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\author{Maria Arbatskaya, Kaushik Mukhopadhaya, and Eric Rasmusen}
\title{The Parking Lot Problem}
\date{24 May 2006/25 May 2007}
\maketitle

\begin{abstract}
\noindent If property rights are not assigned over individual goods such
as
parking spaces, competition for them can eat up the entire surplus. We
find
that when drivers are homogeneous, there is a discontinuity in social
welfare
between \textquotedblleft enough\textquotedblright\ and
\textquotedblleft not
enough.\textquotedblright\ Building slightly too small a parking lot is
worse
than building much too small a parking lot, since both have zero net
benefit
and larger lots cost more to build. More generally, the welfare losses
from
undercapacity and overcapacity are asymmetric, and parking lots should
be
\textquotedblleft overbuilt.\textquotedblright\ That is, the optimal
parking
lot size can be well in excess of mean demand. Uncertainty over the
number of
drivers, which is detrimental in the first-best, actually increases
social
welfare if the parking lot size is too small.

\vspace{0.1in}

\noindent\emph{Keywords:} Property rights; commons; planning; rent-
seeking;
all-pay auction; timing game; capacity; queue.

\noindent\emph{JEL classification numbers: }R4; L91; D72; C7.

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{\small Maria Arbatskaya, Department of Economics, Emory University,
Atlanta,
GA 30322-2240. Phone: (404) 727 2770. Fax: (404) 727 4639. Email:
marbats@emory.edu.\smallskip}

{\small Kaushik Mukhopadhaya, Department of Economics, Emory University,
Atlanta, GA 30322-2240. Phone: (404) 712 8253. Fax: (404) 727 4639.
Email:
kmukhop@emory.edu\smallskip.}

{\small Eric Rasmusen, 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.
Phone: (812) 855 9219. Fax: (812) 855 3354. Erasmuse@indiana.edu.
http://www.rasmusen.org.}

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\noindent\textbf{Acknowledgments }

\noindent We would like to thank Michael Alexeev, Richard Arnott,
Theodore
Bergstrom, Robert Deacon, Milton Kafoglis, Daniel Kovenock, John Lott,
Donald
Shoup, and seminar participants at Michigan State, Purdue, Virginia
Polytechnic Institute, the 2001 Public Choice Meetings, the 2003
Econometric
Society Summer Meetings, the 2004 World Congress of the Game Theory
Society,
and the 2005 International Industrial Organization Conference for
helpful
comments. Rasmusen thanks Harvard Law School's Olin Center and the
University
of Tokyo's Center for International Research on the Japanese Economy for
their
hospitality while this paper was being written. Copies of this paper can
be
found at http://www.rasmusen.org/papers/parking-rasmusen.pdf.





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\section{Introduction}

The problem of the commons shows up in many forms. In the classic
version,
citizens of the village graze too many animals in the communal green,
wrecking
its value for all. Overuse reduces the value of the property. The
problem is
that no single person owns the green. Dividing it up into private plots
would
solve the overgrazing, but would also eliminate economies of scale, and
so
villages turn to other solutions such as formal or informal regulation.

The version of the commons problem of this paper is different. A parking
lot
is divided into separate plots and access lanes in which parking is
forbidden,
an organizational form that solves the ``overgrazing'' problem and
maximizes
the benefit of the property. The problem we will address arises when the
plots
are demarked, but it is impractical to make each plot separately owned,
controlled entirely by its owner and used either by himself or by
someone who
pays him a fee during the time he is not there. In this version,
inefficiency
will arise not from behavior that reduces the benefit of the property,
but
from behavior that creates costs as it determines the allocation of the
property.

The context of parking is familiar to all of us from personal
experience.
Finding a parking space at all is a perennial source of frustration on
college
campuses, and tough competition for the limited number of parking spaces
compels drivers to arrive well in advance of their preferred time.
\textquotedblleft Survival of the earliest\textquotedblright\ is the
rule. On
the other hand, shopping malls display an apparently useless excess of
parking
spaces. In contrast to universities, malls seem to go to ridiculous
extremes,
with acres of parking lots that are never full except at Christmas.
Oddly
enough, it is the non-profits that seem to spend too little on parking
lots
and the for-profits that seem to spend too much.

Parking studies recognize the importance of planning for a sufficient
\textquotedblleft cushion\textquotedblright\ in excess of necessary
spaces.
However, these studies are concerned mostly with the smooth flow of
vehicles
in and out of the parking area, or unforeseen circumstances such as
disruptive
repair. A 1987 study by \textit{Walker Parking Consultants} (
commissioned by
Indianapolis to develop a Regional Center Parking Plan) reports:
\textquotedblleft Thus, a supply of parking operates at peak efficiency
when
occupancy is 85\% to 90\%. When occupancy exceeds this level, there may
be
delays and frustration in finding a space. The parking supply may be
perceived
as inadequate even though there are spaces available in the
system\textquotedblright\ (http://www.bts.gov/NTL/DOCS/rc.html).

In this paper we suggest that an equally important concern is strategic
behavior by drivers. If the parking lot is built even slightly too
small,
rent-seeking competition by drivers can dissipate all of the rents from
parking there, eliminating every trace of the lot's social usefulness
and
making its construction entirely wasteful. Contrary to first-best
engineering
concerns, a parking lot should be large enough to forestall wasteful
competition. It can easily be socially optimal to have parking lots that
on
average are half empty, as we will show below by example. Shopping malls
seem
to realize this better than nonprofit institutions.

It is natural to view empty spaces as a reason to reduce the amount of
available parking. Based on the finding that the parking supply count
exceeded
demand count by 30\%, a Seattle report recommends reducing the parking
supply,
though the authors admit that \textit{\textquotedblleft A major policy-
related
issue is how much allowance to provide over the design-level demand in
setting
the size of a given parking facility.\textquotedblright}\footnote{See
the
\textit{1991 Parking Utilization Study} undertaken by the Research and
Market
Strategy Division of the Transit Department in the Municipality of
Metropolitan Seattle.} In a comprehensive book on parking, Shoup (2005)
advocates against free and plentiful parking, which has various social
costs
including construction and maintenance costs, road congestion and
environmental pollution. He argues that minimum off-street parking
requirements for new construction should be abolished. Instead, he
recommends
charging for curb parking, at a price at which 15 percent of spaces are
vacant.

In our paper, we will take the fact that parking is unpriced (or
underpriced)
as given, and we will argue that if parking is to be free, it ought to
be
plentiful. Strategic behavior of drivers must be considered when
deciding on
parking lot size. Having 30\% of the spaces empty, on average, may well
indicate too \textit{small} a parking lot size, given uncertainty in the
demand for parking. We will show that there is an asymmetry in the
welfare
effects of over- and under-supply of parking. Extra parking spaces
increase
costs gradually, but even a minor shortage can result in a discontinuous
and
huge social loss, as rent-seeking eats up the entire value of whatever
parking
is available. Moreover, the welfare loss can be more severe with a small
shortage than with a large shortage, for which there are fewer rents to
dissipate.

In our model, drivers face a trade-off between the disutility of
arriving
earlier than their preferred time and the increased probability of
securing a
space in the parking lot. Since the cost of arriving early is incurred
regardless of whether the driver is successful in finding a parking
space, the
contest is a multi-unit all-pay auction. Given the dynamic nature of the
players' decisions about \textit{when} to arrive at the lot, their
strategies
can be quite complex. Nevertheless, we are able to show that full (or
almost
full) rent dissipation occurs in any equilibrium when the size of the
lot is
too small. When the number of drivers is known to the planners, the
optimal
size of the parking lot is equal to the number of drivers. When the
number is
not known, the lot should be made so large that on average a large
proportion
of the spots will be unoccupied. The analysis indicates that perhaps
universities have it wrong, while the malls have it right.

\bigskip

\section{The Literature}

The problem of managing congested facilities is a long-standing one in
economics, a problem usually analyzed taking the size of the facility as
given
and studying the effects of various allocation systems. Queueing models
with
tolls, such as Naor (1969), assume an exogenous stationary customer
arrival
process and random service times. Customers benefit from the service,
but they
incur a constant cost per unit of time from queueing. In equilibrium, a
consumer joins a queue if its length is below a threshold level. The
last
consumer in the queue is just indifferent between joining and staying
out. The
equilibrium outcome is inefficient, as a result of the negative
externality
that a customer imposes on those arriving later. Nevertheless, rent
dissipation through queueing is not full because of the randomness in
the
arrival and departure processes. Naor shows how tolls can remedy the
problem.

The large literature on traffic congestion focuses on the effects of
different
pricing systems. Arnott et al. (1993), for example, compare alternative
toll
regimes in a model with a single traffic bottleneck and identical
commuters
incurring linear time inconvenience costs. Part of this literature deals
with
capacity, looking especially at the interaction between different
transportation modes: road capacity (and congestion) and rail capacity.
Arnott
and Yan (2000) survey this literature, mainly dating from the 1970's and
add
their own analysis of the problem. One of the elements in the capacity
decision is what we will look at in the parking context: that more
capacity
will attract more users. Thus, Arnott and Small (2001) note that
traditional
approaches to dealing with congestion can be counterproductive;
increased road
capacity can attract drivers from alternative routes until the road is
as
congested as before. Recent papers by Anderson and de Palma (2004) and
Arnott
and Rowse (1999) study parking congestion and its pricing in a model
with
drivers cruising for parking on two distinct spatial structures:\ a line
with
side streets leading to a common desired destination and a circular road
with
uniformly distributed destinations, respectively. The optimal pricing of
parking internalizes the congestion externality in parking. Arnott
(2001)
provides a good overview of the literature and offers comments on its
future direction.

