Dynamic Programming

Unlike the methods we’ve discussed so far, dynamic programming takes a step back and considers an entire family of related problems rather than a single optimization problem. This approach, while seemingly more complex at first glance, can often lead to efficient solutions.

Dynamic programming leverage the solution structure underlying many control problems that allows for a decomposition it into smaller, more manageable subproblems. Each subproblem is itself an optimization problem, embedded within the larger whole. This recursive structure is the foundation upon which dynamic programming constructs its solutions.

To ground our discussion, let us return to the domain of discrete-time optimal control problems (DOCPs). These problems frequently arise from the discretization of continuous-time optimal control problems. While the focus here will be on deterministic problems, these concepts extend naturally to stochastic problems by taking the expectation over the random quantities.

Consider a typical DOCP of Bolza type:

Rather than considering only the total cost from the initial time to the final time, dynamic programming introduces the concept of cost from an arbitrary point in time to the end. This leads to the definition of the “cost-to-go” or “value function” $J_k(\mathbf{x}_k)$:

This function represents the total cost incurred from stage $k$ onwards to the end of the time horizon, given that the system is initialized in state $\mathbf{x}_k$ at stage $k$. Suppose the problem has been solved from stage $k+1$ to the end, yielding the optimal cost-to-go $J_{k+1}^\star(\mathbf{x}_{k+1})$ for any state $\mathbf{x}_{k+1}$ at stage $k+1$. The question then becomes: how does this information inform the decision at stage $k$?

Given knowledge of the optimal behavior from $k+1$ onwards, the task reduces to determining the optimal action $\mathbf{u}_k$ at stage $k$. This control should minimize the sum of the immediate cost $c_k(\mathbf{x}_k, \mathbf{u}_k)$ and the optimal future cost $J_{k+1}^\star(\mathbf{x}_{k+1})$, where $\mathbf{x}_{k+1}$ is the resulting state after applying action $\mathbf{u}_k$. Mathematically, this is expressed as:

This equation is known as Bellman’s equation, named after Richard Bellman, who formulated the principle of optimality:

An optimal policy has the property that whatever the previous state and decision, the remaining decisions must constitute an optimal policy with regard to the state resulting from the previous decision.

In other words, any sub-path of an optimal path, from any intermediate point to the end, must itself be optimal. This principle is the basis for the backward induction procedure which computes the optimal value function and provides closed-loop control capabilities without having to use an explicit NLP solver.

Dynamic programming can handle nonlinear systems and non-quadratic cost functions naturally. It provides a global optimal solution, when one exists, and can incorporate state and control constraints with relative ease. However, as the dimension of the state space increases, this approach suffers from what Bellman termed the “curse of dimensionality.” The computational complexity and memory requirements grow exponentially with the state dimension, rendering direct application of dynamic programming intractable for high-dimensional problems.

Fortunately, learning-based methods offer efficient tools to combat the curse of dimensionality on two fronts: by using function approximation (e.g., neural networks) to avoid explicit discretization, and by leveraging randomization through Monte Carlo methods inherent in the learning paradigm. Most of this course is dedicated to those ideas.

Backward Recursion¶

The principle of optimality provides a methodology for solving optimal control problems. Beginning at the final time horizon and working backwards, at each stage the local optimization problem given by Bellman’s equation is solved. This process, termed backward recursion or backward induction, constructs the optimal value function stage by stage.

Upon completion of this backward pass, we now have access to the optimal control to take at any stage and in any state. Furthermore, we can simulate optimal trajectories from any initial state and applying the optimal policy at each stage to generate the optimal trajectory.

Example: Optimal Harvest in Resource Management¶

Dynamic programming is often used in resource management and conservation biology to devise policies to be implemented by decision makers and stakeholders : for eg. in fishereries, or timber harvesting. Per Conroy & Peterson (2013), we consider a population of a particular species, whose abundance we denote by $x_t$, where $t$ represents discrete time steps. Our objective is to maximize the cumulative harvest over a finite time horizon, while also considering the long-term sustainability of the population. This optimization problem can be formulated as:

Here, $F(\cdot)$ represents the immediate reward function associated with harvesting, $h_t$ is the harvest rate at time $t$, and $F_\mathrm{T}(\cdot)$ denotes a terminal value function that could potentially assign value to the final population state. In this particular problem, we assign no terminal value to the final population state, setting $F_\mathrm{T}(x_{t_f}) = 0$ and allowing us to focus solely on the cumulative harvest over the time horizon.

In our model population model, the abundance of a specicy $x$ ranges from 1 to 100 individuals. The decision variable is the harvest rate $h$, which can take values from the set $D = \{0, 0.1, 0.2, 0.3, 0.4, 0.5\}$. The population dynamics are governed by a modified logistic growth model:

where the 0.3 represents the growth rate and 125 is the carrying capacity (the maximum population size given the available resources). The logistic growth model returns continuous values; however our DP formulation uses a discrete state space. Therefore, we also round the the outcomes to the nearest integer.

Applying the principle of optimality, we can express the optimal value function $J^\star(x_t,t)$ recursively:

with the boundary condition $J^*(x_{t_f}) = 0$.

It’s worth noting that while this example uses a relatively simple model, the same principles can be applied to more complex scenarios involving stochasticity, multiple species interactions, or spatial heterogeneity.

import numpy as np

# Parameters
r_max = 0.3
K = 125
T = 20  # Number of time steps
N_max = 100  # Maximum population size to consider
h_max = 0.5  # Maximum harvest rate
h_step = 0.1  # Step size for harvest rate

# Create state and decision spaces
N_space = np.arange(1, N_max + 1)
h_space = np.arange(0, h_max + h_step, h_step)

# Initialize value function and policy
V = np.zeros((T + 1, len(N_space)))
policy = np.zeros((T, len(N_space)))

# Terminal value function (F_T)
def terminal_value(N):
    return 0

# State return function (F)
def state_return(N, h):
    return N * h

# State dynamics function
def state_dynamics(N, h):
    return N + r_max * N * (1 - N / K) - N * h

# Backward iteration
for t in range(T - 1, -1, -1):
    for i, N in enumerate(N_space):
        max_value = float('-inf')
        best_h = 0

        for h in h_space:
            if h > 1:  # Ensure harvest rate doesn't exceed 100%
                continue

            next_N = state_dynamics(N, h)
            if next_N < 1:  # Ensure population doesn't go extinct
                continue

            next_N_index = np.searchsorted(N_space, next_N)
            if next_N_index == len(N_space):
                next_N_index -= 1

            value = state_return(N, h) + V[t + 1, next_N_index]

            if value > max_value:
                max_value = value
                best_h = h

