Analiza strategija i pripadnih vjerojatnosti u igri "Rizik"

master's thesis

• Title (english): Analysis of strategies and associated probabilities in the game "Risk"
• Author: Alen Lančić
• Mentor: Miljenko Huzak (mentor)
• Committee member: Miljenko Huzak (predsjednik povjerenstva)
• Committee member: Filip Najman (član povjerenstva)
• Committee member: Igor Ciganović (član povjerenstva)
• Committee member: Eduard Marušić-Paloka (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-03-03, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu smo analizirali neke elemente igre ”Rizik”. Izračunali smo vjerojatnosti ishoda jedne iteracije napada i obrane. Zatim smo igru promatrali kao Markovljev lanac s diskretnim vremenom, našli optimalne strategije obrane za svako stanje i vidjeli kako se skup stanja s obzirom na optimalne strategije obrane particionira na šest particija. Na kraju smo izračunali apsorbirajuće vjerojatnosti lanca počevši iz svakog neapsorbirajućeg stanja te to iskoristili da izračunamo funkcije gustoće vjerojatnosti konačnog ishoda napada teritorija i vjerojatnost njegovog osvajanja počevši iz različitih stanja Markovljevog lanca.

Abstract (english)

In this work, some elements of the game ”Risk” were analyzed. Outcome probabilities of a single iteration of attack and defense were obtained. Then, the game was considered as a Markov chain with discrete time, the optimal defense strategies for each state were found and the way in which the state set partitions into six partitions with regard to the optimal defense strategies was determined. In the end, absorbing probabilities of the chain from each source state were obtained, which were used for obtaining the probability mass functions of the final outcome of an attack of a territory and the probability of its capture starting from various states of the Markov chain.

• Keywords: Markovljev lanac; optimalne strategije obrane; apsorbirajuće vjerojatnosti lanca
• Keywords (english): Markov chain; optimal defense strategies; absorbing chain probabilities
• Language: croatian
• URN:NBN: urn:nbn:hr:217:887318
• Study programme: Title: Pure Mathematics; 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: 2022-03-28 11:50:00