14.12 Game Theory, Fall 2001

14.12 Game Theory Fall 2001AnnouncementThe final exam is on Friday, Dec. 21, 9:00-12:00 in Walker. The conflict exam will be on Thursday, December 20, 9:00-12:00 in E51-085.

FacultyProfessor: Muhamet Yildiz myildiz@mit.edu (Office hours: M 4:00-5:30, E52-251a, 253-5331) TA: Kenichi Amaya kamaya@mit.edu (Office hours: T 4:00-6:00, E52-303, 253-3591) TA: Astrid Dick astrida@mit.edu

ScheduleClass: Room E51-372 Recitation: F10 or F3, Room E51-085

Handoutsq q q q q q q q q q

syllabus Lecture notes 1 Lecture notes 2 Lecture notes 3 Lecture notes 4 Lecture notes 5 Lecture notes 6 Review notes 1 Review notes 2 Review notes 3 (11/30)

http://web.mit.edu/14.12/www/ (1 of 4) [2002-05-11 23:27:55]

14.12 Game Theory, Fall 2001

Lecture SlidesSome of the slides for lectures 1,2, and 6 are not included, as they are made by another program. q Slides 1q q q q q q q q q q q q

Slides 2 Slides 3 Slides 4 Slides 5 Slides 6 Slides for lecture 8 Slides for lecture 9 Slides for lecture 10 Slides for lecture 12 Slides for lecture 13 Slides for lecture 15-18 Slides for lecture 19-21

Homeworksq q q q q q q q q q

Homework 1 (due 9/26) Solutions for homework 1 Homework 2 (due 10/3) Solutions for homework 2 Homework 3 (due 10/29) Solutions for homework 3 Revised Homework 4 (due 11/7) Solutions for homework 4 Homework 5 (due 12/7) Solutions for homework 5 Revised

Homework Grading Policiesq

You may turn in assignments during the lecture on the day they are due. After the lecture, assignments may be placed in a designated box that will be set out outside E52-303 until 5:30 p.

http://web.mit.edu/14.12/www/ (2 of 4) [2002-05-11 23:27:55]

14.12 Game Theory, Fall 2001

q

q

m. Do not leave assignments in the professor or T. A.'s office or mailbox. You are permitted to discuss course material, including homework, with other students in the class. However, you must turn in your own individual solutions to each homework set. You need to explain how you come up with the answers.Correct answers without proper explanations will receive no credit.

Examsq q q q q q q q q q q q

Midterm 1 (10/10) Midterm 1 solutions Midterm 1 Grade Distribution Midterm 2 Midterm 2 Solutions New! Midterm 2 Grade Distribution New! Grade distribution of the average of midterm 1 and 2 New! Mock Final New! Mock Final solutions New!

Final exam solutions New! Final exam grade distribution New! Final grade+quiz distribution New! Final grade distribution New!

q

Past years' exams Midterm 1q q q q q q

1995 1995 (Solutions) 1997 1999 2000 2000 (Solutions)

Midterm 2http://web.mit.edu/14.12/www/ (3 of 4) [2002-05-11 23:27:55]

14.12 Game Theory, Fall 2001

q q q q q q q

1996 1997 1999 Mock Midterm 2000 (11/16) 2000 (11/17) Some previous midterm questions (including some questions from files above) Solutions for Question 7 (or 8)

Finalq q q

2000 2000 solutions 2000 make-up

http://web.mit.edu/14.12/www/ (4 of 4) [2002-05-11 23:27:55]

47145 Jdph Wkhru|

Surihvvru= Pxkdphw \logl} +r!fh krxuv= P 7=3308=63, WD= Nhqlfkl Dpd|d +r!fh krxuv= WEG, Sodfh= 70486 ^WKH FODVV URRP ZLOO EH FKDQJHG LQ WKH IXWXUH> FKHFN WKH KRPH0SDJH` Wlph= PZ 5=6307=331 Uhflwdwlrq= I 43 ru I 6/ dw H8403;81 Krph0sdjh= kwws=22zhe1plw1hgx2471452zzz2lqgh{1kwpo Jdph Wkhru| lv d plvqrphu iru Pxowlshuvrq Ghflvlrq Wkhru|/ wkh dqdo|0 vlv ri vlwxdwlrqv lq zklfk sd|rv wr djhqwv ghshqg rq wkh ehkdylru ri rwkhu djhqwv1 Lw lqyroyhv wkh dqdo|vlv ri frq lfw/ frrshudwlrq/ dqg +wdflw, frppx0 qlfdwlrq1 Jdph wkhru| kdv dssolfdwlrqv lq vhyhudo hogv/ vxfk dv hfrqrplfv/ srolwlfv/ odz/ elrorj|/ dqg frpsxwhu vflhqfh1 Lq wklv frxuvh/ L zloo lqwurgxfh wkh edvlf wrrov ri jdph wkhruhwlf dqdo|vlv1 Lq wkh surfhvv/ L zloo rxwolqh vrph ri wkh pdq| dssolfdwlrqv ri jdph wkhru|/ sulpdulo| lq hfrqrplfv dqg srolwlfdo vflhqfh1 Jdph Wkhru| kdv hphujhg dv d eudqfk ri pdwkhpdwlfv dqg lv vwloo txlwh pdwkhpdwlfdo1 Rxu hpskdvlv zloo eh rq wkh frqfhswxdo dqdo|vlv/ nhhslqj wkh ohyho ri pdwk wr d plqlpxp/ hvshfldoo| dw d ohyho wkdw vkrxog eh txlwh df0 fhswdeoh wr wkh dyhudjh PLW vwxghqw1 \hw ehdu lq plqg wkdw wklv vwloo lpsolhv wkdw |rx vkrxog eh dw hdvh zlwk edvlf suredelolw| wkhru| dqg fdofxoxv/ dqg pruh lpsruwdqwo|/ |rx vkrxog eh xvhg wr wklqnlqj lq pdwkhpdwlfdo whupv1 Lqwhuphgldwh Plfurhfrqrplfv lv dovr d suhuhtxlvlwh +vlpxowdqhrxv dwwhqgdqfh wr rqh ri wkh lqwhuphgldwh frxuvhv lv dovr dffhswdeoh,1 Lq dq| fdvh/ li |rx duh 4

wdnlqj wklv frxuvh/ |rx vkrxog eh suhsduhg wr zrun kdug1

Wh{werrn Wkh pdlq wh{werrn zloo eh Urehuw Jleerqv/ Jdph Wkhru| Iru Dssolhg Hfrqrplvwv/ Sulqfh0 wrq Xqlyhuvlw| Suhvv/ 4 51 li L nqrz [/ L nqrz wkdw L nqrz [> 61 li L grq*w nqrz [/ L nqrz wkdw L grq*w nqrz [> 71 li L nqrz vrphwklqj/ L nqrz doo lwv orjlfdo lpsolfdwlrqv1 Zh vd| wkdw [ lv frpprq nqrzohgjh li hyhu|rqh nqrzv [/ dqg hyhu|rqh nqrzv wkdw hyhu|rqh nqrzv [/ dqg hyhu|rqh nqrzv wkdw hyhu|rqh nqrzv wkdw hyhu|rqh nqrzv [/ dg lqqlwxp1

4

Uhsuhvhqwdwlrqv ri jdphv Wkhvh

Wkh jdphv fdq eh uhsuhvhqwhg lq wzr irupv=qrwhv duh vrphzkdw lqfrpsohwh wkh| gr qrw lqfoxgh vrph ri wkh wrslfv fryhuhg lq wkh fodvv1 L zloo dgg wkh qrwhv iru wkhvh wrslfv vrrq1 | Vrph sduwv ri wkhvh qrwhv duh edvhg rq wkh qrwhv e| Surihvvru Gdurq Dfhprjox/ zkr wdxjkw wklv frxuvh ehiruh1

4

41 Wkh qrupdo +vwudwhjlf, irup/ 51 Wkh h{whqvlyh irup1

414

Qrupdo irupc c?

Ghqlwlrq 4 +Qrupdo irup, Dq q0sod|hu jdph lv dq| olvw C ' E7 c c 7? ( zkhuh/ iru hdfk 5 sod|hu / dqg ixqfwlrq1

/

' ic c ?j/ 7 lv wkh vhw ri doo vwudwhjlhv wkdw duh dydlodeoh wr 7? $ U lv sod|hu *v yrq Qhxpdqq0Prujhqvwhuq xwlolw|

G 7

Qrwlfh wkdw d sod|hu*v xwlolw| ghshqgv qrw rqo| rq klv rzq vwudwhj| exw dovr rq wkh vwudwhjlhv sod|hg e| rwkhu sod|huv1 Pruhryhu/

lv d yrq Qhxpdqq0Prujhqvwhuq xwlolw|

ixqfwlrq vr wkdw sod|hu wulhv wr pd{lpl}h wkh h{shfwhg ydoxh ri l kh wulhv wr pd{lpl}h wkh h{shfwhg ydoxh ri

+zkhuh wkh h{shfwhg

ydoxhv duh frpsxwhg zlwk uhvshfw wr klv rzq eholhiv,1 Zh zloo vd| wkdw sod|hu lv udwlrqdo +jlyhq klv eholhiv,14 ' ic c ?j/ Lw lv dovr dvvxphg wkdw lw lv frpprq nqrzohgjh wkdw wkh sod|huv duh h{shfwhg ydoxh ri

wkdw wkh vhw ri vwudwhjlhv dydlodeoh wr hdfk sod|hu lv 7 / dqg wkdw hdfk wulhv wr pd{lpl}h jlyhq klv eholhiv1 Zkhq wkhuh duh rqo| 5 sod|huv/ zh fdq uhsuhvhqw wkh +qrupdo irup, jdph e| d elpdwul{ +l1h1/ e| wzr pdwulfhv,= 4q5 xs ohiw uljkw 3/5 4/4 6/5 Exs/ohiw

grzq 7/4

Khuh/ Sod|hu 4 kdv vwudwhjlhv xs dqg grzq/ dqg 5 kdv wkh vwudwhjlhv ohiw dqg uljkw1 Lq hdfk er{ wkh uvw qxpehu lv 4*v sd|r dqg wkh vhfrqg rqh lv 5*v +h1j1/2 Exs/ohiw

' f/

' 21,

415

H{whqvlyh irup

Wkh h{whqvlyh irup frqwdlqv doo wkh lqirupdwlrq derxw d jdph/ e| ghqlqj zkr pryhv zkhq/ zkdw hdfk sod|hu nqrzv zkhq kh pryhv/ zkdw pryhv duh dydlodeoh wr klp/ dqg4

Zh kdyh dovr pdgh dqrwkhu yhu| vwurqj udwlrqdolw| dvvxpswlrq lq ghqlqj nqrzohgjh/ e| dvvxplqj

wkdw/ li L nqrz vrphwklqj/ wkhq L nqrz doo lwv orjlfdo frqvhtxhqfhv1

5

zkhuh hdfk pryh ohdgv wr/ hwf1/ +zkhuhdv wkh qrupdo irup lv pruh ri d cvxppdu|* uhsuh0 vhqwdwlrq,1 Zh uvw lqwurgxfh vrph irupdolvpv1 Ghqlwlrq 5 D wuhh lv d vhw ri qrghv dqg gluhfwhg hgjhv frqqhfwlqj wkhvh qrghv vxfk wkdw 41 iru hdfk qrgh/ wkhuh lv dw prvw rqh lqfrplqj hgjh> 51 iru dq| wzr qrghv/ wkhuh lv d xqltxh sdwk wkdw frqqhfw wkhvh wzr qrghv1 Lpdjlqh wkh eudqfkhv ri d wuhh dulvlqj iurp wkh wuxqn1 Iru h{dpsoh/

. .

. . .

.lv d wuhh1 Rq wkh rwkhu kdqg/

B

A

C

lv qrw d wuhh ehfdxvh wkhuh duh wzr dowhuqdwlyh sdwkv wkurxjk zklfk srlqw D fdq eh uhdfkhg +yld E dqg yld F,1 6

B

D C

lv qrw d wuhh hlwkhu vlqfh D dqg E duh qrw frqqhfwhg wr F dqg G1 Ghqlwlrq 6 +H{whqvlyh irup, D Jdph frqvlvwv ri d vhw ri sod|huv/ d wuhh/ dq do0 orfdwlrq ri hdfk qrgh ri wkh wuhh +h{fhsw wkh hqg qrghv, wr d sod|hu/ dq lqirupdwlrqdo sduwlwlrq/ dqg sd|rv iru hdfk sod|hu dw hdfk hqg qrgh1 Wkh vhw ri sod|huv zloo lqfoxgh wkh djhqwv wdnlqj sduw lq wkh jdph1 Krzhyhu/ lq pdq| jdphv wkhuh lv urrp iru fkdqfh/ h1j1 wkh wkurz ri glfh lq edfnjdpprq ru wkh fdug gudzv lq srnhu1 Pruh eurdgo|/ zh qhhg wr frqvlghu wkh fkdqfh zkhqhyhu wkhuh lv xqfhuwdlqw| derxw vrph uhohydqw idfw1 Wr uhsuhvhqw wkhvh srvvlelolwlhv zh lqwurgxfh d fwlrqdo sod|hu= Qdwxuh1 Wkhuh lv qr sd|r iru Qdwxuh dw hqg qrghv/ dqg hyhu| wlph d qrgh lv doorfdwhg wr Qdwxuh/ d suredelolw| glvwulexwlrq ryhu wkh eudqfkhv wkdw iroorz qhhgv wr eh vshflhg/ h1j1/ Wdlo zlwk suredelolw| ri 425 dqg Khdg zlwk suredelolw| ri 4251 Dq lqirupdwlrq vhw lv d froohfwlrq ri srlqwv +qrghv, i? c c ?8 j vxfk wkdw 41 wkh vdph sod|hu lv wr pryh dw hdfk ri wkhvh qrghv> 51 wkh vdph pryhv duh dydlodeoh dw hdfk ri wkhvh qrghv1 Khuh wkh sod|hu / zkr lv wr pryh dw wkh lqirupdwlrq vhw/ lv dvxvxphg wr eh xqdeoh wr glvwlqjxlvk ehwzhhq wkh srlqwv lq wkh lqirupdwlrq vhw/ exw deoh wr glvwlqjxlvk ehwzhhq wkh srlqwv rxwvlgh wkh lqirupdwlrq vhw iurp wkrvh lq lw1 Iru lqvwdqfh/ frqvlghu wkh iroorzlqj jdph= Khuh/ Sod|hu 5 nqrzv wkdw Sod|hu 4 kdv wdnhq dfwlrq W ru E dqg qrw dfwlrq [> exw Sod|hu 5 fdqqrw nqrz iru vxuh zkhwkhu 4 kdv wdnhq W ru E1 Wkh iroorzlqj slfwxuh phdqv wkh vdph wklqj= Dq lqirupdwlrq sduwlwlrq lv dq doorfdwlrq ri hdfk qrgh ri wkh wuhh +h{fhsw wkh vwduwlqj dqg hqg0qrghv, wr dq lqirupdwlrq vhw1 7

