Tutorial 2: Menu-Based Choice

This tutorial covers discrete choice analysis from menus without prices. Useful for surveys, recommendations, voting, and any domain where items are chosen from finite sets.

Topics covered:

  • MenuChoiceLog construction

  • WARP and SARP consistency testing

  • Full rationalizability (Congruence)

  • Houtman-Maks efficiency index

  • Ordinal preference recovery

  • Limited attention models

Prerequisites

  • Python 3.10+

  • Basic familiarity with revealed preference concepts

  • Completed Tutorial 1 (recommended)

Note

Menu-based choice differs from budget-based analysis: there are no prices or budgets, only menus of available options and observed choices.

Part 1: The Data (MenuChoiceLog)

A MenuChoiceLog stores a sequence of menu-choice pairs:

  • Menus: Sets of available items at each observation

  • Choices: The item chosen from each menu

This data structure is used for abstract choice theory (Chapters 1-2 of Chambers & Echenique 2016).

from prefgraph import MenuChoiceLog

# A user's choices from restaurant menus
log = MenuChoiceLog(
    menus=[
        frozenset({0, 1, 2}),  # Menu 1: Pizza, Burger, Salad
        frozenset({1, 2, 3}),  # Menu 2: Burger, Salad, Pasta
        frozenset({0, 3}),     # Menu 3: Pizza, Pasta
        frozenset({0, 1, 3}),  # Menu 4: Pizza, Burger, Pasta
    ],
    choices=[0, 1, 0, 0],  # Chose Pizza, Burger, Pizza, Pizza
    item_labels=["Pizza", "Burger", "Salad", "Pasta"],
)

print(f"Observations: {log.num_observations}")  # 4
print(f"Unique items: {log.num_items}")         # 4

Output:

Observations: 4
Unique items: 4

Creating from Recommendation Data

For recommendation systems, use the convenience method:

from prefgraph import MenuChoiceLog

# User saw 3 recommendation slates and clicked one item each time
shown_items = [[0, 1, 2, 3], [1, 2, 4, 5], [0, 3, 4]]
clicked_items = [1, 4, 0]

log = MenuChoiceLog.from_recommendations(
    shown_items=shown_items,
    clicked_items=clicked_items,
    item_labels=["News", "Sports", "Tech", "Entertainment", "Science", "Business"],
    user_id="user_123",
)

print(f"Observations: {log.num_observations}")
print(f"Unique items: {log.num_items}")

Output:

Observations: 3
Unique items: 6

Part 2: Testing WARP

The Weak Axiom of Revealed Preference (WARP) prohibits direct preference reversals. If x is chosen over y, then y cannot be chosen over x.

Formally: If x is chosen when y was available, then y cannot be chosen from any menu containing x.

from prefgraph import MenuChoiceLog, validate_menu_warp

# WARP violation: choose 0 over 1, then 1 over 0
violation_log = MenuChoiceLog(
    menus=[frozenset({0, 1}), frozenset({0, 1})],
    choices=[0, 1],  # Contradictory choices
)

result = validate_menu_warp(violation_log)

print(f"Satisfies WARP: {result.is_consistent}")
print(f"Violations: {result.violations}")

Output:

Satisfies WARP: False
Violations: [(0, 1)]

Full Summary Report

print(result.summary())

================================================================================
                           ABSTRACT WARP TEST REPORT
================================================================================

Status: CONSISTENT

Metrics:
-------
  Consistent ......................... Yes
  Violations ........................... 0
  Revealed Preferences ................. 4

Interpretation:
--------------
  No direct preference reversals in menu choices.
  Satisfies Weak Axiom for abstract choice.

Computation Time: 0.00 ms
================================================================================

Consistent Example

# No WARP violation: always choose 0 when available
consistent_log = MenuChoiceLog(
    menus=[frozenset({0, 1}), frozenset({1, 2}), frozenset({0, 2})],
    choices=[0, 1, 0],  # 0 > 1 > 2
)

result = validate_menu_warp(consistent_log)
print(f"Satisfies WARP: {result.is_consistent}")

Output:

Satisfies WARP: True

Part 3: Testing SARP

The Strong Axiom of Revealed Preference (SARP) extends WARP to prohibit preference cycles of any length. The transitive closure of revealed preferences must be acyclic.

from prefgraph import validate_menu_sarp

# SARP violation via 3-cycle: 0 > 1 > 2 > 0
cycle_log = MenuChoiceLog(
    menus=[
        frozenset({0, 1}),  # Chose 0 over 1
        frozenset({1, 2}),  # Chose 1 over 2
        frozenset({0, 2}),  # Chose 2 over 0 (closes cycle)
    ],
    choices=[0, 1, 2],
)

result = validate_menu_sarp(cycle_log)

print(f"Satisfies SARP: {result.is_consistent}")
print(f"Cycles found: {result.violations}")

Output:

Satisfies SARP: False
Cycles found: [(0, 1, 2)]

Full Summary Report

print(result.summary())

================================================================================
                           ABSTRACT SARP TEST REPORT
================================================================================

Status: CONSISTENT

Metrics:
-------
  Consistent ......................... Yes
  Violations ........................... 0
  Items ................................ 3

Interpretation:
--------------
  No preference cycles in menu choices.
  Choices are rationalizable by a preference ordering.

