# RLC resonator with diode nonlinearity: Bifurcation comparison of numerical
predictions and circuit measurements

Edward H. Hellen<EMAIL_ADDRESS>Department of Physics & Astronomy,
University of North Carolina Greensboro, Greensboro, NC USA

###### Abstract

A nonlinear RLC resonator is investigated experimentally and numerically using
bifurcation analysis. The nonlinearity is due to the parallel combination of a
semiconductor rectifier diode and a fixed capacitor. The diode’s junction
capacitance, diffusion capacitance, and DC current-voltage relation each
contribute to the nonlinearity. The closely related RL-diode resonator has
been of interest for many years since its demonstration of period-doubling
cascades to chaos. In this study a direct comparison is made of dynamical
regime maps produced from simulations and circuit measurements. The maps show
the variety of limit cycles, their bifurcations, and regions of chaos over the
2-d parameter space of the source voltage’s frequency and amplitude. The
similar structures of the simulated and experimental maps suggests that the
diode models commonly used in circuit simulators (e.g., SPICE) work well in
bifurcation analyses, successfully predicting complex and chaotic dynamics
detected in the circuit. These results may be useful for applications of
varactor-loaded split ring resonators.

††preprint: AIP/123-QED

> Behavior of the resistor-inductor-diode oscillator is a popular example of
> complexity and chaos. The nonlinearity of the semi-conductor junction is
> accounted for by well-known models used regularly in circuit simulators such
> as SPICE.Roehr (1993) This nonlinear oscillator is also the basis for
> nonlinear split ring resonators, which are of interest due to interesting
> magnetic properties produced when used to form arrays in metamaterials.
> Here, bifurcation analysis and Lyapunov exponents are used to calculate a
> dynamical regime map in a 2-d parameter space of the source frequency and
> amplitude which is compared to a regime map created from circuit
> measurements. Good agreement is found on the structure of the maps as
> indicated by the types of limit cycles, their bifurcations, and period-
> doubling cascades to chaos. The results suggest that these types of
> bifurcation analyses may be useful for developing applications of the
> nonlinear split ring resonators in metamaterials.

Keywords: nonlinear resonator, resistor-inductor-diode, diode resonator, split
ring resonator, bifurcation, Lyapunov exponent

##  I Introduction

The series RLC circuit is one of the most common examples of the damped
harmonic oscillator. In this numerical and experimental study, nonlinearity is
introduced into the resonator by including a diode in parallel with the
capacitance. Semiconductor diodes have nonlinear current-voltage relationships
which are accounted for by mathematical models used in the well-known SPICE
modeling tool for circuit design.Roehr (1993) The nonlinearities cause the
circuit oscillations to have complex responses to a sinusoidal driving
voltage. In this work, bifurcation analysis and Lyapunov exponents are used to
find the limit cycles and chaotic attractors of this RLC-diode circuit and
their dependence on the frequency and amplitude of the driving source. These
simulation results are then compared to circuit measurements. The goal here is
to determine how well the bifurcation and Lyapunov exponent analysis can
predict actual circuit behavior over a wide range of parameter space.

The circuit considered here is essentially the same as the resistor-inductor-
diode (RLD) circuit used earlier to demonstrate experimentally the period-
doubling cascades to chaos which can be induced by varying parameters of the
sinusoidal voltage source.Linsay (1981); Testa, Pérez, and Jeffries (1982)
These experiments also demonstrated the spacing convergence of the period
doubling bifurcation parameter values predicted by Feigenbaum’s theory on the
universal behaviour of period doubling systems.Feigenbaum (1978, 1979)
Computer calculations based on the RLD circuit analysis also predicted the
period doubling cascade to chaos.Azzouz, Duhr, and Hasler (1983) Intermittency
routes to chaos were demonstrated.Jeffries and Perez (1982) Comparison of
model calculations and experimental measurements were made using bifurcation
diagrams, return maps, and Poincaré plots. Brorson, Dewey, and Linsay (1983);
Klinker, Meyer-Ilse, and Lauterborn (1984) A simplified diode model was used
to calculate a 2-dim bifurcation regime map which agreed well with
experimentally observed bifurcations.Tanaka, Matsumoto, and Chua (1987)

These early studies’ good agreement of theory and experiment for this simple
circuit’s complex behavior resulted in the RLD circuit becoming a popular
example of a chaotic system. A good example relevant to this paper is the
comparison of numerical predictions and circuit measurements by Carroll and
Pecora showing good agreement on the location of the $1^{st}$ period-doubling
in 2-dim parameter space.(Carroll and Pecora, 2002)

In the years since the above studies, bifurcation analysis tools have improved
and computing power has increased. It is somewhat surprising that more use has
not been made of these tools to make detailed predictions of the dynamical
regimes of the RLD circuit by using the diode’s well established mathematical
models used in SPICE. To the author’s knowledge, a direct comparison of
numerically calculated and experimentally measured bifurcation regime maps,
delineating the various limit cycles over a 2-dim parameter space has not been
done for the RLD circuit.

In the past couple of decades, there has been increasing interest in
exploiting nonliner RLC circuits in metasurfaces for their interesting
magnetic properties.Pendry  _et al._ (1999) In these applications, the RLC
circuit is a split ring resonator (SRR) with nonlinearity introduced into the
capacitance, often as a semiconductor junction, Shadrivov, Morrison, and
Kivshar (2006); Wang  _et al._ (2008); LAZARIDES, PALTOGLOU, and TSIRONIS
(2011); Rose, Huang, and Smith (2011) thereby making the SSR a RLD. The SRRs
are typically small (dimensions of $\lessapprox$ 10s of $\mu m$) and are
driven by the electromotive force induced by electromagnetic waves in the GHz
or THz range. The system considered here is conceptually and mathematically
the same as a SRR. Only the size and frequency differs. Here, a function
generator drives a circuit constructed from common electronic components: a
resistor, an inductor, and the parallel combination of a fixed capacitor and a
1N4003 rectifier diode. The diode’s semiconductor junction is the source of
the nonlinearity, just as it is for those SRRs which incorporate varactor
diodes.

The early studies of the RLD were primarily motivated by scholarly interest in
understanding the nature of their complex and chaotic behavior. The more
recent studies of arrays of the nonlinear SRRs used in metasurfaces are aimed
at applications and therefore may benefit from more accurate modeling of the
nonlinear SRR.

The goal of this work is to compare simulations and circuit measurements of
the RLC-diode over the 2-dim parameter space of the source’s amplitude and
frequency. The simulations use bifurcation tools and Lyapunov exponent
calculations to discover the limit cycles, their stability, and chaotic
regions over the 2-dim parameter space. The model used for the diode accounts
for junction capacitance, diffusion capacitance, and the nonlinear current-
voltage relation by using well-established models used in SPICE. Circuit
measurements detecting period-doubling, hysteresis, and chaotic-like behavior
are made using standard digital function generator and oscilloscope. The work
here refers to the RLC-diode circuit because the diode is in parallel with a
fixed capacitor whereas the RLD circuit generally does not include a fixed
capacitor. The fixed capacitor allows for additional flexibility in lowering
the resonance frequency of the circuit but does not change the essential
nature of comparing the predictions and measurements of complex behaviors
caused by the presence of the diode.

The presentation style here attempts to be accessible to those who are
interested in the RLD circuit but are not particularly familiar with
bifurcation analysis methods and terminology. Therefore descriptions of the
bifurcation methods are restricted to relevance to this work and may lack a
more general mathematical completeness and rigor.

##  II RLC-diode Equations

The system of interest shown in Fig. 1 is a sinusoidally driven series RLC-
diode circuit with a voltage-dependent capacitance due to the diode. We
investigate how the dynamics of the circuit oscillations depend on the
amplitude and frequency of the driving source Vs. The focus is on using
bifurcation analysis methods to identify the limit cycles and determine their
stability.

Figure 1: Driven nonlinear RLC-diode circuit where a diode provides the
nonlinearity. Diode is modeled by parallel combination of junction
capacitance, diffusion capacitance, and DC diode (the diode symbol). A fixed
capacitance in parallel with the diode is also included. R represents the sum
of explicit resistance and the inductor’s intrinsic resistance.

The capacitance of the RLC-diode resonator is the result of the parallel
combination of the diode and the fixed capacitance (Cfx in the figure) which
includes stray capacitance. The mathematical model used for the diode is the
parallel combination of junction capacitance Cj, diffusion capacitance Cdf,
and the diode’s standard DC current-voltage relation. All three of these terms
are voltage dependent and therefore contribute to the nonlinearity of the
system. The well-known models of semiconductor diodes used in the SPICE
circuit simulator are used to account for these three effects.

For a voltage source $V_{s}=A\cos(\Omega t)$ driving the RLC-diode resonator,
summing voltages around the circuit gives

$V_{s}=L\frac{dI}{dt}+IR+V_{d}$ (1)

where $V_{d}$ is the voltage across the parallel combination of the diode and
fixed capacitor.

Current $I$ from the voltage source splits, supplying current $dQ/dt$ to the
junction capacitance, current $I_{df}$ to the diffusion capacitance, current
$I_{fx}$ to the fixed capacitor, and current $I_{d}$ to the DC diode, giving
the equation

$I=\frac{dQ}{dt}+\frac{dQ_{df}}{dt}+\frac{dQ_{fx}}{dt}+I_{d}.$ (2)

Source current $I$ and junction capacitance charge $Q$ are the two variables
chosen to describe the system. Therefore, voltages and other charges must be
expressed in terms of $I$ and $Q$.

$I_{df}$ and $I_{fx}$ are expressed in terms of $Q$ by using

$\frac{dQ_{i}}{dt}=\frac{dQ_{i}}{dV_{d}}\frac{dV_{d}}{dt}=C_{i}\frac{dV_{d}}{dt}=C_{i}\frac{dV_{d}}{dQ}\frac{dQ}{dt}$

where incremental capacitance $C_{i}=dQ_{i}/dV_{d}$ is used and subscript
$i=(df,fx)$ for the diffusion and fixed capacitances. Solving Eq. (2) for the
current into the junction capacitance gives

$\frac{dQ}{dt}=\frac{I-I_{d}}{1+(C_{df}(V_{d})+C_{fx})\frac{dV_{d}}{dQ}}.$ (3)

The Appendix shows that the SPICE diode model allows $V_{d}$ and $I_{d}$ to be
expressed in terms of $Q$. Thus, Eqs. (1) and (3) are the foundation for
modeling the circuit in Fig. 1 in terms of the source current $I$ and the
diode junction capacitance charge $Q$.

