In the world of competitive games, whether it’s chess, esports, or AI agents battling it out in virtual arenas, measuring skill accurately is vital. One name dominates this space: the Elo rating system. Originally designed for chess, Elo has become a core tool in artificial intelligence, particularly in reinforcement learning (RL) and self-play environments.
In this blog, we’ll break down what Elo ratings really are, how the formula works, and why the concept of a zero-sum rating system is so important in both human and machine learning competition.
What Is the Elo Rating System?
The Elo rating system was developed by physicist Arpad Elo as a way to calculate the relative skill levels of players. It’s widely used in chess, esports rankings, and increasingly in AI research, especially where agents are trained through competition or self-play.
Unlike fixed scoring systems, Elo is dynamic and comparative — your score changes based on who you play and how well you perform.
What Is a Zero-Sum Rating System?
Before we dive into the Elo formula, it’s important to understand zero-sum systems.
Definition:
A zero-sum system is one in which the total amount of points exchanged between players always adds up to zero.
That means:
- If Player A gains 10 points, Player B must lose 10 points.
- The net change is zero, hence the term zero-sum.
This principle keeps the rating pool balanced, making Elo a fair and effective system for competitive environments.
Elo Rating Formula Explained
Now let’s explore the formula behind Elo scores. At its core, the Elo system updates a player’s rating based on the expected outcome versus the actual result. The formula is:
New Rating = Old Rating + K × (Actual Score — Expected Score)
Where:
- K is a constant determining the sensitivity of rating changes.
- Actual Score is 1 for a win, 0.5 for a draw, and 0 for a loss.
- Expected Score is calculated using the difference in ratings between two players.
The expected score for Player A against Player B is:
Expected Score = 1 / (1 + 10^((Rating_B — Rating_A)/400))
This formula ensures that beating a higher-rated opponent yields a significant rating increase, while losing to a lower-rated one results in a notable decrease.
Let’s Define the Terms:
R₁ = Rating of Player 1 (e.g., Alice)
R₂ = Rating of Player 2 (e.g., Bob)
E₁ = Expected score for Player 1
S₁ = Actual result for Player 1
- Win = 1
- Draw = 0.5
- Loss = 0
K = Constant (controls how fast ratings change; common value = 32)
Step-by-Step Formula
1. Compute Expected Score for Player 1:

2. Update Rating:

Example: Alice vs Bob
Let’s say:
- Alice has a rating of 1600
- Bob has a rating of 1500
- K = 32
and Alice wins the match.
Step 1: Calculate Expected Scores
For Alice:

So Alice is expected to win 64% of the time.
Step 2: Calculate New Ratings
Since Alice won, her actual score S=1

Result:

Notice how the net change is 0? That’s what zero-sum means.
So,
Total change = 0 → Zero-sum confirmed
Why This Makes Sense?
- If a stronger player beats a weaker one, they gain only a few points — it was expected.
- If a weaker player beats a stronger one, they gain many points — it was an upset!
Elo formula ensures that beating a higher-rated opponent yields a significant rating increase, while losing to a lower-rated one results in a notable decrease.
Why Elo Scores Matter in AI
Elo ratings aren’t just for humans. They’re a powerful tool in AI system evaluation, particularly in:
1. Reinforcement Learning (RL)
In RL, agents learn by trial and error. Elo scores allow researchers to:
- Measure progress over time
- Compare current agents with past versions
- Select the best-performing policies
This is heavily used in self-play systems like AlphaGo, MuZero, and OpenAI Five.
2. Game AI and Agent Evaluation
In environments like chess, Go, StarCraft, or Dota 2, Elo ratings help rank multiple AI agents efficiently without needing absolute performance benchmarks.
3. Tournament-Style Testing
When running simulations or agent competitions, Elo allows for fair and dynamic matchups. You don’t need to evaluate against all players — just a few can give you a meaningful ranking.
Key Takeaways
- Elo rating is a dynamic, zero-sum scoring system that ranks players or agents based on their performance against each other.
- It’s fair, scalable, and statistically grounded.
- In AI, Elo ratings are essential for evaluating agents in competitive or game-like environments.
- The zero-sum nature ensures that the system remains stable and meaningful over time.
Conclusion
Whether you’re building a chess bot or training a reinforcement learning agent, understanding and implementing the Elo rating system provides a robust, interpretable, and fair way to measure skill. As AI continues to evolve, especially in competitive environments, Elo remains one of the most trusted tools for evaluation — and for good reason.