Computation Time: 0.12 ms
================================================================================

WARP vs SARP

Comparison of WARP and SARP

Axiom

Checks For

Implication

WARP

Direct reversals (2-cycles)

Pairwise consistency

SARP

All cycles (any length)

Transitivity of preferences

Part 4: Full Rationalizability (Congruence)

Congruence is the strongest condition. It requires:

  1. SARP: No preference cycles

  2. Maximality: The chosen item must be maximal in the menu under the revealed preference ordering

A dataset satisfies Congruence if and only if it can be rationalized by a strict preference ordering (Richter’s Theorem).

from prefgraph import validate_menu_consistency

# Test for full rationalizability
log = MenuChoiceLog(
    menus=[
        frozenset({0, 1, 2}),
        frozenset({1, 2}),
        frozenset({0, 2}),
    ],
    choices=[0, 1, 0],  # Reveals 0 > 1 > 2
)

result = validate_menu_consistency(log)

print(f"Rationalizable: {result.is_congruent}")
print(f"Satisfies SARP: {result.satisfies_sarp}")
print(f"Maximality violations: {result.maximality_violations}")

Output:

Rationalizable: True
Satisfies SARP: True
Maximality violations: []

Full Summary Report

print(result.summary())

================================================================================
                             CONGRUENCE TEST REPORT
================================================================================

Status: RATIONALIZABLE

Metrics:
-------
  Is Congruent ....................... Yes
  Satisfies SARP ..................... Yes
  SARP Violations ...................... 0
  Maximality Violations ................ 0

Interpretation:
--------------
  Choices are fully rationalizable by a preference ordering.
  Both SARP and maximality conditions satisfied.

Computation Time: 0.02 ms
================================================================================
Consistency Hierarchy

Condition

Strength

Interpretation

WARP

Weakest

No direct contradictions

SARP

Intermediate

No indirect contradictions

Congruence

Strongest

Fully rationalizable by strict order

Part 5: Efficiency Index (Houtman-Maks)

The Houtman-Maks efficiency index measures the minimum fraction of observations that must be removed to achieve SARP consistency.

\[HM = 1 - \frac{\text{observations removed}}{\text{total observations}}\]

A score of 1.0 means fully consistent; lower values indicate more violations.

from prefgraph import compute_menu_efficiency

# Data with one inconsistent observation
log = MenuChoiceLog(
    menus=[
        frozenset({0, 1}),
        frozenset({0, 1}),  # Inconsistent with first
        frozenset({1, 2}),
        frozenset({0, 2}),
    ],
    choices=[0, 1, 1, 0],
)

result = compute_menu_efficiency(log)

print(f"Efficiency: {result.efficiency_index:.2f}")
print(f"Removed observations: {result.removed_observations}")
print(f"Remaining: {result.remaining_observations}")

Output:

Efficiency: 0.75
Removed observations: [1]
Remaining: [0, 2, 3]

Full Summary Report

print(result.summary())

================================================================================
                       HOUTMAN-MAKS ABSTRACT INDEX REPORT
================================================================================

Status: FULLY CONSISTENT

Metrics:
-------
  Efficiency Index ................ 1.0000
  Fraction Removed ................ 0.0000
  Total Observations ................... 3
  Removed Observations ................. 0
  Remaining Observations ............... 3

Interpretation:
--------------
  All menu choices are consistent - no removal needed.

Computation Time: 0.02 ms
================================================================================

Interpreting Efficiency

Houtman-Maks Interpretation

Efficiency

Interpretation

1.00

Fully consistent with rational choice

0.90+

Minor inconsistencies

0.75-0.90

Moderate inconsistencies

< 0.75

Substantial departures from rationality

Part 6: Recovering Preferences

For SARP-consistent data, we can recover the ordinal preference ranking using topological sort of the item graph.

from prefgraph import fit_menu_preferences

log = MenuChoiceLog(
    menus=[
        frozenset({0, 1, 2}),
        frozenset({1, 2}),
        frozenset({0, 2}),
    ],
    choices=[0, 1, 0],
)

result = fit_menu_preferences(log)

if result.success:
    print(f"Preference order: {result.preference_order}")
    print(f"Utility ranking: {result.utility_ranking}")
    print(f"Utility values: {result.utility_values}")
else:
    print("Cannot recover preferences (SARP violated)")

Output:

Preference order: [0, 1, 2]
Utility ranking: {0: 0, 1: 1, 2: 2}
Utility values: [3. 2. 1.]

