Lecture 3: Social Choice Theory & Influences

Topics: Social Choice, Influences, Effects, Derivatives

O'Donnell Chapters: 2.1-2.2
Based on lecture notes by: Vivien Goyal
Notebook by: Gabriel Taboada

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Key Concepts

1. Social Choice Functions: Boolean functions as voting systems
2. Variable Influences: $\text{Inf}_i[f] = \Pr[f(x) \neq f(x^{\oplus i})]$
3. Derivative: $D_i f(x) = f(x^{i \to 1}) - f(x^{i \to 0})$
4. Banzhaf Power Index: Voting power from Boolean analysis

# Install/upgrade boofun (required for Colab)
# This ensures you have the latest version with all features
!pip install --upgrade boofun -q

import boofun as bf
print(f"BooFun version: {bf.__version__}")

[notice] A new release of pip is available: 25.2 -> 26.0
[notice] To update, run: /Library/Developer/CommandLineTools/usr/bin/python3 -m pip install --upgrade pip

/Users/gabrieltaboada/dev/Boofun/boofun/src/boofun/core/errormodels.py:21: UserWarning: uncertainties library not available - some error models disabled
  warnings.warn("uncertainties library not available - some error models disabled")

/Users/gabrieltaboada/dev/Boofun/boofun/src/boofun/quantum/__init__.py:22: UserWarning: Qiskit not available - quantum features limited
  warnings.warn("Qiskit not available - quantum features limited")

BooFun version: 1.1.1

import numpy as np
import matplotlib.pyplot as plt
import boofun as bf
from boofun.visualization import BooleanFunctionVisualizer, plot_function_comparison

np.random.seed(42)
plt.rcParams['figure.figsize'] = (12, 5)

1. Social Choice Functions

A social choice function (SCF) is a Boolean function $f: \{-1, +1\}^n \to \{-1, +1\}$ where:

• Each input $x_i \in \{-1, +1\}$ represents voter $i$'s preference
• Output $f(x)$ is the "society's decision"

Properties of Good Voting Rules

• Unanimous: $f(1,1,...,1) = 1$ and $f(-1,-1,...,-1) = -1$
• Odd: $f(-x) = -f(x)$ (no bias toward either outcome)
• Monotone: If more voters prefer $+1$, outcome shouldn't switch to $-1$
• Symmetric: All voters treated equally

# Common social choice functions
n = 5

functions = {
    "Majority": bf.majority(n),      # Most "fair" symmetric rule
    "Dictator": bf.dictator(n, 0),   # One voter decides everything
    "AND": bf.AND(n),                # Unanimous required
    "OR": bf.OR(n),                  # At least one supporter needed
    "Parity": bf.parity(n),          # XOR - balanced but not monotone!
}

print("Social Choice Functions (n=5 voters):")
print(f"{'Function':15} | Balanced | Monotone | Symmetric")
print("-" * 55)
for name, f in functions.items():
    print(f"{name:15} | {str(f.is_balanced()):8} | {str(f.is_monotone()):8} | {str(f.is_symmetric()):8}")

Social Choice Functions (n=5 voters):
Function        | Balanced | Monotone | Symmetric
-------------------------------------------------------
Majority        | True     | True     | True    
Dictator        | True     | True     | False   
AND             | False    | True     | True    
OR              | False    | True     | True    
Parity          | True     | False    | True    

2. Variable Influences

The influence of variable $i$ measures how often flipping voter $i$'s vote changes the outcome:

$$\text{Inf}_i[f] = \Pr_{x \sim \{-1,+1\}^n}[f(x) \neq f(x^{\oplus i})]$$

Fourier Formula

$$\text{Inf}_i[f] = \sum_{S: i \in S} \hat{f}(S)^2$$

# Compare influences across different voting rules (reusing functions from above)
fig, axes = plt.subplots(1, len(functions), figsize=(16, 4))

for ax, (name, f) in zip(axes, functions.items()):
    influences = f.influences()
    
    ax.bar(range(n), influences, color='steelblue', alpha=0.7)
    ax.set_title(f"{name}\nI[f] = {f.total_influence():.3f}")
    ax.set_xlabel("Voter")
    ax.set_ylabel("Influence")
    ax.set_ylim(0, 1.1)
    ax.set_xticks(range(n))

plt.tight_layout()
plt.show()

print("\nKey Observations:")
print("  Majority: Equal influence (symmetric, monotone)")
print("  Dictator: One voter has all influence")
print("  AND/OR:   Equal but low influence (need many flips to change outcome)")
print("  Parity:   Equal, high influence, but NOT monotone (bad voting rule!)")