A difference between the congestion literature and the present paper is
that
we take a game-theoretic approach rather than the more common aggregate
approach. Moreover, the road congestion cost rises gradually, whereas
the
rent-seeking cost rises sharply when the number of drivers just exceeds
the
parking lot size. Vickrey's seminal 1969 article on traffic bottlenecks
is a
good illustration of this. In his example, if capacity is below 120,
congestion causes delays, while if it is above 120 there is no delay. As
a
result, some drivers arrive at the bottleneck early to avoid the delay.
Vickrey notes a \textquotedblleft sharp discontinuity\textquotedblright\
in
the amount of delay at the level of capacity just sufficient to
accommodate
the traffic, and points out that optimal investments in capacity
extension
differ in the first-best and the second-best situations. In the first-
best,
using the optimal price structure (a toll fee that leads to efficient
facility
use), the benefits of capacity extension are not as \textquotedblleft
capricious\textquotedblright\ as in the second-best, when access cannot
be
restricted by fees. In the second-best, \textquotedblleft Expansion
inadequate
to take care of the entire traffic demand...may turn out to be hardly
worthwhile.\textquotedblright\ In Vickrey's model, however, any capacity
extension does reduce delays and is beneficial to travelers. We will
show that
an insufficient increase in capacity might not have any benefit
whatsoever to
offset construction costs, so that the facility is a pure waste.

Underpricing has also been studied in the context of shopping. The
strategic
incentives of people to adjust their purchase schedules are analyzed by
Deacon
and Sonstelie (1991). Consumers choose the size of purchases to minimize
the
total cost of shopping for an underpriced good, which includes shopping
and
storage costs. The waiting time in a queue increases until the market
clears.
A price ceiling makes consumers no better off, though suppliers are
worse off,
which thus generates a deadweight loss in rationing by waiting. (See
also
Deacon and Sonstelie [1985, 1989] and Deacon [1994].)

In the theory of waiting lists proposed by Lindsay and Feigenbaum
(1984),
market clearing occurs as a result of depreciation of product value over
time.
Since delivery of a service in the future can be less valuable than
immediate
delivery, potential consumers are discouraged from putting their names
on a
waiting list when it is too long. By this argument the authors explain
the
persistence of long waiting lists for non-emergency in-patient care at
Britain's National Health Service and the fruitlessness of short-term
measures
aimed at a substantial reduction of the waiting list. (See, however, the
critique by Cullis and Jones [1986].)

In our model, there is no waiting list, congestion, or waiting in line.
Rather, rent dissipation comes in the form of costly schedule adjustment
by
the travelers. In an effort to secure a parking spot, they arrive well
in
advance of their preferred time and dissipate nearly all rents whenever
the
number of drivers in known to exceed the number of parking spaces.

\bigskip

\section{The Parking Game}

\subsection{The Model}

A set of players -- the drivers -- is indexed by $I\equiv\left\{
1,...,N\right\}  $. The drivers are workers who must show up for work no
later
than time $t=T$. Each driver demands one space in the parking lot, and
his
value for it is $v>0$; e.g., if he cannot find a spot, he must park
somewhere
further away and walk in at cost $v$.\footnote{The players in our model
are
identical; we will later show what happens when players are
heterogeneous.
Also, we assume that the value $v$ is independent of the number of
drivers
unable to secure a parking space. More realistically, it would increase,
as
they would have to walk from more and more distant lots. That could be
modelled as a sequence of limited-capacity lots of the kind we model
here.}
Let $K>0$ be the size of the parking lot, and let $c<v$ be the cost of
increasing capacity by one space. If $N\leq K$, each driver is
guaranteed a
parking space, and all drivers choose to arrive at their preferred time,
$T$.
If $N>K$, however, the drivers compete for spaces, and they may wish to
arrive
early to secure spots. Assume that a driver who arrives earlier than
time $T$
incurs a cost of $w>0$ per unit of time, so his cost of arrival at time
$t$ is
$L(t)=(T-t)w$. We will model time as discrete, with the interesting case
being
what happens as the time interval $\Delta>0$ shrinks to zero. Thus, a
time
grid includes times $t=k\Delta\in\lbrack0,T]$ where
$k\in\{0,...,T/\Delta\}$.

What matters to a player deciding whether to rush to the parking lot at
time
$t$ is the inconvenience of arrival that early and the number of parking
spaces still unoccupied. We assume that the size of the parking lot,
$K$, is
common knowledge and compare two alternative assumptions on whether a
driver
knows the number of spaces still open in the parking lot at time $t$.
When a
player making his arrival decision at time $t$ observes all arrivals
prior to
$t$, this is called \textit{full observability}; when he must decide
without
knowing if the parking lot is full, this is called \textit{
unobservability}.
For any time $t$, define $N_{t}$ as the number of unarrived drivers and
$K_{t}$ as the number of unoccupied parking spaces; $N_{t}=N-K+K_{t}$.
At time
$t$, the remaining drivers simultaneously decide whether to arrive at
the
parking lot at that instant. Under full observability, the decision can
be
conditioned on the number of parking spaces still available, $K_{t}$.
Once a
parking spot is taken by the player, the spot remains occupied until the
end
of the game. If player $i$ arrives at $t$ he obtains a spot with
probability
$p_{i,t}=\min\left\{  K_{t}/n_{t},1\right\}  $, where $n_{t}\leq N_{t}$
denotes the number of players arriving at exactly time $t$.

If player $i$ arrives at time $t$, given that $K_{t}\in\{0,...,K\}$
parking
spaces are unoccupied at $t$ and $n_{t}\in\{0,...,N_{t}\}$ players
arrive at
$t$, his payoff is
\begin{equation}
u_{i,t}=p_{i,t}v-L(t). \tag{1}%
\end{equation}


\noindent A player who arrives at time $T$ and finds no parking space
must
park in a less convenient lot, obtaining payoff zero. This is the same
payoff
structure as in an all-pay auction with multiple prizes. Everyone bids,
a
bidder pays what he bids, and the prizes are distributed to the top
bidders,
with a coin toss breaking ties.

Let us define the \textit{indifference arrival time}, $t^{\ast}$, as the
time
at which the arrival cost, $L(t^{\ast})$, equals the prize value, $v$.
It
follows that
\begin{equation}
t^{\ast}\equiv T-v/w. \tag{2}%
\end{equation}
A player who arrives at $t^{\ast}$ and finds a parking space receives a
payoff
of zero since the disutility of the early arrival equals the value of
the
parking space. When looking for equilibria, we can restrict our
attention to
$t\geq t^{\ast}$ since arriving before $t^{\ast}$ yields a negative
payoff and
is strictly dominated by arriving at $T$.

Similarly, define $t_{p}^{\ast}$ as the time at which a player receives
a zero
payoff from participating in a lottery with probability $p$ of winning a
parking space; $t_{p}^{\ast}$ is determined from equation
$pv-L(t_{p}^{\ast
})=0$. It follows that
\begin{equation}
t_{p}^{\ast}\equiv T-pv/w. \tag{3}%
\end{equation}


Finally, let $\underline{t}$ and $\overline{t}$ denote the earliest and
the
latest time any player arrives with a positive probability along the
equilibrium path.

As we are interested in characterizing equilibria that exist for any
time
grid, even a very fine one, we will assume that time periods are short.

\bigskip

\noindent\textbf{Definition 1.}\textit{\ A time grid is fine if }
\begin{equation}
\Delta<\frac{v}{Nw}. \tag{4}%
\end{equation}


\medskip

When a time grid is fine, there are more than $N\geq K+1$\ periods
between
$t^{\ast}$\ and $T$ since $T-t^{\ast}=v/w>N\Delta$. No more than $K$
players
can profitably enter at time $t=t^{\ast}+\Delta$ when the time grid is
fine.
Otherwise, even at the highest odds of obtaining a space, $K/(K+1)$, the
payoff of a player is negative at $t=t^{\ast}+\Delta$ for a fine time
grid.
When a player is not guaranteed to obtain a parking spot, the player
would
choose to arrive one period earlier to secure a parking space, because
for a
fine time grid the cost of one-period earlier arrival, $w\Delta$, is
justified
by the gain associated with the higher probability of obtaining
parking.\bigskip

\subsection{Pure-Strategy Equilibria under Full Observability}

Under full observability, player $i$'s strategy specifies the
probability of
his arrival, $\sigma_{i,t}\in\lbrack0,1]$ at any time
$t=k\Delta\in\lbrack
t^{\ast},T]$ and history $K_{t}\in\{0,...,K\}$. In the first-best
outcome, all
$N$ players arrive at $T$, and $K$ of them get to park in the parking
lot:
$\sigma_{i,t}=0$ for $t<T$ and $\sigma_{i,T}=1$. However, this cannot be
part
of an equilibrium strategy profile when the time grid is fine, since
each
player would prefer to deviate by arriving just before $T$ and increase
his
probability of finding a spot to 100\%. Although it will turn out that
the
details of the equilibrium strategies are more various and intricate
than one
might expect in such a simple game, the equilibrium payoffs converge to
zero
as time becomes closer to continuous. Thus, the parking lot will have no
benefit for the drivers.