        V[t, i] = max_value
        policy[t, i] = best_h

# Function to simulate the optimal policy with conversion to Python floats
def simulate_optimal_policy(initial_N, T):
    trajectory = [float(initial_N)]  # Ensure first value is a Python float
    harvests = []

    for t in range(T):
        N = trajectory[-1]
        N_index = np.searchsorted(N_space, N)
        if N_index == len(N_space):
            N_index -= 1

        h = policy[t, N_index]
        harvests.append(float(N * h))  # Ensure harvest is a Python float

        next_N = state_dynamics(N, h)
        trajectory.append(float(next_N))  # Ensure next population value is a Python float

    return trajectory, harvests

# Example usage
initial_N = 50
trajectory, harvests = simulate_optimal_policy(initial_N, T)

print("Optimal policy:")
print(policy)
print("\nPopulation trajectory:", trajectory)
print("Harvests:", harvests)
print("Total harvest:", sum(harvests))

Handling Continuous Spaces with Interpolation¶

In many real-world problems, such as our resource management example, the state space is inherently continuous. Dynamic programming, however, is usually defined on discrete state spaces. To reconcile this, we approximate the value function on a finite grid of points and use interpolation to estimate its value elsewhere.

In our earlier example, we acted as if population sizes could only be whole numbers: 1 fish, 2 fish, 3 fish. But real measurements don’t fit neatly. What do you do with a survey that reports 42.7 fish? Our reflex in the code example was to round to the nearest integer, effectively saying “let’s just call it 43.” This corresponds to nearest-neighbor interpolation, also known as discretization. It’s the zeroth-order case: you assume the value between grid points is constant and equal to the closest one. In practice, this amounts to overlaying a grid on the continuous landscape and forcing yourself to stand at the intersections. In our demo code, this step was carried out with numpy.searchsorted.

While easy to implement, nearest-neighbor interpolation can introduce artifacts:

1. Decisions may change abruptly, even if the state only shifts slightly.

2. Precision is lost, especially in regimes where small variations matter.

3. The curse of dimensionality forces an impractically fine grid if many state variables are added.

To address these issues, we can use higher-order interpolation. Instead of taking the nearest neighbor, we estimate the value at off-grid points by leveraging multiple nearby values.

Backward Recursion with Interpolation¶

Suppose we have computed $J_{k+1}^\star(\mathbf{x})$ only at grid points $\mathbf{x} \in \mathcal{X}_\text{grid}$. To evaluate Bellman’s equation at an arbitrary $\mathbf{x}_{k+1}$, we interpolate. Formally, let $I_{k+1}(\mathbf{x})$ be the interpolation operator that extends the value function from $\mathcal{X}_\text{grid}$ to the continuous space. Then:

For instance, in one dimension, linear interpolation gives:

where $x_l$ and $x_u$ are the nearest grid points bracketing $x$. Linear interpolation is often sufficient, but higher-order methods (cubic splines, radial basis functions) can yield smoother and more accurate estimates. The choice of interpolation scheme and grid layout both affect accuracy and efficiency. A finer grid improves resolution but increases computational cost, motivating strategies like adaptive grid refinement or replacing interpolation altogether with parametric function approximation which we are going to see later in this book.

In higher-dimensional spaces, naive interpolation becomes prohibitively expensive due to the curse of dimensionality. Several approaches such as tensorized multilinear interpolation, radial basis functions, and machine learning models address this challenge by extending a common principle: they approximate the value function at unobserved points using information from a finite set of evaluations. However, as dimensionality continues to grow, even tensor methods face scalability limits, which is why flexible parametric models like neural networks have become essential tools for high-dimensional function approximation.

Example: Optimal Harvest with Linear Interpolation¶

Here is a demonstration of the backward recursion procedure using linear interpolation.

import numpy as np

# Parameters
r_max = 0.3
K = 125
T = 20  # Number of time steps
N_max = 100  # Maximum population size to consider
h_max = 0.5  # Maximum harvest rate
h_step = 0.1  # Step size for harvest rate

# Create state and decision spaces
N_space = np.arange(1, N_max + 1)
h_space = np.arange(0, h_max + h_step, h_step)

# Initialize value function and policy
V = np.zeros((T + 1, len(N_space)))
policy = np.zeros((T, len(N_space)))

# Terminal value function (F_T)
def terminal_value(N):
    return 0

# State return function (F)
def state_return(N, h):
    return N * h

# State dynamics function
def state_dynamics(N, h):
    return N + r_max * N * (1 - N / K) - N * h

# Function to linearly interpolate between grid points in N_space
def interpolate_value_function(V, N_space, next_N, t):
    if next_N <= N_space[0]:
        return V[t, 0]  # Below or at minimum population, return minimum value
    if next_N >= N_space[-1]:
        return V[t, -1]  # Above or at maximum population, return maximum value
    
    # Find indices to interpolate between
    lower_idx = np.searchsorted(N_space, next_N) - 1
    upper_idx = lower_idx + 1
    
    # Linear interpolation
    N_lower = N_space[lower_idx]
    N_upper = N_space[upper_idx]
    weight = (next_N - N_lower) / (N_upper - N_lower)
    return (1 - weight) * V[t, lower_idx] + weight * V[t, upper_idx]

# Backward iteration with interpolation
for t in range(T - 1, -1, -1):
    for i, N in enumerate(N_space):
        max_value = float('-inf')
        best_h = 0
        
        for h in h_space:
            if h > 1:  # Ensure harvest rate doesn't exceed 100%
                continue
            
            next_N = state_dynamics(N, h)
            if next_N < 1:  # Ensure population doesn't go extinct
                continue
            
            # Interpolate value for next_N
            value = state_return(N, h) + interpolate_value_function(V, N_space, next_N, t + 1)
            
            if value > max_value:
                max_value = value
                best_h = h
        
        V[t, i] = max_value
        policy[t, i] = best_h

# Function to simulate the optimal policy using interpolation
def simulate_optimal_policy(initial_N, T):
    trajectory = [initial_N]
    harvests = []

    for t in range(T):
        N = trajectory[-1]
        
