DIVISION OF THE HUMANITIES AND SOCIAL SCIENCES
CALIFORNIA INSTITUTE OF TECHNOLOGY
PASADENA, CALIFORNIA 91125
General Equilibrium Methodology Applied to the
Design, Implementation and Performance Evaluation of
Large, Multi
-
Market and
Multi
-
Unit Policy Constrained
Auctions
Charles R. Plott, California Institute of Technology
Timothy Cason, Purdue University
Benjamin Gillen, Claremont McKenna College
Hsing
-
Yang Lee, California Institute of Technology
Travis Maron, California
Institute of Technology
SOCIAL SCIENCE WORKING PAPER 1447
January 2022
1
General Equilibrium Methodology Applied to the Design, Implementation and Performance
Evaluation of Large,
Multi
-
Market and Multi
-
Unit
Policy Constrained Auctions
1
Charles R
. Plott, California Institute of Technology
Timothy Cason, Purdue University
Benjamin Gillen, Claremont McKenna College
Hsing
-
Y
ang Lee, California Institute of Technology
Travis Maron, California Institute of Technology
January 202
2
Abstract
The paper reports on the
methodology,
design
and
outcome
of a large
auction with
multiple
,
interdependent
market
s
constructed from
principle
s
of
general equilibrium
as opposed to game
theoretic auction theory
.
It
distributed
18,788 entitlements
to operate
electronic gaming machines
in 176 interconnected markets to 363 potential buyers representing gaming establishments
subject to
multiple
policy constraints
on
the allocation
.
The
multi
-
round
auction
,
conducted in
one day
, produced
over $600M in revenue.
All
policy constraints were satisfied.
R
evealed
dynamics
of interim allocations
and new statistical tests
provide evidence
of
multiple market
convergence
hypothesized by
classical theories of general equilibrium
.
Results
support
the use of
computer supported,
“
tâtonnement
–
like” market adjustments
as
a
reliable
empirical process
es
and not
a
s
purely theoretical construct
s
.
1
The financial support for analyzing the data and developing this report provided to Plott by the Rising Tide
Foundation (Grant Number:
RTF
-
19
-
500) and the John Templeton Foundation (Grant Number: 58067) are
gratefully acknowledged
.
We also thank
the Calt
ech Laboratory for Experimental Economic and Political Science
for technical support and two anonymous referees for valuable comments
.
The Victorian government allowed
l
imited data analysis for scientific purposes.
The cooperation and help of William Steve
nson of Intelligent Market
Systems LLC were fundamental.
2
Section 1: Introduction
The paper reports on the design and field implementation of a large, multiple market and policy
constrained auction
bas
ed on
principles from
general equilibrium
theory
as opposed to
a
traditional theory of auctions
.
The auction involved the sale of
18,788
ten
-
year entitlements for
the use of electronic gaming machines in Victoria Australia, in May, 2
010.
Policy
goals
dictated
the operation of
176
interconnected markets to allocate sales of these licenses to
363
potential
buyers representing licensed gaming establishments.
The auction
outcomes satisfied
all
policy
constraints,
was conducted in one day
,
and produced
over $600M
in revenue.
The
auction
architecture
rested on principles of
an
exchange economy
in which bidders
are assumed to
have
well
-
formed preferences
and
make
choices
similar to those
guided by the classical tâtonnement
model of
market adjustment
.
The multiple market interdependencies created by policy
constraints, the size of the problem
,
and policy constrained
limitations on
timing
challenged
any
obvious application of traditional forms of auctions.
2
The theoretical framework
u
sed to
interpret
the results is informed by competitive market principles that demonstrate convergence to an
equilibrium with many features predicted by the classical theory.
The paper reviews the policy
background, the theoretical architecture,
some
key f
eatures of the laboratory experimental
testbedding
,
and
detailed
discussions of results and dynamic performance
, providing the first
field demonstration of
a
tâtonnement
-
like adjustment
as an empirical model of price formation
.
We
introduce and expand
two
broad,
overriding questions
used
to
evaluate the
mechanism’s
performance
relative
to
policy
-
focused
market designs.
3
The first question is a
form of proof of principle
evaluating
a
basic
question about the policy
’s implementation
.
(1)
“
Was the
implementation successful in satisfying the policy goals? Or, equi
v
alently, did the
implementation do what it was supposed to do
?
”
The second question
investigates
consistency
in
relation to the underlying theory
. (2) “Were the models used to guide the des
ign successful in the
sense that the
observed
market behaviors are consistent with the principles used in the design?
Or, equivalently, did it do what it did for the right reasons?”
2
Methodology and strategies for using alternative auction formats are open questions beyond the scope of our
analysis. We present an auction that addressed the given allocation problem. In doing so
, we demonstrate the
successful application of the theoretical principles underlying the auction’s design and empirical properties of price
formation and general equilibration across multiple markets.
3
The
se
questions were first posed by Plott (1994) as
a method
ology for
capturing the relationship between
experimental testbeds and policy implementation.
Subsequent literatures addressing design methodology (see
Milgrom (2000) or Roth (2008)) played no role in the design.
