DIVISION OF THE HUMANITIES AND SOCIAL SCIENCES
CALIFORNIA INSTITUTE OF TECHNOLOGY
PASADENA, CALIFORNIA 91125
The Design, Experimental Laboratory Testing and
Implementation of a Large, Multi-Market, Policy
Constrained, State Gaming Machines Auction
Charles R. Plott
,
California Institute of Technology
Timothy Cason, Purdue University
Benjamin Gillen, Claremont McKenna College
Hsingyang Lee, California Institute of Technology
Travis Maron, California Institute of Technology
SOCIAL SCIENCE WORKING PAPER
1447
September
201
9
1
The Design,
Experimental
Laboratory Testing and Implementation of a Large, Multi
-Market,
Policy Constrained
, State Gaming Machines
Auction
Charles R. Plott, California Institute of Technology
Timothy Cason, Purdue University
Benjamin Gillen, Claremont McKenna College
Hsingyang Lee, California Institute of Technology
Travis Maron, California Institute of Technology
(September 2019)
A
bstract:
The paper reports on the theory, design, laboratory experimental testing, field
implementation and results of a
large, multiple market and policy constrained auction. The
auction involved the sale of 18,788 ten-
year entitlements for the use of electronic gaming
machines in 176 interconnected markets to 363 potential buyers representing licensed gaming
establishment
s. The auction was conducted in one day and produced over $600M in revenue.
The experiments and revealed dynamics of the multi
-round auction provide evidence about
basic principles of multiple market convergence found in classical theories of general
equ
ilibrium using new statistical tests of the abstract
properties of tatonnement.
Section 1: Intro
duction
1
The paper reports on the design, laboratory experimental testing, and field implementation of a
large
, multiple market and policy constrained
auction.
The auction involved the sale of
18,788
ten
-year entitlements for the use of electronic gaming machines in Victoria Australia, in
May,
201
0. Policy issues dictated the operation of 176
interconnected markets to allocate
sales
of
these licenses to 363
potential
buyers representing licensed gaming establishments. The
auction was conducted in one day
and
produced
over $600M
in revenue. The design rested on
basic
principles of
competitive economics for
a general equilibrium exchange economy guided
by the
classical tatonnement model of market adjustment. The theoretical framework in which
we 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, a discussion of key features of the
laboratory experimental testing and discussions of results and dynamic performance.
Two overriding
research
questions are
addressed b
y the paper
. The questions are
posed
as broad guidelines for assessing the
success of policy related institutional and market designs
(Plott, 1994). First, was the
implementation
successful
in
satisfying
the goals
the policy
was
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 as is the technical s
upport of the Caltech Laboratory for Experimental Economic and
Political Science. Limited data analysis for scientific purposes was allowed by the Victorian government. The
cooperation and help of William Stevenson of Intelligent Market Systems LLC were fundamental.
2
designed to meet? Secondly, was the science applied to
guide
the design
successful in
attributing
the outcomes to
the principles used in the
design?
As it turns out the auction
outcomes satisfied all policy constraints. Furthermore, t
he auction provides substantial suppo
rt
for the broad reliability of
the
basic
principles
of classical, competitive economic theory
that
support the auction architecture
and guided the design.
The auction was the result of the Victorian government’s efforts
to implement a major
change in the
policies regarding gambling operations in the state of Victoria. 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, the distribution of gaming machines was managed by two large
corporations
associated with the machine manufacturers
. The machines were allocated to
businesses consistent with local policies governing the
ir use. Fi
nance was 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 concerns
with the
historical
policy
reflected a desire for better control
over gambling and concomitant
social problems, gambling related government public
finance
,
and a desire for conformity with
frameworks used for economic regulation. The transition
policy chosen was based on an auction
intended to accommodate several broad economic
objectives. The policy objectives did not include revenue maximization, which would have
justified
a monopolist
ic supply determination
that
was deemed outside the auction design
problem. The demand for entitl
ements was to be discovered by the auction process and the
aggregate
supply was dictated
to be near historical levels by policy. The auction was intended
to allow fluid price discovery and thus allocate entitlements smoothly and efficiently in a
manner
con
sistent
with
a competitive market. The goal was to create minimal economic
dislocations and a climate where future regulatory efforts could be based on competitive
principles of decentralized competition and 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
being
based on historical administrative practices.
The resulting mechanism and its implementation present a remarkable success for the
decad
es of abstract theorizing about general equilibrium in classical economics. As
demonstrated in Section 2’s presentation of the auction structure, this theory proved to be
quite useful in practice when defining the mechanism. Further, Section 3 demonstrates how
lessons learned from testbedding
experiments
in a laboratory environment were able to scale
to field implementations. Partial equilibrium analysis in Section 4 illustrates
the operation of
market principles supporting an efficient allocation of lic
enses within each market.
Evaluating the overall allocation of licenses across markets
and verifying the auction
reached an efficient general equilibrium
presents a challenging empirical exercise. To this end,
3
we introduce an important principle, “exces
s demand revealed at the margin,”
that can be
readily measured from observed bidding behavior. Section 5
demonstrate
s that this excess
demand is exhausted through the auction mechanism’s bid revision process. We further
explore the dynamic properties of the auction mechanism in Section 6, characterizing the total
revenues and surplus generated by the auction mechanism. Section 7 investigates the
relationship of price dynamics across markets to the revealed excess demand in all other
markets
. This analys
is verifies
the conditions for stability that would lead to an efficient
multi
-
market
allocation and general equilibrium across all market segments
given policy cons
traints
.