        # Interpolate optimal harvest rate
        if N <= N_space[0]:
            h = policy[t, 0]
        elif N >= N_space[-1]:
            h = policy[t, -1]
        else:
            lower_idx = np.searchsorted(N_space, N) - 1
            upper_idx = lower_idx + 1
            weight = (N - N_space[lower_idx]) / (N_space[upper_idx] - N_space[lower_idx])
            h = (1 - weight) * policy[t, lower_idx] + weight * policy[t, upper_idx]
        
        harvests.append(float(N * h))  # Ensure harvest is a Python float
        next_N = state_dynamics(N, h)
        trajectory.append(float(next_N))  # Ensure next population value is a Python float

    return trajectory, harvests

# Example usage
initial_N = 50
trajectory, harvests = simulate_optimal_policy(initial_N, T)

print("Optimal policy:")
print(policy)
print("\nPopulation trajectory:", trajectory)
print("Harvests:", harvests)
print("Total harvest:", sum(harvests))

Due to pedagogical considerations, this example is using our own implementation of the linear interpolation procedure. However, a more general and practical approach would be to use a built-in interpolation procedure in Numpy. Because our state space has a single dimension, we can simply use scipy.interpolate.interp1d which offers various interpolation methods through its kind argument, which can take values in ‘linear’, ‘nearest’, ‘nearest-up’, ‘zero’, ‘slinear’, ‘quadratic’, ‘cubic’, ‘previous’, or ‘next’. ‘zero’, ‘slinear’, ‘quadratic’ and ‘cubic’.

Here’s a more general implementation which here uses cubic interpolation through the scipy.interpolate.interp1d function:

import numpy as np
from scipy.interpolate import interp1d

# Parameters
r_max = 0.3
K = 125
T = 20  # Number of time steps
N_max = 100  # Maximum population size to consider
h_max = 0.5  # Maximum harvest rate
h_step = 0.1  # Step size for harvest rate

# Create state and decision spaces
N_space = np.arange(1, N_max + 1)
h_space = np.arange(0, h_max + h_step, h_step)

# Initialize value function and policy
V = np.zeros((T + 1, len(N_space)))
policy = np.zeros((T, len(N_space)))

# Terminal value function (F_T)
def terminal_value(N):
    return 0

# State return function (F)
def state_return(N, h):
    return N * h

# State dynamics function
def state_dynamics(N, h):
    return N + r_max * N * (1 - N / K) - N * h

# Function to create interpolation function for a given time step
def create_interpolator(V_t, N_space):
    return interp1d(N_space, V_t, kind='cubic', bounds_error=False, fill_value=(V_t[0], V_t[-1]))

# Backward iteration with interpolation
for t in range(T - 1, -1, -1):
    interpolator = create_interpolator(V[t+1], N_space)
    
    for i, N in enumerate(N_space):
        max_value = float('-inf')
        best_h = 0

        for h in h_space:
            if h > 1:  # Ensure harvest rate doesn't exceed 100%
                continue

            next_N = state_dynamics(N, h)
            if next_N < 1:  # Ensure population doesn't go extinct
                continue

            # Use interpolation to get the value for next_N
            value = state_return(N, h) + interpolator(next_N)

            if value > max_value:
                max_value = value
                best_h = h

        V[t, i] = max_value
        policy[t, i] = best_h

# Function to simulate the optimal policy using interpolation
def simulate_optimal_policy(initial_N, T):
    trajectory = [initial_N]
    harvests = []

    for t in range(T):
        N = trajectory[-1]
        
        # Create interpolator for the policy at time t
        policy_interpolator = interp1d(N_space, policy[t], kind='cubic', bounds_error=False, fill_value=(policy[t][0], policy[t][-1]))
        
        h = policy_interpolator(N)
        harvests.append(float(N * h))  # Ensure harvest is a Python float

        next_N = state_dynamics(N, h)
        trajectory.append(float(next_N))  # Ensure next population value is a Python float

    return trajectory, harvests

# Example usage
initial_N = 50
trajectory, harvests = simulate_optimal_policy(initial_N, T)

print("Optimal policy (first few rows):")
print(policy[:5])
print("\nPopulation trajectory:", trajectory)
print("Harvests:", harvests)
print("Total harvest:", sum(harvests))

Stochastic Dynamic Programming and Markov Decision Processes¶

While our previous discussion centered on deterministic systems, many real-world problems involve uncertainty. Stochastic Dynamic Programming (SDP) extends our framework to handle stochasticity in both the objective function and system dynamics. This extension naturally leads us to consider more general policy classes and to formalize when simpler policies suffice.

Decision Rules and Policies¶

Before diving into stochastic systems, we need to establish terminology for the different types of strategies a decision maker might employ. In the deterministic setting, we implicitly used feedback controllers of the form $u(\mathbf{x}, t)$. In the stochastic setting, we must be more precise about what information policies can use and how they select actions.

A decision rule is a prescription for action selection in each state at a specified decision epoch. These rules can vary in their complexity based on two main criteria:

1. Dependence on history: Markovian or History-dependent

2. Action selection method: Deterministic or Randomized

Markovian decision rules depend only on the current state, while history-dependent rules consider the entire sequence of past states and actions. Formally, a history $h_t$ at time $t$ is:

The set of all possible histories at time $t$, denoted $H_t$, grows exponentially with $t$:

• $H_1 = \mathcal{S}$ (just the initial state)

• $H_2 = \mathcal{S} \times \mathcal{A} \times \mathcal{S}$

• $H_t = \mathcal{S} \times (\mathcal{A} \times \mathcal{S})^{t-1}$

Deterministic rules select an action with certainty, while randomized rules specify a probability distribution over the action space.

These classifications lead to four types of decision rules:

1. Markovian Deterministic (MD): $\pi_t: \mathcal{S} \rightarrow \mathcal{A}_s$

2. Markovian Randomized (MR): $\pi_t: \mathcal{S} \rightarrow \mathcal{P}(\mathcal{A}_s)$

3. History-dependent Deterministic (HD): $\pi_t: H_t \rightarrow \mathcal{A}_s$

4. History-dependent Randomized (HR): $\pi_t: H_t \rightarrow \mathcal{P}(\mathcal{A}_s)$

where $\mathcal{P}(\mathcal{A}_s)$ denotes the set of probability distributions over $\mathcal{A}_s$.

A policy $\boldsymbol{\pi}$ is a sequence of decision rules, one for each decision epoch:

The set of all policies of class $K$ (where $K \in \{HR, HD, MR, MD\}$) is denoted as $\Pi^K$. These policy classes form a hierarchy:

The largest set $\Pi^{HR}$ contains all possible policies. We ask: under what conditions can we restrict attention to the much simpler set $\Pi^{MD}$ without loss of optimality?