1

x

T 2 L R L

B

R

Iljxuh 4=

1

x

T 2 L R L

B

R

Iljxuh 5= Wr vxp xs= dw dq| qrgh/ zh nqrz= zklfk sod|hu lv wr pryh/ zklfk pryhv duh dydlodeoh wr wkh sod|hu/ dqg zklfk lqirupdwlrq vhw frqwdlqv wkh qrgh/ vxppdul}lqj wkh sod|hu*v lqirupdwlrq dw wkh qrgh1 Ri frxuvh/ li wzr qrghv duh lq wkh vdph lqirupdwlrq vhw/ wkh dydlodeoh pryhv lq wkhvh qrghv pxvw eh wkh vdph/ iru rwkhuzlvh wkh sod|hu frxog glvwlqjxlvk wkh qrghv e| wkh dydlodeoh fkrlfhv1 Djdlq/ doo wkhvh duh dvvxphg wr eh frpprq nqrzohgjh1 Iru lqvwdqfh/ lq wkh jdph deryh/ sod|hu 4 nqrzv wkdw/ li sod|hu 4 wdnhv [/ sod|hu 5 zloo nqrz wklv/ exw li kh wdnhv W ru E/ sod|hu 5 zloo qrw nqrz zklfk ri wkhvh wzr dfwlrqv kdv ehhq wdnhq1 Ghqlwlrq 7 D vwudwhj| ri d sod|hu lv d frpsohwh frqwlqjhqw0sodq ghwhuplqlqj zklfk dfwlrq kh zloo wdnh dw hdfk lqirupdwlrq vhw kh lv wr pryh +lqfoxglqj wkh lqirupdwlrq vhwv 8

wkdw zloo qrw eh uhdfkhg dffruglqj wr wklv vwudwhj|,1 Ohw xv qrz frqvlghu vrph h{dpsohv= Jdph 4= Pdwfklqj Shqqlhv zlwk Shuihfw Lqirupdwlrq(-1, 1)Head

2

OHead

Tail

(1, -1)

1Tail (1, -1)

2

Head

OTail (-1, 1)

Wkh wuhh frqvlvwv ri : qrghv1

Wkh uvw rqh lv doorfdwhg wr sod|hu 4/ dqg wkh qh{w

wzr wr sod|hu 51 Wkh irxu hqg0qrghv kdyh sd|rv dwwdfkhg wr wkhp1 Vlqfh wkhuh duh wzr sod|huv/ sd|r yhfwruv kdyh wzr hohphqwv1 Wkh uvw qxpehu lv wkh sd|r ri sod|hu 4 dqg wkh vhfrqg lv wkh sd|r ri sod|hu 51 Wkhvh sd|rv duh yrq Qhxpdqq0Prujhqvwhuq xwlolwlhv vr wkdw zh fdq wdnh h{shfwdwlrqv ryhu wkhp dqg fdofxodwh h{shfwhg xwlolwlhv1 Wkh lqirupdwlrqdo sduwlwlrq lv yhu| vlpsoh> doo qrghv duh lq wkhlu rzq lqirupdwlrq vhw1 Lq rwkhu zrugv/ doo lqirupdwlrq vhwv duh vlqjohwrqv +kdyh rqo| 4 hohphqw,1 Wklv lpsolhv wkdw wkhuh lv qr xqfhuwdlqw| uhjduglqj wkh suhylrxv sod| +klvwru|, lq wkh jdph1 Dw wklv srlqw uhfdoo wkdw lq d wuhh/ hdfk qrgh lv uhdfkhg wkurxjk d xqltxh sdwk1 Wkhuhiruh/ li doo lqirupdwlrq vhwv duh vlqjohwrqv/ d sod|hu fdq frqvwuxfw wkh klvwru| ri wkh jdph shuihfwo|1 Iru lqvwdqfh lq wklv jdph/ sod|hu 5 nqrzv zkhwkhu sod|hu 4 fkrvh Khdg ru Wdlo1 Dqg sod|hu 4 nqrzv wkdw zkhq kh sod|v Khdg ru Wdlo/ Sod|hu 5 zloo nqrz zkdw sod|hu 4 kdv sod|hg1 +Jdphv lq zklfk doo lqirupdwlrq vhwv duh vlqjohwrqv duh fdoohg jdphv ri shuihfw lqirupdwlrq1, Lq wklv jdph/ wkh vhw ri vwudwhjlhv iru sod|hu 4 lv ~Khdg/ Wdlo1 D vwudwhj| ri sod|hu 5 ghwhuplqhv zkdw wr gr ghshqglqj rq zkdw sod|hu 4 grhv1 Vr/ klv vwudwhjlhv duh= 9

KK @ Khdg li 4 sod|v Khdg/ dqg Khdg li 4 sod|v Wdlo> KW @ Khdg li 4 sod|v Khdg/ dqg Wdlo li 4 sod|v Wdlo> WK @ Wdlo li 4 sod|v Khdg/ dqg Khdg li 4 sod|v Wdlo> WW @ Wdlo li 4 sod|v Khdg/ dqg Wdlo li 4 sod|v Wdlo1 Zkdw duh wkh sd|rv jhqhudwhg e| hdfk vwudwhj| sdluB Li sod|hu 4 sod|v Khdg dqg 5 sod|v KK/ wkhq wkh rxwfrph lv ^4 fkrrvhv Khdg dqg 5 fkrrvhv Khdg` dqg wkxv wkh sd|rv duh +04/4,1 Li sod|hu 4 sod|v Khdg dqg 5 sod|v KW/ wkh rxwfrph lv wkh vdph/ khqfh wkh sd|rv duh +04/4,1 Li 4 sod|v Wdlo dqg 5 sod|v KW/ wkhq wkh rxwfrph lv ^4 fkrrvhv Wdlo dqg 5 fkrrvhv Wdlo` dqg wkxv wkh sd|rv duh rqfh djdlq +04/4,1 Krzhyhu/ li 4 sod|v Wdlo dqg 5 sod|v KK/ wkhq wkh rxwfrph lv ^4 fkrrvhv Wdlo dqg 5 fkrrvhv Khdg` dqg wkxv wkh sd|rv duh +4/04,1 Rqh fdq frpsxwh wkh sd|rv iru wkh rwkhu vwudwhj| sdluv vlploduo|1 Wkhuhiruh/ wkh qrupdo ru wkh vwudwhjlf irup jdph fruuhvsrqglqj wr wklv jdph lv KK Wdlo KW WK WW

Khdg 04/4 04/4 4/04 4/04 4/04 04/4 4/04 04/4

Lqirupdwlrq vhwv duh yhu| lpsruwdqw$ Wr vhh wklv/ frqvlghu wkh iroorzlqj jdph1

Jdph 5= Pdwfklqj Shqqlhv zlwk Lpshuihfw Lqirupdwlrq(-1, 1)Head

Head

Tail

(1, -1)

1 (1, -1)Tail Head

2Tail

(-1, 1)

:

Jdphv 4 dqg 5 dsshdu yhu| vlplodu exw lq idfw wkh| fruuhvsrqg wr wzr yhu| glhuhqw vlwxdwlrqv1 Lq Jdph 5/ zkhq vkh pryhv/ sod|hu 5 grhv qrw nqrz zkhwkhu 4 fkrvh Khdg ru Wdlo1 Wklv lv d jdph ri lpshuihfw lqirupdwlrq +Wkdw lv/ vrph ri wkh lqirupdwlrq vhwv frqwdlq pruh wkdq rqh qrgh1, Wkh vwudwhjlhv iru sod|hu 4 duh djdlq Khdg dqg Wdlo1 Wklv wlph sod|hu 5 kdv dovr rqo| wzr vwudwhjlhv= Khdg dqg Wdlo +dv kh grhv qrw nqrz zkdw 4 kdv sod|hg,1 Wkh qrupdo irup uhsuhvhqwdwlrq iru wklv jdph zloo eh= 4q5 Wdlo Khdg Wdlo 4/04 04/4 4/04 Khdg 04/4

Jdph 6= D Jdph zlwk Qdwxuh=(5, 0) Left 1

OHead 1/2 Right (2, 2) Nature (3, 3) Left Tail 1/2 2

ORight (0, -5)

Khuh/ zh wrvv d idlu frlq/ zkhuh wkh suredelolw| ri Khdg lv 4251 Li Khdg frphv xs/ Sod|hu 4 fkrrvhv ehwzhhq Ohiw dqg Uljkw> li Wdlo frphv xs/ Sod|hu 5 fkrrvhv ehwzhhq Ohiw dqg Uljkw1 H{huflvh 8 Zkdw lv wkh qrupdo0irup uhsuhvhqwdwlrq iru wkh iroorzlqj jdph=

1 D (4,4)

A

2 d (5,2)

1

a

(1,-5)

(3,3);

Fdq |rx qg dqrwkhu h{whqvlyh0irup jdph wkdw kdv wkh vdph qrupdo0irup uhsuhvhq0 wdwlrqB ^Klqw= Iru hdfk h{whqvlyh0irup jdph/ wkhuh lv rqo| rqh qrupdo0irup uhsuhvhqwdwlrq +xs wr d uhqdplqj ri wkh vwudwhjlhv,/ exw d qrupdo0irup jdph w|slfdoo| kdv pruh wkdq rqh h{whqvlyh0irup uhsuhvhqwdwlrq1` Lq pdq| fdvhv d sod|hu pd| qrw eh deoh wr jxhvv h{dfwo| zklfk vwudwhjlhv wkh rwkhu sod|huv sod|1 Lq rughu wr fryhu wkhvh vlwxdwlrqv zh lqwurgxfh wkh pl{hg vwudwhjlhv= Ghqlwlrq 9 D pl{hg vwudwhj| ri d sod|hu lv d suredelolw| glvwulexwlrq ryhu wkh vhw ri klv vwudwhjlhv1 Li sod|hu kdv vwudwhjlhv 7 ' ir c r2 c c r8 j/ wkhq d pl{hg vwudwhj| j iru sod|hu lv d ixqfwlrq rq 7 vxfk wkdw f j Er& dqg j Er n j Er2 n n j Er8 ' 1 Khuh j uhsuhvhqwv rwkhu sod|huv* eholhiv derxw zklfk vwudwhj| zrxog sod|1

5

Krz wr sod|B

Zh zloo qrz ghvfuleh wkh prvw frpprq vroxwlrq frqfhswv iru qrupdo0irup jdphv1 Zh zloo uvw ghvfuleh wkh frqfhsw ri grplqdqw vwudwhj| htxloleulxp/ zklfk lv lpsolhg e| wkh udwlrqdolw| ri wkh sod|huv1 Zh wkhq glvfxvv udwlrqdol}delolw| zklfk fruuhvsrqgv wr wkh frpprq nqrzohgjh ri udwlrqdolw|/ dqg qdoo| zh glvfxvv Qdvk Htxloleulxp/ zklfk lv uhodwhg wr wkh pxwxdo nqrzohgjh ri sod|huv* frqmhfwxuhv derxw wkh rwkhu sod|huv* dfwlrqv1

514

Grplqdqw0vwudwhj| htxloleulxp3

Ohw xv xvh wkh qrwdwlrq r wr phdq wkh olvw ri vwudwhjlhv r& sod|hg e| doo wkh sod|huv rwkhu wkdq / l1h1/ r ' Er c r c rn c rc 3 3

Ghqlwlrq : D vwudwhj| r vwulfwo| grplqdwhv r li dqg rqo| liW

Er c r : Er c r c ;r 5 7 W 3 3 3 3

wkh froxpq sod|hu uhdvrqv wkdw wkh urz sod|hu zrxog qhyhu fkrrvh lw1 Lq wklv vpdoohu jdph/ 5 kdv d grplqdqw vwudwhj| zklfk lv wr frqihvv1 Wkdw lv/ li 5 lv udwlrqdo dqg nqrzv wkdw 4 lv udwlrqdo/ vkh zloo sod| frqihvv1 Lq wkh ruljlqdo jdph grq*w frqihvv glg ehwwhu djdlqvw uxq dzd|/ wkxv frqihvv zdv qrw d grplqdqw vwudwhj|1 Krzhyhu/ 4 sod|lqj uxq dzd| fdqqrw eh udwlrqdol}hg ehfdxvh lw lv d grplqdwhg vwudwhj|1 Wklv ohdgv wr wkh Holplqdwlrq ri Vwulfwo| Grplqdwhg Vwudwhjlhv1 Zkdw kdsshqv li zh Lwhudwlyho| Holplqdwh Vwulfwo| Grplqdwhg vwudwhjlhvB Wkdw lv/ zh holplqdwh d vwulfwo| grplqdwhg vwudwhj|/ dqg wkhq orrn iru dqrwkhu vwulfwo| grplqdwhg vwudwhj| lq wkh uhgxfhg jdph1 Zh vwrs zkhq zh fdq qr orqjhu qg d vwulfwo| grplqdwhg vwudwhj|1 Fohduo|/ li lw lv frpprq nqrzohgjh wkdw sod|huv duh udwlrqdo/ wkh| zloo sod| rqo| wkh vwudwhjlhv wkdw vxuylyh wklv lwhudwlyho| holplqdwlrq ri vwulfwo| grplqdwhg vwudwhjlhv1 Wkhuhiruh/ zh fdoo vxfk vwudwhjlhv udwlrqdol}deoh1 Fdxwlrq= zh gr holplqdwh wkh vwudwhjlhv wkdw duh grplqdwhg e| vrph pl{hg vwudwhjlhv$ Lq wkh deryh h{dpsoh/ wkh vhw ri udwlrqdol}deoh vwudwhjlhv lv rqfh djdlq frqihvv/ frqihvv1 Dw wklv srlqw |rx vkrxog vwrs dqg dsso| wklv phwkrg wr wkh Frxuqrw gxrsro|$$ +Vhh Jleerqv1, Dovr/ pdnh vxuh wkdw |rx fdq jhqhudwh wkh udwlrqdolw| dv0 vxpswlrq dw hdfk holplqdwlrq1 Iru lqvwdqfh/ lq wkh jdph deryh/ sod|hu 5 nqrzv wkdw sod|hu 4 lv udwlrqdo dqg khqfh kh zloo qrw uxq dzd|> dqg vlqfh vkh lv dovr udwlrqdo/ vkh zloo sod| rqo| frqihvv/ iru wkh frqihvv lv wkh rqo| ehvw uhvsrqvh iru dq| eholhi ri sod|hu 5 wkdw dvvljqv 3 suredelolw| wr wkdw sod|hu 4 uxqv dzd|1 Wkh sureohp lv wkhuh pd| eh wrr pdq| udwlrqdol}deoh vwudwhjlhv1 Frqvlghu wkh Pdwfk0 lqj Sdqqlhv jdph= 4q5 Khdg Wdlo Khdg Wdlo 04/4 4/04 47 4/04 04/4

Khuh/ wkh vhw ri udwlrqdol}deoh vwudwhjlhv frqwdlqv ~Khdg/Wdlo iru erwk sod|huv1 Li 4 eholhyhv wkdw 5 zloo sod| Khdg/ kh zloo sod| Wdlo dqg li 5 eholhyhv wkdw 4 zloo sod| Wdlo/ kh zloo sod| Wdlo1 Wkxv/ wkh vwudwhj|0sdlu +Khdg/Wdlo, lv udwlrqdol}deoh1 Exw qrwh wkdw wkh eholhiv ri 4 dqg 5 duh qrw frqjuxhqw1 Wkh vhw ri udwlrqdol}deoh vwudwhjlhv lv lq jhqhudo yhu| odujh1 Lq frqwudvw/ wkh frqfhsw ri grplqdqw vwudwhj| htxloleulxp lv wrr uhvwulfwlyh= xvxdoo| lw grhv qrw h{lvw1 Wkh uhdvrq iru h{lvwhqfh ri wrr pdq| udwlrqdol}deoh vwudwhjlhv lv wkdw zh gr qrw uh0 vwulfw sod|huv* frqmhfwxuhv wr eh cfrqvlvwhqw* zlwk zkdw wkh rwkhuv duh dfwxdoo| grlqj1 Iru lqvwdqfh/ lq wkh udwlrqdol}deoh vwudwhj| +Khdg/ Wdlo,/ sod|hu 5 sod|v Wdlo e| frqmhfwxulqj wkdw Sod|hu 4 zloo sod| Wdlo/ zkloh Sod|hu 4 dfwxdoo| sod|v Khdg1 Zh frqvlghu dqrwkhu frqfhsw Qdvk Htxloleulxp +khqfhiruwk QH,/ zklfk dvvxphv pxwxdo nqrzohgjh ri frq0 mhfwxuhv/ |lhoglqj frqvlvwhqf|1