The mathematical structure of the parking game is similar to preemption
games
and games of timing such as those of Fudenberg and Tirole (1985) and
Simon and
Stinchcombe (1989), though those games usually have just one winner
rather
than the many successful drivers we have here. It is also similar to a
Dutch
auction, in which the price gradually falls, observed by all bidder, and
the
auction ends when a bidder agrees to pay the price fearing that someone
else
will claim the prize first. The complexity in the parking game will
arise
because if some driver fails to arrive at the time specified in the
equilibrium, we must determine the reactions of all the other drivers to
the
unforeseen opportunity to claim his spot.\medskip

\newpage

\noindent\textbf{The Simplest Case: Two Drivers}

We start by illustrating the logic of what happens in the simplest case:
two
drivers who compete for one parking space. The equilibrium arrival
schedule of
Claim $1$ below, illustrated in Figure $1$, specifies contingencies
under
which players arrive provided they have not arrived earlier.

\begin{center}
$\left[  \mathbf{Figure\ 1}\right]  $
\end{center}

\noindent\textbf{Claim 1.} \textit{Consider the parking game on a fine
time
grid under full observability when there are two drivers and one parking
spot
(}$N=2$\textit{ and }$K=1$\textit{). The following arrival strategies
constitute a subgame-perfect pure-strategy equilibrium:\ contingent on
the
parking lot not being full at time }$t$\textit{, (i) at }$t=t^{\ast}%
+k\Delta\in\lbrack t^{\ast},t_{1/2}^{\ast})$,\textit{ player} $1$\textit
{
arrives if} $\ k=0,2,4,...$\textit{ and player }$2$\textit{ arrives if
}$k=1,3,5,...$;\textit{ (ii) at }$t\in\lbrack
t_{1/2}^{\ast},T)$\textit{, both
players arrive. }$\medskip\medskip$

\noindent The proof of Claim 1 is in the Appendix.

When Claim 1's strategies are played out, player 1 arrives at
$t^{\ast}$,
player 2 arrives at $T$, and both players obtain zero payoffs. Thus, the
parking lot yields zero social benefit; and if it is costly to construct
or
the land has other uses, the lot has a negative social value.

Most of the description of the equilibrium is needed to specify what the
players do off the equilibrium path, of one of them were to deviate from
equilibrium. In particular, suppose player 1 fails to arrive at
$t^{\ast}$.
Player 2 then enters at $t^{\ast}+\Delta$ and obtains a positive payoff.
Player 1 waits until $T$ rather than trying to enter at
$t^{\ast}+\Delta$,
because he knows that Player 2 will be entering at $t^{\ast}+\Delta$,
and a
fifty percent probability of getting a spot is not enough to justify
showing
up at $t^{\ast}+\Delta$. A player's arrival at any $t\in\lbrack t^{\ast
},t_{1/2}^{\ast})$ is sustained by the threat of arrival by the
competitor at
$t+\Delta$. For $t\in\lbrack t_{1/2}^{\ast},T)$, arrival at $t$ weakly
dominates any other time-$t$ decision of a player.

The equilibrium is not unique. There is another pure-strategy
equilibrium in
which players alternate their out-of-equilibrium threats of arrival
starting
from time $t^{\ast}+\Delta$. The two equilibria yield similar outcomes
in
which player $1$ arrives at $t^{\ast}$ or $t^{\ast}+\Delta$ and player
$2$
arrives at $T$, and they are the only equilibria in the game, up to the
identity of the players.

\bigskip

\noindent\textbf{The General Case: $N$ Drivers}

Let us now describe the pure-strategy subgame-perfect Nash equilibria
for the
game with $N$ players. Denote by $\widehat{n}_{t}$ the critical number
of
players for whom entry yields a nonnegative payoff at time $t$, given
that
$K_{t}$ spaces remain unclaimed by time $t$. The \emph{free-entry}
number of
players, $\widehat{n}_{t}$, for given $K_{t}$ and $t\geq t^{\ast}$, is
determined as the largest $n_{t}$ that satisfies the positive entry
payoff
condition $u_{i,t}=\left(  K_{t}/n_{t}\right)  v-L(t)\geq0$. By
definition,
$u_{i,t^{\ast}}=v-L(t^{\ast})=0$, and therefore for a given $K_{t}$
there
exists a unique
\begin{equation}
\widehat{n}_{t}\equiv\max\left\{  n_{t}\in Z_{+}:u_{i,t}\geq0\right\}  .
\tag{5}%
\end{equation}


Claim 2 describes all pure-strategy subgame-perfect equilibrium for the
parking game on a fine time grid.$\medskip\medskip$

\noindent\textbf{Claim 2.} \textit{Consider the parking game on a fine
time
grid under full observability and with more drivers than parking spaces
(}$N>K$\textit{).}

\noindent\textbf{A.}\textit{\ In any pure-strategy equilibrium to the
game,
the parking lot is full after }$t^{\ast}+\Delta$\textit{. The
equilibrium
outcome is for }$K_{0}\in\{0,...,K\}$\textit{\ players arrive at }
$t^{\ast}$;
$K-K_{0}$ \textit{players to arrive at} $t^{\ast}+\Delta$\textit{, and }
$N-K$
\textit{players to arrive at }$T$.\textit{\smallskip}

\noindent\textbf{B.}\ \textit{All the existing pure-strategy equilibria
have
the following arrival schedule: (i) at }$t=t^{\ast}$\textit{, }$K_{0}%
\in\{0,...,K_{t}\}$\textit{\ players arrive; (ii) at }$t=t^{\ast}+\Delta
$\textit{, }$K_{t}$ \textit{players arrive; if player }$i\in I$\textit{
deviated by not arriving at }$t^{\ast}$,\ \textit{the set of arriving
players
excludes player }$i$\textit{; (iii) at }$t\in\lbrack t^{\ast}+2\Delta
,t_{1/2}^{\ast})$\textit{, if player }$i\in I$\textit{ deviated by not
arriving at }$t-\Delta$\textit{, one other player arrives; (iv) at }
$t\in\lbrack t_{1/2}^{\ast},T)$\textit{, if player }$i\in I$\textit{
deviated
by not arriving at }$t-\Delta$, \textit{then }$\min\left\{  \widehat{n}%
_{t},N_{t}\right\}  \geq2$\textit{\ players arrive at }$t$\textit{; (v)
at
}$t=T:$\textit{\ all players arrive who have not yet arrived.}$\medskip
\medskip$

Claim 2 shows that there are multiple equilibria under full
observability,
even if we restrict our attention to pure-strategy equilibria. Some
players
might arrive at the indifference arrival time $t^{\ast}$. This yields
them
zero payoffs and has the same effect on the remaining players as if the
parking lot size $K$ had shrunk and the game were started at
$t^{\ast}+\Delta
$. There are many sizes of this initial shrinkage that support
equilibria. For
a fine time grid, the equilibrium outcome in any pure-strategy
equilibrium is
for $K_{0}\in\{0,...,K\}$ parking spaces to be filled at $t^{\ast}$ and
the
rest to be filled at $t^{\ast}+\Delta$. For example, there is an
equilibrium
outcome in which one group of $K$ players arrive at time $t^{\ast}$ and
they
park in the lot, while a second group of $N-K$ players arrive at $T$ and
park
elsewhere. Both groups receive the same payoff of zero. Another polar
equilibrium outcome is for no drivers to arrive at $t^{\ast}$ and for
$K$
players to arrive at $t^{\ast}+\Delta$. In all pure-strategy equilibria,
the
parking lot fills up no later than $t^{\ast}+\Delta$ and no players
arrive
between $t^{\ast}+\Delta$ and $T$.

An important corollary of Claim 2 concerns the extent of rent
dissipation in
the parking game under observability.

\bigskip

\noindent\textbf{Corollary.} \textit{As the time grid becomes infinitely
fine,
drivers dissipate all the rents from parking in any pure-strategy
equilibrium
of the parking game under full observability when there are more drivers
than
parking spaces.}

\medskip$\medskip$

In the limit, as $\Delta\rightarrow0$, players dissipate the entire
value of
the parking lot:\ each driver has an expected payoff of zero and the
total sum
of all the losses incurred by drivers is equal to the total value of
parking,
$vK$. The parking lot is still costly, however, so the social payoff is
negative and equal to the cost of building the parking lot.

We will not attempt to fully describe the mixed-strategy equilibria due
to
their complexity. However, in Section 4 we discuss the extent of rent
dissipation in any equilibrium to the parking game, including those in
mixed strategies.

