# Pythonic Markov Decision Process (MDP)

I’ve been studying a subject called *probabilistic methods for decisions*.

In this course there are a lot of interesting topics like Bayesian Networks (BN), inference and querying in BNs, probabilistic reasoning over time etc. and, in the book (Artificial Intelligence: A Modern Approach – Russell, Norvig) there are a lot of interesting pseudocode ;) to implement.

The simplest algorithm to implement is -IMHO- the *Value-Iteration algorithm* and my goal is to reproduce the graph that shows the evolution of the utilities shown in Figure 17.5.

### What is a MDP

Wikipedia has an article about MDP.

Simplifying, we need three things:

- an initial state:
*S*_{0} - a transition model:
*T(s, a, s’)* - a reward function:
*R(s)*

In this example I assume the environment is *fully observable* and I consider the horizon to be *infinite*.

The **scope** is to find an *optimal policy*, namely a solution that specifies what we should do for *any* state other than the goal.

#### tools & libraries

I used

- python 2.5
- matplotlib 0.90.1
- scipy 0.5.2

#### code for an MDP interface

In my toy problem I need (but is not mandatory) an interface to determine the MDP:

class MarkovDecisionProcess: def transition(self, from_state, action, to_state): raise NotImplementedError def initial_state(self): raise NotImplementedError def reward(self, state): raise NotImplementedError def discount(self, state): raise NotImplementedError

#### code for the modified dict class

I need also a slightly modified dict class, up ahead you will understand why:

class SumDict(dict): def __setitem__(self, key, value): if self.has_key(key): value += self.get(key) dict.__setitem__(self, key, value)

#### code for the Toy MDP

After this I can define the world for the agent, it is identical to the one in the book (Section 17.1):

import numpy as np class ToyMDP(MarkovDecisionProcess): def __init__(self): #this is the same world of the book self.world = np.array([ [-0.04, -0.04, -0.04, 1], [-0.04, None, -0.04, -1], [-0.04, -0.04, -0.04, -0.04], ]) self.initial_state = (0, 0) self.finals = [(0,3), (1,3)] self.actions = ('l', 'r', 'u', 'd') def __iter__(self): class Iterator: def __init__(self, iterator, finals): self.iterator = iterator self.finals = finals def next(self): while True: coords = self.iterator.coords val = self.iterator.next() if val and coords not in self.finals: break return coords, val return Iterator(self.world.flat, self.finals) def _move(self, state, action): """calculates the next state given an action""" shape = self.world.shape next = list(state) if state in self.finals: pass elif action == 'l' and \ (state[1] > 0 and self.world[state[0]][state[1]-1] != None): next[1] -= 1 #... other three elif for up, down and right return tuple(next) def successors(self, state, action): """this function returns a dict with all the successors of a state and the relative probability given an action, for example if you are in (1,0) and you want to go right the function returns {(1,0):0.8, (0,0):0.1, (2,0):0.1}""" #I'm using SumDict because if two or more "_move" #return the same state (and the state is the key of the dict) #we need to sum out the values not overwrite the old value d = SumDict() if action == 'l': d[self._move(state, 'l')] = 0.8 d[self._move(state, 'u')] = 0.1 d[self._move(state, 'd')] = 0.1 #... other three elif for up, down and right return d # from_state, to_state: a tuple # action: l, r, u, d (left, right, up, down) def transition(self, from_state, action, to_state): return self.successors(from_state, action)[to_state] def initial_state(self): return self.initial_state def reward(self, state): return self.world[state[0]][state[1]] def discount(self): return 1

#### code for the *Value-Iteration algorithm*

Now we are ready to compute the utility of every state. The function is a little bit more complex if compared with another one that I have *just found* (really, just 5 minutes ago) :/

There is a little difference between my implementation and the one in the book.

The book uses ; the difference is in the summation: the book formula iterates over *all* the states, including the unreachable ones, while, using the *successor*, I iterate *only* over the reachable states.

I think this is correct: from a state *s* I can go only to *successors(s)* and I can iterate over them instead of over *every* state in the world.

def value_iteration(mdp, e, hook=lambda u, mdp: 1):

_u = np.zeros(mdp.world.shape)

for final in mdp.finals:

_u[final[0]][final[1]] = mdp.world.item(final)

while True:

u = _u.copy()

hook(u, mdp)

d = 0

for state, reward in mdp:

summ = max([sum([prob * u.item(next_state) for next_state,prob in

mdp.successors(state, action).items()]) for action in

mdp.actions])

_u[state[0]][state[1]] = reward + mdp.discount() * summ

diff = fabs(_u.item(state) – u.item(state))

if diff > d:

d = diff

if d <= e*(1-mdp.discount())/mdp.discount(): break
return u
[/sourcecode]

#### results

The output of the *utility of the states matrix* is equal to the one in the book:

vrde@pandora:~/other/blog/mdp$ python mdp.py [[ 0.81155822 0.86780822 0.91780822 1. ] [ 0.76155822 0. 0.66027397 -1. ] [ 0.70530822 0.65530822 0.61141553 0.38792491]]

and also the graph:

That’s all folks, if you want to try it at home you can download here: mdp.tar.bz2 (or, if you want to use wget, here).

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