Second-price (Vickrey)

Highest bid wins and pays the highest losing bid. Truthful bidding is dominant.

Bidding your true value is a dominant strategy here: it is at least as good as anything else you could do, whatever the other bidders do.

A worked example

Three bidders want one item. A values it at 100, B at 72, C at 41, and each of them simply says so — nobody shades, nobody bluffs.

What the mechanism does, 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.

   100 >= 72 >= 41

3. pick winner The highest eligible bid wins the item.

   argmax b_i = A at 100

4. price rule The winner pays the highest competing bid, not their own. Bidding above your value can only make you overpay, and shading below it can only lose the item at a price you would have been happy with, so honesty is the dominant strategy.

   p = max(max(b_-i), r) = max(72, 0) = 72

5. payments A pays 72; the losers pay nothing.

   revenue = 72

How it came out

A wins and pays 72: the highest bid that lost, not the 100 they said. That gap is the whole mechanism. Because the price A pays is set by somebody else's bid, A cannot lower it by bidding less — they can only lose an auction they wanted to win. Bidding your true value is a dominant strategy, which is a stronger claim than an equilibrium: it is the best you can do whatever anybody else does.

Run this auction step by step

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

Settings

Read next

• First-price sealed bid — Highest bid wins and pays its own bid, which rewards shading below your value.
• English (ascending clock) — Price rises until one bidder is left; equivalent to a second-price auction.
• VCG position auction — The same slots to the same bidders, but each winner pays the externality they impose. Truthful.