Parrondov paradoks za homeomorfizme

master's thesis

• Title (english): Parrond's paradox for homeomorphisms
• Author: Luka Rabar
• Mentor: Sonja Štimac (mentor)
• Committee member: Sonja Štimac (predsjednik povjerenstva)
• Committee member: Nikola Sandrić (član povjerenstva)
• Committee member: Goran Radunović (član povjerenstva)
• Committee member: Sanja Varošanec (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-02-25, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu opisan je Parrondov paradoks i njegova dinamička inačica. U prvom dijelu konstruiramo Parrondove igre. Ponašanje kapitala uzastopnim igranjem igara modelirano je pomoću slučajnih šetnji na slučajnom okruženju. Poštenost igre definirana je pomoću graničnog ponašanja kapitala. Pokazano je da je moguće konstruirati dvije gubitničke igre za koje je mješovita igra tih igara dobitna. Dodatno, pokazujemo uz koje je uvjete moguće postići Parrondov paradoks. U drugom dijelu definiran je Parrondov pardoks u kontekstu dinamičkih sustava, na način da na privlačne odnosno odbojne fiksne točke gledamo kao na gubitne odnosno dobitne igre. Pokazano je da u slučaju kada je \(\mathcal{C}\) klasa funkcija sa zajedničkom hiperboličnom fiksnom točkom nije moguće postići Parrondov paradoks. U glavnom dijelu rada pokazano je da se dinamički Parrondov paradoks može ostvariti kada je \(\mathcal{C}\) klasa homeomorfnih preslikavanja na \(\mathbb{R}^k\to\mathbb{R}^k\) za \(k\geq2\), dok za \(k=1\)nije moguće ostvariti paradoks.

Abstract (english)

This paper describes Parrondo’s paradox and its dynamic version. In the first part of the paper, we construct Parrondo’s games. The behavior of capital is modeled by successively playing games using random walks in a random environment. We define the fairness of the game by the marginal behavior of capital. We show that it is possible to construct two losing games for which the mixed game is a winning game. In addition, we show under which conditions Parrondo’s paradox is achievable. In the second part of the paper, we define Parrondo’s paradox in the context of dynamic systems by looking at attractive or repellent fixed points as losing or winning games. We also show that when there is a class of functions \(\mathcal{C}\) with a common hyperbolic fixed point, it is not possible to carry out Parrondo’s paradox. In the central part of the paper, we show that the dynamic Parrondo’s paradox is realized when \(\mathcal{C}\) is the class of homeomorphisms on \(\mathbb{R}^k\to\mathbb{R}^k\) for \(k\geq2\). On the contrary, paradox does not appear when \(k=1\).

• Keywords: Parrondove igre; granično ponašanje kapitala; dobitne igre
• Keywords (english): Parrond's games; marginal behavior of capital; winning games
• Language: croatian
• URN:NBN: urn:nbn:hr:217:296463
• Study programme: Title: Mathematical Statistics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2022-03-30 10:31:03