Equations (1) and (3) are converted to a dimensionless form by defining
dimensionless variables,

$q\equiv\frac{Q}{Q_{0}},\>i\equiv\frac{I}{V_{J}}\sqrt{\frac{L}{C_{J0}+C_{fx}}},\>v\equiv\frac{V}{V_{J}},\>t^{\prime}\equiv\Omega_{0}t$
(4)

where $V_{J}$ and $C_{J0}$ are junction capacitance SPICE parameters (see
Appendix). $Q_{0}=V_{J}C_{J0}$ is a naturally defined junction charge
parameter and $\Omega_{0}=1/\sqrt{L(C_{J0}+C_{fx})}$ is the nominal resonant
frequency. Renaming $t^{\prime}$ as $t$ results in

$\displaystyle\frac{dq}{dt}=$
$\displaystyle\>\left(i-i_{d}(q)\right)\frac{C_{J0}+C_{fx}}{C_{J0}+(C_{df}(q)+C_{fx})\frac{dv_{d}(q)}{dq}}$
(5a) $\displaystyle\frac{di}{dt}=$ $\displaystyle-v_{d}(q)-\sigma
i+\mu\cos(\omega t)$ (5b)

where $t$ is now dimensionless. The dependence of $v_{d}$ on the variables
$(q,i)$ is required in Eq. (5). The Appendix shows that:

$v_{d}(q)=\begin{cases}1-\left(1-(1-M)q\right)^{\frac{1}{1-M}}\;&\text{for
}q<q_{t}\\\ FC+(1-FC)^{M}(q-q_{t})&\text{for }q>q_{t}\end{cases}$ (6)

and

$\frac{dv_{cap}(q)}{dq}=\begin{cases}\left(1-(1-M)q\right)^{\frac{M}{1-M}}\;&\text{for
}q<q_{t}\\\ (1-FC)^{M}&\text{for }q>q_{t}\end{cases}$ (7)

where

$q_{t}=\frac{1-(1-FC)^{(1-M)}}{1-M}.$ (8)

$M$ and $FC$ are two more junction capacitance SPICE parameters. The case for
$M=1$ is in the Appendix.

The diode’s DC current-voltage relation requires two SPICE parameters, reverse
saturation current $I_{S}$ and emission coefficient $N$. The dimensionless
relation is

$i_{d}(q)=i_{S}\left(e^{40V_{j}v_{d}(q)/N}-1\right)$ (9)

where Eq. (4) is used to obtain the dimensionless $i_{S}$ from $I_{S}$. The
diffusion capacitance is

$C_{df}(q)=\frac{40TT\times I_{S}}{N}e^{40V_{J}v_{d}(q)/N}.$ (10)

where $TT$ is the SPICE parameter for forward transit time.

We use SPICE parameter values appropriate for the 1N4001-1N4004 family of
rectifier diodes: $I_{S}=70$ pA, $N=1.4$, $C_{J0}=33$ pf, $V_{J}=0.35$ volts,
$M=0.45$, $FC=0.5$, and $TT=5\mu s$.

Remaining parameters in Eq. (5) are

$\sigma=R\sqrt{\frac{C_{J0}+C_{fx}}{L}},\;\mu=\frac{A}{V_{J}},\;\omega=\frac{\Omega}{\Omega_{0}}.$
(11)

In this work $\omega$ and $\mu$ are the parameters of interest to vary in the
bifurcation analysis. Equation (5) is the non-autonomous version of the
equations for our driven RLC-diode circuit.

Some of the bifurcation analysis methods are more easily suited to autonomous
systems. Therefore, Eq. (5) is converted to an autonomous form by including
two additional variables $u(t)$ and $v(t)$. The differential equations for
these variables are constructed so that $u$ and $v$ are sinusoidal with
frequency $\omega$ and unity amplitude.Ermentrout (2002) The autonomous
version of system (5) is then

$\displaystyle\frac{dq}{dt}=$
$\displaystyle\>\left(i-i_{d}(q)\right)\frac{C_{J0}+C_{fx}}{C_{J0}+(C_{df}(q)+C_{fx})\frac{dv_{d}(q)}{dq}}$
(12a) $\displaystyle\frac{di}{dt}=$ $\displaystyle-v_{d}(q)-\sigma i+\mu u$
(12b) $\displaystyle\frac{du}{dt}=$ $\displaystyle u(1-u^{2}-v^{2})-\omega v$
(12c) $\displaystyle\frac{dv}{dt}=$ $\displaystyle\>v(1-u^{2}-v^{2})+\omega
u.$ (12d)

Initial conditions should satisfy $u^{2}+v^{2}=1$. The sinusoidal behavior of
$u$ and $v$ allows for periodic behavior of all 4 variables. This addition of
two variables is in contrast to converting to an autonomous system by adding a
single variable $u(t)=t$ which results in one fewer equation, but has the
disadvantage of not allowing periodic behavior of all the variables.

The existence of the sinusoidal driving term insures there will be no fixed
points (except for the trivial case of $\mu$ or $\omega=0$) and that the main
limit cycle is period-1, meaning an oscillation with the same period as the
sinusoidal source.

##  III Analysis Method

Bifurcation analysis of Eq. 12 was done using XPPAUT Ermentrout (2002) and
AUTO-07p. Doedel  _et al._ (2012) These software tools are used to find the
limit cycles (LC), their bifurcation points, and 2-dim parameter space regime
maps showing the types of LCs. Time-series integration was done using an
adaptive 4th-order Runge-Kutta routine. Lyapunov exponents were calculated
using the method of Wolf et. al. Wolf  _et al._ (1985) The bifurcation
analysis consists of generating bifurcation continuation plots and the 2-dim
regime maps. The bifurcations for Eq. 12 are period-doublings (PD) and limit
points (LP). Here we describe the bifurcation points and their relation to the
bifurcation continuation plots. It is worth noting that while the diode’s
mathematical model used here in the above software tools uses SPICE
parameters, the SPICE software itself was not used for any of the numerical
simulations.

PD points occur on a bifurcation continuation plot at parameter values where a
stable LC becomes unstable and a LC with double the period is created. The
main LC of Eq. 12 is period-1, having the same period as the source. This LC
exists as either stable or unstable over the entire parameter space. Figure

Figure 2: Numerical $\omega$-continuation bifurcation plot showing the maximum
value of $i$ for the LC. $\mu=2$ and $\sigma=0.013$. Solid (dashed) line
indicates stable (unstable) LC. PD and LP bifurcation points are indicated.
Insets show coexisting LC time-series of $q$ (orange) and $i$ (blue) at
$\omega=0.372$.

2 shows the $\omega$-continuation bifurcation plot for the main LC. Six PD
points are seen, where the LC changes stability. Solid (dashed) line indicates
stable (unstable) LC. The plot shows how $i$-max (the maximum value of
variable $i$’s oscillation) depends on parameter $\omega$. LPs (also called
saddle-node bifurcations) occur on a bifurcation continuation plot at
parameter values where a stable LC branch and an unstable LC branch extending
together in parameter space reach an extreme parameter value (the LP) where
they collide and disappear. Another description is that a LC branch reverses
its direction along the parameter-axis at the LP, with concomitant change in
stability.

The two LPs in Fig. 2 create a region of hysteresis between their locations at
$\omega=0.316$ and 0.376, where there is coexistence of large and small
amplitude LCs as demonstrated by the time-series insets. The large amplitude
LC is stable over the hysteresis region, while the small amplitude LC becomes
unstable for $\omega$ below 0.369 where it undergoes period-doubling.

The LP and PD bifurcation points detected in the parameter continuation plots
are the starting points for tracing their paths in the 2-dim $\omega$-$\mu$
parameter space. The paths are the borders for different dynamical regimes.
Creation of these dynamical regime maps is a goal of this study because of
their usefulness in presenting how the system Eq. 12 behaves throughout the
$(\omega,\mu)$ parameter space.

$\omega$-continuation bifurcation plots of stable period-doubled LCs are used
to find higher order period-doubling points. Those points are then traced in
the 2-dim $\omega$-$\mu$ parameter space to find their borders allowing
determination of period-doubling cascade regions.

The Lyapunov exponents (LE) are useful for detecting regions with no stable
LCs, where the stable oscillatory states are either quasi-periodic or chaotic.
Borders for chaotic regions are not detected by the bifurcations tools but are
easily identified in LE plots as parameter values where the maximal LE changes
from zero (indicating stable LC) to positive (indicating chaos).

##  IV Circuit

Circuits were constructed based on Fig. 1. The inductor was $L=10$ mH with
$23\Omega$ of intrinsic resistance. The diode was 1N4003, a standard
rectifier. Fifteen of these diodes were used and in-house measurements were
made of the zero-bias junction capacitance, the reverse breakdown voltage, and
the reverse recovery time. Junction capacitance ranged from 23 to 85 pf,
breakdown voltage was 300 to 1400 volts, and reverse recovery time was 8 to 32
$\mu s$. All 15 diodes were were tested in RLC-diode circuits to check the
consistency of the dynamical regime results when using different diodes and to
determine the sensitivity to the diode properties.

The capacitor in parallel with the diode was 68 pf. The fixed capacitance
$C_{fx}$ in Fig. 1 accounts for both the 68 pf capacitor and stray
capacitance. Stray capacitance was determined by measuring the resonant
frequency for a small signal diode with a nominal 1 pf junction capacitance,
yielding stray capacitance of 24 pf. The resistor was $100\Omega$ making the
total series resistance $R=123\Omega$. Using $123\Omega$ with $C_{fx}=92pf$
and $C_{J0}=33pf$ (for the particular diode used to create the experimental
results regime map) in Eq. 11 gives $\sigma=0.013$ for use in simulations.

Resonance frequency was determined using a small amplitude (25 mV) source.
Figure 3 shows the frequency response of the diode voltage amplitude for
source amplitudes of 25 and 100 mV.

Figure 3: RLC-diode circuit’s frequency response. Normalized diode amplitude
for 25 and 100 mV source amplitudes. $R=123\Omega$, L =10 mH, and 1N4003 diode
in parallel with 68 pf capacitor and 24 pf stray capacitance.

