Cascading Position Bias

Definition

Cascading Position Bias is a refinement of Position Bias that models more realistic user behavior: users examine items sequentially (top-to-bottom) and the probability of examining an item at position $k$ depends on whether they found satisfactory items at positions $1,...,k-1$.

Formally:

$P({\text{Exam}}_k)={\prod }_{j=1}^{k-1}(1-P({\text{Rel}}_j|q))$

Examination at position $k$ = product of irrelevance of all previous items.

Intuition

Sequential Scanning

Real users don’t independently evaluate every item. They scan top-to-bottom and stop when satisfied:

User sees ranking: [A, B, C, D, E]

Step 1: Examine A
        Is A relevant? 
        YES → Click & Leave
        NO → Continue

Step 2: Examine B (only if A was irrelevant)
        Is B relevant?
        YES → Click & Leave
        NO → Continue

Step 3: Examine C (only if A & B irrelevant)
        ...

This creates dependence: whether you examine item $k$ depends on items $1,...,k-1$.

Contrast with Position-Based Model

PBM (Position-Based Model):

• Examination probability is independent of item relevance
• $P({\text{Exam}}_k)=0.8$ regardless of what’s above

Even if A, B, C are all irrelevant,
C still has 0.8 chance of examination

Cascade Model:

• Examination probability depends on previous items
• If A & B are relevant, user stops; C never examined
• If A & B are irrelevant, C has high exam probability

Mathematical Formulation

The Cascade Model (CM)

Click probability under cascading:

$P({\text{Click}}_{d,k})=P({\text{Exam}}_k)\cdot P({\text{Rel}}_d|q)$

where:

$P({\text{Exam}}_k)={\prod }_{j=1}^{k-1}(1-P({\text{Rel}}_j|q))$

Key: Examination at position $k$ is item-specific and ranking-specific (not just position-specific).

Example

Ranking: [A, B, C, D]
True relevances: $P({\text{Rel}}_A|q)=0.8$, $P({\text{Rel}}_B|q)=0.6$, $P({\text{Rel}}_C|q)=0.7$, $P({\text{Rel}}_D|q)=0.4$

Examination probabilities:

• $P({\text{Exam}}_A)=1.0$ (always examined first)
• $P({\text{Exam}}_B)=1-0.8=0.2$ (20% reach B)
• $P({\text{Exam}}_C)=(1-0.8)(1-0.6)=0.2\times 0.4=0.08$ (8% reach C)
• $P({\text{Exam}}_D)=0.2\times 0.4\times 0.3=0.024$ (2.4% reach D)

Click probabilities:

• $P({\text{Click}}_A)=1.0\times 0.8=0.80$
• $P({\text{Click}}_B)=0.2\times 0.6=0.12$
• $P({\text{Click}}_C)=0.08\times 0.7=0.056$
• $P({\text{Click}}_D)=0.024\times 0.4=0.0096$

Note: Items lower in ranking have much lower click probabilities because of cascading, not just position bias.

Dependent Click Model (DCM)

A specific cascade model parameterization:

$P({\text{Click}}_{d,k})=P({\text{Exam}}_k)\cdot P({\text{Rel}}_d|q)\cdot P({\text{Satisfy}}_d|q)$

Additional factor:

• $P({\text{Satisfy}}_d|q)$ = probability the user is satisfied after clicking (decides to leave)

Parameter ${\lambda }_k$ = rank-dependent satisfaction probability.

Dynamic Bayesian Network (DBN) Model

Another cascade variant:

$P({\text{Exam}}_k|\text{history})=P({\text{Exam}}_{k-1})\cdot (1-\gamma )$

where:

• $\gamma$ = probability of session abandonment (user leaves without finding anything)

Key Differences from PBM

Aspect	PBM	Cascade Model
Exam depends on position	Yes	Yes
Exam depends on relevance of previous items	No	Yes
Ranking-specific exams	No	Yes
Identifiability	Easier	Harder
Empirical fit	Good for some queries	Better overall
Computational complexity	Low	Higher

Empirical Evidence

When Does Cascading Dominate?

Cascading behavior is observed more in:

• Navigational queries: “find this specific item” → user stops once found
• Informational queries in some contexts: “learn about X” → user might scan exhaustively

Less pronounced in:

• Exploratory queries: “show me options” → user might browse many items

Empirical Studies

Research shows:

1. Navigational queries: Cascade model fit better than PBM
2. Commercial queries: Mixed—sometimes cascade, sometimes PBM dominates
3. Informational queries: Varies by context and result quality

Challenges: Session-Dependent Propensities

The critical problem: Under cascading, examination propensities are ranking-specific.

