Definition:Sealed-Bid Auction/First Price

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Definition

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.


Analysis

Let $G$ denote the game under analysis, that is, the first 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 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 $v_i - b_i$.

For the other players the payoff is $0$.


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

$\map {p_i} b = \begin{cases} v_i − b_i & : i = \map m b \\ 0 & : \text{otherwise} \end{cases}$

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


$b$ is a Nash equilibrium if and only if:

$\ds \max_{j \mathop \ne 1} v_j \le \max_{j \mathop \ne 1} b_j = b_1 \le v_1$


That is:

$(1): \quad b_1 \le v_1$ (that is, player $1$ does not suffer from the winner's curse)
$(2): \quad \ds \max_{j \mathop \ne 1} v_j \le b_1$ (that is, player $1$'s bid was high enough to win)
$(3): \quad \ds b_1 = \max_{j \mathop \ne i} b_j$ (that is, another player submitted the same bid as player $1$)

and so player $1$ wins the auction.


Sources