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           \vspace*{-1.3in} \noindent \textbf{Getting Carried Away in an
Auction as Imperfect Value Discovery }

\noindent November 7, 2004. Eric Rasmusen\footnote{
Erasmuse@indiana.edu, http://www.rasmusen.org,
http://www.rasmusen.org/papers/carried.pdf.}

The two possible bidders in an auction, both risk-neutral, have private
values which are statistically independent. The auction is open and
ascending.

Bidder 1's value is $v_1$, which has three components: $v_1= \mu+u+ \epsilon$%
. He knows the value of $\mu$, and he knows that that $u$ and ${\epsilon}$
are independently distributed with mean zero and differentiable densities on
$[-\overline{u},\overline{u} ]$ and $[- \overline{\epsilon}, \overline{%
\epsilon}]$, where $\mu-\overline{u}- \overline{\epsilon}>0$ so $v_1$ is
never negative. If he wishes, at any time he can pay $c$ and learn the value
of $u$ immediately. He cannot discover the other component, ${\epsilon}$,
however, until after the auction.

Bidder 2's value, $v_2$, is $\underline{v_2}$ with probability $\theta$ and $%
\overline{v_2} $ with probability $(1-\theta)$, with $\theta \in (0,1)$; and
with $\underline{v_2} \in (\mu-\overline{u} ,m)$ and $\overline{v_2} \in (m,
\mu+\overline{u})$. Let us assume, for reasons to be explained later, that $%
\overline{v_2}$ is closer to $\mu$ than is $\underline{v_2}$: $\overline{v_2}%
-\mu > \mu-\underline{v_2}$. Bidder 2 knows the value of $v_2$ but not $v_1$.

 

Bidder 1 has three value discovery strategies that might be optimal in
equilibrium: early discovery, late discovery, and no discovery. The early
discovery strategy is to pay to discover $u$ when the bid level reaches some
value $b^* \in [0,\underline{v_2}) $, most simply at the start of the
auction, so $b^*=0$. The late discovery strategy is to pay to discover $u$
if the bid level reaches some level $b^* \in [ \underline{v_2}, \mu+
\overline{u}] $ and Bidder 2 has failed to drop out, most simply if the
bidding reaches Bidder 1's initial bid ceiling, so $b^*= \mu$. The strategy
of no discovery is to refuse to pay to discover $u$ regardless of what
happens.

\noindent \textbf{Continuous Density Model.} Now assume that Bidder 2's
value, $v_2$, is distributed according to an atomless and differentiable
density $g(v_2)$ on $[0,k]$, where $k>\mu$ and where $g(v_2)>0$ for all $v_2$
on that interval. Bidder 2 does not know $v_1$, but he does know $v_2$.

  Bidder 2's optimal strategy is to choose a bid ceiling of $%
v_2$, Bidder 1's optimal bid ceiling is $Ex$, which will be either $\mu$ or $%
\mu+u$, depending on whether he has paid $c$ to discover $u$. Bidder 1 must
also decide at what bid level $p$ to pay $c$ to discover $u$, where possibly
$p=0$ (early discovery) or $p = k$ (no discovery).

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