\bigskip

\subsection{Nonexistence of Pure-Strategy Equilibria under
Unobservability}

Recall that \textquotedblleft unobservability\textquotedblright\ means
that a
player does not know how full the parking lot is when he chooses the
time at
which to arrive. Claim 2 said that under full observability some players
arrive at $t\leq t^{\ast}+\Delta$ and others arrive at $T$, both having
nearly
zero expected payoffs. Under unobservability, Claim 2 does not apply
because
one of the players who is supposed to arrive at $t\leq t^{\ast}+\Delta$
could
deviate and arrive at $T-\Delta$ instead, reducing his early-arrival
cost. The
other players would not observe that he had failed to arrive as
scheduled at
$t$, so they would be unable to respond by taking \textquotedblleft
his\textquotedblright\ spot before $T-\Delta$.\footnote{Of the
equilibria in
Claim 2, one survives, but only when the time grid is \textit{not} fine.
The
following is an equilibrium for the parking game under unobservability
for
$N>K$ and the time grid that is not fine, \noindent(i) At
$t=t^{\ast}+\Delta$,
$\min(\widehat{n}_{t},N_{t})>K$\ players arrive; \noindent(ii) At $T$,
all
players arrive who have not yet done so. If any player deviated to
arrive at
$t^{\ast}$, he would not improve on his equilibrium payoff. If any
player
deviated to arrive after $t^{\ast}+\Delta$ but before $T$, he would not
find a
parking spot and would receive a negative payoff. This establishes that
the
specified strategies form an equilibrium.}

\medskip\medskip

\noindent\textbf{Claim 3.}\textit{ There does not exist a pure-strategy
Nash
equilibrium under unobservability when the time grid is fine and there
are
more drivers than parking spaces}.\medskip\medskip

\noindent\textbf{Proof of Claim 3. }Denote by $t^{\prime}$ the time when
the
parking lot becomes full in a pure-strategy equilibrium. If at
$t^{\prime}$
not all arriving drivers obtain parking, then each of them would find it
profitable to deviate by arriving a period earlier. If parking is
guaranteed
at $t^{^{\prime}}$, then there are drivers who arrive at $T$ because
arriving
between $t^{\prime}$ and $T$ yields a negative payoff. Each of these
drivers
benefits from arriving shortly before $t^{\prime}$ unless
$t^{\prime}\leq
t^{\ast}+\Delta$. A fine time grid implies that only $K$ drivers would
arrive
at $t^{\prime}=t^{\ast}$ or $t^{\prime}=t^{\ast}+\Delta$. Then, each of
them
can arrive later and still obtain the parking space. Hence, no pure-
strategy
equilibrium exists in this case either. Q.E.D.\medskip$\medskip$

This parking game is a particular form of a variety of rent-seeking game
most
intensely studied in the context of multi-prize auctions. The arrival
times
are like bids, and the empty parking spaces are like prizes. The reason
for
the nonexistence of a pure-strategy Nash equilibrium under
unobservability
when the time grid is fine is essentially the same as in a multi-prize
all-pay
auction with continuous strategy space (see Barut and Kovenock, 1998,
Clark
and Riis, 1998, or the general characterization by Baye et al. [1996]).
Auction models focus on bidding rules rather than capacity choice, but
as in
the parking game, the question is how a limited number of goods are
allocated
to a large number of players.\footnote{Holt and Sherman (1982) even use
auction theory to analyze waiting times in a queuing model. They use
auction
theory's revenue equivalence theorem to show that various rules for
determining the length of the wait all lead to the same amount of
waiting
time.} In a pure-strategy equilibrium, everyone knows which bids win and
which
ones lose. Bidders who know they would lose would bid zero. But if they
did
that, winning bidders would bid no more than slightly above zero, in
which
case the losers would want to deviate to bids slightly higher than that
and
win instead.

In a pure-strategy equilibrium of this parking game, players who did not
obtain parking would arrive at the preferred time, so other players
would have
no incentive to arrive more than slightly earlier. This, however, cannot
be an
equilibrium because the players arriving at the preferred time would
deviate
and arrive even earlier for a guaranteed empty space.

Mixed-strategy equilibria under unobservability are complicated to
describe
fully. In the two-driver game there is, for example, an alternating
equilibrium in which for $k\in\{0,1,2...\}$ and $t\leq T$ player $1$
arrives
at $t=t_{0}+2k\Delta$ with probability $(v/(2w\Delta)-k)^{-1}$ and
player $2$
arrives at $t=t_{0}+(2k+1)\Delta$ with the same probability. In the case
of
$w=1$, $v=10$, $T=10$, and $\Delta=1$, the probability of player $1$
arriving
at $t=1,3,5,7,9$ and player $2$ arriving at $t=2,4,6,8,10$ is 1/5, 1/4,
1/3,
1/2, and $1$. After $t=T-\Delta=9$ the parking lot is full with
probability
one. What we can more easily (and usefully) characterize, however, is
the
nature of the payoffs that result from the equilibrium mixed strategies.
In
the next section we will show that almost complete rent dissipation
occurs in
any equilibrium.

\bigskip

\section{Full Rent Dissipation}

We first establish that in any equilibrium, players arriving at
$\underline
{t}$\textit{\ }and\textit{\ } $\overline{t}$ (the earliest and the
latest
times any player arrives with a positive probability along the
equilibrium
path) obtain nearly zero payoffs whenever there are more drivers than
parking
spaces. Then, we will show that all the players in between must almost
fully
dissipate the value from parking. Our propositions will apply under
either
full observability or unobservability.

\medskip$\medskip$

\noindent\textbf{Claim 4. }\textit{Consider the parking game on a fine
time
grid with more drivers than parking spaces. In any subgame-perfect
equilibrium, the probability that the parking lot is full at time }
$\overline{t}\leq T$\textit{ is one under full observability and
approaches
one as the time grid becomes infinitely fine under unobservability. The
equilibrium payoffs of players who arrive at }$\overline{t}$ \textit{
with a
positive probability are zero under full observability and tend to zero
under
unobservability as the time grid becomes infinitely fine. Any player who
arrives at }$\underline{t}$ \textit{has a payoff that tends to zero as
the
time grid becomes infinitely fine.}$\medskip\medskip$

The proof is given in the Appendix. Claim $4$ says that the parking lot
is
almost surely full at $\overline{t}$, and the payoffs of players who
arrive at
$\overline{t}$ and $\underline{t}$ are close to zero. It is nearly
impossible
to find parking at $\overline{t}$, because otherwise a player arriving
at
$\overline{t}$\ could deviate by arriving at $\overline{t}-\Delta$ (
while
following the same strategy before $\overline{t}-\Delta$) and increase
his
odds of obtaining parking. An incentive to deviate exists whenever there
is a
parking space available at $\overline{t}-\Delta$ (under full
observability) or
when there is a nontrivial probability that parking is available at
$\overline{t}-\Delta$ (under unobservability). Since there is almost no
chance
that a parking spot is available at $\overline{t}$, players' payoffs at
$\overline{t}$ are nearly zero. It also follows that no parking spaces
are
available at time $T\geq\overline{t}$ with a probability approaching
one.

\medskip\medskip

\noindent\textbf{Corollary.} \textit{The parking lot is almost surely
full by
time} $T$\textit{.}

\medskip\medskip

Moreover, we can easily prove that the earliest and the latest time
anyone
arrives with a positive probability in an equilibrium approaches
$t^{\ast}$
and $T$, respectively, as the time grid becomes infinitely
fine.\footnote{First, we show that $\underline{t}\rightarrow t^{\ast}$
as
$\Delta\rightarrow0$. If $\underline{t}\leq t^{\ast}+\Delta$, then
$\underline{t}\rightarrow t^{\ast}$. Suppose that $\underline{t}>t^{\ast
}+\Delta$. At $\underline{t}$, all parking spaces are available, and the
probability of obtaining a space at $\underline{t}$ converges to one
because
otherwise a player assigned to arrive at $\underline{t}$ would arrive at
$\underline{t}-\Delta$. Since the payoff at $\underline{t}$ approaches
zero,
$\underline{t}\rightarrow t^{\ast}$ in this case as well. Second, since
it is
nearly impossible to obtain parking at $\overline{t}$ in an equilibrium
and
players receive nonnegative payoffs, the cost of arriving at
$\overline{t}$
must be zero in the limit as well. Hence, $\overline{t}\rightarrow
T$\textit{\ }as $\Delta\rightarrow0$.} We next show that players tend to
dissipate all the rents from parking in any equilibrium of the parking
game.\medskip\medskip

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

\medskip

\noindent\textbf{Proof of Proposition 1.} The proof is by contradiction.
Suppose player $i$ is earning a positive payoff bounded away from zero.
Let
$t\in(\underline{t},\overline{t})$ be the earliest moment the player
arrives
with a positive probability; $t>\underline{t}$. Consider a player (
player $j$)
who arrives at $\overline{t}$ (with pure strategy) and earns an almost
zero
payoff (by Claim 4). If player $j$ follows his equilibrium strategy
until
$t-\Delta$ and arrives with probability one at $t-\Delta$, he would
obtain a
positive payoff bounded away from zero. The costs of arriving at $t$ and
$t-\Delta$ differ by $\Delta w$, a difference that becomes negligible as
$\Delta\rightarrow0$. It suffices to prove that player $j$'s odds of
obtaining
parking at $t-\Delta$ are no worse than player $i$'s odds of obtaining
parking
at $t$. Note that by the definition of time $t$, player $i$ has not
attempted
to arrive before $t$. Player $j$ may have been mixing before $t$.
Therefore,
player $j$ has the expected payoff from arriving at $t-\Delta$ that is
no less
than that of player $i$. Since there is a profitable deviation, there
cannot
exist an equilibrium with a player obtaining a positive payoff at a time
$t$
between $\underline{t}$ and $\overline{t}$. Hence, all players earn
almost
zero payoffs. Q.E.D.

\bigskip

That the driver payoffs approach zero if the parking lot is even
slightly too
small is the important finding for deciding on the optimal size of the
parking
lot. The nature of the equilibrium strategies that lead to this unhappy
outcome is not only less determinate, but less useful.