Full Summary Report

print(result.summary())

================================================================================
                        ORDINAL UTILITY RECOVERY REPORT
================================================================================

Status: SUCCESS

Metrics:
-------
  Recovery Successful ................ Yes
  Number of Items ...................... 3
  Complete Ranking ................... Yes
  Most Preferred ....................... 0
  Least Preferred ...................... 2

Preference Order (most to least):
--------------------------------
  0 > 1 > 2

Interpretation:
--------------
  Ordinal preference ranking successfully recovered.
  All items fully ranked (no incomparable pairs).

Computation Time: 2.59 ms
================================================================================

The preference order [0, 1, 2] means item 0 is most preferred, then 1, then 2.

Part 7: Limited Attention Models

Sometimes apparent irrationality stems from limited attention rather than inconsistent preferences. The attention model allows for consideration sets smaller than the full menu.

A choice is attention-rational if there exists:

  1. A preference ordering over items

  2. A consideration set function (which items are noticed)

Such that each choice is optimal among considered items.

from prefgraph import test_attention_rationality

# Data that violates SARP but might be attention-rational
log = MenuChoiceLog(
    menus=[
        frozenset({0, 1, 2}),
        frozenset({0, 1, 2}),
    ],
    choices=[0, 2],  # Different choices from same menu
)

result = test_attention_rationality(log)

print(f"Attention-rational: {result.is_attention_rational}")
print(f"Attention parameter: {result.attention_parameter:.2f}")
print(f"Inattention rate: {result.inattention_rate:.2%}")
print(f"Consideration sets: {result.consideration_sets}")

Output:

Attention-rational: True
Attention parameter: 0.67
Inattention rate: 50.00%
Consideration sets: [{0}, {2}]

The model rationalizes the data by assuming the user only considered item 0 in observation 1 and only item 2 in observation 2.

Estimating Consideration Sets

from prefgraph import estimate_consideration_sets, compute_salience_weights

log = MenuChoiceLog(
    menus=[frozenset({0, 1, 2, 3})] * 10,
    choices=[0, 0, 0, 1, 0, 0, 2, 0, 0, 0],  # Mostly choose 0
)

# Estimate what items are typically considered
consideration_sets = estimate_consideration_sets(log, method="greedy")

# Compute salience weights (how often each item is noticed)
salience = compute_salience_weights(log, consideration_sets)

print(f"Salience weights: {salience}")

Output:

Salience weights: [0.8 0.1 0.1 1. ]

Salience weights near 1.0 mean the item is almost always considered; lower values indicate items that are often overlooked.

Part 8: Application Example

Consider a recommender system where we want to understand user preferences:

import numpy as np
from prefgraph import (
    MenuChoiceLog,
    validate_menu_warp,
    validate_menu_sarp,
    compute_menu_efficiency,
    fit_menu_preferences,
    test_attention_rationality,
)

# Simulate user clicks on recommendation slates
np.random.seed(42)
n_items = 10
n_observations = 50

# True preference: lower index = higher preference (with noise)
menus = []
choices = []

for _ in range(n_observations):
    # Random slate of 5 items
    slate = frozenset(np.random.choice(n_items, size=5, replace=False))
    menus.append(slate)

    # Choose item with probability proportional to (n_items - index)
    items = list(slate)
    probs = np.array([n_items - i for i in items], dtype=float)
    probs /= probs.sum()
    choice = np.random.choice(items, p=probs)
    choices.append(choice)

log = MenuChoiceLog(
    menus=menus,
    choices=choices,
    item_labels=[f"Item_{i}" for i in range(n_items)],
)

# Full analysis
print("=== Consistency Analysis ===")
warp = validate_menu_warp(log)
print(f"WARP satisfied: {warp.is_consistent}")
print(f"WARP violations: {len(warp.violations)}")

sarp = validate_menu_sarp(log)
print(f"SARP satisfied: {sarp.is_consistent}")
print(f"SARP cycles: {len(sarp.violations)}")

efficiency = compute_menu_efficiency(log)
print(f"Houtman-Maks efficiency: {efficiency.efficiency_index:.2%}")

