Teoria e Lojes! Ekuilibri Nash!........

[Dita]

Ideja per te nisur kete teme me erdhi nga filmi fitues i cmimit Oskar te ketij viti "A beautiful mind" dhe nga bukuria e mendjes se protagonistit te tij, [color=}blue]John Nash[/color].


Ne teorine e finances, ekuilibri Nash eshte nje ingredient shume i rendesishem, ashtu sikurse eshte edhe ne teorine e vendimmarrjes, apo ne ate te negociatave.


Ju ftoj te diskutojme mbi Game Theory dhe mbi ekuilibrin Nash, zbatimet e tyre ne praktike, pikat ku juve ne studim ju kane dale veshtiresi, apo vendet se ku keni arritur t'i gjeni zbatim ne praktike.


Per nder te nobelistit dhe meqe filmin e kam akoma te fresket ne kujtese, po e nis me nje biografi(autobiografi) te John Nash.


Pershendetje!
Dita

[Dita]

John F. Nash, Jr. – Autobiography



My beginning as a legally recognized individual occurred on June 13, 1928 in Bluefield, West Virginia, in the Bluefield Sanitarium, a hospital that no longer exists. Of course I can't consciously remember anything from the first two or three years of my life after birth. (And, also, one suspects, psychologically, that the earliest memories have become "memories of memories" and are comparable to traditional folk tales passed on by tellers and listeners from generation to generation.) But facts are available when direct memory fails for many circumstances.

My father, for whom I was named, was an electrical engineer and had come to Bluefield to work for the electrical utility company there which was and is the Appalachian Electric Power Company. He was a veteran of WW1 and had served in France as a lieutenant in the supply services and consequently had not been in actual front lines combat in the war. He was originally from Texas and had obtained his B. S. degree in electrical engineering from Texas Agricultural and Mechanical (Texas A. and M.).

My mother, originally Margaret Virginia Martin, but called Virginia, was herself also born in Bluefield. She had studied at West Virginia University and was a school teacher before her marriage, teaching English and sometimes Latin. But my mother's later life was considerably affected by a partial loss of hearing resulting from a scarlet fever infection that came at the time when she was a student at WVU.

Her parents had come as a couple to Bluefield from their original homes in western North Carolina. Her father, Dr. James Everett Martin, had prepared as a physician at the University of Maryland in Baltimore and came to Bluefield, which was then expanding rapidly in population, to start up his practice. But in his later years Dr. Martin became more of a real estate investor and left actual medical practice. I never saw my grandfather because he had died before I was born but I have good memories of my grandmother and of how she could play the piano at the old house which was located rather centrally in Bluefield.

A sister, Martha, was born about two and a half years later than me on November 16, 1930.

I went to the standard schools in Bluefield but also to a kindergarten before starting in the elementary school level. And my parents provided an encyclopedia, Compton's Pictured Encyclopedia, that I learned a lot from by reading it as a child. And also there were other books available from either our house or the house of the grandparents that were of educational value.

Bluefield, a small city in a comparatively remote geographical location in the Appalachians, was not a community of scholars or of high technology. It was a center of businessmen, lawyers, etc. that owed its existence to the railroad and the rich nearby coal fields of West Virginia and western Virginia. So, from the intellectual viewpoint, it offered the sort of challenge that one had to learn from the world's knowledge rather than from the knowledge of the immediate community.

By the time I was a student in high school I was reading the classic "Men of Mathematics" by E. T. Bell and I remember succeeding in proving the classic Fermat theorem about an integer multiplied by itself p times where p is a prime.

I also did electrical and chemistry experiments at that time. At first, when asked in school to prepare an essay about my career, I prepared one about a career as an electrical engineer like my father. Later, when I actually entered Carnegie Tech. in Pittsburgh I entered as a student with the major of chemical engineering.

