No stable bid at all
Every profile in an all-pay auction has somebody who wishes they had bid differently. That is not the absence of an equilibrium — it is one made of dice, and here is what the dice look like.
Run with All-pay auction. The link below sweeps every bid, not just these ones, and reports the profile nobody wants to leave. Here is what happens at the bids as typed.
| Bidder | Worth to them | Bid |
|---|---|---|
| A | 100 | 100 |
| B | 80 | 80 |
Step by step
- collect bids Each bidder submits one sealed bid; private values are never revealed.
- 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 >= 80
- pick winner The highest eligible bid wins the item.
argmax b_i = A at 100
- 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 100
- payments A pays 100 and takes the item; the losers pay their bids and get nothing: B loses 80.
revenue = 100 + 80 = 180
How it came out
| Winner | A |
|---|---|
| Price paid by the winner | 100 |
| Seller revenue | 180 |
| Value created | 100 |
| Went to whoever valued it most | yes |
| Bidder | Paid | Ended up with |
|---|---|---|
| A | 100 | 0 |
| B | 80 | -80 |
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.