Merge branch 'main' of https://git.mercurio.moe/Mercury/dth-pingpong-backend

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+# Hybrid Ranking System: Matchmaking Suggestions
+
+## Overview
+
+This proposal outlines a hybrid ranking system for matchmaking in a ping pong app. The system integrates **Elo ratings** for 1v1 matches and **OpenSkill (OSK)** ratings for 2v2 matches. The goal is to use Elo as a supporting metric for matchmaking suggestions in 2v2 scenarios, providing fair and balanced matches for players.
+
+---
+
+## Problem Statement
+
+OpenSkill is ideal for tracking player ratings in 2v2 matches due to its ability to handle uncertainty (![](https://latex.codecogs.com/svg.image?\sigma)) and dynamic team compositions. However, it does not directly account for individual player skill history from 1v1 matches.
+
+The problem lies in matchmaking for 2v2 games when:
+1. Player data is sparse (e.g., fewer 2v2 matches played).
+2. Players' individual skill levels (from Elo) are significantly mismatched within or between teams.
+
+By combining Elo and OpenSkill, we can make more informed matchmaking suggestions.
+
+---
+
+## Proposed Solution
+
+### Approach
+
+We propose using Elo ratings from 1v1 matches as an additional weighting factor when suggesting 2v2 matchups. This involves the following steps:
+
+1. **Team Strength Calculation**:
+   - Use Elo to calculate the **average team Elo**:
+
+![](https://latex.codecogs.com/svg.image?\bg{white}\text{Team&space;Elo}=\frac{\text{Elo}_{\text{Player1}}&plus;\text{Elo}_{\text{Player2}}}{2})
+
+2. **Weigh Elo into OpenSkill Predictions**:
+   - Combine OpenSkill predictions with Elo to generate an **Adjusted Compatibility Score**:
+![](https://latex.codecogs.com/svg.image?\bg{white}\text{Adjusted&space;Score}=w_1\cdot\text{OSK&space;Prediction}&plus;w_2\cdot\text{Elo&space;Difference&space;Factor})
+
+   - *w_1* and *w_2* are weights balancing the influence of OpenSkill and Elo.
+
+3. **Match Balance Assessment**:
+   - Predict the expected win probability for each team using OpenSkill and adjust it with Elo to suggest fairer matches.
+
+4. **Rank Match Suggestions**:
+   - Generate a list of potential matchups ranked by their balance score. Favor matches with close Elo and OpenSkill ratings.
+
+---
+
+### Example Scenario
+
+#### Player Data
+
+| Player | Elo  | OSK (![](https://latex.codecogs.com/svg.image?\mu),![](https://latex.codecogs.com/svg.image?\sigma)) |
+|--------|-------|-----------------------|
+| A      | 1600  | (30, 5)              |
+| B      | 1500  | (28, 6)              |
+| C      | 1550  | (29, 4)              |
+| D      | 1400  | (27, 5)              |
+
+#### Step 1: Form Teams and Calculate Team Elo
+- **Team 1**: A + B → Team Elo: ![](https://latex.codecogs.com/svg.image?\bg{white}\frac{1600&plus;1500}{2}=1550)
+- **Team 2**: C + D → Team Elo: ![](https://latex.codecogs.com/svg.image?\bg{white}\frac{1550&plus;1400}{2}=1475)
+
+#### Step 2: OpenSkill Predictions
+- Team 1: Combined ![](https://latex.codecogs.com/svg.image?\mu)= 29, ![](https://latex.codecogs.com/svg.image?\sigma) = 5.5$
+- Team 2: Combined ![](https://latex.codecogs.com/svg.image?\mu) = 28, ![](https://latex.codecogs.com/svg.image?\sigma) = 4.5
+
+Predicted win probability using OSK: **Team 1 is favored.**
+
+#### Step 3: Adjust with Elo
+- Elo gap: 1550 - 1475 = 75
+- Adjust prediction to slightly favor Team 1 more based on the Elo difference.
+
+#### Step 4: Suggest Matches
+Rank all potential 2v2 matchups based on their **Adjusted Compatibility Score**.
+
+---
+
+## Benefits of the Hybrid System
+
+- **Fairness**: Prevents unbalanced teams by considering both 1v1 and 2v2 performance.
+- **Adaptability**: Handles sparse data effectively by leveraging Elo as a fallback metric.
+- **Dynamic Suggestions**: Offers matchmaking suggestions based on both team composition and player history.
+
+---
+
+## Next Steps
+
+1. Implement the matchmaking algorithm:
+   - Calculate compatibility scores for all possible 2v2 matchups.
+   - Use Elo to filter and refine suggestions.
+2. Define weights *w_1* and *w_2* for Elo and OpenSkill integration.
+3. Test the system with real player data to evaluate balance and fairness.