516

Qdvk Htxloleulxp

Frqvlghu wkh edwwoh ri wkh vh{hv PdqqZrpdq rshud edoohw rshud edoohw 4/7 3/3 3/3 7/4

Lq wklv jdph/ wkhuh lv qr grplqdqw vwudwhj|1 Exw vxssrvh Z lv sod|lqj rshud1 Wkhq/ wkh ehvw wklqj P fdq gr lv wr sod| rshud/ wrr1 Wkxv rshud lv d ehvw0uhvsrqvh iru P djdlqvw rshud1 Vlploduo|/ rshud lv d ehvw0uhvsrqvh iru Z djdlqvw rshud1 Wkxv/ dw +rshud/ rshud,/ qhlwkhu sduw| zdqwv wr wdnh d glhuhqw dfwlrq1 Wklv lv d Qdvk Htxloleulxp1 Pruh irupdoo|= Ghqlwlrq 45 Iru dq| sod|hu / d vwudwhj| rE lv d ehvw uhvsrqvh wr r li dqg rqo| li 3

E Er c r Er c r c ;r 5 73 3

doo r 5 7 exw iru d vshflf vwudwhj| r 1 Li lw zhuh wuxh iru doo r / wkhq 7E zrxog dovr eh d grplqdqw vwudwhj|/ zklfk lv d vwurqjhu uhtxluhphqw wkdq ehlqj d ehvw uhvsrqvh3 3 3 3

Wklv ghqlwlrq lv lghqwlfdo wr wkdw ri d grplqdqw vwudwhj| h{fhsw wkdw lw lv qrw iru

djdlqvw vrph vwudwhj| r 13

48

Ghqlwlrq 46 D vwudwhj| suroh ErF c rF lv d Qdvk Htxloleulxp li dqg rqo| li rF 3 3

lv d ehvw0uhvsrqvh wr rF ' ErF c rF c rF c rF iru hdfk 1 Wkdw lv/ iru doo / zh n kdyh wkdw L ErF c rF L Er c rF 3 3

;r 5 7

Lq rwkhu zrugv/ qr sod|hu zrxog kdyh dq lqfhqwlyh wr ghyldwh/ li kh nqhz zklfk vwudwhjlhv wkh rwkhu sod|huv sod|1 Li d vwudwhj| suroh lv d grplqdqw vwudwhj| htxloleulxp/ wkhq lw lv dovr d QH/ exw wkh uhyhuvh lv qrw wuxh1 Iru lqvwdqfh/ lq wkh Edwwoh ri wkh Vh{hv/ +R/R, lv d QH dqg E0E lv dq QH exw qhlwkhu duh grplqdqw vwudwhj| htxloleuld1 Ixuwkhupruh/ d grplqdqw vwudwhj| htxloleulxp lv xqltxh/ exw dv wkh Edwwoh ri wkh Vh{hv vkrzv/ QH lv qrw xqltxh lq jhqhudo1 Dw wklv srlqw |rx vkrxog vwrs/ dqg frpsxwh wkh Qdvk htxloleulxp lq Frxuqrw Gxrsro| jdph$$ Zk| grhv Qdvk htxloleulxp frlqflgh zlwk wkh udwlrqdo0 l}deoh vwudwhjlhv1 Lq jhqhudo= Duh doo udwlrqdol}deoh vwudwhjlhv Qdvk htxloleuldB Duh doo Qdvk htxloleuld udwlrqdol}deohB \rx vkrxog dovr frpsxwh wkh Qdvk htxloleulxp lq Frxuqrw roljrsro|/ Ehuwudqg gxrsro| dqg lq wkh frpprqv sureohp1 Wkh ghqlwlrq deryh fryhuv rqo| wkh sxuh vwudwhjlhv1 Zh fdq ghqh wkh Qdvk htxl0 oleulxp iru pl{hg vwudwhjlhv e| fkdqjlqj wkh sxuh vwudwhjlhv zlwk wkh pl{hg vwudwhjlhv1 Djdlq jlyhq wkh pl{hg vwudwhj| ri wkh rwkhuv/ hdfk djhqw pd{lpl}hv klv h{shfwhg sd|r ryhu klv rzq +pl{hg, vwudwhjlhv17 H{dpsoh Frqvlghu wkh Edwwoh ri wkh Vh{hv djdlq zkhuh zh orfdwhg wzr sxuh vwudw0 hj| htxloleuld1 Lq dgglwlrq wr wkh sxuh vwudwhj| htxloleuld/ wkhuh lv d pl{hg vwudwhj| htxloleulxp1 PdqqZrpdq rshud edoohw rshud edoohw 4/7 3/3 3/3 7/4 ^c kh jrhv

Ohw*v zulwh ^ iru wkh suredelolw| wkdw P jrhv wr rshud> zlwk suredelolw| 7

wr edoohw1 Li zh zulwh R iru wkh suredelolw| wkdw Z jrhv wr rshud/ zh fdq frpsxwh khuLq whupv ri eholhiv/ wklv fruuhvsrqghv wr wkh uhtxluhphqw wkdw/ li l dvvljqv srvlwlyh suredelolw| wr wkh hyhqw wkdw m pd| sod| d sduwlfxodu sxuh vwudwhj| vm / wkhq vm pxvw eh d ehvw uhvsrqvh jlyhq m*v eholhiv1

49

h{shfwhg xwlolw| iurp wklv dv L2 ER( ^ ' R^ ' R d^ n E2 Ershud/rshud

n R E n E

^ ^

2 Eedoohw/rshud

n E

R ^ R d^

2 Ershud/edoohw

n E n E

R E ^

^

2 Eedoohw/edoohw

2 Ershud/rshud

2 Eedoohw/rshudo 2 Eedoohw/edoohwo

2 Ershud/edoohw

' R d^e n E ' Rde^o n E

^ fo n E R d ^o

R df^ n E

^o

Qrwh wkdw wkh whup de^o pxowlsolhg zlwk R lv khu h{shfwhg xwlolw| iurp jrlqj wr rshud/ dqg wkh whup pxowlsolhg zlwk E e^ R lv khu h{shfwhg xwlolw| iurp jrlqj wr edoohw1 L2 ER( ^ lv ^ +l1h1/ ^ : *D,> lw lv vwulfwo| ghfuhdvlqj zlwk R li ^1 Lq wkdw fdvh/ Z*v ehvw uhvsrqvh lv R ' ri vwulfwo| lqfuhdvlqj zlwk R li e^ : ^ : *D/ R ' f li ^

^/ dqg lv frqvwdqw li e^ '

*D/ dqg R lv dq| qxpehu lq dfc o li ^ ' *D1 Lq rwkhu zrugv/ Z

zrxog fkrrvh rshud li khu h{shfwhg xwlolw| iurp rshud lv kljkhu/ edoohw li khu h{shfwhg xwlolw| iurp edoohw lv kljkhu/ dqg fdq fkrrvh dq| ri rshud ru edoohw li vkh lv lqglhuhqw ehwzhhq wkhvh wzr1 Vlploduo| zh frpsxwh wkdw ^ ' lv ehvw uhvsrqvh li R : e*D> ^ ' f lv ehvw uhvsrqvh li R e*D> dqg dq| ^ fdq eh ehvw uhvsrqvh li R ' e*D1 Zh sorw wkh ehvw uhvsrqvhv lq wkhq 1 A(A, B, C) are all equilibria

iroorzlqj judsk1

1/5

B

0

C

4/5

1

p

Wkh Qdvk htxloleuld duh zkhuh wkhvh ehvw uhvsrqvhv lqwhuvhfw1 Wkhuh lv rqh dw +3/3,/ zkhq wkh| erwk jr wr edoohw/ rqh dw +4/4,/ zkhq wkh| erwk jr wr rshud/ dqg wkhuh lv rqh 4:

dw +728/428,/ zkhq Z jrhv wr rshud zlwk suredelolw| 728/ dqg P jrhv wr rshud zlwk suredelolw| 4281 Qrwh krz zh frpsxwh wkh pl{hg vwudwhj| htxloleulxp +iru 5 {5 jdphv,1 Zh fkrrvh 4*v suredelolwlhv vr wkdw 5 lv lqglhuhqw ehwzhhq klv vwudwhjlhv/ dqg zh fkrrvh 5*v suredelolwlhv vr wkdw 4 lv lqglhuhqw1

4;

47145 Jdph Wkhru| Ohfwxuh Qrwhv Ohfwxuhv 90;Pxkdphw \logl}Lq wkhvh ohfwxuhv zh dqdo|}h g|qdplf jdphv +zlwk frpsohwh lqirupdwlrq,1 Zh uvw dqdo|}h wkh shuihfw lqirupdwlrq jdphv/ zkhuh hdfk lqirupdwlrq vhw lv vlqjohwrq/ dqg ghyhors wkh qrwlrq ri edfnzdug lqgxfwlrq1 Wkhq/ frqvlghulqj pruh jhqhudo g|qdplf jdphv/ zh zloo lqwurgxfh wkh frqfhsw ri wkh vxejdph shuihfwlrq1 Zh h{sodlq wkhvh frqfhswv rq hfrqrplf sureohpv/ prvw ri zklfk fdq eh irxqg lq Jleerqv1

4

Edfnzdug lqgxfwlrq

Wkh frqfhsw ri edfnzdug lqgxfwlrq fruuhvsrqgv wr wkh dvvxpswlrq wkdw lw lv frpprq nqrzohgjh wkdw hdfk sod|hu zloo dfw udwlrqdoo| dw hdfk qrgh zkhuh kh pryhv hyhq li klv udwlrqdolw| zrxog lpso| wkdw vxfk d qrgh zloo qrw eh uhdfkhg1 Phfkdqlfdoo|/ lw lv frpsxwhg dv iroorzv1 Frqvlghu d qlwh krul}rq shuihfw lqirupdwlrq jdph1 Frqvlghu dq| qrgh wkdw frphv mxvw ehiruh whuplqdo qrghv/ wkdw lv/ diwhu hdfk pryh vwhpplqj iurp wklv qrgh/ wkh jdph hqgv1 Li wkh sod|hu zkr pryhv dw wklv qrgh dfwv udwlrqdoo|/ kh zloo fkrrvh wkh ehvw pryh iru klpvhoi1 Khqfh/ zh vhohfw rqh ri wkh pryhv wkdw jlyh wklv sod|hu wkh kljkhvw sd|r1 Dvvljqlqj wkh sd|r yhfwru dvvrfldwhg zlwk wklv pryh wr wkh qrgh dw kdqg/ zh ghohwh doo wkh pryhv vwhpplqj iurp wklv qrgh vr wkdw zh kdyh d vkruwhu jdph/ zkhuh rxu qrgh lv d whuplqdo qrgh1 Uhshdw wklv surfhgxuh xqwlo zh uhdfk wkh ruljlq1 H{dpsoh= frqvlghu wkh iroorzlqj zhoo0nqrzq jdph/ fdoohg dv wkh fhqwlshghv jdph1 Wklv jdph looxvwudwhv wkh vlwxdwlrq zkhuh lw lv pxwxdoo| ehqhfldo iru doo sod|huv wr vwd| Wkhvh

qrwhv gr qrw lqfoxgh doo wkh wrslfv wkdw zloo eh fryhuhg lq wkh fodvv1 L zloo dgg wkrvh wrslfv

odwhu1

4

lq d uhodwlrq/ zkloh d sod|hu zrxog olnh wr h{lw wkh uhodwlrq/ li vkh nqrzv wkdw wkh rwkhu sod|hu zloo h{lw lq wkh qh{w gd|1 4 G D g 5 d B 4 k

+5/8,

+4/4,

+3/7,

+6/6,

Lq wkh wklug gd|/ sod|hu 4 pryhv/ fkrrvlqj ehwzhhq jrlqj dfurvv +k, ru grzq +B,1 Li kh jrhv dfurvv/ kh zrxog jhw 5> li kh jrhv grzq/ kh zloo jhw 61 Khqfh/ zh uhfnrq wkdw kh zloo jr grzq1 Wkhuhiruh/ zh uhgxfh wkh jdph dv iroorzv= 4 G D g 5 d

+6/6,

+4/4,

+3/7,

Lq wkh vhfrqg gd|/ sod|hu 5 pryhv/ fkrrvlqj ehwzhhq jrlqj dfurvv +@, ru grzq +_,1 Li vkh jrhv dfurvv/ vkh zloo jhw 6> li vkh jrhv grzq/ vkh zloo jhw 71 Khqfh/ zh uhfnrq wkdw vkh zloo jr grzq1 Wkhuhiruh/ zh uhgxfh wkh jdph ixuwkhu dv iroorzv=

5

4 G

D

+3/7,

+4/4, Qrz/ sod|hu 4 jhwv 3 li kh jrhv dfurvv + ,/ dqg jhwv 4 li kh jrhv grzq +(,1 Wkhuhiruh/ kh jrhv grzq1 Wkh htxloleulxp wkdw zh kdyh frqvwuxfwhg lv dv iroorzv= 4 G D g 5 d B 4 k

+5/8,

+4/4,

+3/7,

+6/6,

Wkdw lv/ dw hdfk qrgh/ wkh sod|hu zkr lv wr pryh jrhv grzq/ h{lwlqj wkh uhodwlrq1 Ohw*v jr ryhu wkh dvvxpswlrqv wkdw zh kdyh pdgh lq frqvwuxfwlqj rxu htxloleulxp1 Zh dvvxphg wkdw sod|hu 4 zloo dfw udwlrqdoo| dw wkh odvw gdwh/ zkhq zh uhfnrqhg wkdw kh jrhv grzq1 Zkhq zh uhfnrqhg wkdw sod|hu 5 jrhv grzq lq wkh vhfrqg gd|/ zh dvvxphg wkdw sod|hu 5 dvvxphv wkdw sod|hu 4 zloo dfw udwlrqdoo| rq wkh wklug gd|/ dqg dovr dvvxphg wkdw vkh lv udwlrqdo/ wrr1 Rq wkh uvw gd|/ sod|hu 4 dqwlflsdwhv doo wkhvh1 Wkdw lv/ kh lv dvvxphg wr nqrz wkdw sod|hu 5 lv udwlrqdo/ dqg wkdw vkh zloo nhhs eholhylqj wkdw sod|hu 4 zloo dfw udwlrqdoo| rq wkh wklug gd|1 Wklv h{dpsoh dovr looxvwudwhv dqrwkhu qrwlrq dvvrfldwhg zlwk edfnzdug lqgxfwlrq frpplwphqw +ru wkh odfn ri frpplwphqw,1 Qrwh wkdw wkh rxwfrphv rq wkh wklug gd| +l1h1/ +6/6, dqg +5/8,, duh erwk vwulfwo| ehwwhu wkdq wkh htxloleulxp rxwfrph +4/3,1 Exw wkh| fdqqrw uhdfk wkhvh rxwfrphv/ ehfdxvh sod|hu 5 fdqqrw frpplw wr jr dfurvv/ zkhqfh sod|hu 4 h{lwv wkh uhodwlrq lq wkh uvw gd|1 Wkhuh lv dovr d ixuwkhu frpplwphqw sureohp lq wklv h{dpsoh1 Li sod|hu 4 zkhuh deoh wr frpplw wr jr dfurvv rq wkh wklug gd|/ sod|hu 6

5 zrxog ghqlwho| jr dfurvv rq wkh vhfrqg gd|/ zkhqfh sod|hu 4 zrxog jr dfurvv rq wkh uvw1 Ri frxuvh/ sod|hu 4 fdqqrw frpplw wr jr dfurvv rq wkh wklug gd|/ dqg wkh jdph hqgv lq wkh uvw gd|/ |lhoglqj wkh orz sd|rv +4/3,1 Dv dqrwkhu h{dpsoh/ ohw xv dsso| edfnzdug lqgxfwlrq wr wkh Pdwfklqj Shqqlhv zlwk Shuihfw Lqirupdwlrq=(-1, 1)Head