These
novel statistical tests take advantage of the rich data we have available and present the
first
empirical verification of equilibrating dynamics based on the principles of tatonnemont.
Section 8 concludes, presenting a summary of the findings from the implementation of an
economic mechanism to address the allocation problem at
the heart of a complex government
policy project.
Section 2. Auction Structure
Section 2.1 Policy Constraints
The auction design problem was to simultaneously sell rights subject to many
overlapping policy constraints. Key policy constraints were focused on the nature of the
businesses that were allowed to participate in the auction and acquire entitlements. Half
of the
27,500 entitlements were to be sold to businesses classified as Hotels, which were larger
venues
possessing
substantially greater value for the gaming machines. The other half was to
be sold to smaller venues called Clubs that cater to local populat
ions. This reflected differences
between the economic environment and social purposes of these venues and differing political
bases in the Victorian communities.
For purposes of the allocation Victoria was divided into 88 geographic regions, and each
reg
ion had a maximum number of entitlements based on area population or other regulations.
2
These constraints placed limits on the saturation of machines relative to population and were
motivated, in part, by social and health issues related to gambling. Add
itional policy concerns
related to the geographic distribution of entitlements resulted in the creation of a single set
geographic regions designated as metropolitan and maximum number of entitlements that
could be allocated to the set.
Accounting for these constraints, and neglecting the
metropolitan
designation, each
entitlement has two characteristics: the type of venue (Club or Hotel) and the geographical
2
Additionally
, 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 in the system, we avoid discussing the features
implemented t
o accommodate the constraint should it be binding.
4
region (88 distinct areas) in which that establishment is allowed to operate
. This required
176
simultaneous markets.
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 assumes away
complexities created by underlying integers. The
se complexities
will be
addressed in the later sections
that analyze the data from the auction.
As introductory notation, let
iI
∈
index each establishment
and let
i
aA
∈
denote
the
area in which the establishment seeks to
obtain licenses
.
3
T
he indicator variable
{ }
0,1
i
h
∈
identifies
the
i
th
establishment type
, equal
ing
1 if establishment
i
is a hotel.
S
uppose that each bidder submits a continuous
valuation
schedule
( )
i
Vx
reporting
their total willingness to pay for an allocation of
x
entitlements
. As an additional regularity
condition, suppose further that
(
)
( )
i
i
Vx
Dx
x
∂
=
∂
, representing
the bidder’s marginal willingness
to pay
is non
-negative, monotonic, and (weakly) decreasing
.
4
These conditions ensure that the
representation of
bidders’ demand schedules for licenses is continuous and monotonically
weakly decreasing.
The
system allocates
entitlements to bidders so as to max
imize
the total cumulat
ive
reported value of the allocation
. Define
X
as a
vector
of allocations with the
i
th
entry
x
i
representing the allocation to establishment
i
. We measure
the total market value
,
V
(
X
)
, as the
aggregate
value of bidders’ willingness to pay for their given allocations:
( )
( )
ii
iI
VX
Vx
∈
=
∑
3
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 ve
nue 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 d
esigned to minimize coordination between bidders. All bidders were located in a large room
with cubicles from which other bidders could not be viewed. To constrain the information that bidders receive,
external communication devices were prohibited and m
onitors were tasked with ensuring no unauthorized
communication occurred 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 tha
t venue for
purposes of policy consistency.
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 entit
y. Special rules and monitoring
were imposed for mult
iple venues operating under the same ownership.
4
We will discuss the mechanics for eliciting bidders’ demand and valuations in the next subsection. In discussing
implementation
, we will also review
some complexities that were overcome when mapping
the
co
ntinuous model
onto a discrete price and commodity space.
5
As discussed in the previous section, these
allocations must satisfy the set of policy
constraints defined by the government of Victoria. Administering these constraints is facilitated
by defining the total allocations to each area by venue type:
(
) {
}
{
}
11
1
aC
i
i
i
aH
i i
i
a
aC
aH
iI
iI
x
x h aa x
xhaa x x x
∈∈
= −= =
= =+
∑∑
Imposing these defini
tions 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,
deno
ted
a
x
:
,
aa
x x aA
≤ ∀∈
.
Administering Victoria-
wide constraints on the total allocations to Hotels
and
Clubs,
respectively denoted
HC
xx
and
, is facilitated by similar constraints:
H
aH
C
aC
aA
aA
x
xx x
∈∈
=
=
∑∑
Each of these aggregated allocations must satisfy government
-imposed inequality constraints
so that
H H
CC
x x xx
≤≤
and
.
The Lagrangian for
the constrained allocation problem is:
( ) ( )
(
)
{
}
(
)
{
}
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
11
1
aC
aC
i
i
i
aA
iI
aH
aH
i i
i
aA
iI
a
a
aC
aH
aA
aa a
aA
H
H
aH
C
C
aC
aA
aA
H H H
CC C
O
HC
X VX
x
x h aa
x
xh a
a
xx x
xx
x xxx
x x
xx
xx x
λ
λ
λ
μ
λλ
μμ
μ
∈∈
∈∈
∈
∈
∈∈
=
−
− −=
− −=
−
−−
−−
−− −−
− −− −
− −−
∑∑
∑∑
∑
∑
∑∑
L
Constraints on
area Club and
Hotel allocations
Co
nstraints on
aggregate Club and
Hotel allocations
(1)
Here, the shadow costs denoted by
λ
impose
binding equality constraints for aggregating
allocations within different market segments and
Kuhn
-Tucker
shadow costs denoted by
μ
correspond to non-
negative inequality constraints that may or may not bind on the final
allocation.
