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{\Large \textbf{Getting Carried Away in  Auctions as Imperfect Value Discovery }  
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  WORKING PAPER\\ (Comments welcomed)





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  20 March 2007.   \\ 

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Eric Rasmusen \\ 

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\textit{Abstract}
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  If  estimating his private value is costly for a bidder, he  may wait to do so until the middle of an auction and rationally  revise his reservation price upwards, to his  ex post regret-- apparently ``getting carried away''. 
 
\noindent
 {\small \hspace*{20pt} Dan R. and Catherine M. Dalton Professor, Department  of Business Economics and Public Policy, Kelley School of Business,Indiana University,  BU 456, 1309 E. 10th Street, Bloomington, Indiana, 47405-1701. Office: (812) 855-9219. Fax: 812-855-3354. Erasmuse@indiana.edu, http://rasmusen.org. http://rasmusen.org/papers/carried-rasmusen.pdf.    JEL Code: D44.  }

\noindent {\small \footnote{I thank Michael Baye, Rene Kirkegaard,  Kerry Krutilla, Yacheng Sun, and participants in    seminars at Indiana University's Kelley School of Business, DePaul University,   the University of Tokyo's economics department, and the 2004 International Industrial Organization Conference for their comments, and Harvard Law School's Olin Center and the University of Tokyo's Center for International Research on the Japanese Economy for their hospitality. }




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 Sometimes bidders 
  seem to get carried away in auctions, paying more than the reservation price they entered the auction willing to pay. This might be due  to irrational emotion, as examined in Ku,  Malhotra \&   Murnighan. Or, if     the auction is common-value,  the bidder might revise his own value estimate upwards on seeing other bidders' confidence.  I here suggest an alternative  using the idea in Compte \& Jehiel (2000, 2004)  and Rasmusen (2003)   that even if  an auction is private-value,  other players' bids may stimulate a player to learn more about his own value during the auction, and sometimes his estimate rises.  

  
 
Let there be  two possible bidders, both risk-neutral and  with private values, in   open-exit auction.  The price starts at zero and rises continuously until one player drops out at his ``bid ceiling'' and      the other player wins. 

  Bidder 1's value is $v_1$, which has three components:   $v_1= \mu+u+ \epsilon$. He   knows $\mu$  but not $u$ or  $\epsilon$, which  are independently distributed  according to   symmetric densities $f(u)$ and $g(\epsilon)$ with mean zero and  $Min (\mu+u+\epsilon)\geq 0$.   His initial expectation of $v_1$  thus equals $\mu$. If he wishes, at any time he can pay $c$ and learn $u$'s value, but he cannot discover    $\epsilon$  until after the auction.

  Bidder 2 knows his value, $v_2$, which  is distributed according to the  atomless and   differentiable density $h(v_2)$  where $h(v_2)>0$ on the support   $[\alpha, \beta]$ and    $0 < \alpha < \mu$ and  $\beta>\mu$.   

 


 The optimal bid ceiling for a player equals      his expected value for the object being auctioned. If he bids  less, he might lose when the winning bid was below his  expected value. If he bids  more, he might win when the  price  exceeded  his  expected value.
If Bidder 1 does not acquire any information about his value, his best strategy is to bid  up to $\mu$, the expected value of the object to him. If he does discover $u$, it is to bid up to $(\mu+u)$, his updated estimate of $v_1$. Bidder 2's optimal strategy is    a bid ceiling of  $v_2$.  Bidder 2  will not change    his bid ceiling   to affect the timing of Bidder 1's value discovery, because value discovery is instantaneous,  unlike in Rasmusen (2003a).

\noindent
Bidder 1 has three possible value discovery strategies:

\noindent
 {\bf  Early discovery.}  Discover $u$   when the bid level reaches  the optimal ``discovery level'' $p^* \in [0,\alpha)$.

\noindent
 {\bf Late discovery.} Discover $u$ if the bid level reaches   $p^* \geq \alpha$.

\noindent
 {\bf No discovery}. Never discover $u$.