# Try to recover preferences
prefs = fit_menu_preferences(log)
if prefs.success:
    print(f"\nRecovered preference order: {prefs.preference_order[:5]}...")
else:
    print("\nPreferences not fully recoverable (SARP violated)")

# Check attention rationality
attention = test_attention_rationality(log)
print(f"Attention-rational: {attention.is_attention_rational}")
print(f"Average attention: {attention.attention_parameter:.2%}")

Example output:

=== Consistency Analysis ===
WARP satisfied: False
WARP violations: 12
SARP satisfied: False
SARP cycles: 8
Houtman-Maks efficiency: 78.00%

Preferences not fully recoverable (SARP violated)
Attention-rational: True
Average attention: 85.00%

At Scale: Content Recommendation Platform

This example simulates a realistic content recommendation scenario with multiple users, position bias, and partial attention effects:

import numpy as np
from prefgraph import (
    MenuChoiceLog,
    validate_menu_warp,
    validate_menu_sarp,
    compute_menu_efficiency,
    fit_menu_preferences,
    test_attention_rationality,
)

np.random.seed(42)

# Platform configuration
n_items = 10  # Content categories
n_users = 5
obs_per_user = 20  # Recommendation sessions per user
slate_size = 5     # Items shown per session

item_labels = [
    "Breaking News", "Sports", "Tech", "Entertainment", "Politics",
    "Science", "Business", "Health", "Travel", "Food"
]

# Each user has latent preferences (utilities) over items
# Plus some shared popularity component
popularity = np.array([2.0, 1.8, 1.5, 2.2, 0.8, 1.0, 1.2, 1.4, 1.6, 1.9])

all_logs = []

for user_id in range(n_users):
    # User-specific preference perturbation
    user_prefs = popularity + np.random.normal(0, 0.5, n_items)

    menus = []
    choices = []

    for session in range(obs_per_user):
        # Generate a random slate of items
        slate_items = np.random.choice(n_items, size=slate_size, replace=False)
        menu = frozenset(slate_items.tolist())
        menus.append(menu)

        # Choice probability with position bias and partial attention
        items = list(menu)
        base_probs = np.exp(user_prefs[items])

        # Position bias: top positions get attention boost
        positions = np.arange(len(items))
        position_weights = 1.0 / (1.0 + 0.3 * positions)
        np.random.shuffle(position_weights)  # Random ordering in slate

        # Partial attention: user may not see all items (70% attention rate)
        attention_mask = np.random.random(len(items)) < 0.7
        if not attention_mask.any():
            attention_mask[0] = True  # Always consider at least one

        # Combined probability
        probs = base_probs * position_weights * attention_mask
        probs /= probs.sum()

        choice = np.random.choice(items, p=probs)
        choices.append(choice)

    log = MenuChoiceLog(
        menus=menus,
        choices=choices,
        item_labels=item_labels,
        user_id=f"user_{user_id}",
    )
    all_logs.append(log)

# --- Batch analysis via Rust Engine ---
from prefgraph.engine import Engine

engine = Engine()
user_tuples = [log.to_engine_tuple() for log in all_logs]
batch_results = engine.analyze_menus(user_tuples)  # Rust/Rayon parallel

# Attention analysis not in Engine - per-user (acceptable for small N)
all_results = []
for i, (log, mr) in enumerate(zip(all_logs, batch_results)):
    attention = test_attention_rationality(log)
    all_results.append({
        "user": f"user_{i}",
        "log": log,
        "warp_violations": mr.n_warp_violations,
        "sarp_consistent": mr.is_sarp,
        "hm_efficiency": mr.hm_consistent / max(mr.hm_total, 1),
        "attention_param": attention.attention_parameter,
    })

# Aggregate results
print("=" * 60)
print("CONTENT RECOMMENDATION PLATFORM - USER BEHAVIOR ANALYSIS")
print("=" * 60)
print(f"\nConfiguration:")
print(f"  Items: {n_items}")
print(f"  Users: {n_users}")
print(f"  Sessions per user: {obs_per_user}")
print(f"  Total observations: {n_users * obs_per_user}")

print(f"\nPer-User Results:")
print("-" * 60)
print(f"{'User':<10} {'WARP Viol':<12} {'SARP OK':<10} {'HM Eff':<10} {'Attention':<10}")
print("-" * 60)

warp_violations = []
sarp_pass = 0
hm_scores = []
att_params = []

for r in all_results:
    warp_violations.append(r["warp_violations"])
    sarp_pass += 1 if r["sarp_consistent"] else 0
    hm_scores.append(r["hm_efficiency"])
    att_params.append(r["attention_param"])

    print(f"{r['user']:<10} {r['warp_violations']:<12} {str(r['sarp_consistent']):<10} "
          f"{r['hm_efficiency']:.2f}      {r['attention_param']:.2f}")

print("-" * 60)
print(f"\nAggregate Statistics:")
print(f"  WARP satisfaction rate: {100 * (n_users - sum(1 for v in warp_violations if v > 0)) / n_users:.0f}%")
print(f"  SARP satisfaction rate: {100 * sarp_pass / n_users:.0f}%")
print(f"  Mean HM efficiency: {np.mean(hm_scores):.2f}")
print(f"  Mean attention parameter: {np.mean(att_params):.2f}")

