Markov decision process in marketing intution

Graph
Markov
Author

Fabian

Published

October 6, 2024

Introduction

Markov Decision Processes (MDPs) might sound like a complex mathematical concept, but they’re incredibly useful for making decisions in uncertain environments. In this post, we’ll explore how MDPs can be applied to marketing decisions, making it easier to understand both the concept and its practical applications.

What is a Markov Decision Process?

An MDP is a mathematical framework for modeling decision-making where outcomes are partly random and partly controlled by a decision-maker. Think of it as a formalized way to make decisions when:

  1. You have different states your system can be in
  2. You can take various actions
  3. The outcomes of your actions are somewhat uncertain
  4. You receive rewards (or incur costs) based on your decisions

Let’s visualize this with Python:

import networkx as nx
import matplotlib.pyplot as plt

def create_marketing_mdp():
    G = nx.DiGraph()
    
    # Define states
    states = ['New Lead', 'Engaged', 'Considering', 'Customer']
    
    # Add nodes
    for state in states:
        G.add_node(state)
    
    # Add edges with actions
    edges = [
        ('New Lead', 'Engaged', 'Email Campaign'),
        ('New Lead', 'Considering', 'Direct Call'),
        ('Engaged', 'Considering', 'Product Demo'),
        ('Engaged', 'Customer', 'Special Offer'),
        ('Considering', 'Customer', 'Personalized Proposal'),
        ('Considering', 'Engaged', 'Follow-up'),
    ]
    
    G.add_edges_from([(x[0], x[1]) for x in edges])
    
    # Create layout
    pos = nx.spring_layout(G)
    
    # Draw the graph
    plt.figure(figsize=(8, 8))
    nx.draw_networkx_nodes(G, pos, node_color='lightblue', 
                          node_size=2000)
    nx.draw_networkx_edges(G, pos, edge_color='gray', 
                          arrows=True, arrowsize=20)
    nx.draw_networkx_labels(G, pos)
    
    # Add edge labels
    edge_labels = {(x[0], x[1]): x[2] for x in edges}
    nx.draw_networkx_edge_labels(G, pos, edge_labels, font_size=8)
    
    plt.title("Marketing Customer Journey as an MDP")
    plt.axis('off')
    plt.show()

create_marketing_mdp()

A Marketing Example: Customer Journey Optimization

Let’s break down a concrete marketing example:

States

In our customer journey, we have four main states: 1. New Lead 2. Engaged 3. Considering 4. Customer

Actions

For each state, we have different possible marketing actions: - Email campaigns - Direct calls - Product demos - Special offers - Personalized proposals - Follow-ups

Transition Probabilities

Let’s model some example transition probabilities:

import pandas as pd

# Create transition probability matrix
transitions = {
    'Current State': ['New Lead', 'New Lead', 'Engaged', 'Engaged'],
    'Action': ['Email Campaign', 'Direct Call', 'Product Demo', 'Special Offer'],
    'Next State': ['Engaged', 'Considering', 'Considering', 'Customer'],
    'Probability': [0.3, 0.4, 0.5, 0.6],
    'Cost': [10, 50, 100, 200],
    'Expected Revenue': [0, 0, 0, 1000]
}

df = pd.DataFrame(transitions)
df
Current State Action Next State Probability Cost Expected Revenue
0 New Lead Email Campaign Engaged 0.3 10 0
1 New Lead Direct Call Considering 0.4 50 0
2 Engaged Product Demo Considering 0.5 100 0
3 Engaged Special Offer Customer 0.6 200 1000

Rewards

The rewards in our marketing MDP could include: - Revenue from converted customers - Minus the cost of marketing actions - Long-term customer value considerations

Implementing an MDP Solver

Here’s a simple value iteration implementation for our marketing MDP:

def value_iteration(states, actions, transitions, rewards, discount_factor=0.9):
    # Initialize value function
    V = {state: 0 for state in states}
    theta = 0.01  # Convergence threshold
    
    while True:
        delta = 0
        V_new = V.copy()
        
        for s in states:
            if s == 'Customer':  # Terminal state
                continue
                
            # Find maximum value over all actions
            values = []
            for a in actions:
                v = 0
                # Sum over all possible next states
                for s_next in states:
                    # Simplified transition probability
                    prob = 0.3  # Example probability
                    reward = rewards.get((s, a, s_next), 0)
                    v += prob * (reward + discount_factor * V[s_next])
                values.append(v)
            
            V_new[s] = max(values)
            delta = max(delta, abs(V_new[s] - V[s]))
        
        V = V_new
        if delta < theta:
            break
    
    return V

# Example usage
states = ['New Lead', 'Engaged', 'Considering', 'Customer']
actions = ['Email Campaign', 'Direct Call', 'Product Demo', 'Special Offer']
rewards = {
    ('New Lead', 'Email Campaign', 'Engaged'): -10,
    ('New Lead', 'Direct Call', 'Considering'): -50,
    ('Engaged', 'Product Demo', 'Considering'): -100,
    ('Engaged', 'Special Offer', 'Customer'): 800,
}

optimal_values = value_iteration(states, actions, None, rewards)
print("\nOptimal Values for Each State:")
for state, value in optimal_values.items():
    print(f"{state}: {value:.2f}")

Optimal Values for Each State:
New Lead: 341.01
Engaged: 581.01
Considering: 341.01
Customer: 0.00

Practical Implications

Using MDPs in marketing offers several advantages:

  1. Systematic Decision Making: Instead of gut feelings, decisions are based on data and expected outcomes.

  2. Long-term Optimization: The discount factor helps balance immediate returns with long-term value.

  3. Risk Management: Probability distributions help account for uncertainty in customer behavior.

  4. Resource Allocation: Understanding the value of each state helps optimize marketing budget allocation.

Conclusion

MDPs provide a powerful framework for optimizing marketing decisions. While the math might seem complex, the underlying concept is straightforward: make decisions that maximize expected long-term rewards while accounting for uncertainty.

References

  1. Puterman, M. L. (2014). Markov Decision Processes: Discrete Stochastic Dynamic Programming.
  2. Russell, S., & Norvig, P. (2020). Artificial Intelligence: A Modern Approach.
Back to top