Understanding Game Theory Through Empirical Modelling

Transcription

1 Unerstning Gme Theory Through Empiril Moelling 756 Astrt Gme theory is funmentl onept tht governs our interprettion of the worl roun us. Utilistion s of gme theory rnge from moelling oin flips to nlysing finnil mrkets. With suh wie rnging pplitions, n erly unerstning of its priniples is essentil. The purpose of this pper is to show how, through the use of empiril moelling, moel n e rete tht presents these priniples n reltes them to the rel worl gme. ntroution Empiril moelling is onerne with the moelling of situtions tht n e iretly linke to rel worl senrios. The purpose of this is to rete the mentl link etween the moel n the rel worl. This llows for further thought into the pplition of moels n the priniples tht surroun the stte of rel worl entities. When eveloping moelling using empiril moelling the fous of thought shoul e on how the moel reltes to its rel worl ontext n whether the moel is true representtion of this senrio. This pper isusses the pplition of gme theory to noughts n rosses whih is simple gme with wie rnging gme theory pplitions. Gme Theory. ntroution The onern of gme theory is in the representtion of senrios s iniviul gmes. These gmes n e represente y hoie etween numer of prtiipnts or y proility. To illustrte this point we will initilly look t the si gme of flipping fir oin. This gme simply hs one step where the oin is flippe n n e represente y the following igrm: H.5.5 This igrm (lle Gme Grph or the Extensive Form) shows the initil stte, in this se, the two T sttes tht n follow this, H n T, n the proility of eh stte ourring,.5 n.5 (or 5%). As n e seen from this grph, for the se of the oin flip moel, there re two possile sttes tht oul e entere from the initil stte. This is, however, very trivil exmple s there is urrently no pprent point to this gme. We n therefore inlue pyoffs. y, for exmple tht the gme is eing plye y plyer A n plyer B. We n then sy tht the gme offers the the pyoff of to the plyer tht wins the gme n if they lose. f plyer A ets on H n B ets on T the resulting gme grph woul e rete:.5.5 This now shows tht there is 5% hne of eh plyer hieving or pyoff.. Equiliri Within gmes ontining more tht one plyer stte my exist where neither plyer hs the inentive to hnge their move. This is known s Nsh Equiliri. One of the est known exmples of this is the prisoners ilemm. There re two prisoners, n, who re use of rime, they eh hve the hoie of either remining silent or onfessing. f prisoner hooses to onfess n remins silent, will e let off for o-operting (shown s pyoff of 5) ut will get 5 yers (pyoff of ) n vie-vers for plyer onfessing. f neither onfesses they will reeive lesser sentene (pyoff

2 of 3) for lk of eviene n finlly if oth of them onfess they will oth reeive sentene of yers (pyoff of ). This gme n e represente in the following tle (known s the strtegi form): C C This is n exmple of gme tht hs perfet informtion in tht t oth plyer n plyer s turn eh plyer n tell where in the gme they re through knowing the future moves. Using the strtegi form we n sertin the est responses of eh plyer (shown s highlighte pyoffs) n from these the equiliri (squres where the pyoffs for eh plyer re highlighte). Clulting equiliri for given gme is funmentl onept n n provie plyer with wht oul e the optimum move for gme. This type of equiliri is known s pure equiliri, mening tht they n e hieve through plying single strtegy when the other plyer lso plys their si strtegy. Pure equiliri, however, re not lwys present in gmes leing to the nee to fin mixe equiliri (equiliri tht exists through mixe numer of possile moves) lthough this is out of the sope of this pper. Using solely strtegi form, it is not possile to sertin whether the gme hs perfet informtion. nformtion sets n lso e use in the extensive form to show wht informtion plyer hs t point in the gme. ttes tht exist within single informtion set must hve the sme numer of moves leving them with the sme lels..3 Perfet nformtion n nformtion ets The hoie etween using extensive n strtegi form is more thn simply through preferene, eh hs speifi funtions. Although equiliri n e shown n lulte in oth extensive n strtegi form, informtion sets n whether gme hs perfet informtion n only e shown in extensive form. f gme hs perfet informtion it mens tht plyer n know where they re in the gme through knowing wht their urrent moves re. This gme shows tht oth moves y plyer exist in the sme informtion set n it is therefore not possile for plyer to sertin whih point in the

