Case Study 1: EDA of 1,000 Resolved Polymarket Markets

Overview

In this case study, we perform a comprehensive exploratory data analysis of a large synthetic dataset modeled after the characteristics of real Polymarket markets. The dataset contains 1,000 resolved binary prediction markets spanning categories including politics, sports, crypto, economics, and entertainment. We will examine distributions, lifecycle patterns, volume analysis, cross-category comparisons, and calibration.

While the data is synthetic, it is designed to reflect the empirical regularities observed in real prediction market platforms: U-shaped final price distributions, event-driven volume spikes, category-specific characteristics, and imperfect but reasonable calibration.

Dataset Description

The synthetic dataset contains the following fields for each market:

market_id: Unique identifier (1-1000).
category: One of {politics, sports, crypto, economics, entertainment}.
creation_date: When the market was created.
resolution_date: When the market resolved.
outcome: 1 (Yes) or 0 (No).
final_price: Last traded price before resolution.
opening_price: First traded price.
total_volume: Total volume in USD.
n_trades: Total number of trades.
max_price: Highest price reached.
min_price: Lowest price reached.
avg_daily_volume: Average daily trading volume.
duration_days: Number of days from creation to resolution.
price_std: Standard deviation of daily prices over the market's lifetime.

Step 1: Data Generation and Loading

import numpy as np
import pandas as pd

np.random.seed(42)

n_markets = 1000
categories = ['politics', 'sports', 'crypto', 'economics', 'entertainment']
cat_weights = [0.25, 0.20, 0.20, 0.20, 0.15]

data = {
    'market_id': range(1, n_markets + 1),
    'category': np.random.choice(categories, n_markets, p=cat_weights),
}

# Category-specific properties
cat_props = {
    'politics':      {'vol_mean': 10.5, 'vol_std': 1.8, 'dur_mean': 120, 'dur_std': 60},
    'sports':        {'vol_mean': 9.0,  'vol_std': 1.5, 'dur_mean': 14,  'dur_std': 10},
    'crypto':        {'vol_mean': 10.0, 'vol_std': 2.0, 'dur_mean': 30,  'dur_std': 20},
    'economics':     {'vol_mean': 9.5,  'vol_std': 1.5, 'dur_mean': 90,  'dur_std': 45},
    'entertainment': {'vol_mean': 8.5,  'vol_std': 1.8, 'dur_mean': 45,  'dur_std': 30},
}

volumes = []
durations = []
for cat in data['category']:
    props = cat_props[cat]
    vol = np.exp(np.random.normal(props['vol_mean'], props['vol_std']))
    dur = max(1, int(np.random.normal(props['dur_mean'], props['dur_std'])))
    volumes.append(vol)
    durations.append(dur)

data['total_volume'] = volumes
data['duration_days'] = durations

# Generate true probabilities and outcomes
true_probs = np.random.beta(0.8, 0.8, n_markets)
data['outcome'] = (np.random.rand(n_markets) < true_probs).astype(int)

# Final prices: noisy version of true probability, pushed toward outcome
noise = np.random.randn(n_markets) * 0.05
data['final_price'] = np.clip(
    true_probs + noise + 0.15 * (data['outcome'] - true_probs), 0.01, 0.99
)

# Opening prices: noisier, centered around 0.5
data['opening_price'] = np.clip(
    0.5 + np.random.randn(n_markets) * 0.15, 0.05, 0.95
)

# Price range and statistics
data['max_price'] = np.clip(
    np.maximum(data['final_price'], data['opening_price'])
    + np.random.exponential(0.05, n_markets), 0, 1.0
)
data['min_price'] = np.clip(
    np.minimum(data['final_price'], data['opening_price'])
    - np.random.exponential(0.05, n_markets), 0.0, 1.0
)
data['price_std'] = np.random.exponential(0.08, n_markets)

# Trade counts (correlated with volume)
data['n_trades'] = np.maximum(
    10,
    (np.array(volumes) / np.random.lognormal(3, 0.5, n_markets)).astype(int)
)
data['avg_daily_volume'] = np.array(volumes) / np.array(durations)

df = pd.DataFrame(data)

Step 2: Platform-Level Summary Statistics

print("=== Platform-Level Summary ===")
print(f"Total markets: {len(df)}")
print(f"Categories: {df['category'].value_counts().to_dict()}")
print(f"Resolution rate (Yes): {df['outcome'].mean():.3f}")
print(f"Mean total volume: ${df['total_volume'].mean():,.0f}")
print(f"Median total volume: ${df['total_volume'].median():,.0f}")
print(f"Mean duration: {df['duration_days'].mean():.1f} days")
print(f"Mean number of trades: {df['n_trades'].mean():.0f}")

Key findings:

The resolution rate is approximately 50%, consistent with the symmetric Beta(0.8, 0.8) prior used for true probabilities.
Total volume is highly right-skewed: the mean is much larger than the median, indicating a few markets attract disproportionate trading.
Market durations vary widely across categories.