Given the optimization problem (1) solves for a Pareto efficient allocation of licenses,
the second welfare theorem states that this allocation can be
supported as the competitive
equilibrium of a market mechanism with
asso
ciated
prices. Those prices are approximated by
6
the shadow costs
for the binding constraints from the optimization problem. In practice, the
constraint on aggregate Club allocations
was
not binding whereas the constraint for aggregate
Hotel allocations d
id 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. Accounting for these binding constraints, the relationship between the shadow
costs and approximate prices for different types of licenses is
summarized in Table 1
.
In section 4 we demonstrate the second welfare theorem’s application to the gaming
auction problem. First, we define the market segments for licenses availab
le to each type of
bidder and characterize the derived demand for and supply of each type of license. The
intersection of derived demand and supply in each market segment represents the market
clearing price for that market, corresponding to the shadow co
sts from the constrained
optimization problem and establishing partial equilibrium in each market segment.
Table 1: Lagrangian Shadow Costs and Approximate Prices
Type of License
Binding Constraints
Shadow Costs
Approximate
Price
Club License in
Unconstrained Area
Total Allocation
μ
O
μ
O
Hotel License in
Unconstrained Area
Total Allocation
μ
O
μ
O
+
μ
H
Hotel Allocation
μ
H
Club License in
Constrained Area
a
Total Allocation
μ
O
μ
O
+
μ
a
Area Allocation
μ
a
Hotel License in
Constrained Area
a
Total Allocation
μ
O
μ
O
+
μ
H
+
μ
a
Hotel Allocation
μ
H
Area Allocation
μ
a
Section
3: Elicitation to Determine Allocations and Prices
In practice,
bidders report
valuation
schedules through a bidding mechanism. We defer
a discussion of bidders’
incentives and the conditions under which they can revise bids based
on provisional allocations
until section 5. F
or now
, we
describ
e the submitted bid schedules
under the
simplifying
assumption that reported bid schedules reveal bidders’ true valuations
to
illustrate properties of the allocation
.
3.1
Reported Bid Schedules
and Accumulated Bid Functions
Each establishment submits a bid schedule containing
L
i
entries specifying its quoted
willingness to pay for each marginal good.
The
lists’
entries are sorted
by descending bid and
7
the
entry at the
l
th
level
in the bid schedule
is denoted
(
)
,
il
il
il
B bx
=
. The bid price,
b
il
, reports
the
price the bidder is willing to pay and the
quantity,
x
il
, reports the number of marginal units
the bidder demands at that price
in addition to the units they’d receive from any higher
-priced
bids
. B
idder
i
’s cumulative bid schedule
, denoted
( )
i
Xp
, reports the total quantity of bids in
the list with reported value weakly greater than
p
and
can be computed
by summing
( )
{
}
1
1
i
L
i
il
il
l
Xp
x b p
=
=
≥
∑
.
From the reported bid schedule, let
(
)
ˆ
i
Vx
denote
bidder
i
’s cumulative reported
valuation
for an allocation of
x
licenses
. We calculate
( )
ˆ
i
Vx
by summing the area under the
bidder’s reported bid function up to the quantity of
x
, characterizing the total value the bidder
reportedly assigns to th
e allocation.
5
Since
(
)
ˆ
i
Vx
can be evaluated for any quantity of licenses,
it can also be stated as a function of price evaluated at bidder
i
’s cumulative bid schedule.
Denoted
( )
(
)
ˆ
ii
VX p
, this states the total valuati
on bidder
i
assigns to the number of licenses
they
woul
d bid for if the price were
p
.
Table
2 provides a
hypothetical
example of an individual bid schedule and its translation
into
cumulative
bids
and
reported
valu
ation
s. Panel A presents a schedule with four entries at
four different price points for an establishment that bids for up to 25 units if the price is less
than 80. The Cumulative Bid Schedule in Panel B demonstrates how different prices translate
into total
quantity desired by that bidder at each price
.
Table
2: Sample
Individual and Cumulative
Bid Schedule
s
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 [
( )
(
)
ˆ
ii
VX p
]
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
5
The formula for
( )
ˆ
i
Vx
is a bit convoluted, due to the discrete nature of bids, but can be calculated as:
( )
11
1
1
11
1
ˆ
1
11
il
L
l
ll
l
i
il
il
ij
ij
ij
ij
l
j
jj
j
Vx
bx
xx x x
xx
xx
−−
=
=
=
=
=
=
<+−
≥
<
∑ ∑ ∑∑ ∑
8
3.2
Implemented
Allocation Rule
The
system
approximates the continuous model presented in section 2.2 with the
elicited valuations as reported in section 2.3. Measuring total welfare by the aggregated
reported license valuations,
( )
( )
ˆˆ
ii
iI
VX
Vx
∈
=
∑
, and maintaining all the relevant constraints,
the system determines the allocation by optimizing:
( )
(
)
(
) {
}
(
)
{
}
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
ˆ
ˆ
11
1
aC
aC
i
i
i
aA
iI
aH
aH
i i
i
aA
iI
a
a
aC
aH
aA
aa a
aA
H
H
aH
C
C
aC
aA
aA
H H H
CC C
O
HC
X VX
x
x h aa
x
xh a
a
xx x
xx
x xxx
x x
xx
xx x
λ
λ
λ
μ
λλ
μμ
μ
∈∈
∈∈
∈
∈
∈∈
=
−
− −=
− −=
−
−−
−−
−− −−
− −− −
− −−
∑∑
∑∑
∑
∑
∑∑
L
Constraints on
area Club and
Hotel allocations
Constraints on
aggregate Club and
Hotel allocations
(2)
The discrete nature of the problem introduces
a number of complexities in solving the
optimization problem from equation (2). The complexities 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
. Tied bids are resolved through
a first come, first serve
rule.