3 gme they re (n susequently wht the pyoffs woul e for given move. 3 Current Moels 3. ntroution n orer to unerstn how TKEen works to moel stnr gme of Noughts n Crosses (OXO) will e looking t numer of previously rete moels. Through unerstning these it is possile tht will e le to rete new moel, or even iretly moify n existing one, to improve its ury. 3. oxojoy99 The moel egins y funtioning three possile gmes of stnr OXO. Using ommn line interfe the Plyer is ske to input their move into the TKEen winow, the ommn line lso ts to show the Plyer where moves hve lrey een plye. f the Plyer wishes to ply their O in squre, it woul require them to type: s = O into the TKEen winow. One the move hs een plye, the omputer will ommene its turn. The omputer is esigne to look for ny possile winning moves for the Plyer n ple their X in lotion whih will lok ny vitory. f there is no suh move open to the Plyer, the Computer will ple their X in n ritrry ple. At no point in the gme will the Computer tively ttempt to win. The next lyer involves the numeri vlues the omputer will llote to eh squre eing shown on the sreen llowing the user to see how the omputer lultes its move. The finl lyer shows the entirety of the gme n implements turn se system. t lso sets the governing rules of the gme thus preventing the overwriting of previous turns s foun in the OXO moel esrie previously. This moel works to urtely moel the OXO gme s there n e no ening of the rules, however, it is still possile to et the omputer. This retes sope for improvement within the A of the moel. Moelling. Gme Grph Complexity Through evlution of the OXO moel it is pprent tht the retion of gme grph woul not e fesile ue to the numer of possile pths tht single gme my tke. Therefore, it hs een hosen tht simpler moel shll e use to emonstrte gme theory through empiril moelling. The moel tht hs een hosen is the sissors-pper-stone (P) gme. This gme involves eh plyer hoosing either sissors, pper or stone. Eh omintion of hoies results in either plyer eing elre the winner (sissors ets pper, stone ets sissors n pper ets stone) or, if oth plyers hoose the sme option, in rw. Unlike the OXO moel, the P moel presents no single equiliri: This element is prt of the reson tht the moel oesnt represent true OXO gme s the seon plyer woul lso e iming for vitory. There is lso the ft tht Plyer n overwrite turn whih hs lrey een plye, move whih woul not hppen in true gme. P t 3.3 oxogrner999 The oxogrner moel initilly strts y showing the winning lines for n OXO gme in igrmmti form. The oneptul lyer then follows llowing the plyer to ple noughts n rosses onto the or. This llows the user to unerstn how the gme funtions. P t

4 Rther, the gme emonstrtes the importne of perfet informtion n the effet this n hve on the gme.. nterfe Moelling The first stge of moelling the P gme involves the retion n interfe to show the urrent stte of the gme, the extensive gme grph n the strtegi form. The user of the moel must lso hve the ility to intert with the moel vi set of uttons. The extensive gme grph will show ll possile options for the entire gme to llow the user to fully omprehen the hoies ville n their onsequenes. howing the strtegi form to the user n llow them to relise the lk of equiliri n the seemingly stohsti nture of the gme. Through plying the gme y visulising oth extensive n strtegi forms, the plyer n evelop n test ny possile strtegies for plying the gme n therefore unerstn how their tions effet the gme. ffet the omputers move s this is governe y rnom vrile. This provies n urte moel of P lthough it oes prevent some of the lultions ple s prt of gme theoreti pproh. The moel lso llows for the user to view the pth of the gme through the gme grph to further unerstn how the opponent is plying the gme. 5 Conlusion 5. Evlution As previously stte, the moel tht hs een rete urtely moels the P gme. This oes however, limit the moels usefulness in terms of unerstning the pplitions of gme theory..3 Gme Moelling P is gme with imperfet informtion where the move of eh ply nnot e etermine through nlysis of either the extensive or strtegi form. There re two resons for the ehviour; the gme is one shot gme (where eh plyer plys simultneously) n the gme hs no pure equiliri. This presents oth n iel moel to llow si unerstning of gme theory whilst lso proviing suitle sis for further stuy. The gme hs een moelle in suh wy tht the user nnot iretly Approhing this sujet through empiril moelling hs llowe the unerstning of not simply how tool n e rete to show how the P gme works ut lso unerstning how the P gme funtions n how it oul e pprohe. The purpose of this moel ws to llow users new to gme theory to unerstn its priniples n how they n e pplie to rel-worl situtions. To lrge extent this im hs een hieve s the moel oes llow users to visulise how oth the extensive n strtegi form pplie to the rel-worl sttes. For this to e fully hieve, however, the user must pproh the moel from n empiril perspetive. This implies tht the user will use the moel with the unerstning tht the moel is intene to pply to

5 the rel-worl sitution of plying P gme. Approhing this moel from this perspetive llows the user to evelop eep unerstning of the funmentls of gme theory. 5. Future Applitions The OXO moel, oul still provie the sis for eveloping useful moel to i the unerstning of gme theory. The pplition of suh moel woul require high levels of omputtion to ensure the moel is oth fesile n essile. This oul e hieve through simplifie extensive form where the user is simply show the most likely (or possily verge) pyoff for eh possile move. This woul only show the user the next ouple of lyers in the extensive form whilst still proviing enough informtion to mke informe eision. t woul however, e more vntgeous to simply moel mny simple gmes tht oul provie the user with numer of strightforwr, senrios tht over ommonly fe gme theoreti prolems. Further pplitions oul lso e me to the urrent moel y llowing mnipultion oth the extensive n strtegi form. This oul llow the user to tively hoose their next move iretly from these views. Finlly, the moel oul e extene to mke oth the extensive n strtegi form ynmi, llowing multiple gmes to e quikly moelle n even proviing tools for utomti fining of equiliri. This oul lso llow the omputer plyer to mke plys se on gme theoreti onepts further inresing the sophistition of the A n proviing more urte moel. Aknowlegements This work hs relie on the reserh unertken y Mike Joy (99) n imon Grner (999) n oul not hve een omplete without these originl moels.