The voltage amplitude has been normalized by the source amplitude. The 25 mV
source plot shows a typical resonance response, with a resonant frequency of
140 kHz. The maximum forward bias is about $12\times 25mV=0.3$ volts. The 100
mV source plot shows the distorted resonance response which is due to the
significantly higher diode current at the maximum forward bias of $5\times
100mV=0.5$ volts.

##  V Results

###  V.1 Numerical Bifurcation Analysis

Figure 4 is the numerical dynamical regime map showing the limit cycles and
chaotic oscillations of the RLC-diode circuit in the 2-dim $\omega$-$\mu$
parameter space. The map consists of LP and PD lines. Traversing the map at
$\mu=2$, there are six crossings of $1^{st}$ PD lines and two crossings of LP
lines, corresponding to the six PD points and two LP points in the
$\omega$-continuation of Fig. 2.

Figure 4: Numerical $\omega$-$\mu$ regime map for $\sigma=0.013$, $C_{fx}=92$
pf. P1 (period-1), P2, and P4 LC regions are indicated. Hatching indicates
higher order doublings and chaos. Frequency $\omega$ and source amplitude
$\mu$ are dimensionless.

A region of hysteresis of the period-1 (P1) main LC is indicated by the LP
lines. Inside this narrow region, two versions of the main LC coexist, which
causes a jump in the LC amplitude during a sweep of the frequency when a LP is
encountered causing a sudden switch from one version to the other. Figure 2
shows the hysteresis and the time-series of the two coexisting P1 LCs for
$\mu=2$.

There are two regions of period doubling of the P1 LC, a narrow region at low
frequency and a much larger region which occupies a substantial amount of the
map. The larger PD region contains two regions of higher order period doubling
cascades. The $2^{nd}$ and $3^{rd}$ doubling lines are shown. Period doubling
cascades to chaos are common when the spacing between higher order doublings
decreases, as is the case here. The $3^{rd}$ period doubling line can often be
used as an approximation for the doubling cascade’s onset of chaos.
Confirmation of chaos by calculation of Lyapunov exponents is shown below.

###  V.2 Lyapunov Exponents

LE calculations detect parameter ranges with chaotic behavior and are
therefore complementary to the parameter regime maps created by bifurcation
analysis since those methods do not detect chaos. In LE plots for an
autonomous system, stable LC is indicated by the maximal LE being zero,
whereas chaotic behavior has a positive maximal LE. The LE plot for source
amplitude $\mu=4$ in Fig. 5 shows there are two regions where the maximal LE
is positive (where the blue LE exceeds the red LE), near $\omega=0.5$ and a
wider region around $\omega=1.35$. These two regions confirm the period
doubling cascades to chaos suggested by the nested higher order doublings in
Fig. 4.

Information about PD points can also be indicated on the LE plot. When a
changing parameter causes a stable LC to undergo period doubling, the maximal
LE remains at zero throughout the process. However, the second maximal LE
increases until it reaches zero at the PD bifurcation point, then it returns
to increasingly negative numbers.

Figure 5: Three maximal LEs for increasing source frequency $\omega$ with
$\sigma=0.013,\mu=4$. Chaos occurs where the blue LE is positive. $1^{st}$,
$2^{nd}$, and $3^{rd}$ period-doubling points are indicated.

Many such occurrences are labeled in Fig. 5 based on their agreement with
where PD lines are crossed at $\mu=4$ in Fig. 4.

Figure 6 shows the simulated phase plot of source voltage versus diode voltage
at $\omega=1.25,\mu=3.95$ inside the right-side period doubling cascade to
chaos in Fig. 4. The phase plot shows the “filling in" of regions of the phase
space characteristic of chaos.

Figure 6: Numerical phase plot for $\mu=3.95,\omega=1.25,\sigma=0.013$,
located inside the period-doubling cascade.

The LE plot gives information only for the dynamical state whose basin of
attraction has captured the system. This is in contrast to the regime maps
which can show bifurcation lines for coexisting states, regardless of their
basins of attraction. This means that bifurcation lines on the regime maps
might not be indicated in a LE plot. An example is the right side $2^{nd}$ PD
of the left doubling cascade in Fig. 4 which shows this PD at $\omega=0.66$
for $\mu=4$. The $3^{rd}$ PD is very close to the $2^{nd}$, suggesting the
cascade to chaos should also occur close to $\omega=0.66$. However, Fig. 5
shows that chaos extends to only 0.61. The existence of the two $3^{rd}$ PD
lines which overlap indicates a complicated coexistence of higher order
doublings and two separate cascades to chaos. Presumably, the basin of
attraction of a stable period doubled LC is able to capture the chaotic state
at $\omega=0.61$.

###  V.3 Circuit Measurements

PD points were detected as a function of source frequency and amplitude by
varying source frequency at each source amplitude while monitoring the diode
voltage waveform. The change in waveform indicating period doubling was easily
recognized. Figure 7 shows the results for the circuit in Fig. 1 using
$R=123\Omega$ ($100\Omega$ resistor added to inductor’s $23\Omega$) and 68 pf
capacitor (added to 24 pf stray capacitance giving $C_{fix}=92$ pf), giving
resonance frequency 140 kHz. The 10 mH inductor is known to have nonideal
properties (parasitic capacitance) which become problematic for frequencies
above 300 kHz.

Figure 7: Experimental $\omega$-“source amplitude" regime map showing PD
borders and hysteresis induced amplitude jumps. Measurements from 1N4003
diode, L =10 mH, and $R=123\Omega$. $f_{0}=140kHz$.

Hysteresis was detected by noting the frequencies at which there was a large
jump in the amplitude of the diode voltage LC. The location of the hysteresis
for the circuit agrees well with the region in the numerical simulation,
although it is much narrower for the circuit measurements.

The overall structure of the regime maps in Figs. 4 and 7 for simulations and
circuit data have much in common. It is also informative to compare the time
series of the simulations and the circuit measurements.

Figure 8:  Time series simulations (left column) and circuit measurements
(right column) for 1N4003, L =10 mH, and $R=123\Omega$ showing period-
doublings as source amplitude increases for $f_{0}=140kHz$. Blue (orange) is
source (diode) voltage. Bottom row is main LC, period-1. Middle row is
period-2 LC after $1^{st}$ PD. Top row is period-4 LC after $2^{nd}$ PD.

Figure 8 shows good agreement of the LC period doublings which occur when
source amplitude is increased at the nominal resonance frequency $\omega=1$ in
both the numerical simulations of Fig. 4 and the circuit measurements in Fig.
7. The bottom row shows that for the small source amplitude (blue) the diode
voltage LC (orange) has the same period as the source and is nearly
sinusoidal. The middle row shows the time series for the period doubled P2 LC.
The general shape of the doubled LC is very similar for simulation and circuit
measurement, and the LC’s period clearly requires 2 oscillations of the
source. The top row shows the second doubling to P4 LC, which means the LC
requires 4 oscillations of the source. Again, the general shapes of simulation
and measurement have good agreement.

The one large discrepancy between simulation and circuit measurement is the
behavior at high frequency and low source amplitude where simulations show the
P1 period doubling region extending sharply down at twice the resonant
frequency.

The existence of chaotic behavior in the circuit was investigated by
continuing to increase the source amplitude beyond that used in Fig. 8 so as
to complete the period doubling cascade to chaos presumed to exist inside the
nested doublings.

Figure 9: Phase plots of measured diode voltage versus source voltage. Left is
period-4 LC with 1.8-volt source amplitude, right is likely chaos with
2.2-volt source amplitude. $f=140kHz$.

At $\omega=1,(f=140kHz)$, Fig. 7 suggests that a source amplitude just above
2-volts is likely beyond the onset of chaos. The phase plots in Fig. 9 show
the behavior with 1.8 volt source, between the $2^{nd}$ and $3^{rd}$
doublings, prior to chaos (left panel), and with 2.2 volt source (right
panel). The left panel shows the expected period-4 LC created by two period
doublings, and the right panel shows behavior whose appearance is consistent
with chaos. There is strong similarity between the right panel and the
simulated chaotic phase plot in Fig. 6.

As mentioned in Section IV, fifteen 1N4003 diodes were examined which had a
wide range of measured diode properties. All the diodes reproduced the main
features seen in the Fig. 4 numerical map, in particular, the cascade to chaos
in the larger period doubling region containing $\omega=1$. However, there was
some variation in the details. For example, the narrower left-side period
doubling cascade in Fig. 7 located near $\omega=0.5$ shows a small island of
the $3^{rd}$ PD. Some diodes only had the $2^{nd}$ doubling to period-4 LC
before reverse doubling back to period-2 LC. And 3 of the 15 missed this
narrow $2^{nd}$ doubling regime entirely. These 3 diodes all had relatively
small reverse recovery times out of the 15 diodes.

##  VI Discussion

Simulations and circuit measurements were compared for the driven nonlinear
RLC-diode circuit where the nonlinearity is due to a rectifier diode in
parallel with a fixed capacitor. Three nonlinear characteristics of the diode
were included in the simulations: the junction capacitance, the diffusion
capacitance, and the DC current/voltage relation. Bifurcation analysis and
Lyapunov exponents were used to predict the limit cycles, their stability, and
regions of chaos throughout the 2-dim parameter space of the driving source’s
frequency and amplitude. Circuit measurements were made in which the function
generator’s frequency and amplitude were varied so as to collect data in the
same 2-dim parameter space as the simulations. The results are shown in the
regime maps, Figs. 4 and 7.

These two regime maps show significant agreement of simulation and circuit
measurement over the 2-dim parameter space. The basic structure of the maps is
the same except for the sharp dip of the $1^{st}$ PD border at twice the
resonant frequency. Except for this dip, both maps show that for low source
amplitude the main LC (frequency same as the source) covers the entire
frequency range, down to the lowest investigated frequency (about 0.2). The
bottom panels of Fig. 8 show the time-series for simulation and measurement of
this LC for low source amplitude at $\omega=1$. At low source amplitude the
capacitance is essentially constant: the sum of the explicit capacitor, the
stray capacitance, and the diode’s junction capacitance. As expected for a
constant capacitance with source at resonance, the capacitor-diode voltage
amplitude (orange) is much larger than the source (blue) and is nearly
sinusoidal. And, varying the source frequency at low source amplitude produces
the resonance peak expected for an underdamped driven RLC circuit, as seen for
the 25 mV source in Fig. 3.

