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 \vspace*{-1in} \noindent \textbf{The Parking Lot Problem}

\noindent
 13 November 2007. Maria Arbatskaya, Kaushik Mukhopadhaya, and Eric
Rasmusen.\\
marbats@emory.edu,kmukhop@emory.edu, Erasmuse@Indiana.edu,
http://www.rasmusen.org.


\noindent \textit{Assumptions: } $N$ drivers all have the same
preferred
arrival time, $t= T$ in a parking lot of size $K$. The value of
finding a
spot is $v$, and a player has cost $w$ per period for arriving early,
so the
loss from arriving early is $L(t) = w(T-t)$. All the players arriving
in the
period $t^{\prime}$ when the lot fills have an equal probability $p$
of
getting a spot.   The cost of increasing the parking
lot
size is $c$ per spot  with linear costs (or $C(K)$ more generally).

In the first best, if $N>K$ all $N$ players arrive at $T$ and $K$ of
them park in the parking lot.

Define the \textit{indifference arrival time} as $t^{*}\equiv T-v/w. $,
so a player parking then has a payoff of zero. Assume $t^{*}$  and $T$
are
even multiples of $\Delta$ so they are feasible.



\noindent
  \textit{Full Observability:} A player observes all arrivals up through
period $t-\Delta$ before he makes his own decision at $t$.

\noindent \textit{Unobservability:} No player observes any other
player's
arrival.

\noindent\textbf{Proposition 1. } Under either full observability or
unobservability, as the time grid becomes infinitely fine and there
are more
drivers than parking spots ($N>K$),the players fully dissipate rents
from
the parking lot in any equilibrium.

In other words, if you build a parking lot slightly too small, you
would do
better not to build one at all.



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