  Suppose  (contrary to the assumptions) that Bidder 1 knows $v_2$, and that $p \leq \mu$, so   there is positive probability that Bidder 1 pays $c$ and discovers $u$.

If    $v_2< p  $  Bidder 1 wins  at price $v_2$, for an expected payoff of $(\mu-v_2)$. If    $v_2> p  $   he pays $c$ to discover $u$.  He loses the auction if $\mu + u <p$;  otherwise, he wins. Overall, if $v_2 >p$ his expected payoff is
 \begin{equation} \label{e100}
\displaystyle{  \pi_1(v_2| v_2>p)  = -c +  \int_{u= - \infty }^{v_2- \mu } (0) f(u) du + \int_{u=v_2- \mu    }^{\infty } (\mu + u -v_2 ) f(u) du.  }
\end{equation}
Integrating over the possible values of $v_2$ yields  
 \begin{equation} \label{e101}
  \displaystyle{  pi_1  =  \int_{v_2=\alpha}^{p} (\mu - v_2) h(v_2) dv_2 + \int_{v_2=p}^{\beta} \left( -c + \int_{u=v_2- \mu}^{\infty } (\mu + u -v_2 ) f(u) du   \right) h(v_2) dv_2  }
\end{equation}

If, on the other hand, $p > \mu$, then Bidder 1 is following the policy of no discovery, and his expected  payoff is   the first part of equation (\ref{e101}):
 \begin{equation} \label{e101a}
\displaystyle{ \pi_1 (p>\mu)  =  \int_{v_2=\alpha}^{\mu} (\mu - v_2) h(v_2) dv_2.  }
\end{equation}



\noindent
 {\bf Proposition:}     {\it   The optimal discovery level,  $p^*$,     rises with $c$, rising strictly  if $p^* \in (\alpha, \mu)$. Bidder 1 will  follow a policy of early discovery ($p^* \in [0, \alpha)$)  if $c$ is low enough, late  discovery ($p^* \in [  \alpha,\mu]$)  for higher levels of  $c$, and   no discovery ($p^* \in (  \mu, \infty]$) if $c$ is sufficiently high.      }


\noindent
 {\it Proof:}
 Differentiating  equation (\ref{e101}) with respect to $p$ yields
 \begin{equation} \label{e101b}
\begin{array}{ll}
\frac{d \pi_1 }{dp} & =  \displaystyle{ (\mu - p) h(p)   - \left( -c + \int_{u=p- \mu}^{\infty
} (\mu + u -p ) f(u) du   \right) h(p)} \\
 & \\
 & = \displaystyle{ \left[  c  + (\mu - p) - \int_{u=p- \mu}^{\infty } (\mu   -p  + u ) f(u) du
\right] h(p)  } \\
 & \\
  & \displaystyle{= \left[  c  +     -\int_{u=-\infty}^{\infty } (\mu - p+u) f(u) du   -
\int_{u=p- \mu}^{\infty } (\mu   -p  +  u ) f(u) du \right] h(p) }  \\
 & \\
  & = \displaystyle{ \left[c  +   \int_{u=-\infty}^{p- \mu } (\mu   -p  +  u ) f(u) du  
\right] h(p)  } \\
 \end{array}
\end{equation}
   If $h(p)>0$   (true between $\alpha$ and $\mu$) and $c$ is small enough,    derivative  (\ref{e101b}) is negative. If $c$ is small enough,   $\frac{d \pi_1 }{dp}<0$  for   $p \in [\alpha, \mu]$, and the payoff  rises if $p$ is reduced to below  $\alpha$ -- that is, to early discovery. If $p<\alpha$, then $h(p)=0$, so further reductions are unimportant.