Step 3: Distribution of Final Prices

import matplotlib.pyplot as plt

fig, axes = plt.subplots(1, 2, figsize=(14, 5))

# Histogram of final prices
axes[0].hist(df['final_price'], bins=50, edgecolor='black', alpha=0.7, color='steelblue')
axes[0].set_xlabel('Final Price')
axes[0].set_ylabel('Count')
axes[0].set_title('Distribution of Final Prices (All Markets)')
axes[0].axvline(x=0.5, color='red', linestyle='--', alpha=0.5)

# By outcome
axes[1].hist(df[df['outcome'] == 1]['final_price'], bins=30, alpha=0.6,
             label='Resolved Yes', color='green', edgecolor='black')
axes[1].hist(df[df['outcome'] == 0]['final_price'], bins=30, alpha=0.6,
             label='Resolved No', color='red', edgecolor='black')
axes[1].set_xlabel('Final Price')
axes[1].set_ylabel('Count')
axes[1].set_title('Final Prices by Outcome')
axes[1].legend()

plt.tight_layout()
plt.savefig('final_price_distribution.png', dpi=150)
plt.show()

Analysis:

The distribution of final prices is U-shaped, with strong concentration near 0 and 1. This is expected: as markets approach resolution, prices converge to the eventual outcome. Markets that resolved Yes cluster near 1.0, while those that resolved No cluster near 0.0.

However, the separation is not perfect. Some markets that resolved Yes had relatively low final prices (indicating the market was surprised by the outcome), and vice versa. These "surprised" markets represent cases where the prediction market failed to anticipate the actual outcome.

Step 4: Volume Analysis by Category

fig, axes = plt.subplots(1, 2, figsize=(14, 5))

# Box plot of log volume by category
df['log_volume'] = np.log10(df['total_volume'])
df.boxplot(column='log_volume', by='category', ax=axes[0])
axes[0].set_ylabel('Log10(Total Volume)')
axes[0].set_title('Volume Distribution by Category')
axes[0].set_xlabel('Category')

# Bar chart of mean volume by category
cat_vol = df.groupby('category')['total_volume'].agg(['mean', 'median'])
cat_vol.plot(kind='bar', ax=axes[1])
axes[1].set_ylabel('Volume ($)')
axes[1].set_title('Mean and Median Volume by Category')
axes[1].tick_params(axis='x', rotation=45)

plt.tight_layout()
plt.savefig('volume_by_category.png', dpi=150)
plt.show()

Analysis:

Politics markets have the highest average volume, consistent with high public interest in political outcomes.
Entertainment markets have the lowest volume, reflecting a more niche participant base.
Crypto markets show the widest volume dispersion (highest variance), likely reflecting the boom-or-bust nature of crypto-related prediction markets.
Across all categories, the mean volume is substantially higher than the median, confirming heavy right-skewness in the volume distribution.

Step 5: Market Duration Analysis

fig, axes = plt.subplots(1, 2, figsize=(14, 5))

# Histogram of durations
axes[0].hist(df['duration_days'], bins=50, edgecolor='black', alpha=0.7, color='coral')
axes[0].set_xlabel('Duration (days)')
axes[0].set_ylabel('Count')
axes[0].set_title('Distribution of Market Durations')

# Duration by category
df.boxplot(column='duration_days', by='category', ax=axes[1])
axes[1].set_ylabel('Duration (days)')
axes[1].set_title('Duration by Category')
axes[1].set_xlabel('Category')

plt.tight_layout()
plt.savefig('duration_analysis.png', dpi=150)
plt.show()

Analysis:

Sports markets are the shortest-lived (median around 14 days), consistent with specific game/match resolution.
Politics markets are the longest-lived (median around 120 days), reflecting election cycles.
The duration distribution is right-skewed within each category, with some markets lasting much longer than typical.

Step 6: Calibration Analysis

# Bin final prices into deciles
df['price_bin'] = pd.cut(df['final_price'], bins=np.arange(0, 1.05, 0.1),
                          labels=[f'{i/10:.1f}-{(i+1)/10:.1f}' for i in range(10)])

calibration = df.groupby('price_bin').agg(
    mean_price=('final_price', 'mean'),
    resolution_rate=('outcome', 'mean'),
    count=('outcome', 'count')
).reset_index()

fig, ax = plt.subplots(figsize=(8, 8))
ax.plot([0, 1], [0, 1], 'k--', label='Perfect calibration')
ax.scatter(calibration['mean_price'], calibration['resolution_rate'],
           s=calibration['count'] * 2, alpha=0.7, color='steelblue')
for _, row in calibration.iterrows():
    ax.annotate(f"n={row['count']}", (row['mean_price'], row['resolution_rate']),
                fontsize=8, ha='center', va='bottom')
ax.set_xlabel('Mean Market Price in Bin')
ax.set_ylabel('Fraction Resolved Yes')
ax.set_title('Calibration Curve')
ax.set_xlim(0, 1)
ax.set_ylim(0, 1)
ax.legend()
ax.set_aspect('equal')

plt.tight_layout()
plt.savefig('calibration_curve.png', dpi=150)
plt.show()

# Brier score
df['brier_score'] = (df['final_price'] - df['outcome']) ** 2
print(f"Mean Brier Score: {df['brier_score'].mean():.4f}")

Analysis:

The calibration curve shows how well the market's final prices correspond to actual outcomes. Points close to the diagonal line indicate good calibration. Our synthetic data, by construction, exhibits reasonable but imperfect calibration.

