+ ü + ÿ ù ù · 2016. 6. 8. · Kefàlaio1 Eisagwg† 1.1Ti e–nai h jewr–a paign–wn 'Ena qarakthristikÏ pou parathro‘me se pollà oikonomikà,biologikà,koinw-nikà,klp.fainÏmena

Συγγραφή

Κριτικός αναγνώστης

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Copyright © ΣΕΑΒ, 2015

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PerieqÏmena

PrÏlogoc i

1 Eisagwg† 11.1 Ti e–nai h jewr–a paign–wn . . . . . . . . . . . . . . . . . . . . 11.2 Istorik† anadrom† . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 'Allec exel–xeic . . . . . . . . . . . . . . . . . . . . . . 61.3 SkopÏc tou bibl–ou . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Bibliograf–a . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Statikà pa–gnia me pl†rh plhrofÏrhsh 112.1 Eisagwg† . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 BasikÏ pla–sio . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Ep–lush . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 Apaloif† kuriarqo‘menwn strathgik∏n . . . . . . . . . 132.3.2 Isorrop–a Nash . . . . . . . . . . . . . . . . . . . . . . 162.3.3 Eidikà jËmata . . . . . . . . . . . . . . . . . . . . . . . 212.3.4 Pa–gnia me àpeirec strathgikËc . . . . . . . . . . . . . 25

2.4 MiktËc StrathgikËc . . . . . . . . . . . . . . . . . . . . . . . . 262.4.1 Arq† thc adiafor–ac . . . . . . . . . . . . . . . . . . . 30

2.5 EfarmogËc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.5.1 Duop∏lio . . . . . . . . . . . . . . . . . . . . . . . . . 312.5.2 Pa–gnia diagwnism∏n . . . . . . . . . . . . . . . . . . . 322.5.3 SumbatÏthta logismik∏n . . . . . . . . . . . . . . . . . 332.5.4 Dhmopras–a de‘terhc tim†c . . . . . . . . . . . . . . . . 35

2.6 Istorik† anadrom† . . . . . . . . . . . . . . . . . . . . . . . . 372.7 Ask†seic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.8 Orolog–a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.9 Bibliograf–a . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3 Dunamikà pa–gnia 443.1 Eisagwg† . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.2 EktetamËnh morf† . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2.1 StrathgikËc . . . . . . . . . . . . . . . . . . . . . . . . 49

i

ii PERIEQOMENA

3.2.2 Isorrop–a . . . . . . . . . . . . . . . . . . . . . . . . . 523.2.3 Kàpoia paràdoxa apotelËsmata . . . . . . . . . . . . . 58

3.3 Suneq† s‘nola energei∏n . . . . . . . . . . . . . . . . . . . . 613.4 EfarmogËc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.4.1 UpÏdeigma Stackelberg . . . . . . . . . . . . . . . . . . 653.4.2 Endogenopo–hsh qronik†c dom†c . . . . . . . . . . . . . 663.4.3 Diadoqik† diapragmàteush . . . . . . . . . . . . . . . . 68

3.5 Tuqa–a gegonÏta . . . . . . . . . . . . . . . . . . . . . . . . . 693.6 Istorik† anadrom† . . . . . . . . . . . . . . . . . . . . . . . . 723.7 Ask†seic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.8 Orolog–a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.9 Bibliograf–a . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4 EpanalambanÏmena pa–gnia 784.1 Eisagwg† . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.2 BasikÏ pla–sio . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.2.1 Istor–ec kai termatikËc istor–ec . . . . . . . . . . . . . 804.2.2 DiaqronikËc protim†seic . . . . . . . . . . . . . . . . . 814.2.3 Ta pa–gnia GT (�) kai G1(�) . . . . . . . . . . . . . . . 834.2.4 StrathgikËc . . . . . . . . . . . . . . . . . . . . . . . . 844.2.5 Isorrop–ec gia peperasmËnwc epanalambanÏmena pa–gnia 864.2.6 Isorrop–ec gia ape–rwc epanalambanÏmena pa–gnia . . . 88

4.3 EfarmogËc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.3.1 Sumpaign–a sto oligop∏lio . . . . . . . . . . . . . . . . 954.3.2 Epilog† poiÏthtac . . . . . . . . . . . . . . . . . . . . 97


5 Statikà pa–gnia me ellip† plhrofÏrhsh 1065.1 Eisagwg† . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.2 BasikÏ pla–sio . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.3 Ep–lush . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.3.1 Metatrop† se pa–gnio atelo‘c plhrofÏrhshc . . . . . . 1115.4 Ermhne–a mikt∏n strathgik∏n . . . . . . . . . . . . . . . . . . 1125.5 EfarmogËc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.5.1 Duop∏lio me ellip† plhrofÏrhsh . . . . . . . . . . . . 1165.5.2 Dhmopras–a pr∏thc tim†c . . . . . . . . . . . . . . . . 1185.5.3 Dipl† dhmopras–a . . . . . . . . . . . . . . . . . . . . . 1225.5.4 DhmÏsia agajà . . . . . . . . . . . . . . . . . . . . . . 125

5.6 Istorik† anadrom† . . . . . . . . . . . . . . . . . . . . . . . . 1265.7 Paràrthma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.8 Ask†seic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

PERIEQOMENA iii

5.9 Orolog–a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1325.10 Bibliograf–a . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6 Pa–gnia sunergas–ac 1346.1 Eisagwg† . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.2 BasikÏ pla–sio . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.2.1 IdiÏthtec . . . . . . . . . . . . . . . . . . . . . . . . . . 1376.3 O pur†nac . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.3.1 Pa–gnia me mh kenÏ pur†na . . . . . . . . . . . . . . . . 1446.4 H katanom† Shapley . . . . . . . . . . . . . . . . . . . . . . . 148

6.4.1 SqËsh metax‘ katanom†c Shapley kai pur†na . . . . . . 1536.5 Istorik† anadrom† . . . . . . . . . . . . . . . . . . . . . . . . 1546.6 Ask†seic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1556.7 Orolog–a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1586.8 Bibliograf–a . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

7 Diapragmate‘seic: axiwmatik† prosËggish 1607.1 Eisagwg† . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1607.2 BasikÏ pla–sio . . . . . . . . . . . . . . . . . . . . . . . . . . 1617.3 L‘sh katà Nash . . . . . . . . . . . . . . . . . . . . . . . . . . 1647.4 L‘sh katà Kalai-Smorodinsky . . . . . . . . . . . . . . . . . 1727.5 'Allec l‘seic . . . . . . . . . . . . . . . . . . . . . . . . . . . 1747.6 EfarmogËc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

7.6.1 Diapragmate‘seic ergodos–ac-ergazomËnwn . . . . . . . 1767.6.2 Diapragmate‘seic kratiko‘ qrËouc . . . . . . . . . . . 177


8 Diapragmate‘seic: mh sunergatik† prosËggish 1838.1 Eisagwg† . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1838.2 PeperasmËnoc qronikÏc or–zontac . . . . . . . . . . . . . . . . 1848.3 'Apeiroc qronikÏc or–zontac . . . . . . . . . . . . . . . . . . . 187

8.3.1 Epektàseic . . . . . . . . . . . . . . . . . . . . . . . . 1908.4 Istorik† anadrom† . . . . . . . . . . . . . . . . . . . . . . . . 1958.5 Ask†seic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1958.6 Orolog–a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1978.7 Bibliograf–a . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

9 Dunamikà pa–gnia me ellip† plhrofÏrhsh 1999.1 Eisagwg† . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1999.2 TËleia isorrop–a katà Bayes . . . . . . . . . . . . . . . . . . . 2009.3 Pa–gnia shmatodÏthshc . . . . . . . . . . . . . . . . . . . . . . 211

PERIEQOMENA i

9.3.1 BasikÏ pla–sio . . . . . . . . . . . . . . . . . . . . . . 2119.3.2 ShmatodÏthsh sthn agorà ergas–ac . . . . . . . . . . . 218

9.4 Istorik† Anadrom† . . . . . . . . . . . . . . . . . . . . . . . . 2329.5 Ask†seic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2329.6 Orolog–a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2369.7 Bibliograf–a . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

PrÏlogoc

To parÏn bibl–o apotele– mia eisagwg† sth jewr–a twn paign–wn. Ja pa-rousiasto‘n se autÏ oi kuriÏterec kathgor–ec twn mh sunergatik∏n kai twnsunergatik∏n paign–wn. SugkekrimËna, ja analujo‘n statikà kai dunamikàpa–gnia, tÏso upÏ pl†rh † tËleia plhrofÏrhsh, Ïso kai upÏ ellip† † atel†plhrofÏrhsh, kai ja melethjo‘n oi basikËc Ënnoiec ep–lushc touc. Gia kàjekathgor–a paign–wn, ja dojo‘n antiproswpeutikËc oikonomikËc efarmogËc.To bibl–o apeuj‘netai se proptuqiako‘c foithtËc oikonomik∏n sqol∏n

pou epijumo‘n na apokt†soun mia pr∏th gnwrim–a me th jewr–a twn paign–wn.O anagn∏sthc anamËnetai na Ëqei stoiqei∏deic, mÏno, gn∏seic diaforiko‘ lo-gismo‘, beltistopo–hshc sunart†sewn kai pijanot†twn. Oi oikonomikËc efar-mogËc estiàzoun se upode–gmata ta opo–a didàskontai sta pr∏ta Ëth spoud∏nenÏc tupiko‘ oikonomiko‘ tm†matoc.Ja †jela na euqarist†sw jermà ton Ep–kouro Kajhght† Kwnstant–no

PapadÏpoulo, o opo–oc wc kritikÏc anagn∏sthc bo†jhse me tic leptomere–cparathr†seic tou sth belt–wsh tou bibl–ou. Ep–shc, euqarist∏ idiaitËrwc tonDr. Paraskeuà LekËa gia tic pol‘timec sumbolËc tou se jËmata teqnik†cepimËleiac, kaj∏c kai th filÏlogo Nektar–a Kladà gia th glwssik† epimËleiatou suggràmmatoc.

Ge∏rgioc StamatÏpouloc RËjumno, DekËmbrioc 2015

Ep–kouroc Kajhght†cTm†ma Oikonomik∏n Episthm∏nPanepist†mio Kr†thc

i

Kefàlaio 1

Eisagwg†

1.1 Ti e–nai h jewr–a paign–wn

'Ena qarakthristikÏ pou parathro‘me se pollà oikonomikà, biologikà, koinw-nikà, klp. fainÏmena e–nai h diadikas–a l†yewc apofàsewn. OntÏthtec Ïpwcepiqeir†seic, kubern†seic, biologiko– organismo–, klp, suqnà prËpei na epi-lËxoun kàpoia enËrgeia me krit†rio thn megistopo–hsh tou atomiko‘ ofËlouctouc, kai màlista se peribàllonta Ïpou to Ïfeloc autÏ exartàtai Ïqi mÏnoapÏ th dik† touc enËrgeia allà kai apÏ tic enËrgeiec pou epilËgoun oi upÏ-loipoi. Me àlla lÏgia, h diadikas–a l†yewc apofàsewn lambànei q∏ra entÏcenÏc plais–ou allhlep–drashc: kàje àtomo ephreàzei kai ephreàzetai apÏ taàlla àtoma.Kàje àtomo sunep∏c ja prËpei, prin epilËxei kàpoia enËrgeia, na anal‘sei

tic parapànw allhlepidràseic. To gegonÏc autÏ kajistà th l†yh apofàsewnËna mh tetrimmËno, kai tautÏqrona endiafËron, prÏblhma. H jewr–a paign–wne–nai o klàdoc eke–noc o opo–oc meletà susthmatikà tËtoiou e–douc probl†ma-ta. H jewr–a aut† prosfËrei mia enia–a gl∏ssa, me thn opo–a mporo‘me namorfopoi†soume, na katano†soume kai na anal‘soume me susthmatikÏ trÏpoth diadikas–a l†yewc apofàsewn se peript∏seic Ïpou upàrqei allhlep–drashmetax‘ twn lhpt∏n apofàsewn.Ac do‘me orismËna aplà parade–gmata ta opo–a proskalo‘n th qr†sh thc

jewr–ac paign–wn. 'Estw Ïti h kubËrnhsh m–ac q∏rac epijume– na pwl†sei miaradiosuqnÏthta mËsw dhmopras–ac. H diadikas–a Ëqei wc ex†c. Kàje upoy†-fioc agorast†c ja upobàllei thn prosforà tou (tim† agoràc) se Ënan fàkelo.Nikht†c e–nai o upoy†fioc me th megal‘terh prosforà. O nikht†c ja apokt†-sei th radiosuqnÏthta, plhr∏nontac thn tim† pou Ëqei prote–nei. H tim† pouupobàllei kàje upoy†fioc agorast†c († alli∏c, h strathgik† tou) sqet–zetaime to pÏso apotimà o –dioc th radiosuqnÏthta, allà kai me tic ektim†seic touanaforikà me tic timËc twn àllwn upoyhf–wn. To prÏblhma epilog†c tim∏ng–netai pio endiafËron an upojËsoume Ïti kàje agorast†c gnwr–zei mÏno thdik† tou apot–mhsh, kai Ïqi twn àllwn. Poiec e–nai oi strathgikËc pou ja

1

2 KEFALAIO 1. EISAGWGüH

epilËxoun oi upoy†fioi agorastËc sth dhmopras–a aut†;Oi q∏rec A kai B Ëqoun th dunatÏthta apÏkthshc purhnik∏n Ïplwn. To

ideatÏ gia m–a q∏ra, pq, thn A, ja †tan eke–nh mÏno na apokt†sei purhnikÏoplostàstio, kai Ïqi h àllh q∏ra B. PeraitËrw, h q∏ra A ja protimo‘sekamm–a q∏ra na mhn Ëqei purhnikà Ïpla apÏ to na Ëqoun kai oi d‘o. TËloc, toqeirÏtero dunatÏ senàrio gia thn A e–nai na apokt†sei purhnikÏ oplostàsiomÏno h B. Anàlogec parathr†seic isq‘oun gia th q∏ra B. Ti ja kànei telikàkàje q∏ra anaforikà me thn apÏkthsh purhniko‘ oplostas–ou;Ta parapànw e–nai merikà mÏno parade–gmata problhmàtwn pou meletà h

jewr–a paign–wn. H melËth sthr–zetai sthn kataskeu† enÏc plais–ou anàlu-shc, † alli∏c, enÏc paign–ou, to opo–o perilambànei ta ex†c qarakthristikà:

• touc l†ptec apofàsewn, † alli∏c touc pa–ktec

• tic diajËsimec enËrgeiec twn paikt∏n, † alli∏c tic strathgikËc touc

• thn plhrofÏrhsh pou diajËtoun oi pa–ktec anaforikà me Ïla ta shmantikàstoiqe–a tou probl†matoc

• tic apodÏseic † wfËleiec pou apokom–zoun oi pa–ktec wc sunàrthsh twnstrathgik∏n pou epilËgoun tÏso oi –dioi, Ïso kai oi àlloi pa–ktec

Ta pa–gnia mporo‘n na taxinomhjo‘n se diàforec kathgor–ec. Ta k‘ria kri-t†ria kathgoriopo–hshc Ëqoun na kànoun me ton qrÏno, thn plhrofÏrhsh twnpaikt∏n, to jesmikÏ peribàllon, klp. Oi shmantikÏterec kathgor–ec e–nai oiex†c:

1. Statikà † dunamikà pa–gnia'Ena pa–gnio onomàzetai statikÏ Ïtan den upeisËrqetai o qrÏnoc se autÏ,upÏ thn Ënnoia Ïti Ïloi oi summetËqontec epilËgoun enËrgeiec m–a foràkai sto –dio qronikÏ shme–o. AntijËtwc, Ëna pa–gnio onomàzetai dunamikÏÏtan oi epilogËc twn paikt∏n g–nontai se diaforetikà qronikà shme–a (kaiendeqomËnwc me epanalambanÏmeno trÏpo).