Key Observations:
  Majority: Equal influence (symmetric, monotone)
  Dictator: One voter has all influence
  AND/OR:   Equal but low influence (need many flips to change outcome)
  Parity:   Equal, high influence, but NOT monotone (bad voting rule!)

3. Total Influence and Spectral Formula

The total influence is: $$I[f] = \sum_{i=1}^n \text{Inf}_i[f] = \sum_{S} |S| \cdot \hat{f}(S)^2$$

This measures the overall "sensitivity" of the function.

# Verify: I[f] = Σ Inf_i[f] = Σ |S| · f̂(S)²
print("Total Influence: I[f] = Σ Inf_i[f] = Σ|S|·f̂(S)²")
print("=" * 60)
print(f"{'Function':15} | {'Σ Inf_i':<10} | {'Σ|S|·f̂(S)²':<12} | {'f.total_influence()':<10}")
print("-" * 60)

for name, f in functions.items():
    inf_sum = sum(f.influences())
    fourier = f.fourier()
    spectral = sum(bin(s).count('1') * fourier[s]**2 for s in range(2**n))
    direct = f.total_influence()
    
    print(f"{name:15} | {inf_sum:<10.4f} | {spectral:<12.4f} | {direct:<10.4f}")

Total Influence: I[f] = Σ Inf_i[f] = Σ|S|·f̂(S)²
============================================================
Function        | Σ Inf_i    | Σ|S|·f̂(S)²  | f.total_influence()
------------------------------------------------------------
Majority        | 1.8750     | 1.8750       | 1.8750    
Dictator        | 1.0000     | 1.0000       | 1.0000    
AND             | 0.3125     | 0.3125       | 0.3125    
OR              | 0.3125     | 0.3125       | 0.3125    
Parity          | 5.0000     | 5.0000       | 5.0000    

4. Weighted Majority

A weighted majority assigns different weights to voters: $f(x) = \text{sign}(\sum_i w_i \cdot x_i)$

This models systems like shareholder voting or the Electoral College.

# Weighted majority interpolates between majority and dictator
# As we concentrate weight on x_0, influence shifts from equal to concentrated

voting_rules = {
    "Equal [1,1,1,1,1]": bf.weighted_majority([1, 1, 1, 1, 1]),
    "Mild [2,1,1,1,1]": bf.weighted_majority([2, 1, 1, 1, 1]),
    "Strong [5,1,1,1,1]": bf.weighted_majority([5, 1, 1, 1, 1]),
    "Extreme [9,1,1,1,1]": bf.weighted_majority([9, 1, 1, 1, 1]),
    "Dictator": bf.dictator(n, 0),
}

fig = plot_function_comparison(voting_rules, metric="influences")

print("\nAs weight concentrates on x_0:")
print("  Equal weights → uniform influence (majority)")
print("  More weight on x_0 → more influence for x_0")
print("  Extreme weights → approaches dictator")

As weight concentrates on x_0:
  Equal weights → uniform influence (majority)
  More weight on x_0 → more influence for x_0
  Extreme weights → approaches dictator

Summary

Key Takeaways

1. Social Choice as Boolean Functions: Voting rules are monotone, odd Boolean functions

2. Influence = Voting Power: $\text{Inf}_i[f]$ measures how often voter $i$ is "pivotal"

3. Spectral Formula: $\text{Inf}_i[f] = \sum_{S \ni i} \hat{f}(S)^2$

4. Total Influence: $I[f] = \sum_S |S| \cdot \hat{f}(S)^2$

boofun Usage (Direct API)

# Compute influences - direct methods!
influences = f.influences()
total_inf = f.total_influence()
fourier = f.fourier()

# Create voting functions
maj = bf.majority(n)
dict_f = bf.dictator(n, i)