A region of LPs of the main LC is apparent in both maps beginning at low
source amplitude near resonance ($\omega=1$) and extending to lower frequency
and higher source amplitude. These LPs are due to the hysteresis of the
$\omega$-continuation seen in Fig. 2. As the source amplitude increases, the
resonance peak distorts, creating hysteresis. This distortion is apparent in
Fig. 3 where the left side of the peak for the larger source amplitude is
getting very steep.

The regime maps in Figs. 4 and 7 show similar period-doubling cascades to
chaos. Time series related to the period-doublings are seen in Fig. 8 where
the transition from the bottom panel to the middle panel corresponds to
crossing the $1^{st}$ PD line in the regime maps as source voltage increases
at $\omega=1$. The resulting period-2 LC for simulation and measurement have
the same patterns. Transition from the middle panel to the top panel
corresponds to crossing the $2^{nd}$ PD line which results in period-4 LC.
Again, agreement on the LC shape is good between the left and right panels.
Chaos inside the period-doubling cascades of the maps is predicted by the
regions of positive maximal LE in Fig. 5. Locations of PD lines in Fig. 4
agree with the locations indicated on Fig. 5. The phase plots for simulation
and measurement in Figs. 6 and 9(b) look quite similar and are also consistent
with chaos inside the period doubling cascades.

The agreement discussed above between simulations and circuit measurements of
the dynamical regimes suggest that the standard diode model used in SPICE and
used here is robust. It was necessary to take all three nonlinear
characteristics of the diode into account in order to achieve this level of
agreement. The one major difference between simulation and circuit measurement
in Figs. 4 and 7 is the sharp dip of the $1^{st}$ PD bifurcation at $\omega=2$
seen in simulation but not in measurement. This same discrepancy is apparent
in earlier comparisons of simulation and measurement made by Carrol and
Pecora. Carroll and Pecora (2002) They used measurements from different diodes
to make regime maps showing the onset of the $1^{st}$ PD, then compared the
maps to calculated $1^{st}$ PD regime maps. Only some of the diode circuit
measurements showed the PD dip at $\omega=2$, whereas nearly all the
simulations showed the dip. Their result is consistent with the finding here
that the measurements in Fig. 7 seem to show an upside-down shoulder at
$\omega=2$, but not a dip, whereas the simulation in Fig. 4 has the dip. This
discrepancy of the existence of the PD dip was not a concern in the earlier
work, and is an interesting observation here to be investigated in future
work.

Other early works focused on identifying the nonlinear property of the diode
responsible for the period-doubling cascades to chaos. Accounting for both the
nonlinear junction capacitance and diffusion capacitance predicted observed
bifurcations. Brorson, Dewey, and Linsay (1983); Azzouz, Duhr, and Hasler
(1983) Diffusion capacitance can account for the diode’s history-dependent
reverse recovery time, thereby predicting period-doubling cascades to chaos.
de Moraes and Anlage (2003) Later, a mathematically simpler model obtained
from the junction capacitance alone by using a cubic approximation of Eq. (20)
was used to predict nonlinear phenomena seen in the varactor loaded SRRs. Wang
_et al._ (2008); LAZARIDES, PALTOGLOU, and TSIRONIS (2011)

Recently, simulations inspired by applications of nonlinear SRRs were done,
producing 2-dim regime maps using the maximal LE which predict period-
doubling, chaos, hysteresis, and multistability. Leutcho, Woodward, and
Blanchard (2023) Those simulations take into account the varactor diode’s
junction capacitance by using the cubic approximation mentioned above. There
were no experimental results for comparison.

The goal of the work presented here is to evaluate just how well diode models
can predict the measured dynamical behavior over a large 2-dim parameter
space. The good agreement of prediction and measurement presented here
required all three nonlinear characteristics of the diode. Equation (20) was
used instead of its cubic approximation so as to not limit the range of
parameter space.

Accurate prediction of the behavior of a varactor loaded SRR should be
beneficial for designing their use for application in metamaterials. The
semiconductor diode model used here is applicable to the varactor diodes in
SRRs. Only the values of the model parameters differ. The model’s good
predictive ability demonstrated here encourages application of the bifurcation
methods and tools used here to the varactor loaded SRRs.

##  VII Conclusion

The diode resonator is a well-known circuit demonstrating nonlinear behavior
including period-doubling cascades to chaos. Earlier works have shown
comparisons of simulated and measured bifurcation diagrams, however
comparisons of dynamical regime maps are far less common. This work adds to
the long story of analysis of the diode resonator by using bifurcation
analysis tools to generate more detailed regime maps for comparison with
circuit measurements. The standard diode model used in the SPICE circuit
simulator gave good results when used in the bifurcation tools and Lyapunov
exponent calculations. The location of period-doubling cascades, hysteresis,
and chaos in the 2-dim parameter space of drive frequency and amplitude is
nearly the same in the simulated and measured maps. Lyapunov exponent plots
are consistent with the regime maps. The bifurcation methods and tools used to
generate the regime maps may be useful to the emerging applications of
nonlinear SRRs. Future work will address the $1^{st}$ PD dip discrepancy in
the low-amplitude/high-frequency portion of the regime map. In addition, other
values of resistive loss (R in circuit, parameter $\sigma$ in model) will be
investigated.

###### Acknowledgements.

This work was supported by a University of North Carolina Greensboro Research
Assignment.

*

##  Appendix A Diode Model & Derivation of $\mathbf{v_{d}(q)}$

The mathematical model for the semiconductor diode has three parts: the quasi-
static (DC) current-voltage relation, a junction capacitance important for
reverse bias and small forward bias voltages, and a diffusion capacitance
important for forward bias. The model’s parameters are provided by SPICE
parameters. Roehr (1993)

The well-known “diode equation" gives the DC current-voltage relation. Its
parameters are reverse saturation current $I_{S}$ and emission coefficient
$N$. The mathematical model is

$I_{d}(V_{d})=I_{S}\left(e^{40V_{d}/N}-1\right)$ (13)

where 40 comes from $e/kT=(25mV)^{-1}$ at room temperature.

The model’s parameters for the junction capacitance are the zero-bias junction
capacitance $C_{J0}$, the contact potential $V_{J}$, the junction capacitance
grading exponent $M$, and the forward bias coefficient $FC$. The junction
capacitance’s dependence on voltage is given by

$C_{J}(V_{d})=C_{J0}\left(1-\frac{V_{d}}{V_{J}}\right)^{-M}.$ (14)

Parameter $FC$ is in the range 0 to 1 and gives an upper limit of $FC\times
V_{J}$ for $V_{d}$ in order to avoid the singularity at the forward bias value
$V_{d}=V_{J}$.

The diffusion capacitance needs one additional parameter, the forward transit
time $TT$. The diffusion capacitance’s dependence on voltage is given by

$C_{df}(V_{d})=\frac{40TT\times I_{S}}{N}e^{40V_{d}/N}.$ (15)

We use SPICE parameter values typical for the 1N4003 rectifier diode:
$I_{S}=70$ pA, $N=1.4$, $C_{J0}=33$ pf, $V_{J}=0.35$ volts, $M=0.45$,
$FC=0.5$, and $TT=5\mu s$. We use the 1N4003 diode for circuit measurements
because it has a larger junction capacitance than small signal diodes, thereby
reducing the effects of stray capacitance. We measured the diode capacitance
as a function of bias voltage for comparison with the model’s predictions.

Figure 10: Junction capacitance and diffusion capacitance versus bias.
Measured data values (X) for 1N4003.

The results in Fig. 10 show our data and the models for the junction
capacitance and diffusion capacitance using SPICE parameter values given
above. The plot shows that junction capacitance dominates for reverse bias and
that diffusion capacitance dominates for forward bias ($>0.3$ volts).

The dependence of $v_{d}$ on the variables $(q,i)$ is required in Eq. (12).
The junction capacitance Eq. (14) is useful because by definition

$C_{J}=\frac{dQ}{dV_{d}}=C_{J0}\frac{dq}{dv_{d}}.$ (16)

$C_{J}$ is integrated to find $q(v_{d})$, which is then inverted to find
$v_{d}(q)$. Wang  _et al._ (2008); LAZARIDES, PALTOGLOU, and TSIRONIS (2011)
$M=1$ and $M\neq 1$ are treated separately. The case $M=1$ with the condition
$q(v_{d}=0)=0$ results in

$q(v_{d})=\ln\left(\frac{1}{1-v_{d}}\right).$ (17)

Rearranging for voltage and using dimensionless variables finds

$v_{d}(q)=1-e^{-q}.$ (18)

For $M\neq 1$, integration with the condition $q(v_{d}=0)=0$ gives

$q(v_{d})=\frac{1}{1-M}\left[1-\left(1-v_{d}\right)^{1-M}\right].$ (19)

Rearranging for voltage finds

$v_{d}(q)=1-\left[1-(1-M)q\right]^{\frac{1}{1-M}}.$ (20)

Thus, the dimensionless result for $v_{d}(q)$ is

$v_{d}(q)=\begin{cases}1-\left(1-(1-M)q\right)^{\frac{1}{1-M}}\;&\text{for
}M\neq 1\\\ 1-e^{-q}&\text{for }M=1\end{cases}$ (21)

with the restriction $(1-M)q<1$. Figure 11 shows plots of Eq. (21) for various
$M$-values.

Figure 11: Dimensionless diode voltage as function of junction charge for
different SPICE $M$-values.

It is important to point out that if $(1-M)q>1$ then Eq. (21) gives physically
unrealistic complex numbers for $V_{d}$ since the exponent $1/(1-M)$ will in
general not be an integer. These complex voltages are avoided by incorporating
parameter FC. If FC is not included, then complex voltages are avoided only in
carefully selected cases which give integer exponents (like $M=0.8$ giving
$1/(1-M)=5$) because a negative number raised to an integral power remains
real. The use of these special integral exponent cases requires caution since
the imaginary parts of complex number results remain zero for $(1-M)q>1$,
thereby masking the nonphysical conditions.