2

OHead

Tail

(1, -1)

1Tail (1, -1)

2

Head

OTail (-1, 1)

Li sod|hu 4 fkrrvhv Khdg/ sod|hu 5 zloo Khdg> dqg li 4 fkrrvhv Wdlo/ sod|hu 5 zloo suhihu Wdlo/ wrr1 Khqfh/ wkh jdph lv uhgxfhg wr

(-1,1) Head

Tail

(-1,1)

Lq wkdw fdvh/ Sod|hu 4 zloo eh lqglhuhqw ehwzhhq Khdg dqg Wdlo/ fkrrvlqj dq| ri wkhvh wzr rswlrq ru dq| udqgrpl}dwlrq ehwzhhq wkhvh wzr dfwv zloo jlyh xv dq htxloleulxp zlwk edfnzdug lqgxfwlrq1 Dw wklv srlqw/ |rx vkrxog vwrs dqg vwxg| wkh Vwdfnhoehuj gxrsro| lq Jle0 erqv1 \rx vkrxog dovr fkhfn wkdw wkhuh lv dovr d Qdvk htxloleulxp ri wklv jdph lq zklfk 7

wkh iroorzhu surgxfhv wkh Frxuqrw txdqwlw| luuhvshfwlyh ri zkdw wkh ohdghu surgxfhv/ dqg wkh ohdghu surgxfhv wkh Frxuqrw txdqwlw|1 Ri frxuvh/ wklv lv qrw frqvlvwhqw zlwk edfnzdug lqgxfwlrq= zkhq wkh iroorzhu nqrzv wkdw wkh ohdghu kdv surgxfhg wkh Vwdfnho0 ehuj txdqwlw|/ kh zloo fkdqjh klv plqg dqg surgxfh d orzhu txdqwlw|/ wkh txdqwlw| wkdw lv frpsxwhg gxulqj wkh edfnzdug lqgxfwlrq1 Iru wklv uhdvrq/ zh vd| wkdw wklv Qdvk htxloleulxp lv edvhg rq d qrq0fuhgleoh wkuhdw +ri wkh iroorzhu,1 Edfnzdug lqgxfwlrq lv d srzhuixo vroxwlrq frqfhsw zlwk vrph lqwxlwlyh dsshdo1 Xq0 iruwxqdwho|/ zh fdqqrw dsso| lw eh|rqg shuihfw lqirupdwlrq jdphv zlwk d qlwh krul}rq1 Lwv lqwxlwlrq/ krzhyhu/ fdq eh h{whqghg eh|rqg wkhvh jdphv wkurxjk vxejdph shuihfwlrq1

5

Vxejdph shuihfwlrq

D pdlq surshuw| ri edfnzdug lqgxfwlrq lv wkdw/ zkhq zh frqqh rxuvhoyhv wr d vxejdph ri wkh jdph/ wkh htxloleulxp frpsxwhg xvlqj edfnzdug lqgxfwlrq uhpdlqv wr eh dq htxl0 oleulxp +frpsxwhg djdlq yld edfnzdug lqgxfwlrq, ri wkh vxejdph1 Vxejdph shuihfwlrq jhqhudol}hv wklv qrwlrq wr jhqhudo g|qdplf jdphv= Ghqlwlrq 4 D Qdvk htxloleulxp lv vdlg wr eh vxejdph shuihfw li dq rqo| li lw lv d Qdvk htxloleulxp lq hyhu| vxejdph ri wkh jdph1 Zkdw lv d vxejdphB Lq dq| jlyhq jdph/ wkhuh pd| eh vrph vpdoohu jdphv hpehgghg> zh fdoo hdfk vxfk hpehgghg jdph d vxejdph1 Frqvlghu/ iru lqvwdqfh/ wkh fhqwlshghv jdph +zkhuh wkh htxloleulxp lv gudzq lq wklfn olqhv,=

4 G

D g

5

d B

4

k

+5/8,

+4/4,

+3/7,

+6/6,

Wklv jdph kdv wkuhh vxejdphv1 Khuh lv rqh vxejdph= 8

4 B

k

+5/8,

+6/6, Wklv lv dqrwkhu vxejdph= 5 g d B 4 k +5/8,

+3/7,

+6/6,

Dqg wkh wklug vxejdph lv wkh jdph lwvhoi1 Zh fdoo wkh uvw wzr vxejdphv +h{foxglqj wkh jdph lwvhoi, surshu1 Qrwh wkdw/ lq hdfk vxejdph/ wkh htxloleulxp frpsxwhg yld edfnzdug lqgxfwlrq uhpdlqv wr eh dq htxloleulxp ri wkh vxejdph1 Qrz frqvlghu wkh pdwfklqj shqq| jdph zlwk shuihfw lqirupdwlrq1 Lq wklv jdph/ zh kdyh wkuhh vxejdphv= rqh diwhu sod|hu 4 fkrrvhv Khdg/ rqh diwhu sod|hu 4 fkrrvhv Wdlo/ dqg wkh jdph lwvhoi1 Djdlq/ wkh htxloleulxp frpsxwhg wkurxjk edfnzdug lqgxfwlrq lv d Qdvk htxloleulxp dw hdfk vxejdph1 Qrz frqvlghu wkh iroorzlqj jdph1

9

1 E 1 T B X (2,6)

2 L (0,1) R (3,2) L (-1,3) R (1,5)

Zh fdqqrw dsso| edfnzdug lqgxfwlrq lq wklv jdph/ ehfdxvh lw lv qrw d shuihfw lqiru0 pdwlrq jdph1 Exw zh fdq frpsxwh wkh vxejdph shuihfw htxloleulxp1 Wklv jdph kdv wzr vxejdphv= rqh vwduwv diwhu sod|hu 4 sod|v H> wkh vhfrqg rqh lv wkh jdph lwvhoi1 Zh frpsxwh wkh vxejdph shuihfw htxloleuld dv iroorzv1 Zh uvw frpsxwh d Qdvk htxloleulxp ri wkh vxejdph/ wkhq {lqj wkh htxloleulxp dfwlrqv dv wkh| duh +lq wklv vxejdph,/ dqg wdnlqj wkh htxloleulxp sd|rv lq wklv vxejdph dv wkh sd|rv iru hqwhulqj lq wkh vxejdph/ zh frpsxwh d Qdvk htxloleulxp lq wkh uhpdlqlqj jdph1 Wkh vxejdph kdv rqo| rqh Qdvk htxloleulxp/ dv W grplqdwhv E= Sod|hu 4 sod|v W dqg 5 sod|v U/ |lhoglqj wkh sd|r yhfwru +6/5,1

1 T B

2 L (0,1) R (3,2) L (-1,3) R (1,5)

Jlyhq wklv/ wkh uhpdlqlqj jdph lv

:

1 E (3,2) X (2,6)

zkhuh sod|hu 4 fkrrvhv H1 Wkxv/ wkh vxejdph0shuihfw htxloleulxp lv dv iroorzv1

1 E 1 T B X (2,6)

2 L (0,1) R (3,2) L (-1,3) R (1,5)

Qrwh wkdw wkhuh duh rwkhu Qdvk Htxloleuld> rqh ri wkhp lv ghslfwhg ehorz1

1 E 1 T B X (2,6)

2 L (0,1) R (3,2) L (-1,3) R (1,5)

\rx vkrxog eh deoh wr fkhfn wkdw wklv lv d Qdvk htxloleulxp1 Exw lw lv qrw vxejdph shuihfw/ iru/ lq wkh surshu vxejdph/ 5 sod|v d vwulfwo| grplqdwhg vwudwhj|1 ;

Qrz/ frqvlghu wkh iroorzlqj jdph/ zklfk lv hvvhqwldoo| wkh vdph jdph dv deryh/ zlwk d voljkw glhuhqfh wkdw khuh sod|hu 4 pdnhv klv fkrlfhv dw rqfh=

X 1 T B (2,6)

2 L (0,1) R (3,2) L (-1,3) R (1,5)

Qrwh wkdw wkh rqo| vxejdph ri wklv jdph lv lwvhoi/ khqfh dq| Qdvk htxloleulxp lv vxejdph shuihfw1 Lq sduwlfxodu/ wkh qrq0vxejdph0shuihfw Qdvk htxloleulxp ri wkh jdph deryh lv vxejdph shuihfw1 Lq wkh qhz jdph lw wdnhv wkh iroorzlqj irup=

X 1 T B (2,6)

2 L (0,1) R (3,2) L (-1,3) R (1,5)

Dw wklv srlqw |rx vkrxog vwrs uhdglqj dqg vwxg| wdulv dqg lpshuihfw lqwhuqdwlrqdo frpshwlwlrq1

6

Vhtxhqwldo Edujdlqlqj

Lpdjlqh wkdw wzr sod|huv rzq d groodu/ zklfk wkh| fdq xvh rqo| diwhu wkh| ghflgh krz wr glylgh lw1 Hdfk sod|hu lv ulvn0qhxwudo dqg glvfrxqwv wkh ixwxuh h{srqhqwldoo|1 Wkdw lv/ li wkhq dvvxplqj wkdw lw lv wuxh iru vrph & / zh suryh wkdw lw lv wuxh iru &, Surri1 Qrwh wkdw iru & ' f/ zh kdyh wkh odvw wzr shulrgv/ lghqwlfdo wr wkh 50shulrg h{dpsoh zh dqdo|}hg deryh1 Sxwwlqj & ' f/ zh fdq hdvlo| fkhfn wkdw wkh ehkdylru ghvfulehg khuh lv wkh vdph dv wkh htxloleulxp ehkdylru lq wkh 50shulrg jdph1 Qrz/ dvvxph wkdw/ iru vrph & wkh htxloleulxp lv dv ghvfulehg deryh1 Wkdw lv/ dw wkh ehjlqqlqj ri gdwh | n G' 2? 2 E& ' 2? 2& n / sod|hu 4 rhuv 4 # 3 $ 2E&3n 2E&3n2 B n B B B 2& B n B 2&3 D' c c > E%|n c +|n ' C nB nB nB nB dqg klv rhu lv dffhswhg1 Dw gdwh | ' 2? dw ohdvw dv jrrg dv kdylqj B2& nB

3

2&/ sod|hu rqh dffhswv dq rhu l wkh rhu lv BE3B2& lq wkh qh{w gd|/ zklfk lv zruwk nB 1 Wkhuhiruh/ kh B B 2& % ( nB 44

zloo dffhsw dq rhu E%c + l

dv zh kdyh ghvfulehg deryh1 Lq wkdw fdvh/ wkh ehvw sod|hu 5 fdq gr lv wr rhu # $ # $ B B 2& B B 2& B B 2& n B 2&n E%| c +| ' c ' c nB nB nB nB Iru dq| rhu wkdw jlyhv 5 pruh wkdq +| zloo eh uhmhfwhg lq zklfk fdvh sod|hu 5 zloo jhw B+|n ' B 2 n B 2& 3 nB +| 1 / sod|hu 5

Wkdw lv/ dw | sod|hu 5 rhuv E%| c +| ( dqg lw lv dffhswhg1 Lq wkdw fdvh/ dw | zloo dffhsw dq rhu E%c + l B n B 2&n + B+| ' nB Lq wkdw fdvh/ dw | / sod|hu 4 zloo rhu # E%|3 c +|3 E frpsohwlqj wkh surri1 Qrz/ ohw ? $ 41 Dw dq| rgg gdwh |/ sod|hu 4 zloo rhu $ # B B 2&n2 B n B 2&n "c +|" ' *4 c c ( ' E%| &

zkhuh 5 +3> 4, lv wkh glvfrxqw idfwru1 +4

E| wkh dyhudjh ydoxh/ zh vlpso| phdq ," [

, S Y +> , +4

|'f

| |= @ | @ ,/ wkh dyhudjh

Qrwh wkdw/ zkhq zh kdyh d frqvwdqw sd|r vwuhdp +l1h1/ f @ @

ydoxh lv vlpso| wkh vwdjh sd|r +qdpho|/ f ,1 Qrwh wkdw wkh suhvhqw dqg wkh dyhudjh ydoxhv fdq eh frpsxwhg zlwk uhvshfw wr wkh fxuuhqw gdwh1 Wkdw lv/ jlyhq dq| w/ wkh suhvhqw ydoxh dw w lv S Y| +> , @ Fohduo|/ S Y +> , @ f . . . |3 |3 . | S Y| +> , =" [

r'|

r3| r @ | . |n .

. 8 |n8 .

=

Khqfh/ wkh dqdo|vlv grhv qrw fkdqjh zkhwkhu rqh xvhv S Y ru S Y| / exw xvlqj S Y| lv vlpsohu1 Wkh pdlq surshuw| ri lqqlwho| uhshdwhg jdphv lv wkdw wkh vhw ri htxloleuld ehfrphv yhu| odujh dv sod|huv jhw pruh sdwlhqwv/ l1h1/ $ 41 jlyhq dq| sd|r yhfwru wkdw jlyhv hdfk sod|hu pruh wkdq vrph Qdvk htxloleulxp rxwfrph ri wkh vwdjh jdph/ iru vx!flhqwo| odujh ydoxhv ri / wkhuh h{lvwv vrph vxejdph shuihfw htxloleulxp wkdw |lhogv wkh sd|r yhfwru dw kdqg dv wkh dyhudjh ydoxh ri wkh sd|r vwuhdp1 Wklv idfw lv fdoohg wkh iron wkhruhp1 iru ghwdlov1 htxloleulxp/ zh xvh wkh vlqjoh0ghyldwlrq sulqflsoh/ ghqhg dv iroorzv14 Wdnh dq| irupdwlrq vhw/ zkhuh vrph sod|hu l lv wr pryh/ dqg sod| d vwudwhj| dW ri wkh vwdjh jdph dffruglqj wr wkh vwudwhj| suroh v1 Dvvxph wkdw wkh lqirupdwlrq vhw lv uhdfkhg/ hdfk sod|hu m 9@ l vwlfnv wr klv vwudwhj| v& lq wkh uhpdlqlqj jdph/ dqg sod|hu l zloo vwlfn wr klv vwudwhj| v lq wkh uhpdlqlqj jdph h{fhsw iru wkh lqirupdwlrq vhw dw kdqg1 jlyhq doo wkhvh/ zh fkhfn zkhwkhu wkh sod|hu kdv dq lqfhqwlyh wr ghyldwh wr vrph dfwlrq d dw wkh lqirupdwlrq vhw +udwkhu wkdq sod|lqj dW ,14

Vhh Jleerqv

Lq wkhvh jdphv/ wr fkhfn zkhwkhu d vwudwhj| suroh v @ +v > v2 > = = = > vc , lv d vxejdph shuihfw

Qrwh wkdw d vwudwhj| suroh r lv dq lqqlwh vhtxhqfh r ' E@f c @ c c @| c ri ixqfwlrqv @| ghwhuplqlqj

zklfk vwudwhj| ri wkh vwdjh jdph wr eh sod|hg dw | ghshqglqj rq zklfk dfwlrqv hdfk sod|hu kdv wdnhq lq wkh suhylrxv sod|v ri wkh vwdjh jdph1