3
In the light of those questions t
hree
results stand out. First,
the aucti
on outcomes satisfied
all
the complex
policy constraints.
Second,
the auction provides substantial
field
support
, the first
of its kind,
for basic principles of classical
general equilibrium
and
competitive economic theory
including price discovery
, inhere
nt randomness
,
and convergence dynamics
.
The mechanism was
not narrowly Walrasian
because bidders submitted value functions and did not simply report
quantities demanded at announced prices
. The value functions were used to compute
s
urplus
maximizing allocations and
determine
equilibrium supporting prices that served as the basis for
additional bidding
. Nevertheless, the process was very “tâtonnement
-
like” as it employed an
algorithm for rou
nd
-
to
-
round dynamic adjustments of prices that responded to revealed excess
demand at previous prices. This process continued
until it “approximately” converged. The
assignment
process
was also “t
â
tonnement
-
like” because no trading occurred until prices re
ached
the approximate equilibrium. Moreover, t
he data show how the market reached equilibrium as
opposed to simply assuming that the outcomes correspond to an equilibrium outcome.
T
hird, the
data support the
application of the auction architecture that gui
ded the gaming machines design.
The auction was the result of the Victorian government’s efforts to change policies
regarding gambling operations
and regulations
in the state. In 2008, the government initiated a
reorganization of this industry by changing
the method of allocating the entitlements to operate
electronic gaming machines (e.g. poker and slot machines) and the method of finance.
Historically,
two large corporations managed
the distribution of gaming machines. The machines
were allocated to busi
nesses consistent with local policies governing their use. Finance
had been
based on the revenue produced by the machines with roughly a third going to the local
establishment, a third to the managing company, and a third to the government.
Governmental c
oncerns with the historical policy reflected a desire
to
better control
gambling and concomitant social problems,
gambling
-
related government public finance, and a
desire for conformity with frameworks used for economic regulation.
A
uction
-
style mechanism
s
emerged as
possible
tool
s
with
t
he
aggregate supply
quantity
dictated to be near historical levels.
Authorities wished
to create minimal economic dislocations and a climate where future
regulatory efforts could be based on principles of decentralized competition and
operator
profits.
The auction was also implemented to allow for possible entry and shifting of entitlements
from
past use to reflect underlying economic value rather than historical administrative practices.
4
The resulting
auction
mechanism and its implementation present a remarkable success for
the decades of abstract theorizing about general equilibrium in cl
assical economics.
It shows that
t
â
tonnement can be considered an empirical concept that can be used in market design, and
whose properties in reality can be estimated statistically.
As demonstrated in Section 2’s
presentation of the
formal, multiple marke
t
structure,
general equilibrium
theory proved quite
useful in practice when defining the mechanism.
Further, Section 3
and the
associated Appendix
A
demonstrate
the
auction
procedures and
illustrate
how lessons learned from testbed
experiments in a labora
tory environment
provided insights
for its
field implementation.
4
Partial
equilibrium analysis
of individual markets
in Section 4
and Appendix C
illustrates the operation
of
competitive
market principles supporting an efficient allocation of licenses with
in each market
and demonstrates bidders act as price
-
takers in these markets
.
Empirically
e
valuat
ing
the overall allocation of licenses across markets and verify
ing
the
auction
followed the principles used in the theory of
its
design
to
reach an efficient general
equilibrium
presents a challenging exercise
. To
approach these questions
, we
focus on the
dynamics of equilibration
as driven by excess demand dynamics
interpreted throu
gh
an
important principle, “excess demand revealed at the margin
.
”
This
excess demand
,
readily
measured from observed bidding behavior
and
interim allocations,
was
first
observed in the
testbed experiments
.
Section 5 demonstrates that this excess demand
be
comes
exhausted through
the auction mechanism’s bid revision process.
Subsequent sections focus on coordinated
equilibration of multiple markets.
We
describe
the dynamic
behavior
of the auction mechanism
in Section 6, characterizing the total revenues and
surplus generated
as bidding rounds
progressed
.
Section 7 investigates the relationship of price dynamics across markets to the
revealed excess demand in all other markets.
This analysis verifies the conditions for stability
that would lead to an efficient
multi
-
market allocation and general equilibrium across all market
segments given
the
policy constraints.
T
aking advantage of the rich data available, t
hese novel
statistical tests present the first empirical verification of equilibrating dynamics based on
the
principles of t
â
tonnem
e
nt.
Section 8 concludes
with
a summary of the findings from the
4
The
se
testbed experiments are des
igned to develop intuition about bidder behavior and support judgements about
auction
procedures
. The exploratory nature of the experiments stands in contrast to traditional applications of
experimental economics to
evaluate
a fully articulated theory agai
nst alternative hypothes
e
s
using
large sample sizes
to provide power for classical statistical
test
ing
.
5
implementation of an economic mechanism to address the allocation problem at the heart of a
complex government policy project.