\bigskip

\section{Welfare and Parking Lot Size}

Sections $3$ and $4$ described and analyzed the rent-seeking competition
among
drivers for a fixed number of parking spaces. We next characterize
welfare
more fully by accounting for the cost of building the parking lot. The
drivers
have a value $v>0$ for each of the parking spaces in use. Let us denote
the
cost of providing $K$ parking spaces by an increasing function $C(K)$.
The
welfare from a parking lot of size $K$ is
\begin{equation}
W(K)=\left\{
\begin{tabular}
[c]{lll}%
$vN-E_{\sigma(N,K)}\left(  \sum_{i=1}^{N}L(t_{i})\right)  -C(K)$ & if &
$K\geq
N$\\
$vK-E_{\sigma(N,K)}\left(  \sum_{i=1}^{N}L(t_{i})\right)  -C(K)$ & if &
$K<N,$%
\end{tabular}
\ \ \ \ \ \right.  \tag{6}%
\end{equation}
where $E_{\sigma(N,K)}\left(  \sum_{i=1}^{N}L(t_{i})\right)  $ denotes
the
expected equilibrium cost to drivers of arriving early for given $K$ and
$N$.
Welfare can be decomposed into three parts: the \textit{total value} of
the
parking lot (which is either $vN$ or $vK$, depending on whether
$K\geq N$ or
not), the \textit{rent-seeking loss}, and the \textit{construction cost}
$C(K)$. The next two subsections deal with the problem of finding the
optimal
parking lot size in cases of a certain and an uncertain number of
drivers for
an infinitely fine time grid.\medskip

\subsection{Optimal Capacity Under Certainty}

Section 4 showed that in competition for a limited number of parking
spaces,
drivers tend to dissipate all the rents from parking. In any
equilibrium, they
are indifferent between arriving early enough to secure a spot and not
parking
in the parking lot at all. The rent-seeking loss from earlier arrivals
equals
zero when $K\geq N$ and is $vK$ when $K<N$. Hence, welfare is%

\begin{equation}
W(K)=\left\{
\begin{tabular}
[c]{lll}%
$vN-C(K)$ & if & $K\geq N$\\
&  & \\
$-C(K)$ & if & $K<N$\\
&  &
\end{tabular}
\ .\right.  \tag{7}%
\end{equation}


Full rent dissipation occurs if the number of players exceeds the size
of the
lot. To avoid wasteful rent-seeking activity, parking lots should be
designed
to accommodate 100\% of the people who want parking. When the number of
such
people, $N$, is known with certainty, the parking lot should have $N$
parking
spaces; having slightly too few spaces is disastrous, a result
summarized in
Proposition $2$ and illustrated in Figure $2$.

\medskip

\noindent\textbf{Proposition 2.} \textit{The optimal size of the parking
lot
under certainty equals the number of users: }$K^{\ast}=N$\textit{. All
smaller
sizes have negative welfare, with the minimum welfare being at }%
$K=N-1$\textit{. All greater sizes have the same total value from
parking
equal to }$vN$\textit{, but increasingly high construction costs. This
is true
under both full observability and unobservability.}

\medskip

\noindent\textbf{Proof of Proposition 2. }If $K<N$, then by Proposition
1, in
any equilibrium almost all rents are dissipated and $W(K)=-C(K)$. Thus,
the
minimum welfare for $K\in\{0,1,2,...,N-1\}$ is at $K=N-1$. If $K\geq N$,
each
player is guaranteed a parking space and arrives at the preferred time
$T$;
therefore $W(K)=vN-C(K)$ and welfare falls gradually in $K$. It follows
that
welfare is maximized at $K=N$ as long as it is at all socially
beneficial to
build a parking lot, i.e. when $vN\geq C(K)$. Q.E.D.

According to Proposition 2, the size of the parking lot should be set at
$K=N$, as long as it is at all socially beneficial to build a parking
lot.
This result is clear from the shape of the welfare function in Figure 2.
There
is a sharp discontinuity in social welfare between \textquotedblleft
enough\textquotedblright\ and \textquotedblleft not enough,
\textquotedblright%
\ with the minimum social welfare being at slightly too small a parking
lot.

\begin{center}
$\left[  \mathbf{Figure\ 2}\right]  $
\end{center}

Instead of finding the optimal capacity for a given number of drivers (
fixing
$N$ and choosing $K$), we could consider the problem of choosing a
number of
permits for a parking lot of a given size (fixing $K$ and choosing $N$).
This
interpretation is appropriate when capacity is a chosen in the long run,
while
access to the parking lot can be regulated in the short run. Proposition
2
implies that the optimal number of permits is equal to the lot's
capacity. The
next subsection deals with the case of uncertain number of drivers. We
will
see that uncertainty over the number of drivers actually increases
social
welfare if the parking lot size is too small.

\medskip

\subsection{Optimal Capacity under Uncertainty}

Assume that planners must decide on the size of the parking lot before
they
know the number of drivers, $N$, whereas drivers know both $N$ and the
parking
lot capacity, $K$. Thus, drivers know $N$ but planners only know that it
is
drawn from a known probability distribution, $f(N)$, with support
$\{0,1,...,\overline{N}\}$.\footnote{We do not consider here the case
where
drivers are equally uncertain about the number of people seeking
parking. We
conjecture that this case would result in a lower optimal capacity.}
What is
the optimal parking lot size? We will next show that maximization of the
expected welfare would require a much bigger parking lot than the
expected
number of drivers because the loss function in Figure $2$ is asymmetric.

When there is competition for parking spots, $K<N$, the welfare from the
parking lot is negative, $W=-C\left(  K\right)  $. When the size of the
lot is
large enough, $K\geq N$, the welfare is $W=vN-C\left(  K\right)  $.
Therefore,
for $K\leq\overline{N}$, the expected welfare is%

\begin{equation}
EW(K)=\sum_{N=0}^{K}(vN)f(N)-C\left(  K\right)  =vE(N\text{
%TCIMACRO{\TEXTsymbol{\vert}}%
%BeginExpansion
$\vert$%
%EndExpansion
}N\leq K)-C\left(  K\right)  . \tag{8}%
\end{equation}
The size of the parking lot should be increased from $K-1$ to $K$ as
long as
the change in the expected welfare is nonnegative: $EW(K)-EW(K-1)\geq0$.
Thus,
the optimal $K$ is the biggest $K$ such that $K \leq\bar{N}$ and
\begin{equation}
vKf(K)-\left(  C\left(  K\right)  -C\left(  K-1\right)  \right)  \geq0.
\tag{9}\label{e9}%
\end{equation}
The expression on the left-hand side of inequality (\ref{e9}) has
intuitive
meaning. Drivers benefit from the $K$th parking space only if $N=K$
exactly.
If $N<K$ the marginal space is unused, and if $N>K$ the benefit is
dissipated
by rent-seeking so it does not matter whether there are $K$ or $K-1$
spaces.
Since the $K$th space matters only if $N=K$, the change in the expected
welfare is the probability that there are exactly $K$ drivers multiplied
by
the benefit from eliminating rent-seeking behavior, $vK$, net of the
change in
construction costs. Whether the marginal benefit of a parking space is
decreasing or increasing depends on the relative strength of two
effects. On
the one hand, at larger parking lot sizes, it is more important to have
a
sufficiently big parking lot because there are more people who could get
benefit from it. On the other hand, it could be less likely that larger
parking lots are filled up. When the first effect dominates and $Kf(K)$
increases in $K$, the marginal benefit increases with $K$ too. For a
constant-marginal-cost technology, $C(K)=cK$, this implies a corner
solution
to the problem: the parking lot should accommodate all potential
drivers, even
if that is much greater than the expectation of the number of drivers,
or not
be built at all.

To illustrate, consider a discrete uniform distribution for the number
of
drivers on the support $\{0,...,\overline{N}\}$. For the uniform
distribution,
$Kf(K)$ does increase in $K$, so we have a corner solution. Under the
uniform
distribution, it is equally likely at any capacity that the parking
demand
will be barely met. At a larger capacity benefits accrue to more people,
so at
larger capacity levels, the benefit from building an additional space is
higher, and planners should design the parking lot for the
\textquotedblleft
peak demand,\textquotedblright\ accommodating all potential
drivers.\footnote{Designing for peak demand is not always the optimal
choice.
For example, in a case of the binomial distribution of $N$, if each of
$100$
drivers is in need of parking with probability $0.5$ and $c/v=1/5$, the
optimal parking size is $K^{\ast}=58$. Only about $8$ out of $58$ spaces
are
empty, on average. This corresponds to about $86\%$ utilization level.}
Figure
$3$ shows the expected welfare for the uncertainty case and welfare for
the
certainty case at different capacity levels when $c=1$, $w=1$, $v=5$,
and
$\overline{N}=100$.

\begin{center}
\noindent$\left[  \mathbf{Figure\ 3}\right]  $
\end{center}

At the optimal capacity level, only $50\%$ of parking spaces are
occupied on
average. Compare the consequences of a limited capacity for certain and
uncertain $N$. Uncertainty over the number of drivers actually increases
social welfare if the parking lot size is too small. While the optimal
capacity under uncertainty is $K^{\ast}=100$, welfare is positive even
if
$K=40$. This is a big difference from the case without uncertainty,
where
welfare would be negative and large in magnitude at $K=40$. The reason
is that
under uncertainty, even with very few parking spaces, it may happen that
very
few people need to park, and so there is no wasteful rent-seeking and
the
parking spaces are valuable.