Stochastic System Dynamics¶

In the stochastic setting, our system evolution takes the form:

Here, $\mathbf{w}_t$ represents a random disturbance or noise term at time $t$ due to the inherent uncertainty in the system’s behavior. The stage cost function may also incorporate stochastic influences:

In this context, our objective shifts from minimizing a deterministic cost to minimizing the expected total cost:

where the expectation is taken over the distributions of the random variables $\mathbf{w}_t$. The principle of optimality still holds in the stochastic case, but Bellman’s optimality equation now involves an expectation:

In practice, this expectation is often computed by discretizing the distribution of $\mathbf{w}_k$ when the set of possible disturbances is very large or even continuous. Let’s say we approximate the distribution with $K$ discrete values $\mathbf{w}_k^i$, each occurring with probability $p_k^i$. Then our Bellman equation becomes:

Optimality Equations in the Stochastic Setting¶

When dealing with stochastic systems, a central question arises: what information should our control policy use? In the most general case, a policy might use the entire history of observations and actions. However, as we’ll see, the Markovian structure of our problems allows for dramatic simplifications.

Let $h_t = (s_1, a_1, s_2, a_2, \ldots, s_{t-1}, a_{t-1}, s_t)$ denote the complete history up to time $t$. In the stochastic setting, the history-based optimality equations become:

where we now explicitly use the transition probabilities $p_t(j|s_t,a)$ rather than a deterministic dynamics function.

Intuition: This formalizes Bellman’s principle of optimality: “An optimal policy has the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision.” The recursive structure means that optimal local decisions (choosing the best action at each step) lead to global optimality, even with uncertainty captured by the transition probabilities.

A simplification occurs when we examine these history-based equations more closely. The Markov property of our system dynamics and rewards means that the optimal return actually depends on the history only through the current state:

Intuition: The Markov property means that the current state contains all information needed to predict future evolution. The past provides no additional value for decision-making. This result allows us to work with value functions $v_t^*(s)$ indexed only by state and time, dramatically simplifying both theory and computation.

This state-sufficiency result, combined with the fact that randomization never helps when maximizing expected returns, leads to a dramatic simplification of the policy space:

Intuition: Even in stochastic systems, randomization in the policy doesn’t help when maximizing expected returns: you should always choose the action with the highest expected value. Combined with state sufficiency, this means simple state-to-action mappings are optimal.

These results justify focusing on deterministic Markov policies and lead to the backward recursion algorithm for stochastic systems:

While SDP provides us with a framework to for handling uncertainty, it makes the curse of dimensionality even more difficult to handle in practice. Both the state space and the disturbance space must be discretized. This can lead to a combinatorial explosion in the number of scenarios to be evaluated at each stage.

However, just as we tackled the challenges of continuous state spaces with discretization and interpolation, we can devise efficient methods to handle the additional complexity of evaluating expectations. This problem essentially becomes one of numerical integration. When the set of disturbances is continuous (as is often the case with continuous state spaces), we enter a domain where numerical quadrature methods could be applied. But these methods tend to scale poorly as the number of dimensions grows. This is where more efficient techniques, often rooted in Monte Carlo methods, come into play. Two ingredients tackle the curse of dimensionality:

1. Function approximation (through discretization, interpolation, neural networks, etc.)

2. Monte Carlo integration (simulation)

These two elements essentially distill the key ingredients of machine learning, which is the direction we’ll be exploring in this course.

Example: Stochastic Optimal Harvest in Resource Management¶

Building upon our previous deterministic model, we now introduce stochasticity to more accurately reflect the uncertainties inherent in real-world resource management scenarios Conroy & Peterson, 2013. As before, we consider a population of a particular species, whose abundance we denote by $x_t$, where $t$ represents discrete time steps. Our objective remains to maximize the cumulative harvest over a finite time horizon, while also considering the long-term sustainability of the population. However, we now account for two sources of stochasticity: partial controllability of harvest and environmental variability affecting growth rates. The optimization problem can be formulated as:

Here, $F(\cdot)$ represents the immediate reward function associated with harvesting, and $h_t$ is the realized harvest rate at time $t$. The expectation $\mathbb{E}[\cdot]$ over both harvest and growth rates, which we view as random variables. In our stochastic model, the abundance $x$ still ranges from 1 to 100 individuals. The decision variable is now the desired harvest rate $d_t$, which can take values from the set $D = {0, 0.1, 0.2, 0.3, 0.4, 0.5}$. However, the realized harvest rate $h_t$ is stochastic and follows a discrete distribution:

By expressing the harvest rate as a random variable, we mean to capture the fact that harvesting is a not completely under our control: we might obtain more or less what we had intended to. Furthermore, we generalize the population dynamics to the stochastic case via:

$$

x_{t+1} = x_t + r_tx_t(1 - x_t/K) - h_tx_t $$

where $K = 125$ is the carrying capacity. The growth rate $r_t$ is now stochastic and follows a discrete distribution:

where $r_{\text{max}} = 0.3$ is the maximum growth rate. Applying the principle of optimality, we can express the optimal value function $J^\star(x_t, t)$ recursively:

where the expectation is taken over the harvest and growth rate random variables. The boundary condition remains $J^*(x_{t_f}) = 0$. We can now adapt our previous code to account for the stochasticity in our model. One important difference is that simulating a solution in this context requires multiple realizations of our process. This is an important consideration when evaluating reinforcement learning methods in practice, as success cannot be claimed based on a single successful trajectory.

import numpy as np
from scipy.interpolate import interp1d

# Parameters
r_max = 0.3
K = 125
T = 30  # Number of time steps
N_max = 100  # Maximum population size to consider
h_max = 0.5  # Maximum harvest rate
h_step = 0.1  # Step size for harvest rate

# Create state and decision spaces
N_space = np.linspace(1, N_max, 100)  # Using more granular state space
h_space = np.arange(0, h_max + h_step, h_step)

# Stochastic parameters
h_outcomes = np.array([0.75, 1.0, 1.25])
h_probs = np.array([0.25, 0.5, 0.25])
r_outcomes = np.array([0.85, 1.05, 1.15]) * r_max
r_probs = np.array([0.25, 0.5, 0.25])

# Initialize value function and policy
V = np.zeros((T + 1, len(N_space)))
policy = np.zeros((T, len(N_space)))

# State return function (F)
def state_return(N, h):
    return N * h

# State dynamics function (stochastic)
def state_dynamics(N, h, r):
    return N + r * N * (1 - N / K) - h * N

# Function to create interpolation function for a given time step
def create_interpolator(V_t, N_space):
    return interp1d(N_space, V_t, kind='linear', bounds_error=False, fill_value=(V_t[0], V_t[-1]))