Section 2
:
Auction Structure
The economics literature on auctions reports several alternative
auction
designs that were
considered but seemed inapplicable given
the
structure of
multiple
policy constraints
,
the
scale
required
and the overriding government goal of ef
ficiency
.
O
bvious
auction types
include
d
sealed bid auctions similar to U.S. Treasury auctions and the sequential forms of auctions such as
the simultaneous, multi
-
round, ascending prices auction
(SMRA)
used to allocate the
electromagnetic spectrum (
Milgro
m, 2000;
Bichler
and
Goeree, 2017
).
Both
types of auctions
have the capacity to
accom
m
odate
large
numbers of bidders competing for large numbers of
items.
However,
t
he
governmental
policy constraint
s
limit
ed
the allocations of available units to
different subsets of market
s
. T
ogether with limitations on
which
markets individuals can
participate
,
this
led
to structural
complexities
not addressed in traditional applications of those
forms of auctions.
For examp
le,
in
the traditional
,
multi
-
market
(SMRA)
auction
the number of
units for sale
in each
market is fixed
.
M
issing
from the traditional auctions
and created in the
auction analyzed here,
are m
arket
institutions
that
facilitate
efficiency
-
improving
movement
of
resources
and arbitrage
among markets
, typical of general equilibrium
.
Efficiency issues related
to
the limited
bidder
feedback of
sealed bid processes are exacerbated by the heterogeneous
preferences and limited information available to bidders about preferences and numbers of
other
bidders. Time allowed for the auction presents a
further
challenge
, as the
Victorian
machine
auction w
as permitted only one day
.
In contrast, s
imultaneous ascending price auction
s
can
require months from start to
finish
and
it is unclear
how
such auctions
might be
conducted more
quickly
without
bidder errors and
compromis
ed
efficiency
.
Given the auction s
cale (numbers of units and markets), human information processing,
decision speeds and error correction delays, open questions exist regarding how to make
traditional auction systems work while guaranteeing that all governmental constraints would be
satisf
ied.
W
e were (
and still
are) un
a
ware of
appropriate
modifications
,
s
o
both classes of auction
architectures were ruled out by
scale and
policy constraints.
Of course, in the absence of an
impossibility theorem, it
is conceivable
that modified versions of t
he
traditional
auction
architectures
might
be
developed and
used
. Currently, none exist.
6
2.1 Policy Constraints
The auction design problem was to sell
entitlements
simultaneously
,
subject to many
overlapping policy constraints.
P
olicy constraints were focused on the nature of the businesses
that were allowed to participate in the auction. Half of the 27,500 entitlements were to be sold to
businesses classified as Hotels
.
The other
half was to be sold to smaller venues called Clubs that
cater to local populations.
This reflected differences between the economic environment and
social purposes of these venues and differing political bases in
various
Victorian communities.
For purpose
s of the allocation Victoria was divided into 88 geographic regions, and each
region had a maximum number of entitlements based on area population or other regulations.
These constraints
, established by legislation,
placed limits on the saturation of machi
nes relative
to population and were motivated, in part, by social
, community,
and health issues related to
gambling.
Additional policy concerns
regarding
the geographic distribution of entitlements
resulted in the creation of a single set
of
geographic reg
ions designated as metropolitan and
maximum number of entitlements that could be allocated to the set.
5
Accounting for these constraints, and neglecting the metropolitan designation, each
entitlement has two characteristics: the type of venue (Club or Ho
tel) and the geographical
region (88 distinct areas) in which that establishment is allowed to operate. This required 176
simultaneous markets.
While the underlying resource in all markets is an “entitlement
,
” the
policy restrictions differ across clubs, h
otels and areas
.
So, from an economic and modeling
perspective
,
the items sold in these
176
markets are
completely
different commodities
even
though all are “entitlements
.
”
2.2
Determining Allocations and Prices in a Continuous Model
The basic auction design can best be understood in the context of a continuous model that
seeks an efficient allocation given the values revealed by bidders but
assumes away complexities
created by underlying integers.
The model is built from the assumptio
n that the efficient
allocation and supporting prices can be found through a constrained optimization process.
Such
complexities will be addressed in the later sections that analyze the data from the auction.
5
In particular
, geographical
areas designated as “metropolitan” were limited to obtain no more than 80 percent of all
entitlements.
Since this constraint did not bind at any point
during the auction
, we
do not
discuss the features
implemented to accommodate the constraint should it b
e binding.
7
The model begins with a classical economic post
ulate. Bidders are assumed to have well
-
formed preferences and objectives that
form the foundation of economic efficiency and
are
revealed only through actions taken in the market. Assume
that each bidder submits a continuous
valuation schedule
reporting their total willingness to pay for an allocation of
x
entitlements.
Assume
that
, representing the bidder’s marginal willingness to pay
is non
-
negative, monotonic, and (weakly) decreasing
, so
that the rep
resentation of bidders’
demand schedules for licenses is continuous and monotonically weakly decreasing.