2. Pa–gnia me pl†rh † ellip† plhrofÏrhsh'Otan se Ëna pa–gnio Ïloi oi summetËqontec gnwr–zoun Ïla ta qarakth-ristikà tou paign–ou tÏte lËme Ïti to pa–gnio Ëqei pl†rh plhrofÏrhsh.Se ant–jeth per–ptwsh, Ïtan dhlad† Ëna toulàqiston àtomo den gnwr–-zei kàpoio † kàpoia apÏ ta qarakthristikà autà, Ëqoume thn per–ptwshpaign–ou me ellip† plhrofÏrhsh.

3. Pa–gnia me tËleia † atel† plhrofÏrhsh'Otan se Ëna dunamikÏ pa–gnio Ïloi oi summetËqontec parathro‘n sekàje qronikÏ shme–o Ïlec tic epilogËc pou Ëqoun g–nei mËqri tou shme–ouauto‘, tÏte lËme Ïti to pa–gnio Ëqei tËleia plhrofÏrhsh. Se ant–jethper–ptwsh, Ëqoume Ëna pa–gnio me atel† plhrofÏrhsh.

1.2. ISTORIKüH ANADROMüH 3

4. Sunergatikà kai mh sunergatikà pa–gniaSunergatikà e–nai ta pa–gnia sta opo–a oi summetËqontec Ëqoun th du-natÏthta na epilËxoun enËrgeiec oi opo–ec megistopoio‘n ta ofËlh m–acomàdac atÏmwn. Paràllhla, h sunergatik† aut† sumperiforà e–nai de-smeutik† gia Ïla ta mËlh tou sunÏlou. Mh sunergatikà e–nai ta pa–gniasta opo–a oi summetËqontec epilËgoun enËrgeiec me gn∏mona th megisto-po–hsh tou atomiko‘ sumfËrontoc touc † akÏmh kai th megistopo–hsh thcwfËleiac enÏc sunÏlou atÏmwn, qwr–c Ïmwc h sunergatik† sumperiforàna e–nai desmeutik†.

5. Pa–gnia stajero‘ † mh stajero‘ ajro–smatoc'Ena pa–gnio onomàzetai stajero‘ ajro–smatoc Ïtan oi apodÏseic (a-moibËc) twn paikt∏n ajro–zoun ston –dio arijmÏ gia kàje sunduasmÏstrathgik∏n. Se Ëna pa–gnio mh stajero‘ ajro–smatoc, h idiÏthta aut†den isq‘ei. H pio sunhjismËnh morf† paign–wn stajero‘ ajro–smatoc e–-nai aut† tou mhdeniko‘ ajro–smatoc me d‘o summetËqontec: se m–a tËtoiaper–ptwsh, to kËrdoc tou enÏc e–nai apl∏c h zhmià tou àllou.

Ta upode–gmata pou anapt‘ssei h jewr–a paign–wn sthr–zontai sthn upÏjeshÏti oi pa–ktec e–nai orjologikà àtoma. Me autÏ ennoo‘me Ïti kàje pa–kthcgnwr–zei tic enallaktikËc epilogËc tou, Ëqei xekàjarec protim†seic sqetikà meautËc, sqhmat–zei prosdok–ec gia Ïsa stoiqe–a tou e–nai àgnwsta, kai epilËgeitic enËrgeiec eke–nec oi opo–ec megistopoio‘n th qrhsimÏthta tou (Osborne kaiRubinstein 1994).

1.2 Istorik† anadrom†

H pr∏th susthmatik† paignio-jewrhtik† melËth parousiàsthke sta oikono-mikà apÏ ton Cournot (1838). Sth melËth aut† o Cournot anËluse th lei-tourg–a miac agoràc sthn opo–a droun d‘o epiqeir†seic. Oi pr∏tec anal‘seicgenik∏n paign–wn mhdeniko‘ ajro–smatoc Ëginan apÏ ton Borel (1921) kai tonvon Neumann (1928). To 1944 oi von Neumann kai Morgestern exËdwsan toklasikÏ bibl–o touc Theory of Games and Economic Behavior, to opo–o e–qeep–drash se arketo‘c tome–c thc jewr–ac paign–wn (pa–gnia mhdeniko‘ ajro–-smatoc, sunergatikà pa–gnia, klp). Pollo– ereunhtËc jewro‘n Ïti to bibl–oautÏ shmatodote– thn afethr–a thc jewr–ac paign–wn.Ta epÏmena Ëth, †tan h seirà tou Nash na suneisfËrei me kajoristikÏ

trÏpo sthn exËlixh thc jewr–ac paign–wn (Nash 1950, 1951, 1953). Kat’arqàc, o Nash morfopo–hse thn Ënnoia thc isorrop–ac paign–wn mh mhdeni-ko‘ ajro–smatoc, h opo–a e–nai gnwst† s†mera wc isorrop–a Nash.1 Katàde‘teron, sunËbale sth jewr–a twn sunergatik∏n paign–wn diapragmàteushc,dhlad† paign–wn sta opo–a oi pa–ktec diapragmate‘ontai th katanom† enÏc

1H Ënnoia aut† proÙp†rqe, se Ëna ligÏtero genikÏ pla–sio, sthn ergas–a tou Cournot.


pleonàsmatoc, apodeqÏmenoi Ïti h katanom† aut† ja prËpei na ikanopoie– ËnanarijmÏ axiwmàtwn.H parapànw qronik† per–odoc apode–qthke idia–tera gÏnimh, kaj∏c tÏte

anapt‘qjhkan d‘o apÏ tic pio shmantikËc Ënnoiec ep–lushc twn paign–wn su-nergas–ac: o pur†nac, o opo–oc parousiàsthke apÏ ton Gilles (1953) kai hkatanom† († ax–a) katà Shapley, h opo–a parousiàsthke ep–shc thn –dia qronià(Shapley 1953). O Nash xek–nhse eke–nh thn per–odo Ëna ereunhtikÏ prÏgram-ma, stÏqoc tou opo–ou †tan h gef‘rwsh tou keno‘ metax‘ sunergatik∏n kai mhsunergatik∏n paign–wn. BasikÏ sustatikÏ aut†c thc prospàjeiac e–nai h mhsunergatik† jemel–wsh († dikaiolÏghsh) twn ennoi∏n ep–lushc twn paign–wnsunergas–ac. To prÏgramma autÏ, to opo–o argÏtera onomàsthke prÏgrammatou Nash, moiàzei, Ïpwc anafËrei o Serrano (2005), me th prospàjeia mikrooi-konomik†c jemel–wshc thc makrooikonomik†c, h opo–a prospaje– na fËrei piokontà touc d‘o basiko‘c auto‘c tome–c thc oikonomik†c jewr–ac.Stic arqËc tou 20ou ai∏na o Zermelo anËluse pa–gnia sta opo–a d‘o pa–-

ktec epilËgoun enËrgeiec diadoqikà, gnwr–zontac se kàje qronikÏ shme–o oËnac tic epilogËc tou àllou (Zermelo 1913). H pio sunhjismËnh morf† parou-s–ashc twn paign–wn aut∏n e–nai h ektetamËnh morf†, h opo–a eis†qjei to 1944apÏ touc von Neumann kai Morgestern, sto proanaferjËn bibl–o touc. Giata pa–gnia autà, o Selten eis†gage katà ta mËsa thc dekaet–ac tou 1960 thntËleia katà upopa–gnio isorrop–a (Selten 1965), h Ïpoia Ëmelle na exeliqje– sem–a apÏ tic shmantikÏterec Ënnoiec isorrop–ac sth jewr–a paign–wn.Proc ta tËlh thc dekaet–ac tou 1950 anapt‘qjhkan ta epanalambanÏmena

pa–gnia. Ta pa–gnia autà, ta opo–a, Ïpwc kai ta pa–gnia diadoqik∏n epilog∏n,an†koun sth kathgor–a twn dunamik∏n paign–wn, meleto‘n fainÏmena pou e-panalambànontai auto‘sia ston qrÏno.2 Shme–o anaforàc sta pa–gnia autàapotele– to gegonÏc Ïti Ënac pol‘ megàloc arijmÏc apotelesmàtwn mpore–na apotelËsei isorrop–a, upÏ thn proÙpÏjesh Ïti h diàrkeia touc e–nai àpei-rh. To apotËlesma autÏ pist∏netai ston Friedman (1971). ParÏla autà, toapotËlesma †tan ousiastikà gnwstÏ arketà qrÏnia prin, kaj∏c pollo– apÏtouc ereunhtËc pou ergàzontan ston tomËa autÏ, e–te to gn∏rizan, e–te aplàupoyiàzontan Ïti Ëna tËtoio apotËlesma prËpei na isq‘ei.3

Thn per–odo 1966-1968, oi Aumann kai Maschler melËthsan epanalam-banÏmena pa–gnia me ellip† plhrofÏrhsh, dhlad† epanalambanÏmena pa–gniasta opo–a oi summetËqontec den gnwr–zoun pl†rwc Ïla ta qarakthristikà thcallhlep–drashc h opo–a epanalambànetai ston qrÏno (Aumann kai Masch-ler 1995). Katà thn –dia per–pou per–odo, o Harsanyi (1967-1968) anËptuxesusthmatikà th jewr–a twn statik∏n paign–wn me ellip† plhrofÏrhsh. Tapa–gnia autà purodÏthsan sth sunËqeia thn anàptuxh twn oikonomik∏n thc

2Shmei∏noume Ïti nwr–tera, o Shapley (1953) anËptuxe ta legÏmena stoqastikà pa–gnia,ta opo–a anal‘oun katastàseic oi opo–ec epanambànontai ston qrÏno me tuqa–o (stoqasti-kÏ) trÏpo.

3UpÏ aut† thn Ënnoia to apotËlesma autÏ apotelo‘se m! Ëroc tou folklÏr thc jewr–acpaign–wn. Gia to lÏgo autÏ to sqetikÏ je∏rhma Ëmeine gnwstÏ wc Folk theorem.

1.2. ISTORIKüH ANADROMüH 5

plhrofÏrhshc.Sta mËsa thc dekaet–ac tou 1970 xek–nhse m–a bibliograf–a skopÏc thc

opo–ac †tan h melËth paign–wn ta opo–a diajËtoun perissÏterec apÏ m–a isor-rop–ec. StÏqoc thc †tan h anàptuxh krithri∏n ta opo–a epitrËpoun na apor-rifjo‘n kàpoiec apÏ autËc tic isorrop–ec wc e‘logec l‘seic enÏc paign–ou.Sta pla–sia auto‘ tou ereunhtiko‘ progràmatoc, to opo–o onomàzetai exeuge-nismÏc † di‘lish twn isorrop–wn, o Selten (1975) (xanà-) suz†thse to krit†riothc teleiÏthtac. To krit†rio autÏ Ëtuqe perissÏterhc epexergas–ac apÏ tonMyerson (1978), o opo–oc prÏteine to legÏmeno krit†rio thc katallhlÏthtac.'Allec prosegg–seic prÏteinan krit†ria pou sqet–zontai me omadikËc -kai ÏqiatomikËc- epilogËc strathgik∏n. Sthn kathgor–a aut† emp–ptei, gia paràdeig-ma, h prosËggish twn Bernheim, Peleg kai Whinston (1987) per– isorropi∏npou epibi∏noun tou sqhmatismo‘ summaqi∏n metax‘ twn paikt∏n, kaj∏c kai h(progenËsterh) Ënnoia thc textitisqur†c isorrop–ac, pou anËptuxe o Aumann(1959).Se m–a kontin† me thn parapànw bibliograf–a, oi Harsanyi kai Selten (1988)

parous–asan th jewr–a touc per– epilog†c isorrop–ac se pa–gnia me pollËc i-sorrop–ec. Oi suggrafe–c auto– anËlusan d‘o krit†ria gia thn epilog† m–acmÏno isorrop–ac wc l‘shc enÏc mh sunergatiko‘ paign–ou: to Ëna krit†rioepikentr∏netai sto r–sko pou emperiËqei mia isorrop–a, en∏ to de‘tero epiken-tr∏netai stic apodÏseic pou prok‘ptoun gia touc pa–ktec apÏ thn isorrop–a.H bibliograf–a tou exeugenismo‘ (di‘lishc) twn isorropi∏n Nash epiken-

tr∏netai sto na epilËxei sugkekrimËna upos‘nola twn isorrop–wn enÏc paign–-ou. Sthn ant–pera Ïqjh, o Aumann (1974) prÏteine m–a gen–keush thc isor-rop–ac Nash, th legÏmenh susqetizÏmenh isorrop–a. H isorrop–a katà Nashsthr–zetai sthn upÏjesh Ïti oi pa–ktec epilËgoun strathgikËc anexàrthta oËnac apÏ ton àllo. O Aumann tropopo–hse thn upÏjesh aut†, epitrËpontacËna e–doc susqËtishc metax‘ twn epilog∏n twn paikt∏n. Gia paràdeigma, oipa–ktec parathro‘n diaforetikËc ekfànseic tou –diou fainomËnou, kai katÏpinepilËgoun tic strathgikËc touc bàsei twn parathr†sewn touc.ApÏ tic arqËc thc dekaet–ac tou 1980 àrqise na anapt‘ssetai h mh su-

nergatik† prosËggish sth jewr–a diapragmàteushc, se antidiastol† me thsunergatik† prosËggish tou Nash. BasikÏ shme–o anaforàc aut†c thc pro-sËggishc up†rxe to upÏdeigma enallassÏmenwn prosfor∏n tou Rubinstein(1982). To upÏdeigma autÏ, emploutismËno sth sunËqeia kai me stoiqe–a Ïpwch ellip†c plhrofÏrhsh, br†ke pollËc efarmogËc sta oikonomikà, en∏ de–qthkekai h sqËsh tou me to upÏdeigma sunergatik†c diapragmàteushc tou Nash.Stic arqËc thc dekaet–ac tou 1980 kai tou 1990 parousiàsthkan d‘o apÏ

tic basikÏterec Ënnoiec ep–lushc twn dunamik∏n paign–wn ellipo‘c plhrofÏrh-shc: h diadoqik† isorrop–a twn Kreps kaiWilson (1982) kai h tËleia isorrop–akatà Bayes (bl. Fudenberg kai Tirole 1991). Oi d‘o autËc Ënnoiec isorrop–ac,oi opo–ec e–nai arketà kontinËc metax‘ touc, p†ran stoiqe–a apÏ thn ep–lushtwn dunamik∏n paign–wn, kaj∏c kai apÏ thn ep–lush twn paign–wn ellipo‘cplhrofÏrhshc. M–a qarakthristik† efarmog† twn dunamik∏n paign–wn elli-


po‘c plhrofÏrhshc apotËlesan ta legÏmena pa–gnia shmatodÏthshc, ta opo–ae–nai pa–gnia diadoqik∏n epilog∏n, sta opo–a o pa–kthc pou kine–tai pr∏tocdiajËtei kal‘terh plhrofÏrhsh apÏ touc upÏloipouc pa–ktec.4

