Ads: four slots, one ladder

Search ads sell position, not the page. Higher slots get more clicks, so what each bidder pays depends on who is standing below them.

Run with Generalized second-price (GSP). Here is that auction, run once.

Step by step

1. collect bids Each bidder submits one sealed bid; private values are never revealed.
2. sort Bids are ranked from highest to lowest. Equal bids keep the order the bidders were listed in, so a tie goes to the first-listed bidder.

   92 >= 78 >= 66 >= 51

3. click-through rates A bid here is a price per click, and the 3 slots differ only in how often they are clicked: slot i receives 0.5^i of the clicks slot 0 gets. A slot is therefore worth its holder's value per click times that rate, so the same bidder is worth less further down the page.

   ctr_0 = 1, ctr_1 = 0.5, ctr_2 = 0.25

4. assign slots The highest bid takes the most clicked slot, the next bid the one below it, and so on down: A takes slot 0 at ctr 1, B takes slot 1 at ctr 0.5, C takes slot 2 at ctr 0.25. Both payment rules allocate exactly this way — only the bill that follows differs.

   slot 0 -> A (92/click) | slot 1 -> B (78/click) | slot 2 -> C (66/click)

5. price rule Each winner pays the next bidder's bid for every click their slot gets, never their own bid. That is the rule that breaks truthfulness: your own bid decides which slot you land in, but the bidder below you decides what that slot costs, so deliberately bidding low enough to drop a slot can buy cheaper clicks and leave you better off than bidding your value.

   p_i = ctr_i x b_(i+1)

6. slot price A holds slot 0 and is charged B's bid of 78 on every one of that slot's clicks.

   p_A = ctr_0 x b_1 = 1 x 78 = 78

7. slot price B holds slot 1 and is charged C's bid of 66 on every one of that slot's clicks.

   p_B = ctr_1 x b_2 = 0.5 x 66 = 33

8. slot price C holds slot 2 and is charged D's bid of 51 on every one of that slot's clicks.

   p_C = ctr_2 x b_3 = 0.25 x 51 = 12.75

9. payments A tops the ladder and pays 78. Every winner is billed at whatever sits one rung below them — the next bidder's bid, or the reserve once the ladder runs out — and anybody who won no slot pays nothing.

   revenue = 78 + 33 + 12.75 = 123.75

How it came out

Open this example in the walkthrough

The link carries the whole setup, so you can change a bid and watch what moves — and send the result to somebody else.

Read next

• VCG position auction — The same slots to the same bidders, but each winner pays the externality they impose. Truthful.
• Second-price (Vickrey) — Highest bid wins and pays the highest losing bid. Truthful bidding is dominant.