Description
Finding the Optimal State-Value Function
Problem
Description
The game DieN is played in the following way:
1. You will be given a die with N sides. You will know the size of N, and can assume that N is a value greater than 1 and less than or equal to 30.
2. You will be given a boolean mask vector isBadSide where the value of 1 indicates the sides of the die that will make you lose. The vector will be of size N, and 1 indexed. (there is no 0 side)
3. You start with 0 dollars.
4. At any time you have the option to roll the die or to quit the game
a. If you decide to roll:
i. And you roll a number not in isBadSide, you receive that many dollars. (eg. if you roll the number 2 and 2 is not a bad side — meaning the second element of the vector is 0 — in isBadSide, then you receive 2 dollars)
Repeat step 4.
ii. And you roll a number in isBadSide, then you lose all money obtained in previous rolls and the game ends.
b. If you decide to quit:
i. You keep all the money gained from previous rolls and the game ends.
Procedure
● For this problem, determine an optimal policy for playing the game DieN for N sides. You will be given N and the array isBadSide which indicates which sides are bad. As you will see, the optimal policy for this game will depend on your current bankroll.
● You can try solving this problem either by creating an MDP of the game (state, action, transition, reward function, and assume a gamma of 1) and then calculating the optimal
1
state-value function or you can plug-in values and solve directly using the Bellman Equations.
● What is the expected amount of dollars for this game if you follow an optimal policy? That is, what is the optimal state-value function for the initial state of the game? Provide answers for the problems you are given on Canvas. Your answer must be correct to 3 decimal places.
Examples
The following examples can be used to verify your calculation is correct.
● Input: N = 21, isBadSide = [1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0], Output: 7.379
● Input: N = 22, isBadSide = [1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0], Output: 6.314 ● Input: N = 6, isBadSide = [1, 1, 1, 0, 0, 0], Output: 2.583
Resources
The concepts explored in this homework are covered by:
● Lectures
○ Lesson 1: Smoov & Curly’s Bogus Journey
● Readings
○ Littman (1996) (chapters 1-2)
● Libraries
○ https://pymdptoolbox.readthedocs.io/en/latest/api/mdptoolbox.html
Submission Details
To complete the assignment calculate answers to the specific problems given and submit the results on Canvas. You have a maximum of 10 attempts.
2
