# Markov Decision Processes

This session is a precursor to reinforcement learning.

• Reading: Russel and Norvig, Chapter 16.1 and Chapter 23.1
• A video demo
• Eirik’s lecture notes from 2022

# Exercises

Recall the requirements to model a problem with the MDP framework:

• Sequential decision problem
• Stochastic Environment
• Fully observable
• with a Markovian Transition model

And the properties:

• A set of states $$s \in S$$
• A set of actions $$a \in A$$
• A transition function $$T(s,a,s')$$
• A reward function $$R(s,a,s')$$
• A start state $$S_0$$

### Part A

Find a problem on CodinGame (preferably one you have worked on), and check if it fulfills the requirements above. If not, can you think of how you can change the problem (e.g. by adding randomness to actions)? If this is not possible, try with another problem.

### Part B

Using the properties of an MDP:

Can you make a graphical representation of the modified problem from Task A? You can chose a subset of the states (and actions if necessary) to reduce the size of the representation. Use either the Dynamic Decision Network from the book (ch 16.1.4), or a simple representation as was done on the slides. (E.g. from here or here )

### Part C

• Does the problem have a finite or infinite horizon?
• If you were to attempt to solve the MDP, could the current horizon pose a problem, why/why not?

### Part D

• Does the problem have a discounted reward?
• If you were to attempt to solve the MDP, what discount factor would make sense to use for the utility function?

## Exploring the Frozen Lake (Task 2)

We will be using the Gymnasium framework to test concepts and ideas from reinforcement learning. You may want to consult the documentation, but you should try playing with it first.

Install with these statements. (I am not sure if you need pygame or not.)

pip install gymnasium
pip install pygame

### Part A

Familiarize yourself with the FrozenLake environment You can import and start the simulation of the Frozen Lake like so:

import gymnasium as gym
env = gym.make('FrozenLake-v1', desc=None, map_name="4x4", is_slippery=True,render_mode="human")
obs, info = env.reset(return_info=True)

You should now hopefully see a render of the environment.

Try out some of these functions and see what they do:

env.action_space.sample()
observation, reward, terminated, truncated, info = env.step(env.action_space.sample())
env.P # The MDP
env.P[s] # Transition matrix of state s
env.P[s][a] # Transitions from state s given action a

Try to make a custom Frozen Lake map

When creating a frozen-lake environment you can add a custom-map with the desc argument, e.g:

fl_map = ["HFFFS", "FHHFF", "FFFFH", "HFFHG"]
env = gym.make('FrozenLake-v1', desc=fl_map, is_slippery=True)

For a custom 5x4 map.

Recall the value/utility-function:

$U(s) = \mathop\max\limits_{a \in A(s)} \sum\limits_{s'}P(s'|s,a)[R(s,a,s') + \gamma U(s')]$

The Q-Function:

$Q(s,a) = \sum\limits_{s'}P(s'|s,a)[R(s,a,s') + \gamma \mathop\max\limits_{a'}Q(s',a')]$

And the function to extract an optimal policy from the Q-Function:

$\pi^*(s) = \mathop{\mathrm{argmax}}\limits_aQ(s,a)$

### Part A

Implement the above functions in Python

### Part B

Given a FrozenLake map, and a list of pre-calculated expected utilities, (e.g.:

utilities = [0.41,0.38,0.35,0.34,0.43,0,0.12,0,0.45,0.48,0.43,0,0,0.59,0.71,1]

for the default FrozenLake 4x4 map)

• Test out the utility-function, and see if it makes sense/work as it should.
• Use the function to extract an optimal policy to move around the map (discount factor can be e.g. 0.99).