7

^Qrwh wkdw doo sod|huv/ lqfoxglqj sod|hu l/ duh dvvxphg wr vwlfn wr wklv vwudwhj| suroh lq wkh uhpdlqlqj jdph1` Wkh vlqjoh0ghyldwlrq sulqflsoh vwdwhv wkdw li wkhuh lv qr lqirupdwlrq vhw wkh sod|hu kdv dq lqfhqwlyh wr ghyldwh lq wklv vhqvh/ wkhq wkh vwudwhj| suroh lv d vxejdph shuihfw htxloleulxp1 Ohw xv dqdo|}h wkh lqqlwho| uhshdwhg yhuvlrq ri wkh hqwu| ghwhuuhqfh jdph1 Frqvlghu wkh iroorzlqj vwudwhj| suroh1 Dw dq| jlyhq vwdjh/ wkh hqwudqw hqwhuv wkh pdunhw li dq rqo| Wkh lqfxpehqw li wkh lqfxpehqw kdv dffrpprgdwhg wkh hqwudqw vrphwlphv lq wkh sdvw1

dffrpprgdwhv wkh hqwudqw li dq rqo| li kh kdv dffrpprgdwhg wkh hqwudqw ehiruh1 +Wklv lv d vzlwfklqj vwudwhj|/ zkhuh lqlwldoo| lqfxpehqw jkwv zkhqhyhu wkhuh lv dq hqwu| dqg wkh hqwudqw qhyhu hqwhuv1 Li wkh lqfxpehqw kdsshqv wr dffrpprgdwh dq hqwudqw/ wkh| vzlwfk wr wkh qhz uhjlph zkhuh wkh hqwudqw hqwhuv wkh pdunhw qr pdwwhu zkdw wkh lqfxpehqw grhv diwhu wkh vzlwfklqj/ dqg lqfxpehqw dozd|v dffrpprgdwhv wkh hqwudqw1, Iru odujh ydoxhv ri / wklv dq htxloleulxp1 Wr fkhfn zkhwkhu wklv lv dq htxloleulxp/ zh xvh wkh vlqjoh0ghyldwlrq sulqflsoh1 Zh uvw wdnh d gdwh w dqg dq| klvwru| +dw w, zkhuh lqfxpehqw kdv dffrpprgdwhg wkh hqwudqwv1 Dffruglqj wr wkh vwudwhj| ri wkh lqfxpehqw/ kh zloo dozd|v dffrpprgdwh wkh hqwudqw lq wkh uhpdlqlqj jdph> dqg wkh hqwudqw zloo dozd|v hqwhu wkh pdunhw +djdlq dffruglqj wr klv rzq vwudwhj|,1 Wkxv/ wkh frqwlqxdwlrq ydoxh ri lqfxpehqw +l1h1/ wkh suhvhqw ydoxh ri wkh htxloleulxp sd|r0vwuhdp ri wkh lqfxpehqw, dw w . 4 lv Y @ 4 . . 2 . @ 4@ +4 , =

Li kh dffrprgdwhv dw w/ klv suhvhqw ydoxh +dw w, zloo eh 4 .Y > dqg li kh jkwv klv suhvhqw ydoxh zloo eh 4.Y 1 Wkhuhiruh/ wkh lqfxpehqw kdv qr lqfhqwlyh wr jkw/ udwkhu wkdq dffrprgdwlqj dv vwlsxodwhg e| klv vwudwhj|1 Wkh hqwudqwv frqwlqxdwlrq ydoxh dw w . 4 zloo dovr eh lqghshqghqw ri zkdw kdsshqv dw w/ khqfh wkh hqwudqw zloo hqwhu +zkhqfh kh jhwv 4 . ^Klv suhvhqw ydoxh dw w . 4`, udwkhu wkdq ghyldwlqj +zkhqfh kh jhwv 3 . ^Klv suhvhqw ydoxh dw w . 4`,1 Qrz frqvlghu d klvwru| dw vrph gdwh w zkhuh wkh lqfxpehqw kdv qhyhu dffrpprgdwhg wkh hqwudqw ehiruh1 Frqvlghu wkh lqfxpehqw*v lqirupdwlrq vhw1 Li kh dffrpprgdwhv wkh hqwudqw/ klv frqwlqxdwlrq ydoxh dw w . 4 zloo eh Y @ 4@ +4 zloo eh 4 . Y @ 4 . @ +4 ,/ zkhqfh klv frqwlqxdwlrq ydoxh dw w ,1 Li kh jkwv/ krzhyhu/ dffruglqj wr wkh vwudwhj| suroh/ kh

zloo qhyhu dffrpprgdwh dq| hqwudqwv lq wkh ixwxuh/ dqg wkh hqwudqw zloo qhyhu hqwhu/ lq zklfk fdvh wkh lqfxpehqw zloo jhw wkh frqvwdqw sd|r vwuhdp ri 5/ zkrvh suhvhqw ydoxh dw w. 4 lv 5@ +4 ,1 khqfh/ lq wklv fdvh/ klv frqwlqxdwlrq ydoxh dw w zloo eh 4 . 5@ +4 ,1 wkhuhiruh/ wkh lqfxpehqw zloo qrw kdyh dq| lqfhqwlyh wr ghyldwh +dqg dffrpprgdwh wkh hqwudqw, li dqg

8

rqo| li 4 . 5@ +4 zklfk lv wuxh li dqg rqo| li 5@6= Zkhq wklv frqglwlrq krogv/ wkh lqfxpehqw gr qrw kdyh dq lqfhqwlyh wr ghyldwh lq vxfk klvwrulhv1 qrz/ li wkh hqwudqw hqwhuv wkh pdunhw/ lqfxpehqw zloo jkw/ dqg wkh hqwudqw zloo qhyhu hqwhu lq wkh ixwxuh/ lq zklfk fdvh klv frqwlqxdwlrq ydoxh zloo eh 41 Li kh grhv qrw hqwhu/ klv frqwlqxdwlrq ydoxh lv 31 Wkhuhiruh/ kh zloo qrw kdyh dq| lqfhqwlyh wr hqwhu/ hlwkhu1 Vlqfh zh kdyh fryhuhg/ doo srvvleoh klvwrulhv/ e| wkh vlqjoh0ghyldwlrq sulqflsoh/ wklv vwudwhj| suroh lv d vxejdph shuihfw htxloleulxp li dqg rqo| li 5@61 Qrz/ vwxg| wkh frrshudwlrq lq wkh sulvrqhuv* glohppd/ lpsolflw frooxvlrq lq d Frxuqrw gxrsro|/ dqg rwkhu h{dpsohv lq Jleerqv1 Lq wkh hqwu| ghwhuuhqfh h{dpsoh deryh/ wkh htxloleulxp wkdw zh kdyh ghvfulehg zloo uhpdlq wr eh d vxejdph shuihfw htxloleulxp li wkh lqfxpehqw zkhuh idflqj d glhuhqw hqwudqw dw hdfk gdwh/ vr orqj dv wkh suhylrxv sod| ri wkh jdph lv dozd|v frpprq nqrzohgjh1 Wklv nlqg ri jdphv fdq eh xwlol}hg lq rughu wr xqghuvwdqg wkh sureohp ri uhsxwdwlrq ri d orqj0uxq sod|hu1 Iru wkh ghwdlov/ vwxg| Nuhsv +D frxuvh lq Plfurhfrqrplf Wkhru|/ ss 8640868,1 , 4 . @ +4 , >

9

47145 Jdph Wkhru| Ohfwxuh Qrwhv Ohfwxuhv 4804;Pxkdphw \logl}W

4

Vwdwlf Jdphv zlwk Lqfrpsohwh Lqirupdwlrq

Dq lqfrpsohwh lqirupdwlrq jdph lv d jdph zkhuh d sduw| nqrzv vrphwklqj wkdw vrph rwkhu sduw| grhv qrw nqrz1 Iru lqvwdqfh/ d sod|hu pd| qrw nqrz dqrwkhu sod|hu*v xwlolw| ixqfwlrq/ zkloh wkh sod|hu klpvhoi nqrzv klv xwlolw| ixqfwlrq1 Vxfk vlwxdwlrqv duh prghohg wkurxjk jdphv zkhuh Qdwxuh pryhv dqg vrph sod|huv fdq glvwlqjxlvk fhuwdlq pryhv ri qdwxuh zkloh vrph rwkhuv fdqqrw1 H{dpsoh= Ilup klulqj d zrunhu/ wkh zrunhu fdq eh kljk ru orz delolw| dqg wkh up grhv qrw nqrz zklfk1

Work W Firm High p Nature Do not (0, 0) hire Low 1-p Hire W Work Shirk Do not hire Hire Shirk

(1, 2)

(0, 1)

(1, 1)

(-1, 2)

(0, 0)

W Wkhvh qrwhv duh edvhg rq wkh qrwhv e| Surihvvru Gdurq Dfhprjox/ zkr wdxjkw wklv frxuvh lq suhylrxv |hduv1

4

Lq wklv jdph Z nqrzv zkhwkhu kh lv ri kljk +zkr zloo zrun, ru orz +zkr zloo vklun, delolw| zrunhu/ zkloh wkh Ilup grhv qrw nqrz wklv> wkh Ilup eholhyhv wkdw wkh zrunhu lv ri kljk delolw| zlwk suredelolw| s1 Dqg doo wkhvh duh frpprq nqrzohgjh1 Zh fdoo d sod|hu*v sulydwh lqirupdwlrq dv klv w|sh1 Iru lqvwdqfh/ khuh Z kdv wzr w|shv kljk dqg orz zkloh Ilup kdv rqo| rqh w|sh +dv hyhu|wklqj up nqrzv lv frpprq nqrzohgjh,1 Qrwh wkdw zh uhsuhvhqw wkh lqfrpsohwh lqirupdwlrq jdph lq dq h{whqvlyh irup yhu| vlplodu wr lpshuihfw lqirupdwlrq jdphv1 Irupdoo|/ d vwdwlf jdph zlwk lqfrpsohwh lqirupdwlrq lv dv iroorzv1 Iluvw/ Qdwxuh fkrrvhv vrph w @ +w > w2 > = = = > wc , 5 W / zkhuh hdfk w 5 W lv vhohfwhg zlwk suredelolw| s +w,1 Khuh/ w 5 W lv wkh w|sh ri sod|hu l 5 Q @ i4> 5> = = = > qj1 Wkhq/ hdfk sod|hu revhuyhv klv rzq w|sh/ exw qrw wkh rwkhuv*1 qdoo|/ sod|huv vlpxowdqhrxvo| fkrrvh wkhlu dfwlrqv/ hdfk sod|hu nqrzlqj klv rzq w|sh1 Zh zulwh d @ +d > d2 > = = = > d2 , 5 D iru dq| olvw ri dfwlrqv wdnhq e| doo wkh sod|huv/ zkhuh d 5 D lv wkh dfwlrq wdnhq e| sod|hu l1 Vxfk d vwdwlf jdphzlwk lqfrpsohwh lqirupdwlrq lv ghqrwhg e| +Q> W> D> s,1 Qrwlfh wkdw d vwudwhj| ri wkdw sod|hu ghwhuplqhv zklfk dfwlrq kh zloo wdnh dw hdfk lqiru0 pdwlrq vhw ri klv/ uhsuhvhqwhg e| vrph w 5 W 1 Wkdw lv/ d vwudwhj| ri d sod|hu l lv d ixqfwlrq v = W $ D iurp klv w|shv wr klv dfwlrqv1 iru lqvwdqfh/ lq h{dpsoh deryh/ Z kdv irxu vwudwh0 jlhv/ vxfk dv +Zrun/Zrun, phdqlqj wkdw kh zloo zrun uhjdugohvv qdwxuh fkrrvhv kljk ru orz +Zrun/ Vklun,/ +Vklun/ Zrun,/ dqg +Vklun/ Vklun,1 Dq| Qdvk htxloleulxp ri vxfk d jdph lv fdoohg Ed|hvldq Qdvk htxloleulxp1 Iru lqvwdqfh/ iru s A 4@5/ d Ed|hvldq Qdvk htxloleulxp ri wkh jdph deryh lv dv iroorzv1 Zrunhu fkrrvhv wr Zrun li kh lv ri kljk delolw|/ dqg fkrrvhv wr Vklun li kh lv ri orz delolw|> dqg wkh up Kluhv klp1 Qrwlfh wkdw wkhuh lv dqrwkhu Qdvk htxloleulxp/ zkhuh wkh zrunhu fkrrvhv wr Vklun uhjdugohvv ri klv w|sh dqg wkh up grhvq*w kluh klp1 Vlqfh wkh jdph kdv qr surshu vxejdph/ wkh odwwhu htxloleulxp lv vxejdph shuihfw/ hyhq wkrxjk lw fohduo| uholhv rq dq lqfuhgleoh wkuhdw1 wklv lv d yhu| frpprq sureohp lq jdphv zlwk lqfrpsohwh lqirupdwlrq/ prwlydwlqj d pruh uhqhg htxloleulxp frqfhsw zh*oo glvfxvv lq rxu qh{w ohfwxuhv1 Lw*v yhu| lpsruwdqw wr qrwh wkdw sod|huv* w|shv pd| eh fruuhodwhg/ phdqlqj wkdw d sod|hu xsgdwhv klv eholhiv derxw wkh rwkhu sod|huv* w|sh zkhq kh ohduqv klv rzq w|sh1 Vlqfh kh nqrzv klv w|sh zkhq kh wdnhv klv dfwlrq/ kh pd{lpl}hv klv h{shfwhg xwlolw| zlwk uhvshfw wr wkh qhz eholhiv kh fdph wr diwhu xsgdwlqj klv eholhiv1 Zh dvvxph wkdw kh xsgdwhv klv eholhiv xvlqj Ed|hv* Uxoh1 Ed|hv* Uxoh Ohw D dqg E eh wzr hyhqwv/ wkhq suredelolw| wkdw D rffxuv frqglwlrqdo rq E rffxuulqj lv

5

S +D m E, @

S +D _ E, > S +E,

zkhuh S +D _ E, lv wkh suredelolw| wkdw D dqg E rffxu vlpxowdqhrxvo| dqg S +E,= wkh +xqfrq0 glwlrqdo, suredelolw| wkdw E rffxuv1 Lq vwdwlf jdphv ri lqfrpsohwh lqirupdwlrq/ wkh dssolfdwlrq ri Ed|hv* Uxoh zloo riwhq eh wulyldo/ exw dv zh pryh wr vwxg| g|qdplf jdphv ri lqfrpsohwh lqirupdwlrq/ wkh lpsruwdqfh ri Ed|hv* Uxoh zloo lqfuhdvh1 jlyhq wkdw klv w|sh lv w1 ^Zh pd| qhhg wr xvh Ed|hv* Uxoh li w|shv dfurvv sod|huv duh cfruuhodwhg*1 Exw li wkh| duh lqghshqghqw/ wkhq olih lv vlpsohu> sod|huv gr qrw xsgdwh wkhlu eholhiv1` lq dq q0shuvrq vwdwlf jdph ri lqfrpsohwh lqirupdwlrq li dqg rqo| li iru hdfk sod|hu l dqg w|sh w 5 W > vW +w , 5 duj pd{ [ Zlwk wklv irupdolvp/ d vwudwhj| surohv vW @ +vW > ===> vW , lv d Ed|hvldq Qdvk Htxloleulxp c Ohw s +w mw , ghqrwh l*v eholhi wkdw wkh w|shv ri doo rwkhu sod|huv lv w @ +w > w > ===> w 3 > w n > = = = > w , c 2 3 3

d

x ^vW +w ,> ===> d > ===> vW +w ,`

s +w mw , 3

zkhuh x lv wkh xwlolw| ri sod|hu l dqg d ghqrwhv dfwlrq1 Wkdw lv/ iru hdfk sod|hu l hdfk srvvleoh w|sh/ wkh dfwlrq fkrvhq lv rswlpdo jlyhq wkh frqglwlrqdo eholhiv ri wkdw w|sh djdlqvw wkh rswlpdo vwudwhjlhv ri doo rwkhu sod|huv1 Wkh uhpdlqlqj qrwhv duh derxw wkh dssolfdwlrqv dqg yhu| vnhwfk|> iru wkh ghwdlov vhh Jle0 erqv1 H{dpsoh= Frxuqrw zlwk Lqfrpsohwh Lqirupdwlrq1 Zh kdyh d olqhdu ghpdqg ixqfwlrq= S +T, @ d T