Regarding the circumstances of my studies at Carnegie (now Carnegie Mellon U.), I was lucky to be there on a full scholarship, called the George Westinghouse Scholarship. But after one semester as a chem. eng. student I reacted negatively to the regimentation of courses such as mechanical drawing and shifted to chemistry instead. But again, after continuing in chemistry for a while I encountered difficulties with quantitative analysis where it was not a matter of how well one could think and understand or learn facts but of how well one could handle a pipette and perform a titration in the laboratory. Also the mathematics faculty were encouraging me to shift into mathematics as my major and explaining to me that it was not almost impossible to make a good career in America as a mathematician. So I shifted again and became officially a student of mathematics. And in the end I had learned and progressed so much in mathematics that they gave me an M. S. in addition to my B. S. when I graduated.

I should mention that during my last year in the Bluefield schools that my parents had arranged for me to take supplementary math. courses at Bluefield College, which was then a 2-year institution operated by Southern Baptists. I didn't get official advanced standing at Carnegie because of my extra studies but I had advanced knowledge and ability and didn't need to learn much from the first math. courses at Carnegie.

When I graduated I remember that I had been offered fellowships to enter as a graduate student at either Harvard or Princeton. But the Princeton fellowship was somewhat more generous since I had not actually won the Putnam competition and also Princeton seemed more interested in getting me to come there. Prof. A. W. Tucker wrote a letter to me encouraging me to come to Princeton and from the family point of view it seemed attractive that geographically Princeton was much nearer to Bluefield. Thus Princeton became the choice for my graduate study location.

But while I was still at Carnegie I took one elective course in "International Economics" and as a result of that exposure to economic ideas and problems, arrived at the idea that led to the paper "The Bargaining Problem" which was later published in Econometrical. And it was this idea which in turn, when I was a graduate student at Princeton, led to my interest in the game theory studies there which had been stimulated by the work of von Neumann and Morgenstern.

As a graduate student I studied mathematics fairly broadly and I was fortunate enough, besides developing the idea which led to "Non-Cooperative Games", also to make a nice discovery relating to manifolds and real algebraic varieties. So I was prepared actually for the possibility that the game theory work would not be regarded as acceptable as a thesis in the mathematics department and then that I could realize the objective of a Ph. D. thesis with the other results.

But in the event the game theory ideas, which deviated somewhat from the "line" (as if of "political party lines") of von Neumann and Morgenstern's book, were accepted as a thesis for a mathematics Ph. D. and it was later, while I was an instructor at M.I.T., that I wrote up Real Algebraic Manifolds and sent it in for publication.

I went to M.I.T. in the summer of 1951 as a "C.L.E. Moore Instructor". I had been an instructor at Princeton for one year after obtaining my degree in 1950. It seemed desirable more for personal and social reasons than academic ones to accept the higher-paying instructorship at M.I.T.

I was on the mathematics faculty at M.I.T. from 1951 through until I resigned in the spring of 1959. During academic 1956 - 1957 I had an Alfred P. Sloan grant and chose to spend the year as a (temporary) member of the Institute for Advanced Study in Princeton.

During this period of time I managed to solve a classical unsolved problem relating to differential geometry which was also of some interest in relation to the geometric questions arising in general relativity. This was the problem to prove the isometric embeddability of abstract Riemannian manifolds in flat (or "Euclidean") spaces. But this problem, although classical, was not much talked about as an outstanding problem. It was not like, for example, the 4-color conjecture.

So as it happened, as soon as I heard in conversation at M.I.T. about the question of the embeddability being open I began to study it. The first break led to a curious result about the embeddability being realizable in surprisingly low-dimensional ambient spaces provided that one would accept that the embedding would have only limited smoothness. And later, with "heavy analysis", the problem was solved in terms of embeddings with a more proper degree of smoothness.

While I was on my "Sloan sabbatical" at the IAS in Princeton I studied another problem involving partial differential equations which I had learned of as a problem that was unsolved beyond the case of 2 dimensions. Here, although I did succeed in solving the problem, I ran into some bad luck since, without my being sufficiently informed on what other people were doing in the area, it happened that I was working in parallel with Ennio de Giorgi of Pisa, Italy. And de Giorgi was first actually to achieve the ascent of the summit (of the figuratively described problem) at least for the particularly interesting case of "elliptic equations".