1.2.1 'Allec exel–xeic

H jewr–a paign–wn paradosiakà sthr–qjhke sthn idËa Ïti ta àtoma e–nai pl†-rwc orjologikà, upÏ thn Ënnoia Ïti mporo‘n pàntote na epit‘qoun ton stÏqothc megistopo–hshc thc apÏdoshc touc. ApÏ palià pàntwc, kai sugkekrimËnaapÏ th dekaet–a tou 1950, up†rqe h àpoyh Ïti h upÏjesh tou pl†rouc orjo-logismo‘ e–nai mh realistik†, kai Ïti ja prËpei na antikatastaje– apÏ aut†tou fragmËnou orjologismo‘, h opo–a jËtei Ïria sto ti mporo‘n na pet‘qounoi pa–ktec. K‘rioc ekprÏswpoc aut†c thc tàshc up†rxe o Simon (1955).M–a megàlh bibliograf–a anapt‘qjhke, h opo–a skopÏ e–qe na enswmat∏sei

me ton Ënan † ton àllo trÏpo thn upÏjesh tou fragmËnou orjologismo‘sta pa–gnia. 'Ena kommàti thc bibliograf–ac aut†c uiojËthse thn upÏjeshÏti ta àtoma akoloujo‘n aplo‘c kanÏnec epilog†c strathgik∏n, Ëna àlloupËjese Ïti ta àtoma te–noun na epilËgoun tic kalËc strathgikËc pio suqnàapÏ tic ligÏtero kalËc (qwr–c anagkastikà na epitugqànetai megistopo–hshthc qrhsimÏthtac) † Ïti epilËgoun tic strathgikËc touc periorizÏmenoi seËna upos‘nolo aut∏n. Genikà, den upàrqei Ëna koinÏ upÏdeigma fragmËnouorjologismo‘ sta pa–gnia, kai katà ton Aumann (1997) den ja upàrxei potË.Paràllhla me thn anàptuxh thc jewr–ac twn paign–wn, oi oikonomolÏgoi

Ëqoun asqolhje– me ton sqediasmÏ peiramatik∏n paign–wn gia na elËgxoun ticproblËyeic thc jewr–ac. TËtoia peiràmata pragmatopoio‘ntai ed∏ kai arketËcdekaet–ec. M–a episkÏphsh twn pr∏twn, qronikà, peiramàtwn dhmosie‘thkeapÏ touc Rapoport kai Orwant (1962). Suqnà, ta apotËlesmata tËtoiwnpeiramàtwn de–qnoun Ïti oi epilogËc twn atÏmwn ephreàzontai apÏ stoiqe–-a Ïpwc h dikaios‘nh, o altrouismÏc, to pla–sio anaforàc (pa–gnia me –diecepilogËc/apodÏseic odhgo‘n touc pa–ktec se diaforetik† sumperiforà an pa-rousiasto‘n me diaforetikÏ trÏpo to Ëna apÏ ta àlla). Ta eur†mata autàod†ghsan sthn anàptuxh twn sumperiforik∏n paign–wn, dhlad† paign–wn taopo–a enswmat∏noun sthn anàlush touc stoiqe–a Ïpwc ta parapànw.Kàpoiec àllec shmantikËc exel–xeic pou Ëlaban q∏ra katà ta teleuta–ec

dekaet–ec Ëqoun na kànoun me th s‘ndesh thc isorrop–ac Nash me Ënnoiec thcexeliktik†c biolog–ac, thn anàptuxh twn exeliktik∏n paign–wn, thn anàlushtwn paign–wn apÏ gnwsiologik† pleurà (poia kai pÏsh gn∏sh apaite–tai ekmËrouc twn paikt∏n gia na ftàsoume se m–a isorrop–a), thn anàptuxh upodeig-màtwn màjhshc, dhlad† upodeigmàtwn sta opo–a Ëna pa–gnio epanalambànetaiston qrÏno, me sunËpeia oi pa–ktec na maja–noun gia th sumperiforà twn àl-lwn paikt∏n, kai na katal†goun Ëtsi makroqrÏnia se epilogËc isorrop–ac,klp.

4Gia na e–maste akribe–c, orismËnec apÏ tic efarmogËc prohg†jhkan thc jewr–ac, Ïpwcp.q. sunËbh me to upÏdeigma shmatodÏthshc sthn agorà ergas–ac tou Spence (1973).

1.3. SKOPüOS TOU BIBLüIOU 7

TËloc, apÏ pleuràc susqËtishc me àlla episthmonikà ped–a, oi tome–cstouc opo–ouc h jewr–a paign–wn Ëqei brei efarmog† e–nai pàmpolloi (me dià-forouc bajmo‘c epituq–ac). Oi tome–c auto– perilambànoun ta oikonomikà (p.q,biomhqanik† orgànwsh, jewr–a diejno‘c empor–ou, makrooikonomik† politik†,anàlush agor∏n me as‘mmetrh plhrofÏrhsh, politik† oikonom–a, klp), tic koi-nwnikËc epist†mec (politik† epist†mh, yuqolog–a), th biolog–a, thn epist†mhtwn upologist∏n, thn epiqeirhsiak† Ëreuna, kai àllouc.

1.3 SkopÏc tou bibl–ou

To parÏn egqeir–dio Ëqei skopÏ na eisagàgei ton anagn∏sth sta basikà stoi-qe–a thc jewr–ac paign–wn. Ja parousiasto‘n kai ja analujo‘n oi basi-kÏterec kathgor–ec paign–wn pou sunant∏ntai sthn oikonomik† anàlush. Jaxekin†soume me ta statikà pa–gnia pl†rouc plhrofÏrhshc kai th basik† Ënnoiaep–lushc touc, thn isorrop–a katà Nash (Kefàlaio 2). KatÏpin, ja parousia-sto‘n ta dunamikà pa–gnia. Sto Kefàlaio 3 parousiaste– h ektetamËnh morf†enÏc paign–ou, h tËleia katà upopa–gnio isorrop–a kai o upologismÏc thc mËswthc mejÏdou thc opisjogeno‘c epagwg†c. Sto Kefàlaio 4 ja melethjo‘n taepanalambanÏmena pa–gnia me peperasmËno kai àpeiro qronikÏ or–zonta. 'Em-fash ja doje– sto legÏmeno Dhm∏dec Je∏rhma, to opo–o prosdior–zei tos‘nolo twn isorropi∏n twn epanalambanÏmenwn paign–wn ape–rou or–zonta.Me to Kefàlaio 5 ja epanËljoume sta statikà pa–gnia, allà me elli-

p† plhrofÏrhsh. Ja eisaqje– h isorrop–a katà Bayes kai ja parousiaste–,metax‘ àllwn, h sqËsh metax‘ thc isorrop–ac enÏc paign–ou ellipo‘c plhrofÏ-rhshc kai thc isorrop–ac Nash se miktËc stathgikËc tou ant–stoiqou paign–oupl†rouc plhrofÏrhshc. Sth sunËqeia ja strafo‘me sta sunergatikà pa–-gnia. Ja parousiasto‘n ta pa–gnia sunergas–ac me metabibàsimh qrhsimÏthtakai ja melethjo‘n oi d‘o basikÏterec Ënnoiec ep–lushc touc, o pur†nac kai hkatanom† Shapley (Kefàlaio 6).Ta Kefàlaia 7 kai 8 estiàzoun sta pa–gnia diapragmàteushc. Arqikà, ja

parousiaste– h axiwmatik† prosËggish twn paign–wn diapragmàteushc. 'Emfa-sh ja doje– sth l‘sh diapragmàteushc tou Nash, allà ja parousiasto‘n kaiàllec l‘seic, Ïpwc p.q. h l‘sh twn Kalai kai Smorodinsky. Sth sunËqeia,ja analuje– h mh sunergatik† prosËggish, me shme–o anaforàc to upÏdeigmadiapragmàteushc enallassÏmenwn prosfor∏n tou Rubinstein. To teleuta–okefàlaio tou egqeirid–ou (Kefàlaio 9) ja parousiàsei ta dunamikà pa–gnia meellip† plhrofÏrhsh. Ja parousiaste– h tËleia isorrop–a katà Bayes kai jaanaluje– m–a eidik† kathgor–a dunamik∏n paign–wn me ellip† plhrofÏrhsh, tapa–gnia shmatodÏthshc.To egqeir–do apeuj‘netai ston proptuqiakÏ foitht† twn tmhmàtwn oiko-

nomik∏n sqol∏n. H prosËggish pou akolouje– sthn parous–ash twn jemàtwne–nai arketà eisagwgik†, kai Ïqi austhr†. Kàje kefàlaio perilambànei arketàparade–gmata, ta opo–a bohjo‘n sthn katanÏhsh twn ennoi∏n. TËloc, den


apaito‘ntai apÏ ton anagn∏sth idia–terec gn∏seic majhmatik∏n (p.q. oi majh-matikËc apode–xeic twn pio poll∏n apotelesmàtwn Ëqoun paralhfje– apÏ thnparous–ash).

1.4 Bibliograf–a

1. Aumann, R. (1959), Acceptable points in general cooperative n-persongames, Contributions to the Theory of Games vol. IV, Tucker, A.W.,Luce, R.D. (eds). Princeton University Press.

2. Aumann, R., Maschler, M. (1995), Repeated Games with IncompleteInformation. MIT Press.

3. Aumann, R. (1974), Subjectivity and correlation in randomized strategies,Journal of Mathematical Economics 1, 67-96.

4. Aumann, R. (1997), Rationality and bounded rationality, Games and E-conomic Behavior 21, 2-14.

5. Bernheim, B.D., Peleg, B., Whinston, M. (1987), Coalition-proof Nashequilibria I. Concepts, Journal of Economic Theory 42, 1-12.

6. Borel, E. (1921), La theorie du jeu et les equation integrales a noyausymetrique gauche, Comptes Rendus de l’ Academie des Sciences 173,1304-1308.

7. Cournot, A. (1838), Researches into the Mathematical Principles of theTheory of Wealth, aggl. mtfr. Bacon, N.T. MacMillan 1929.

8. Friedman, J. (1971), A non-cooperative equilibrium for supergames, Re-view of Economics Studies 38, 1-12.

9. Fudenberg, D., J. Tirole, J. (1991), Perfect bayesian equilibrium and se-quential equilibrium, Journal of Economic Theory 53, 236-260.

10. Gilles, D. B. (1953), Some theorems on N -person games, Ph.D thesis,Department of Mathematics, Princeton University.

11. Harsanyi, J. (1967-68), Games with incomplete information played bybayesian players, Parts I,II, III, Management Science 14, 159-182, 320-334, 486-502.

12. Harsanyi, J., Selten, R. (1967-68), A General Theory of Equilibrium Se-lection in Games. MIT Press.

13. Kreps, D., Wilson, R. (1982), Sequential equilibria, Econometrica 50,863-894.

1.4. BIBLIOGRAFüIA 9

14. Mage–rou, E. (2012), Pa–gnia kai Apofàseic: M–a Eisagwgik† ProsËggi-sh. EkdÏseic Kritik† (kefàlaio 7).

15. Myerson, R. (1978), Refinements of the Nash equilibrium concept, Inter-national Journal of Game Theory 7, 73-80.

16. Nash, J.F. (1950), The bargaining problem, Econometrica 18, 155-162.

17. Nash, J.F. (1950), Equilibrium points in N -person games, Proceedings ofthe National Academy of Sciences of the United States of America 36,48-49.

18. Nash, J.F. (1951), Non-cooperative games, Annals of Mathematics 54,286-295.

19. Nash, J.F. (1953), Two person cooperative games, Econometrica 21, 128-140.

20. Osborne J.M, Rubinstein, A. (1994), A Course in Game Theory, MITPress (kefàlaio 1).

21. Osborne, J.M. (2010), Eisagwg† sth Jewr–a Paign–wn. EkdÏseic Kleidà-rijmoc (kefàlaio 1).

22. Rapoport, A., Orwant, C. (1962), Experimental games: A review, Sy-stems Research and Behavioral Science 7, 1-37.

23. Rubinstein, A. (1982), Perfect equilibrium in a bargaining model, Econo-metrica 50, 97-109.

24. Selten, R. (1965), Spieltheoretische Behandlung eines Oligopolmodellsmit Nachfragetragheit, Zeitschrift fur die gesamte Staatswissenschaft 121,301-324, 667-689.

25. Selten, R. (1975), Re-examination of the perfectness concept for equili-brium points in extensive games, International Journal of Game Theory4, 22-55.

26. Serano, R. (2005), Fifty years of the Nash program, 1953-2003, Investi-gaciones Economicas 29, 219-258.

27. Shapley, L. (1953), A value for n-person games, Contribution to the the-ory of games, vol. II, Kuhn H.W., Tucker, A.W. (eds.), Princeton Uni-versity Press.

28. Shapley, L. (1953), Stochastic games, Proceedings of the National Aca-demy of Sciences of the United States of America, 1095-1100.

