}lø�Òí,�»ç

S{˘,{˘íà¤uÑ7X@G/‘J£^û\X|¸þ}9‚\µ, {ê˘(

íç� (,ÝÀ,ç) nádÑ˘Àí}ê,4|(yOÎ.°ÝáíÝçªW sy�

ì}º5,5 FJ.°ÝáíÝÀ,ç Syuâ�Ò�»ç S� ˘0 τJ£ÝçªW

2F‚í˘‰#ÏuN�Òí�»çÓOÝÀß�¾íTò7�×ÙÓÅíÛï�3Ç4.1 Ñ«É �42² 6� —N˘ívÍ’e, n‚* 2002� 1~ 22Uƒ 2005� 1~ 21U,u 314°’e, ’eVÄÑ

�2M¬Åtï˘‰� æ¦, æ� http://www.roclotto.com.tw�4ÄÑ\*,�»ç2¦•7ø¶M, FJv˘0¢˚Ѧ|0 (Take-out Rate)�5Jü—NÑW, ª}êí,ÝÀ,çdìÑ�Ò�»çí56%, Ê�ÝÝÀ}ºÑ5(, ”ìíÝÀOÎ38% � 12% �

15%¸35%íªWYå}ƒåÝBûÝ�

4.1. HÜA!˘‰�Ò 71

0

5

10

15

20

25

30

22-

Jan-

02

22-

Apr-

02

22-

Jul-

02

22-

Oct-

02

22-

Jan-

03

22-

Apr-

03

22-

Jul-

03

22-

Oct-

03

22-

Jan-

04

22-

Apr-

04

22-

Jul-

04

22-

Oct-

04

億銷售額

獎金順移

頭獎

Ç 4.1: ˘‰�»çDÝÀß� (from Taiwan Data)

syú6u°²ìí, É[�ªJŸA,

Sy = sy(1 − τ)S

∑x0 sy = 1

y = 0, 1, . . . , x

‡Þ�Üê7˘‰�ÒX#«Tíj�5(, Ék�Òí=1qì´”-ø_Þ

²Ûbªø¥²ì, 6ÿu �˘‰�Òª?íÛ°¾� � ¬ d.Tܢ‰Û°í

j�×_}Ñs�, �ø�j¶u‚àø<þ}%ȉbC6u˘‰�ÒíõÒ’

eVR,�˘‰�ÒíÛ°ƒb,6 7�ù�j¶†u*H[4_Aí²µÑ|ê

õ, eH[4_Aí˘‰Û°ÑÜ4²µ-í|_!‹, Bk˘‰�ÒíÛ°¾†

6¥6uñ‡d.,œÖç6Sàí}&j¶, �E�íè6ªJªø¥¡5 Mikesell (1994)�Walker (1998)� Farrell

and Walker (1999) ¸ Farrell et al. (1999) ¥�¹�

72 CHAPTER 4. _�Dõðql

uH[4_A˘‰Û°í‹,,7 HÜA!l��_í}&j�ªœ�¡k(6, O

uJæ”4¸�ÌÜ4¦H_ñíWÑ� OÎ¥šíhõql, ˘‰�ÒíÛ°¾

ÿ²ìÊk˛�˘‰íAbJ£F˛�íÀç¥s_ÄÖ7� BbI�Ò,,u�

i_A (i = 1, 2, . . . , N), †©_AíF)à Ii[ý, Bk_ñ˛�˘‰í�œJ

£²µ¬˙, BbøG&4.1.2� (Ü72) H ÜA�˙í¶}yÌ�zp, 7Ñ7

}&,íZ‚4, ‹,_�qlí3bñíuÑ7ªœ.°Æ�¶íÍTô^, B

bªø¥cq©‚©_AíF)îÑÕÞ#ì7/ó°, Ii = I, ∀i��Ék˘‰�

Ò=1qìí3b¡b, J-BbJø²¾M,!cÜ5,

M = (x, X, r̄, τ, s0, . . . , sx, N, I)

4.1.2 HÜA�˙

ÑBóA} �˘‰? ¹Uø−AÐÞúíuø_.t� (unfair)í �, ÑBó

ÖbA´uø�â˼ª˘‰�Ò³ ‘DAÐí�-? ˛�˘‰íWÑ, ˝§u

%¬Ü425-í|_!‹? ´uÉuq-Q�íI¦7˛? ¥<½æí��øò

uû˝˘‰�Òíç6|>E�í¶}, 7Bb6ó]�à_ñ˛�˘‰í�œç

Í.}É�ø_, 6ÿuz, HÜA�˙íql6æ�Ö�íª?,Ñ7Ó‹HÜA

�˙íª]�, Bb}�謠.°ç6‡ú˘‰˛�WÑFT|í.°j„, J

Ü�}&Dõ„û˝1½íj�, ql|¯_íHÜAWÑ�

%Èçð*'oJ‡ÿÇátÇ�Aé ²íWÑd¯Ü“íj„, o‚%Èç

ðuJã‚^àÜ�Vj„AéÞú.üì4-í²µWÑ, FJ_ñu´}¡D

² (xÔ) ÿ²ìÊk_ñíêÔG�J£FÞúí �¥s_ÄÖ§ ¯(

7¥jÞíd.óú,ÿV)ý, ªœhíd.è6ªJ¡5 Hartley and Lanot (2003) ød�

4.1. HÜA!˘‰�Ò 73

í!‹, çFÞúíu.t�í �, ã‚^àÜ�µsBb{êÔ�f6".}

¡‹� êÔ=ß6†.øì, Í7úÛõÞºíhôBbªJêÛ, Ab¡D.t

� � (�˘‰) íä0Ýò, 7/%%ÖbA˛�˘‰í°v6}˛�\Ô,

Ñ7?D·HAb°vË�êÔ�fDêÔ=ßíWÑ, Friedman and Savage

(1948) þtJ �Ýo4� (non-convexity) ^àƒbVj„5, FbwÑøOAb

Ê‹óÕ”óúQJ£óúò®ÄvÑêÔ�f6, ‹óÕ”�ks65Èív`

†˘kêÔ=ß6, âkÝo4^àƒbÖÔí¼¨�êÔRß, U)v^àƒb

üõªJj„°v˛�˘‰D\Ô¥s�êÔG�úíWÑ, .¬¹Zà¤, Ý

o4^àƒbEÍæÊ@à,í˚Ø, WàÝo4^àƒbÿ̶j„õ„,Ab

M/˛�˘‰íÛï (¹U›¸û_‹óò®), 7˛�˘‰A¨í‹ó$l}06

1Ýv^àƒbFéýÖÕ2Ê2¨í8$� Kaplan (1987) Ê1Ÿ‹“×�ê

W˘‰íb_˚ªW‚š|ŒÿêÛò®ýBÿAí˚A¨{%.ÉøŸ˛�¬˘

‰, Kitchen and Powells (1991) hô‹“×ýô˘‰X|íðl|Œ’eêÛ,

‹óúkw2ûôíðê©�˘‰X|�éOí£²�à�8

e/.�Ýo4^àƒbu´ÿuAéêÔG�íöõ$�, h©t%ÈçJã

‚^àÜ�$ÇVj„AbÞú.üìí8”, !…,ÿuwìAbíLS²µW

Ñ·u¯¯Ü41/‡(ø_í, ‹,ã‚^àÜ�F˙8í#×l�?‰, U)

AbÊLSv`ÞúLS �·ªJ£üËwø �íÝÀ!ZJ£2Ýœ0, 6

ªJ/qËl�|ã‚^àM, Ĥ_ñu´}¡D �ÿêr²ìÊv_ñŒÞ

^àíRß˙�,7, FJÊÞú˘‰¥é.t� �ív`, h©t%ÈçÉ?