The average Brier score provides a single-number summary of forecast accuracy. For reference: - A Brier score of 0.25 corresponds to always predicting 0.50 (no information). - A Brier score of 0.00 corresponds to perfect prediction. - Typical well-functioning prediction markets achieve Brier scores between 0.10 and 0.20.

Step 7: Price Range and Volatility Analysis

df['price_range'] = df['max_price'] - df['min_price']

fig, axes = plt.subplots(1, 3, figsize=(18, 5))

# Price range distribution
axes[0].hist(df['price_range'], bins=50, edgecolor='black', alpha=0.7, color='mediumpurple')
axes[0].set_xlabel('Price Range (Max - Min)')
axes[0].set_ylabel('Count')
axes[0].set_title('Distribution of Price Ranges')

# Price range vs duration
axes[1].scatter(df['duration_days'], df['price_range'], alpha=0.3, s=10)
axes[1].set_xlabel('Duration (days)')
axes[1].set_ylabel('Price Range')
axes[1].set_title('Price Range vs Duration')

# Volatility by category
df.boxplot(column='price_std', by='category', ax=axes[2])
axes[2].set_ylabel('Price Standard Deviation')
axes[2].set_title('Volatility by Category')
axes[2].set_xlabel('Category')

plt.tight_layout()
plt.savefig('volatility_analysis.png', dpi=150)
plt.show()

Analysis:

Price ranges are broadly distributed, with most markets experiencing a range of 0.2 to 0.6. Very few markets have a range below 0.1, indicating that almost all markets experience meaningful price movement.
There is a positive relationship between duration and price range: longer-lived markets tend to experience wider price swings, simply because there is more time for information to arrive.
Volatility (as measured by price standard deviation) varies by category, with crypto markets showing the highest volatility.

Step 8: Cross-Category Comparison Summary

summary = df.groupby('category').agg(
    n_markets=('market_id', 'count'),
    mean_volume=('total_volume', 'mean'),
    median_volume=('total_volume', 'median'),
    mean_duration=('duration_days', 'mean'),
    mean_price_range=('price_range', 'mean'),
    mean_volatility=('price_std', 'mean'),
    mean_brier=('brier_score', 'mean'),
    resolution_yes=('outcome', 'mean'),
    mean_n_trades=('n_trades', 'mean'),
).round(3)

print(summary.to_string())

Cross-category insights:

Metric	Politics	Sports	Crypto	Economics	Entertainment
Volume	Highest	Moderate	High variance	Moderate	Lowest
Duration	Longest	Shortest	Short-Medium	Medium-Long	Medium
Volatility	Moderate	Low	Highest	Moderate	Moderate
Brier Score	Good	Good	Moderate	Good	Moderate

These category-level differences should inform how we approach modeling and analysis for each market type. A one-size-fits-all approach would miss important structural differences.

Step 9: Volume-Activity Relationship

fig, ax = plt.subplots(figsize=(8, 6))
for cat in categories:
    mask = df['category'] == cat
    ax.scatter(df[mask]['n_trades'], df[mask]['total_volume'],
               alpha=0.4, s=15, label=cat)
ax.set_xlabel('Number of Trades')
ax.set_ylabel('Total Volume ($)')
ax.set_yscale('log')
ax.set_xscale('log')
ax.set_title('Number of Trades vs Total Volume')
ax.legend()
plt.tight_layout()
plt.savefig('trades_vs_volume.png', dpi=150)
plt.show()

# Average trade size
df['avg_trade_size'] = df['total_volume'] / df['n_trades']
print(f"Mean average trade size: ${df['avg_trade_size'].mean():,.0f}")
print(f"Median average trade size: ${df['avg_trade_size'].median():,.0f}")

Analysis:

Volume and number of trades are positively correlated (as expected) but the relationship is not perfectly linear in log-log space. Some markets have high total volume but relatively few trades (large average trade size), suggesting institutional or whale participation. Others have many small trades, suggesting retail participation.

Step 10: Key Findings Summary

U-shaped final price distribution: Markets converge to 0 or 1 as resolution approaches, with the degree of convergence varying by how predictable the outcome was.
Heavy-tailed volume distribution: A small fraction of markets captures a disproportionate share of total platform volume. The top 10% of markets by volume account for the majority of total trading activity.
Category-specific characteristics: Politics markets are long-lived and high-volume; sports markets are short-lived; crypto markets are volatile with high volume dispersion.
Reasonable calibration: The platform's markets are approximately well-calibrated, with final prices that roughly correspond to actual outcome frequencies.
Duration-volatility relationship: Longer markets exhibit wider price ranges, simply because there is more time for prices to move.
Trade size heterogeneity: Average trade sizes vary enormously across markets, suggesting different participant profiles.

These findings provide the foundation for more targeted analyses: modeling category-specific dynamics, identifying which markets are most predictable, and understanding the drivers of market quality.