zkhuh T @ t . t2 lv wkh wrwdo ghpdqg/ dqg frqvwdqw pdujlqdo frvwv1 Wkh pdujlqdo frvw ri Ilup 4 lv f dqg frpprq nqrzohgjh/ zkloh Ilup 5*v pdujlqdo frvw lv lwv sulydwh lqirupdwlrq1 Lwv pdujlqdo frvw +w|sh, fdq eh hlwkhu fw zlwk suredlolw| / ru fu zlwk suredelolw| 4 up pd{lpl}hv lwv h{shfwhg surw1 Krz wr qg wkh Ed|hvldq Qdvk HtxloleulxpB Ilup 5 kdv wzr srvvleoh w|shv> dqg glhuhqw dfwlrqv zloo eh fkrvhq iru wkh wzr glhuhqw w|shv1 6 1 Hdfk

it2 +fu ,> t2 +fw ,jW Vxssrvh up 5 lv w|sh kljk1 Wkhq/ jlyhq wkh txdqwlw| t fkrvhq e| sod|hu/ lwv sureohp lv

pd{+S

2

fw ,t2 @ ^d

t

t2

fw ` t2 =

Khqfh/ dW t 5

W t2 +fw , @

fw

+-,

Vlploduo|/ vxssrvh up 5 lv orz w|sh=

pd{ ^d

2

W t

t2

fu ` t2 /

khqfh dW t 5

W t2 +fu , @

fu

=

+--,

Lpsruwdqw Uhpdun= Wkh vdph ohyho ri t lq erwk fdvhv1 Zk|BB Ilup 4*v sureohp=

pd{ ^d

t

W t2 +fw ,

f` t . +4

, ^d

t

W t2 +fu ,

f` t

W t @

^d

W t2 +fw ,

f` . +4 5

, ^d

W t2 +fu ,

f`

+---,

W W W Vroyh -/ --/ dqg --- iru t > t2 +fu ,> t2 +fw ,1

W t2 +fw , @

d

5fw . f +4 . 6 d 5fu . f 6

,+fw 9 +fw fu ,

fu ,

W t2 +fu , @

9

W t @

d

5f . fw . +4 6 7

,fu

Kduvdu|l*v Mxvwlfdwlrq iru Pl{hg Vwudwhjlhv Frqvlghu wkh jdph R R I 5 . w > 4 3> 3 I 3> 3 4> 5 . w2

zkhuh w > w2 sulydwh lqirupdwlrq ri sod|huv 4 dqg 5/ uhvshfwlyho|/ dqg duh lqghshqghqw gudzq iurp xqlirup glvwulexwlrq ryhu ^ > 5`1 Fkhfn wkdw + R li w 3 vW +w , @ I rwkhuzlvh/ dqg v2 +w2 , @W

+

I li w2 3 R rwkhuzlvh

irup d Ed|hvldq Qdvk htxloleulxp1 Qrwh wkdw sod|hu 4 sod|v 3 zlwk suredelolw| 526 dqg sod|hu 5 sod|v I zlwk suredelolw| 5261 Dv $ 3 +l1h1/ dv xqfhuwdlqw| glvdsshduv,/ wklv vwudwhj| suroh frqyhujhv wr d pl{hg vwudwhj| htxloleulxp lq zklfk sod|hu 4 sod|v 3 zlwk suredelolw| 526 dqg sod|hu 5 sod|v I zlwk suredelolw| 5261 Vhh Jleerqv iru ghwdlov1

Dxfwlrqv Wzr elgghuv iru d xqltxh jrrg1 y = ydoxdwlrq ri elgghu l1 Ohw xv dvvxph wkdw y *v duh gudzq lqghshqghqwo| iurp d xqlirup glvwulexwlrq ryhu ^3> 4`1 y lv sod|hu l*v sulydwh lqirupdwlrq1 Wkh jdph wdnhv wkh irup ri erwk elgghuv vxeplwwlqj d elg/ wkhq wkh kljkhvw elgghu zlqv dqg sd|v khu elg1 Ohw e eh sod|hu l*v elg1 y +e > e2 > y > y2 , @ y e li e A e& u li e @ e& 2 3 li e ? e&3

pd{+y

u

4 e ,S ureie A e& +y& ,mjlyhq eholhiv ri sod|hu l, . +y 5

e ,S ureie @ e& +y& ,m===,

Ohw xv uvw frqmhfwxuh wkh irup ri wkh htxloleulxp= Frqmhfwxuh= V|pphwulf dqg olqhdu htxloleulxp

8

e @ d . fy= Wkhq/ +y 2 e ,S ureie @ e& +y& ,m===, @ 31 Khqfh/

pd{+y

u

e ,S ureie d . fy& j @ e ,S ureiy& e d f j @ +y e , +e d, f

+y IRF=

e @

y . d li y d 5 @d li y ? d

+4,

Wkh ehvw uhvsrqvh e fdq eh d olqhdu vwudwhj| rqo| li d @ 31 Wkxv/ 4 e @ y = 5 Grxeoh Dxfwlrq Vlpxowdqhrxvo|/ Vhoohu qdphv Sr dqg Ex|hu qdphv Su 1 Li Su ? Sr / wkhq qr wudgh> li Su Sr wudgh dw sulfh s @

' n' 1 2K r

Ydoxdwlrqv duh sulydwh lqirupdwlrq= Yu xqlirup ryhu +3> 4, Yr xqlirup ryhu +3> 4, dqg lqghshqghqw iurp Yu Vwudwhjlhv Su +Yu , dqg Sr +Yr ,1 Wkh ex|hu*v sureohp lv Su . Sr +Yr , = Su Sr +Yr , @ 'K 5 Su . H^Sr +Yr ,mSu Sr +Yr ,` S ureiSu Sr +Yr ,j 5 pd{H Yu

pd{ Yu

'

K

Sr +Yr ,=

zkhuh H^Sr +Yr ,mSu Sr +Yr ,` lv wkh h{shfwhg vhoohu elg frqglwlrqdo rq Su ehlqj juhdwhu wkdq Vlploduo|/ wkh vhoohu*v sureohp lv 9

Sr . Su +Yu , pd{H 'r 5 Sr . H^Su +Yu ,mSu +Yu , Sr pd{ 'r 5 Sr +Yr ,1

Yr = Su +Yu , Sr ` @ Yr ` S ureiSu +Yu , Sr j

Htxloleulxp lv zkhuh Sr +Y& , lv d ehvw uhvsrqvh wr Su +Yu , zkloh Su +Yu , lv d ehvw uhvsrqvh wr Wkhuh duh pdq| Ed|hvldq Qdvk Htxloleuld1 Khuh lv rqh1 Sr @ [ Su @ [ li li Yr [ Yu [=

Dq htxloleulxp zlwk {hg sulfh1 Zk| lv wklv dq htxloleulxpB Ehfdxvh jlyhq Sr @ [ li Yr [> wkh ex|hu grhv qrw zdqw wr wudgh zlwk Yu ? [ dqg zlwk Yu A [> Su @ [ lv rswlpdo1

vb v b /v s Trade X Efficient not to trade

0 Inefficient lack of tradeQrz frqvwuxfw dq htxloleulxp zlwk olqhdu vwudwhjlhv= su @ du . fu yu sr @ dr . fr yr >

vS

zkhuh du / dr / fu / dqg fr duh wr eh ghwhuplqhg1 Qrwh wkdw su sr +yr , @ dr . fr yr l yr su fr dr =

:

Olnhzlvh/ sr su +yu , @ du . fu yu l yu Wkhq/ wkh ex|hu*v sureohp lv4 pd{H yu su . sr +yr , = su sr +yr , 5 ] RK 3@r Sr @ pd{ yu sr fu du =

K

su . sr +yr , gyr K f 5 ] RK 3@r Sr su . dr . fr yr gyr yu @ pd{ K f 5 ] RK 3@r Sr su dr su . dr fr @ pd{ yr gyr yu K fr 5 5 f su dr su . dr fr su dr 2 @ pd{ yu K fr 5 7 fr su dr su . dr su dr yu @ pd{ K fr 5 7 su dr yu @ pd{ K fr 4 fr yu 6su . dr 7 6+su dr , @3 7fr

6su . dr 7

=

I1R1F1=

l1h1/ 5 4 su @ yu . dr = 6 6 Vlploduo|/ wkh vhoohu*v sureohp lv4

+5,

Wkhuh lv vrphzkdw vlpsohu zd| lq wr jhw wkh vdph rxwfrph> vhh Jleerqv1

;

sr . su +yu , pd{H r 5

sr . du . fu yu yr = su +yu , sr @ pd{ R 3@ yr gyu r rS K 5 K ] sr du sr . du fu @ pd{ 4 yr . 3 yu gyu r fu 5 5 RrS @K K # $ sr du sr . du fu sr du 2 @ pd{ 4 yr . 4 r fu 5 7 fu fu sr du sr du sr . du yr . . @ pd{ 4 r fu 5 7 7 sr du 6sr . du @ pd{ 4 r fu 7 ]

fu yr . 7

I1R1F1

4 6sr . du fu 7 6sr . du 7

4 6 yr . . 4 7 7 6 fu yr . . +fu 7 7

sr

fu

du

@3

l1h1/ +sr du ,, @ 3>

l1h1/ 6sr @ 5 l1h1/ 5 du . fu sr @ yr . = 6 6 E| +5,/ du @ dr @6/ dqg e| +6,/ dr @ du @ 4@451 Wkh htxloleulxp lvK

du . yr 7

fu 6 du . fu . +fu . du , @ yr . 7 7 5

+6,

d . 2 1 Khqfh/ 0 also represents .

3

Attitudes toward risk

Suppose now the prizes of lotteries are money. Lottery A gives you 50 dollars for sure (a degenerated lottery). Lottery B gives you 100 dollars with probability 1 1 2 and 0 with probability 2 . Which do you prefer? The two lotteries give you the same expected amount of money, 50 dollars, so your choice reects whether you like risk or not. If you prefer Lottery A, then we can say you are risk averse. Your vNM utility function satises u(50) > 1 1 u(0) + u(100). 2 2

This is true if your vNM utility function is concave (u < 0 if u is twice dierentiable). In general, if your vNM utility function is concave, you always prefer getting the avarage for sure to getting some risky lottery. In contrast, if your vNM utility function is concave, you always prefer getting some risky lottery to getting the avarage for sure. In this case, you are called risk loving. If your vNM utility function is linear, you are indierent between getting some risky lottery and getting the avarage for sure. In this case, you are called risk neutral. 3

4

Modelling games

When we write down a game which represents some economic environment, the payos of the game are vNM utilities of the players, not the actual money amount received. In this way, we can assume the players maximize their expected payos. The only case where the payos of a player is equal to the actual money amount received is the case where the player is risk neutral. Actually, in many textbook modelling we nd the players are assumed to be risk neutral and therefore maximize expected revenue.

4

14.12 Review Notes Dynamic games with incomplete informationKenichi Amaya November 30, 2001

1

Perfect Bayesian equilibrium s = (s1 , , sn ): strategy prole. A belief at an information set is a probability distribution over decision nodes in the information set. A belief system is the collection of beliefs at all information sets.

Denition A pair of strategy prole and a belief system (s, ) is a Perfect Bayesian (Nash) equilibrium (PBE) if 1. Given their beliefs, the players strategies must be sequentially rational. That is, at each information set the action taken by the player with the move (and the players subsequent strategy) must be optimal given the players belief at that information set and the other players subsequent strategies. 2. Beliefs are determined by Bayes rule and the players equilibrium strategies wherever possible. Note 1: When you are asked to describe a PBE, you need to show not only the strategy prole but also the belief system. (However, you usually dont need to describe the belief at a singleton information set because it is obvious). Note 2: PBE is a stronger concept than subgame perfect Nash equilibrium. Therefore, the strayegy prole of a PBE is a subgame perfect Nash equilibrium. E52-303,

kamaya@mit.edu

1

2

How do we nd PBE?1. Using sequential rationality and Bayes rule, determine strategies and beliefs wherever possible. If you can determine strategies and beliefs everywhere, thats it. 2. When you cant do more, make an assumption about a strategy at any information set. Then, using sequential rationality and Bayes rule, determine strategies and beliefs wherever possible. 3. You may be able to determine strategies and beliefs everywhere, i.e., to nd an equilibrium. Then, change your assumption and look for another equilibrium. 4. You may reach a contradiction. In this case, the assumption was wrong. Change your assumption. 5. When you cant do more only with the assumption you made, make a further assumption.

Here is an abstract procedure of looking for PBE.

The idea we use in nding a mixed strategy Nash equilibrium applies here too: The mixing probability must be such that the other player is indierent between the strategies he is mixing.

2

14.12 Economic Applications of Game TheoryProfessor: Muhamet Yildiz Lecture: MW 2:30-4:00 @4-153? Office Hours: M 4-5:30 @E52-251a TA: Kenichi Amaya F 10,3 @E51-85 Office Hours:TBA Web: http://web.mit.edu/14.12/www/

Name of the gameGame Theory = Multi-person decision theory The outcome is determined by the actions independently taken by multiple decision makers. Strategic interaction. Need to understand what the others will do what the others think that you will do

1

Hawk-Dove game

V c V c , 2 2

(V,0) (V/2,V/2)

(0,V)

Chicken

(-1,-1) (0,1)

(1,0) (1/2,1/2)

2

Stag Hunt

(2,2) (0,4)

(4,0) (6,6)

Quiz Problem Without discussing with anyone, each student is to write down a real number xi between 0 and 100 on a paper and submit it to the TA. The TA will then compute the averagex= x1 + x2 + + xn . n

The students who submit the number that is closest to x / 3 will share 100 points equally;the others will get 0.

3

14.12 Game TheoryLecture 2: Decision Theory Muhamet Yildiz

Road Map1. Basic Concepts (Alternatives, preferences,) 2. Ordinal representation of preferences 3. Cardinal representation Expected utility theory 4. Applications: Risk sharing and Insurance

5. Quiz

1

Basic Concepts: Alternatives Agent chooses between the alternatives X = The set of all alternatives Alternatives are Mutually exclusive, and Exhaustive Example: Options = {Tea, Coffee} X = {T, C, TC, NT} where T= Tea, C = Coffee, TC = Tea and Coffee, NT = Neither Tea nor Coffee

Basic Concepts: Preferences TeX TeX ordinal representation

2

Examples Define a relation among the students in this class by x T y iff x is at least as tall as y; x M y iff xs final grade in 14.04 is at least as high as ys final grade; x H y iff x and y went to the same high school; X Y y iff x is strictly younger than y; x S y iff x is as old as y;

Exercises Imagine a group of students sitting around a round table. Define a relation R, by writing x R y iff x sits to the right of y. Can you represent R by a utility function? Consider a relation } among positive real 2 numbers represented by u with u ( x ) = x . Can this relation be represented byu * ( x ) = x ? What about u **( x) = 1/ x ?

3

TeX OR Theorem TeX Cardinal representation VNM Axioms Theorem

A Lottery

4

Two Lotteries$1000 .3.3

$1M .00001

$10 .99999 $0 $0

.4

Exercise Consider an agent with VNM utility 2 function u with u ( x ) = x . Can his preferences be represented by VNM utility function u * ( x ) = x ? What about u **( x) = 1/ x ?

5

Attitudes towards Riskp

A fair gamble:1-p

x y

px+(1-p)y = 0.

An agent is said to be risk neutral iff he is indifferent towards all fair gambles. He is said to be (strictly) risk averse iff he never wants to take any fair gamble, and (strictly) risk seeking iff he always wants to take fair gambles.

A utility functionEU u A u(pW1+(1- p)W2) C EU(Gamble) B

W1

pW1+(1-p)W2

W2

6

An agent is risk-neutral iff he has a linear utility function, i.e., u(x) = ax + b. An agent is risk-averse iff his utility function is concave. An agent is risk-seeking iff his utility function is convex.