29. Simon, H. (1955), A behavioral model of rational choice, Quarterly Jour-nal of Economics 64, 99-118.


30. Soldàtoc, G. (2005), Jewr–a Paign–wn gia OikonomolÏgouc. EkdÏseicPanepisthm–ou Makedon–ac (kefàlaio 1).

31. Spence, M. (1973), Job market signaling, Quarterly Journal of Economics87, 355-374.

32. von Neumann, J. (1928), On the theory of games of strategy, Contribu-tions to the Theory of Games, vol. IV, Tucker, A.W., Luce, R.D. (eds).Princeton University Press.

33. von Neumann, J., Morgenstern, O. (1944), Theory of Games and Econo-mic Behavior. Princeton University Press.

34. Zermelo, E. (1913), On an application of set theory to the theory of thegame of chess, Proceedings of the Fifth International Congress of Mathe-maticians, 501-504.

Kefàlaio 2

Statikà pa–gnia me pl†rh

plhrofÏrhsh

2.1 Eisagwg†

H pio apl†, allà kai jemeli∏dhc, kathgor–a paign–wn e–nai aut† twn statik∏npaign–wn me pl†rh plhrofÏrhsh. Sta pa–gnia autà oi summetËqontec epilË-goun enËrgeiec tautÏqrona o Ënac me ton àllo. BasikÏ stoiqe–o aut∏n twnpaign–wn e–nai Ïti Ïloi oi pa–ktec Ëqoun pl†rh plhrofÏrhsh gia Ïla ta qa-rakthristikà tou paign–ou. Me àlla lÏgia, gnwr–zoun Ïla ta qarakthristikàtou peribàllontoc mËsa sto opo–o kino‘ntai. EpiplËon, h plhrofÏrhsh aut†apotele– koin† gn∏sh: Ïloi oi pa–ktec gnwr–zoun Ïti Ïloi oi pa–ktec gnwr–-zoun ta stoiqe–a tou paign–ou, kai epiplËon Ïloi gnwr–zoun Ïti Ïloi gnwr–zoun,k.o.k.Wc paràdeigma, ac do‘me m–a statik† oligopwliak† agorà, dhlad† m–a a-

gorà sthn opo–a oi epiqeir†seic antagwn–zontai m–a forà. Sthn agorà aut†,kàje epiqe–rhsh epilËgei thn tim† sthn opo–a ja pwl†sei to proÏn thc. OiepilogËc twn tim∏n g–nontai tautÏqrona apÏ Ïlec tic epiqeir†seic. StÏqockàje epiqe–rhshc e–nai h megistopo–hsh tou atomiko‘ kËrdouc thc. H upÏjeshthc pl†rouc plhrofÏrhshc shma–nei Ïti kàje epiqe–rhsh gnwr–zei Ïla ta qa-rakthristikà thc agoràc: gia paràdeigma, gnwr–zei tic sunj†kec z†thshc kaikÏstouc gia kàje m–a apÏ tic epiqeir†seic pou leitourgo‘n sthn agorà.Sunoy–zontac ta parapànw, ta tr–a sustatikà stoiqe–a enÏc paign–ou me

pl†rh plhrofÏrhsh e–nai:

• summetËqontec sto pa–gnio: pa–ktec

• diajËsimec enËrgeiec summeteqÏntwn: strathgikËc paikt∏n

• qrhsimÏthtec summeteqÏntwn: apodÏseic paikt∏n

To basikÏ prÏblhma pou ja mac apasqol†sei e–nai na prosdior–soume poiastrathgik† († poiËc strathgikËc) ja epilËxei kàje pa–kthc se Ëna pa–gnio

11

12 KEFALAIO 2. STATIKüA PAüIGNIA

thc parapànw morf†c. 'Opwc ja do‘me, kàje pa–kthc epilËgei th strathgik†tou Ëtsi ∏ste na megistopoi†sei thn apÏdosh tou, Ëqontac sqhmat–sei m–aprÏbleyh anaforikà me to ti ja epilËxoun kai oi àlloi pa–ktec. H epilog† tousunep∏c, ja prËpei na e–nai bËltisth dedomËnhc thc prÏbleyhc aut†c. Ep–shc,h prÏbleyh pou kànei kàje pa–kthc ja prËpei e–nai swst†. Oi paradoqËc autËce–nai h bàsh thc ep–lushc twn mh sunergatik∏n paign–wn (Ïpou wc ep–lushor–zoume ton prosdiorismÏ twn strathgik∏n pou ja epilËxoun oi pa–ktec).To kefàlaio Ëqei thn ex†c morf†. Arqikà ja parousiaste– to basikÏ ma-

jhmatikÏ pla–sio twn statik∏n paign–wn. KatÏpin ja oriste– kai ja analuje–h basik† Ënnoia ep–lushc touc, h isorrop–a katà Nash, tÏso se kajarËc Ïsokai se miktËc strathgikËc. Anaforà ja g–nei ep–shc stic isorrop–ec se kur–ar-qec strathgikËc. To kefàlaio ja oloklhrwje– me thn parous–ash orismËnwnendeiktik∏n efarmog∏n twn statik∏n paign–wn sthn oikonomik† anàlush.

2.2 BasikÏ pla–sio

'Ena pa–gnio pl†rouc plhrofÏrhshc se strathgik† morf† e–nai m–a sullog†� = {N, (Xi, ui)i2N} Ïpou:

• N = {1, 2, . . . , n} e–nai to s‘nolo twn paikt∏n

• Xi e–nai to s‘nolo twn strathgik∏n tou pa–kth i

• ui : X ! R e–nai h sunàrthsh apÏdoshc tou pa–kth i

Ïpou to s‘nolo X e–nai to kartesianÏ ginÏmeno twn sunÏlwn strathgik∏nÏlwn twn paikt∏n, dhlad† X = X1 ⇥X2 ⇥ . . .⇥Xn.Ac do‘me Ëna paràdeigma. 'Estw to s‘nolo paikt∏n N = {1, 2}. To

s‘nolo strathgik∏n tou pa–kth 1 e–nai to X1 = {A1, B1,�1} kai tou pa–kth2 to X2 = {A2, B2,�2}. Gia na perigràyoume tic apodÏseic twn paikt∏nja qrhsimopoi†soume ton p–naka apodÏsewn. Oi grammËc tou p–naka auto‘antistoiqo‘n stic strathgikËc enÏc apÏ touc d‘o pa–ktec, Ëstw tou pa–kth 1.Oi st†lec tou p–naka antistoiqo‘n stic strathgikËc tou àllou pa–kth, toupa–kth 2.

A2 B2 �2A1 2, 0 0, 1 0, 2B1 0, 0 1, 1 1, 0�1 3, 2 1, 3 1, 1

Sq†ma 2.1: Pa–gnio me d‘o pa–ktec

Oi apodÏseic twn paikt∏n pou antistoiqo‘n stic diàforec epilogËc strathgi-k∏n anagràfontai sta kelià tou p–naka. Gia paràdeigma, an oi pa–ktec 1 kai 2

2.3. EPüILUSH 13

epilËxoun tic strathgikËc A1 kai A2 ant–stoiqa, tÏte h apÏdosh tou pa–kth1 e–nai 2 kai tou pa–kth 2 e–nai 0. Paromo–wc, eàn oi pa–ktec 1 kai 2 epilËxounA1 kai B2 ant–stoiqa, tÏte h apÏdosh tou pa–kth 1 e–nai 0 kai tou pa–kth 2e–nai 1, k.o.k.Wc x = (x1, x2, . . . , xn) or–zoume Ëna sunduasmÏ strathgik∏n Ïlwn twn

paikt∏n.1 Suqnà ja gràfoume x = (xi, x�i), Ïpou o sunduasmÏc x�i em-periËqei tic strathgikËc Ïlwn twn paikt∏n plhn tou i. Dhlad†, x�i =(x1, . . . , xi�1, xi+1, . . . , xn). Parathro‘me Ïti x�i 2 X�i, Ïpou X�i e–nai tokartesianÏ ginÏmeno twn sunÏlwn strathgik∏n Ïlwn twn paikt∏n plhn toui. Dhlad† X�i = X1 ⇥ . . .⇥Xi�1 ⇥Xi+1, . . .⇥Xn.

2.3 Ep–lush

2.3.1 Apaloif† kuriarqo‘menwn strathgik∏n

'Opwc Ëqoume pei, wc ep–lush enÏc paign–ou or–zoume ton prosdiorismÏ twnepilog∏n pou ja kànoun oi pa–ktec sto pa–gnio. Xekino‘me me thn pio apl†Ënnoia ep–lushc. H Ënnoia aut† qrhsimopoie– tic legÏmenec kur–arqec kai ku-riarqo‘menec strathgikËc, tic opo–ec ja eisàgoume mËsw enÏc parade–gmatoc.'Estw pa–gnio me touc pa–ktec 1 kai 2, Ïpou o 1 epilËgei grammËc kai o 2

st†lec. O p–nakac apodÏsewn Ëqei wc ex†c:

A2 B2 �2A1 2, 0 2, 1 1, 2B1 0, 0 1, 1 0, 0�1 3, 2 0, 1 1, 1

Sq†ma 2.2: Austhr† kuriarq–a

Ston parapànw p–naka parathro‘me Ïti anexàrthta thc strathgik†c tou pa–-kth 2, h strathgik† A1 tou pa–kth 1 apofËrei megal‘terh apÏdosh apÏ thstrathgik† B1. Se aut† thn per–ptwsh ja lËme Ïti h strathgik† A1 kuriar-qe– austhrà ep– thc strathgik†c B1 (alli∏c, h strathgik† B1 e–nai austhràkuriarqo‘menh apÏ thn A1).DedomËnou tou parapànw parade–gmatoc mporo‘me na d∏soume ton orismÏ

thc austhr†c kuriarq–ac gia kàje pa–gnio.

OrismÏc 1 'Estw Ëna pa–gnio � = {N, (Xi, ui)i2N}. H strathgik† xitou pa–kth i kuriarqe– austhrà ep– thc strathgik†c x̃i eàn ui(xi, x�i) >ui(x̃i, x�i) gia kàje x�i 2 X�i.

ApÏ ed∏ kai sto ex†c ja deqto‘me Ïti oi pa–ktec den qrhsimopoio‘n austhràkuriarqo‘menec strathgikËc. Aut† h arq† ja mac epitrËyei na anapt‘xoume

1 'Enac tËtoioc sunduasmÏc kale–tai kai prof–l strathgik∏n.


m–a mËjodo me thn opo–a ja mporo‘me na epil‘oume me e‘kolo trÏpo orismËneckathgor–ec paign–wn. H mËjodoc aut† onomàzetai mËjodoc thc diadoqik†capaloif†c austhrà kuriarqo‘menwn strathgik∏n.

Paràdeigma 1'Estw to ex†c pa–gnio d‘o paikt∏n, Ïpou o pa–kthc 1 epilËgei grammËc kai opa–kthc 2 epilËgei st†lec:

A2 B2 �2A1 3, 2 1, 1 1, 0B1 1, 3 0, 2 0, 4�1 2,�1 �1, 3 2, 0

Sq†ma 2.3: ArqikÏ pa–gnio

Wc pr∏to b†ma, parathro‘me Ïti h strathgik† A1 tou pa–kth 1 kuriarqe–austhrà ep– thc B1. AutÏ isq‘ei diÏti:

u1(A1, A2) = 3 > 1 = u1(B1, A2)

u1(A1, B2) = 1 > 0 = u1(B1, B2)

u1(A1,�2) = 1 > 0 = u1(B1,�2)

O pa–kthc 1 den ja qrhsimopoi†sei, sunep∏c, th strathgik† B1. AutÏ apote-le– koin† gn∏sh. Dhlad†, o pa–kthc 2 gnwr–zei Ïti o 1 den ja qrhsimopoi†seithn B1. EpiprosjËtwc, o 1 gnwr–zei Ïti o 2 gnwr–zei thn mh epilog† thc B1,k.o.k. To pa–gnio, metà th diagraf† thc strathgik†c B1, ousiastikà mei∏netaisto:

A2 B2 �2A1 3, 2 1, 1 1, 0�1 2,�1 �1, 3 2, 0

Sq†ma 2.4: To pa–gnio metà thn apaloif† thc B1

Sto mikrÏtero pa–gnio, h strathgik† B2 tou pa–kth 2 kuriarqe– austhrà ep–thc strathgik†c �2 (aut† h sqËsh den –sque prin th diagraf† thc strathgik†cB1 tou pa–kth 1). Sunep∏c, h �2 apalo–fetai. To gegonÏc autÏ kai pàlisunistà koinÏ tÏpo. 'Ara Ëqoume:

A2 B2A1 3, 2 1, 1�1 2,�1 �1, 3

Sq†ma 2.5: To pa–gnio metà thn apaloif† twn B1 kai �2

2.3. EPüILUSH 15

A2 B2A1 3, 2 1, 1

Sq†ma 2.6: To pa–gnio metà thn apaloif† twn B1, �2 kai �1

PlËon, h strathgik† A1 kuriarqe– austhrà ep– thc �1. Sunep∏c Ëqoume:Wc teleuta–o b†ma, apalo–fetai h strathgik† B2. 'Ara h mËjodoc thc dia-doqik†c apaloif†c austhrà kuriarqo‘menwn strathgik∏n odhge– sto ze‘gocstrathgik∏n (A1, A2).

Paràdeigma 2'Estw to ex†c pa–gnio tri∏n paikt∏n, Ïpou o pa–kthc 1 epilËgei grammËc, opa–kthc 2 epilËgei st†lec kai o pa–kthc 3 epilËgei p–nakec (se kàje kel– opr∏toc arijmÏc e–nai h apÏdosh tou pa–kth 1, o de‘teroc arijmÏc h apÏdoshtou pa–kth 2 kai o tr–toc arijmÏc h apÏdosh tou pa–kth 3):

A2 B2 �2A1 2, 0, 2 0, 1, 0 1, 2, 1B1 0, 0, 2 �1, 1, 0 0, 0, 2�1 0, 2, 2 �1, 3, 1 �1, 1, 1

A3

A2 B2 �2A1 2, 3, 1 0, 4,�1 0, 3, 0B1 3, 1, 1 1, 2,�1 0, 0, 1�1 3, 1, 0 1, 2, 0 1, 1, 0

B3

Sq†ma 2.7: Pa–gnio me treic pa–ktec

Wc pr∏to b†ma parathro‘me Ïti h strathgik† A3 tou pa–kth 3 kuriarqe– au-sthrà ep– thc strathgik†c B3: anexart†twc twn epilog∏n twn paikt∏n 1 kai2, h strathgik† (o p–nakac) A3 d–nei megal‘terh apÏdosh ston pa–kth 3 apÏ thstrathgik† (ton p–naka) B3. Mporo‘me sunep∏c na diagràyoume ton p–nakaB3 kai na estiàsoume ston p–naka A3:

A2 B2 �2A1 2, 0, 2 0, 1, 0 1, 2, 1B1 0, 0, 2 �1, 1, 0 0, 0, 2�1 0, 2, 2 �1, 3, 1 �1, 1, 1

A3

Sq†ma 2.8: To pa–gnio metà thn diagraf† thc B3

DedomËnhc thc parapànw diagraf†c, parathro‘me Ïti h strathgik† A1 ku-riarqe– austhrà tÏso ep– thc B1, Ïso kai ep– thc �1. Mporo‘me sunep∏c nadiagràyoume tic strathgikËc autËc. Odhgo‘maste loipÏn ston p–naka 2.9.TËloc, o pa–kthc 2 epilËgei th strathgik† �2. Sunep∏c, h mËjodoc thc dia-doqik†c apaloif†c odhge– ston sunduasmÏ (A1,�2, A3).