Since all bids
are time
-stamped, if multiple bids are submitted at
the market quoted price, then allocations are made to the bids according to the
ir arrival time.
The market clearing prices need not be unique if the quantity demanded at a price exactly
equals the supply
to that market and thus multiple prices can clear the market. This was
addressed by adding a very small quantity to every bid so
the
quantity demanded at a price is
always slightly above the integer parts. Finally, though
multiple constraints could bind
and thus
create multiple price solutions
, such multiplicity could only arise from
relationships between
metropolitan constraints and area constraints. The problem never surfaced because the
metropolitan constraint was never binding.
Given the adjustments to the optimization problem necessary to address these practical
considerations, the shadow costs from the optimization problem only approximate the shadow
costs from the continuous Lagrangian in equation (1). As we will see in s
ection 4, these
approximations do not induce disequilibrium in any individual market’s allocation.
3.3 Preview of Dynamic Auction Features and Bidding Revisions
9
Th
e system arrived at its final allocation after progressing through a series of 63 bidding
rounds
. Section 5 presents
a detailed discussion of bidding dynamics and general equilibrium
convergence
. Each round begins with establishments submitting provisional bid functions.
The auction algorithm computes the allocation and prices
based on these bid functions
. T
hese
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.
Give
n this information, bidders are aware of the prices that revised bids
need to meet or
beat if they wish to obtain additional licenses. Before beginning the next round, bidders can
take advantage of this information and revise the
ir submitted bid functions
, subject to the
restriction that they increase their original bid by at least a minimal increment. This restriction
induces an ascending auction format and excluded bid pricing format
as bidding progresses
from round to round.
The system initiates an ending process based on the numbers of significant revisions
(according to rules
from
testbed
experiments
) in
individually
submitted bid schedules
and
related patterns of market price changes
. At this point, bidders are notified
about the num
ber
of
rounds
the market will remain open terminating with the announcement
that the market will
close in the subsequent round and given a final
opportunity to revise their bid schedules. If
there are still no significant revisions in this next
round, the
auction closes. After this last
round, the bidder
s pa
y the announced price for their market for each entitlement awarded.
3.4 Testbedding Designs to Determine Parameters
The
mechanism design exercise began with
a theoretical sketch of an auction where
: (i) each
bidder preferences
were
limited to a single establishment; (ii) agents submit “truthful” demand
functions; (iii) the auction winner is chosen by maximizing the revealed value of the allocation;
and (iv)
policy
limits
regarding multiple mark
ets exist as constraints. The theoretical sketch did
not address
behavior and implementation, even at an experimental, testbed level, but
identified the
analytical
principles guiding the mechanism and technical challenges
that
to be
addressed in
that impl
ementa
tion. Complex auction experiments regarding call markets (Plott
and Pogorelskiy
, 2017),
power grids (Chao and Plott,
2009)
and combinatorial auctions
(Lee
,
Maron
, and
Plott
, 2014
) offered prototypes that address important aspects of the theory
necessary for implementation. However, other experiments with pure tatonnement (Plott
,
2001)
and
on Natural gas pipelines
(Plott
, 1988)
served as a warning about the incomplete
nature
of the
ory
as a model of actual behavior. A key behavioral
feature, demand revelation at
the margin, had also been identified in an unpublished working paper
on fuel efficiency under
the
CAFE
constraint
(Katz and
Plott
, 2009)
.
10
Experimental work recommended an
auction based on dynamic
equilibration and
convergence as opposed to a one shot, sealed bid computation. Early
experimentation with
auction architectures
focused on continuous processes using tatonnement type process
or
double aucti
ons
as opposed to sealed bid system. T
he 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 individual units.
Competitive theory as app
lied to
smooth
demand
functions identified relationships among bids, prices (as Lagrangian
multipliers), equilibrium (as a competitive equilibrium), allocations and efficiency.
The actual
auction
required
solving
an integer
-constrained
, linear program
for
the
allocation
problem.
The
size of the problem and ability of bidders to deal with information flow
was best
accommodated by
a round structure as opposed to continuous bidding that had proved
successful in both experiments and the field.
3.4.1 Testbedd
ing Designs at Scale
The scale of the design problem derives from the
size o
f markets,
the
number of markets,
the
number of units
, and number of bidders.
The testbedding exercise
focused on establishing the
economic performance
and
technical
control
of the system
. The scaling technique relied on
homogeneity properties of economies.
If agents have linear demand functions
, increases in the
number of demanders of each type
, accompanied by appropriate supply increases
, will leave
prices and individual allocations of a given type unchanged.
The experimental scaling started
with
only
two areas (small and large), a fixed number of hotels and clubs in each and a fixed
number of entitlements to be equally split between clubs and hotels.
More complex designs
replicated this module, increasing the number of markets, bidders, and license
s to be allocated.
Th
is scaling method
tested
key aspects of convergence, efficiency
, and computation time
in
relation to
the size of the eco
nomic problem.
It did so without a need to undertake complex
computational challenges that can exist with large numbers of markets.
Two performance measures were
useful
tools
to refine the rules of the auction. The first
was market efficiency measured as c
onsumer surplus as developed by Plott and Smith (
1978)
.
This
measure is 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 with 8 participants, 4 regions and 8 markets.
