All-pay auction

Highest bid wins, but every eligible bidder pays their bid — losing bids are sunk.

Bidding your true value is not dominant here. A bidder who knows something about their rivals can do better by bidding something else, and the worked example below shows what and why.

A worked example

One prize, and every bid is spent whether it wins or not. Think of a research grant competition, a patent race, or a lobbying contest: the effort is sunk the moment it is made.

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.

   40 >= 35 >= 20

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

   argmax b_i = A at 40

4. price rule Every bidder who cleared the reserve pays their own bid, win or lose. A losing bid buys nothing, which is why this models lobbying, contests and R&D races.

   p_i = b_i for every eligible i; winner pays 40

5. payments A pays 40 and takes the item; the losers pay their bids and get nothing: B loses 35, C loses 20.

   revenue = 40 + 35 + 20 = 95

How it came out

A wins and pays 40. B and C pay 35 and 20 and receive nothing at all — their utilities are negative, which no sealed-bid auction above can produce. The seller collects 95 for an item one person got. That is why all-pay contests raise so much and why, with no pure equilibrium to settle into, the equilibrium view of this mechanism is a mixed one.

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.
• Second-price (Vickrey) — Highest bid wins and pays the highest losing bid. Truthful bidding is dominant.