A2 B2 �2A1 2, 0, 2 0, 1, 0 1, 2, 1

A3

Sq†ma 2.9: To pa–gnio metà thn diagraf† twn B3, B1 kai �1

2.3.2 Isorrop–a Nash

Ac do‘me xanà ta d‘o teleuta–a parade–gmata, xekin∏ntac me to Paràdeigma1. An melet†soume to apotËlesma sto opo–o katal†xame, dhlad† ton sun-duasmÏ strathgik∏n (A1, A2), ja diapist∏soume Ïti oi emplekÏmenec stra-thgikËc qarakthr–zontai apÏ m–a sqËsh amoiba–ac beltistÏthtac: dedomËnhcthc strathgik†c A2 tou pa–kth 2, h bËltisth strathgik† tou pa–kth 1 e–naih A1. Paromo–wc, dedomËnhc thc strathgik†c A1 tou pa–kth 1 , h bËltisthstrathgik† tou pa–kth 2 e–nai h A2. Se mia tËtoia per–ptwsh ja lËme Ïti oistrathgikËc autËc e–nai amoiba–a bËltistec. An prote–name stouc d‘o pa–ktecton sunduasmÏ strathgik∏n (A1, A2), oude–c ja e–qe k–nhtro na apokl–nei mo-nomer∏c apÏ thn proteinÏmenh strathgik† tou. Me àlla lÏgia, o sunduasmÏc(A1, A2) sunistà isorrop–a (katà Nash, Ïpwc ja thn onomàsoume argÏtera).Ac do‘me t∏ra to Paràdeigma 2 kai to sunduasmÏ (A1,�2, A3). Kai pàli

isq‘ei h –dia idiÏthta: dedomËnwn twn strathgik∏n �2 kai A3, h bËltisthstrathgik† tou pa–kth 1 e–nai h A1. Paromo–wc, dedomËnwn twn strathgik∏nA1 kai A3, h bËltisth strathgik† tou pa–kth 2 e–nai h �2. TËloc, dedomËnwntwn A1 kai �2, h bËltisth strathgik† tou pa–kth 3 e–nai h A3. Sunep∏c,o sunduasmÏc (A1,�2, A3) e–nai tËtoioc ∏ste oude–c pa–kthc Ëqei lÏgo naapokl–nei monomer∏c apÏ th strathgik† pou antistoiqe– se autÏn.Oi parapànw d‘o sunduasmo– sunisto‘n parade–gmata auto‘ pou onomà-

zetai isorrop–a katà Nash. M–a isorrop–a katà Nash e–nai Ënac sunduasmÏcstrathgik∏n o opo–oc emperiËqei amoiba–a bËltistec strathgikËc, upÏ thn Ën-noia pou exhg†same parapànw. H isorrop–a aut† apotele– thn basik† Ënnoiaep–lushc twn mh sunergatik∏n paign–wn, kai se aut†n ja estiàsei Ëna megàlomËroc thc parous–ashc mac.

OrismÏc 2 'Estw Ëna pa–gnio � = {N, (Xi, ui)i2N}. O sunduasmÏc strath-gik∏n x⇤ = (x⇤1, x

⇤2, . . . , x

⇤n) sunistà isorrop–a katà Nash eàn gia kàje i 2 N ,

ui(x⇤i , x

⇤�i) � ui(xi, x⇤�i), gia kàje xi 2 Xi (2.1)

O parapànw orismÏc mac lËei akrib∏c Ïti o sunduasmÏc strathgik∏n x⇤ =(x⇤1, x

⇤2, . . . , x

⇤n) emperiËqei amoiba–a bËltistec strathgikËc. DedomËnwn twn

epilog∏n x⇤�i (dhlad† twn epilog∏n Ïlwn twn paikt∏n plhn tou i), h bËltisthepilog† tou i e–nai h x⇤i , Ïpwc de–qnei h (2.1). EfÏson anàlogh sqËsh isq‘eigia kàje pa–kth, o sunduasmÏc x⇤ sunistà sunduasmÏ amoiba–a bËltistwnstrathgik∏n.

2.3. EPüILUSH 17

O orismÏc thc isorrop–ac Nash Ëqei d‘o mËrh. Katà pr∏ton, kàje pa–kthcepilËgei th strathgik† tou me bËltisto trÏpo, dedomËnwn twn pepoij†sewntou gia tic epilogËc twn àllwn paikt∏n. Katà de‘teron, oi pepoij†seic autËce–nai swstËc, upÏ thn Ënnoia Ïti e–nai sunepe–c me tic pragmatikËc epilogËctwn paikt∏n.E–nai h isorrop–a Nash e‘logo apotËlesma se Ëna pa–gnio; Me àlla lÏgia,

gia poio lÏgo ja perimËname oi pa–ktec na epilËxoun strathgikËc bàsei thcisorrop–ac Nash; Diàforec ermhne–ec Ëqoun protaje–.

• H isorrop–a Nash wc auto-epiballÏmenh sumfwn–aAc upojËsoume Ïti prin xekin†sei to pa–gnio, oi pa–ktec epikoinwno‘n metax‘touc o‘twc ∏ste na epit‘qoun m–a -mh desmeutik†- sumfwn–a sqetikà me topoiec strathgikËc ja epilËxoun. An Ïntwc epiteuqje– m–a tËtoia sumfwn–a,e–nai arketÏ logikÏ na upojËsoume Ïti h sumfwn–a aut† ja epilËgei m–aisorrop–a Nash. Giat– autÏ; DedomËnou Ïti h sumfwn–a e–nai mh desmeutik†,kàje pa–kthc Ëqei th dunatÏthta na mhn thn efarmÏsei allà na epilËxei thnkatallhlÏterh gia ton –dio enËrgeia, dedomËnwn twn prosdoki∏n tou gia ticepilogËc twn àllwn paikt∏n. An h sumfwn–a epilËxei m–a isorrop–a, tÏteoude–c pa–kthc Ëqei k–nhtro na mhn thn thr†sei. An antijËtwc, h sumfwn–aepilËxei m–a mh isorrop–a, tÏte Ënac toulàqiston pa–kthc Ëqei k–nhtro naapokl–nei monomer∏c apÏ aut†n.

• H isorrop–a Nash wc apotËlesma endoskÏphshcOi orjologiko– pa–ktec suqnà mporo‘n na problËpoun tic epilogËc twn àl-lwn paikt∏n kànontac m–a endoskÏphsh tou paign–ou. Dhlad†, prospajo‘nna problËyoun ti ja sumbe– se Ëna pa–gnio qrhsimopoi∏ntac th gn∏sh toucanaforikà me tic sunart†seic apÏdoshc kai thn orjologikÏthta twn àllwnpaikt∏n, th gn∏sh touc Ïti kàje àlloc pa–kthc gnwr–zei Ïti Ïloi oi àlloignwr–zoun autà ta stoiqe–a, klp.

• H isorrop–a Nash wc prÏtash epilog†c strathgik∏nAn Ëna tr–to prÏswpo Ëdine sumboulËc stouc pa–ktec anaforikà me thnepilog† strathgik∏n tÏte ja prÏteine m–a isorrop–a Nash. Se ant–jethper–ptwsh, Ënac toulàqiston pa–kthc ja e–qe k–nhtro na mhn uiojet†sei,monomer∏c, th sumboul†.

• H isorrop–a Nash wc apotËlesma màjhshcAn oi pa–ktec summetËqoun sto –dio pa–gnio († se kontinËc parallagËc tou)xanà kai xanà tÏte kàje Ënac ja te–nei na màjei tic strathgikËc pou a-koloujo‘n oi àlloi, me apotËlesma oi pa–ktec na odhghjo‘n, upÏ kàpoiecproÙpojËseic, se epilogËc pou sunisto‘n isorrop–a.

Ac melet†soume tic isorrop–ec katà Nash gia orismËna aplà parade–gmatapaign–wn.


Paràdeigma 3'Estw o ex†c p–nakac apodÏsewn enÏc paign–ou d‘o paikt∏n:

A2 B2 �2A1 0, 0 0, 1 1, 2B1 0, 2 1, 1 0, 0�1 1, 2 0, 1 0, 3

Sq†ma 2.10: Isorrop–a sto (A1,�2)

Ja exetàsoume poia ze‘gh strathgik∏n sunisto‘n isorrop–a. Ac do‘me pr∏tato ze‘goc (A1, A2). To shme–o autÏ den sunistà shme–o isorrop–ac, diÏti Ënactoulàqiston pa–kthc Ëqei lÏgo na apokl–nei monomer∏c se mia strathgik†diaforetik† apÏ aut†n pou prote–netai sto diànusma autÏ. P.q, ac pàroume tonpa–kth 1. Sto shme–o (A1, A2) o 1 Ëqei apÏdosh 0. Eàn apokl–nei monomer∏csth strathgik† �1, h apÏdosh tou ja e–nai 1 > 0. Sunep∏c, o sunduasmÏc(A1, A2) den sunistà sunduasmÏ amoiba–a bËltistwn strathgik∏n.

O sunduasmÏc (B1, A2), ep–shc den e–nai sunduasmÏc isorrop–ac. Stoshme–o autÏ o pa–kthc 1 Ëqei apÏdosh 0. Eàn kai pàli apokl–nei monomer∏csth strathgik† �1 h apÏdosh tou ja e–nai 1 > 0. O sunduasmÏc (�1, A2) densunistà isorrop–a, diÏti o pa–kthc 2 Ëqei k–nhtro monomero‘c apÏklishc: eànant– gia A2 epËlege �2, h apÏdosh tou ja a‘xane apÏ 2 se 3 (dedomËnhc pàntathc strathgik†c �1 tou pa–kth 1).

O sunduasmÏc (�1, B2) d–nei k–nhtro monomero‘c apÏklishc tÏso stonpa–kth 1, Ïso kai ston pa–kth 2. Gia paràdeigma, o pa–kthc 2 Ëqei k–nhtro naapokl–nei monomer∏c sth strathgik† �2 kerd–zontac 3 ant– gia 1. Ant–stoiqok–nhtro Ëqei o 2 se sqËsh me to diànusma (B1, B2). Sto shme–o autÏ kerd–zei1, en∏ an apokl–nei monomer∏c sth strathgik† A2, ja kerd–sei 2.

O sunduasmÏc (A1, B2) d–nei k–nhtro monomero‘c apÏklishc tÏso ston 1,Ïso kai ston 2. P.q. o 2 mpore– na apokl–nei sth strathgik† �2 kerd–zontac2 > 1. Ant–stoiqo sumpËrasma isq‘ei gia touc sunduasmo‘c (�1,�2) kai(B1,�2), afo‘ kai oi d‘o pa–ktec Ëqoun k–nhtro monomero‘c apÏklishc (giaparàdeigma, kai stic d‘o peript∏seic o 1 apokl–nei sth strathgik† A1).

Ac do‘me tËloc, ton sunduasmÏ (A1,�2). Gia thn per–ptwsh aut† denupàrqei k–nhtro monomero‘c apÏklishc. DedomËnhc thc strathgik†c �2, hbËltisth enËrgeia tou pa–kth 1 e–nai h epilog† thc strathgik†c A1. Kai dedo-mËnhc thc strathgik†c A1, h bËltisth enËrgeia tou pa–kth 2 e–nai h epilog† thcstrathgik†c �2. O sunduasmÏc (A1,�2) e–nai sunep∏c sunduasmÏc amoiba–abËltistwn strathgik∏n, e–nai dhlad† isorrop–a katà Nash.

Paràdeigma 4'Estw to ex†c pa–gnio d‘o paikt∏n:

2.3. EPüILUSH 19

A2 B2 �2A1 0, 0 0, 1 1, 2B1 0, 2 1, 1 0, 0�1 1, 2 0, 1 0, 1

Sq†ma 2.11: Isorrop–ec sto (A1,�2) kai (�1, A2)

Sto pa–gnio autÏ upàrqoun d‘o prof–l amoiba–a bËltistwn strathgik∏n: toprof–l (A1,�2) kai to prof–l (�1, A2). Ac do‘me to pr∏to. DedomËnhcthc strathgik†c �2, h bËltisth strathgik† tou pa–kth 1 e–nai h A1. Toant–strofo ep–shc isq‘ei, dhlad†, dedomËnhc thc strathgik†c A1, h bËltisthstrathgik† tou pa–kth 2 e–nai h �2. Anàloga sumperàsmata isq‘oun gia toprof–l (�1, A2). TËloc, Ïpwc e‘kola mporo‘me na elËgxoume, Ïla ta upÏloipaprof–l d–noun k–nhtra monomero‘c apÏklishc.