It seems conceivable that if either de Giorgi or Nash had failed in the attack on this problem (of a priori estimates of Holder continuity) then that the lone climber reaching the peak would have been recognized with mathematics' Fields medal (which has traditionally been restricted to persons less than 40 years old).

Now I must arrive at the time of my change from scientific rationality of thinking into the delusional thinking characteristic of persons who are psychiatrically diagnosed as "schizophrenic" or "paranoid schizophrenic". But I will not really attempt to describe this long period of time but rather avoid embarrassment by simply omitting to give the details of truly personal type.

While I was on the academic sabbatical of 1956 - 1957 I also entered into marriage. Alicia had graduated as a physics major from M.I.T. where we had met and she had a job in the New York City area in 1956 - 1957. She had been born in El Salvador but came at an early age to the U.S. and she and her parents had long been U.S. citizens, her father being an M. D. and ultimately employed at a hospital operated by the federal government in Maryland.

The mental disturbances originated in the early months of 1959 at a time when Alicia happened to be pregnant. And as a consequence I resigned my position as a faculty member at M.I.T. and, ultimately, after spending 50 days under "observation" at the McLean Hospital, travelled to Europe and attempted to gain status there as a refugee.

I later spent times of the order of five to eight months in hospitals in New Jersey, always on an involuntary basis and always attempting a legal argument for release.

And it did happen that when I had been long enough hospitalized that I would finally renounce my delusional hypotheses and revert to thinking of myself as a human of more conventional circumstances and return to mathematical research. In these interludes of, as it were, enforced rationality, I did succeed in doing some respectable mathematical research. Thus there came about the research for "Le Probleme de Cauchy pour les E'quations Differentielles d'un Fluide Generale"; the idea that Prof. Hironaka called "the Nash blowing-up transformation"; and those of "Arc Structure of Singularities" and "Analyticity of Solutions of Implicit Function Problems with Analytic Data".

But after my return to the dream-like delusional hypotheses in the later 60's I became a person of delusionally influenced thinking but of relatively moderate behavior and thus tended to avoid hospitalization and the direct attention of psychiatrists.

Thus further time passed. Then gradually I began to intellectually reject some of the delusionally influenced lines of thinking which had been characteristic of my orientation. This began, most recognizably, with the rejection of politically-oriented thinking as essentially a hopeless waste of intellectual effort.

So at the present time I seem to be thinking rationally again in the style that is characteristic of scientists. However this is not entirely a matter of joy as if someone returned from physical disability to good physical health. One aspect of this is that rationality of thought imposes a limit on a person's concept of his relation to the cosmos. For example, a non-Zoroastrian could think of Zarathustra as simply a madman who led millions of naive followers to adopt a cult of ritual fire worship. But without his "madness" Zarathustra would necessarily have been only another of the millions or billions of human individuals who have lived and then been forgotten.

Statistically, it would seem improbable that any mathematician or scientist, at the age of 66, would be able through continued research efforts, to add much to his or her previous achievements. However I am still making the effort and it is conceivable that with the gap period of about 25 years of partially deluded thinking providing a sort of vacation my situation may be atypical. Thus I have hopes of being able to achieve something of value through my current studies or with any new ideas that come in the future.

From Les Prix Nobel 1994.

[Dita]

Disa shembuj nga historia ku kjo teori ka gjetur zbatim, ndonese ende nuk ishte ngritur si e tille.