Risk Sharing Two agents, each having a utility function u with u ( x ) = x and an asset: .5$100 $0

For each agent, the value of the asset is Assume that the value of assets are independently distributed.

.5

7

If they form a mutual fund so that each agent owns half of each asset, each gets$100 1/4 1/2 1/4 $0 $50

Insurance We have an agent with u(x ) = x and.5 .5 $1M $0

And a risk-neutral insurance company with lots of money, selling full insurance for premium P.

8

Lecture 3 Decision Theory/Game Theory14.12 Game Theory Muhamet Yildiz

Road Map1. 2. Basic Concepts (Alternatives, preferences,) Ordinal representation of preferences

3. Cardinal representation Expected utility theory

4. Applications: Risk sharing and Insurance

5. Quiz 6. Representation of games in strategic and extensive forms 7. Quiz?

1

A Lottery

Two Lotteries$1000 .3.3

$1M .00001

$10 .99999 $0 $0

.4

2

Exercise Consider an agent with VNM utility 2 function u with u ( x ) = x . Can his preferences be represented by VNM utility function u * ( x ) = x ? What about u **( x) = 1/ x ?

Attitudes towards Risk A fair gamble:p 1-p x y

px+(1-p)y = 0.

An agent is said to be risk neutral iff he is indifferent towards all fair gambles. He is said to be (strictly) risk averse iff he never wants to take any fair gamble, and (strictly) risk seeking iff he always wants to take fair gambles.

3

A utility functionEU u A u(pW1+(1- p)W2) C EU(Gamble) B

W1

pW1+(1-p)W2

W2

An agent is risk-neutral iff he has a linear utility function, i.e., u(x) = ax + b. An agent is risk-averse iff his utility function is concave. An agent is risk-seeking iff his utility function is convex.

4

Risk Sharing Two agents, each having a utility function u with u ( x ) = x and an asset: .5$100 $0

For each agent, the value of the asset is Assume that the value of assets are independently distributed.

.5

If they form a mutual fund so that each agent owns half of each asset, each gets$100 1/4 1/2 1/4 $0 $50

5

Insurance We have an agent with u(x ) = x and.5 .5 $1M $0

And a risk-neutral insurance company with lots of money, selling full insurance for premium P.

Quiz Problem Without discussing with anyone, each student is to write down a real number xi between 0 and 100 on a paper and submit it to the TA. The TA will then compute the averagex= x1 + x2 + + xn . n

The students who submit the number that is closest to x / 3 will share 100 points equally;the others will get 0.

6

Multi-person Decision Theory Who are the players? Who has which options? Who knows what? Who gets how much?

Knowledge1. 2. 3. 4. If I know something, it must be true. If I know x, then I know that I know x. If I dont know x, then I know that I dont know x. If I know something, I know all its logical implications. Common Knowledge: x is common knowledge iff Each player knows x Each player knows that each player knows x Each player knows that each player knows that each player knows x Each player knows that each player knows that each player knows that each player knows x ad infinitum

7

Representations of games

Normal-form representationDefinition (Normal form): A game is any listG = (S1 , l , S n ; u1 , l , u n )

where, for each i N = {1,2,l, n}, Si is the set of all strategies available to i, ui : S1 m Sn is the VNM utility function of player i.

Assumption: G is common knowledge.Definition: A player i is rational iff he tries to maximize the expected value of ui given his beliefs.

8

Chicken

(-1,-1) (0,1)

(1,0) (1/2,1/2)

Extensive-form representationDefinition: A tree is a set of nodes connected with directed arcs such that 1. For each node, there is at most one incoming arc; 2. each node can be reached through a unique path;

9

A tree?

A tree??BA B

A CD C

10

A treeNon-terminal nodes Terminal Nodes

Extensive form definitionDefinition: A game consists of a set of players a tree an allocation of each non-terminal node to a player an informational partition (to be made precise) a payoff for each player at each terminal node.

11

Information setAn information set is a collection of nodes such that 1. The same player is to move at each of these nodes; 2. The same moves are available at each of these nodes. An informational partition is an allocation of each non-terminal node of the tree to an information set.

A game1 L (2,2) l 1 (1,3) (3,1) (3,3) R 2 r 1 u (1,1) (0,0)

12

Another Game1 T 2 L R L R x

B

The same game1 T 2 L R L R B x

13

StrategyA strategy of a player is a complete contingent-plan, determining which action he will take at each information set he is to move (including the information sets that will not be reached according to this strategy).

Matching pennies with perfect information1 Head 2 head (-1,1) tail (1,-1) 2 head (1,-1) tail (-1,1) Tail 2s Strategies: HH = Head if 1 plays Head, Head if 1 plays Tail; HT = Head if 1 plays Head, Tail if 1 plays Tail; TH = Tail if 1 plays Head, Head if 1 plays Tail; TT = Tail if 1 plays Head, Tail if 1 plays Tail.

14

Matching pennies with perfect information2

1

HH

HT

TH

TT

Head

Tail

Matching pennies with Imperfect information1 Head 2 head (-1,1) tail (1,-1) head (1,-1) tail (-1,1) Tail 1 2

Head (-1,1) (1,-1)

Tail (1,-1) (-1,1)

Head Tail

15

A game with natureLeft 1 Head 1/2 Nature 1/2 Tail 2 Left Right (2, 2) (3, 3) (5, 0)

Right (0, -5)

A centipede game1 D (4,4) A 2 (5,2) d (3,3) 1 a (1,-5)

16

Lecture 4 Representation of Games & Rationalizability14.12 Game Theory Muhamet Yildiz

Road Map1. Representation of games in strategic and extensive forms 2. Dominance 3. Dominant-strategy equilibrium 4. Rationalizability

1

Normal-form representationDefinition (Normal form): A game is any listG = (S1 , l , S n ; u1 , l , u n )

where, for each i N = {1,2,l, n}, Si is the set of all strategies available to i, ui : S1 m Sn is the VNM utility function of player i.

Assumption: G is common knowledge.Definition: A player i is rational iff he tries to maximize the expected value of ui given his beliefs.

Chicken

(-1,-1) (0,1)

(1,0) (1/2,1/2)

2

Extensive-form representationDefinition: A tree is a set of nodes connected with directed arcs such that 1. For each node, there is at most one incoming arc; 2. each node can be reached through a unique path;

A treeNon-terminal nodes Terminal Nodes

3

Extensive form definitionDefinition: A game consists of a set of players a tree an allocation of each non-terminal node to a player an informational partition (to be made precise) a payoff for each player at each terminal node.

Information setAn information set is a collection of nodes such that 1. The same player is to move at each of these nodes; 2. The same moves are available at each of these nodes. An informational partition is an allocation of each non-terminal node of the tree to an information set.

4

A game1 L (2,2) l 1 (1,3) (3,1) (3,3) R 2 r 1 u (1,1) (0,0)

Another Game1 x

T 2 L R L

B

R

5

The same game1 x

T 2 L R L

B

R

StrategyA strategy of a player is a complete contingent-plan, determining which action he will take at each information set he is to move (including the information sets that will not be reached according to this strategy).

6

Matching pennies with perfect information1 Head 2 head (-1,1) tail (1,-1) 2 head (1,-1) tail (-1,1) Tail 2s Strategies: HH = Head if 1 plays Head, Head if 1 plays Tail; HT = Head if 1 plays Head, Tail if 1 plays Tail; TH = Tail if 1 plays Head, Head if 1 plays Tail; TT = Tail if 1 plays Head, Tail if 1 plays Tail.

Matching pennies with perfect information2 1

HH

HT

TH

TT

Head

Tail

7

Matching pennies with Imperfect information1 1 Head 2 head (-1,1) tail (1,-1) head (1,-1) tail Tail 2

Head (-1,1)

Tail (1,-1)

Head

Tail(-1,1)

(1,-1)

(-1,1)

A game with natureLeft 1 Head 1/2 Nature 1/2 Tail 2 Right (0, -5) Left (3, 3) Right (2, 2) (5, 0)

8

Mixed StrategyDefinition: A mixed strategy of a player is a probability distribution over the set of his strategies. Pure strategies: Si = {si1,si2,,sik} A mixed strategy: i: S [0,1] s.t. i(si1) + i(si2) + + i(sik) = 1. If the other players play s-i =(s1,, si-1,si+1,,sn), then the expected utility of playing i is i(si1)ui(si1,s-i) + i(si2) ui(si2,s-i) + + i(sik) ui(sik,s-i).

How to play

9

Dominances-i =(s1,, si-1,si+1,,sn) Definition: A pure strategy si* strictly dominates si if and only if

u i ( si* , s i ) > u i ( si , s i )

si .

A mixed strategy i* strictly dominates si iff i ( si1 )ui ( si1 , si ) + m + i ( sik )ui ( sik , si ) > ui ( si , si ) si

A rational player never plays a strictly dominated strategy.

Prisoners Dilemma2 1

Cooperate (5,5) (6,0)

Defect (0,6) (1,1)

Cooperate Defect

10

A game1 2

L (3,0) (1,0) (0,3)

m (1,1) (0,10) (1,1)

R (0,3) (1,0) (3,0)

T M B

Weak DominanceDefinition: A pure strategy si* weakly dominates si if and only if

u i ( si* , s i ) u i ( si , s i )

si .

and at least one of the inequalities is strict. A mixed strategy i* weakly dominates si iff

i ( si1 )ui ( si1 , si ) + + i ( sik )ui ( sik , si ) > ui ( si , si ) siand at least one of the inequalities is strict. If a player is rational and cautious (i.e., he assigns positive probability to each of his opponents strategies), then he will not play a weakly dominated strategy.

11

Dominant-strategy equilibriumDefinition: A strategy si* is a dominant strategy iff si* weakly dominates every other strategy si. Definition: A strategy profile s* is a dominant-strategy equilibrium iff si* is a dominant strategy for each player i. If there is a dominant strategy, then it will be played, so long as the players are

Prisoners Dilemma2 1

Cooperate (5,5) (6,0)

Defect (0,6) (1,1)

Cooperate Defect

12

Second-price auction N = {1,2} buyers; The value of the house for buyer i is vi; Each buyer i simultaneously bids bi; i* with bi* = max bi gets the house and pays the second highest bid p = maxjibj.

QuestionWhat is the probability that an nxn game has a dominant strategy equilibrium given that the payoffs are independently drawn from the same (continuous) distribution on [0,1]?

13

(1/n) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

(2n-2)

1

2

3

4

5

6

7

8

9

10

A game2 1

Assume: Players are rational and player 2 knows that 1 is rational. 1 is rational and 2 knows this:

L (3,0) (1,0) (0,3)

m (1,1) (0,10) (1,1)

R T (0,3) (1,0) (3,0) T B B

L

m

R

(3,0) (1,1) (0,3) (0,3) (1,1) (3,0)

T M B

And 2 is rational:

L

R

(3,0) (0,3) (0,3) (3,0)

14

RationalizabilityEliminate all the strictly dominated strategies.

Yes

Any dominated strategy In the new game? No Rationalizable strategies

The play is rationalizable, provided that

Simplified price-competitionFirm 2 Firm 1

High 6,6 10,0 8,0

Medium 0,10 5,5 8,0

Low 0,8 0,8 4,4

High Medium Low

Dutta

15

A strategy profile is rationalizable when Each players strategy is consistent with his rationality, i.e., maximizes his payoff with respect to a conjecture about other players strategies; These conjectures are consistent with the other players rationality, i.e., if i conjectures that j will play sj with positive probability, then sj maximizes js payoff with respect to a conjecture of j about other players strategies; These conjectures are also consistent with the other players rationality, i.e., Ad infinitum

16

Lecture 5 Nash equilibrium & Applications14.12 Game Theory Muhamet Yildiz

Road Map1. Rationalizability summary 2. Nash Equilibrium 3. Cournot Competition1. Rationalizability in Cournot Duopoly

4. 5. 6. 7.

Bertrand Competition Commons Problem Quiz Mixed-strategy Nash equilibrium

1

Dominant-strategy equilibriums-i =(s1,, si-1,si+1,,sn) Definition: si* strictly dominates si iff si* weakly dominates si iff u i ( s , s i ) u i ( si , s i ) si and at least one of the inequalities is strict. Definition: A strategy si* is a dominant strategy iff si* weakly dominates every other strategy si. Definition: A strategy profile s* is a dominantstrategy equilibrium iff si* is a dominant strategy for each player i. Examples: Prisoners Dilemma; Second-Price auction.* i

u i ( si* , s i ) > u i ( si , s i )

si ;

QuestionWhat is the probability that an nxn game has a dominant strategy equilibrium given that the payoffs are independently drawn from the same (continuous) distribution on [0,1]?

2

(1/n) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

(2n-2)

1

2

3

4

5

6

7

8

9

10

RationalizabilityEliminate all the strictly dominated strategies.

Yes

Any dominated strategy In the new game? No Rationalizable strategies

The play is rationalizable, provided that

3

Simplified price-competitionFirm 2 Firm 1

High 6,6 10,0 8,0

Medium 0,10 5,5 8,0

Low 0,8 0,8 4,4

High Medium Low

Dutta

A strategy profile is rationalizable when Each players strategy is consistent with his rationality, i.e., maximizes his payoff with respect to a conjecture about other players strategies; These conjectures are consistent with the other players rationality, i.e., if i conjectures that j will play sj with positive probability, then sj maximizes js payoff with respect to a conjecture of j about other players strategies; These conjectures are also consistent with the other players rationality, i.e., Ad infinitum

4

Stag Hunt

(2,2) (0,4)

(4,0) (6,6)

A summary If players are rational (and cautious), then they play the dominant-strategy equilibrium whenever it exists But, typically, it does not exist

If it is common knowledge that players are rational, then they will play a rationalizable strategy-profile Typically, there are too many rationalizable strategies

Now, a stronger assumption: The players are rational and their conjectures are mutually known.

5

Nash EquilibriumDefinition: A strategy-profile s* =(s1*,,sn*) is a Nash Equilibrium iff, for each player i, and for each strategy si, we have* u i ( s1* , l , si*1 , si* , si*+1 , l , s n ) * * u i ( s1 , l , si*1 , si , si*+1 , l , s n ),

i.e., no player has any incentive to deviate if he knows what the others play. ??If players are rational, and their conjectures about what the others play are mutually known, then they must be playing a Nash equilibrium.

Stag Hunt

(2,2) (0,4)

(4,0) (6,6)

6

Economic Applications1. Cournot (quantity) Competition1. Nash Equilibrium in Cournot duopoly 2. Nash Equilibrium in Cournot oligopoly 3. Rationalizability in Cournot duopoly

2. Bertrand (price) Competition 3. Commons Problem

Cournot Oligopoly N = {1,2,,n} firms; Simultaneously, each firm i produces qi units of a good at marginal cost c, and sells the good at price P = max{0,1-Q} where Q = q1++qn. Game = (S1,,Sn; 1,,n) where Si = [0,), P 1

Q 1

i(q1,,qn) = qi[1-(q1++qn)-c] if q1++qn < 1, -qic otherwise.

7

Cournot Duopoly -- profitqj=0.2 Profit 0 qi(1-qj-c) -cqi c=0.2

-0.2

0

(1-qj-c)/2

1-qj-c

1

C-D best responses qiB(qj) = max{(1-qj-c)/2,0};q2 q1=q1B(q2)

Nash Equilibrium q*: q1* = (1-q2*-c)/2; q2* = (1-q1*-c)/2; q1* = q2* = (1-c)/3

q*1 c 2

q2=q2B(q1) q1

1-c

8

Cournot Oligopoly --Equilibrium q>1-c is strictly dominated, so q 1-c. i(q1,,qn) = qi[1-(q1++qn)-c] for each i. FOC: ( q , , q ) [ qi (1 q1 qn c )] 1 i n

qi

=

q=q

*

qi

q = q*

* * = (1 q1 qn c ) qi* = 0.