It is
important to note that while bids are observable, the underlying preferences that might produce a
valuation function are not observable. Such pre
ferences exist in the analysis as a basic
assumption of economic theory
and are a foundation for measures of economic efficiency
. All
inquiries that might have yielded information about the existence and form of the preferences of
bidders were prevented by
government probity policies, which also strictly prohibited all
communications between policy makers or policy implementers and bidders.
Thus, auction
failure could
result
from the failure of this basic postulate of underlying theory.
Regulations regarding bidding establishments depend upon the type of establishment, a
“club” or a “hotel” and the area in which the establishment is located.
As introductory notation,
let
index each establishment and let
denote the area in which the establishment
seeks to obtain licenses.
6
The indicator variable
identifies the
i
th
establishment type,
equaling 1 if
establishment
i
is a hotel.
The system
is designed to allocate
entitlements to bidders to maximize the total
cumulative reported value of the allocation
consistent with all policy constraints
.
In that sense an
efficient allocation is one that optimizes total value subject to constraints.
Define
as a vector
6
Each establishment corresponds to a policy
-
defined “venue” meaning the location where the machines would be
housed and operated.
Bids are submitted by the venue and the entitlement is issued to the venue where the machine
must be located and
counts against the area constraints.
A business might own more than one venue and employ a
representative bidder authorized to submit bids for more than one venue that the business might own.
Auction rules
were designed to minimize coordination between bid
ders.
All bidders were located in a large
convention hall
with
cubicles from which other bidders could not be viewed
(see Figure 5)
.
E
xternal communication devices
(e.g., mobile
phones)
were prohibited and monitors ensur
ed
no unauthorized communication occ
urred amongst the bidders.
While
a representative bidder might tender bids for all of the venues from the same terminal
,
each bid is attached to a
specific venue and recorded as made by that venue. As such, even though a single business might own more than
one venue even in one particular area, the system records and treats each venue as a separate entity. Special rules
and monitoring were imposed for multiple venues operating under the same ownership.
(
)
i
Vx
(
)
(
)
i
i
Vx
Dx
x
¶
=
¶
iI
Î
i
aA
Î
{
}
0,1
i
h
Î
X
8
of allocations with the
i
th
entry
x
i
representing the allocation to establishment
i
.
T
he total market
value,
V
(
X
),
i
s
the aggregate value of bidders’ willingness to pay for their given allocations:
Allocations must satisfy the set of policy constraints facilitated by defining the total allocations
to each area by venue type:
Imposing these definitions as equality constraints on the maximization problem identifies the
shadow costs for allocating a marginal license to each area and bidder type.
The total allocation
to an area,
x
a
, must satisfy the constraint defined by the Victoria government, denoted
:
.
Administering Victoria
-
wide constraints on the total allocations to Hotels and Clubs,
respectively denoted
, is facilitated by similar constraints:
Each of these aggregated allocations must satisfy government
-
imposed inequality constraints
,
.
The Lagrangian for the constrained allocation problem is:
(1)
Here, the shadow costs denoted by
μ
impose binding equality constraints for aggregating
allocations within different market segments and Kuhn
-
Tucker shadow costs denoted by
l
correspond to non
-
negative inequality constraints that may or may not bind on the final
(
)
(
)
ii
iI
VXVx
Î
=
å
(
)
{
}
{
}
11 1
aCiiiaHiiiaaCaH
iIiI
xxhaaxxhaaxxx
ÎÎ
=
-
====+
åå
a
x
,
aa
xxaA
£"Î
HC
xx
and
HaHCaC
aAaA
xxxx
ÎÎ
==
åå
and
HHCC
xxxx
££
(
)
(
)
(
)
{
}
(
)
{
}
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
11
1
aCaCiii
aAiI
aHaHiii
aAiI
aaaCaH
aA
aaa
aA
HHaHCCaC
aAaA
HHHCCC
OHC
XVX
xxhaa
xxhaa
xxx
xx
xxxx
xxxx
xxx
l
l
l
μ
ll
μμ
μ
ÎÎ
ÎÎ
Î
Î
ÎÎ
=
ü
---
=
ï
ï
--
=
ï
ý
---
ï
ï
--
ï
þ
ü
----
ï
ï
----
ý
ï
---
ï
þ
åå
åå
å
å
åå
L
Constraints on
area Club and
Hotel allocations
Co
nstraints on
aggregate Club and
Hotel allocations
9
allocation.
For theoretical purposes the shadow costs
,
μ
a
,
μ
C
,
and
μ
H
translate into the
prices for
licenses associated with different
area
s
, clubs and hotels.
Given
sufficient regularity conditions,
the optimization problem (1) solves for a Pareto
efficient allocation of licenses
and
the second welfare theorem states that this allocation can be
suppor
ted as the competitive equilibrium of a market mechanism with associated prices.
Those
prices are approximated by the shadow costs for the binding constraints.
7
Section 3:
Auction Procedures
to Determine Allocations and Prices
In practice, bidders
submit
schedules
reporting their willingness to pay for different
allocations
through a bidding mechanism
described in
S
ection 5.
Here
, we describe the submitted
bid schedules under the simplifying assumption that reported bid schedules reveal bidders’ true
valu
ations
,
while
recognizing that bidders’ reported willingness to pay and valuations may
diverge in practice
and that the auction need not have a “preference revelation” property
.