$P({\text{Exam}}_k|\text{ranking }y)$

depends on $y$, not just on $k$.

Example Problem

Scenario 1: Ranking [Relevant, Irrelevant, ...]
  P(Exam_2) = 1 - P(Rel_1) = small (user likely satisfied)

Scenario 2: Ranking [Irrelevant, Relevant, ...]
  P(Exam_2) = 1 - P(Rel_1) = large (user continues scanning)

Same position (position 2), but very different examination probability!

This breaks the standard IPS formula which assumes position-specific propensities.

Solution: Session-Dependent Probabilities

Use clicks in the current session to estimate propensities:

$\hat{P}({\text{Exam}}_k|\text{session})=\frac{\#\text{seen item }k}{\#\text{sessions where user reached }k}$

This requires:

• Logging which positions users examined
• Per-session adaptation
• More data per session

Cascade Model Estimation

EM for Cascade Models

Classic approach: Expectation-Maximization with latent examination variables.

E-step: Given current parameters, infer which items were examined.

M-step: Update parameters to maximize likelihood.

Complication: Must jointly estimate relevance and satisfaction parameters.

RegressionEM with Cascade

Extend cascade model to use features:

$P({\text{Rel}}_d|q)=\text{sigmoid}(f(x_d,x_q;\theta ))$

Fits regression model to cascade-inferred relevance.

IPS Breaks Down with Cascading

The Problem

If you use PBM-based IPS on cascade-generated data:

${\text{IPS}}_{\text{PBM}}={\sum }_k\frac{{\text{Click}}_k}{P({\text{Exam}}_k^{\text{PBM}})}$

where P(\text{Exam}_k^{\text{PBM})}} is position-specific (ignoring cascading):

Result: Severe bias.

Why?

• PBM assumes high examination at rank 3
• Cascade model implies low examination at rank 3 (if ranks 1-2 were relevant)
• IPS weights rank-3 clicks too lightly
• Underestimates relevance of rank-3 items

Empirical Impact

Studies show:

• IPS with PBM on cascade data: biased, noisy estimates
• Specially designed cascade-based IPS: better, but still challenging

Solutions for Cascading

Solution 1: Cascade-Based Propensity Estimation

Estimate propensities accounting for cascading:

$P({\text{Exam}}_k|\text{ranking})={\prod }_{j<k}(1-\hat{P}({\text{Rel}}_j))$

Then apply IPS with cascade propensities.

Advantage: Theoretically sound
Disadvantage: Requires accurate relevance estimation first (circular dependency)

Solution 2: Click Models Instead of IPS

Fit a cascade-based click model directly:

${\max }_{\theta }{\sum }_{\text{logs}}\log P(\text{clicks}|\text{cascade model}(\theta ))$

Advantage: Avoids IPS variance
Disadvantage: Identifiability issues (multiple solutions possible)

Solution 3: Doubly Robust with Cascade Model

Combine cascade-based DM + IPS:

$\text{DR}={\text{DM}}_{\text{cascade}}-{\text{IPS}}_{\text{cascade correction}}$

Advantage: Low variance + unbiased if either component correct
Disadvantage: Complex to implement

Solution 4: Online Randomization

The nuclear option: randomize with random ranking probabilities to break cascading.

$P({\text{Exam}}_k)=\text{known}$ (not confounded by previous relevance)

Advantage: Clean propensity estimates
Disadvantage: Harms user experience; requires new data collection

Real-World Implications

When You Should Care

• Web search: Moderate cascading (users often scan multiple results)
• E-commerce search: Strong cascading (users stop after finding good product)
• Recommendations: Moderate (depends on context)
• Ads: Less cascading (users might ignore ad regardless)

When You Can Ignore It

• Item-level feedback (not position-based)
• Conversion data (not click data)
• Systems where users always examine all items

Related Models

Variants of Cascade

1. Dependent Click Model: Adds satisfaction probability
2. Dynamic Bayesian Network: Adds abandonment probability
3. Cascade with attractiveness: Items attract examination independent of position

Different User Behaviors

• Trust Bias: Position affects relevance perception (different from cascade)
• Item Selection Bias: Hard cutoff (items below fold never seen)
• Outlier Bias: Distinctive items break cascade (attract examination out of order)

Connections

• Extends: Position Bias (PBM is a special case)
• Alternative: Click Models provide joint estimation
• Related: Examination Hypothesis (same assumption but different model)
• Impact on: Inverse Propensity Weighting (breaks standard IPS)
• Solution: Doubly Robust Estimation, cascade-based click models

Appears In

• Click Models
• Unbiased Learning to Rank
• User behavior modeling in Information Retrieval