Paràdeigma 5'Estw to ex†c pa–gnio tri∏n paikt∏n (Ïpou o pa–kthc 1 epilËgei grammËc, opa–kthc 2 epilËgei st†lec kai o pa–kthc 3 epilËgei p–nakec):

A2 B2 �2A1 2, 0, 0 0, 1, 0 1, 2, 1B1 0, 0, 2 �1, 1, 0 0, 0, 2�1 0, 2, 2 �1, 3, 1 �1, 1, 1

A3

A2 B2 �2A1 2, 3, 1 0, 4, 1 0, 3, 2B1 3, 1, 1 2, 2, 1 0, 0, 1�1 3, 1, 0 1, 2, 0 1, 1, 0

B3

Sq†ma 2.12: Isorrop–a sto (B1, B2, B3)

Parathro‘me Ïti to prof–l strathgik∏n (B1, B2, B3) sunistà isorrop–a katàNash. Gia na bebai∏soume thn parat†rhsh aut†, arke– na de–xoume Ïti oude–cpa–kthc Ëqei k–nhtro na apokl–nei monomer∏c apÏ thn proteinÏmenh se autÏnstrathgik†. Ac do‘me pr∏ta ton pa–kth 1. DedomËnwn twn strathgik∏n B2kai B3, h apÏdosh tou pa–kth 1 megistopoie–tai eàn epilËxei B1, Ïpwc kai stoproteinÏmeno prof–l. 'Ara o 1 den Ëqei k–nhtro monomero‘c apÏklishc.Ac do‘me t∏ra ton pa–kth 2. DedomËnwn twn strathgik∏n B1 kai B3

twn paikt∏n 1 kai 3, h apÏdosh tou pa–kth 2 megistopoie–tai eàn epilËxei B2.Sunep∏c, o‘te o 2 Ëqei k–nhtro na apokl–nei. ParÏmoio sumpËrasma isq‘ei kaigia ton pa–kth 3: an oi 1 kai 2 epilËxoun B1 kai B2 ant–stoiqa, h apÏdosh tou3 megistopoie–tai eàn epilËxei B3.To pa–gnio den Ëqei àllec isorrop–ec. Ac exetàsoume endeiktikà kàpoia

prof–l strathgik∏n, xekin∏ntac me to (A1, A2, A3). Sto shme–o autÏ o pa–-kthc 3 Ëqei apÏdosh 0. Eàn apokl–nei monomer∏c sth strathgik† B3, h apÏdo-sh tou ja e–nai 1. Sunep∏c, to prof–l (A1, A2, A3) den e–nai prof–l amoiba–abËltistwn strathgik∏n. Paromo–wc, to prof–l (�1, A2, B3) den e–nai shme–oisorrop–ac: dojËntwn twn �1 kai B3, o pa–kthc 2 Ëqei k–nhtro na epilËxei th


strathgik† B2, kai Ïqi thn A2. ParÏmoioi sullogismo– isq‘oun kai gia taupÏloipa prof–l strathgik∏n.

O orismÏc kai upologismÏc twn isorropi∏n Nash emperiËqei thn Ënnoia thcbËltisthc ant–drashc. H sunàrthsh bËltisthc ant–drashc enÏc pa–kth d–neitic strathgikËc eke–nec pou megistopoio‘n thn apÏdosh tou dedomËnwn twnstrathgik∏n twn àllwn paikt∏n.Ac upolog–soume tic sunart†seic autËc gia to Paràdeigma 3. Sumbol–-

zoume me b1(x2) th sunàrthsh bËltisthc ant–drashc tou pa–kth 1, Ïpou x2sumbol–zei kàpoia strathgik† tou pa–kth 2, dhlad† x2 2 {A2, B2,�2}. Hsunàrthsh aut† parousiàzetai ston epÏmeno p–naka:

x2 b1(x2)A2 �1B2 B1�2 A1

P–nakac 2.1: BËltistec antidràseic pa–kth 1 sto Paràdeigma 3

Sumbol–zoume me b2(x1) th sunàrthsh bËltisthc ant–drashc tou pa–kth 2,Ïpou x1 sumbol–zei th strathgik† tou pa–kth 1, dhlad† x1 2 {A1, B1,�1}.Sqhmatikà Ëqoume:

x1 b2(x1)A1 �2B1 A2�1 �2

P–nakac 2.2: BËltistec antidràseic pa–kth 2 sto Paràdeigma 3

Upenjum–zoume Ïti h isorrop–a epitugqànetai se kàje prof–l amoiba–a bËlti-stwn strathgik∏n. ApÏ touc parapànw p–nakec blËpoume Ïti b1(�2) = A1kai b2(A1) = �2, dhlad† oi strathgikËc A1 kai �2 e–nai amoiba–a bËltistec.GenikÏtera, Ëstw Ïti Ëqoume Ëna pa–gnio thc morf†c � = {N, (Xi, ui)i2N}.

H sunàrthsh bËltisthc ant–drashc tou pa–kth i or–zetai wc:

bi(x�i) = argmaxxi

ui(xi, x�i), i = 1, 2, . . . , n (2.2)

Ac pàroume to s‘sthma twn n exis∏sewn pou dhmiourge–tai apÏ tic n bËltistecantidràseic:

x1 = b1(x�1)

x2 = b2(x�2)

2.3. EPüILUSH 21

.

.

.

xn = bn(x�n)

H l‘sh tou parapànw sust†matoc mac d–nei thn (tic) isorrop–a(–ec) Nashtou paign–ou �. Shmei∏noume Ïti oi sunart†seic bËltisthc ant–drashc ja macqrhsime‘soun idia–tera sthn melËth paign–wn Ïpou oi pa–ktec diajËtoun àpeirecston arijmÏ strathgikËc, to opo–o e–nai to antike–meno thc epÏmenhc enÏthtac.Ja oloklhr∏soume thn paro‘sa enÏthta epanerqÏmenoi sthn Ënnoia twn

kur–arqwn strathgik∏n. Sthn arq† thc enÏthtac epil‘same pa–gnia me thnmËjodo thc diadoqik†c apaloif†c twn austhrà kuriarqo‘menwn strathgik∏n.Ac eisagàgoume t∏ra m–a parapl†sia Ënnoia, thn asjen† kuriarq–a.

OrismÏc 3 H strathgik† xi tou pa–kth i kuriarqe– asjen∏c ep– thc stra-thgik†c x̃i eàn ui(xi, x�i) � ui(x̃i, x�i) gia kàje x�i 2 X�i kai ui(xi, x�i) >ui(x̃i, x�i) gia Ëna toulàqiston x�i.

To epÏmeno paràdeigma ja mac bohj†sei na katalàboume thn Ënnoia thc asje-no‘c kuriarq–ac. 'Estw o ex†c p–nakac apodÏsewn:

A2 B2 �2A1 0, 0 1, 1 1, 2B1 0, 2 1, 1 0, 0�1 1, 2 0, 1 0, 1

Sq†ma 2.13: Asjen†c kuriarq–a

Parathro‘me Ïti h strathgik† A1 d–nei ston pa–kth 1 e–te –dia apÏdosh me thnB1 (gia tic peript∏seic Ïpou o pa–kthc 2 epilËgei A2 † B2) e–te megal‘terh(gia thn per–ptwsh Ïpou o 2 epilËgei �2). Sunep∏c, h strathgik† A1 toupa–kth 1 kuriarqe– asjen∏c ep– thc B1. H strathgik† A1 sunep∏c kuriarqe–asjen∏c ep– thc B1.

2.3.3 Eidikà jËmata

Ta pa–gnia pou ja parousiasto‘n se aut†n thn enÏthta Ëqoun wc stÏqo nakatade–xoun sugkekrimËna zht†mata pou sqet–zontai me ta mh sunergatikà pa–-gnia kai thn isorrop–a Nash. Kàje m–a kathgor–a paign–wn ja parousiaste–mËsw m–ac apl†c istor–ac, h opo–a Ïmwc e–nai dhlwtik† genikÏterwn fainomË-nwn.

A. D–lhmma fulakismËnwnD‘o àtoma sullambànontai wc ‘popta gia Ëna paràptwma. H astunom–a den


Ëqei epark† stoiqe–a gia na ta katadikàsei, ektÏc eàn Ëna toulàqiston àtomoomolog†sei. H astunom–a kratà touc d‘o upÏptouc se qwristà kelià kai toucanakoin∏nei ta ex†c: eàn kai oi d‘o omolog†soun, tÏte kai oi d‘o ja kata-dikasto‘n se Ëxi m†nec fulàkish. An oude–c omolog†sei, ja katadikasto‘nkai oi d‘o gia Ëna aplÏ pta–sma me poin† fulàkishc enÏc mhnÏc. Eàn ËnacmÏno omolog†sei, tÏte autÏc pou omolÏghse ja afeje– ele‘jeroc kai o àllocja katadikaste– se ennËa m†nec fulàkish (Ëxi m†nec gia to Ëgklhma kai treicm†nec gia mh sunergas–a me tic arqËc).Oi strathgikËc kàje krato‘menou e–nai: na omolog†sei, O, † na me–nei siw-

phlÏc, ⌃. O p–nakac apodÏsewn tou paign–ou Ëqei wc ex†c (gia aplo‘steush,jewro‘me Ïti h zhmià apÏ m–a poin† taut–zetai me th diàrkeia thc):

⌃ O⌃ -1,-1 -9, 0O 0,-9 -6,-6

Sq†ma 2.14: Anapotelesmatik† isorrop–a

To pa–gnio Ëqei m–a isorrop–a Nash: kai oi d‘o pa–ktec omologo‘n, me sunËpeiana fulakisto‘n gia Ëxi m†nec. To endiafËron e–nai Ïti h isorrop–a aut† kuriar-qe–tai katà Pareto apÏ to ze‘goc strathgik∏n (⌃, ⌃). To ze‘goc autÏ Ïmwcden e–nai isorrop–a Nash (an Ënac pa–kthc siwp†sei, h bËltisth ant–drash touàllou e–nai na omolog†sei kai na afeje– ele‘jeroc).Telei∏nontac to paràdeigma, ja prËpei na ton–soume Ïti pa–gnia twn o-

po–wn oi isorrop–ec Nash e–nai mh apotelesmatikËc katà Pareto emfan–zontaisuqnÏtata sthn oikonomik† (kai Ïqi mÏno) anàlush. OrismËna gnwstà oikono-mikà parade–gmata me anapotelesmatikËc isorrop–ec emfan–zontai sthn jewr–atou oligopwl–ou, se pa–gnia paroq†c dhmos–wn agaj∏n, klp.

B. PÏlemoc twn f‘lwnO 1 kai h 2 sqediàzoun thn bradun† touc Ëxodo. Mporo‘n na epilËxoun metax‘m–ac paràstashc Ïperac O kai enÏc ag∏na podosfa–rou ⇧. Ta d‘o àtomaËqoun diaforetikËc protim†seic, kaj∏c h 2 protimà thn Ïpera kai o 1 protimàton ag∏na podosfa–rou. ApÏ thn àllh, kai oi d‘o protimo‘n na kànoun kàpoiadrasthriÏthta apÏ koino‘. ProkeimËnou na apofas–soun pwc ja peràsountelikà to bràdu, oi d‘o pa–ktec ja gràyoun tautÏqrona se Ëna qart– thnepilog† touc (anexàrthta o Ënac apÏ ton àllo). O p–nakac apodÏsewn toupaign–ou e–nai o ex†c (o 1 epilËgei grammËc kai h 2 st†lec):

P OP 2,1 0, 0O 0, 0 1, 2

Sq†ma 2.15: Pa–gnio suntonismo‘

2.3. EPüILUSH 23

To pa–gnio autÏ Ëqei d‘o isorrop–ec: e–te kai oi d‘o pa–ktec epilËgoun thnÏpera e–te kai oi d‘o epilËgoun ton ag∏na podosfa–rou. To pa–gnio autÏe–nai Ëna paràdeigma paign–ou suntonismo‘, dhlad† enÏc paign–ou sto opo–ooi pa–ktec protimo‘n epilËxoun († na suntonisto‘n se) Ïmoiec strathgikËc.To (dunhtikÏ) prÏblhma e–nai Ïti oi pa–ktec endËqetai na apot‘qoun stonmetax‘ touc suntonismÏ, epilËgontac anÏmoiec strathgikËc, to opo–o e–nai toqeirÏtero dunatÏ apotËlesma kai gia touc d‘o.

G. Kun†giD‘o kunhgo–, oi 1 kai 2, Ëqoun thn epilog† e–te na kunhg†soun apÏ koino‘ Ënamegàlo j†rama, pq, elàfi, e–te na kunhg†sei o kàje Ënac mÏnoc Ëna mikrÏteroj†rama, pq, lagÏ. H epituq†c Ëkbash thc pr∏thc epilog†c apaite– aparait†twcth sunergas–a twn kunhg∏n, kàti pou den isq‘ei gia th de‘terh per–ptwsh(Ënac kunhgÏc arke– apÏ mÏnoc tou gia na piàsei Ëna lagÏ). To megàlo j†ramaax–zei shmantikà perissÏtero apÏ to mikrÏ. O p–nakac apodÏsewn tou paign–ousto opo–o kàje kunhgÏc apofas–zei eàn ja kunhg†sei elàfi, E, † lagÏ, L,e–nai o ex†c:

E LE 7,7 0, 3L 3, 0 3, 3

Sq†ma 2.16: Pa–gnio kajaro‘ suntonismo‘

'Opwc parathro‘me to pa–gnio Ëqei d‘o isorrop–ec: e–te kai oi d‘o pa–ktecepilËgoun to kun†gi elafio‘ e–te kai oi d‘o epilËgoun to kun†gi lago‘. SthsugkekrimËnh per–ptwsh Ëqoume Ëna pa–gnio kajaro‘ suntonismo‘. 'Opwc kaista pa–gnia suntonismo‘, oi pa–ktec protimo‘n na epilËxoun Ïmoiec kai ÏqidiaforetikËc strathgikËc. EpiplËon Ïmwc, upàrqei m–a isorrop–a h opo–a ku-riarqe– katà Pareto ep– thc àllhc: kai oi d‘o pa–ktec protimo‘n thn isorrop–a(E, E) apÏ thn isorrop–a (L, L). Par' Ïla autà, h de‘terh isorrop–a emperiËqeiqamhlÏtero r–sko: m–a endeqÏmenh apotuq–a suntonismo‘ sto kun†gi lago‘den ja mei∏sei thn apÏdosh enÏc kunhgo‘, se ant–jesh me autÏ pou ja sumbe–se per–ptwsh apotuq–ac suntonismo‘ sto kun†gi elafio‘.H istor–a twn kunhg∏n, h opo–a parousiàsthke apÏ ton filÏsofo Rousse-

au sto pla–sio tou Koinwniko‘ Sumbola–ou tou, qwr–c o –dioc na kànei qr†shthc jewr–a twn paign–wn, apotele– Ëna tupikÏ paràdeigma epilog†c metax‘ m–acstrathgik†c qamhlo‘ r–skou (kun†gi lago‘) kai m–ac pio prosodofÏrou stra-thgik†c (kun†gi elafio‘), h opo–a Ïmwc apaite– th sunergas–a twn paikt∏n.

