Markov Decision Processes

  • Markov Decisions Problems; Bellman’s Equation; Two examples


A Markov Decision Process is a Dynamic Program where the state evolves in a random/Markovian way.

As in the post on Dynamic Programming, we consider discrete times t=0,1,...,T, states x\in\mathcal{X}, actions a\in\mathcal{A}_t and rewards r_t(a_t,x_t). However, the plant equation and definition of a policy are slightly different.

Def 1 [Plant Equation] The state evolves according to functions f_t: \mathcal{X}\times \mathcal{A}_t \times [0,1] \rightarrow \mathcal{X}. Here

where (U_t)_{t\geq 0} are IIDRVs uniform on [0,1]. This is called the Plant Equation. As noted in the equivalence above, we will often suppress dependence on U_t. Also, we will use the notation

Def 2 [Policy] A policy \pi choses an action \pi_t at each time t as a function of past states x_0,...,x_t and past actions \pi_0,...,\pi_{t-1}. We let \mathcal{P} be the set of policies.

A policy, a plant equation, and the resulting sequence of states and rewards describe a Markov Decision Process. Objective is to find a process that optimizes the following objective.

Def 3 [Markov Decision Problem] Given initial state x_0, a Markov Decision Problem is the following optimization

Further, let R_\tau({x_\tau, \Pi}) (Resp. W_\tau(x_\tau)) be the objective (Resp. optimal objective) for when the summation is started from time t=\tau and state x_\tau, rather than t=0 and x_0.

Ex 1 [the Bellman Equation]Setting W_T(x)=r_T(x) for t=T-1,T-2,...,0

The above equation is Bellman’s equation for a Markov Decision Process.

Ex 2 You need to sell a car. At every time t=0,...,T-1, you set a price p_t and a customer then views the car. The probability that the customer buys a car at price p is D(p). If the car isn’t sold be time T then it is sold for fixed price W_T, W_T<1. Maximize the reward from selling the car and find the recursion for the optimal reward when D(p) = (1-p)_+.

 

Ex 3 You own a call option with strike price p. Here you can buy a share at price p making profit x_t-p where x_t is the price of the share at time t. The share must be exercised by time T. The price of stock x_t satisfies for \epsilon_t IIDRV with finite expectation. Show that there exists a decreasing sequence \{a_t\}_{0\leq t \leq T} such that it is optimal to exercise whenever x_s \geq a_s occurs.


Answers

Ans 1 \mathcal{P}_t, the set policies that can be implemented from time t to T, is the product actions at time t and the set of policies from time t+1 onward. That is \mathcal{P}_t = \{ (\pi_t,\Pi): \Pi \in\mathcal{P}_{t+1}, \pi_t: \mathcal{X}^t \times \prod_{\tau = 0}^{t-1}\mathcal{A}_\tau \rightarrow \mathcal{A}_t \}.

2nd equality uses structure of \mathcal{P}_t, takes the r_t term out and then takes conditional expectations. 3rd equality takes the supremum over \mathcal{P}_{t+1}, which does not depend on \pi_t, inside the expectation and notes the supremum over \pi_t is optimized at a fixed action a\in\mathcal{A} (i.e. the past information did not help us.)

Ans 2  The optimal reward from time t onward (assuming you still have the car) satisfies:

If D(p)=(1-p)_+ then W_t= (1+W_{t+1})^2/4. (Note the relation is monotone and W_0 converges to fixed point if T\rightarrow \infty.)

Ans 3 Taking the and subtracting x gives

1) W_t(x)\geq W_{t+1}(x), because all policies implemented from time t+1 are subset of policies from time t.

2) W_t(x)-x is decreasing and cts in x because W_T(x)= \max \{x-p, 0\} is and, since expectation and maximum preserve these properties, so is W_t by induction on .

Thus the point a_t where p=\mathbb{E}\left[W_{t+1}(a_t+\epsilon_t) -a_t\right] must decrease.

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