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In a simplified version of a popular online game, a team of Champions move between locations to protect their own Nexus, whilst attempting to destroy the opponent team's Nexus. We know the following about the locations and rules for movement: A team consists of 5 Champions • There are five locations: Home (Nexus), Enemy (Nexus), Safe lane, Mid lane, and Off lane. • Home is adjacent to the Safe lane, Mid lane, and Off lane • Enemy is adjacent to the Safe lane, Mid lane, and Off lane . No other connections exist • All connections are bi-directional (Champions can move in both directions) • For safety reasons, Champions must move to an adjacent location at each time step and cannot stay in the same location • Champions move to adjacent locations with equal probabilities at each time step and cannot stay in the same location. If you wish, you can simplify the location using their respective starting letters. I.e. H,E,S,M,O. Stylistic note: If you have a strong preference, you may use Ancient/Hero in place of Nexus/Champion. | (a) Construct the Markov diagram for this game and show that the associate transition matrix T can be written as the following. T= 0 0 co|wo|tw|100 1|21|2OO 12120 12120 000 00 0 Hint: Be sure to state your ordering of the states and justify the entries in T (b) Show that λ = 1 is an eigenvalue of T (c) Determine an eigenvector associated with λ = 1 and hence find the stable distribution

In a simplified version of a popular online game, a team of Champions move between locations to protect their own Nexus, whilst attempting to destroy the opponent team's Nexus. We know the following about the locations and rules for movement: A team consists of 5 Champions • There are five locations: Home (Nexus), Enemy (Nexus), Safe lane, Mid lane, and Off lane. • Home is adjacent to the Safe lane, Mid lane, and Off lane • Enemy is adjacent to the Safe lane, Mid lane, and Off lane . No other connections exist • All connections are bi-directional (Champions can move in both directions) • For safety reasons, Champions must move to an adjacent location at each time step and cannot stay in the same location • Champions move to adjacent locations with equal probabilities at each time step and cannot stay in the same location. If you wish, you can simplify the location using their respective starting letters. I.e. H,E,S,M,O. Stylistic note: If you have a strong preference, you may use Ancient/Hero in place of Nexus/Champion. | (a) Construct the Markov diagram for this game and show that the associate transition matrix T can be written as the following. T= 0 0 co|wo|tw|100 1|21|2OO 12120 12120 000 00 0 Hint: Be sure to state your ordering of the states and justify the entries in T (b) Show that λ = 1 is an eigenvalue of T (c) Determine an eigenvector associated with λ = 1 and hence find the stable distribution

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 20T
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Answer (c) only. Determine the eigenvector and the stable distribution.

In a simplified version of a popular online game, a team of Champions move between locations to protect their own Nexus, whilst attempting to destroy the opponent team's Nexus. We know the following about the locations and rules for movement: • A team consists of 5 Champions • There are five locations: Home (Nexus),Enemy (Nexus), Safe lane, Mid lane, and Off lane. • Home is adjacent to the Safe lane, Mid lane, and Off lane • Enemy is adjacent to the Safe lane, Mid lane, and Off lane • No other connections exist • All connections are bi-directional (Champions can move in both directions) • For safety reasons, Champions must move to an adjacent location at each time step and cannot stay in the same location • Champions move to adjacent locations with equal probabilities at each time step and cannot stay in the same location. If you wish, you can simplify the location using their respective starting letters. I.e. H,E,S,M,O. Stylistic note: If you have a strong preference, you may use Ancient/Hero in place of Nexus/Champion. | (a) Construct the Markov diagram for this game and show that the associate transition matrix T can be written as the following. 1 1 T = 0 0 Hint: Be sure to state your ordering of the states and justify the entries in T (b) Show that )= 1 is an eigenvalue of T (c) Determine an eigenvector associated with A = 1 and hence find the stable distribution HINHIN O O0 HINHIN O O o O O H/3 313 O H3-10 3
Transcribed Image Text:In a simplified version of a popular online game, a team of Champions move between locations to protect their own Nexus, whilst attempting to destroy the opponent team's Nexus. We know the following about the locations and rules for movement: • A team consists of 5 Champions • There are five locations: Home (Nexus),Enemy (Nexus), Safe lane, Mid lane, and Off lane. • Home is adjacent to the Safe lane, Mid lane, and Off lane • Enemy is adjacent to the Safe lane, Mid lane, and Off lane • No other connections exist • All connections are bi-directional (Champions can move in both directions) • For safety reasons, Champions must move to an adjacent location at each time step and cannot stay in the same location • Champions move to adjacent locations with equal probabilities at each time step and cannot stay in the same location. If you wish, you can simplify the location using their respective starting letters. I.e. H,E,S,M,O. Stylistic note: If you have a strong preference, you may use Ancient/Hero in place of Nexus/Champion. | (a) Construct the Markov diagram for this game and show that the associate transition matrix T can be written as the following. 1 1 T = 0 0 Hint: Be sure to state your ordering of the states and justify the entries in T (b) Show that )= 1 is an eigenvalue of T (c) Determine an eigenvector associated with A = 1 and hence find the stable distribution HINHIN O O0 HINHIN O O o O O H/3 313 O H3-10 3
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