Compute GSP vs VCG payments with reserves

Suppose there is one query with two ad slots. The position click-through rates are $[0.10, 0.06]$. Three advertisers arrive with bids $b$ and quality scores $q$: A has $b=3, q=2.0$; B has $b=4, q=1.5$; C has $b=5, q=1.0$. The reserve price is $r=1.0$. Ads are ranked by $b \cdot q$.

Answer:

1. For GSP with quality scores, find the slot allocation and the per-click prices. Apply the usual payment rule $p_i = \frac{\text{next advertiser's } b \cdot q}{q_i}$, while respecting the reserve $r$. Then compute expected revenue and welfare, where welfare is the sum over slots of $\text{CTR}_\text{slot} \times \text{value per click}$ and value is approximated by the true bid; for this part, assume bids equal values.
2. Under VCG with quality scores and the same reserve, calculate each winning advertiser’s per-click price and the platform’s expected revenue, then compare the result with GSP.
3. Assuming A and C bid truthfully, demonstrate a profitable bid-shading deviation for B under GSP if one exists. Give a shading interval that keeps B in the same slot but lowers its price.
4. If every advertiser has a daily budget cap that creates pacing through an effective bid multiplier $\alpha \in (0,1]$, explain how pacing affects the quality-adjusted ranking, and state which mechanism—GSP or VCG—better preserves truthful reporting when budgets constrain bids.

Overview: This problem tests auction theory and mechanism design skills, specifically revenue and welfare analysis with quality-adjusted bids, reserve prices, and strategic behaviors such as bid shading and budget pacing.