Title Igre s ponavljanjem Author Ivan Petrunić Mentor Lavoslav Čaklović (mentor)Committee member Lavoslav Čaklović (predsjednik povjerenstva)Committee member Andrej Dujella (član povjerenstva)Committee member Boris Muha (član povjerenstva)Committee member Zvonimir Bujanović (član povjerenstva)Granter University of Zagreb Defense date and country 2015-07-13, Croatia Scientific / art field, NATURAL SCIENCES Abstract Saznanja teorije igara danas se koriste u ekonomiji, sociologiji, psihologiji, političkim znanostima i brojnim drugim područjima. U ovom radu proučavamo ponovljene igre, pomoću kojih možemo modelirati situacije u kojima se ista igra ponavlja više puta. Kod ponovljenih igara, javljaju se fenomeni koji nisu prisutni kod igara u jednom koraku. Promatramo tri varijante modela ponovljenih igara. U prvoj varijanti, igra traje konačan broj koraka T, te svaki igrač nastoji maksimizirati svoj prosječan dobitak. U drugoj varijanti, igra se sastoji od beskonačno mnogo koraka, te svaki igrač želi maksimizirati svoj dugoročan prosječni dobitak. U trećoj varijanti, igra se sastoji od beskonačno mnogo koraka, te svaki igrač želi maksimizirati sadašnju vrijednost svojeg ukupnog dobitka. Kako se broj strategija koje neki igrač ima na raspolaganju jako brzo povećava s brojem koraka igre, nije nam bio cilj odrediti ravnoteže ponovljenih igara, nego okarakterizirati skup isplata koje igrači dobivaju pri tim ravnotežama. Kod igara ponovljenih T puta, pokazali smo da, ako u svakom koraku igramo ravnotežu osnovne igre, dolazimo do jedne ravnoteže ponovljene igre. No, također smo pokazali da ponovljena igra ima i drugih ravnoteža. Vidjeli smo da je pri ravnoteži dobitak svakog igrača veći ili jednak njegovoj minmax vrijednosti, što je najmanji dobitak koji igrač može sigurno ostvariti. Na kraju smo dokazali Folk teorem za T puta ponovljene igre, koji daje karakterizaciju skupa ravnotežnih isplata tih igara. Kod igara ponovljenih beskonačno puta smo primjerom pokazali da imaju veći skup ravnotežnih isplata nego konačne igre čija duljina je puštena u beskonačnost. Pomoću očekivanja limesa prosječnih isplata definirali smo ravnotežu beskonačno puta ponovljenih igara, a zatim smo iskazali Folk teorem za takve igre. Konačno, kod diskontiranih igara smo uzeli u obzir vremensku vrijednost novca, odnosno dobitka u igrama. Uočili smo da dugačke ponovljene igre možemo modelirati pomoću diskontiranih igara kod kojih se diskontni faktor približava 1. Vidjeli smo da kod diskontiranih igara ne vrijedi analogon Folk teorema za beskonačno puta ponovljene igre. Konačno, uz dodatne tehničke uvjete, iskazali smo Folk teorem za diskontirane igre.
Abstract (english) Insights from game theory are used today in economics, sociology, psychology, political science and in many other areas. In this thesis we study repeated games, which could be used to model situations where a game is played repeatedly. In repeated games, we can observe phenomena which are not present in one-stage games. We analyze three variants of the repeated game model. In the first variant, the game lasts a finite number of stages T, and each player wants to maximize his average payoff. In the second variant, the game lasts an infinite number of stages, and each player wants to maximize his long-run average payoff. In the third variant, the game lasts an infinite number of stages, and each player wants to maximize his discounted payoff. Since the number of strategies available to each player increases very quickly as the number of stages increases, we have not tried to find all the equilibria of repeated games. Instead, our aim was to characterize the sets of equilibrium payoffs. For T-stage repeated games, we have shown that playing the base game equilibrium at every stage leads to an equilibrium of the repeated game. But we have also shown that a repeated game has other equilibria as well. We have seen that, at every equilibrium of the repeated game, the payoff to each player is at least equal to his minmax value. Finally, we have proved the Folk Theorem, which characterizes the set of equilibrium payoffs of the T-stage repeated game. For infinitely repeated games, we have shown that they lead to equilibrium payoffs which cannot be obtained as equilibrium payoffs of finite games whose lengths increase to infinity. Using the expectation of the limit of average payoffs, we have defined an equilibrium of infinitely repeated games; we have then stated the Folk Theorem for such games. Finally, for discounted games, we have taken into account the time value of money. We have seen that we can model long repeated games using discounted games where the discount factor approaches 1. We have observed that the analogue of the Folk Theorem for infinitely repeated games does not hold for discounted games. Finally, using additional technical conditions, we have stated the Folk Theorem for discounted games.
Keywords
ponovljene igre
ravnoteža
Folk teorem
diskontirane igre
Keywords (english)
repeated games
equilibrium
Folk theorem
discounted games
Language croatian URN:NBN urn:nbn:hr:217:411293 Study programme Title: Finance and Business Mathematics Type of resource Text File origin Born digital Access conditions Access restricted to students and staff of home institution Terms of use Created on 2019-01-29 13:17:27