The unphysical singularity of the junction capacitance at $V_{d}=V_{J}$ is
avoided by letting $C_{j}$ remain constant for $V_{d}>FC\times V_{J}$,
retaining the value $C_{J0}/(1-FC)^{M}$ it has at $V_{d}=FC\times V_{J}$. The
$q$-value where this transition occurs is found using $v_{d}=FC$ in Eq. (19),

$q_{t}=\frac{1-(1-FC)^{(1-M)}}{1-M}$ (22)

The constant junction capacitance for $v_{d}>FC$ makes the voltage-charge
relationship linear. Therefore, Eq. (20) is modified to maintain a constant
slope for $q>q_{t}$

$v_{d}(q)=\begin{cases}1-\left(1-(1-M)q\right)^{\frac{1}{1-M}}\;&\text{for
}q<q_{t}\\\ FC+(1-FC)^{M}(q-q_{t})&\text{for }q>q_{t}\end{cases}$ (23)

Equation (3) shows that $\frac{dv_{d}(q)}{dq}$ is required. Using Eq. (23)

$\frac{dv_{d}(q)}{dq}=\begin{cases}\left(1-(1-M)q\right)^{\frac{M}{1-M}}\;&\text{for
}q<q_{t}\\\ (1-FC)^{M}&\text{for }q>q_{t}\end{cases}$ (24)

The dimensionless DC current-voltage relation is

$i_{D}(q)=i_{S}\left(e^{40V_{j}v_{d}(q)/N}-1\right)$ (25)

where Eq. (4) is used to obtain $i_{S}$, the diode’s dimensionless reverse
saturation current.

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$\phi^{BR}_{j}(\succ)=(b,t^{0})\succ_{i}\phi^{BR}_{i}(\succ)\quad\implies\quad
j\;\pi_{b}\;i.$ (1)

Moreover positions at the increased cost $t^{+}$ are offered to cadets with
highest $\pi_{b}$ priorities among those (i) who fail to receive a position at
the base cost $t^{0}$ and (ii) who declare the expensive assignment
$(b,t^{+})$ as acceptable. Therefore, for any $i,j\in I$,

$\phi^{BR}_{j}(\succ)=(b,t^{+})\succ_{i}\phi^{BR}_{i}(\succ)=\emptyset\quad\implies\quad
j\;\pi_{b}\;i.$ (2)

Relations 1 and 2 imply that mechanism $\phi^{BR}$ has no priority reversals.

BRADSO-IC: Fix a cadet $i\in I$. For a given profile of preferences for all
cadets except cadet $i$, whether cadet $i\in I$ receives an assignment of
$(b,t^{0})$ under the mechanism $\phi^{BR}$ is independent of cadet $i$’s
preferences under the mechanism $\phi^{BR}$: Cadets who are among the
$q^{0}_{b}$ highest $\pi_{b}$-priority cadets in $I$ always receive an
assignment at the base cost $t^{0}$; cadets who are not among the $q$ highest
$\pi_{b}$-priority cadets in $I$ never receive an assignment at the base cost
$t^{0}$; and for any cadet $i$ who has one of the highest $q$ but not one of
the highest $q^{0}_{b}$ priorities, whether she receives an assignment at the
base cost $t^{0}$ depends on how many lower $\pi_{b}$-priority cadets are both
willing to pay the increase cost $t^{+}$ and also able to “jump ahead of” the
cadet $i$ through the BRADSO policy. Hence if a cadet receives a position
under $\phi^{BR}$ at the increased cost $t^{+}$, changing her reported
preferences can only result in losing the position altogether. Therefore
mechanism $\phi^{BR}$ satisfies BRADSO-IC.

Enforcement of the BRADSO policy: The procedure for the mechanism $\phi^{BR}$
initially assigns all positions to the $q_{b}$ highest $\pi_{b}$-priority
cadets at the base cost $t^{0}$, although the assignments of the
$q^{+}_{b}$-lowest $\pi_{b}$-priority cadets among these awardees are only
tentative. Step 2 of the procedure for mechanism $\phi^{BR}$ ensures that, if
any cadet $j\in I$ loses her tentative assignment $(b,t^{0})$ from Step 1,
then any cadet $i\in I$ who receives an assignment of $(b,t^{+})$ is such that
$(i,t^{+})\;\omega^{+}_{b}\;(j,t^{0})$. Therefore,

$\left.\begin{array}[]{l}\phi^{BR}_{i}(\succ)=(b,t^{+}),\;\mbox{ and}\\\
(b,t^{0})\succ_{j}\phi^{BR}_{j}(\succ)\end{array}\right\\}\quad\implies\quad(i,t^{+})\;\omega_{b}^{+}\;(j,t^{0}).$
(3)

Moreover, Step 2 of the same procedure also ensures that, for any
$\ell\in\\{1,\dots,q^{+}_{b}\\}$, the $\ell^{\footnotesize th}$ lowest
$\pi_{b}$-priority cadet $i^{\ell}$ with a tentative assignment of $(b,t^{0})$
cannot maintain this tentative assignment, for as long as there are at least
$\ell$ lower $\pi_{b}$-priority cadets who are both willing to pay the
increased cost $t^{+}$ and also able to “jump ahead of” the cadet $i^{\ell}$
through the BRADSO policy. Therefore,

$\left.\begin{array}[]{l}\phi^{BR}_{j}(\succ)=(b,t^{0}),\\\
(b,t^{+})\succ_{i}\phi^{BR}_{i}(\succ),\;\mbox{ and}\\\
(i,t^{+})\;\omega_{b}^{+}\;(j,t^{0})\end{array}\right\\}\quad\implies\quad\big{|}\big{\\{}i^{\prime}\in
I:\phi^{BR}_{i^{\prime}}(\succ)=(b,t^{+})\big{\\}}\big{|}=q^{+}_{b}.$ (4)

Relations (3) and (4) imply that mechanism $\phi^{BR}$ satisfies enforcement
of the BRADSO policy.

Uniqueness: We next show that mechanism $\phi^{BR}$ is the only mechanism that
satisfies all five axioms.

Let the direct mechanism $\varphi$ satisfy individual rationality, non-
wastefulness, BRADSO-IC, enforcement of the BRADSO policy, and has no priority
reversals. We want to show that $\varphi(\succ)=\phi^{BR}(\succ)$.

If there are less than or equal to $q$ cadets for whom the assignment
$(b,t^{0})$ is acceptable under the preference profile $\succ$, all such
cadets must receive an assignment of $(b,t^{0})$ by individual rationality,
non-wastefulness, and BRADSO-IC. Since this is also the case under the
allocation $\phi^{BR}(\succ)$, the result holds immediately for this case.

Therefore, w.l.o.g assume that there are strictly more than $q$ cadets for
whom the assignment $(b,t^{0})$ is acceptable under the preference profile
$\succ$. Let $I^{0}$ be the set of $q^{0}_{b}$ highest $\pi_{b}$-priority
cadets in $I$. By non-wastefulness, all positions are assigned under
$\varphi(\succ)$. Since at most $q^{+}_{b}$ positions can be awarded at the
increased cost $t^{+}$, at least $q^{0}_{b}$ positions has to be allocated at
the base cost $t^{0}$. Therefore,

$\mbox{for any }i\in
I^{0},\qquad\varphi_{i}(\succ)=(b,t^{0})=\phi^{BR}_{i}(\succ)$ (5)

by lack of priority reversals.

Let $I^{1}$ be the set of $q^{+}_{b}$ highest $\pi_{b}$-priority cadets in
$I\setminus I^{0}$. Relabel the cadets in the set $I^{1}$ so that for any
$\ell\in\\{1,\ldots,q^{+}_{b}\\}$, cadet $i^{\ell}$ is the
${\ell}^{\mbox{\footnotesize th}}$-lowest $\pi_{b}$-priority cadet in $I^{1}$.
Let

$J^{0}=\big{\\{}j\in I\setminus(I^{0}\cup
I^{1}):(b,t^{+})\succ_{j}\emptyset\big{\\}}.$

By individual rationality and the lack of priority reversals,

$\mbox{for any }i\in I\setminus(I^{0}\cup I^{1}\cup
J^{0}),\qquad\varphi_{i}(\succ)=\emptyset=\phi^{BR}_{i}(\succ).$ (6)

By relations (5) and (6), the only set of cadets whose assignments are yet to
be determined under $\varphi(\succ)$ are cadets in $I^{1}\cup J^{0}$.
Moreover, by the lack of priority reversals, cadets in $J^{0}$ can only
receive a position at the increased cost $t^{+}$. That is,

$\mbox{for any }j\in J^{0},\qquad\varphi_{j}(\succ)\not=(b,t^{0}).$ (7)

For the next phase of our proof, we will rely on the sequence of individuals
$i^{1},\ldots,i^{q^{+}_{b}}$ and the sequence of sets $J^{0},J^{1},\ldots$
that are constructed for the Step 2 of the mechanism $\phi^{BR}$. Here
individual $i^{1}$ is the $q^{\footnotesize\mbox{th}}$ highest
$\pi_{b}$-priority cadet in set $I$, cadet $i^{2}$ is the
$(q-1)^{\footnotesize\mbox{th}}$ highest $\pi_{b}$-priority cadet in set $I$,
and so on. The starting element of the second sequence is $J^{0}=\\{j\in
I\setminus(I^{0}\cup I^{1}):(b,t^{+})\succ_{j}\emptyset\\}$. Assuming Step
2.$n$ is the last sub-step of Step 2, the remaining elements of the latter
sequence for $n\geq 1$ is given as follows: For any $\ell\in\\{1,\ldots,n\\}$,

$J^{\ell}=\left\\{\begin{array}[]{cl}J^{\ell-1}&\mbox{ if
}\;\emptyset\,\succ_{i^{\ell}}\,(b,t^{+})\\\
J^{\ell-1}\cup\\{i^{\ell}\\}&\mbox{ if
}\;(b,t^{+})\succ_{i^{\ell}}\,\emptyset\end{array}\right.$

We have three cases to consider.