6
Participants and
numbers of markets were scaled up such that prices
and allocation per type remained
approximately the same to study process characteristics in response to scale. Over 40 different
experiments were performed and often repeated to explore any problems exposed. The
6
In some of the early experiments, each market contained only one bidder. Competition emerged from the
interconnection of markets as any excess supply in a market could be absorbed by other markets.
11
largest testbed featuring human subject participants operated with 50 markets and 160
participants at a cost of $8,866. Larger scales were simulated with multiple computers
programmed to place bids to test network configurations, processing speeds
, and computation
reliability and speed.
3.4.
2 T
estbedding
Parameters for the Mechanism
Experiments revealed the ability of the mechanism to coordinate convergence
over
multiple
market
s, including derived demand and prices. Allocations and prices typically ended near the
predictions of the general comp
etitive 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.
Figure 1 illustrates the rapid rise of total auction revenue and social value (gains from trade) in
the first few rounds for one experiment.
7
When the auction closed in the 17
th
round, revenue
almost exactly matched the theoretical equilibrium, and over 90 percent of the optimal social
value was realized.
Figure 1: Total Revenue and Social Welfare for an Example Experiment
7
This experiment focused on five areas. Each area had a
club market and a hotel market. Thus there were ten
markets and thirty subjects. Each subject had a linear induced demand.
0
10000000
20000000
30000000
40000000
50000000
60000000
70000000
80000000
90000000
100000000
110000000
120000000
130000000
140000000
150000000
160000000
170000000
180000000
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
100223 Total Revenue and Revealed Social Welfare
Observed Social value
Total Revenue
theory total rev
Optimal Social Value
12
While each experiment examined multiple dimensions of performance, a narrow focus
was on two areas. The first related to real time control of price movement and ending the
auction. Previous experimental work revealed that biddi
ng incentives and stopping rules are
important for performance. The second broad area included performance, efficiency and
reliability in both a software and behavioral sense.
The timing of
the auction rounds
needed
adjustments
to account for
the reactio
n speed
of bidders.
Bidder behavior reflected
their own
information
and preferences as well as a
second, strategic
, dimension
affecting
the degree to which bidders sh
ould “reveal” the
willingness to pay by increasing bids.
An increment
al
requirement defi
ning the minimal
allowable increase had an obvious role
ensuring revisions were economically meaningful.
We
adopted a
two
-clock methodology for controlling bidder behavior.
8
Experiments suggested
the
use of
discrete
rounds
as opposed to continuous time to
control information
reporting and
allow bidders
time
to
process price and allocation information. How scale, in term of numbers
of market and bidders involved
, would affect mechanism performance involved
unknowns
that
were revealed through the testbed proc
ess.
Requ
iring only a
single bid
revision
for auction continuation is not practical because
various levels of randomness in bidding and bidding timing are always present. The practical
question becomes “how many” bids or price changes in a round does it
take to justify
keep
ing
markets open
for additional rounds
. Testing in experiments with different controls led to the
number of bidders that attempted to increase their holdings 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
thresholds
. Changes were well publicized. New bid increment requirements were announced as
a percentage over existing prices
and these
were enforced beginning in
a specified round
in the
future. Specific hypotheses about responses to these controls were impossible to test given the
size, time
, and expense of experiments. Nevertheless, the experiments provided substantia
l
experience with how
the
auction would respond to the
chosen
parameters
.
Adjustment speed depended on the bidding response to price increments. The
responses were the primary source of information about the likely auction ending time. How
long the auction might take was important given the government’s decision to limit it to
one
day. Experiments demonstrated that bidding followed a principle of revelation at the margin.
8
In continuous time actions one clock counts down in seconds and resets with the submission of a new bid in any
market. If no bid a
ppears in any market, the 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. Such possibilities are avoided by the use of a
second clock that counts down and resets if a bid results in new winners, thus exerting pressure to place bids that
cause prices to evolve.
13
Announced
prices were accompanied by the increment requirement. All
new
bids
or changed
bids
must be no less than the existing price plus increments.
Unch
anged bids remain in the
system.
The upcoming price is not known and price could remain unchanged. N
ew bids are
automatically integrated with the bidder’s existing bids to form a revised bid function. The new
bid function is a type of “revealed demand” b
ut
it is not fully revealing. The revealed demand
function always falls short of the limit values
(demand prices)
of infra
-marginal units. Since the
actual demand functions are known in testbed
experiments
the size of
this “shortfall” can be
observed
direc
tly
.
While the revealed demand curve falls under the
actual
demand curve an important
element
– demand at the margin –
is accurately revealed. The
experiments indicated that the
quantity demanded at the announced price is always very
close to the quantity demanded
according to the induced preferences. Given the rules
, the bidder can adjust the bid price to
assure the purchase of the marginal unit 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
subject
merely
needs to express a willingness to pay for it by tendering a bid price for the unit
above the stated market price.
Th
us, the value of the marginal demand is revealed.
The
demand function becomes traced out a
s price moves up following the required
bid increments.
In particular, the slope is use
ful
information
revealing
the state of demand relative to prices and
thus when the auction is near a competitive equilibrium.
Thi
s key feature is discussed
in more
detail in Section 5
and illustrated in Figure
s 5 and 6
.
3.4.
3 Testbedding
Parameters for the Mechanism
The experiments were the only source of information about bidder behaviors and perceptions.