3.1
Reported Bid Schedules and Accumulated Bid Functions
Each establishment
submits a bid schedule containing
L
i
entries specifying its willingness
to pay for each marginal
unit
.
The lists’ entries are sorted by descending bid and the entry at the
l
th
level in the bid schedule is denoted
.
The bid price
b
i
l
reports the price the bidder
is willing to pay and the quantity
x
i
l
reports the number of marginal units the bidder demands at
that price in addition to the units they
woul
d receive from any higher
-
priced bids.
Bidder
i
’s
cumulative bid s
chedule, denoted
, reports the total quantity of bids with reported value
weakly greater than
p
,
computed by summing
.
From the reported bid schedule, let
denote bidder
i
’s cumulative reported valuation
for an allocation of
x
licenses.
We calculate
by summing the area under the bidder’s
reported bid function up to the quantity of
x
.
8
Since
can be evalua
ted
at
any quantity, it
7
In practice, the constraint on aggregate Club allocat
ions was not binding whereas the constraint for aggregate Hotel
allocations did bind.
Further, not all areas’ allocation constraints were binding, so these constraints only affected the
prices paid for licenses within those areas facing binding constraints
.
8
The formula for
is a bit convoluted, due to the discrete nature of bids, but can be calculated as:
(
)
,
ililil
Bbx
=
(
)
i
Xp
(
)
{
}
1
1
i
L
iilil
l
Xpxbp
=
=
³
å
(
)
ˆ
i
Vx
(
)
ˆ
i
Vx
(
)
ˆ
i
Vx
(
)
ˆ
i
Vx
10
can also be stated as a function of price evaluated at bidder
i
’s cumulative bid schedule.
Denoted
, this
represents
the total valuation bidder
i
assigns to the licenses they would bid for
if the price were
p
.
Table
1
provides a hypothetical example of an individual bid schedule and its translation
into cumulative bids and reported valuations.
Panel A presents a schedule with four entries at
four different price points for an establishment that bids for up to 25
entitlements
if the price is
no greater
than 80.
The Cumulative Bid Schedule in Panel B demonstrates how different prices
translate int
o total quantity desired by that bidder at each price.
Table
1
: Sample Individual and Cumulative Bid Schedules
Panel A: Reported Bid Schedule
Panel B: Cumulative Bid Schedule
List
Entry
[
l
]
Bid
[
b
il
]
Bid Quantity
[
x
il
]
Price
[
p
]
Cumulative Bid
[
X
i
(
p
)]
Cumulative
Reported Value
[
]
1
100
5
100
5
500
2
95
5
96
5
500
3
90
10
95
10
975
4
80
5
90
20
1,875
80
25
2,275
3.2
Implemented Allocation Rule
The
auction allocation
system approximates the continuous model presented in section
2.2
using
the elicited valuations.
The system determines the allocation by maximizing the
“m
easur
ed”
total welfare by the aggregated reported license valuations,
,
main
taining all the relevant constraints
indicated in (1).
The discrete nature of the problem
complicates
solving th
is
optimization problem.
The
se
are well known features of integer programming optimization, requiring tie
-
breaking rules, non
-
uniqueness of shadow costs, and potential for multiple solutions due to overlapping constraints.
We resolved t
ied bids through a first come, first serve rule.
A
ll bids are time
-
stamped,
so
bids are
rationed
at the market price
(if needed)
according to their arrival time.
The market clearing prices
need not be unique if the quantity demanded at a price exactly equals the supply to that market.
(
)
(
)
ˆ
ii
VXp
(
)
(
)
ˆ
ii
VXp
(
)
(
)
ˆˆ
ii
iI
VXVx
Î
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11
This was addressed b
y adding a very small quantity to every bid
(essentially picking the best
price for the seller)
so the quantity demanded at a price is always slightly above the integer parts.
Finally,
al
though multiple constraints could bind and thus create multiple price
solutions,
this
could only arise from relationships between metropolitan and area constraints.
The problem
never surfaced because the metropolitan constraint was never binding.
Given the adjustments to
the optimization problem necessary
due
to these pract
ical considerations, the shadow costs from
the optimization only approximate the shadow costs from the continuous Lagrangian in equation
(1)
.
B
ut
these approximations do not induce disequilibrium in any individual market’s allocation.
3.3
Preview of Dyna
mic Auction Features and Bidding Revisions
The system arrived at its final allocation after progressing through a series of 63 bidding
rounds.
Each round beg
a
n with establishments submitting provisional bid functions.
The auction
algorithm computes the al
location and prices based on these bid functions.
These provisional
prices and allocations are then announced at the conclusion of the round.
Thus bidders observe
the quantity of entitlements that they would purchase, and the price they would pay per unit,
if
the auction were to stop in that round.
Given this information, bidders are aware of the prices revised bids need to meet or beat
to obtain additional licenses.
Before the next round
begins
, bidders can
use
this information
to
revise their submitted b
id functions, subject to the restriction that they increase their original bid
by at least a
specified
minimal increment.
