Definition:Sealed-Bid Auction/Second Price

Definition

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.

Let $G$ denoted the game under analysis, that is, the second price auction under discussion.

Let $F$ be the object being bid for.

As in the formal definition of the sealed-bid auction, let the players be labelled in order of their valuations of $F$:

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

In this context, a move by player $i$ is the bid that $i$ places on $F$.

Let $b$ be the profile of moves made by all players.

Let $\map m b$ be defined as the lowest $j$ such that $\ds b_j = \max_{k \mathop \in \set {1, 2, \ldots, n} } b_k$.

That is, $\map m b$ is the index of the player who has the lowest index of all players who place the highest bid on $F$.

Let player $i$ win the auction by move $b_i$.

By hypothesis, player $i$'s valuation for $F$ is $v_i$.

Hence $i$'s payoff is $v_i - \max_{j \mathop \ne 1} b_j$.

For the other players the payoff is $0$.

Thus the payoff function of player $i$ is defined as follows:

$\map {p_i} b = \begin{cases} v_i − \max_{j \mathop \ne i} b_j & : i = \map m b \\ 0 & : \text{otherwise} \end{cases}$

$\ds \max_{j \mathop \ne i} v_j \le \max_{j \mathop \ne i} b_j = b_i \le v_1$

That is:

$(1): \quad \ds \max_{j \mathop \ne i} v_j \le b_i$ (that is, player $i$'s bid was high enough to win)

$(2): \quad \ds \max_{j \mathop \ne i} b_j \le v_i$ (that is, player $i$'s valuation was high enough to avoid suffering the winner's curse).

A second price auction is also known as a Vickrey auction, for William Spencer Vickrey.