Our assessment of instructions, bid submission screens, bidder feedback, tools for bidder
expression
, and other subtle variables related to bidder perceptions and r
eaction came from
experiments. The experiments and experimental subjects shaped all instructions and
the
technical aspects of the human-
to-auction interface
. Probity issues placed governmental
constraints
that limited
how the auction designers could
inte
rface with individual bidders. The
government prohibited all interactions with potential bidders or anyone else that might interact
with bidders
, due to fears that such
interactions might create unfair advantages among bidders.
Consequently, computer scr
eens, instructions, explanations of bid functions versus
bids,
increment requirements, stopping rules, price determination, the
time taken for individual
decisions and many other variables related to preconceptions, perceptions and skills of the
bidd
er pool
could not be
studied directly. The designers had no contact with actual bidders.
The designer
’s understanding of these variables was acquired entirely through laboratory
experiment
s using
college students as subjects
.
Other important issues such as how to deal with typos or slow computational speeds
were
studied
through experiments. Concerns about hackers, power reliability, and local
14
computers and network reliability were based on theory and experiments with limited
op
portunity to test the actual hardware
system that was used.
Section 4:
Partial Equilibrium Properties of the Final Allocation
This section relate
s the system’s algorithm for determining the allocation and prices to a
competitive model of demand under ine
lastic supply with segmented markets. In doing so, we
describe what might be modeled as
partial equilibrium properties of the final allocations in
each market segment.
We discuss e
ach segment and its partial equilibrium model separately.
Section 5
presen
ts more details of the bid revision process and dynamics that
determine total
supply across each of the markets, including
bidders’ incentives to truthfully
reveal their
demand through these revisions.
We begin by defining the different market segments based on the prices paid for
different licenses and aggregate the derived demand for establishments competing
to obtain
these licenses based on firms’ reported bid schedules. Within each of the market seg
ment
s,
the derived supply is price
-inelastic at a fixed quantity.
9
The price at which derived demand
equals derived supply in each market is approximated by the price premia for different types of
licenses from the constrained optimization problem determi
ning the allocation. As such, the
final allocation and prices in each market segment are consistent with an ex
-post partial
equilibrium given bidders’ unwillingness to submit revisions to their reported bid schedules.
We identify the different market se
gments by the inequality constraints in the
optimization problem that bind on any given allocation. Throughout our analysis, we assume
three stylized facts that applied to all allocations in the system that allow us to identify these
segments. First, we partition the set of areas
CU
AA A
=
into constrained areas
C
A
(where
,
aa
C
x x aA
= ∀∈
) and unconstrained areas
U
A
(where
,
aa
U
xxaA
< ∀∈
). Second, hotel
establishments
we
re allocated the maximum number of licenses available based on their venue
type (
HH
xx
=
) while club establishments’ allocation did not meet this maximum (
CC
xx
<
).
Third, all licenses available to the system were allocated (i.e.,
HC
xx x
= +
). These assumptions
give rise to two large market segments for unconstrained clubs and unconstrained hotels and a
number of smaller market segments for each constrained area in
C
A
commanding a local price
premium.
4.1 Partial Equilibrium in t
he Market for Unconstrained Clubs
9
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 section. We return to discuss of the general
equilibrium properties of the market system below when analyzing stability of excess demand functions and a
model of convergence to an equilibrium.
15
Consider the market for club licenses in unconstrained areas under an allocation in
which
CC
xx
<
. 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 Demand
for Unconstrained Clubs,
which
can b
e calculated as:
( )
( )(
) {
}
11
U
UC
i
i
i
aA iI
D p
Xp
h a a
∈∈
=
−=
∑∑
.
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:
( )
C
t
UC
H
aC
aA
S p xx
x
∈
=−−
∑
.
Finally, def
ine the “Club Base Price,”
*
UC
p
, as the derived market clearing price where
( )
( )
**
UC
UC
UC
UC
Dp Sp
=
.
Figure
2: Derived Demand and Supply for Unconstrained Clubs
Panel A: Full Demand and Supply
Panel B: Equilibrium Detail
Figure 2 presents the derived demand and supply for unconstrained clubs in the final
round of the system’s operation. The residual supply available to this market consisted of 3,693
units, which matched Derived Demand to set the Club Base P
rice of $5,500. Due to a large
mass of demand at $5,500,
the quantity demanded
at this price exceeded the available supply.
The proportion of demand met at this price was determined by priority based on when
establishments submitted their bids. The mass at this price point has two important
implications for the system. First, any establishment that did not receive an allocation at the
$5,500 price could have increased their bid to receive additional licenses with no impact on
16
price. Given the q
uantity supplied at this margin, an establishment could have obtained up to
61 additional licenses without having any impact on price, a number of licenses that was
greater than the number acquired by any establishment. Second, the unmet demand of 64
units at the external margin suggests that no establishment would have any power to reduce
prices by lowering their bids while still receiving the same allocation. If any bidder were to
reduce their stated value for licenses allocated at this external margin,
they would lose those
licenses to the unmet demand.
Finally, we connect the price for this market segment to the optimization problem (1).
Note that the allocation of all other license types are at their constrained maxima
, whether due
to individual ar
ea constraints or the aggregate hotel maximum license constraint.
Consequently, any additional licenses made available to the market in total (i.e., an increase in
x
) would be sold in the unconstrained club market at price
*
UC
p
. As the potential increase in
the value function from relaxing the total license quantity constraint, this price must then equal
the shadow cost of the constraint so that
*
5, 500
UC
O
p
μ
= =
.
This analysis demonstrates that the market for unconstrained club licenses cleared in a
classical sense. Bidders for unconstrained club licenses could unilaterally revise their bid
schedules to increase their provisional allocation without changing the prices they would pay
for the allocate
d licenses. Bidders could cancel bids that were not provisional winners and
avoid the possibility that the bids would be filled on subsequent rounds.