This restriction induces an ascending auction format
in
markets defined by constraints,
with
excluded bid pricing as bidding
progresses from round to
round
with the quantity demanded at the announced price revealed
to the auction system but not
revealed to participants
.
While individual bidders cannot decrease leading bids,
prices in
some markets can go
down while the prices in other markets go up
due to
the interdependence of the markets
, the role
of “consumer surplus”
and shifting supplies
.
9
Price decreases can occur i
n other circumstances
9
A simple example illustrates this unintuitive feature
.
Suppose there are three units to be allocated in two markets,
{commodity 1 in mkt1 and commodity 2 in mkt2} and six agents, {a,b,c,d,e,f}. Each wants only one unit and can
place bids only in that mark
et. Suppose agents a,b,c have values (4, 0.05, 0.05), respectively, for commodity 1 and
agents d,e,f have values (1,1,1) for commodity 2 and all agents reveal their value in their bids. Prices, which equal
the bid of the last accepted unit in a market, are
(4,1) for markets mkt1 and mkt 2 and volumes are (1,2). Total
revenue is 4+2 = 6 and (maximized) total “
buyer
surplus” is 6 = 4+1+1. Now, suppose agent b increases assessed
value from 0.05 to 2 and raises her bid to 2. As a result of this demand increase,
the surplus maximizing feature of
the process automatically shifts an entitlement unit from the supply in market 2 to the supply in market 1. Prices as
12
that involve multiple market coordination and derived demand.
Also notice
that
unlike traditional
forms of auctions
the system response need not be related to
revealed
excess demands at existing
prices
.
Furthermore,
potential price de
creases reflect
movement of items across markets and
i
llustrate a fundamental departure from traditional ascending price auctions in which the number
of items available for sale in a market is fixed.
The system initiates a
two
-
clock
ending process based on
the number of significant
revisions
(attempts to acquire more units)
in individually submitted bid schedules
(clock one)
and
the resulting
patterns
(numbers)
of market price changes
(clock two)
.
When the ending
process is initiated,
bidders are notified about the number of
potential
rounds the market will
remain open
. This process
terminat
es
with
an
announcement that the market
may
close in the
subsequent round and
bidders are
given a final opportunity to revise their bid schedules.
Absent
any
significant revisions in this next round, the auction closes.
After this last round, the bidders
pay the announced price for their market for each entitlement awarded.
3.4 Design Decisions and Testbed Methods
The auction design was
guided
by t
estbed experiments outlined in Appendix
A
, which
focus
ed
on
evaluating
selected behavioral feature
s
as well as
expanded
scale
.
The
objectives of
testbed experiments
are to
inform
judgements
regarding
behavior and
possible
outcomes
in a
wide range of environments
where theory is incomplete (or absent)
and stress
-
test those
predictions in
very unlikely or extreme events.
While t
he policy objectives of the machines auction were clearly stated
,
the operational
design and
the
process
i
tself
faced many challenges aside from the size and technical complexity
of the allocation problem
.
First, probity concerns about information advantages among bidders
prevented all inquiry, feedback, discussions with or exercises involving potential bidder
s or the
circumstances they faced
, both during the design phase and while the auction was open
.
Second,
the time of the auction was limited to one day.
Third, no comparable auction processes existed in
practice or in theory
to provide any useful
history of
applications.
determined by the marginal bid, decrease to 2 in mkt 1 and remain 1 in mkt 2 while volume increases to
2 in mkt 1
and decreases to 1 in mkt 2. Total revenue is 5 = 2+2+1 and total
buyer surplus
is 7 = 4+2+1. Notice that prices and
total revenue have both decreased but
buyer surplus
has increased. Bids increased but the prices decreased.
Such
instances are a
pparent later when we present the time
-
series of market prices and area price premia in Figure 6.
13
The design exercise began with a theoretical sketch of an auction
with emphasis on
structure
where: (i) bidder preferences were limited to a single establishment; (ii) agents submit
“truthful” demand functions; (iii) the auction winner is ch
osen by maximizing the revealed value
of the allocation; and (iv) policy limits regarding multiple markets exist as constraints
on final
allocations
.
This
sketch
suggested the theory of general equilibrium as a possible source of
principles.
The
appendix d
iscusses
subsequent and evolving design judgements
.
Laboratory experimental studies of multiple markets lend strong support to the basic
principle that market dynamics are driven by excess demands
,
which
guide the system to a
general equilibrium.
In spite
of theoretical discussions that raise doubts about applicability of
general equilibrium,
such as
Ackerman (2002)
, the
Sonnenshein
-
Mantel
-
Debreu Theorem
(
Mas
-
Colell,
Whinston
, Green,
1995)
,
papers in Bridel (2011) and
additional
general
equilibrium
critics
listed
(
and challenged
)
in
Mukherji (2019)
,
the design decisions rested on the fact that the
convergence principles are evident in a
wide range of
experiments. Th
e excess demand driven
convergence
is found in the multiple markets of int
ernati
onal trade
in
Noussair
et
al
.