Ne vepren e tij Republika, Platoni ne nje pike ndan shqetesimin e Sokratit per situaten qe pershkruhet ne vazhdim.
Nje ushtar gjendet ne front dhe pret per luftetaret e tjere per t’iu kunderpergjigjur sulmit te armikut. Ai mund te mendoje si me poshte: nese mbrojtja do te jete e suksesshme atehere contributi i tij me probabilitet te larte nuk do te jete aq domethenes. Por nese kjo ndodh, pra ai qendron per te luftuar, atehere ai prballet me rrezikun qe te vritet ose te plagoset.
Nga ana tjeter, nese armiku fiton betejen, atehere mundesia qe ai te vritet ose te mbetet i plagosur eshte akoma me e larte. Duke u bazar ne kete arsyetim duket se alternativa me e mire per ushtarin eshte te largohet nga fusha e betejes pa e vrare mendjen se cila nga palet ka me shume gjasa ta fitoje betejen.


Nese te gjithe ushtaret arsyetojne ne kete menyre (ne fakt kjo eshte menyra ne te cilen duhet te llogjikojne perderisa te gjithe gjenden ne te njejten situate), atehere rezultati do te jete I sigurte, beteja do te humbet. Ne kete pike erdhem ne duke marre rolin e analisteve, por ne kete pike mund te vijne fare mire edhe ushtaret vete. A ka mundesi atehere qe ata ta kuptojne se cfare do te shkaktojne dhe te ndryshojne mendim (pra te qendrojne)?
Ne fakt ndodh e kunderta: sa me e madhe te jete firka se beteja do te humbet aq me e madhe do te jete shtysa qe ata te ikin tutje. Dhe sa me e madhe te jete bindja se beteja do te fitohet, pa nje kontribut te konsiderueshem nga ana e tyre, aq me pak arsye do te kene ata qe te luftojne. Nese cdo ushtar mendn ne kete menyre, atehere shpejt ata do te arsyetojne nje gjendje paniku dhe beteja do te jete e humbur qe pa filluar.



Nje tjeter shembull.

Konkuistatori spanjoll Cortez gjate zbarkimit ne Meksike me nje trupe te vogel ushtarake.
Ai kishte arsye te besonte se trupa e tij e frikesuar, duke u munduar ta vinte veten ne rolin e vendasve, do te arrinte ne perfundimin se duhej te largohej perpara se t’u digjnin anijet qe qene mjeti I vetem me te cilin mund te shpetonin. Duke e parashikuar kete mendim, Kortez dha urdher te digjeshin anijet. Largimi fizik u be i pamundur dhe ushtaret spanjolle u detyruan te luftonin me sa fuqi qe kishin.


Llogjikimi i Cortez shkoi edhe me tutje. Ai u kujdes qe t’i digjte anijet ne menyre shume te dukshme, ne menyre qe te ishte i sigurte se azteket e pane c’kishte bere. Llogjika e tij ndoqi kete linje:
Cdo komandant qe ne menyre te vullnetshme eliminon rrugen e vetme te shpetimit nese do te perballej me humbjen ne beteje, duhet te kete arsye shume te mira per kete optimizem. S’eshte pune me mend te sulmosh nje kundershtar qe ka nje arsye te mire per te qene i bindur se nuk do te humbase. Azteket si pasoje u terhoqen nga fusha e betejes dhe Cortez fitoi pa derdhur gjak.


Llogjika mbas Platonit dhe Cortez-it

Ushtaret nuk e kane si qellim paresor dhe si interes te vetin te largohen nga fusha e betejes. Ata gjejne arsye per te ikur kur fillojne te perfshijne ne logjikimin e tyre dhe ate se cfare do te ishte me kuptim per te tjeret qe te benin. Dhe kjo perfshirje ne logjikim eshte e dukshme edhe per te tjeret. Edhe nje luftetar trim do te largohej, por pa dashur ai me veprimin e vet po nxit dhe pjesen tjeter te largohet. Si perfundim mund te imagjinohet nje situate ne te cilen nje ushtri e tere e perbere prej luftetaresh trima zhduket nga syte kembet perballe armiqve. Nese luftetare jane trima atehere mund te kuptohet fare mire qe kjo qe ndodhi nuk ka qene qellimi i tyre. Secili prej tyre vec e vec do te kish preferuar te qendronte dhe te luftonte.
Dukuria qe paraqitet ketu eshte nje shembull I nderveprimit te proceseve vedimmarrese raciaonale te shume individeve. – nje process per cdo luftetar – rezultati qe arrihet nuk e kish pasur asnjeri per qellim.