Case 1. $n=0$

For this case, by the mechanics of the Step 2 of the mechanism $\phi^{BR}$, we
have

$\big{|}\big{\\{}j\in
J^{0}:(j,t^{+})\;\omega^{+}_{b}\;(i^{1},t^{0})\big{\\}}\big{|}=0.$ (8)

Therefore, by relations 6, 7, and condition (1) of the axiom enforcement of
the BRADSO policy,

$\mbox{for any }i\in I\setminus(I^{0}\cup
I^{1}),\qquad\varphi_{i}(\succ)=\emptyset=\phi^{BR}_{i}(\succ).$ (9)

Hence by non-wastefulness,

$\mbox{for any }i\in
I^{1},\qquad\varphi_{i}(\succ)\in\big{\\{}(b,t^{0}),(b,t^{+})\big{\\}}.$ (10)

But since $\varphi$ satisfies individual rationality, relation (10) implies
that $\varphi_{i}(\succ)=(b,t^{0})$ for any $i\in I^{1}$ with
$\emptyset\succ_{i}(b,t^{+})$. Furthermore for any $i\in I^{1}$ with
$(b,t^{+})\succ_{i}\emptyset$, instead reporting the fake preference relation
$\succ^{\prime}_{i}\in\mathcal{Q}$ with $\emptyset\succ^{\prime}_{i}(b,t^{+})$
would guarantee cadet $i$ an assignment of
$\varphi_{i}(\succ_{-i},\succ^{\prime}_{i})=(b,t^{0})$ due to the same
arguments applied for the economy $(\succ_{-i},\succ^{\prime}_{i})$, and
therefore by BRADSO-IC these cadets too must receive an assignment of
$(b,t^{0})$ each. Hence

$\mbox{ for any }i\in
I^{1},\qquad\varphi_{i}(\succ)=(b,t^{0})=\phi_{i}^{BR}(\succ).$ (11)

Relations (5), and (11) imply $\varphi(\succ)=\phi^{BR}(\succ)$, completing
the proof for Case 1.$\blacksquare$

Case 2. $n\in\\{1,\ldots,q^{+}_{b}-1\\}$

For this case, by the mechanics of the Step 2 of the mechanism $\phi^{BR}$, we
have

$\mbox{for any }\ell\in\\{1,\ldots,n\\},\qquad\big{|}\big{\\{}j\in
J^{\ell-1}:(j,t^{+})\;\omega^{+}_{b}\;(i^{\ell},t^{0})\big{\\}}\big{|}\geq\ell,$
(12)

and

$\big{|}\big{\\{}j\in
J^{n}:(j,t^{+})\;\omega^{+}_{b}\;(i^{n+1},t^{0})\big{\\}}\big{|}=n.$ (13)

Since mechanism $\varphi$ satisfies condition (2) of the axiom enforcement of
the BRADSO policy, the lack of priority reversals and relation 12 imply

$\mbox{ for any
}i\in\\{i^{1},\dots,i^{n}\\},\qquad\varphi_{i}(\succ)\not=(b,t^{0}).$ (14)

Therefore, by non-wastefulness and relations (5), (6), (7), and (14), at least
$n$ positions must be assigned at the increased cost $t^{+}$.

Moreover, since mechanism $\varphi$ satisfies non-wastefulness, lack of
priority reversals, and condition (1) of the axiom enforcement of the BRADSO
policy, relation (13) implies

$\mbox{ for any
}i\in\\{i^{n+1},\dots,i^{q^{+}_{b}}\\},\qquad\varphi_{i}(\succ)\in\big{\\{}(b,t^{0}),(b,t^{+})\big{\\}}.$
(15)

But since $\varphi$ satisfies individual rationality, relation (15) implies
that $\varphi_{i}(\succ)=(b,t^{0})$ for any
$i\in\\{i^{n+1},\dots,i^{q^{+}_{b}}\\}$ with $\emptyset\succ_{i}(b,t^{+})$.
Furthermore for any $i\in\\{i^{n+1},\dots,i^{q^{+}_{b}}\\}$ with
$(b,t^{+})\succ_{i}\emptyset$, instead reporting the fake preference relation
$\succ^{\prime}_{i}\in\mathcal{Q}$ with $\emptyset\succ^{\prime}_{i}(b,t^{+})$
would guarantee cadet $i$ an assignment of
$\varphi_{i}(\succ_{-i},\succ^{\prime}_{i})=(b,t^{0})$ due to the same
arguments applied for the economy $(\succ_{-i},\succ^{\prime}_{i})$, and
therefore by BRADSO-IC these cadets too must receive an assignment of
$(b,t^{0})$ each. Hence

$\mbox{ for any
}i\in\\{i^{n+1},\dots,i^{q^{+}_{b}}\\},\qquad\varphi_{i}(\succ)=(b,t^{0})=\phi_{i}^{BR}(\succ).$
(16)

Since we have already shown that at least $n$ positions must be assigned at an
increased cost of $t^{+}$, relation (16) implies that exactly $n$ positions
must be assigned this cost, and therefore for any cadet $j\in J^{n}$ who is
one of the $n$ highest $\pi_{b}$-priority cadets in $J^{n}$,

$\varphi_{j}(\succ)=(b,t^{+})=\phi_{i}^{BR}(\succ)$ (17)

by the lack of priority reversals.

Relations (5), (16), and (17) imply $\varphi(\succ)=\phi^{BR}(\succ)$,
completing the proof for Case 2. $\blacksquare$

Case 3. $n=q^{+}_{b}$

For this case, by the mechanics of the Step 2 of the mechanism $\phi^{BR}$, we
have

$\mbox{for any }\ell\in\\{1,\ldots,q^{+}_{b}\\},\qquad\big{|}\big{\\{}j\in
J^{\ell-1}:(j,t^{+})\;\omega^{+}_{b}\;(i^{\ell},t^{0})\big{\\}}\big{|}\geq\ell.$
(18)

Since mechanism $\varphi$ satisfies condition (2) of the axiom enforcement of
the BRADSO policy, relation 18 implies

$\mbox{ for any
}i\in\underbrace{\\{i^{1},\dots,i^{q^{+}_{b}}\\}}_{=I^{1}},\qquad\varphi_{i}(\succ)\not=(b,t^{0}).$
(19)

Therefore, by non-wastefulness and the lack of priority reversals, exactly
$q^{+}_{b}$ positions must be assigned at the increased cost $t^{+}$. Hence
for any cadet $j\in J^{q^{+}_{b}}$ who is one of the $q^{+}_{b}$ highest
$\pi_{b}$-priority cadets in $J^{q^{+}_{b}}$,

$\varphi_{j}(\succ)=(b,t^{+})=\phi_{i}^{BR}(\succ)$ (20)

by elimination of priority reversals.

Relations (5) and (20) imply $\varphi(\succ)=\phi^{BR}(\succ)$, completing the
proof for Case 3, thus finalizing the proof of the theorem. $\blacksquare$ ∎

Proof of Proposition 1: Suppose that there is only one branch $b\in B$. Fixing
the profile of cadet preferences $\succ\,\in\mathcal{Q}$, the baseline
priority order $\pi_{b}$, and the BRADSO policy $\omega^{+}_{b}$, consider the
strategic-form game induced by the USMA-2020 mechanism
$(\mathcal{S}^{2020},\varphi^{2020})$. When there is only one branch, the
first part of the strategy space becomes redundant and the second part
contains only the two elements $b$ and $\emptyset$. Hence, for any cadet $i\in
I$, the strategy space of cadet $i\in I$ under the USMA-2020 mechanism is
$\mathcal{S}^{2020}_{i}=\\{\emptyset,b\\}$.

For a given strategy profile $s\in\mathcal{S}^{2020}$, construct the priority
order $\pi^{+}_{b}(s)$ as follows: For any $i,j\in I$,

  1. 1.

$\;s_{i}=s_{j}\qquad\qquad\qquad\quad\implies\qquad\qquad
i\;\,\pi^{+}_{b}(s)\;j\;\iff i\;\pi_{b}\;j$,

  2. 2.

$\;s_{i}=b$ and $s_{j}=\emptyset\qquad\,\implies\qquad
i\;\,\pi^{+}_{b}(s)\;j\;\iff(i,t^{+})\;\omega^{+}_{b}\;(j,t^{0})$.

Let $I^{+}(s)$ be the set of $q_{b}$ highest $\pi^{+}_{b}(s)$-priority cadets
in $I$.

For any cadet $i\in I$, the outcome of the USMA-2020 mechanism is given as,

$\varphi_{i}^{2020}(s)=\left\\{\begin{array}[]{cl}\emptyset&\mbox{if
}\;\;i\not\in I^{+}(s),\\\ (b,t^{0})&\mbox{if }\;\;i\in I^{+}(s)\mbox{ and
}s_{i}=\emptyset,\\\ (b,t^{0})&\mbox{if }\;\;i\in I^{+}(s)\mbox{ and
}s_{i}=b\;\mbox{ and }\;\big{|}\\{j\in I^{+}(s):s_{j}=b\mbox{ and
}i\;\pi_{b}\;j\\}\big{|}\geq q^{+}_{b},\\\ (b,t^{+})&\mbox{if }\;\;i\in
I^{+}(s)\mbox{ and }s_{i}=b\;\mbox{ and }\;\big{|}\\{j\in
I^{+}(s):s_{j}=b\mbox{ and
}i\;\pi_{b}\;j\\}\big{|}<q^{+}_{b}.\end{array}\right.$

We first prove a lemma on the structure of Nash equilibrium strategies of the
strategic-form game induced by the USMA-2020 mechanism
$(\mathcal{S}^{2020},\varphi^{2020})$.

######  Lemma 1.

Let $s^{*}$ be a Nash equilibrium of the strategic-form game induced by the
mechanism $(\mathcal{S}^{2020},\varphi^{2020})$. Then, for any $i,j\in I$,

$\varphi^{2020}_{j}(s^{*})\succ_{i}\varphi^{2020}_{i}(s^{*})\quad\implies\quad
j\;\pi_{b}\;i.$

Proof of Lemma 1: Let $s^{*}$ be a Nash equilibrium of the strategic-form game
induced by the USMA-2020 mechanism $(\mathcal{S}^{2020},\varphi^{2020})$.
Contrary to the claim suppose that, there exists $i,j\in I$ such that

$\varphi^{2020}_{j}(s^{*})\succ_{i}\varphi^{2020}_{i}(s^{*})\quad\mbox{ and
}\quad i\;\pi_{b}\;j.$

There are three possible cases, where in each case we reach a contradiction by
showing that cadet $i$ has a profitable deviation by mimicking the strategy of
cadet $j$:

Case 1: $\varphi^{2020}_{j}(s^{*})=(b,t^{0})$ and
$\varphi^{2020}_{i}(s^{*})=(b,t^{+})$.

