Generalized Second Price Auction

Overview

GSP allocates K ad slots to N bidders, assigning the highest bidder the top slot, second-highest the second slot, etc. Each winner pays the bid of the advertiser ONE POSITION BELOW them (per-slot second price). Used by Google Ads and Bing Ads. Runs in O(N log N) for sorting bids.

When to Use

Trigger conditions:

• Understanding search engine ad auction mechanics
• Computing ad position and cost-per-click from bid and quality data
• Analyzing bidding strategy in sponsored search

When NOT to use:

• When you need incentive-compatible truthful bidding (use VCG mechanism)
• When analyzing display/programmatic ad auctions (typically use first-price)

Algorithm

IRON LAW: GSP Is NOT Incentive-Compatible
Unlike Vickrey (single-item second-price) auctions, truthful bidding
is NOT a dominant strategy in GSP. Bidders may strategically shade
bids below their true value. The equilibrium depends on competitor bids.
Ad Rank = Bid × Quality Score (Google's variant adds format/extensions).

Phase 1: Input Validation

Collect: bids, quality scores (or ad rank scores) for all competing advertisers. Define available slot positions and their click-through rate multipliers. Gate: All bids positive, quality scores in valid range.

Phase 2: Core Algorithm

1. Compute Ad Rank for each advertiser: AdRank_i = Bid_i × QualityScore_i
2. Sort advertisers by Ad Rank descending
3. Assign top-K to slots 1 through K
4. Compute payment: CPC_i = AdRank_{i+1} / QualityScore_i (price to maintain position)
5. Last slot winner pays the minimum bid threshold

Phase 3: Verification

Check: all payments ≤ bids, positions ordered by Ad Rank, no advertiser pays more than their bid. Gate: Payment ≤ bid for all winners, positions consistent.

Phase 4: Output

Return slot assignments with positions, CPCs, and estimated clicks.

Output Format

{
  "slots": [{"advertiser": "A", "position": 1, "ad_rank": 8.5, "cpc": 2.10, "est_clicks": 100}],
  "metadata": {"total_bidders": 15, "slots_available": 4, "auction_type": "gsp"}
}

Examples

Sample I/O

Input: Bidders: A(bid=3, QS=8), B(bid=4, QS=5), C(bid=2, QS=9). Slots: 2. Expected: Ranks: A=24, C=18, B=20. Order: A(1st), B(2nd). CPC_A = 20/8 = 2.50, CPC_B = 18/5 = 3.60.

Edge Cases

Input	Expected	Why
Tie in Ad Rank	Platform tiebreaker (historical CTR, etc.)	GSP needs strict ordering
One bidder	Wins slot 1, pays minimum CPC	No competition → floor price
Bid below threshold	Not eligible	Minimum bid requirement enforced

Gotchas

• Quality Score is opaque: Google's QS includes expected CTR, ad relevance, and landing page experience. The exact formula is proprietary.
• Strategic bid shading: Since GSP isn't truthful, sophisticated advertisers shade bids. This means observed bids don't reflect true willingness to pay.
• Position ≠ value: Higher position gets more clicks but at higher CPC. The most profitable position may be #2 or #3, not #1.
• Budget constraints: GSP doesn't account for daily budgets. Budget-constrained advertisers must pace bids throughout the day.
• Broad match expansion: The auction includes query-expanded matches, which may have different conversion rates than exact matches.

References

• For Nash equilibrium analysis of GSP, see references/gsp-equilibrium.md
• For comparison with VCG mechanism, see references/gsp-vs-vcg.md