J_ñíŒÞ^àÑRß]Ó (êÔ=ß6) íÜâj„5, Ou*õ„,Bbº

õƒrÖDã‚^àÜ�ó�peíÛï, ÊÊéýŒÞ^àÞj„?‰.—íø

8BkF)òQúk˘‰Û°�àíû˝†uóç.ø_, Wà Rubenstein and Scafidi (2002)� Herring and Bledsoe

(1994) ¸ Stranahan and Borg (1998) ÿ}�×Û|£²� Š²J£ÌéOÉ[ú�.°íû˝!‹�

74 CHAPTER 4. _�Dõðql

Þ, FJJ*ÝŒÞ^àÞ (-Üçíi�) íhõVj„AbÑS˛�˘‰íW

Ñ@v}y¯_�

�˘‰wõÿu_ñÞúüì¸Ü ( ·)D.üì‚) (ª?2ÝíÀç) 5È

Çgí!‹, Ê Kahneman and Tversky (1982)íû˝³å, wéýøOA1³

�?‰ªW�Kíœ0R�, ¦7H5íuSàø<¡íj¶ (heuristics) C6

˝/<%ð¶†VªWÇg, Ĥû__ñÊœ0í‡i,ñqßÞýíRÏ

(biases), û˝N|øOAñqò,êÞœ0Qí9Kœ0 (Wà=)˘‰ííÝ)

ºQ,êÞœ0òí9Kœ0 (WàÊò§t˜,§\Îó), 7_ñ…™6}¬

�A] (overconfidence) kÇg(í!‹, Ÿ…ࡶ†VTÜøOí9Ó…

™ÿx�óçí¡N4DZ‚4, OuçÞúƒ˘‰¥é.t� �ív`, Çg

RÏFßÞíA…ÿ�}íú<7�FJ*wø-Üçíi�Võ, �úk2Ýœ0

í˜Ïwø� nuAbÑBó˛�˘‰íõ”ŸÄ, 72Ýœ0ÇgRÏí!‹†

uAbʲµ¬˙2§ƒ¡¶†í�àF¨Aí, Rogers (1998) J£ Griffiths

and Wood (1999) ÊFbdı³åÿ�è7rÖAb�˘‰vq"íÍ$4RÏ

J£FðÞ|Ví.Ü4WÑ, 3b¨�� �−„í˜g� (illusion of control)�

�H[4¶†� (Representativeness Heuristics) ¸ �¡Z4¶†� (availability

heuristics) �W�

�−„í˜g� uâ Langer (1975)FT|, ÊF‡ú ²WÑíû˝³åêÛ,

N¬_A9‡í›‰, AbúkAŠœ0í3hã‚øòkîhíœ0, _ñ}w

ÑAЪJ�â*3íj�®ƒZ‰C−„!‹íñí� Ê¥ší]1XM-, _

ñ¿]FbË�ªJHÝÍ$í�x, FJAÐV²Unªœ�œ}=, 7˘‰�

Ò,y�F‚íùðu�âTXù“}&í˘‰p�Ñ“, ,HíÛï·¥ø|A

4.1. HÜA!˘‰�Ò 75

bÊ˛�˘‰ív`, ‡ú.°íU{� �AB<…²U� (conscious selection),

7¥<r¶·u �−„í˜g� Fù–í‡iÏÏ��H[4¶†� ¸ �¡Z4¶†�

†uâ Tversky and Kahneman (1974) FT|, F‚H[4¶†uNAbʇ

iš…u´â/‚ñFßÞíœ0v, %%}â‡6D(6íH[4CóN�dÑ

©¾í™Ä, 7F‚¡Z4¶††uNAbÊÇ,/9KêÞíœ0v, ¦}J

9KêÞíÕ”u´ñq�;чií!Ä, 6ÿuzBñq;ƒC%ªJõƒ

=ƒí9K, ÿ}wÑéN9KêÞíœ0Bò�

*H[4¶†Ô�ù.|VøOA"íø_!…˜Ï¢˚Ñ �üb¶†�(law

of small number), NíuAbñqYW¬ ýbíš…%ðV‡i9Kœ0í£

ü4, WàJ¬ �‚ÖŸÇ|©Uí8$, µóAbwÑ-ø‚¢Ç|©U

íœ0ø}Ó×, Çø_óÉíWäuF‚í � �í#Ï� (gambler’s fallacy),

�í �¥7}ã‚�ùŸÓœ‚ší!‹D�øŸu×ÛŠóÉ, 6ÿuz �

¥7}wÑ-ø‚¢Ç|©Uíœ0.Ó¥Á, }ßÞ¥s�ÇgRÏíWÑ·

uÄÑH[¶†cq¬ í%ðúköõ9Kí‚ñx�/�˙�íH[4, FJ

nû__ñ;Wüš…ÿd¬}íR�, 9õ,, ¥<üš…í9K1̶êrH

[‚ñdчiœ0í²µ!�� ´�ø_�àAbÇgöõ2Ýœ0íRÏWÑ

ÿu¡Z4¶†, ¸.�2ím7óª,�2ím7Ê7½2\�;íä0œò,7

©ŸÉ�ýb�_A2íÝím7%âýñ…<½µÈ£â2(, U)Ab/qË

wÑ2ÝuøK'�7/ñqí9, ¥7�I�Ò,æÊbJNl�ðí9õ,

7˘‰�Ò5FJ}ßÞ �˘‰#Ï� íÛï, |3b6uÄÑýñRšŒ`í^

‹, Ú+íÝÀß�ÊýñM/.iË#|-, û_u´˛�˘‰AÑAbilT

Üí98� °š§ƒ¡Z4¶†�àí´¨�F‚í �(3À:� (aversion to re-

76 CHAPTER 4. _�Dõðql

gret) Ûï,9 vÛïN|ÓOýñòÕËf×)ÎçÝÀí2ÝAím7, Ÿ…

³�˛�˘‰íAbñq>ƒ(3 (ÄÑà‹ç���, 2ÝíAÿª?uAÐ),

ª7ª?Ó‹-Ÿ˛�í<è�

FJ, ã¯,H®�úkAb-ܲµíõ„hô, BbFqlíHÜA�˙ø

*�šú_Ab�ZÊ˘‰�Òíu¦WÑÇá,10 }�u,

• ˘‰#Ï,

• AB<…²U,

• (3À:�

˘‰#Ï

˘‰�Ò|½b/cíõ„ÛïÿuAb˛�˘‰íÀçÖ›¸åÝÀçíd_

×ü (jackpot size) pé×Û£²íÉ[ (¡5Ç4.1), 6ÿuF‚í˘‰#ÏÛ

ï, 7˘‰#Ï5FJ}|Û, 3bu¦Äkýñ��7þ}×Ví·<‰ƒ˘‰

�ÒßÞÝÀß�íuæ,Þ, ‹,Abñqò,AÐ2åÝí9Kœ0, ĤB

bql_ñ˛�˘‰í¡D˙� (participation level) }�§ƒåÝÀçíd_

×üJ£úk2åÝœ0í3hwøs_ÄÖí�à,11

αi = φ(J, pi)∂αi

∂J> 0,

∂αi

∂pi> 0 (4.1)

9¡© Statman (2002)�10¥³HÜA�˙íql3b6u•/ Chen and Chie (2003) v¹íqì, |×íÏæÊk_ñ²ì˛�˘‰íj�

.°�11Clotfelter and Cook (1989) Êwz2 (�71Ü) 6Tƒ, ˘‰5FJ}¥ó§¡CíŸÄ, Î7åÝÀçÜùA5Õ,

_ñ…™úk2Ýœ0í˜Ïwø6�.üí�à, ŸÄuÄÑ_ñó]AЪJ�â/<Ôyí}&j¶Z¾2Ýí

œ0�

4.1. HÜA!˘‰�Ò 77

αiu_ñ˛�˘‰2F)íX|ªW, φu_ñ¡D˙�íƒb$� (participa-

tion function), 7 JD pi†}�uåÝÀçJ£_ñúk2åÝœ0í3hw

ø,12 s6úk˘‰íX| αiI·u£²í�à�

φ íü~$�ÓOFn�_�-Z�-_ñTܽæí?‰òQ7�.°íq

ì, Êh©t%Èçí}&j¶³, _ñÄÑ\eÑêrÜ4/Ë�#×íl�?

