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:

1. 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).
2. VCG with quality scores (same reserve)
   • Compute each winner’s per-click price and the platform’s expected revenue per impression. Compare to GSP.
3. 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.
4. 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.