 If $c$ is greater,   $\frac{d \pi_1 }{dp} >0$ at $p=\alpha$, and the optimal $p$ exceeds $\alpha$. At the optimal $p$, 
 \begin{equation} \label{e101c}
\begin{array}{ll}
\frac{d^2 \pi_1 }{dp^2}      &\displaystyle{ = \left( (\mu   -p  +   [p-\mu] ) f(u)   +
\int_{u=-\infty}^{p- \mu } (-1) f(u) du \right) h(p) +  \left[ c+
\int_{u=-\infty}^{p- \mu } (\mu   -p  +  u ) f(u) du    \right] h'(p) }  \\
  &\\
 & = \displaystyle{ 0 -  \left( \int_{u=-\infty}^{p- \mu }   f(u) du \right)  h(p) + (0)h'(p) <0}
\\
 \end{array}
\end{equation}
  where we use the fact that   $\frac{d \pi_1 }{dp}=0$  at the optimum to obtain
the term $(0)h'(p)$. Since  
\begin{equation} \label{e101d}
\begin{array}{ll}
\frac{d^2 \pi_1 }{dp dc}      & =  (1)  h(p) >0,     \\
 \end{array}
\end{equation}
 the implicit function theorem tells us that $\frac{d p }{ dc}  >0$ when $h(p) > 0$,  i.e., the optimal discovery level rises with the   discovery cost.  Thus, there  exist levels of $c$ such that the optimal discovery level lies within  $(\alpha, \mu)$ and  late discovery is optimal. As $c$ increases,  the optimal discovery level exceeds $\mu$, so   ``no discovery'' becomes optimal. $\blacksquare$

\bigskip

  If the discovery cost is low enough, early discovery is best, because the bidder averts the possibility that he  might pay more than his value by winning even at the other bidder's lowest possible value. If the discovery cost is somewhat higher, it is not worth avoiding that possibility--thus, late discovery.  How late depends on the size of the discovery cost, and the optimal discovery level rises smoothly with the discovery cost. If the discovery cost is too high, then no discovery becomes optimal.

 

 
This model  provides an interpretation for ``getting carried away''.   Suppose we see a bidder winning an auction at a price higher than his initial reservation price, and   he later regrets having won---  an ``unhappy victory.'' Here is the model's interpretation.   At the start of the auction, $\mu$ was the most he  intended to bid. The auction began, and the bidding rose to $\mu$. he  reconsidered, at  cost $c$,   and raised his bid ceiling to $(\mu+u)$. This new ceiling exceeded $\beta$, the most    other bidders would pay, so he won,     at price $\beta$. After the auction is over, however, he  learned $\epsilon$ and found that $\mu+u+\epsilon<\beta $.  His reaction is, ``I got carried away and bid too much. I wish I'd stuck with my original ceiling of $\mu$.'' This will not happen in every auction for every bidder, but   the winning bidder is likely to be one who revised his value upwards, and if the final estimate is unbiased with a symmetric error it is as likely to be too high as too low. Thus, roughly half the time the revising bidder will be willing to pay too much, and often competition will lead him to do so. Observationally, this will look like getting carried away by emotion. An empirical test would be whether bidders who revise their reservation prices upwards regret doing so {\it on average} (emotion)  or not (rational value discovery). 

    

   

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\noindent
\textbf{References} 

 Compte, Olivier \& Philippe Jehiel (2000) ``On the Virtues of the Ascending Price Auction: New Insights in the Private Value Setting,'' working paper, CERAS-ENPC, CNRS, Paris, http://www.enpc.fr/ceras/jehiel/ascend.pdf (December 2000). 


Compte, Olivier \& Philippe Jehiel (2004) ``Auctions and Information Acquisition: Sealed-Bid or Dynamic Formats?'' working paper, CERAS-ENPC, CNRS, Paris,  http://www.enpc.fr/ceras/compte/ascendR.pdf (May 2004) 

Ku, Gillian, Deepak Malhotra \& J. Keith Murnighan (2005): ``Towards a Competitive Arousal Model of Decision Making: A Study of Auction Fever in Live and Internet Auctions,'' {\it  Organizational Behavior and Human Decision Processes},  26:   89-103.
 


Rasmusen, Eric (2003) ``Strategic Implications of Uncertainty Over One's Own Private Value in Auctions,'' January 26, 2003, \\ http://rasmusen.org/papers/auction.pdf.

 


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