(
1995
),
disequilibrium dynamics
in
Gillen
et
al.
(202
1
)
,
Anderson
et
al.
(2004)
, and
Hatfield
et
al.
(2016
)
,
and finance
in
Asparouhova
et
al.
(
2003
)
.
Importantly, the fundamental role of excess
demand in driving price adjustments exists when bidding is expressed as demand functions
(
Goeree and Lindsay
,
2016).
Convergence
is also seen in
call markets (Plott and Pogorelskiy,
2017)
, and c
omplex auctions ex
hibit the same tendency. A prime example is the simultaneous,
ascending price auction used for the auction of the
broadcast
spectrum that was successfully
demonstrated in the U.S. and other
countries with convergence properties clearly demonstrated
in expe
riments
(
Plott and Salmon
,
2004)
.
The principle is observed in complex networks such as
power grids (Chao and Plott, 2009) and combinatorial auctions (Lee
et
al
.
, 2014)
.
However,
experiments
have
demonstrate
d
the unreliability of the
pure
tâtonnement insti
tution
with no
trading at announced prices
(Plott, 1988
;
2001)
and
served as a warning about the incomplete
nature of theory as a model of actual behavior.
Still
,
in other experiments exploring institutional
variations a
key behavioral feature, demand revelation at the margin, had been identified in an
unpublished working paper on fuel efficiency under the CAFE constraint (Katz and Plott, 200
8
).
The use of r
epeate
d rounds as a design foundation,
as opposed to a one shot, sealed bid
computation
,
is based on findings from
general equilibrium related experiments
that
consistently
reflect
increasing efficiency,
dynamic equilibration
,
and convergence
over
repeated round
s of
14
bidding
.
Even in more simplified environments with established incentives to report true
valuations, such as second
-
price sealed bid auctions, for a single unit, dynamic (English
ascending price) versions lead to
increasingly
truthful revelations (Kag
el and Levin, 2015).
The
scale of the gaming machines auction and nature of the multi
-
unit demand for licenses required
analyzing functions (inverse demand functions) as opposed to separate bids on
(thousands of)
individual units.
Competitive theory as app
lied to smooth demand functions identified
theoretical
and technical
relationships among bids, prices (as Lagrangian multipliers), equilibrium (as a
competitive equilibrium), allocations and efficiency.
The actual auction required solving an
integer
-
constr
ained, linear program for the allocation problem
each round
.
The scale of the design problem derives from the size of markets
and
the number of
markets, units, and bidders.
The testbed exercise
s
outlined in Appendix
A
establish
ed
the
economic performance
and technical control of the system
in relation to scale
. Two performance
measures were useful tools to refine the rules of the auction. The first was market efficiency
measured as consumer surplus as developed by Plott and Smith (1978).
Since the cost of
the
entitlements is ignored in this government allocation problem, t
his measure is
simply
the sum of
observed willingness to pay divided by the maximum sum of willingness to pay given
experimentally induced preferences.
The second was speed of convergence
measured in terms of
number of rounds required for equilibration.
The scale testing experiments started
small and were scaled up.
Over 40 different
experiments were
conducted
and
some were
repeated to explore problems
they
exposed.
The
largest testbed
employing
human subject participants operated with 50 markets and 160
participants at a
subject payment
cost of $8,866.
Larger scales were simulated with multiple
computers programmed to place bids to test network configurations, proces
sing speeds, and
computation reliability and speed.
Experiments revealed the
auction
could
coordinate convergence
for prices and allocations
across
multiple markets,
equilibrating
derived demand
with the available supply
.
Allocations and
prices typically e
nded near the predictions of the general competitive equilibrium and thus
efficiency tended to be in the high 90% levels and often near 100%. Such high performance
occurred at all tested levels of scale.
While each experiment examined multiple dimensions
of performance,
many
focus
ed
on
two
specific
areas.
The first related to real time control of price movement and
procedures for
15
ending the auction. Previous experimental work revealed that bidding incentives and stopping
rules are important for performance
.
The second broad area included
market
performance,
efficiency and reliability in both software and
bidding
behavior.
The timing of the auction rounds needed adjustments to account for the reaction speed of
bidders.
An increment requirement defining the
minimal allowable increase had
the
obvious role
of
ensuring revisions were economically meaningful. We adopted a two
-
clock methodology for
controlling bidder behavior.
10
Requiring only a
single bid revision for auction continuation is not
practical because
randomness in bidding and bid timing.
The question becomes “how many” bids
or price changes in a round justif
ies
keeping markets open for additional rounds. Testing in
experiments with different controls led to
a decision to use
the number of bidders that
attempted
to increase their
allocation
as the controlling measurement for the first clock and number of
markets that changed prices as the controlling measure for the second clock.
Time was measured
in number of rounds required before a change in these th
resholds. New bid increment
requirements were announced as a percentage over existing prices and these were enforced
beginning in a specified
future
round.
E
xperiments provided substantial experience with how the auction would respond to the
chosen
parameters.
