Definition:Sealed-Bid Auction

Definition

A sealed-bid auction is a game in the following format:

An object $F$ is to be assigned to a player in a set $\left\{ {1, 2, \ldots, n}\right\}$ in exchange for a payment.

The valuation of $F$ by player $i$ is $v_i$, and:

$v_1 > v_2 > \cdots > v_n > 0$

Each player simultaneously submits a non-negative number, called a bid.

$F$ is then given to the player with the lowest index among those who submit the highest bid, in exchange for a payment.

A first price auction is a sealed-bid auction in which the payment made by the winner is the price which is bid by that player.

A second price auction is a sealed-bid auction in which the payment made by the winner is the highest bid made by the players who did not win.

So if only one player submits the highest bid, the price paid is the second highest bid.