Thus, uncertainty over the number of drivers actually increases social
welfare
if the parking lot size is too small. Although under uncertainty the
welfare
is positive when the parking lot is slightly too small, under certainty
it
would be near its minimum and negative. Uncertainty creates the
possibility
that $N<K$, so costly rent-seeking is less necessary. This advantage of
uncertainty is somewhat paradoxical. In the first-best, when people are
allocated to parking spots, welfare would be highest under certainty
since no
parking spot would ever go unused and no driver would ever fail to find
a
parking spot. The same is true when drivers are not strategically
changing
their arrival schedules in an attempt to secure a spot. Uncertainty
would then
make some excess capacity optimal, but would reduce welfare.

If people are strategic, however, the consequences of mistaken policy
lead to
quite different outcomes. Having a slightly too small parking lot would
be
disastrous under certainty, with zero total value from parking, but
under
uncertainty the total value would still be positive. The problem of
rent-seeking certainly does not disappear under uncertainty, however,
and
optimal capacity is much larger under uncertainty than under certainty.
Importantly, empty parking spaces are not an indication that the parking
lot
is too big. In a stochastic model, there will usually be many empty
spaces in
a lot of optimal size. Planners should not be tempted to reduce capacity
(or
increase the number of parking permits), although some observers may
decry the
wastefulness of building parking lots that are too large (or of cruelly
limiting the number of permits despite the presence of unused parking
spaces).

Finally, we present the analysis of the parking problem when the number
of
drivers is large and calculus methods can be employed. Consider the
production
technology with a constant marginal cost, $C(K)=cK$ where $c<v$. The
expected
welfare from a parking lot of size $K$ is then%

\begin{equation}
EW(K)=v\int_{0}^{K}Nf(N)dN-cK. \tag{10}\label{e10}%
\end{equation}
The optimal size of the parking lot is the solution to the first-order
condition
\begin{equation}
\frac{\partial EW(K)}{\partial K}=vKf(K)-c=0, \tag{11}%
\end{equation}
which can be written as
\begin{equation}
Kf(K)=\frac{c}{v}. \tag{11'}\label{e11}%
\end{equation}


Expansion of the parking lot improves welfare if the probability that
exactly
$K$ drivers compete for $K$ parking spaces times the size of the parking
lot
exceeds the relative cost of building a parking space. We can use (\ref{
e11})
to find the optimal level of $K$ as an internal solution if the maximand
in
(\ref{e10}) is concave. Unfortunately, that seems unlikely. The second-
order
condition, $\partial^{2}EW(K)/\partial K^{2}<0$, can be written as
$Kf^{\prime}(K)+f(K)<0$, or $Kf^{\prime}/f<-1$. To guarantee the
existence of
the interior solution, the density must be declining and elastic in the
relevant range of $K$. Otherwise, a parking lot should be built for the
peak
demand (the highest $N$ possible).\footnote{For example, consider a
family of
continuous density functions $f(N)=\alpha N^{-\beta}$ defined on the
support
$[\underline{N},\overline{N}]$, where $\alpha,\beta>0$, and $[\underline
{N},\overline{N}]\subset\lbrack0,\infty)$ are such that
$\int_{\underline{N}%
}^{\overline{N}}\alpha N^{-\beta}dN=1$. The first-order condition
implies the
optimal capacity $K^{\ast}=\left(  \frac{1}{\alpha}\frac{c}{v}\right)
^{-1/(\beta-1)}$, and the second-order condition is satisfied if and
only if
$\beta>1$. For $\beta\leq1$, the parking lot should have $\overline{N}$
spaces, if it is built. The mean utilization of the parking lot of
optimal
size can be measured as a ratio of the mean number of parking spots
taken,
$E(X)$, to the optimal capacity size, $K^{\ast}$. If $N<K$, then all $N$
drivers find parking; if $N\geq K$, then $K$ out of $N$ drivers find
parking.
Hence, $E(X)=\int_{\underline{N}}^{K^{\ast}}Nf(N)dN+\int_{K^{\ast}}%
^{\overline{N}}K^{\ast}f(N)dN$. For example, 50\% of the parking spots
will
remain unoccupied on average when $c/v=0.5$, $\beta=1.5$, $K^{\ast}=4$,
and
$E(X)=2$, and the 50\%-utilized parking lot is socially optimal.{}}

\bigskip

\section{Heterogeneous Drivers}

Above, we analyzed the extreme case in which all players have the same
parking
value $v$, early arrival cost $w$, and preferred arrival time $T$. What
happens if players differ in these parameters? It turns out that the
extreme
result of full rent dissipation disappears but rents are still partially
dissipated, and parking lots should be built large because the welfare
loss
from having capacity of a wrong size is still asymmetric: the welfare
loss
from a slightly too small capacity is much larger than from a slightly
too
large capacity. We will establish these results in the framework of the
parking game under full observability, for which the pure-strategy
subgame-perfect equilibrium continues to exist under driver
heterogeneity.

The effects of heterogeneity in parking value, $v$, or early arrival
cost,
$w$, are the same. Suppose drivers have heterogenous values for parking.
Let
$N$ be the number of drivers with positive values and let $v_{i}$ denote
the
value of $i$'th-highest-valuing driver, so $v_{1}\geq v_{2}\geq...\geq
v_{N}>0$. For example, in the case of linear demand, $v_{i}=a-bi$ for
some
constants $a$ and $b$. Assume that there are also drivers who have zero
value
for parking and hence arrive at their preferred time, $T$. As before,
the cost
of constructing a parking space is $c$, and the cost per unit time of
early
arrival is $w$.

In the first-best case, both the parking lot capacity and access to
parking
can be regulated. The first-best policy is to choose capacity to
accommodate
all players with values higher (or weakly higher) than $v_{i}=c$ and to
give
only them access to the parking lot.

When access to parking cannot be restricted, however, planners should
design a
larger parking lot. Intuitively, the second-best policy is to set a
higher
than the first-best capacity in order to reduce rent-seeking losses that
occur
when access to parking cannot be restricted. The simple policy
prescription to
equate the marginal benefit to the marginal construction cost is not
valid
when drivers can engage in costly schedule adjustments. In the second-
best
world, the optimally designed parking lot provides parking even to
drivers who
value it less than the construction cost. In many cases, including that
of
linear demand for parking, the second-best policy is to choose $K=N$ and
guarantee parking to all players with positive values.

To establish these surprising results, we need to derive the welfare for
a
parking lot of size $K$ when drivers are heterogeneous. When a parking
lot has
$N$ or more spaces, all players with $v>0$ arrive at $T$ and find a
parking
space. There is no rent-seeking loss since nobody arrives before $T$ and
there
is no \textquotedblleft unserved demand loss\textquotedblright\ because
all
players with $v>0$ find spaces. For parking lots of size $K>N$, welfare
falls
due to \textquotedblleft excess construction cost \textquotedblright\ at
a
rate equal to the marginal cost of construction. For example, if $K=N+1$
instead of $K=N$ spaces are built, construction cost is higher by $c$
because
one extra space is constructed.

When a parking lot has fewer than $N$ spots, in any subgame-perfect
pure-strategy equilibrium to the parking game under observability,
rent-seeking will occur in the equilibrium. If $K<N$, the equilibrium
outcome
to the parking game on an infinitely fine time grid dictates that the
$K$
players with the highest values arrive at
$t_{K+1}^{\ast}\equiv T-v_{K+1}/w$
and find a space, and players with values $v_{K+1}$ or less arrive at
$T$. The
player with value $v_{K+1}$ will not deviate because his benefit from
arriving
at $t_{K+1}^{\ast}$ would be equal to his cost. The strategies that
sustain
this equilibrium outcome are similar to those given in Claim 2, and
arguments
similar to those given in Proposition 1 imply that the rent-seeking
losses are
equal to $K\cdot v_{K+1}$, because the $K$ players with the highest
values
must incur disutility $v_{K+1}$ of arriving at $t_{K+1}^{\ast}$ to deter
players with value $v_{K+1}$ or less from arriving early. There is
always
partial rent dissipation when the parking capacity cannot accommodate
all
drivers with positive value for parking, that is, when $v_{K+1}>0$.

The welfare from a parking lot of size $K\geq N$ is equal to the total
value
minus construction costs. For any $K<N$, the welfare also includes
rent-seeking losses. Thus, welfare is
\begin{equation}
W(K)=\left\{
\begin{tabular}
[c]{lll}%
$\sum\limits_{i=1}^{N}v_{i}-cK$ & if & $K\geq N$\\
&  & \\
$\sum\limits_{i=1}^{K}v_{i}-K\cdot v_{K+1}-cK$ & if & $K<N,$%
\end{tabular}
\ \ \ \ \ \ \right.  \tag{12}%
\end{equation}
and the change in welfare from adding an extra parking spot for
$K\leq N$ is
\begin{equation}
W(K)-W(K-1)=K\left(  v_{K}-v_{K+1}\right)  -c. \tag{13}%
\end{equation}
Expansion of capacity is warranted whenever $v_{K}-v_{K+1}\geq$ $c/K$,
or the
value from parking does not decline at $K$ faster than $c/K$. For linear
demand, there is a corner solution: a planner designs the parking lot to
accommodate all $N$ players with positive values, as long as it is
worthwhile
to build the parking lot at all, that is, as long as
$W(N)=\sum\limits_{i=1}%
^{N}v_{i}-cN\geq0$.