Observations from testbed experiments were the
primary
source of
information
about the likely auction ending time
, which
was important given the government’s
decision to limit
the auction
to one day.
Experiments demonstrated that bidding follow
ed a
principle of revelation at the margin.
Announced prices were accompanied by the increment
requirement.
All new bids or changed bids
had to
be no less than the existing price plus
increments. Unchanged bids remain
ed
in the system.
The upcoming price
wa
s not known and
price
often
remain unchanged
in many individual markets
. New bids
we
re automatically
integrated with the bidder’s existing bids to form a revised bid function.
The new bid function is a type of “revealed demand” but it is not fully reveali
ng. The
revealed demand function always falls short of the limit values (demand prices) of infra
-
marginal
units
but
an important element
–
demand at the margin
–
i
s accurately revealed.
As illustrated in
10
In continuous time a
u
ctions one clock counts down in seconds and resets with the submission of a new bid in any
market.
In the absence of additional bids
, t
he clock counts down to zero and the auction ends.
A second clock is
employed in auctions with complex bids, such as bid functions, because new bids need not result in price changes
and can become cheap talk that simply keeps the auction open.
We avoid s
uc
h possibilities by us
ing
a second clock
that counts down and resets if a bid results in new winners, thus exerting pressure to place bids that
affect
prices.
16
the
experimental background from Appendix A
, t
he qua
ntity demanded at the announced price
wa
s
typically
very close to the quantity demanded according to the induced preferences.
11
Section 4: Partial Equilibrium Properties of the Final Allocation
A prominent feature of the theory of general equilibrium is a
suggestion that a
complex
economic system can be viewed as
a collection of separate, identifiable markets each of which
rests at its own equilibrium given that all other markets are in equilibri
um
.
For the machines
auction such a pattern is described by market equilibrium equations (
1
), where the
constellation
of partial equilibrium relationships
is modeled as having emerged from a pattern of binding
policy constraints
.
Described by the theoretic
al equations are
two large markets
and a set of
smaller markets. One large market consists
of
“unconstrained” clubs
, a second consists of
“unconstrained” hotels
, while and the others are
smaller market segments associated with the
several constrained area
s
commanding a local price premium
.
The broad question posed for
testing is the compatibility of the data with the idea that complex general equilibrium theory can
be decomposed into a pattern of multiple markets and separate partial equilibrium theories. I
n
some respects, that question is at the heart of microeconomic theory.
We agg
egate the derived demand for establishments competing to obtain licenses based
on firms’ reported bid schedules.
Throughout
this
analysis, we assume three stylized facts
for
all
allocations in the systems.
First, we partition the set of areas
into constrained areas
(where
) and unconstrained areas
(where
).
Second,
hotel establishments were allocated the maximum number
possible given the regulations
(
) while club establishments’ allocation did not meet this maximum (
).
Third, all
licenses available to the system were allocated (i.e.,
).
W
i
thin each of the market segments, the derived supply is price
-
inelastic at a fixed
quantity.
12
The price at which demand equals derived supply in each market is approximated by
11
Given the rules, the bidder could adjust the bid price to ensure the purchase of the marginal u
nit given the
announced price and do so without directly influencing that price. To express a preference for an additional unit at
the stated price the bidder merely needed to express a willingness to pay for it by tendering a bid price for the unit
above
the stated market price. Thus, the value of the marginal demand is revealed. The demand function becomes
traced out as price moves up following the required bid increments. In particular, the slope is useful information
revealing the state of demand relati
ve to prices and thus when the auction is near a competitive equilibrium.
12
The derived quantity supplied may depend on allocations to other market segments, though this feature of the
market system doesn’t impact the partial equilibrium analysis in this s
ection.
We return to discuss the general
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17
the price premia for different types of licenses.
As such, the final allocation and prices in each
market segment are consistent with an ex
-
post partial equilibrium given bid
ders’ unwillingness
to submit revisions to their reported bid schedules.
4.1
Partial Equilibrium
in
the Market
for Club Licenses
in
Unconstrained Areas
Consider the market for club licenses in unconstrained areas under an allocation in which
.
B
idders
in these markets
compete with each other for the pool of licenses that are not
allocated to hotels or
to
any of the constrained areas.
Collectively, the aggregated bid schedules
for bidders in these markets identify the
Derived Deman
d
for Unconstrained Club
s
calculated as:
.
The supply available to these bidders is determined after all constrained area markets for both
clubs and hotels, and the unconstrained hotel market have already cleared:
.
Finally, the “Club Base Price,”
,
i
s
the market clearing price where
.
Figure
1
: Derived Demand and Supply for
Licenses for
Clubs
in Unconstrained Areas
Panel A: Full Theoretical Demand and Supply
Panel B: Equilibrium Detail
Figure
1
presents the derived demand and supply for unconstrained clubs in the final
round of the
auction relative to the minimum permissible club price of $5,500
.
The residual
supply available to this market consisted of 3,693 units, which matched Derived Demand to set
equilibrium properties of the market system below when analyzing stability of excess demand functions and a model
of convergence to an equilibrium.
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