Always Have a Plan: Monte Carlo Simulation

🔮 Hello Fellow Magicians,

As we continue our journey through Optimization in the ML Typologies series, this week’s edition dives into a fascinating approach: Monte Carlo Simulation 🎲, a useful method to solve certain types of optimization problems.

✨ Optimization at a Glance

Optimization is all about maximizing or minimizing a specific objective within a system. A few examples:

• 📉 Minimizing risk in an investment portfolio

• 🌍 Maximizing greenhouse gas reductions in building renovations

This week, we'll explore how Monte Carlo Simulation can help us understand and tackle these complex optimization problems.

🎲 Monte Carlo Simulation Explained

While Reinforcement Learning learns through directed trial and error 🧪 and Mathematical Optimization relies on mathematically defining constraints and leveraging solver algorithms 📐, Monte Carlo Simulation is different. In Monte Carlo we generate many scenarios, which through analysis unearth system dynamics and strategies for achieving your objective. Named for its application to partially random systems (like roulette or dice games), Monte Carlo is ideal for understanding relatively simple environments with an element of chance.

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🏆 What’s the Best Property on a Monopoly Board?

To illustrate, let’s build a Monte Carlo Simulation for Monopoly! The question: Which properties are best to purchase? 🏠

Spoiler: Mastering this technique might give you an edge at family game night, so wield your power responsibly! 😏

🛠️ Steps for Running a Monte Carlo Simulation

1. Set Up the Simulation 🛤️: Our first step is to build a simplified Monopoly simulation where each player:

   • Rolls and moves, following the rules 🎲

   • Buys properties based on chance if they land on an unowned space

   • Pays rent when landing on owned properties 💸

1. Tally Property Earnings 💰: Track how much money each property generates over time.

2. Vary the Scenario 🎭: Try different numbers of opponents to see how it impacts the game.

3. Run the Simulation (A Lot!) 🔄: Run the simulation thousands or millions of times to get reliable data. You want to look for the output data descriptive statistics to stabilize. 

4. Analyze the Results 📊: Calculate the mean earnings for each property to find out which has the highest expected return.

In the end, you’ll have a dataset showing each property’s potential game value:

The highest mean value reveals the best properties to purchase—but to get the answer, you'll have to run the simulation yourself or check out the awesome video linked here for more insight!

🔍 Going Further: Strategy Analysis with Monte Carlo Data

Let’s take it up a notch! Output a game log alongside property earnings, showing moves like:

• Player 1 rolls an 8 and buys Vermont Ave.

• Player 2 rolls an 8 and pays rent to Player 1.

This game log allows us to compare similar games and adjust strategies based on real-time dynamics. Imagine you start with one strategy but need to pivot mid-game. By searching for similar scenarios in your data, you can analyze alternative strategies on the fly! ⚡

Monte Carlo Simulations offer flexibility by giving insights into alternative strategies for unexpected situations. Or as Mike Tyson might say if he were a data scientist: Everyone has an optimal strategy until they get punched in the mouth. Luckily, with Monte Carlo, we have millions of alternative strategies ready for just that scenario. 💥

📝 Notes from the Author:

I've found Monte Carlo Simulations invaluable, especially when helping clients understand how many factors impact relatively simple systems. For example, planning timelines to build new buildings or renovate them, given budget and time constraints. You can build a simulation to test different schedules, generating insights to find the best solutions within various constraints. 📅

Let me know if you’d like me to develop a Monopoly Monte Carlo to share with you all!

Happy Simulating! 🧙