When $K$\ and $N$\ are large we could treat them as continuous variables
and
use calculus. Using $v=v(x)$ to denote the demand for parking, welfare
can be
written as
\begin{equation}
W(K)=\left\{
\begin{tabular}
[c]{lll}%
$\int_{0}^{N}v(x)dx-cK$ & if & $K\geq N$\\
&  & \\
$\int_{0}^{K}v(x)dx-K\cdot v(K)-cK$ & if & $K<N.$%
\end{tabular}
\ \ \ \ \right.  \tag{14}%
\end{equation}
Taking the derivative, we obtain
$\partial W(K)/\partial K=-K\cdot v^{\prime
}(K)-c$ for $K<N$. An interior solution exists when $v$ is sufficiently
convex
to ensure that $v^{\prime}(K)+K\cdot v^{\prime\prime}(K)>0$. In other
words,
allowing all drivers with positive values to park is not optimal when
$K\cdot
v^{\prime\prime}(K)/\left(  -v^{\prime}(K)\right)  >1$, that is, when
the
slope of the inverse demand is elastic.

\begin{center}
$\left[  \mathbf{Figure\ 4}\right]  $
\end{center}

Figure $4$ depicts the welfare from parking lots of various sizes for a
linear
demand. It illustrates how welfare can be decomposed into total value,
rent-seeking loss, and construction cost. In this example, we assume
that
$N=50$ drivers have positive values from $10$ to $0.2$ (player $1$ has
the
highest value, $v_{1}=10$, player $2$ has $v_{2}=9.8$, and so forth till
$v_{50}=0.2$), and some players have value $v=0$. The discrete demand
for
parking can be written as $v_{i}=\left(  51-i\right)  /5$; $i=1,...,50$.
As
before, we assume that $c=1$ and $w=1$. As is apparent in Figure $4$,
the
linear demand model has partial but not complete rent dissipation
(rent-seeking losses are lower than the total value). The biggest
difference
is that minimum welfare is at $K=4$ or $K=5$ rather than $K=N-1$. It
remains
true, however, that the biggest drop in welfare results from going from
the
optimum, $K=N$, to a slightly smaller lot, $K=N-1$. What is most
important for
planning purposes is that (i)\ as in the homogeneous-player model, the
loss
function from choosing the wrong capacity is asymmetric near the
optimum, and
choosing too small capacity is worse than choosing too large;\footnote{
If
$K=N=50$, the welfare is the sum of drivers' values for parking net of
construction cost, $W(N)=205$. If $K>N$, the welfare is reduced linearly
by
$c=1$ for each extra space constructed. On the other hand, if $K<N$,
construction costs are saved, but there is a loss from unserved demand
and
rent-seeking. For instance, if $K=N-1$, the welfare loss relative to
$K=N$ is
$9$ since $1$ is saved in construction, $0.2$ is lost due to unserved
demand,
and $9.8$ is lost in rent-seeking. Similarly, if $K=N-2$, the welfare
loss is
$17.8$, and for $K=N-3$ it is $26.4$. This is in contrast with the
welfare
loss from overprovision, which is $1$ for each extra unit constructed.}
and
(ii) as in the case of uncertain $N$, the conclusion that parking lots
should
be overbuilt holds.

To see this, think about the first-best capacity when the planner can
restrict
access to parking. In the first-best, the planner does not need to worry
about
rent-seeking losses; they will equal zero. All he needs to look at are
the
marginal construction cost of $c=1$ and the marginal benefit of $v$.
Thus, he
will choose either $K=45$ or $K=46$, which both yield the same welfare
since
$v_{46}=c=1$. If the social planner cannot regulate access to parking,
however, then $K=45$ is not optimal. All $N=50$ players with positive
values
will attempt to park, even though it is inefficient for those with $v<1$
to do
so, and it will be necessary for $45$ players to arrive early to avoid
losing
spaces to the $5$ low-valuing players. The second-best capacity is
$K=N=50$.

Another way to look at the problem is that if capacity is chosen at the
first-best optimum of $K=45$ there will be a discontinuous loss from
allowing
more than $N=45$ players access to the parking lot. Consider the problem
of
choosing the number of parking permits, $n$, for a lot of a given size,
$K$,
which arises when capacity is chosen in the long run while permits are
allocated in the short run. Assume that permits are rationed efficiently
to
the highest-value players. Clearly, the optimal number of permits cannot
be
less than capacity. For any $n>K$, the welfare is $K\cdot v_{K+1}$ lower
than
at $n=K$. Hence, the welfare loss from having too few or too many
parking
permits is not symmetric. Welfare jumps down by $K\cdot v_{K+1}$ when
one too
many parking permit is issued, but welfare falls gradually when too few
drivers are served. At the first-best capacity level $K=45$, welfare
drops due
to rent-seeking loss by $45\cdot5=380$ when $46$ permits (or more) are
allocated. In contrast, welfare drops by $1.2$ when $n=44$ and one
driver with
value $1.2$ is unserved. There is a discontinuity in welfare at the
optimal
number of permits, $n=K$; oversupply of permits is more dangerous than
undersupply.

\bigskip

\section{Concluding Remarks}

We have constructed various versions of model of strategic parking.
Suppose
1,001 drivers want to arrive at the same time and have the same costs of
arriving early, and each derives a benefit of \$250 from parking in a
particular lot during the year. If the cost of a parking space is \$200
per
year, it is obvious that 1,001 spaces should be built, for a net payoff
of
\$50,050 per year. What is not so obvious, and what has been the theme
of this
paper, is that if 1,000 spaces are built instead, the net payoff is not
\$50,000, but -\$200,000. Competition in the form of early arrival for
the
scarce spots eats up the entire benefit of the parking lot. Of course,
the
implications of a shortage are not as dramatic when, for example, the
number
of drivers is uncertain, but the extreme case shows the nature of the
problem.

Strategic incentives are an essential element in planning capacity for
an
underpriced good -- as important, or perhaps more important, than the
obvious
decision-theory problem of predicting uncertain demand and engineering
problem
of predicting capacity cost. If for some reason direct pricing is
impractical,
and the planner is aware that, with some probability, demand for the
good will
exceed the supply, he should realize that the damage from such
situations in
not limited to a few people being left unable to find a parking space.
People's actions to forestall being shut out vastly increase the social
loss.
This idea may extend to other settings as well.\footnote{The parking
problem
calls to mind matching games such as those studied by Roth and Xing (
1994) and
Avery et al. (2001). In Avery et al., for example, federal judges must
each
select one clerk from graduating law students, and students can work for
no
more than one judge. What has happened in recent years is that clerks
and
judges pair up earlier and earlier, rather towards the end of the
student's
last year of law school. A judge who waits too long to hire would not be
able
to find any good clerks available, so judges hire clerks early even
though a
lot more is known about an older student's quality. The top students are
analogous to the parking spaces in our model, and the possibility of
mistakenly hiring an incompetent student is the cost of arriving early.}
What
is special about this model compared to congestion models is that there
is a
sharp break in players' equilibrium payoffs when the number of people
seeking
to use the facility equals its capacity.

\newpage

\noindent{\Large Appendix: Proofs}$\medskip\medskip$

\noindent\textbf{Proof of Claim 1. }To prove that the listed strategies
are
part of a subgame-perfect Nash equilibrium we must show that there are
no
profitable deviations for any player at any point in time, given the
strategy
of his rival.

Both players have equilibrium payoffs of zero. Time $t^{\ast}$ is such
that
player $1$ is indifferent between arriving at $t^{\ast}$ and not parking
in
the lot at all. Arriving earlier than $t^{\ast}$ (at $t<t^{\ast}$)
yields a
negative payoff, $v-L(t)<0$. If player $1$ deviates at $t^{\ast}$ by
delaying
his arrival, player $2$ arrives at $t^{\ast}+\Delta$. Arriving at
$t^{\ast
}+\Delta$, player $1$ obtains $v/2-L(t^{\ast}+\Delta)<0$. Hence, player
$1$
cannot do any better than to arrive at $T$ and receive a zero payoff.

Similarly, a player assigned to arrive at
$t\in(t^{\ast},t_{1/2}^{\ast})$ will
do so, because otherwise the rival arrives next period. A one-period
delay in
player $i$'s arrival would save the player the cost of arriving one
period
earlier, $w\Delta$, but would reduce his odds of obtaining parking from
$1$ to
$1/2$. For a fine grid, player $i$ would prefer a secure space at $t$ to
a
competition for the parking space at $t+\Delta$ with fifty-fifty odds.
The
parking lot is full in later periods. A player assigned to arrive at
$t\in\lbrack t_{1/2}^{\ast},T)$ will do so when the parking lot is not
full at
$t$, because arrival at these times results in a nonnegative payoff even
when
the player's rival is certain to arrive at $t$.