# Backward iteration with stochastic dynamics
for t in range(T - 1, -1, -1):
    interpolator = create_interpolator(V[t+1], N_space)
    
    for i, N in enumerate(N_space):
        max_value = float('-inf')
        best_h = 0

        for h in h_space:
            if h > 1:  # Ensure harvest rate doesn't exceed 100%
                continue

            expected_value = 0
            for h_factor, h_prob in zip(h_outcomes, h_probs):
                for r_factor, r_prob in zip(r_outcomes, r_probs):
                    realized_h = h * h_factor
                    realized_r = r_factor

                    next_N = state_dynamics(N, realized_h, realized_r)
                    if next_N < 1:  # Ensure population doesn't go extinct
                        continue

                    # Use interpolation to get the value for next_N
                    value = state_return(N, realized_h) + interpolator(next_N)
                    expected_value += value * h_prob * r_prob

            if expected_value > max_value:
                max_value = expected_value
                best_h = h

        V[t, i] = max_value
        policy[t, i] = best_h

# Function to simulate the optimal policy using interpolation (stochastic version)
def simulate_optimal_policy(initial_N, T, num_simulations=100):
    all_trajectories = []
    all_harvests = []

    for _ in range(num_simulations):
        trajectory = [initial_N]
        harvests = []

        for t in range(T):
            N = trajectory[-1]
            
            # Create interpolator for the policy at time t
            policy_interpolator = interp1d(N_space, policy[t], kind='linear', bounds_error=False, fill_value=(policy[t][0], policy[t][-1]))
            
            intended_h = policy_interpolator(N)
            
            # Apply stochasticity
            h_factor = np.random.choice(h_outcomes, p=h_probs)
            r_factor = np.random.choice(r_outcomes, p=r_probs)
            
            realized_h = intended_h * h_factor
            harvests.append(N * realized_h)

            next_N = state_dynamics(N, realized_h, r_factor)
            trajectory.append(next_N)

        all_trajectories.append(trajectory)
        all_harvests.append(harvests)

    return all_trajectories, all_harvests

# Example usage
initial_N = 50
trajectories, harvests = simulate_optimal_policy(initial_N, T)

# Calculate average trajectory and total harvest
avg_trajectory = np.mean(trajectories, axis=0)
avg_total_harvest = np.mean([sum(h) for h in harvests])

print("Optimal policy (first few rows):")
print(policy[:5])
print("\nAverage population trajectory:", avg_trajectory)
print("Average total harvest:", avg_total_harvest)

# Plot results
import matplotlib.pyplot as plt

plt.figure(figsize=(12, 6))
plt.subplot(121)
for traj in trajectories[:20]:  # Plot first 20 trajectories
    plt.plot(range(T+1), traj, alpha=0.3)
plt.plot(range(T+1), avg_trajectory, 'r-', linewidth=2)
plt.title('Population Trajectories')
plt.xlabel('Time')
plt.ylabel('Population')

plt.subplot(122)
plt.hist([sum(h) for h in harvests], bins=20)
plt.title('Distribution of Total Harvest')
plt.xlabel('Total Harvest')
plt.ylabel('Frequency')

plt.tight_layout()
plt.show()

Linear Quadratic Regulator via Dynamic Programming¶

We now examine a special case where the backward recursion admits a closed-form solution. When the system dynamics are linear and the cost function is quadratic, the optimization at each stage can be solved analytically. Moreover, the value function itself maintains a quadratic structure throughout the recursion, and the optimal policy reduces to a simple linear feedback law. This result eliminates the need for discretization, interpolation, or any function approximation. The infinite-dimensional problem collapses to tracking a finite set of matrices.

Consider a discrete-time linear system:

where $\mathbf{x}_t \in \mathbb{R}^n$ is the state and $\mathbf{u}_t \in \mathbb{R}^m$ is the control input. The matrices $A_t \in \mathbb{R}^{n \times n}$ and $B_t \in \mathbb{R}^{n \times m}$ describe the system dynamics at time $t$.

The cost function to be minimized is quadratic:

where $Q_T \succeq 0$ (positive semidefinite), $Q_t \succeq 0$, and $R_t \succ 0$ (positive definite) are symmetric matrices of appropriate dimensions. The positive definiteness of $R_t$ ensures the minimization problem is well-posed.

What we now have to observe is that if the terminal cost is quadratic, then the value function at every earlier stage remains quadratic. This is not immediately obvious, but it follows from the structure of Bellman’s equation combined with the linearity of the dynamics.

We claim that the optimal cost-to-go from any stage $t$ takes the form:

for some positive semidefinite matrix $P_t$. At the terminal time, this is true by definition: $P_T = Q_T$.

Let’s verify this structure and derive the recursion for $P_t$ using backward induction. Suppose we’ve established that $J_{t+1}^\star(\mathbf{x}_{t+1}) = \frac{1}{2}\mathbf{x}_{t+1}^\top P_{t+1} \mathbf{x}_{t+1}$. Bellman’s equation at stage $t$ states:

Substituting the dynamics $\mathbf{x}_{t+1} = A_t\mathbf{x}_t + B_t\mathbf{u}_t$ and the quadratic form for $J_{t+1}^\star$:

Expanding the last term:

The expression inside the minimization becomes:

Collecting terms involving $\mathbf{u}_t$:

This is a quadratic function of $\mathbf{u}_t$. To find the minimizer, we take the gradient with respect to $\mathbf{u}_t$ and set it to zero:

Since $R_t + B_t^\top P_{t+1} B_t$ is positive definite (both $R_t$ and $P_{t+1}$ are positive semidefinite with $R_t$ strictly positive), we can solve for the optimal control:

Define the gain matrix:

so that $\mathbf{u}_t^\star = -K_t\mathbf{x}_t$. This is a linear feedback policy: the optimal control is simply a linear function of the current state.

Substituting $\mathbf{u}_t^\star$ back into the cost-to-go expression and simplifying (by completing the square), we obtain:

where $P_t$ satisfies the discrete-time Riccati equation:

Putting everything together, the backward induction procedure under the LQR setting then becomes:

Markov Decision Process Formulation¶

Rather than expressing the stochasticity in our system through a disturbance term as a parameter to a deterministic difference equation, we often work with an alternative representation (more common in operations research) which uses the Markov Decision Process formulation. The idea is that when we model our system in this way with the disturbance term being drawn indepently of the previous stages, the induced trajectory are those of a Markov chain. Hence, we can re-cast our control problem in that language, leading to the so-called Markov Decision Process framework in which we express the system dynamics in terms of transition probabilities rather than explicit state equations. In this framework, we express the probability that the system is in a given state using the transition probability function:

This function gives the probability of transitioning to state $\mathbf{x}_{t+1}$ at time $t+1$, given that the system is in state $\mathbf{x}_t$ and action $\mathbf{u}_t$ is taken at time $t$. Therefore, $p_t$ specifies a conditional probability distribution over the next states: namely, the sum (for discrete state spaces) or integral over the next state should be 1.

