Consider a single query with 2 ad slots. Position CTRs are [0.10, 0.06]. Three advertisers have bids b and quality scores q as: A: b=3, q=2.0; B: b=4, q=1.5; C: b=5, q=1.0. Reserve price r=1.0. Ranking is by b·q. Answer: 1) Under GSP with quality scores, determine the allocation and per-click prices. Use the standard payment formula p_i = (next advertiser’s b·q)/q_i (respecting reserve r). Compute expected revenue and welfare (sum over slots of CTR_slot × value per click where value≈true bid; assume bids reflect values for this part). 2) Under VCG with quality scores and the same reserve, compute each winner’s per-click price and the platform’s expected revenue; compare to GSP. 3) Show a profitable bid shading deviation for B (if one exists) under GSP given A and C bid truthfully. Identify a shading range that preserves B’s slot while reducing price. 4) If each advertiser has a daily budget cap that induces pacing (effective bid multiplier α∈(0,1]), explain how pacing interacts with the quality-adjusted ranking and which mechanism (GSP vs VCG) better preserves truthful reporting under budget constraints.

This question evaluates competency in auction theory and mechanism design, focusing on revenue and welfare analysis with quality-adjusted bids, reserve pricing, and strategic behavior such as bid shading and budget pacing.

How do I approach Statistics & Math interview questions?

Statistics & Math questions require understanding of core concepts and practice. PracHub provides solutions with explanations to help you master statistics & math interviews.

What difficulty level is this interview question?

This is a medium difficulty Statistics & Math question, commonly asked during HR Screen rounds at Roblox.

What role is this question designed for?

This question is commonly asked for Data Scientist candidates at Roblox during technical interviews.

Compute GSP vs VCG payments with reserves | Roblox Interview Question

Sponsored Search Auction with Quality Scores and Reserve

You are analyzing a single-query auction with 2 ad slots. Position CTR multipliers are [0.10, 0.06]. Three advertisers have bids b and quality scores q:

A: b=3, q=2.0
B: b=4, q=1.5
C: b=5, q=1.0

Reserve price r = 1.0. Ads are ranked by the quality-adjusted score s_i = b_i · q_i. Assume ties are broken in favor of higher quality q (A over B). Assume the standard separable-click model: the expected click probability for advertiser i in slot s is CTR_s × q_i.

Answer the following:

GSP with quality scores
- Determine the allocation and per-click prices using the standard payment formula p_i = max(r, (next advertiser’s b·q)/q_i).
- Compute the platform’s expected revenue per impression and total welfare per impression, where welfare = sum over filled slots of (CTR_slot × q_i × value per click) and value per click ≈ bid (assume bids reflect values for this part).
VCG with quality scores (same reserve)
- Compute each winner’s per-click price and the platform’s expected revenue per impression. Compare to GSP.
Bid shading for B under GSP
- Given A and C bid truthfully, identify whether a profitable bid shading deviation exists for B under GSP. Specifically, is there a shading range that preserves B’s slot while reducing B’s per-click price? If not, explain why; if yes, give the range and the impact on B’s utility.
Budgets and pacing
- Suppose each advertiser has a daily budget cap that induces pacing via an effective bid multiplier α ∈ (0, 1], so the effective rank score is α·b·q. Explain how pacing interacts with quality-adjusted ranking and which mechanism (GSP vs. VCG) better preserves truthful reporting under budget constraints.

Sponsored Search Auction with Quality Scores and Reserve

You are analyzing a single-query auction with 2 ad slots. Position CTR multipliers are [0.10, 0.06]. Three advertisers have bids b and quality scores q:

A: b=3, q=2.0

B: b=4, q=1.5

C: b=5, q=1.0

Answer the following:

GSP with quality scores

Determine the allocation and per-click prices using the standard payment formula p_i = max(r, (next advertiser’s b·q)/q_i).
Compute the platform’s expected revenue per impression and total welfare per impression, where welfare = sum over filled slots of (CTR_slot × q_i × value per click) and value per click ≈ bid (assume bids reflect values for this part).

VCG with quality scores (same reserve)

Compute each winner’s per-click price and the platform’s expected revenue per impression. Compare to GSP.

Bid shading for B under GSP

Given A and C bid truthfully, identify whether a profitable bid shading deviation exists for B under GSP. Specifically, is there a shading range that preserves B’s slot while reducing B’s per-click price? If not, explain why; if yes, give the range and the impact on B’s utility.

Budgets and pacing

Suppose each advertiser has a daily budget cap that induces pacing via an effective bid multiplier α ∈ (0, 1], so the effective rank score is α·b·q. Explain how pacing interacts with quality-adjusted ranking and which mechanism (GSP vs. VCG) better preserves truthful reporting under budget constraints.

Compute GSP vs VCG payments with reserves

Quick Overview

Sponsored Search Auction with Quality Scores and Reserve

Solution

Comments (0)

Compute GSP vs VCG payments with reserves

Quick Overview

Sponsored Search Auction with Quality Scores and Reserve

Solution

Comments (0)