That is,

* * * 2q1 + q2 + + qn = 1 c * * * q1 + 2q2 + + qn = 1 c

* * * q1 + q2 + + nqn = 1 c

Therefore, q1*==qn*=(1-c)/(n+1).

Cournot oligopoly comparative staticsP

1

n=1 n=2 n=3 n=4 Q c 1

9

Rationalizability in Cournot Duopolyq2 1-c Assume that players are rational.

1 c 2

q11 c 2

1-c

Players are rational:q2 1-c Assume that players know this.

1 c 2

q11 c 2

1-c

10

Players are rational and know that players are rationalq2 1-c Assume that players know this.

1 c 2

q11 c 2

1-c

Players are rational; players know that players are rational; players know that players know that players are rational q2

1-c Assume that players know this.

1 c 2

q11 c 2

1-c

11

Rationalizability in Cournot duopolyIf i knows that qj q, then qi (1-c-q)/2. If i knows that qj q, then qi (1-c-q)/2. We know that qj q0 = 0. Then, qi q1 = (1-c-q0)/2 = (1-c)/2 for each i; Then, qi q2 = (1-c-q1)/2 = (1-c)(1-1/2)/2 for each i; Then, qn qi qn+1 or qn+1 qi qn where qn+1 = (1-c-qn)/2 = (1-c)(1-1/2+1/4-+(-1/2)n)/2. As n, qn (1-c)/3.

Bertrand (price) competition N = {1,2} firms. Simultaneously, each firm i sets a price pi; If pi < pj, firm i sells Q = max{1 pi,0} unit at price pi; the other firm gets 0. If p1 = p2, each firm sells Q/2 units at price p1, where Q = max{1 p1,0}. The marginal cost is 0. if p1 < p2 p1 (1 p1 ) 1 ( p1 , p2 ) = p1 (1 p1 ) / 2 if p1 = p2 0 otherwise.

12

Bertrand duopoly -- EquilibriumTheorem: The only Nash equilibrium in the Bertrand game is p* = (0,0). Proof: 1. p*=(0,0) is an equilibrium. 2. If p = (p1,p2) is an equilibrium, then p = p*.1. If p = (p1,p2) is an equilibrium, then p1 = p2... If pi > pj= 0, for sufficiently small >0, pj = is a better response to pi for j. If pi > pj> 0, pi = pj is a better response for i. If p1 = p2>0, for sufficiently small >0, pj = pj - is a better response to pj for i.

2. Given any equilibrium p = (p1,p2) with p1 = p2, p = p*.

Commons Problem N = {1,2,,n} players, each with unlimited money; Simultaneously, each player i contributes xi 0 to produce y = x1+xn unit of some public good, yielding payoff Ui(xi,y) = y1/2 xi.

13

QuizEach student i is to submit a real number xi. We will pair the students randomly. For each pair (i,j), if xi xj, the student who submits the number that is closer to (xi+xj)/4 gets 100; the other student gets 20. If xi = xj, then each of i and j gets 50.

14

Lectures 6-7Nash Equilibrium & Backward Induction14.12 Game Theory Muhamet Yildiz

Road Map1. 2. 3. 4. 5. 6. 7. Bertrand Competition Commons Problem Mixed-strategy Nash equilibrium Bertrand competition with costly search Backward Induction Stackelberg Competition Sequential Bargaining

1

Bertrand (price) competition N = {1,2} firms. Simultaneously, each firm i sets a price pi; If pi < pj, firm i sells Q = max{1 pi,0} unit at price pi; the other firm gets 0. If p1 = p2, each firm sells Q/2 units at price p1, where Q = max{1 p1,0}. The marginal cost is 0. if p1 < p2 p1 (1 p1 ) 1 ( p1 , p2 ) = p1 (1 p1 ) / 2 if p1 = p2 0 otherwise.

Bertrand duopoly -- EquilibriumTheorem: The only Nash equilibrium in the Bertrand game is p* = (0,0). Proof: 1. p*=(0,0) is an equilibrium. 2. If p = (p1,p2) is an equilibrium, then p = p*.1. If p = (p1,p2) is an equilibrium, then p1 = p2... If pi > pj= 0, for sufficiently small >0, pj = is a better response to pi for j. If pi > pj> 0, pi = pj is a better response for i. If p1 = p2>0, for sufficiently small >0, pj = pj - is a better response to pj for i.

2. Given any equilibrium p = (p1,p2) with p1 = p2, p = p*.

2

Commons Problem N = {1,2,,n} players, each with unlimited money; Simultaneously, each player i contributes xi 0 to produce y = x1+xn unit of some public good, yielding payoff Ui(xi,y) = y1/2 xi.

Stag Hunt

(2,2) (0,4)

(4,0) (5,5)

3

Equilibrium in Mixed StrategiesWhat is a strategy? A complete contingent-plan of a player. What the others think the player might do under various contingency.

What do we mean by a mixed strategy? The player is randomly choosing his pure strategies. The other players are not certain about what he will do.

Stag Hunt

(2,2) (0,4)

(4,0) (5,5)

4

Mixed-strategy equilibrium in Stag-Hunt game Assume: Player 2 thinks that, with probability p, Player 1 targets for Rabbit. What is the best probability q she wants to play Rabbit? His payoff from targeting Rabbit: U2(R;p) = 2p + 4(1-p) = 4-2p. From Stag: U2(R;p) = 5(1-p) She is indifferent iff 4-2p = 5(1-p) iff p = 1/3.5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 5(1-p) 4 - 2p

0

0.2

0.4

0.6

0.8

1

if p < 1/3 0 q ( p ) = q [0,1] if p = 1/3 1 if p > 1/3 BR

Best responses in Stag-Hunt gameq

1/3 p 1/3

5

Bertrand Competition with costly search N = {F1,F2,B}; F1, F2 are firms; B is buyer B needs 1 unit of good, worth 6; Firms sell the good; Marginal cost = 0. Possible prices P = {1,5}. Buyer can check the prices with a small cost c > 0. Game: 1. Each firm i chooses price p i; 2. B decides whether to check the prices; 3. (Given) If he checks the prices, and p1p2, he buys the cheaper one; otherwise, he buys from any of the firm with probability .

Bertrand Competition with costly searchF2 F1 High F2 High Low F1 High High Low

Low

Low

Check

Dont Check

6

Mixed-strategy equilibrium Symmetric equilibrium: Each firm charges High with probability q; Buyer Checks with probability r. U(check;q) = q21 + (1-q2)5 c = 5 - 4 q2 c; U(Dont;q) = q1 + (1-q)5 = 5 - 4 q; Indifference: 4q(1-q) = c; i.e., U(high;q,r) = 0.5(1-r(1-q))5; U(low;q,r) = qr1 + 0.5(1-qr) Indifference = r = 4/(5-4q).

Dynamic Games of Perfect Information & Backward Induction

7

DefinitionsPerfect-Information game is a game in which all the information sets are singleton. Sequential Rationality: A player is sequentially rational iff, at each node he is to move, he maximizes his expected utility conditional on that he is at the node even if this node is precluded by his own strategy. In a finite game of perfect information, the common knowledge of sequential rationality gives Backward Induction outcome.

A centipede game1 D (4,4) A 2 (5,2) d (3,3) 1 a (1,-5)

8

Backward InductionTake any pen-terminal node Pick one of the payoff vectors (moves) that gives the mover at the node the highest payoff Assign this payoff to the node at the hand; Eliminate all the moves and the terminal nodes following the node Yes Any non-terminal node No The picked moves

Battle of The Sexes with perfect information1

T

B

2 L R L

2 R

(2,1)

(0,0)

(0,0)

(1,2)

9

Note There are Nash equilibria that are different from the Backward Induction outcome. Backward Induction always yields a Nash Equilibrium. That is, Sequential rationality is stronger than rationality.

Matching Pennies (wpi)1 Head 2 head tail 2 head (1,-1) tail (-1,1)

Tail

(-1,1)

(1,-1)

10

Stackelberg DuopolyGame: P N = {1,2} firms w MC = 0; 1. Firm 1 produces q1 units 1 2. Observing q1, Firm 2 produces q2 units 3. Each sells the good at price P = max{0,1-(q1+q2)}. i(q1, q2) = qi[1-(q1+q2)] if q1+ q2 < 1, 0 otherwise.1

Q

Stackelberg equilibrium If q1 > 1, q2*(q1) = 0. If q1 1, q2*(q1) = (1-q1)/2. Given the function q2*, if q1 1 = q1 (1-q1)/2; 0 otherwise. q1* = . q2*(q1*) = .1 P 1

1(q1;q2*(q1)) = q1[1-(q1+ (1-q1)/2)]

11

Sequential Bargaining1 D

N = {1,2} X = feasible expected-utility pairs (x,y X ) Ui(x,t) = itxi d = (0,0) D disagreement payoffs1

Timeline 2 periodAt t = 1, Player 1 offers some (x1,y1), Player 2 Accept or Rejects the offer If the offer is Accepted, the game ends yielding (x1,y1), Otherwise, we proceed to date 2.

At t = 2, Player 2 offers some (x2,y2), Player 1 Accept or Rejects the offer If the offer is Accepted, the game ends yielding payoff (x2,y2). Otherwise, the game end yielding d = (0,0).

1

(x1,y1)

2

2 Reject

(x2,y2)

1

(0,0) Reject (x2,y2)

Accept

Accept

(x1,y1)

12

1

(x1,y1)

2

2 Reject

(x2,y2)

1

(0,0) Reject (x2,y2)

Accept

Accept

(x1,y1)

At t = 2, Accept iff y2 0. Offer (0,1). At t = 1, Accept iff x2 . Offer (1,).

Timeline 2n periodT = {1,2,,2n-1,2n} If t is odd, Player 1 offers some (xt,yt), Player 2 Accept or Rejects the offer If the offer is Accepted, the game ends yielding t(xt,yt), Otherwise, we proceed to date t+1.

If t is even Player 2 offers some (xt,yt), Player 1 Accept or Rejects the offer If the offer is Accepted, the game ends yielding payoff (xt,yt), Otherwise, we proceed to date t+1, except at t = 2n, when the game end yielding d = (0,0).

13

Equilibrium Scientific Word

14

Lectures 8Subgame-perfect Equilibrium & Applications14.12 Game Theory Muhamet Yildiz

Road Map1. Subgame-perfect Equilibrium1. 2. 3. 4. Motivation What is a subgame? Definition Example

2. Applications1. Bank Runs 2. Tariffs & Intra-industry trade

3. Quiz

1

A game1 E 1 T B X (2,6)

2 L (0,1) R (3,2) L (-1,3) R (1,5)

Sequential Bargaining1 D

N = {1,2} X = feasible expected-utility pairs (x,y X ) Ui(x,t) = itxi d = (0,0) D disagreement payoffs1

2

Timeline periodT = {1,2,, n-1,n,} If t is odd, Player 1 offers some (xt,yt), Player 2 Accept or Rejects the offer If the offer is Accepted, the game ends yielding t(xt,yt), Otherwise, we proceed to date t+1.

If t is even Player 2 offers some (xt,yt), Player 1 Accept or Rejects the offer If the offer is Accepted, the game ends yielding payoff (xt,yt), Otherwise, we proceed to date t+1.

Backward induction Can be applied only in perfect information games of finite horizon. How can we extend this notion to infinite horizon games, or to games with imperfect information?

3

A subgameA subgame is part of a game that can be considered as a game itself. It must have a unique starting point; It must contain all the nodes that follow the starting node; If a node is in a subgame, the entire information set that contains the node must be in the subgame.

A game1 D (4,4) A 2 (5,2) d (3,3) 1 a (1,-5)

4

And its subgames1 a d (3,3) (1,-5) 2 (5,2) 1 a d (1,-5)

(3,3)

A game1 E 1 T B X (2,6)

2 L (0,1) R (3,2) L (-1,3) R (1,5)

5

DefinitionsA substrategy is the restriction of a strategy to a subgame. A subgame-perfect Nash equilibrium is a Nash equilibrium whose substrategy profile is a Nash equilibrium at each subgame.

Example1 E 1 T B X (2,6)

2 L (0,1) R (3,2) L (-1,3) R (1,5)

6

A Backward-Induction-like methodTake any subgame with no proper subgame Compute a Nash equilibrium for this subgame Assign the payoff of the Nash equilibrium to the starting node of the subgame Eliminate the subgame Yes Any non-terminal node No The moves computed as a part of any (subgame) Nash equilibrium

TheoremIn a finite, perfect-information game, the set of subgame-perfect equilibria is the set of strategy profiles that are computed via backward induction.

7

A subgame-perfect equilibrium?X 1 T B (2,6)

2 L (0,1) R (3,2) L (-1,3) R (1,5)

Bank Run1 W 2 W (r,r) DW W W DW 1 DW DW (R,R) DW R > D > r > D/2

(D,2r-D) (2r-D,D)

2 W (R,R) DW W

(2R-D,D) (D,2R-D)

8

Lectures 9 Applications of Subgame-perfectEquilibrium & Forward Induction14.12 Game Theory Muhamet Yildiz

Road Map1. Applications1. Tariffs & Intra-industry trade 2. Infinite horizon bargaining Singledeviation principle

2. Forward Induction Examples 3. Finitely Repeated Games 4. Quiz

1

Single-Deviation principleDefinition: An extensive-form game is continuous at infinity iff, given any > 0, there exists some t such that, for any two path whose first t arcs are the same, the payoff difference of each player is less than . Theorem: Let G be a game that is continuous at infinity. A strategy profile s = (s1,s2,,sn) is a subgame-perfect equilibrium of G iff, at any information set, where a player i moves, given the other players strategies and given is moves at the other information sets, player i cannot increase his conditional payoff at the information set by deviating from his strategy at the information set.

Sequential Bargaining1 D

N = {1,2} D = feasible expected-utility pairs (x,y D ) Ui(x,t) = itxi d = (0,0) D disagreement payoffs1

2

Timeline periodT = {1,2,, n-1,n,} If t is odd, Player 1 offers some (xt,yt), Player 2 Accept or Rejects the offer If the offer is Accepted, the game ends yielding t(xt,yt), Otherwise, we proceed to date t+1.

If t is even Player 2 offers some (xt,yt), Player 1 Accept or Rejects the offer If the offer is Accepted, the game ends yielding payoff t(xt,yt), Otherwise, we proceed to date t+1.

SPE of -period bargainingTheorem: At any t, proposer offers the other player /(1+), keeping himself 1/(1+), while the other player accept an offer iff he gets /(1+). Proof: Single-deviation principle: Take any date t, at which i offers, j accepts/rejects. According to the strategies in the continuation game, at t+1, j will get 1/(1+). Hence, j accepts an offer iff she gets at least /(1+). i must offer /(1+).

3

Forward InductionStrong belief in rationality: At any history of the game, each agent is assumed to be rational if possible. (That is, if there are two strategies s and s of a player i that are consistent with a history of play, and if s is strictly dominated but s is not, at this history no player j believes that i plays s.)

Table for the bidding gameUi = 20(2+2minjbidj bidi)

min bid 1 2 3

1 60 40 20

2 80 60

3 100

4

Nash equilibria of bidding game 3 equilibria: s1 = everybody plays 1; s2 = everybody plays 2; s3 = everybody plays 3. Assume each player trembles with probability < 1/2, and plays each unintended strategy w.p. /2, e.g., w.p. /2, he thinks that such other equilibrium is to be played. s3 is an equilibrium iff s2 is an equilibrium iff s1 is an equilibrium iff

1 0.8 0.6 0.4 0.2 0 -0.2 (1-/2) -1/2 -0.4 -0.6 -0.8 -1 0.032 0.05 (1-) +(1-/2) -1 0 0.1 0.15 0.2n n n

0.047

5

Bidding game with entry feeEach player is first to decide whether to play the bidding game (E or X); if he plays, he is