Given the control theory formulation of our problem via a deterministic dynamics function and a noise term, we can derive the corresponding transition probability function through the following relationship:

Here, $q_t(\mathbf{w})$ represents the probability density or mass function of the disturbance $\mathbf{W}_t$ (assuming discrete state spaces). When dealing with continuous spaces, the above expression simply contains an integral rather than a summation.

For a system with deterministic dynamics and no disturbance, the transition probabilities become much simpler and be expressed using the indicator function. Given a deterministic system with dynamics:

The transition probability function can be expressed as:

With this transition probability function, we can recast our Bellman optimality equation:

Here, ${c}(\mathbf{x}_t, \mathbf{u}_t)$ represents the expected immediate reward (or negative cost) when in state $\mathbf{x}_t$ and taking action $\mathbf{u}_t$ at time $t$. The summation term computes the expected optimal value for the future states, weighted by their transition probabilities.

This formulation offers several advantages:

1. It makes the Markovian nature of the problem explicit: the future state depends only on the current state and action, not on the history of states and actions.

2. For discrete-state problems, the entire system dynamics can be specified by a set of transition matrices, one for each possible action.

3. It allows us to bridge the gap with the wealth of methods in the field of probabilistic graphical models and statistical machine learning techniques for modelling and analysis.

Notation in Operations Reseach¶

The presentation above was intended to bridge the gap between the control-theoretic perspective and the world of closed-loop control through the idea of determining the value function of a parametric optimal control problem. We then saw how the backward induction procedure was applicable to both the deterministic and stochastic cases by taking the expectation over the disturbance variable. We then said that we can alternatively work with a representation of our system where instead of writing our model as a deterministic dynamics function taking a disturbance as an input, we would rather work directly via its transition probability function, which gives rise to the Markov chain interpretation of our system in simulation.

Note that the notation used in control theory tends to differ from that found in operations research communities, in which the field of dynamic programming flourished. We summarize those (purely notational) differences in this section.

In operations research, the system state at each decision epoch is typically denoted by $s \in \mathcal{S}$, where $S$ is the set of possible system states. When the system is in state $s$, the decision maker may choose an action $a$ from the set of allowable actions $\mathcal{A}_s$. The union of all action sets is denoted as $\mathcal{A} = \bigcup_{s \in \mathcal{S}} \mathcal{A}_s$.

The dynamics of the system are described by a transition probability function $p_t(j | s, a)$, which represents the probability of transitioning to state $j \in \mathcal{S}$ at time $t+1$, given that the system is in state $s$ at time $t$ and action $a \in \mathcal{A}_s$ is chosen. This transition probability function satisfies:

It’s worth noting that in operations research, we typically work with reward maximization rather than cost minimization, which is more common in control theory. However, we can easily switch between these perspectives by simply negating the quantity. That is, maximizing a reward function is equivalent to minimizing its negative, which we would then call a cost function.

The reward function is denoted by $r_t(s, a)$, representing the reward received at time $t$ when the system is in state $s$ and action $a$ is taken. In some cases, the reward may also depend on the next state, in which case it is denoted as $r_t(s, a, j)$. The expected reward can then be computed as:

Combined together, these elemetns specify a Markov decision process, which is fully described by the tuple:

where $\mathrm{T}$ represents the set of decision epochs (the horizon).

What is an Optimal Policy?¶

Let’s go back to the starting point and define what it means for a policy to be optimal in a Markov Decision Problem. For this, we will be considering different possible search spaces (policy classes) and compare policies based on the ordering of their value from any possible start state. The value of a policy $\boldsymbol{\pi}$ (optimal or not) is defined as the expected total reward obtained by following that policy from a given starting state. Formally, for a finite-horizon MDP with $N$ decision epochs, we define the value function $v^{\boldsymbol{\pi}}(s, t)$ as:

where $S_t$ is the state at time $t$, $A_t$ is the action taken at time $t$, and $r_t$ is the reward function. For simplicity, we write $v^{\boldsymbol{\pi}}(s)$ to denote $v^{\boldsymbol{\pi}}(s, 1)$, the value of following policy $\boldsymbol{\pi}$ from state $s$ at the first stage over the entire horizon $N$.

In finite-horizon MDPs, our goal is to identify an optimal policy, denoted by $\boldsymbol{\pi}^*$, that maximizes total expected reward over the horizon $N$. Specifically:

We call $\boldsymbol{\pi}^*$ an optimal policy because it yields the highest possible value across all states and all policies within the policy class $\Pi^{\text{HR}}$. We denote by $v^*$ the maximum value achievable by any policy:

In reinforcement learning literature, $v^*$ is typically referred to as the “optimal value function,” while in some operations research references, it might be called the “value of an MDP.” An optimal policy $\boldsymbol{\pi}^*$ is one for which its value function equals the optimal value function:

This notion of optimality applies to every state. Policies optimal in this sense are sometimes called “uniformly optimal policies.” A weaker notion of optimality, often encountered in reinforcement learning practice, is optimality with respect to an initial distribution of states. In this case, we seek a policy $\boldsymbol{\pi} \in \Pi^{\text{HR}}$ that maximizes:

where $P_1(S_1 = s)$ is the probability of starting in state $s$.

The maximum value can be achieved by searching over the space of deterministic Markovian Policies. Consequently:

This equality significantly simplifies the computational complexity of our algorithms, as the search problem can now be decomposed into $N$ sub-problems in which we only have to search over the set of possible actions. This is the backward induction algorithm, which we present a second time, but departing this time from the control-theoretic notation and using the MDP formalism:

Note that the same procedure can also be used for finding the value of a policy with minor changes;

This code could also finally be adapted to support randomized policies using:

Example: Sample Size Determination in Pharmaceutical Development¶

Pharmaceutical development is the process of bringing a new drug from initial discovery to market availability. This process is lengthy, expensive, and risky, typically involving several stages:

1. Drug Discovery: Identifying a compound that could potentially treat a disease.

2. Preclinical Testing: Laboratory and animal testing to assess safety and efficacy. . Clinical Trials: Testing the drug in humans, divided into phases:

   • Phase I: Testing for safety in a small group of healthy volunteers.