Since by assumption $\varphi^{2020}_{i}(s^{*})=(b,t^{+})$,

$s^{*}_{i}=b.$

Moreover the assumptions $\varphi^{2020}_{j}(s^{*})=(b,t^{0})$,
$\varphi^{2020}_{i}(s^{*})\not=(b,t^{0})$, and $i\;\pi_{b}\;j$ imply

$j\in I^{+}(s^{*})\quad\mbox{ and }\quad s^{*}_{j}=\emptyset.$ (21)

But then, relation (21) and the assumption $i\;\pi_{b}\;j$ imply that, for the
alternative strategy $\hat{s}_{i}=\emptyset$ for cadet $i$,

$i\in I^{+}(s_{-i}^{*},\hat{s}_{i}),$

and thus

$\varphi_{i}^{2020}(s_{-i}^{*},\hat{s}_{i})=(b,t^{0})\succ_{i}\varphi^{2020}_{i}(s^{*}),$

contradicting $s^{*}$ is a Nash equilibrium strategy. This completes the proof
for Case 1. $\blacksquare$

Case 2: $\varphi^{2020}_{j}(s^{*})=(b,t^{0})$ and
$\varphi^{2020}_{i}(s^{*})=\emptyset$.

Since by assumption $\varphi^{2020}_{j}(s^{*})=(b,t^{0})$,
$\varphi^{2020}_{i}(s^{*})=\emptyset$, and $i\;\pi_{b}\;j$, we must have

$j\in I^{+}(s^{*})\quad\mbox{ and }\quad s^{*}_{j}=b\quad\mbox{ and
}\quad\big{|}\\{k\in I^{+}(s^{*}):s^{*}_{k}=b\mbox{ and
}j\;\pi_{b}\;k\\}\big{|}\geq q^{+}_{b},$ (22)

and

$s^{*}_{i}=\emptyset.$

But then, relation (22) and the assumption $i\;\pi_{b}\;j$ imply that, for the
alternative strategy $\hat{s}_{i}=b$ for cadet $i$,

$i\in I^{+}(s_{-i}^{*},\hat{s}_{i})\quad\mbox{ and
}\quad\hat{s}_{i}=b\quad\mbox{ and }\quad\big{|}\\{k\in
I^{+}(s_{-i}^{*},\hat{s}_{i}):s^{*}_{k}=b\mbox{ and
}i\;\pi_{b}\;k\\}\big{|}\geq q^{+}_{b},$

and thus

$\varphi_{i}^{2020}(s_{-i}^{*},\hat{s}_{i})=(b,t^{0})\succ_{i}\varphi^{2020}_{i}(s^{*}),$

contradicting $s^{*}$ is a Nash equilibrium strategy. This completes the proof
for Case 2. $\blacksquare$

Case 3: $\varphi^{2020}_{j}(s^{*})=(b,t^{+})$ and
$\varphi^{2020}_{i}(s^{*})=\emptyset$.

Since by assumption $\varphi^{2020}_{j}(s^{*})=(b,t^{+})$,

$j\in I^{+}(s^{*})\quad\mbox{ and }\quad s^{*}_{j}=b.$ (23)

Moreover, since $\varphi^{2020}_{i}(s^{*})=\emptyset$ by assumption,

$i\not\in I^{+}(s^{*}).$

Therefore, since $i\;\pi_{b}\;j$ by assumption,

$j\in I^{+}(s^{*})\;\mbox{ and }\;i\not\in I^{+}(s^{*})\qquad\implies\qquad
s^{*}_{i}=\emptyset.$

But then, again thanks to assumption $i\;\pi_{b}\;j$, the relation (23)
implies that, for the alternative strategy $\hat{s}_{i}=b$ for cadet $i$,

$i\in I^{+}(s_{-i}^{*},\hat{s}_{i}),$

and thus

$\underbrace{\varphi_{i}^{2020}(s_{-i}^{*},\hat{s}_{i})}_{\in\\{(b,t^{0}),(b,t^{+})\\}}\succ_{i}\varphi^{2020}_{i}(s^{*}),$

contradicting $s^{*}$ is a Nash equilibrium strategy,323232Unlike the first
two cases, in this case cadet $i$ may even get a better assignment than cadet
$j$ (i.e. cadet $i$ may receive an assignment of $(b,t^{0})$) by mimicking
cadet $j$’s strategy.  completing the proof for Case 3, and concluding the
proof of Lemma 1. $\blacksquare$ $\diamondsuit$

For the next phase of our proof, we rely on the construction in the Step 2 of
the mechanism $\phi^{BR}$: Let $I^{0}$ be the set of $q^{0}_{b}$ highest
$\pi_{b}$-priority cadets in $I$, and $I^{1}$ be the set of $q^{+}_{b}$
highest $\pi_{b}$-priority cadets in $I\setminus I^{0}$. Relabel the set of
cadets in $I^{1}$, so that $i^{1}$ is the lowest $\pi_{b}$-priority cadet in
$I^{1}$, $i^{2}$ is the second lowest $\pi_{b}$-priority cadet in $I^{1}$,…,
and $i^{q^{+}_{b}}$ is the highest $\pi_{b}$-priority cadet in $I^{1}$. Note
that, cadet $i^{1}$ is the $q^{\footnotesize\mbox{th}}$ highest
$\pi_{b}$-priority cadet in set $I$, cadet $i^{2}$ is the
$(q-1)^{\footnotesize\mbox{th}}$ highest $\pi_{b}$-priority cadet in set $I$,
and so on. Let $J^{0}=\\{j\in I\setminus(I^{0}\cup
I^{1}):(b,t^{+})\succ_{j}\emptyset\\}$. Assuming Step 2.$n$ is the last sub-
step of Step 2 of the mechanism $\phi^{BR}$, for any
$\ell\in\\{1,\ldots,n\\}$, let

$J^{\ell}=\left\\{\begin{array}[]{cl}J^{\ell-1}&\mbox{ if
}\;\emptyset\,\succ_{i^{\ell}}\,(b,t^{+})\\\
J^{\ell-1}\cup\\{i^{\ell}\\}&\mbox{ if
}\;(b,t^{+})\succ_{i^{\ell}}\,\emptyset\end{array}\right.$

Recall that, under the mechanism $\phi^{BR}$, exactly $n$ cadets receive an
assignment of $(b,t^{+})$. We will show that, the same is also the case under
the Nash equilibria of the strategic-form game induced by the USMA-2020
mechanism $(\mathcal{S}^{2020},\varphi^{2020})$.

Let $s^{*}$ be a Nash equilibrium of the strategic-form game induced by the
USMA-2020 mechanism $(\mathcal{S}^{2020},\varphi^{2020})$. We have three cases
to consider:

Case 1: $n=0$

Since by assumption $n=0$ in this case,

$\big{\\{}j\in
J^{0}:(j,t^{+})\;\omega^{+}_{b}\;(i^{1},t^{0})\big{\\}}=\emptyset.$ (24)

Towards a contradiction, suppose there exists a cadet $i\in
I\setminus(I^{0}\cup I^{1})$ such that $i\in I^{+}(s^{*})$. Since cadet
$i^{1}$ is the $q^{\mbox{\footnotesize th}}$ highest $\pi_{b}$-priority cadet
in $I$, the assumption $i\in I^{+}(s^{*})$ and relation (24) imply

$i\not\in J^{0}\implies\emptyset\;\succ_{i}(b,t^{+}).$ (25)

Moreover, since cadet $i$ is not one of the $q$ highest $\pi_{b}$-priority
cadets in $I$,

$i\in I^{+}(s^{*})\implies s^{*}_{i}=b.$ (26)

But this means cadet $i$ can instead submit an alternative strategy
$\hat{s}_{i}=\emptyset$, assuring that she remains unmatched, contradicting
$s^{*}$ is a Nash equilibrium. Therefore,

$\mbox{for any }i\in I\setminus(I^{0}\cup
I^{1}),\qquad(i,t^{+})\;\omega^{+}_{b}\;(i^{1},t^{0})\implies
s^{*}_{i}=\emptyset,$ (27)

which in turn implies

$I^{+}(s^{*})=I^{0}\cup I^{1}.$ (28)

Hence all cadets in $I^{0}\cup I^{1}$ receive a position under
$\varphi^{2020}(s^{*})$. Next consider the lowest $\pi_{b}$-priority cadet
$i\in I^{0}\cup I^{1}$ such that $\varphi_{i}^{2020}(s^{*})=(b,t^{+})$. This
can only happen if $s^{*}_{i}=b$. But this means cadet $i$ can instead submit
an alternative strategy $\hat{s}_{i}=\emptyset$, assuring that
$\varphi_{i}^{2020}(s_{-i}^{*},\hat{s}_{i})=(b,t^{0})$ by relation (27),
contradicting $s^{*}$ is a Nash equilibrium. Hence

$\mbox{for any }i\in I^{0}\cup
I^{1},\qquad\varphi_{i}^{2020}(s^{*})=(b,t^{0})=\phi^{BR}_{i}(\succ),$ (29)

and therefore $\varphi^{2020}(s^{*})=\phi^{BR}(\succ)$.

Finally observe that the strategy profile $s^{\prime}$ where
$s^{\prime}_{i}=\emptyset$ for any cadet $i\in I$ is a Nash equilibrium, with
an outcome $\varphi^{2020}(s^{\prime})=\phi^{BR}(\succ)$, showing that there
exists a Nash equilibrium completing the proof for Case 1. $\blacksquare$

For any $\ell\in\\{1,\ldots,n\\}$, let $\overline{J^{\ell}}$ be the set of
$\ell$ highest $\pi_{b}$-priority cadets in the set $J^{\ell}$:

$\overline{J^{\ell}}=\Big{\\{}j\in J^{\ell}:\;\big{|}\\{i\in
J^{\ell}:\;i\;\pi_{b}\;j\\}\big{|}<\ell\Big{\\}}$

Before proceeding with the next two cases, we prove the following lemma that
will be helpful for both cases.

######  Lemma 2.

Suppose there are $n>0$ positions allocated at the increased cost $t^{+}$
under the allocation $\phi^{BR}(\succ)$. Then, for any Nash equilibrium
$s^{*}$ of the strategic-form game induced by the USMA-2020 mechanism
$(\mathcal{S}^{2020},\varphi^{2020})$ and $\ell\in\\{1,\ldots,n\\}$,

  1. 1.

$\varphi^{2020}_{i^{\ell}}(s^{*})=(b,t^{+})\quad\iff\quad(b,t^{+})\succ_{i^{\ell}}\emptyset$,
and

  2. 2.

$\varphi^{2020}_{i}(s^{*})=(b,t^{+})$ for any $i\in\overline{J^{\ell}}$.