D. Pa–gnio jàrroucOi odhgo– 1 kai 2 pa–rnoun mËroc se Ëna paiqn–di ep–deixhc jàrrouc. Sugkekri-mËna, odhgo‘n ta autok–nhta touc se pore–a metwpik†c s‘gkroushc, Ëqontacthn epilog† na allàxoun kate‘junsh l–go prin th s‘gkroush. An oude–c al-làxei kate‘junsh, kai oi d‘o qànoun diÏti ja epËljei s‘gkroush. Eàn Ënac


mÏno odhgÏc str–yei tÏte o –dioc qànei, kaj∏c ja jewrhje– deilÏc, en∏ oàlloc odhgÏc kerd–zei, kaj∏c ja jewrhje– jarralËoc. TËloc, an kai oi d‘ostr–youn tÏte oude–c qànei † kerd–zei.2

Kàje odhgÏc Ëqei sunep∏c d‘o strathgikËc sto pa–gnio: na parame–nei ep–thc pore–ac s‘gkroushc mËqri tËlouc, E, † na str–yei, S. Ja upojËsoume Ïtio p–nakac apodÏsewn tou paign–ou Ëqei wc ex†c:

E SE -10,-10, 5,-5S -5,5 0, 0

Sq†ma 2.17: Pa–gnio anti-suntonismo‘

To parapànw pa–gnio Ëqei d‘o isorrop–ec katà Nash: se kàje m–a apÏ au-tËc, Ënac odhgÏc str–bei kai Ënac paramËnei ep– thc pore–ac s‘gkroushc. Oiisorrop–ec sunep∏c d–nontai apÏ touc sunduasmo‘c (E,S) kai (S,E). Se kàjeisorrop–a, oi pa–ktec epilËgoun diaforetikËc strathgikËc. To pa–gnio autÏ e–-nai Ëna paràdeigma paign–ou anti-suntonismo‘: h epilog† Ïmoiwn strathgik∏nepifËrei kÏstoc parà Ïfeloc.

E. PËtra-yal–di-qart–Oi pa–ktec 1 kai 2 pa–zoun to gnwstÏ paiqn–di pËtra-yal–di-qart– (upenjum–-zoume Ïti sto paiqn–di autÏ oi pa–ktec anakoin∏noun tautÏqrona m–a apÏ tictreic epilogËc me thn pËtra nikà to yal–di kai na qànei apÏ to qart–, kai toqart– na qànei apÏ to yal–di). Oi amoibËc e–nai 1 † -1 (se per–ptwsh n–khc kai†ttac antisto–qwc) kai 0 (se per–ptwsh isopal–ac, dhlad† –diwn epilog∏n). OiapodÏseic sunoy–zontai ston epÏmeno p–naka (Ïpou ⇧ sumbol–zei thn pËtra, to yal–di kai X to qart–):

⇧ X⇧ 0, 0 1,-1 -1,1 -1, 1 0, 0 1,-1X 1,-1 -1, 1 0, 0

Sq†ma 2.18: Pa–gnio qwr–c isorrop–a

'Opwc e‘kola blËpoume to paiqn–di autÏ den Ëqei isorrop–a. Gia kàje pro-f–l strathgik∏n upàrqei pànta Ënac pa–kthc o opo–oc Ëqei k–nhtro monomero‘capÏklishc. To pa–gnio autÏ sunistà paràdeigma paign–ou qwr–c isorrop–a Na-sh (se kajarËc strathgikËc3).

2To pa–gnio autÏ e–nai gnwstÏ wc pa–gnio tou kotÏpoulou. O Ïroc kotÏpoulo qrhsi-mopoie–tai apÏ touc Agglosàxonec wc dhlwtikÏc thc deil–ac enÏc atÏmou.

3Se epÏmenh paràgrafo ja or–soume th diaforà metax‘ kajar†c kai mikt†c strathgi-k†c. Proc to parÏn, wc kajar† strathgik† onomàzoume kàje mËloc tou sunÏlou strath-gik∏n enÏc pa–kth.

2.3. EPüILUSH 25

2.3.4 Pa–gnia me àpeirec strathgikËc

Se pollà apÏ ta pa–gnia pou sunanto‘me sthn oikonomik† anàlush, oi pa–ktecdiajËtoun àpeirec ston arijmÏ strathgikËc (gia na e–maste pio akribe–c, tas‘nola strathgik∏n e–nai suneq† diast†mata). Wc paràdeigma, anafËroumeËna oligop∏lio Ïpou oi epiqeir†seic antagwn–zontai epilËgontac tic posÏth-tec twn proÏntwn pou ja prosfËroun sthn agorà. To s‘nolo twn efikt∏nposot†twn Ëqei àpeira stoiqe–a (h posÏthta mpore– na e–nai opoiosd†pote mharnhtikÏc arijmÏc). O prosdiorismÏc thc isorrop–ac se pa–gnia me àpeirecstrathgikËc sthr–zetai kai pàli ston prosdiorismÏ amoiba–a bËltistwn stra-thgik∏n. IdiaitËrwc t∏ra Ïmwc, oi isorrop–ec mporo‘n na upologisto‘n mËswstoiqeiwd∏n pràxewn diaforiko‘ logismo‘.Ac do‘me kai pàli Ëna pa–gnio thc morf†c � = {N, (Xi, ui)i2N}. Ja

kànoume tic ex†c upojËseic:

A1 Xi = [ai, bi]

A2 H sunàrthsh apÏdoshc tou pa–kth i e–nai diafor–simh kai ko–lh wc procxi

A3 To prÏblhma megistopo–hshc (2.2) Ëqei eswterik† l‘sh

TÏte, h sunàrthshc bËltisthc ant–drashc tou pa–kth i (Ïpou i = 1, 2, . . . , n)prok‘ptei apÏ th l‘sh thc sunj†khc pr∏thc tàxhc:

@ui(xi, x�i)

@xi= 0 (2.3)

Ac do‘me Ëna sqetikÏ paràdeigma me d‘o pa–ktec.

Paràdeigma 6Jewro‘me Ëna pa–gnio me touc pa–ktec 1 kai 2. Oi sunart†seic apÏdoshc twnpaikt∏n e–nai oi:

u1(x1, x2) = 4x1 � x21 � x1x2 � 3

u2(x1, x2) = x2 � x22 � 2x1x2 � 1

Ta s‘nola strathgik∏n e–nai X1 = X2 = [a, b], Ïpou a kai b pragmatiko–arijmo–. EfarmÏzontac th sunj†kh (2.3) gia ton pa–kth 1 pa–rnoume thn ex†csqËsh:

@u1(x1, x2)

@x1= 0 , 4� 2x1 � x2 = 0 (2.4)

H sunàrthsh bËltisthc ant–drashc tou pa–kth 1 prok‘ptei apÏ th l‘sh thc(2.4) wc proc x1. Dhlad†:


b1(x2) = 2�x22

(2.5)

H sunj†kh (2.3) gia ton pa–kth 2 e–nai:

@u2(x1, x2)

@x2= 0 , 1� 2x2 � 2x1 = 0 (2.6)

H sunàrthsh bËltisthc ant–drashc tou pa–kth 2 prok‘ptei apÏ th l‘sh thc(2.6) wc proc x2. Dhlad†:

b2(x1) =1

2� x1 (2.7)

H l‘sh tou sust†matoc twn (2.5) kai (2.7) ja mac d∏sei thn isorrop–a Nashtou paign–ou. Sumbol–zoume th l‘sh me x⇤1 kai x

⇤2. TÏte:

x⇤1 =7

2, x⇤2 = �3

2.4 MiktËc StrathgikËc

H mËqri t∏ra anàlush bas–sthke sthn upÏjesh Ïti oi pa–ktec epilËgoun stra-thgikËc katà nteterministikÏ trÏpo. UpÏ aut† thn Ënnoia, to s‘nolo stra-thgik∏n Xi tou pa–kth i onomàzetai s‘nolo kajar∏n † amig∏n strathgik∏n.Sth paro‘sa enÏthta epekte–noume thn Ënnoia thc strathgik†c epitrËpontacstouc pa–ktec na epilËgoun strathgikËc me tuqa–o † pijanotikÏ trÏpo. Mebàsh th prosËggish aut†, m–a strathgik† tou pa–kth i e–nai m–a katanom†pijanÏthtac pou or–zetai sto s‘nolo Xi.Ac do‘me Ëna aplÏ paràdeigma. 'Estw Ïti o pa–kthc 1 diajËtei to s‘nolo

kajar∏n strathgik∏n X1 = {A1, B1}. M–a mikt† strathgik† tou 1 e–nai Ënadiànusma (p, 1�p), Ïpou p e–nai h pijanÏthta epilog†c thc kajar†c strathgi-k†c A1 kai 1�p e–nai h pijanÏthta epilog†c thc kajar†c strathgik†c B1, kai0 p 1. Eàn gia paràdeigma, p = 1/3, tÏte h ant–stoiqh mikt† strathgik†e–nai to diànusma (1/3, 2/3).To epÏmeno paràdeigma ja mac de–xei pwc prosdior–zontai oi apodÏseic twn

paikt∏n Ïtan qrhsimopoio‘ntai miktËc strathgikËc.

Paràdeigma 7D‘o àtoma, o 1 kai 2, anakoin∏noun tautÏqrona thn Ïyh enÏc nom–smatoc.Dhlad† kàje pa–kthc anakoin∏nei e–te kor∏na K, e–te gràmmata G. Eàn oianakoin∏seic e–nai –diec, o 1 plhr∏nei Ëna eur∏ ston 2. Kai eàn oi anakoin∏seicdiafËroun, o 2 plhr∏nei Ëna eur∏ ston 1. To pa–gnio autÏ, to opo–o onomàzetaipa–gnio twn kermàtwn, parousiàzetai ston epÏmeno p–naka.Jewro‘me tic ex†c miktËc strathgikËc: o pa–kthc 1 qrhsimopoie– th stra-

thgik† (3/4, 1/4) kai o pa–kthc 2 qrhsimopoie– th strathgik† (1/3, 2/3). Dh-lad†, o pa–kthc 1 epilËgei thn kajar† strathgik† K me pijanÏthta 3/4 kai

2.4. MIKTüES STRATHGIKüES 27

K (13) G (23)

(34) K �1, 1 1,�1(14) G 1,�1 �1, 1

Sq†ma 2.19: MiktËc strathgikËc

thn G me pijanÏthta 1/4. Kai o pa–kthc 2 epilËgei thn kajar† strathgik† Kme pijanÏthta 1/3 kai thn G me pijanÏthta 2/3.Me bàsh tic parapànw strathgikËc, oi opo–ec epilËgontai anexàrthta h

m–a apÏ thn àllh, to apotËlesma (K,K), dhlad† to endeqÏmeno Ïti kai oi d‘opa–ktec epilËgoun K lambànei q∏ra me pijanÏthta 34 ·

13 =

14 . To apotËlesma

(K,G) lambànei q∏ra me pijanÏthta 34 ·23 =

12 . Paromo–wc, ta apotelËsmata

(G,K) kai (G,G) lambànoun q∏ra me ant–stoiqec pijanÏthtec 14 ·13 =

112 kai

14 ·

23 =

16 .

H anamenÏmenh apÏdosh tou pa–kth 1, thn opo–a sumbol–zoume me V1, Ïtanoi d‘o pa–ktec qrhsimopoio‘n tic parapànw strathgikËc, or–zetai wc:

V1 =3

4· 13· (�1) + 3

4· 23· 1 + 1

4· 13· 1 + 1

4· 23· (�1) = 1

6(2.8)

Paromo–wc h anamenÏmenh apÏdosh tou pa–kth 2, V2, e–nai:

V2 =3

4· 13· 1 + 3

4· 23· (�1) + 1

4· 13· (�1) + 1

4· 23· 1 = �1

6(2.9)

MËqri stigm†c Ëqoume or–sei isorrop–ec Nash mÏno gia th per–ptwsh twn ka-jar∏n strathgik∏n. Sto epÏmeno paràdeigma ja upolog–soume isorrop–ectÏso se kajarËc Ïso kai se miktËc strathgikËc. H logik† den e–nai diafo-retik†: Ïpwc stic kajarËc strathgikËc, Ëtsi kai stic miktËc, qreiàzetai naupolog–soume amoiba–a bËltistec strathgikËc.

Paràdeigma 8'Estw o p–nakac apodÏsewn:

A2 (q) B2 (1� q)(p) A1 1, 2 0, 0

(1� p) B1 0, 0 2, 1

H anamenÏmenh apÏdosh tou pa–kth 1 Ïtan qrhsimopoie– thn (proc stigm† au-ja–reth) strathgik† (p, 1�p) kai o 2 qrhsimopoie– thn (proc stigm† auja–reth)strathgik† (q, 1� q) e–nai:

V1 = pq + 2(1� p)(1� q)


Ac prosdior–soume thn bËltisth strathgik† tou pa–kth 1. Gia ton skopÏautÏ arke– na estiàsoume sto p. H paràgwgoc thc sunàrthshc anamenÏmenhcapÏdoshc tou pa–kth 1 wc proc p e–nai:

@V1@p

= q � 2(1� q) = 3q � 2

Parathro‘me Ïti eàn 3q � 2 > 0 † alli∏c eàn q > 2/3, tÏte h sunàrthsh V1e–nai gnhs–wc a‘xousa wc proc p. 'Ara o pa–kthc 1 Ëqei k–nhtro na epilËxeiÏso to dunatÏn uyhlÏtero p. DedomËnou Ïti 0 p 1, to bËltisto p seaut† thn per–ptwsh e–nai p = 1. ApÏ thn àllh merià, eàn 3q � 2 < 0 † alli∏ceàn q < 2/3, tÏte h sunàrthsh V1 e–nai gnhs–wc fj–nousa wc proc p. 'Arao pa–kthc 1 Ëqei k–nhtro na epilËxei Ïso to dunatÏn mikrÏtero p. DedomËnouÏti 0 p 1, to bËltisto p e–nai p = 0. TËloc, eàn 3q � 2 = 0 † alli∏ceàn q = 2/3, tÏte h sunàrthsh V1 e–nai anexàrthth tou p. 'Ara, se aut† thnper–ptwsh, o pa–kthc 1 e–nai adiàforoc wc proc to p. Me àlla lÏgia kàje0 p 1 e–nai ex–sou kalÏ.Sunoy–zontac, to prÏshmo thc parag∏gou thc sunàrthshc anamenÏmenhc

apÏdoshc tou pa–kth 1 e–nai:

@V1@p

8<

:

> 0, an q > 2/3= 0, an q = 2/3< 0, an q < 2/3

(2.10)

Sunep∏c h sunàrthsh bËltisthc ant–drashc, b1(q), tou pa–kth 1 Ëqei thn pa-rakàtw morf†:

b1(q) =

8<

:

1, an q > 2/3[0,1], an q = 2/30, an q < 2/3

(2.11)

Paromo–wc, h anamenÏmenh apÏdosh tou pa–kth 2 Ïtan o 1 qrhsimopoie– thnstrathgik† (p, 1� p) kai o 2 qrhsimopoie– thn strathgik† (q, 1� q) e–nai:

V2 = 2pq + (1� p)(1� q)

H paràgwgoc wc proc q thc parapànw sunàrthshc anamenÏmenhc apÏdoshce–nai:

@V2@q

= 2p� (1� p) = 3p� 1

Sunep∏c, to prÏshmo thc parag∏gou (wc sunàrthsh thc strathgik†c toupa–kth 1) e–nai:

2.4. MIKTüES STRATHGIKüES 29

@V2@q

8<

:

> 0, an p > 1/3= 0, an p = 1/3< 0, an p < 1/3

(2.12)

H sunàrthsh bËltisthc ant–drashc, b2(p), tou pa–kth 2 Ëqei sunep∏c thnparakàtw morf† (h ex†ghsh thc sunàrthshc aut†c e–nai anàlogh aut†c poud∏same gia th sunàrthsh bËltisth ant–drashc tou pa–kth 1):

b2(p) =

8<

:

1, an p > 1/3[0,1], an p = 1/30, an p < 1/3

(2.13)

Oi d‘o sunart†seic bËltisthc ant–drashc (2.11) kai (2.13) apeikon–zontai stoparakàtw diàgramma:

q

p

E1

E2

E3

b1(q)

b2(p)

2/3

1/30 1

1

Ta shme–a tom†c twn d‘o kampul∏n bËltisthc ant–drashc mac d–doun tic isor-rop–ec tou paign–ou. Ta shme–a autà e–nai ta E1, E2 kai E3. Oi isorrop–ecE2 kai E3 antistoiqo‘n se kajarËc strathgikËc kai h E1 se miktËc, Ïpwcexhgo‘me euj‘c amËswc.Ac do‘me pr∏ta to shme–o E2. Sto shme–o autÏ Ëqoume p = 0 kai q = 0 †

alli∏c 1� p = 1 kai 1� q = 1. Me àlla lÏgia sto E2, o pa–kthc 1 epilËgei


thn strathgik† B1 me pijanÏthta 1 (upenjum–zoume Ïti p e–nai h pijanÏthtaepilog†c thc A1 kai 1�p e–nai h pijanÏthta epilog†c thc B1). Paromo–wc, stoE2, o pa–kthc 2 epilËgei thn strathgik† B2 me pijanÏthta 1 (upenjum–zoume Ïtiq e–nai h pijanÏthta epilog†c thc A2 kai 1� q e–nai h pijanÏthta epilog†c thcB2). Dhlad†, sthn ous–a sto E2 oi pa–ktec epilËgoun kajarËc strathgikËc.Ac do‘me t∏ra to shme–o E3. Sto shme–o autÏ Ëqoume p = 1 kai q = 1 †

alli∏c 1� p = 0 kai 1� q = 0. Me àlla lÏgia sto E3, o pa–kthc 1 epilËgeithn strathgik† A1 me pijanÏthta 1. Paromo–wc, sto E3, o pa–kthc 2 epilËgeithn strathgik† A2 me pijanÏthta 1. Dhlad†, sto E3 oi pa–ktec epilËgoun kaipàli kajarËc strathgikËc.Ac do‘me tËloc to shme–o E1. Sto shme–o autÏ Ëqoume p = 1/3 (kai àra

1 � p = 2/3) kai q = 2/3 (kai àra 1 � q = 1/3). Sto shme–o E1, loipÏn,kàje pa–kthc epilËgei kai tic d‘o kajarËc strathgikËc tou me austhrà jetik†pijanÏthta (isorrop–a se miktËc strathgikËc).

2.4.1 Arq† thc adiafor–ac

Ac parame–noume sto prohgo‘meno paràdeigma kai ac estiàsoume sthn isor-rop–a se miktËc strathgikËc, dhlad† sta dian‘smata (p, 1 � p) = (1/3, 2/3)kai (q, 1 � q) = (2/3, 1/3). 'Opwc e–nai emfanËc apÏ th sqËsh (2.13), Ïtano pa–kthc 1 epilËgei th mikt† strathgik† (p, 1 � p) = (1/3, 2/3) tÏte o pa–-kthc 2 e–nai adiàforoc metax‘ Ïlwn twn strathgik∏n tou. En prokeimËnw,e–nai adiàforoc metax‘ twn d‘o kajar∏n strathgik∏n A2 kai B2. 'Ontwc, hanamenÏmenh apÏdosh tou pa–kth 2 Ïtan epilËgei A2 kai o pa–kthc 1 epilËgei(p, 1� p) = (1/3, 2/3) e–nai

V2(1/3, 2/3, A2) = 1/3 · 2 + 2/3 · 0 = 2/3,

Ïsh akrib∏c e–nai kai anamenÏmenh apÏdosh tou apÏ thn epilog† thc B2,

V2(1/3, 2/3, A2) = 1/3 · 0 + 2/3 · 1 = 2/3

ParÏmoio sumpËrasma isq‘ei kai gia ton pa–kth 1 se sqËsh me th mikt† stra-thgik† (q, 1 � q) = (2/3, 1/3) tou pa–kth 2: Ïtan o 2 epilËgei th strathgik†aut†, o 1 e–nai adiàforoc metax‘ Ïlwn twn strathgik∏n tou (sqËsh 2.11), kaien prokeimËnw metax‘ twn d‘o kajar∏n stathgik∏n tou:

V1(A1, 2/3, 1/3) = 2/3 · 1 + 1/3 · 0 = 2/3

kai

V1(B1, 2/3, 1/3) = 2/3 · 0 + 1/3 · 2 = 2/3

Ta parapànw shma–noun ta ex†c:

2.5. EFARMOGüES 31

(i) H strathgik† isorrop–ac (p, 1 � p) = (1/3, 2/3) tou pa–kth 1 mpore–na breje– exis∏nontac tic anamenÏmenec apodÏseic tou pa–kth 2 apÏ tickajarËc strathgikËc A2 kai B2. Dhlad† h strathgik† (p, 1 � p) =(1/3, 2/3) l‘nei thn ex–swsh V2(p, 1� p,A2) = V2(p, 1� p,B2).

(ii) H strathgik† isorrop–ac (q, 1 � q) = (2/3, 1/3) tou pa–kth 2 mpore–na breje– exis∏nontac tic anamenÏmenec apodÏseic tou pa–kth 1 apÏ tickajarËc strathgikËc A1 kai B1. Dhlad† h strathgik† (q, 1 � q) =(2/3, 1/3) l‘nei thn ex–swsh V1(A1, q, 1� q) = V1(B1, q, 1� q).

Ta parapànw sunisto‘n th mËjodo thc arq†c thc adiafor–ac gia thn e‘reshtwn isorrop–wn Nash se miktËc strathgikËc.

2.5 EfarmogËc

2.5.1 Duop∏lio

Jewro‘me m–a agorà sthn opo–a leitourgo‘n d‘o epiqeir†seic, oi 1 kai 2. Oiepiqeir†seic paràgoun Ëna omoiogenËc proÏn. H posÏthta thc epiqe–rhshc isumbol–zetai me xi, i = 1, 2. H sunàrthsh kÏstouc thc epiqe–rhshc i e–naiC(xi) = cxi, i = 1, 2. TËloc, h ant–strofh sunàrthsh z†thshc d–netai apÏthn p = a� x1 � x2, Ïpou p h tim† tou proÏntoc kai a > 0.H sunàrthsh apÏdoshc (kËrdouc) thc epiqe–rhshc 1 e–nai:

⇡1(x1, x2) = (a� x1 � x2 � c)x1

en∏ aut† thc epiqe–rhshc 2 e–nai:

⇡2(x1, x2) = (a� x1 � x2 � c)x2

Kàje epiqe–rhsh epidi∏kei na megistopoi†sei to kËrdoc thc epilËgontac thnposÏthta pou ja paràgei kai ja prosfËrei sthn agorà. Ta s‘nola twn stra-thgik∏n twn epiqeir†sewn sunep∏c e–nai X1 = X2 = [0,1). Ja prosdior–-soume thn isorrop–a gia thn agorà aut†, dhlad† to ze‘goc († ta ze‘gh) twnamoiba–a bËltistwn posot†twn.To pr∏to b†ma gia thn e‘resh thc isorrop–ac e–nai o upologismÏc twn

sunart†sewn bËltisthc ant–drashc twn epiqeir†sewn. AutËc ja prok‘younapÏ tic sunj†kec pr∏thc tàxhc gia mËgista kËrdh. Oi sunj†kec autËc e–nai:

@⇡1(x1, x2)

@x1= 0 , ↵� 2x1 � x2 � c = 0 (2.14)

@⇡2(x1, x2)

@x2= 0 , ↵� 2x2 � x1 � c = 0 (2.15)

L‘nontac to parapànw s‘sthma lambànoume tic sunart†seic bËltisthc ant–-drashc twn d‘o epiqeir†sewn:


b1(x2) =

⇢0, eàn x2 � a� ca�x2�c

2 , eàn x2 < a� c

b2(x1) =

⇢0, eàn x1 � a� ca�x1�c

2 , eàn x1 < a� c

Diagrammatikà Ëqoume ta ex†c:

x2

x1

E

0

a� c

↵�c2

a� c↵�c2

b2(x1)

b1(x2)

↵�c3

↵�c3

Sq†ma 2.20: Isorrop–a Nash sto duop∏lio

H isorrop–a thc agoràc epitugqànetai sto shme–o tom†c twn kampul∏n b1(x2)kai b2(x1), dhlad† sto shme–o E. Me aplo‘c upologismo‘c br–skoume Ïti oiposÏthtec isorrop–ac e–nai:

x⇤1 = x⇤2 =

a� c3

2.5.2 Pa–gnia diagwnism∏n

Jewro‘me Ëna s‘nolo d‘o atÏmwn, ta àtoma 1 kai 2, ta opo–a katabàllounprospàjeia gia na apokt†soun Ëna brabe–o ax–ac A. H pijanÏthta kàpoio sug-kekrimËno àtomo na kerd–sei to brabe–o kajor–zetai apÏ thn prospàjeia poukatabàlloun kai ta d‘o àtoma. 'Estw Ïti ei sumbol–zei thn prospàjeia poukatabàllei o i. H pijanÏthta apÏkthshc tou brabe–ou apÏ ton i sumbol–zetai

2.5. EFARMOGüES 33

me pi(e1, e2). H katabol† prospàjeiac ‘youc ei apofËrei kÏstoc gia ton i.To kÏstoc autÏ e–nai Ci(ei). H sunàrthsh apÏdoshc tou i Ëqei th morf†:

ui(e1, e2) = pi(e1, e2)A� Ci(ei), i = 1, 2 (2.16)

Ja upojËsoume gia eukol–a Ïti oi sunart†seic pijanÏthtac kai kÏstouc Ëqounth morf†

pi(e1, e2) =ei

e1 + e2, Ci(ei) = ei (2.17)

Qrhsimopoi∏ntac tic (2.16) kai (2.17), pa–rnoume tic d‘o sunj†kec pr∏thctàxhc oi opo–ec odhgo‘n (metà apÏ aplËc pràxeic) stic sqËseic:

e1A = (e1 + e2)2 (2.18)

e2A = (e1 + e2)2 (2.19)

Ajro–zontac katà mËlh tic d‘o parapànw sqËseic pa–rnoume:

A(e1 + e2) = 2(e1 + e2)2

apÏ thn opo–a prok‘ptei Ïti:

e1 + e2 =A

2(2.20)

TËloc, eisàgontac thn (2.20) stic (2.18)-(2.19) pa–rnoume tic strathgikËcisorop–ac katà Nash:

e⇤1 = e⇤2 =

A

4

2.5.3 SumbatÏthta logismik∏n

Jewro‘me d‘o paragwgo‘c programmàtwn logismiko‘, touc paragwgo‘c 1kai 2. Ta progràmmata autà leitourgo‘n sumplhrwmatikà metax‘ touc. Taqarakthristikà twn programmàtwn pou anapt‘ssoun oi paragwgo– 1 kai 2sunoy–zontai apÏ tic metablhtËc x1 kai x2 ant–stoiqa. O paragwgÏc i epilËgeithn tim† thc metablht† xi me bàsh d‘o krit†ria:

• h tim† thc metablht†c xi ja prËpei na br–sketai Ïso to dunatÏn piokontà se mia ideat† tim† ai

• h tim† thc metablht†c xi ja prËpei na br–sketai Ïso to dunatÏn piokontà sthn ant–stoiqh metablht† tou paragwgo‘ j, xj


Ac do‘me ti ennoo‘me me ta parapànw. Kat’ arqàc, h tim† ai e–nai h tim† touqarakthristiko‘ pou ja epËlege o paragwgÏc i an to logismikÏ tou †tan tomÏno diajËsimo sthn agorà. To logismikÏ tou paragwgo‘ i Ïmwc sunupàrqeime Ëna sumplhrwmatikÏ logismikÏ, to j. H sumplhrwmatikÏthta apaite–, pro-fan∏c, thn anàptuxh programmàtwn me kontinà qarakthristikà. To gegonÏcautÏ exhge– to gia poio lÏgo o paragwgÏc i epijume– h tim† xi na br–sketaiÏso to dunatÏn pio kontà sthn tim† xj . H apÏstash metax‘ xi kai xj metràth sumbatÏthta metax‘ twn d‘o programmàtwn: Ïso mikrÏterh (megal‘terh)h apÏstash tÏso megal‘terh (mikrÏterh) h sumbatÏthta.

Ja upojËsoume Ïti h sunàrthsh apÏdoshc tou paragwgo‘ 1 Ëqei th morf†:

u1(x1, x2) = v � (x1 � a1)2 � (x1 � x2)2 (2.21)

Ïpou v e–nai m–a jetik† paràmetroc. ApÏ th sunàrthsh (2.21) parathro‘meÏti Ïso megal‘terh h apÏstash metax‘ x1 kai a1 tÏso mikrÏterh h apÏdoshtou paragwgo‘ 1. Paromo–wc, Ïso megal‘terh h apÏstash metax‘ x1 kai x2(Ïso megal‘terh h asumbatÏthta twn d‘o programmàtwn) tÏso mikrÏterh, kaipàli, h apÏdosh tou paragwgo‘ 1.

H ant–stoiqh sunàrthsh apÏdoshc gia ton paragwgÏ 2 Ëqei m–a parÏmoiamorf†:

u2(x1, x2) = v � (x2 � a2)2 � (x2 � x1)2 (2.22)

O paragwgÏc 1 epilËgei to x1 o‘twc ∏ste na megistoihje– h sunàrthsh (2.21),kai o 2 epilËgei to x2 o‘twc ∏ste na megistopoihje– h sunàrthsh (2.22). Oisunj†kec pr∏thc tàxhc odhgo‘n stic sunart†seic bËltisthc

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+ ü + ÿ ù ù · 2016. 6. 8. · Kefàlaio1 Eisagwg† 1.1Ti e–nai h jewr–a paign–wn 'Ena qarakthristikÏ pou parathro‘me se pollà oikonomikà,biologikà,koinw-nikà,klp.fainÏmena