Shumica e ushtrive mundohen ta shmangin kete lloj situate duke vepruar sic beri Cortez.
Meqe ata nuk mund ta bejne te pamundur terheqjen FIZIKISHT, ata e bejne EKONOMIKISHT; ata vrasin dezertoret. Ne kete menyre qendrimi dhe luftimi perbejne per luftetaret alternativen me te mire, sepse kostoja e alternatives tjeter (largimit) eshte te pakten po aq e larte sa ajo e qendrimit.


Nje tjeter shembull e gjejme tek Shekspiri.

Tek Henri V, Shekspiri vendos fitimtarin Agincourt qe te shpjegoje vendimin e marre per te masakruar te burgosurit franceze nen vezhgimin e kundershtareve. Masakrimi ndodhi si me poshte: Trupat angleze ndoqen zhvillimin e masakres, njekohesisht ato pane se trupat franceze e pane nje gje te tille, pra secili prej tyre tashme e dinte se c’do ta gjente ne rast e se e humbisnin luften. Ne menyre metaforike kjo korrespondonte me djegjen e anijeve nga Cortez.


Si rrjedhoje te burgosurit franceze vdiqen dhe Henry u dergoi nje sinjal trupave te veta, duke ndikuar ne ndryshimin e nxtijes se tyre te brendshme.


Nje tjeter shembull

Vepra “Leviathan” e filozofit Thomas Hobbe vleresohet si nje veper baze per filozofine politike moderne. Eshte teksti ku gjeti fillesen teresia e analizave per funksionin dhe justifikimin dhe kufizimet e lirise personale.

Thelbi i arsyetimit te Hobbe eshte si me poshte:

Situata me e mire per te gjithe njerezit eshte ajo ku secili eshte i lire te beje ate cfare deshiron.

Njerezit e lire do te ishin vetem atehere te gatshem te bashkepunonin me njeri-tjetrin kur do t'u duhej te realizonin dicka qe secili me vete do ta kishte te pamundur ta arrinte.

P.sh. Dy persona bien dakord per te ndihmuar ne ngritjen e shtepive te njeri-tjetrit. I pari ve ne dispozicion krahun e punes gratis me kushtin qe edhe tjetri te beje te njejten gje pasi te jete ndertuar shtepia e tij. Cfare mund te beje i dyti pas perfundimit te shtepise? Ai mund te mohoje premtimin qe i ka dhene te parit. Por ai njekohesisht arsyeton se duke e lene te parin pa shtepi ky i fundit do te kete nje arsye te mire per t’ia marre shtepine atij vete. Kjo do ta beje te dytin qe te gjendet ne nje gjendje te vazhdueshme frike dhe do ta beje te shpenzoje kohe dhe para duke u perpjekur te mbrohet nga i pari. Keto kosto ky person mund t’i minimizoje duke e vrare te parin. Por te gjithe kete rrjedhe llogjikimi te te dytit e ndjek edhe i pari, keshtu qe ai e di ne avance se cfare do te ndodhe me te dhe se cfare planesh ka i dyti.
I dyti arrin ta kuptoje se i pari e ka kuptuar vijen e tij te logjikimit, keshtu qe frika e tij fillestare ndaj te parit nuk qe e ekzagjeruar; gjithashtu edhe e te parit ndaj te dytit. Nuk eshte e nevojshme qe te jesh immoral qe te arrish te logjikosh ne kete menyre. Mjafton te mendosh se ekziston nje probabilitet I mire qe premtimet e dhena ne fund te marreveshjes mund edhe te mos mbahen. Ne momentin kur ne mendjen tone lind dyshimi se papritur mund te gjendemi ne befasi perballe faktit qe tjetri na ka goditur para se ta godasim ne atehere zhduket baza per bashkepunim. Ne kete menyre duke u lene me mjetet qe kane vete ne dispozicion agjentet racionale nuk do te nxirrnin asnjehere perfitim nga bashkepunimi dhe do te jetonin ne nje gjendje te “nje lufte te te gjitheve kundra te gjitheve”.