   • Phase II: Testing for efficacy and side effects in a larger group with the target condition.

   • Phase III: Large-scale testing to confirm efficacy and monitor side effects.

3. Regulatory Review: Submitting a New Drug Application (NDA) for approval.

4. Post-Market Safety Monitoring: Continuing to monitor the drug’s effects after market release.

This process can take 10-15 years and cost over $1 billion Adams & Brantner (2009). The high costs and risks involved call for a principled approach to decision making. We’ll focus on the clinical trial phases and NDA approval, per the MDP model presented by Chang (2010):

1. States ($S$): Our state space is $S = \{s_1, s_2, s_3, s_4\}$, where:

   • $s_1$: Phase I clinical trial

   • $s_2$: Phase II clinical trial

   • $s_3$: Phase III clinical trial

   • $s_4$: NDA approval

2. Actions ($A$): At each state, the action is choosing the sample size $n_i$ for the corresponding clinical trial. The action space is $A = \{10, 11, ..., 1000\}$, representing possible sample sizes.

3. Transition Probabilities ($P$): The probability of moving from one state to the next depends on the chosen sample size and the inherent properties of the drug. We define:

   • $P(s_2|s_1, n_1) = p_{12}(n_1) = \sum_{i=0}^{\lfloor\eta_1 n_1\rfloor} \binom{n_1}{i} p_0^i (1-p_0)^{n_1-i}$ where $p_0$ is the true toxicity rate and $\eta_1$ is the toxicity threshold for Phase I.

• Of particular interest is the transition from Phase II to Phase III which we model as:

  $P(s_3|s_2, n_2) = p_{23}(n_2) = \Phi\left(\frac{\sqrt{n_2}}{2}\delta - z_{1-\eta_2}\right)$

  where $\Phi$ is the cumulative distribution function (CDF) of the standard normal distribution:

  $\Phi(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-t^2/2} dt$

  This is giving us the probability that we would observe a treatment effect this large or larger if the null hypothesis (no treatment effect) were true. A higher probability indicates stronger evidence of a treatment effect, making it more likely that the drug will progress to Phase III.

  In this expression, $\delta$ is called the “normalized treatment effect”. In clinical trials, we’re often interested in the difference between the treatment and control groups. The “normalized” part means we’ve adjusted this difference for the variability in the data. Specifically $\delta = \frac{\mu_t - \mu_c}{\sigma}$ where $\mu_t$ is the mean outcome in the treatment group, $\mu_c$ is the mean outcome in the control group, and $\sigma$ is the standard deviation of the outcome. A larger $\delta$ indicates a stronger treatment effect.

  Furthermore, the term $z_{1-\eta_2}$ is the $(1-\eta_2)$-quantile of the standard normal distribution. In other words, it’s the value where the probability of a standard normal random variable being greater than this value is $\eta_2$. For example, if $\eta_2 = 0.05$, then $z_{1-\eta_2} \approx 1.645$. A smaller $\eta_2$ makes the trial more conservative, requiring stronger evidence to proceed to Phase III.

  Finally, $n_2$ is the sample size for Phase II. The $\sqrt{n_2}$ term reflects that the precision of our estimate of the treatment effect improves with the square root of the sample size.

• $P(s_4|s_3, n_3) = p_{34}(n_3) = \Phi\left(\frac{\sqrt{n_3}}{2}\delta - z_{1-\eta_3}\right)$ where $\eta_3$ is the significance level for Phase III.

4. Rewards ($R$): The reward function captures the costs of running trials and the potential profit from a successful drug:

   • $r(s_i, n_i) = -c_i(n_i)$ for $i = 1, 2, 3$, where $c_i(n_i)$ is the cost of running a trial with sample size $n_i$.

   • $r(s_4) = g_4$, where $g_4$ is the expected profit from a successful drug.

5. Discount Factor ($\gamma$): We use a discount factor $0 < \gamma \leq 1$ to account for the time value of money and risk preferences.

import numpy as np
from scipy.stats import binom
from scipy.stats import norm

def binomial_pmf(k, n, p):
    return binom.pmf(k, n, p)

def transition_prob_phase1(n1, eta1, p0):
    return np.sum([binomial_pmf(i, n1, p0) for i in range(int(eta1 * n1) + 1)])

def transition_prob_phase2(n2, eta2, delta):
    return norm.cdf((np.sqrt(n2) / 2) * delta - norm.ppf(1 - eta2))

def transition_prob_phase3(n3, eta3, delta):
    return norm.cdf((np.sqrt(n3) / 2) * delta - norm.ppf(1 - eta3))

def immediate_reward(n):
    return -n  # Negative to represent cost

def backward_induction(S, A, gamma, g4, p0, delta, eta1, eta2, eta3):
    V = np.zeros(len(S))
    V[3] = g4  # Value for NDA approval state
    optimal_n = [None] * 3  # Store optimal n for each phase

    # Backward induction
    for i in range(2, -1, -1):  # Iterate backwards from Phase III to Phase I
        max_value = -np.inf
        for n in A:
            if i == 0:  # Phase I
                p = transition_prob_phase1(n, eta1, p0)
            elif i == 1:  # Phase II
                p = transition_prob_phase2(n, eta2, delta)
            else:  # Phase III
                p = transition_prob_phase3(n, eta3, delta)
            value = immediate_reward(n) + gamma * p * V[i+1]
            if value > max_value:
                max_value = value
                optimal_n[i] = n
        V[i] = max_value

    return V, optimal_n

# Set up the problem parameters
S = ['Phase I', 'Phase II', 'Phase III', 'NDA approval']
A = range(10, 1001)
gamma = 0.95
g4 = 10000
p0 = 0.1  # Example toxicity rate for Phase I
delta = 0.5  # Example normalized treatment difference
eta1, eta2, eta3 = 0.2, 0.1, 0.025

# Run the backward induction algorithm
V, optimal_n = backward_induction(S, A, gamma, g4, p0, delta, eta1, eta2, eta3)

# Print results
for i, state in enumerate(S):
    print(f"Value for {state}: {V[i]:.2f}")
print(f"Optimal sample sizes: Phase I: {optimal_n[0]}, Phase II: {optimal_n[1]}, Phase III: {optimal_n[2]}")

# Sanity checks
print("\nSanity checks:")
print(f"1. NDA approval value: {V[3]}")
print(f"2. All values non-positive and <= NDA value: {all(v <= V[3] for v in V)}")
print(f"3. Optimal sample sizes in range: {all(10 <= n <= 1000 for n in optimal_n if n is not None)}")

Infinite-Horizon MDPs¶

It often makes sense to model control problems over infinite horizons. We extend the previous setting and define the expected total reward of policy $\boldsymbol{\pi} \in \Pi^{\mathrm{HR}}$, $v^{\boldsymbol{\pi}}$ as:

One drawback of this model is that we could easily encounter values that are $+\infty$ or $-\infty$, even in a setting as simple as a single-state MDP which loops back into itself and where the accrued reward is nonzero.