‰, FJ_ñí˘‰X| αiIuJåÝÀç JD2åÝíîhœ0 pi(h©t%È

çe_ñí3hwøÑîhíöõ2Ýœ0) Fu°l�|íã‚^àM²ìí,

FJ φʬ FrÆíiHÿuJã‚^àÜ�dÑ}&íj�� Ĥb²ì_ñ

˛�˘‰íX|Àç, ÿõBbuàS õ&Ab˛�˘‰víG�, u˘kÜ4

´u.Ü4? O·Þíz¶, AbÄÑÊd²µvñq§ƒø<¡¶†í�à

7û_wWÑ�O/�˙�í.Ü4, .¬˝§u.Ü4ƒBó˙�, _ñÊÞú

˘‰vu˘kÝí.Ü4á (êr.Ê˛2Ýœ0í×ü)? ´uÉ�¶}í.

Ü4 (ò,7…™2Ýíœ0)? ªJzucCc��7BbwÑAbʲìu´˛

�˘‰ív`, úkv˘‰íã‚Ñ{ (expected value) ´ux�øì˙�íÜ

>4 (}°v5?ÝÀíÜù˙�Dwóúí2Ýœ0u´9�), 6ÿuz, _ñ

øjÞ7jÇ,ã‚Ñ{í½b4, øjÞº¢¬Mò, (—h) A™2Ýíœ},

FJ¥³Bbcì_ñ˛�˘‰íj� (φ) ´uJã‚^àÜ�dÑl�í3W,

Ou_ñºuà3hwøíœ0V‡il�(íã‚Ñ{u´M)˛��13

à‹OÎBbíqì, _ñu´˛�˘‰ÿ«õ_ñ3hã‚í˘‰Ñ{^àM

12¥³í JõÒ,u˘(íåÝÀç, OuÑ7jZBbzŸ, BbJ JH� τJ , à‹³�Ô�zp, %(í J·uN

˘(íåÝÀç�135FJ}²ìé_ñË�.�üíl�?‰, uÄÑÖÍBb¦³ƒ7úk2Ýœ0í¬k—hu_ñ˛�˘‰í

ÉœÄÖ5ø,7/_ñBu—hÿ}�íBÖ, OuàS²ì_ñü~í˘‰˛�¾, Êd.,º´³�ø_™Äíd

¶, ‹,ã‚^àÜ�üõ�w}&íjZ4DÃã4, FJBbn}Jõ„hô!¯Ü�_�íj�Vl�_ñí˘

‰˛�¾�

78 CHAPTER 4. _�Dõðql

expected utilitywith subjective probability pi

Expenditure

Jackpot

Income

monetary unit

utility

Ç 4.2: _ñ˘‰˛�¾í²ìj�

u´òk_ñí�áF)^àM7ì (¡5Ç4.2)� BbcqF�í_ñ·u˘k

êÔ�f6 ui(c) = log c, 7©Ÿ_ñ·uYW-23hí2Ýœ0Vl�炢

‰íã‚Ñ{^àM, FJ_ñí˘‰ã‚Ñ{ EUiÿ[ýA,

EUi = (1 − pi)ui[(1 − αi)I] + piui[(1 − αi)I + J ]

= (1 − pi) log[(1 − αi)I] + pi log[(1 − αi)I + J ]

(4.2)

J_ñí3hã‚Ñ{^àMòkç‚F)^àM EUi > ui(I), †_ñ}wÑ�

‚ªÇª7˛�˘‰, ¥5†.}˛��14

Bk_ñ@v�Öýnßá? âk¥³í_ñ1.dh©t%Èçqìí_ñ

øOË�êríÜ4, FJBb6ÿ.é_ñ…<Ë J°3hã‚Ñ{^àMí

”ד (maximization), .¬_ñb6.} ›&AÐ, ĤBbªø¥cqç

_ñ²ì˛�˘‰v, _ñ|Ö�ƒ3hã‚Ñ{^àM�kç‚F)^àMÑ¢

EUi = ui(I)(Bý.Ï›), ²Æuz, ‘K� EUi > ui(I)H[O_ñíÜ4w

ø, OuwÜ4ºu�Ìí, ¥øÊ‘K� EUi = ui(I)™,� FJ, ‚àv‘K

BbÿªJj|_ñí˘‰X|ªW, 7Ê°j¬˙2Bbš�d7ø<“, 514ÖÍBbø−¥1.uêr£üí˘‰ã‚Ñ{, ÄÑÎ7åÝ5Õ@v´¨�wFíÝáJ£.°Ýáí3h2

Ýœ0nú, .¬d.,6XMåÝ5ÕíÝçúk˘‰íã‚Ñ{1³�Ö×í�à, ĤÖbíT¶?É5?‹p

åÝ7˛, c Walker (1998) C Mason, Steagall and Fabritius (1997)�

4.1. HÜA!˘‰�Ò 79

? J±×k (1− αi)I‹,¢¦7 logƒb, FJBbôI EUi(Þøá2í(1−αi)IJjZ°j, Ĥ°jí‘K�ÿ�Ñ,

EUi ≈ (1 − pi) log[(1 − αi)I] + pi log J

= ui(I) = log I

(4.3)

FJ,

log I = log[(1 − αi)I]1−piJpi

I = [(1 − αi)I]1−piJpi

%â�ácܬ(, BbªJ)ƒ,

(I

Jpi)

11−pi = (1 − αi)I

αi =I − ( I

Jpi)

11−pi

I

²Æuz, �7F) (I)� åÝÀç (J) J£3h2Ýœ0 (pi) J(, Bbÿª

J;W (4.3) �l�|_ñIp˘‰íX|ªW (αi) J£X|Àç (αiI) 7, F

JÊBb_Òí�Ò³, _ñ©‚bIÖýÂÊ˘‰íX|,ÿâ-Þí‘K�V

²ì�

α∗i =

I−( IJpi )

11−pi

I if J ≥ I15

0 otherwise(4.4)

ªø¥�ð,�u´�¯¯ç� (4.1) �íqì, Bb}�õ J¸ pií‰�u

´úk α∗i�£²í�à� íl, ç JÓ‹ív`, α∗

i 6üõ}�OÓ‹� ÇÕ, ç

_ñwÑêr.}2ív` ”pi → 0”,

α∗i = (I − (

I

1)1)/I = 0

15péË, α∗i ük1, 7‹pÌ„�íñíÿuéw?D×k0, ßÞø_ij (corner solution)�

80 CHAPTER 4. _�Dõðql

[ý_ñø.}ILSø„ �˘‰, ó¥Ë, J_ñg)AÐÝ�]-}2

Ýíu ”pi → 1”,

α∗i = (I − (

I

J)∞)/I � 1

†_ñ¥7}zF�í“ �˘‰� FJ piú α∗i 6u£²í�à�

AB<…²U

˛�˘‰í¬˙}As_¼¨, _ñílb²ìk˛�í˘‰b¾, Í(n²Ï;

bíU{C6uÚ7²U, Ĥ�˘‰u˘kù¼¨²µí½æ� AB<…²UÌ

Ou˘‰�ÒÇø_½bíõ„Ûï, 6ÿuNøO¬Vúk.°í˘‰²U ¯

�O.°íRß×ü, Wà¬V�Zÿ.}wÑ123456¥ ²UDwFí²U

¯wõË�ó°íÇ|œ0, C6uXƒÔyí9KC�n, ¬V}JóÉíU{

 ²U, ¥<·uAB<…²UíÛï, Äѧƒ �−„í˜g� í�à, ¬VuJ

ÝÓœíG�V²ÏÓœÇ|í˘‰U{ ¯, 7�Ò,y�F‚íùðŠ�}&

˘‰U{ívÍ*�dÑTX×VíÇø_²Ï (D$��Ò,íxX}&��æ

�°�5\)�

Ñ7�š_ñAB<…²UíÔ4, Bb5?ø_� X&�7/uâ0D1 A

í²¾ bi, Xu5‡ì2˘‰ªX²ÏíU{,b� J bi52�/ø&�u1, †

[ývbåu_ñíAB<…²U, ¥5Ju|Û0, †[ý_ñúwbå³�Ô�

íRß, ²Æuz, biÿu¥ø_ñAB<…²UíU{ÀÀ, FJ*Ç4.3 2Bb

ªJÀUõƒ, Ê¥ší biqì-, v_ñFË�íAB<…²U,u¨��_b

å, }�u1�3�5�6�8�12�14�18J£19�

ÇÕì2ø_hí‰b zi[ý bi³&�Ñ1í_b,¸, FJÊ,ø_Wä2

4.1. HÜA!˘‰�Ò 81

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20

1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 0 1 1 0

01 03 05 06 08 12 14 18 19

TotalNumbers

Selected Numbers

ib→

Ç 4.3: AB<…²Uzp¸W

zi = 9, 7/ 1 ≤ zi ≤ X� ÊBbíqì-, à‹ zi(�Ô�RßíU{_b)

ß� x_, xu5‡ì2©"˘‰.⑲ø U{í_b, †_ñÉ}Jv U

{ ¯˛�˘‰� à‹ zií_b×k x_, †_ñ}*vU{ í§ ¯2Óœ

‘²� ²U ˛�˘‰, J‡ÞíWäVzp, Jcq x = 5, 7ĤªX_ñ

²ÏíU{ ¯ÿ,u� C95(= 126) �bu zií_bük x_, †_ñF‘²í

U{ ¯, ©"Î7Ÿ�Ô�RßíU{5Õ, wìíU{ÿ*pÎ�Ô�RßU

{(íbå (X − zi) ³Óœ‘²�

(3À:

F‚í �(3À:� uNçAbÊdê²µJ(, êÛ!‹.`Ü;ª7wÑà‹

ç�²ÏÇø_µI �6r� }yßí-ÜTà, FJÊ˘‰�Ò³å, (3À:

ÿu·H³�˛�˘‰í_ñ}§ƒ�Ò,u´�A2íÝí!‹�à, ª7Z‰

˛�<èí8$� cà�Ò,�A2íÝ, ª?}éµ<ç�³�˛�˘‰íA>

ƒ(37wÑbuç���, 2ÝíAª?ÿ}uAÐ�ó¥Ë,bu�Ò,³�A

2íÝ, ç�³�˛�˘‰íA¥7}>ƒ/�˙�íC1, ¥uÄÑFb}ßÞ

�´ßB³�� �76.}2� í^k-Ü�

Ñ7l�,íjZ, Bbcq_ñ ií^àƒbÉ°Qø‚í¾‘, FJ_ñí

82 CHAPTER 4. _�Dõðql

^àƒbDã�Ì„�}�ªJŸA,

ui(c) = log c (4.5)

c ≤ (1−αi)I + π (4.6)

,�2, cu¾‘, Iu�áF), αi¸ π† }�u˛�˘‰íX|ªWDF×)í

ÝÀ� Ĥ, ‡ÞFz³�˛�˘‰í_ñw^àíòQ²ìÊk�Ò,ííÝu

´�A×)í8$, †[ýà-,

ui(c) =

log[(1 − θi)I] if α∗i = 0 and Nx = 0

log[(1 + θi)I] if α∗i = 0 and Nx = 0

log[(1 − α∗i )I + π] otherwise

(4.7)

Ê,�í^àƒb2, _ñbu³�˛�˘‰ (α∗i = 0), w^àíòQuÇÕ²

ìÊku´�A×)íÝí‘K,, Ju�A“• (Nx = 0), _ñÄÑ>ƒ(3

7û_^à±Q, ¥5Ju³�A)ƒíÝ (Nx = 0), _ñ†}>ƒ^k7û_

^àÓ‹� θiu©¾_ñ-ÜTà˙�×üíN™, �k0ƒ15È, θià‹B×[

ý.�˘‰í_ñBñq§ƒ�A (u´�A2íÝ) í�à, *Çø_i�Võ,

θiB×6[ý�Ò,_ñD_ñ^à5ÈíÖ4BQ, FJ�â θiíql, Bb

;hô_ñ˛�˘‰íWу|(˝§ub²kÖT“� .} Ê˛�AFË�

í, ´u}‰)yñq§FAF�à� fflœkAÐD�AíÏæ�

Ĥ,ÉkêcíHÜA�˙Fqìí3b‰b, BbcܦÑ1Jø²¾N[ý,

N = (pi, bi, θi)

7.°_ñÈíæ”4ÿéÛÊ¥ú_‰bMíÏæ,�

4.1. HÜA!˘‰�Ò 83

4.1.3 ){j�

ÊÇázp RL J£ GA àS){FqlíHÜA�˙5‡, Bb�.bl 7

ju´©ø�Æ�¶·_¯·H_ñ˛�˘‰íWÑ? OÎ�úıí;¶, .°í

Æ�¶ÄÑ.°íÆ�Ô47®A�w_¯j„í_ñWÑ, ĤBbø�e RL

D GA s652u´Ë�¯¯_ñ˛�˘‰íWÑÔ4�

íl, *_ñYÕm7íi�Võ, �˘‰Î7§ƒ_ñ…™wø‰b (cogni-

tive variables) í�à5Õ, þ}‰b (social variables) 6uø_½bí�àÄ

Ö, F‚íþ}‰buN}�à ²6 (˛�˘‰) WÑíø<ÕÊ=1,16 6ÿu

z, _ñJ™Tñq:;ƒ˘‰í=1ç2 (WàI·¦ƒTŠ), }ĤŒéÓ

‹_ñ˛�˘‰í<è, 7*þ}‰bô.¬Vís_ù˘±È, °b^@ (peer

effect) J£ã¼^@ (bandwagon effect), 6·éýFAúk_ñ˛�˘‰WÑ

�O.üí�à,17 FJ SGA uBb5?íÆ�¶5ø� 7*_ñTÜm7FI

‘-2í˙�×üVõ, °˘_A�ç3í RL D MGA, RL ÄѪœ_¯TÜ

Àí98 (¯¯˘‰íÔ4) J£q§8>˝¬CÉ%âòg²ìíWÑ (DA

b˛�˘‰íWÑø_), FJ RL uBb5?íÇø_Æ�¶, BkÇø_.Sà

MGA íŸÄ, †uÄÑ MGA Ê¥³FqlíHÜA�˙í){,6�j„,

í˚Ø,18FJQOBbø}�J RL D SGA V){‡ÞFqlíHÜA�˙�19

16¡Î˜H¸� ŠNp¸�?+ (2002)�17Browne and Brown (1994) íû˝ÿN|,ã¿×çÞ�˘‰í|7,l‰bÿuw�¤Cð˘¡D˘‰�ÒíϬ

˙��18dBbÿ'Ø j„_ñÑSÊø°vÈqºË�.°í3hœ0J£(3˙��19%(à‹³�Ô�zp, †Tƒ GA ív`·uN SGA�

84 CHAPTER 4. _�Dõðql

J RL ){HÜA�˙

�è�úıBbF�Üí RL «T¼˙ (ªJ¡5Ç3.1, Ü36), RL «TíÔ4

ÿu¬ [ÛóúßíµI-Ÿ\²Ïíœ0ª–wFíµIøóúVí×, 6ÿ

uz, _ñ²ÏµIíYWu²ìÊ.°µIí¬ [Û2wr¶µIíóú×ü,

FJZA RL Æ�¶í|ÉœÄÖÿuY¬ óú[ÛV²µíœ0Ÿ†� Ê�á

“v, .°íµIÄÑË�ó°íb²�b, FJ_ñíµIÕ¯íœ0}º˘k

ÌG}º (uniform distribution), [ý©ø_µI\²Ïƒíœ}Ì�, 7(%

â%ðíÚ+, vµIÕ¯íœ0}ºøƉA®�ª?í$� (ª?˝RC6¬

R), 7P�¼Ý (peak) íµIÿu¬ [Û|ßíµI, 6[ývµIyŸ\²

σíœ0Ñóú|ò�

ÊøOí RL ql,, wqlíj�Ö*ªX_ñ²ÏíµI|ê (*µIÞV

){),20 6ÿuz, ql6íl}d¸ø_�Ì/Ñ×àíµIÕ¯#8_ñ (Ä

Ñ RL F·Hí_ñu˘k\�íç3G�), Í(é_ñº¯œ0²µíŸ†V

þtµIÕ¯2í®�ª?4�¥³Bb.H�¥ód, ŸÄuBbg)ʲ¾N2

F«ní·u˘k-ܵÞí‰b, .�u_ñí3hwøœ0� (3À:˙�Ý

Bk_ñí²URß, w|cíj�@v˘kÚª�í|c (�ßåËâòBQC

âQBò), 1._¯ø¥<‰bòQeѵIøOªÓ-Fkí|c,21 ĤÑ7

b°v\� RL íÆ�Ô4 (Y¬ óú[ÛV²µíœ0Ÿ†) J£uè-܉

bÑÚª�|cí4”, Bb RL í){úïø*-܉b…™�²ƒ-܉bí

|cj², (¡5Ç4.4)�

20¥Dç� Roth and Erev T| RL Æ�¶uÑ7Tܡ�2_ñíç3WÑ�É�21¥ÿuÑSBbg) MGA ._¯“V){íŸÄ, J_A�ç3íi�Võ, ø-܉b){�²AªX²Ïí

µI}‰)̶j„, _ñ.@vu  �‘²� .°˙�í-Ü>§, 7u  �Ç,� b./&M´u^£ñ‡í-܉

b˙��

4.1. HÜA!˘‰�Ò 85

tAt qq ,,A ,−+

Probabilistic choice rule

tti cu ρ−)(

updating

1,1, , ++ −+ tAtA PP

Probabilistic choice rule

updating

iP Xiz

iθ

θ∆±P∆± Xz∆±

Ç 4.4: J RL ){HÜA�˙

_ñílb²ìú_-܉bMí×ü, Í(©__ñÊ.°í-܉b-®A

Ë�s_µIªJ²Ï k (k = A+, A−), FJ_ñíµI˛ÈÕ¯Ñ k × k × k�

µI A+[ý_ñø} �²,|c� ú@í-܉b (Wà,^3h2Ýœ0� (

3À:˙�Ó‹C6yòí²URß), ó¥Ë, µI A−†[ý_ñø} �²-|

c� ú@í-܉b (Wà-^3h2Ýœ0� (3À:˙�±QC6ÁýRß²

U)� s_µIí�áb²�bîÑÕÞ#ì7/ó� qA+,0 = qA−,0,22[ýøÇá

ív`¥s_µI\²Ïƒíœ}Ì�, 7s_b²�bí‹,ÿ˚Ñ,b²^‰

Qt = qA+,t + qA−,t�µI#“íj�à-,

qk,t+1

=

qk,t + (ui(ct) − ρt) if k is chosen in period t, and

leading to utility ui(ct).

qk,t if k is not chosen in period t.

(4.8)

OÎd.,3¼í}&�‘, °šËBb6²Ïé¡5õ};WFõÛíÑ{ª

22ÄÑ˘‰�Ò³�¥£j (close form), FJBb̶°)�Òí�ÌÑ{, Ĥ�á,b²^‰BbÿcqâÕÞ

#ì�

86 CHAPTER 4. _�Dõðql

WAB|c, �áí¡5õ[ýÑ log ρ0,

ρt+1 =

(1 − w+)ρt + w+u(ct) if ui(ct) ≥ ρt.

(1 − w−)ρt + w−u(ct) if ui(ct) < ρt.(4.9)

7_ñÊ©Ÿ²µ¬(ø}AB$Aø_�Ék|cj²íœ0}º, JdÑ-

Ÿ|c-܉bí¡5,

Pk,t =qk,t

Qt(4.10)

cqÚ+ƒñ‡²,|cíª½óúk²-|cV)ò (PA+,t ≥ PA−,t) íu,

¥[ý²,|cíµI¬ ?�#_ñœßíÑ{, FJ_ñÿ�óúòíœ}S

¦²,|cíµI (A+) J&MCÓ#ñ‡-܉bí®Ä�23 Ĥ_ñÊ3hw

øœ0 (pi) í|cD|cµIíÉ[[ýà-,

pi,t+1 =

pi,t + ∆p, if A+ is selected at t,

pi,t − ∆p, if A− is selected at t,(4.11)

FJà‹_ñí²,|cµI�O.˜íYŸ, †�óç×íœ}_ñ}‰)y�

]- pi,t+∆pÍ(Ó‹˘‰í˛�¾, ¥5_ñ†}Áý˛�˘‰í]-˙� pi,t−∆p� ∆p[ý©Ÿ-܉bí|cÙ�, Ñ7l�jZ, BbcqwÑÕÞ#ì, 7

/%,|cD%-|cíÙ�ó° (+∆p = −∆p)�24

°šíÉ[�6PàÊ_ñíAB<…²U ( zi

X ) ¸(3À:í˙� (θi) ,,

zi

X t+1=

zi

X t+ ∆ z

X , if A+ is selected at t,

zi

X t− ∆ z

X , if A− is selected at t,(4.12)

23�øõÛbÔ�·<, PA+,t = 0.6 > 0.4 = PA−,tuóúœ0í–1, [ý_ñ� 0.6íœ0}²,|c, Ou6�

0.4íª?}²-|c, 1.u×kÿ[ýøì}²,|c�24çÍBbwwAé-2�®�.°í|c�� (WàNb|c� úb|c), õ&£Þm7DŠÞm7íG�6.ø

š (ª?æ�¸Üdf), ½æu¥šóúµÆíqìÊ¥³�³�w.b4? ª–(4í|c¢ªJÓ‹Öýj„í˛

Èá? ÄÑ¥³Bb3bíñíuhô_ñç3|cíj², Ĥ|cÙ�íqìÿb²kÀ�

4.1. HÜA!˘‰�Ò 87

J£

θi,t+1 =

θi,t + ∆θ, if A+ is selected at t,

θi,t − ∆θ, if A− is selected at t,(4.13)

Ñ7¯¯|c,íø_4 (piD θi·u0ƒ1–ÈíªW|c), BbZ*){‰b

zi/X(M�RßU{íªW) Vhô_ñAB<…²Uí|cj²,25 FJ_ñÓ‹

Rß²Uí_bÿªJ[ýA zi

X t+ ∆ z

X, ¥5_ñÁýRß²Uí_bÑ zi

X t−

∆ zX

, 7 θi,t + ∆θD θi,t + ∆θ†}�[ý_ñ(3À:í˙�‹¿¸Á/� |(

Bbyì2¡b T[ýç3í,Hb� Ĥ¥ší){j�, .O°v\� RL í

Ô4J£-܉b©/|cíÔ”, ‹,$Aœ0}ºíµIÉ�s�, FJBb

¥³Z.5? RL íÇø�‰$ (r��õðC–��õð) 7�

ĤJ RL V){HÜA�˙, F�−í¡bqløu¨���á,b²^‰

Q0(= qA+,0 + qA−,0)� �á¡5õ log ρ0� ¡5õí|c¡b w+, w−� ú_-Ü

‰bí|cÙ� (6uç3í§�) ∆p, ∆ zX , ∆θJ£ç3í,Hb T �

J SGA ){HÜA�˙

GA |!…íÀPÿu2Hñ7, 7[Û2Hñíj��rÖ�, |cíø�Ñ

ùj){, 6ÿu2Hñuâ0D1íå� ¯7A, .¬[ÛíG¶e.°í½æ

7�óú_¯í){j�, ¥³Ñ°l�,íjZª7T¯l�í§�, Bb²Ï

àõb){òQV[Û‰b piJ£ θi, �¤ôI�²¬˙íl�¾, Bk_ñíR

ß²U bi, ÄÑw…™Êj„,ÿu‚à0D1å�í ¯, FJ¥¶MBb&Mù

j){í$�� ,!Vz, J SGA ){í©__ñÉË�ø_2Hñ, 7©_2

Hñuâú_¶M A, }�u‰b pi� bi¸ θi, w2í piD θiuJõb[ý, 725ÖÍ¥³){íúïš�Z‰, OBb3bíñíu;hô_ñRß²UÓÁíj², FJ�à.×�

88 CHAPTER 4. _�Dõðql

bi†uJùjå�[ý�

²ìßàS){2Hñíj�5(, GA í�ø_�Tÿu�á“|�øHí

‚ñ (C2Hñ_b), FßÞ|í2Hñ_bøòƒ|(øH·&M.‰, BbI

T[ý,uÆ“íHb, 7ÊHDH5È, Š�Ç,2Hñiší†u_¯�ƒb,

_¯�ƒbÓ½æí$�.°7�.°í™Ä, 7ÊBbqlí˘‰�Ò³, _¯

�ƒbíqìÑ max ui(ct),[ýF˘^àMBòí2HñnB�œ}\\GB-

øH, ĤÊ_6Þæí|òNûŸ†-, 2Hñ5È6ñ��²iGí!Ä (>²

m7) n?J}`AͲ�í9‰, -ÞBbøzp2HñÊ˘‰�Ò2ó�>¼

í¬˙qì (ªJ¡5�úıÇ3.2, Ü41)�

2Hñæº- í�ø¥, \²Ï� øO GA àV‘²2Hñíj�3b�s

�, 3b�²ÏD–™ˇ�²Ï, s�·ckd.52, Ou¨ø�j¶ªœ_

¯àVTÜHÜA!�Ò�_Euø_&j²í‡æ (open issue), .¬*Çø_

µÞVõ, .°í²Ï¶w*(6<âO.°íAÒæß, ×_Vz, 3b�²Ï

F¿Öíu©_AD©_A5È·�ó�©!ír��æ˜, 7–™ˇ�²Ï†u

JýbAÑø , â D 5ÈZAí–��æ˜, ʳ�ªø¥„Wéý¨ø�

æ˜_¯˘‰�Ò5‡, ÎÜzs�²Ï¶·@vþtõõnú, OuøjÞÑ7

ù-Ê˘‰�Ò_ñWÑíªœ}&,, ÇøjÞþ}‰b¢N|úk_ñ˛�˘

‰í<è�éO�àÉÊkýbíÉœA™,, Wà°ç� f¤CåÝ)3, FJ

Bb²ìJ–™ˇ�²ÏdÑ‘²2Hñíj¶, «Tí¬˙à-, Bb}Óœ‚

¦ ϕ_2Hñ<pºúŠ2dÑ¡‹¥ø�¯–™ˇí²G, Í(¦w2_¯�|

òí‡s±dÑßÞäHíŸ� (6ÿu‚H), à¤nêAø_�¯í–™ˇ�²

Ï, BkÛbd��¯†eBbuàS�j�éäH¦H‚H, ¥G&(Þzp�

4.1. HÜA!˘‰�Ò 89

Parent 1

Parent 2

Offspring 1

Offspring 2

Characteristic 1 Characteristic 2 Characteristic 3

Crossover point

Ç 4.5: ÀõíAú>ºí¸W

2Hñbæº- í�ù¥, Æ“� GA }�‚à7>ºJ£ˇ‰¥s_!ĉ

bV[®‚HÆ“ƒäHí¬˙, Bk>ºDˇ‰íþ}<2ÊBb�ÜÆ�¶v

˛%zp, ¥³ø.y;H, BbøòQ*¥s_!ĉbÊ¥³íqìÇá� í

l, ‚Hu´ªW>ºCßÞˇ‰uâ>º0 (Pc) J£ˇ‰0 (Pm) }�²ì, b

MBòH[‚HBñqßޓ牓, Bk>ºJ£ˇ‰íj�Bb²Ï|Uà

íÀõ>º¸õˇ‰s�j�, .¬ÄÑ°ø_2Hñq¨Ö7ú�.°íÔ47

/�s�.°í){j�, FJÀõ>ºJ£õˇ‰Ûbdø<|c, J-BbÔ

Wzp�

5?s_\²Ñ‚Hí2Hñ, J>ºíœ„\ó�, ÄÑ©_2Hñ·uâú

�.°Ô4í–¨ A, ĤBblJÓœ‚¦íj�²ìs_2Hñb>²m7

í–È, Í(n²ì>ºõíP0, WàÊÇ4.5 52, >ºõíP0ÿ\Óœ²

ìÊ�ù_Ô4 (AB²U<…) q, 76É�˘k¥¶Mí2Hñn}\~i˛

¤�², ”-ís_Ô4ÿ\MŸVí$�, ¥�j¶¢˚ÑÀõíAú>º (one

point paired crossover)�

90 CHAPTER 4. _�Dõðql

à‹‚¦ƒíÔ4u˘kõb){í¶M (�øD�úÔ4), Bb†*0ƒ1–

È2Óœ‚¦ø_bMøs_2Hñíõb¶Md(4 ¯7ßÞ|äH, cqª

W>ºí¶MurÊ�øÔ4,7s_2Hñí3hwøœ0}�u P1DP2, Bb

ÇÕ*0ƒ1–ÈÓœ‚¦ø_bM a1, ĤäHí3hwøœ0ÿ}�Ñ P3 =

a1P1 + (1 − a1)P2J£ P4 = (1 − a1)P1 + a1P2� Juˇ‰íœ„\ó�, ùj

){íˇ‰j�Dl‡ó°, !ÄM*0‰A1C6*1‰A0, Bkõb){íˇ

‰j�†uÊŸ…íbM,Óœ‹Áø_ß×á σ, UwªJÊ…™Ë¡dü¸ˇ

í��,26 J,ÞíWäzp, ˇ‰(í3hwøœ0ÿ‰AÑ P ′3 = P3 + σ�

ÊêAs_!ĉbí˙å5(, äHøÄe¦H‚H, 7øO GA í¦Hj

�ªJ}Ñs�, øur¶¦H (generational replacement) C6u¶M¦H

(steady-state replacement), ‡6uøH�í‚ñêrâhßÞí2HñF¦H,

(6†uÉ�¦HH�‚ñ2|6íì}5 η, ²Æuz, ‡66u(6íø_Ô

W (η = 100)�FJà‹²Ï‡6dÑäH¦H‚Híj�, µBb,ubd N/2_

�¯–™ˇ�²Ï (F�2Hñ_bíøš) n�Æ“êøH, J²Ï(6, †Ûb

d η100

N2 _�¯�ÖÍÊBbíql³ ηuª|cí¡b5ø, OÑ7ù·kÆ“�

xíªœ,, ¥³Bb6É5? η = 100í8”, 6ÿuzF�Hí2Hñø\h

í2Hñêr¦H�27

FJJà GA(SGA) V){HÜA�˙, F�−í¡bql†¨��Æ“Hb

T � >º0 Pc� ˇ‰0 Pm� ß×á σ� ©�¯–™ˇ�²ÏAb ϕJ£©Ÿ¦H

H�2Hñíì}ª η�

26çÍ σªJqìA®�}ºí�G (GC6Nb��),Ñ7fnÄÑqìRßí.°7û_õð!‹íÏæ4,B

b¥³øz σqlAÑ_Òùj){íˇ‰j�, Ì�íqñøG&õðqlzp�27*Çø_µÞVõ, ¶M¦HÄÑcq_ñÀUËø−AÐu´v|c…™íµIí‘K¬k�p (˘k�µì}

5 ηí_ñÿøìb²µI, ˘k,µì}5 1 − ηíº.à), FJBb.Ñp5?�

4.1. HÜA!˘‰�Ò 91

4.1.4 Æ“¼˙

�Üê7˘‰�Òíql� HÜA�˙íqìJ£Æ�¶í){j�5(, FJc

_�ÒíÆ“¼˙àÇ4.6 F[ý, �ø¥, RLC SGA øl�á“|�Ò, N_

_ñ, Í(c_�Òí«T*Çáƒ!!uâû_×2ü.°íc˛ ¯7A, *

׃üYåuÆ“íHb T � ˘‰©ø‚íȽÙb (_ñí²µ‚È) r̄� ©øÙ

N__ñí²µWÑJ£k˛�˘‰_ñí˛�Àç αiI�

*©øÙÇá, N__ñ};W|híÚ+åÝÀç (jackpot) Yå²ìAÐ

u´b˛�˘‰, à‹_ñ3hã‚í˘‰Ñ{^àMòk_ñí�áF)^àM

(ui(I) < EUi), †_ñø}˛�˘‰ (Q/AB<…²U), ¥5†.� (Q/(

3À:)�˛�˘‰í_ñøes_‘Ku´Å—7T¢˛�, }�Ñ3hã‚í˘

‰Ñ{^àMu´˛%�k_ñí�áF)^àM (ui(I) = EUi) J£X|Àç

u´˛%|F) (αi > 1)�7çN__ñ²ìêçÙí˛�¾5(, �Òø�“

çÙíÚ+åÝÀç1dyh, G&pÙéF�í_ñyŸ½h²µ� °ší¥�

øM/ r̄Ù, |(�Ò}Ç|ç‚í2ÝU{, ¥vBbÿªJl�©__ñí2

Ý8$Dóú@í^àM ui(ct), 1J¤dÑ_ñ|c-܉bíYW, _ñêA

yhJ( (YW.°í){j�) n�!!7¥øH, -øH_ñøJç3¬(í

h-܉b½µ,Hí²µ¬˙, øòƒ� TH!!Ñ¢�

92 CHAPTER 4. _�Dõðql

Start

Proc initialization

T generation

runs(evaluation cycle)

N agents

Access theJackpot

ui(I)<EUi

ui(I)=EUi

Consciousselection

Random selection

Yes

Yes

No

Aversion to Regret

Yes

No

No

Budget=Budget-1

Yes

NextQuantity

i=N?

Next Agent i=i+1

Yes

No

Record each agent's parameter

Yes

End of runs

Drawing the Winning numbers

Evaluate each Agent's

Prize & utility

RL or GAOperators(Updating)

End of generations

End

Update Market

state

Next run

Nextgeneration

r

1i += II iαα

1<iα

Ç 4.6: ˘‰�ÒíÆ“¼˙Ç

4.2. õðql 93

4.2 õðql

4.2.1 ¡bqì

%â4.1.1� (Ü69) D4.1.2� (Ü72) ícÜ, Bb˛%ÀËøHÜA!˘‰

�Ò„ÇJs .°í¡b[ý, }�u�Òí¡b ¯J£_ñWÑ¥@í¡b

¯, ÇÕ4.1.3� (Ü83) † uzp.°íÆ�¶wÊ){_ñWÑvª? ð.

|Ví.°¡b�

íluÉk�Ò¡b ¯í¶M M,

M = (x, X, r̄, τ, s0, . . . , sx, N, I)

ÊÛõÞº2, Î7|(s_¡b NJ£ IÌ¶â˘‰�Éç�Í−5Õ, ”-í

¡b·u˘kYjd†íø¶}� à‹*|_íi�Võ, Cr�A}�E�kÊ

#ì NJ£ Ií8”-, u.u?vƒø |_í�Ò¡b ¯,

(x∗, X∗, r̄∗, τ ∗, s0∗, . . . , sx

∗)

U)˘‰íYï (˘Y) ”ד�28 ªu¥³Bb1.H�¥ód, øjÞuÄÑ,

H¡bí§ ¯�ØÖ�íª?, ÇøjÞuÑ7?ù·ÊÆ�¶í«n,, Ä

¤¥³BbÉ²Ï τdÑBbí−„‰b�

5FJ‘² τíŸÄuÄÑBb�E�ø−˘‰�Òu´6æ�F‚í…ú�

�( (Laffer curve)? ²Æuz, ˘‰í˘Yúk˘‰í˘0u´x�Ü>�? à

‹�, µó|_í˘0@vuÖý? à‹³�, µóµ<.Ü>í˘0F°Qí¸

ˇ¢u¨³ƒ¨³? _Òí!‹úköõí˘‰�Ò¢�Öýj„?‰?287¥6uJ ACE _�}&íßT5ø, ÿu?DéµÆíÍ$C‡æ�l�TÜí˛È�

94 CHAPTER 4. _�Dõðql

[ 4.1: õð¡bqì

Market Parameters

Pick x from X (x/X) 5/16

Lottery Tax Rate (τ) 10%, ..., 90%

s0, s1, ..., s5 0%, 0%, 35%, 15%, 12%, 38%

Drawing Periods (r̄) 3

Number of Agents (N) 5000

Income (I) 200

GA Parameters

Range of pi,0 [0, 0.003]

Periods (Generations) (T ) 500

Crossover Rate (Pc) 90%

Mutation Rate (Pm) 0.1%

Arithmetic Mutation Size (σ) Equation (4.14)

Tournament Size (ϕ) 200

Generation Gap (η) 100

RL Parameters

Range of pi,0 [0, 0.003]

Periods (Generations) (T ) 500

Initial Strength (qA+,0, qA−,0) (1,1)

Initial Reference Point (ρ0) log 200

Memory Effect (ω+, ω−) (0.01, 0.02)

Learning Speed (∆p, ∆ zX , ∆θ) (10−6, 10−3, 10−3)

FJ…õðÎ7}«n˘‰�Ò˘YD˘0íÉ[�5Õ, ÇøjÞ6}hô_

ñÊ.°Æ�¶í«T-F[Û|VíWÑ, J£.°˘0úwWÑí�à� Ñ7

¥_ñí, Êõð2 τíqìø* 10%ƒ 90%(J10%ÑȽ),7wìí�Ò¡b

ÊF�í_Ò¬˙ç2ø&M�ì� õð¬˙2F�¡bíqì (¨�Æ�¶í¡

b) Bb$øcÜÊ[ 4.1 2�

ÄÑql¥_ ACE _�í3bñíuı��â¤}&j�?DéBbúk®

�.°í˘‰WÑ�yªø¥7j, ª7ªœ.°Æ�¶-_ñWÑíÏæ, FJ

�Òí¡b1³�Ô< �Ĭ� .¬, /<¡bíqìBb´u²Ïé…Döõ

4.2. õðql 95

˘‰�Òíqìø_, ñíuı�?±Q¥¶}ÄÑqìíÏæ4F�Vj„,í

ß×, Wà.°ÝáíÝçªW,s0, s1, ..., s5, ÿuYΫÉõÒí}êªW »

˘� ˘‰ÇÝíȽÙb(r̄) °š6uqlAøUÇÝsŸí8”� ªœ}IA˚

;í@vu x/X¥_¶M, !…,, x/XíqìÑSu¥ø|v˘‰|Ûø_×

=ð (N2å˘) íîhœ0ÑÖý, 6ÿuÛb�Öýí˛�¾ �Ü�� ,n}�

ø_A“ƒå˘� à‹ x = 6, X = 49, �ÌÛb1400N" (C496 ) n}�ø_A

2íÝ, Ju x = 6, X = 53, †‰A�ÌÛb2300N" (C536 ) n}�ø_A2

íÝ�

�7¥<îhœ05(, ø_˘‰�Ò=)åÝí‚�Abÿ²ìÊ�Òí,�

»¾ (∑N

i=1 αiI) ,7,29 6ÿuz, x/Xwõ1³�Ÿ¶ÀÖk αi, N, I5Õd²

ì� Walker and Young (2001) ÿN|, à‹FêWí˘‰Ø¬kñq=, µóÝ

Àß�|ÛíŸbøóúV)™ý, U)‡(‚í�»¾³�Ø×íÏæ, 7ñq

û_Abú˘‰ßÞQˇ, Ou˘‰JqlAØØ=íu, Å‚-Vúk˘‰í�

»¾6�.ßí�à, cqÊ”«í8$�-, ©‚í�»¾·ýƒÌ¶X¬ßÞ

ø_×=ðíœ}, FJÖÍ�ÝÀß�ºÌ¶ÜùØÖí·<, 7˘‰¾‘

6Êñwƒ�˛³�œ}=)íÝí8”-, ñq²ÏwFí²ûV¦H˘‰, .

â�ÝÀß�MMÚ+AªhíÀç5(n�ª?Ip˘‰�Ò�

ÖÍ ACE _�…™úk N³�LSíÌ„, Ou‹p¯Ü_ÒvÈíA…5

¾, Ní¸ˇÝÖ�k5000ƒ10000A7˛, ¥_bå±±Qköõ˘‰�Òí¡

DAb, ĤBbÉßú x/Xíqìd_�Ë^£Jw?DDöõí�Ò (Wà

«É) óã@� FJ¥êBbSજüí˘‰d_, ¹ x = 5, X = 16, J£

¯_í3hœ0¸ˇ pi ∈ [0, 0.003], Vº¯BbFqìí�ÒAb¸©AF)

29Bbqìø"˘‰íg�Ñø)Â, FJ,X|Àçn}�k,�»¾�

96 CHAPTER 4. _�Dõðql

N = 5000, I = 200�30

[í�ù¶}uJ GA _Òv}àƒí¡b, Ñ7UBbZk}&, ,Hí¡b

qìÊ_Ò¬˙2ø&M�ì� w2, -܉bÑõb){ (pi, θi) íˇ‰¬˙u

OÎ (4.14) �F²ì,

pinew = pi

old +

16∑i=1

BPm(1

2)i · (−1)

B 12 (4.14)

pioldD pi

new}�[ýˇ‰‡¸ˇ‰(í3hwøœ0, 7(Þµøáÿu_"ù

j){ˇ‰j�íß×á σ, F_"í2Hñ,u�16_!Äb, BPm¸ B 1

2†[

ýAŠœ0}�Ñ Pm(ˇ‰œ0) J£0.5í+›‰Óœ‰b (Bernoulli random

variable)�ÇÕ, ø–™ˇ�²ÏíqìAb…òƒ200(¿Ö_ñ5Èí��yÑ

äõ), ñí6uÑ7¥ø�Ò,_ñÄѧƒýñRšŒ`í�à, 7æËù·

k˘‰�Òí8$�

[ 4.1 í|(ø¶}u�Ék RL _Òví¡bqì, °šÊc__Ò¬˙ç

2, ¡bíqìM6ø\M.‰� s_µIí�áb²�b (qA+,0, qA−,0) BbÕÞ

#ìÑ1, Ĥ�áí,b²^‰�ks65¸ Q0 = 2� �áí¡5õBb²ÏD

.�íF)®Ädªœ, FJρ0 = log 200, 7_ñúk¡5õíp[#ÿ†u•

à Erev and Roth (1996) �Ĭ(qì (ω+, ω−) = (0.01, 0.02), s6F¥ø

|Víu_ñ}�}Ê˛˘‰íô^, 7¥øõ߯¯þ},×ÖbA2˘‰í

G�� Bk-܉bíç3§�, ÄÑAB<…²U¸(3À:øÇáqìí�á

¸ˇu�k0ƒ15È, FJBbé¥s6í‰�Ù� ∆ zX ¸ ∆θ×k ∆p, š�‹

30|3bí5¾ÄÖÿus_�Ò=)åÝí‚�Abu´�ø_� �Ì7k, «É©‚ªJßÞ1.8_å˘)3 (š

…¸ˇ~¡5Ç4.1 ízp), JPàƒBbí�Ò-, †�Ì©AÛb�1.58"˘‰ (1.8 × C165 /5000) nDX¬1.8_å

˘)3, 7�â (4.4) � (α∗i = 0.0079(1.58/200), J = 3004(1.58 × 5000 × 38%)) BbÿªJ�|¯_í3hœ0¸ˇ (×

��k0ƒ0.0065È), .¬©ø‚_ñ¢�úÙívȪJÚ+åÝÀç,�Ÿ¿t(Bb²ì¦wøšdÑ|(í3

hœ0¸ˇ�

4.2. õðql 97

0w|c��

ÄÑ GA D RL s6ç3íj�·u˘kÓœí½©, <âO¹UÊó°í¡

bqì-6}ßÞ.øší!‹, FJÑ7Ó#Bb!�í£ü4, .°í˘0�

-©_Æ�¶ø¥º�25_�¯