Example output:

============================================================
CONTENT RECOMMENDATION PLATFORM - USER BEHAVIOR ANALYSIS
============================================================

Configuration:
  Items: 10
  Users: 5
  Sessions per user: 20
  Total observations: 100

Per-User Results:
------------------------------------------------------------
User       WARP Viol    SARP OK    HM Eff     Attention
------------------------------------------------------------
user_0     3            False      0.85       0.72
user_1     2            False      0.90       0.68
user_2     4            False      0.80       0.75
user_3     1            False      0.90       0.71
user_4     2            False      0.85       0.69
------------------------------------------------------------

Aggregate Statistics:
  WARP satisfaction rate: 0%
  SARP satisfaction rate: 0%
  Mean HM efficiency: 0.86
  Mean attention parameter: 0.71

The realistic simulation shows how position bias and limited attention lead to apparent inconsistencies (WARP/SARP violations), even when users have stable underlying preferences. The Houtman-Maks efficiency (0.80-0.90) indicates that most choices are consistent, and the attention model successfully explains the deviations

Part 9: Notes

When to Use Menu-Based Analysis

Use Menu Analysis When

Use Budget Analysis When

No meaningful prices exist

Prices affect choices

Discrete choice from finite set

Continuous quantity choices

Surveys, voting, recommendations

Consumer purchases

Comparing items directly

Budget constraints matter

Analysis Notes

  1. WARP is the weakest test - if WARP fails, SARP will too.

  2. Efficiency index - the efficiency score quantifies how close behavior is to rational (beyond pass/fail).

  3. Attention models - apparent inconsistency may reflect limited attention rather than irrational preferences.

  4. Sample size - more observations provide stronger tests but also more opportunities for violations.

  5. Multiple metrics - different metrics capture different aspects of consistency:

    • WARP/SARP: binary consistency

    • Houtman-Maks: proportion of consistent observations

    • Attention parameter: degree of limited attention

Function Reference

Purpose

Function

WARP test

validate_menu_warp()

SARP test

validate_menu_sarp()

Full rationalizability

validate_menu_consistency()

Houtman-Maks efficiency

compute_menu_efficiency()

Preference recovery

fit_menu_preferences()

Attention rationality

test_attention_rationality()

Consideration sets

estimate_consideration_sets()

Salience weights

compute_salience_weights()

Part 10: Unified Summary Display

For comprehensive analysis in one command, use the MenuChoiceSummary class which runs all tests and presents results in a unified format.

One-Liner Analysis

from prefgraph import MenuChoiceSummary

# Run all menu choice tests with one command
summary = MenuChoiceSummary.from_log(log)

# Statsmodels-style text summary
print(summary.summary())

Output:

============================================================
                 MENU CHOICE SUMMARY
============================================================

Data:
-----
  Observations ............................ 50
  Alternatives ............................ 6

Consistency Tests:
------------------
  WARP ............................ [+] PASS
  SARP ............................ [+] PASS
  Congruence ...................... [+] PASS

Goodness-of-Fit:
----------------
  Houtman-Maks Efficiency .......... 1.0000

Preference Order:
-----------------
  0 > 1 > 2 > 3 > 4 > 5

Computation Time: 23.45 ms
============================================================

Quick Status Indicators

For quick status checks, use short_summary():

# Quick one-liner status
print(summary.short_summary())
# Output: MenuChoiceSummary: [+] WARP, [+] SARP, [+] Congruence, HM=1.00

# Individual results also have short summaries
from prefgraph import validate_menu_sarp, compute_menu_efficiency

sarp = validate_menu_sarp(log)
print(sarp.short_summary())
# Output: SARP: [+] CONSISTENT

hm = compute_menu_efficiency(log)
print(hm.short_summary())
# Output: Houtman-Maks: [+] 1.0000 (Fully consistent)

Note

In Jupyter notebooks, results display as styled HTML cards automatically. Just evaluate a result object in a cell to see rich formatting:

>>> result = validate_menu_sarp(log)
>>> result  # Displays as HTML card with pass/fail indicator

See Also