No player who is not assigned to arrive at
$t\in\lbrack t^{\ast},t_{1/2}%
^{\ast})$ would arrive at that time because this would yield a negative
payoff
to the player. Q.E.D.$\medskip\medskip$

\noindent\textbf{Proof of Claim 2.} Consider a pure-strategy subgame-
perfect
Nash equilibrium. No player can arrive earlier than $t^{\ast}$. In part
A, we
need to show that the parking lot is full after $t^{\ast}+\Delta$ at the
latest. Let $t^{\prime}$ denote the time when the parking lot becomes
full and
suppose that $t^{\prime}\in\lbrack t^{\ast}+2\Delta,T]$. If the odds of
obtaining a parking space at $t^{\prime}$ are less than one for player
$i$ who
is supposed to arrive at $t^{\prime}$, the player would instead arrive
one
period earlier. If a parking space is guaranteed at $t^{\prime}$, then a
player who is supposed to arrive at $T$ would instead arrive at
$t^{\prime
}-\Delta$. Such players exist, since $N>K$ and the number of arrivals
prior to
$T$ is equal to the number of parking spaces. Hence,
$t^{\prime}\leq t^{\ast
}+\Delta$.

The proof of part B is similar to that of Claim $1$. Arriving earlier
than
$t^{\ast}$ never benefits a player. If player $i$ deviates from his
equilibrium strategy by not arriving at $t^{\ast}$, player $j\neq i$
arrives
at $t^{\ast}+\Delta$, and player $i$ receives a payoff of zero at best.
For
$t\in(t^{\ast},t_{1/2}^{\ast})$, it is always better for a player to
obtain a
parking spot at $t$ then to contest one remaining parking space next
period or
wait till later periods. At $t\in\lbrack t_{1/2}^{\ast},T)$, a delay by
the
player assigned to arrive at $t$ implies that the player does not obtain
parking, as the parking lot is filled at time $t$. The condition on the
fineness of the time grid implies
$\Delta<\frac{v}{\left(  K+1\right)  w}$;
this ensures that if $K_{0}=1$ and the single player assigned to arrive
at
$t^{\ast}$ deviates by arriving at $t^{\ast}+\Delta$, there are enough
other
players to compete for the parking spaces at $t^{\ast}+\Delta$ and make
his
deviation unattractive. Q.E.D.$\medskip\medskip$

\noindent\textbf{Claim 4. }\textit{Consider the parking game on a fine
time
grid with more drivers than parking spaces. In any subgame-perfect
equilibrium, the probability that the parking lot is full at time }%
$\overline{t}\leq T$\textit{ (a) is one under full observability and (b)
approaches one as the time grid becomes infinitely fine under
unobservability.
The equilibrium payoffs of players who arrive at }$\overline{t}$ \textit
{with
a positive probability (c) are zero under full observability and (d)
tend to
zero under unobservability as the time grid becomes infinitely fine. Any
player who arrives at }$\underline{t}$ \textit{has a payoff that (e)
tends to
zero as the time grid becomes infinitely fine.}$\medskip\medskip$

\noindent\textbf{Proof of Claim 4.} Recall that $\underline{t}$\textit{\
}and
$\overline{t}$ are defined as the earliest and the latest time any
player
arrives with a positive probability along an equilibrium path. Consider
a
subgame-perfect equilibrium of the parking game, under either full
observability or unobservability. By definition of $\overline{t}$, some
player
arrives at $\overline{t}$ with a positive probability along an
equilibrium
path and thus does not have a pure strategy of arriving before
$\overline{t}$.
Denote the player as player~$i$.

First, we prove statements a) and c) of Claim 4 under full
observability. Any
unarrived player $i$ who at $\overline{t}-\Delta$ observes $K_{\overline
{t}-\Delta}>0$ parking spaces available, would choose to arrive
immediately at
$\overline{t}-\Delta$ rather than wait to arrive at $\overline{t}$. This
is
true regardless of the number of other players, $n$, who happen to
arrive at
$\overline{t}-\Delta$. Intuitively, the earlier arrival increases player
$i$'s
odds of obtaining parking by a positive amount bounded away from zero
and any
such increase in odds justifies the additional cost of the earlier
arrival,
$w\Delta$, when the grid is fine.

More formally, denote by $p_{i,t}$ player\ $i$'s odds of obtaining
parking at
$t$ conditional on player $i$'s arrival at $t$. Consider two cases which
could
occur along the equilibrium path depending on realizations of
$K_{\overline
{t}-\Delta}>0$ and $n\geq0$. If $0\leq n<K_{\overline{t}-\Delta}$,
parking lot
becomes full only at $\overline{t}$;
$p_{i,\overline{t}}=K_{\overline{t}%
}/(N-\left(  K-K_{\overline{t}}\right)  )<K/N<1$ and $p_{i,\overline{t}%
-\Delta}=1$, and therefore $p_{i,\overline{t}-\Delta}-p_{i,\overline{t}%
}>\left(  1-K/N\right)  \geq1/N$. If $n\geq K_{\overline{t}-\Delta}>0$,
parking lot becomes full at $\overline{t}-\Delta$;
$p_{i,\overline{t}}=0$ and
$p_{i,\overline{t}-\Delta}=K_{\overline{t}-\Delta}/(n+1)>0$, and
therefore
$p_{i,\overline{t}-\Delta}-p_{i,\overline{t}}=K_{\overline{t}-\Delta
}/(n+1)>1/N>0$. By definition of a fine grid, $w\Delta<\frac{v}{N}$.
This
implies that the benefit of earlier arrival
$\left(  p_{i,\overline{t}-\Delta
}-p_{i,\overline{t}}\right)  v>\frac{v}{N}$ is higher than the cost,
$w\Delta
$, regardless of $K_{\overline{t}-\Delta}>0$ and $n\geq0$.

We proved that any unarrived player $i$ who observes
$K_{\overline{t}-\Delta
}>0$ arrives immediately at $\overline{t}-\Delta$. But then no parking
is
available at $\overline{t}$ since all $N$ players arrive before
$\overline{t}%
$. By definition of $\overline{t}$, some player must arrive at
$\overline{t}$
with a positive probability along an equilibrium path. Hence, with the
positive probability $K_{\overline{t}-\Delta}=0$ is realized in the
equilibrium. Any unarrived player who observes
$K_{\overline{t}-\Delta}=0$
parking spaces available at $\overline{t}-\Delta$, chooses to arrive at
$T$.
Hence, with a positive probability some players arrive at $T$, and by
definition of $\overline{t}$, $\overline{t}=T$. Moreover, no parking is
available at $\overline{t}=T$ in the case of $K_{\overline{t}-\Delta}>0$
as
well. To summarize, we proved that under full observability
$\overline{t}=T$
and no parking is available at $\overline{t}$. Any player arriving at
$\overline{t}=T$ along an equilibrium path has a zero probability of
obtaining
a parking space and, therefore, a zero equilibrium payoff. \smallskip

Second, we prove statements b) and d) of Claim 4 under unobservability.
Player
$i$ who arrives at $\overline{t}$ with a positive probability along an
equilibrium path must be unwilling to deviate to a pure strategy of
arriving
at $\overline{t}-\Delta$ if his strategy (which may be mixed) has not
led him
to arrive before that. Player $i$'s deviation is not profitable if
$u_{i,\overline{t}-\Delta}-u_{i,t}=E_{\sigma_{-i}}\left[  p_{i,\overline
{t}-\Delta}-p_{i,\overline{t}}\right]  ~\cdot~v-w\Delta\leq0$, where the
expected value is based on the equilibrium mixed strategies of other
players
(which determine $K_{\overline{t}-\Delta}$ and $n$). The analysis under
full
observability reveals that if $K_{\overline{t}-\Delta}=0$,
$p_{i,\overline
{t}-\Delta}-p_{i,\overline{t}}=0$, whereas if
$K_{\overline{t}-\Delta}>0$,
$\left(  p_{i,\overline{t}-\Delta}-p_{i,\overline{t}}\right)
v>\frac{v}{N}$
for any $n\geq0$. Under unobservability, the expected gain to player $i$
from
the earlier arrival is therefore larger than $\left[  \Pr\left(
K_{\overline{t}-\Delta}=0\right)  \cdot0+\Pr\left(
K_{\overline{t}-\Delta
}>0\right)  \cdot\frac{v}{N}\right]  $. The expression in the brackets
has to
be lower than the expected loss of $w\Delta$ for player $i$ not to
deviate
from the equilibrium strategy. Since the expected loss of $w\Delta$
approaches
zero as the time grid becomes infinitely fine, the expected gain has to
be
converging to zero as well. This is only possible if $\Pr\left(
K_{\overline{t}-\Delta}>0\right)  $ approaches zero as $\Delta$ shrinks
to
zero. Under unobservability we find that almost certainly no parking is
available to player $i$ arriving at $\overline{t}$, which is part (b) of
Claim
4. Player $i$'s payoff must tend to zero too, which is part (d) of Claim
4,
because his probability of finding a parking space tends to zero yet his
payoff cannot be negative (since he could always just arrive at $T$%
).\smallskip

Third, we prove statement e) under both full observability and
unobservability. Suppose a player arriving at $\underline{t}$ with a
positive
probability were to obtain a positive payoff bounded away from zero.
This
would imply that the player with a nearly zero payoff at $\overline{t}$
could
increase his payoff by arriving at $\underline{t}-\Delta$. Hence, the
payoff
of the player arriving at $\underline{t}$ must approach zero as $\Delta$
shrinks to zero, which is part (e) of Claim 4. Thus, we have proved all
five
statements of Claim 4. Q.E.D.

\pagebreak

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