That bidders chose not
to take advantage of such revision opportunities demonstrates the allocation of li
censes among
bidders for unconstrained club licenses represents an ex
-post equilibrium.
4.2 Partial Equilibrium in t
he Market for Unconstrained Hotels
Next consider the market for licenses
for
hotel establishments competing in
unconstrained regions.
Similar to
the market for unconstrained clubs
, these bidders compete
solely with each other for the pool of hotel licenses that are not allocated to any of the
constrained areas. Define the
t
th
round
Derived Demand
and Supply, respectively, for
Unconstrai
ned Hotels as
( )
( ) {
}
( )
1
UC
UH
i
i
i
UH
H
aH
aA iI
aA
D
p
D ph a a
S
p x
x
∈∈
∈
=
=
= −
∑∑
∑
, an d ,
. The
“Hotel Base Price” represents the market clearing price,
*
UH
p
, that balances derived demand
and supply so that
( )
( )
**
UH
UH
UH
UH
Dp Sp
=
.
Figure 3 presents the derived demand and supply for unconstrained hotels in the final
round of the system’s operation. The 10,356 units of supply available in this market matched
Derived Demand to set the Hotel Base Price of $33,350. Derived demand for h
otel licenses
appears relatively elastic compared to demand for unconstrained club licenses, likely as a
17
consequence of greater heterogeneity in values for establishments in this market segment.
Still, market power and market impact for bidders is quite limited, with fourteen units allocated
out of twenty demanded at the Hotel Base Price. Consequently, a bidder in the unconstrained
hotel market would be able to obtain an additional fourteen licenses by stating a higher
willingness to pay without impacting
their actual price paid. Bidders could also reduce their
stated willingness to pay for the non provisional winning bids and thus avoid acquiring units in
subsequent rounds.
Figure
3: Derived Demand and Supply for Unconstrained Hotels
Panel A: Full Demand and Supply
Panel B: Equilibrium Detail
From the optimization problem (1), the Hotel Base Price reflects the marginal revenue
available from selling one additional license to a hotel, effectively increasing
H
x
by one unit.
However, holding
x
constant means this unconstrained hotel license must come from the
supply of unconstrained club licenses. Combined, the Hotel Base Price equals the shadow cost
of the cons
traint on the supply of hotel licenses plus the shadow constraint on the total supply
of licenses, i.e.,
**
UH
O
H
UC
H
pp
μμ
μ
=+= +
. We refer to the margin between the Base Hotel
Cost and Base Club Cost,
27, 850
H
μ
=
, as the Hotel Price Pre
mium.
As is the case with the market for unconstrained club licenses, the analysis
demonstrates that the market for unconstrained hotel licenses cleared in a classical sense.
Bidders in this market retain the unilateral ability to revise their bid sched
ules and alter their
provisional allocation without changing the prices paid. Consequently, bidders’ revealed
unwillingness to make such revisions demonstrates the ex
-post equilibrium nature of the
allocation among bidders for unconstrained hotel licenses
.
4.3 Partial Equilibrium in t
he Market
s for
Licenses in C
onstrained
Areas
18
In areas where the quantity of licenses is constrained, Clubs and Hotels compete with
each other to determine the Area Price Premium. The derived demand for licenses in these
areas aggregates the demand schedules for bidders in the area in excess of the base price
determined by the unconstrained markets for each venue type:
( )
(
)
(
)
(
) {
}
**
11
a
i
UH
i
i
UC
i
i
iI
Dp
X pp h X pp
h a a
∈
= + ++ − =
∑
.
Impounding the venue base price into the demand schedule translates each establishment’s bid
in terms of the premium realized by not allocating the license
to a bidder in an unconstrained
market. In constrained markets, the inelastic derived supply is fixed at the maximum
constraint so
( )
aa
Sp x
=
. The “Area Price Premium,” denoted
*
a
p
, represents the price that
clears the market, so that
( )
( )
**
aa
aa
Dp Sp
=
.
Figure
4: Derived Demand and Supply for Constrained Area 110
Panel A: Full Demand and Supply
Panel B: Equilibrium Detail
Figure
4 presents the derived demand and supply for the final allocation in
area number
05, with the schedules for all other areas appearing in Appendix A
. The total allocation of 494
licenses for this area is much smaller than for the unconstrained markets, with
the market
clearing at an area price premium of $26,900. Total demand at this external margin was 160
units, leaving 19 units of unmet demand at this price. As in the unconstrained markets, this
atom of demand demonstrates bidders’ ability to increase t
heir allocations by revising their bid
schedules without impacting prices
and the
ir unwillingness to cancel non
-provisional winners to
avoid acquiring additional units at the stated market price
.
Key Result 4.0:
The prices and allocations for each licen
se type is determined by the bid
schedules submitted for bidders seeking to acquire that type of license. The solution to the
Victoria Gaming Auction satisfies classical market clearing conditions balancing derived demand
19
and derived supply within each mar
ket segment. This consistency suggests the market for each
license type achieves a partial equilibrium for allocating its inelastic supply.
Section 5:
Bidding Dynamics
, Excess Demand Revelation, and Equilibrium
Convergence
We now present the auction’s dynamic features
, driven by shifting
bid functions that
can be modified, supplemented, or cancelled across rounds.
At the conclusion of each bidding
round, provisional prices
are announced for each market
so bidders a
re aware of the prices
they need
to meet or beat to obtain
a different
allocation
of entitlements.