Therefore, it is often more convenient to work with an alternative formulation which guarantees the existence of a limit: the expected total discounted reward of policy $\boldsymbol{\pi} \in \Pi^{\mathrm{HR}}$ is defined to be:

for $0 \leq \gamma < 1$ and when $\max_{s \in \mathcal{S}} \max_{a \in \mathcal{A}_s}|r(s, a)| = R_{\max} < \infty$, in which case, $|v_\gamma^{\boldsymbol{\pi}}(s)| \leq (1-\gamma)^{-1} R_{\max}$.

Finally, another possibility for the infinite-horizon setting is the so-called average reward or gain of policy $\boldsymbol{\pi} \in \Pi^{\mathrm{HR}}$ defined as:

We won’t be working with this formulation in this course due to its inherent practical and theoretical complexities.

Extending the previous notion of optimality from finite-horizon models, a policy $\boldsymbol{\pi}^*$ is said to be discount optimal for a given $\gamma$ if:

Furthermore, the value of a discounted MDP $v_\gamma^*(s)$, is defined by:

More often, we refer to $v_\gamma$ by simply calling it the optimal value function.

As for the finite-horizon setting, the infinite horizon discounted model does not require history-dependent policies, since for any $\boldsymbol{\pi} \in \Pi^{HR}$ there exists a $\boldsymbol{\pi}^{\prime} \in \Pi^{MR}$ with identical total discounted reward:

Random Horizon Interpretation of Discounting¶

The use of discounting can be motivated both from a modeling perspective and as a means to ensure that the total reward remains bounded. From the modeling perspective, we can view discounting as a way to weight more or less importance on the immediate rewards vs. the long-term consequences. There is also another interpretation which stems from that of a finite horizon model but with an uncertain end time. More precisely:

Let $v_\nu^{\boldsymbol{\pi}}(s)$ denote the expected total reward obtained by using policy $\boldsymbol{\pi}$ when the horizon length $\nu$ is random. We define it by:

Vector Representation in Markov Decision Processes¶

Let V be the set of bounded real-valued functions on a discrete state space S. This means any function $f \in V$ satisfies the condition:

where notation $\|f\|$ represents the sup-norm (or $\ell_\infty$-norm) of the function $f$.

When working with discrete state spaces, we can interpret elements of V as vectors and linear operators on V as matrices, allowing us to leverage tools from linear algebra. The sup-norm ($\ell_\infty$ norm) of matrix $\mathbf{H}$ is defined as:

where $\mathbf{H}_{s,j}$ represents the $(s, j)$-th component of the matrix $\mathbf{H}$.

For a Markovian decision rule $\pi \in \Pi^{MD}$, we define:

For a randomized decision rule $\pi \in \Pi^{MR}$, these definitions extend to:

In both cases, $\mathbf{r}_\pi$ denotes a reward vector in $\mathbb{R}^{|S|}$, with each component $\mathbf{r}_\pi(s)$ representing the reward associated with state $s$. Similarly, $\mathbf{P}_\pi$ is a transition probability matrix in $\mathbb{R}^{|S| \times |S|}$, capturing the transition probabilities under decision rule $\pi$.

For a nonstationary Markovian policy $\boldsymbol{\pi} = (\pi_1, \pi_2, \ldots) \in \Pi^{MR}$, the expected total discounted reward is given by:

Using vector notation, this can be expressed as:

This formulation leads to a recursive relationship:

where $\boldsymbol{\pi}^{\prime} = (\pi_2, \pi_3, \ldots)$.

For a stationary policy $\boldsymbol{\pi} = \mathrm{const}(\pi)$ with constant decision rule $\pi$, the total expected reward simplifies to:

This last expression is called a Neumann series expansion, and it’s guaranteed to exists under the assumptions of bounded reward and discount factor strictly less than one.

Note that for any induced matrix norm $\|\cdot\|$ (i.e., a norm satisfying $\|\mathbf{H}\mathbf{v}\| \leq \|\mathbf{H}\| \cdot \|\mathbf{v}\|$ for all vectors $\mathbf{v}$) and any matrix $\mathbf{H}$, the spectral radius is bounded by:

This inequality provides a practical way to verify the convergence condition $\rho(\mathbf{H}) < 1$ by checking the simpler condition $\|\mathbf{H}\| < 1$ rather than trying to compute the eigenvalues directly.

We can now verify that $(\mathbf{I} - \gamma \mathbf{P}_d)$ is invertible and the Neumann series converges.

1. Norm of the transition matrix: Since $\mathbf{P}_d$ is a stochastic matrix (each row sums to 1 and all entries are non-negative), its $\ell_\infty$-norm is:

   $$\|\mathbf{P}_d\| = \max_{s \in S} \sum_{j \in S} [\mathbf{P}_d]_{s,j} = \max_{s \in S} 1 = 1.$$

   (96)

2. Norm of the scaled matrix: Using the homogeneity property of norms, we have:

   $$\|\gamma \mathbf{P}_d\| = |\gamma| \cdot \|\mathbf{P}_d\| = |\gamma| \cdot 1 = |\gamma|.$$

   (97)

3. Bounding the spectral radius: Since the spectral radius is bounded by the matrix norm:

   $$\rho(\gamma \mathbf{P}_d) \leq \|\gamma \mathbf{P}_d\| = |\gamma|.$$

   (98)

4. Verifying convergence: Since $0 \leq \gamma < 1$ by assumption, we have:

   $$\rho(\gamma \mathbf{P}_d) \leq |\gamma| < 1.$$

   (99)

   This strict inequality guarantees that $(\mathbf{I} - \gamma \mathbf{P}_d)$ is invertible and the Neumann series converges.

Therefore, the Neumann series expansion converges and yields:

Consequently, for a stationary policy, $\mathbf{v}_\gamma^{d^\infty}$ can be determined as the solution to the linear equation:

which can be rearranged to: