Modelování hry Dostihy a Sázky pomocí Markovových řetězců

Abstract

This bachelor thesis deals with homogeneous discrete-time Markov chains on a finite state space and their application to modelling the movement of a player in the board game Betting on Hor- ses. The long-term behaviour of the player on the game board is analysed using stationary and limiting distributions. The model includes key game mechanisms, in particular repeated dice rolls, the Distanc square, and the effect of Chance cards. The resulting stationary distribution is further used, together with the theory of Markov chains with rewards, to analyse the profitability and pa- yback period of individual squares and to determine an optimal investment strategy. The thesis also examines the effect of different game mechanisms on the stationary distribution. First, it in- vestigates how the choice of a k-sided die, for k = 2, . . . , 20, affects the uniformity of the stationary distribution. The basic model is extended to include variants using two dice inspired by the game Monopoly, differing in how repeated rolls are handled, and their impact on the resulting stationary distribution is compared.

Pra ́ce se zaby ́va ́ homogennı ́mi Markovovy ́mi r ̌ete ̌zci s diskre ́tnı ́m c ̌asem a jejich aplikacı ́ na mo- delova ́nı ́ pohybu hra ́c ̌e ve hr ̌e Dostihy a Sa ́zky. Dlouhodobe ́ chova ́nı ́ hra ́c ̌e je analyzova ́no po- mocı ́ staciona ́rnı ́ho a limitnı ́ho rozde ̌lenı ́. Model zahrnuje klı ́c ̌ove ́ hernı ́ mechanismy, zejme ́na opa- kovane ́ hody kostkou, polı ́c ̌ko Distanc a karty Na ́hoda. Staciona ́rnı ́ rozde ̌lenı ́ je da ́le vyuz ̌ito k analy ́ze vy ́nosnosti a na ́vratnosti jednotlivy ́ch polı ́c ̌ek, kde je aplikova ́na teorie Markovovy ́ch r ̌ete ̌zcu ̊ s ocene ̌nı ́m pr ̌echodu ̊, a k urc ̌enı ́ optima ́lnı ́ investic ̌nı ́ strategie. Pra ́ce se rovne ̌z ̌ ve ̌nuje vlivu ru ̊zny ́ch hernı ́ch mechanismu ̊ na staciona ́rnı ́ rozde ̌lenı ́. Nejprve je zkouma ́n vliv volby k-ste ̌nne ́ hracı ́ kostky pro k = 2, . . . , 20 na rovnome ̌rnost staciona ́rnı ́ho rozde ̌lenı ́. Na 'sledne ̌ je za ́kladnı ́ model rozs ̌ı ́r ̌en o varianty se dve ̌ma kostkami inspirovane ́ hrou Monopoly, ktere ́ se lis ̌ı ́zpu ̊sobem zohledne ̌nı ́ opakovany ́ch hodu ̊, a je porovna ́n jejich dopad na vy 'sledne ́ staciona ́rnı ́ rozde ̌lenı ́.