Ne keto kushte, e gjithe jeta njerezore do te ishte e vetmuar, e varfer, e shemtuar dhe e shkurter.

Si zgjidhje per kete problem Hobbe propozon tiranine. Njerezit mund te angazhojne nje Agjent – Qeverine – detyra e se ciles do te jete te ndeshkoje te gjithe ata qe do te mund te shkelin premtimet.
Per aq kohe sa shkelja e premtimeve denohet ne menyre te mjaftueshme – Hobbe mendonte prerjen e kokes si denimin e pershtatshem – njerezit nuk do te jene te interesuar te shkelin premtimet, sepse kostoja e shkeljes do t’i tejkalonte ato te mbajtjes se premtimit.
E njejta llogjike ndiqet edhe me vendimin e qellimit te dezertoreve. Nese te gjithe njerezit e dine se shumica e ketyre nxitjeve vlejne per shumicen e te tjereve, kooperimi mes tyre do te mund te behet i mundshem dhe per me tej do te kthehet ne nje norme.
Lufta e te gjitheve kundra te gjitheve do te kthehet ne nje paqe te pergjithshme.

Per me tej Hobbe arrin ne nje konkluzion te rendesishem duke argumentuar se jo vetem qe duhet nje qeveri me te drejten dhe forcen per te nxitur kooperimin, por per me teper kjo qeveri duhet te jete e pandare dhe e ndertuar ne menyre te tille qe vullneti i nje drejtuesi te vetem te imponoje detyrim te pergjithshem per te tjeret.

Teoria filozofike e Hobbe bazohet ne idene se autoriteti dhe praktikat detyruese te qeverise gjejne justifikim ne nevojen e njerezve per te mbrojtur veten nga ato situata te cilat teoristet e lojes I quajne “dilemma sociale”.
Hobbe nuk argumenton nese tirania eshte dicka e deshirueshme ne vetvete. Struktura e argumentit te tij eshte se llogjika e nderveprimit strategjik le vetem dy alternativa hapur: tiranine dhe anarkine. Agjentet racionale zgjedhin si pasoje tiranine si me e vogla nder dy te keqija.


Pika e perbashket ne llogjikimet e mesiperme:

Ne secilin prej rasteve pika thelbesore mbi te cilen bazohet llogjikimi dhe zhvillimi i situates eshte “teresia e parashikimeve dhe e reaksioneve te mundshme te agjenteve te tjere perballe strategjive vetiake".


Ne kete pike te arsyetimit futet ndryshimi mes veprimit parametrik dhe atij joparametrik.

Parametrik arsyetimi eshte kur ndikohet nga faktore rezultatet e te cileve jane te matshme dhe pasojat e te cileve jane te parashikueshme. Keshtu p.sh. do te behej fjale per perzgjedhjen midis disa aksioneve te cilat kane per objekt sende. Veprimi parametrik do te kryhej ne nje bote pasive.

Joparametrik arsyetimi eshte kur ndikohet nga faktore rezultatet e te cilave jane te paparashikueshme.
Veprimet parametrike zhvillohen ne nje bote e cila kerkon te veproje para se te kryhen veprimet e lartepermendura.
Veprime te tilla jane karakteristike per nje bote qe ka per baze logjikimin, mes te tjerash ne boten e nderveprimit njerezor.



Per te paraqitur veprimin joparametrik si shembull klasik sillet “Dilema e te Burgosurit”


(vijon)

[crackeri]

Pershendetje te gjitheve a kish ndihmue ndokush te me ndihmoje te me jep nje shembull nga perditshmeria perkitazi me teorin e lojrave.

Ju lutem sa me shpejt te me ndihmoni se po me nevojitet shume.