To make the price
determination more transparent we adopted last-
accepted bid rules for the uniform price,
rather than first
-rejected bid rules. Although these uniform price rules do not m
ake value
revelation incentive compatible, as described earlier they nevertheless encourage value
revelation on the margin as the market price rises through successive bidding rounds.
Identifying the demand revealed at the margin as a measure of excess dem
and for licenses
suggests
prices adjusted following a tatonnemont
-like process. Using the rich bidding data, we
estimate the parameters of this price adjustment process to empirically
verify it satisfies well-
known stability conditions for general equilibrium. The theoretical and empirically verified
properties of the
system
demonstrates
the final allocation achieved general equilibrium
conditions to represent
an efficient allocation across markets.
5.1
Demand Revelation and Incentives at the Margin
To track revisions in bid schedules across rounds, superscript bids, prices, and demand
calculations with their associated round. To illustrate,
(
)
,
t
tt
il
il
il
B bx
=
represents the
l
th
entry
from the
i
th
establishment’s bid schedule submitted in the
t
th
round of the mechanism with
associated
cumulative bid schedule
( )
t
i
Xp
. Similarly, let
t
i
x
denote the
t
th
round’s provisional
allocation to establishment
i
, with
t
aH
p
and
t
aC
p
identifying the market clearing prices for this
allocation. Given the uniform pricing rule, these prices identify the point in the demand
schedule at which a bidder’s incentives for truthful demand
have a binding property
.
10
For
bid
schedule entries with prices above or below this margin, a bidder could respectively inflate or
deflate their stated willingness to pay without changing their allocation or the payment
required to receive that allocation.
Round
t
opens after announc
ement of
the provisional prices and quantities allocated
based on the bid functions submitted by all bidders in round
t
-1
. As the solution to the
10
As demonstrated in Section 4, no individual bidder has the power to influence prices, so truthfully reporting their
demand ensures they obtain exactly the quantity they want regardless of the final price. As advice to bidders,
market designers suggested taking this approach when submitting bids. While some bidders followed this
suggestion, the pilot experiments and testbed studies demonstrated the vast majority of bidders responded to
annou
nced prices and provisional allocations in a marginal manner.
20
optimization problem, these provisional allocations and prices each satisfy the partial
equilibrium conditions balancing derived supply and reported demand demonstrated in section
4. Bidders
respond
by submitting
round
t
’
s bid sched
ules. If the bidder is satisfied with their
t
-1
round allocation at the
t
-1
round’s ending price, they would have no incentive to revise their
reported bid schedule. If all bidders are satisfied with their
t
-1
round allocation at the
t
-1
round’s ending price, then no bidders would revise their reported bid schedules
and
the market
would close due to inactivity, determining the final allocations and prices.
11
Given their
potential as final auction outcomes, the provisional allocations and prices
offer ince
ntives for
further revelation due to threat of closing
.
However, if a bidder want
ed
to increase their
t
-1
round allocation, they could do so by
increasing the bid price for some entries in their schedule to be above the
t
-1
round ending
price. B
id revisions arise when bidders decide they want a greater allocation at the announced
price, with the system prompting a bidder to consider whether they want more licenses at the
announced price
. T
he system incentivizes
bidders to report the quantity they wish
to buying at
(or slightly above)
the
publicly announced prices
for the market
. At each
newly announced
price
, bidders reveal the maximum quantity they want at that price, a process we refer to as
“demand revelation at the margin.”
11
When there was an insufficient amount of bidding activity in a round, the auctioneer would publicly announce
that the auction will close if there is insufficient activity in one more round of bidding. If that tentatively “last”
round of bidding featured significant revision activity, the auction would again proceed until revision activity
ceased again. The auctioneer would then repeat their announcement and the process would continue until there
is insufficient revision activity in that “last” round of b
idding. In practice, the auctioneer announced only two
tentative ends to the bidding process, with the second corresponding to the close of the auction.
21
Figure 5:
Illustrating Dynamics of Demand Revelation
Th
rough demand revelation at the margin, rising prices across auction rounds “trace
out” points on the
market’s aggregate
demand curve
as
illustrated in Figure 5
. The market
opens with an initially announced pri
ce
of
P
1
, leading bidders to reveal (approximately) the
actual quantity demanded at this price in the subsequent round, but with potential
under
-
revelation of demand at higher prices as illustrated in
aggregated
bid function
B
1
. Th
e reported
bids generating bid function
B
1
lead to an announced price of
P
2
. The round 2 bid revisions
generate a
new revealed demand curve, with the point
B
2
indicat
ing
the true quantity desired
at this price.
The reported bids generating aggregated
schedule
B
2
lead to a market clearing
price of
P
3
, which then becomes the operative point of demand revelation for the round 3 bid
revisions. As rounds progress and prices continue to rise, demand revelation at the margin will
trace out more points very close to the
true demand curve
until the system converges
.
Figure
6 contains the time path of revealed demand at the margin for the unconstrained
hotels
, discussed above in Section 4.2
. The total number of units allocated to constrained
hotels
changes
very little over periods so the supply of entitlements to constrained hotels
remains relatively constant (
as illustrated in the figure
). Shown for each period is the aggregate
demand revealed at the market price of for unconstrained hotels. Total demand revealed at
the margin appears somewhat inelastic and with
the difference between total
excess demand
at the margin and supply
slowly shrinking to zero where the auction terminates. The
panels
at
the upper right show the details in terms of units of exces
s demand at the end of the auction
[quantity demanded at the margin –
supply = 10509 -
10332
] and also the entire demand curve