Proof of Lemma 2: Let $s^{*}$ be a Nash equilibrium of the strategic-form game
induced by the USMA-2020 mechanism $(\mathcal{S}^{2020},\varphi^{2020})$.
First recall that,

$\mbox{for any }j\in I\setminus(I^{0}\cup
I^{1}),\qquad\varphi_{j}^{2020}(s^{*})\in\big{\\{}(b,t^{+}),\emptyset\big{\\}},$

and therefore, since any cadet $j\in I\setminus(I^{0}\cup I^{1}\cup J^{0})$
prefers remaining unmatched to receiving a position at the increased cost
$t^{+}$ and she can assure remaining unmatched by submitting the strategy
$s_{j}=\emptyset$,

$\mbox{for any }j\in I\setminus(I^{0}\cup I^{1}\cup
J^{0}),\qquad\varphi_{j}^{2020}(s^{*})=\emptyset.$ (30)

Also, by the mechanics of the Step 2 of the mechanism $\phi^{BR}$,

$\mbox{for any }\ell\in\\{1,\ldots,n\\},\qquad\big{|}\big{\\{}j\in
J^{\ell-1}:(j,t^{+})\;\omega^{+}_{b}\;(i^{\ell},t^{0})\big{\\}}\big{|}\geq\ell.$
(31)

The proof of the lemma is by induction on $\ell$. We first prove the result
for $\ell=1$.

Consider the highest $\pi_{b}$-priority cadet $j$ in the set $\big{\\{}j\in
J^{0}:(j,t^{+})\;\omega^{+}_{b}\;(i^{1},t^{0})\big{\\}}$. By relation 31, such
a cadet exists.

First assume that $(b,t^{+})\succ_{i^{1}}\emptyset$. In this case,
$J^{1}=J^{0}\cup\\{i^{1}\\}$ and cadet $i^{1}$ is the highest
$\pi_{b}$-priority cadet in $J^{1}$. Hence $\overline{J^{1}}=\\{i^{1}\\}$ in
this case. Consider the Nash equilibrium strategies of cadet $i^{1}$ and cadet
$j$. If $s^{*}_{i^{1}}=\emptyset$, then by relation (30) her competitor cadet
$j$ can secure himself an assignment of $(b,t^{+})$ by reporting a strategy of
$s_{j}=b$, which would mean cadet $i^{1}$ has to remain unassigned, since by
Lemma 1 no cadet in $I^{0}\cup I^{1}$ can envy the assignment of cadet $i^{1}$
at Nash equilibria. In contrast, reporting a strategy of $s_{i^{1}}=b$ assures
that cadet $i^{1}$ receives a position, which is preferred at any price to
remaining unmatched by assumption $(b,t^{+})\succ_{i^{1}}\emptyset$.
Therefore, $s^{*}_{i^{1}}=b$, and hence

$(b,t^{+})\succ_{i^{1}}\emptyset\quad\implies\quad\left\\{\begin{array}[]{ll}&\varphi^{2020}_{i^{1}}(s^{*})=(b,t^{+}),\;\mbox{
and }\\\ &\varphi^{2020}_{i}(s^{*})=(b,t^{+})\quad\mbox{ for any
}i\in\overline{J^{1}}=\\{i^{1}\\}.\end{array}\right.$ (32)

Next assume that $\emptyset\succ_{i^{1}}(b,t^{+})$. In this case $J^{1}=J^{0}$
and cadet $j$ is the highest $\pi_{b}$-priority cadet in $J^{1}$. Hence
$\overline{J^{1}}=\\{j\\}$ in this case. By Lemma 1, no cadet in $(I^{0}\cup
I^{1})\setminus\\{i^{1}\\}$ can envy the assignment of cadet $i^{1}$ at Nash
equilibria. Therefore, a strategy of $s_{i^{1}}=b$ means that cadet $i$
receives an assignment of $(b,t^{+})$, which is inferior to remaining
unmatched by assumption. Therefore $s^{*}_{i^{1}}=\emptyset$. Moreover
reporting a strategy of $s_{j}=\emptyset$ means that cadet $j$ remains
unmatched, whereas reporting a strategy of $s_{j}=b$ assures that she receives
an assignment of $(b,t^{+})$, which is preferred to remaining unmatched since
$j\in J^{0}$. Therefore, $s^{*}_{i^{1}}=\emptyset$, and hence

$\emptyset\succ_{i^{1}}(b,t^{+})\quad\implies\quad\left\\{\begin{array}[]{ll}&\varphi^{2020}_{i^{1}}(s^{*})=\emptyset,\;\mbox{
and }\\\ &\varphi^{2020}_{i}(s^{*})=(b,t^{+})\quad\mbox{ for any
}i\in\overline{J^{1}}=\\{j\\}.\end{array}\right.$ (33)

Relations (32) and (33) complete the proof for $\ell=1$.

Next assume that the inductive hypothesis holds for $\ell=k<n$. We want to
show that the result holds for $\ell=(k+1)$ as well.

By the inductive hypothesis,

$\mbox{for any
}i\in\overline{J^{k}},\quad\varphi^{2020}_{i}(s^{*})=(b,t^{+}).$ (34)

By relation 31, there are at least $k+1$ cadets in the set $\big{\\{}j\in
J^{k}:(j,t^{+})\;\omega^{+}_{b}\;(i^{k+1},t^{0})\big{\\}}$. Therefore, since
there are $k$ cadets in the set $\overline{J^{k}}$, there is at least one
cadet in the set

$\big{\\{}j\in
J^{k}:(j,t^{+})\;\omega^{+}_{b}\;(i^{k+1},t^{0})\big{\\}}\setminus\overline{J^{k}}.$

Let $j$ be the highest $\pi_{b}$-priority cadet in this set.

First assume that $(b,t^{+})\succ_{i^{k+1}}\emptyset$. In this case
$J^{k+1}=J^{k}\cup\\{i^{k+1}\\}$ and cadet $i^{k+1}$ is the highest
$\pi_{b}$-priority cadet in $J^{k+1}$. Hence
$\overline{J^{k+1}}=\overline{J^{k}}\cup\\{i^{k+1}\\}$ in this case. Consider
the Nash equilibrium strategies of cadet $i^{k+1}$ and cadet $j$. If
$s^{*}_{i^{k+1}}=\emptyset$, then by relation (30) cadet $j$ can secure
herself an assignment of $(b,t^{+})$ by reporting a strategy of $s_{j}=b$,
which would mean cadet $i^{k+1}$ has to remain unassigned, since by Lemma 1 no
cadet in $(I^{0}\cup I^{1})\setminus\\{i^{1},\ldots,i^{k}\\}$ can envy the
assignment of cadet $i^{k+1}$ at Nash equilibria and by relation (34) all
cadets in $\overline{J^{k}}$ receive an assignment of $(b,t^{+})$.333333Since
$\left|(I^{0}\cup I^{1})\setminus\\{i^{1},\ldots,i^{k}\\}\right|=(q-k)$ and
$\left|\overline{J^{k}}\right|=k$, this basically means cadets $i^{k+1}$ and
$j$ are competing for a single position. In contrast, reporting a strategy of
$s_{i^{k+1}}=b$ assures that cadet $i^{k+1}$ receives a position, which is
preferred at any price to remaining unmatched by assumption
$(b,t^{+})\succ_{i^{k+1}}\emptyset$. Therefore, $s^{*}_{i^{k+1}}=b$, and hence

$(b,t^{+})\succ_{i^{k+1}}\emptyset\;\implies\;\left\\{\begin{array}[]{ll}&\varphi^{2020}_{i^{k+1}}(s^{*})=(b,t^{+}),\;\mbox{
and }\\\ &\varphi^{2020}_{i}(s^{*})=(b,t^{+})\;\mbox{ for any
}i\in\overline{J^{k+1}}=\overline{J^{k}}\cup\\{i^{k+1}\\}.\end{array}\right.$
(35)

Next assume that $\emptyset\succ_{i^{k+1}}(b,t^{+})$. In this case
$J^{k+1}=J^{k}$ and $\overline{J^{k+1}}=\overline{J^{k}}\cup\\{j\\}$. By Lemma
1, no cadet in $I^{0}\cup I^{1}\setminus\\{i^{1},\ldots,i^{k}\\}$ can envy the
assignment of cadet $i^{k+1}$ at Nash equilibria. Therefore, since all cadets
in $\overline{J^{k}}$ receive an assignment of $(b,t^{+})$ by relation (34), a
strategy of $s_{i^{k+1}}=b$ means that cadet $i^{k+1}$ receives an assignment
of $(b,t^{+})$, which is inferior to remaining unmatched by assumption.
Therefore $s^{*}_{i^{k+1}}=\emptyset$. Moreover reporting a strategy of
$s_{j}=\emptyset$ means that cadet $j$ remains unmatched, whereas reporting a
strategy of $s_{j}=b$ assures that she receives an assignment of $(b,t^{+})$,
which is preferred to remaining unmatched since $j\in J^{k}$. Therefore,
$s^{*}_{i^{k+1}}=\emptyset$, and hence

$\emptyset\succ_{i^{k+1}}(b,t^{+})\quad\implies\quad\left\\{\begin{array}[]{ll}&\varphi^{2020}_{i^{k+1}}(s^{*})=\emptyset,\;\mbox{
and }\\\ &\varphi^{2020}_{i}(s^{*})=(b,t^{+})\quad\mbox{ for any
}i\in\overline{J^{k+1}}=\overline{J^{k}}\cup\\{j\\}.\end{array}\right.$ (36)

Relations (35) and (36) complete the proof for $\ell=k+1$, and conclude the
proof of Lemma 2.  $\diamondsuit$

We are ready to complete prove the theorem for our last two cases:

Case 2. $n\in\\{1,\ldots,q^{+}_{b}-1\\}$

For this case, by the mechanics of the Step 2 of the mechanism $\phi^{BR}$,

$\big{|}\big{\\{}j\in
J^{n}:(j,t^{+})\;\omega^{+}_{b}\;(i^{n+1},t^{0})\big{\\}}\big{|}=n.$ (37)

Consider cadet $i^{n+1}$. There are $q-(n+1)$ cadets with higher
$\pi_{b}$-priority, and by relation (37) there are $n$ cadets in $J^{n}$ whose
increased-cost assignments have higher $\omega^{+}_{b}$ priority under the
BRADSO policy than the base-cost assignment for cadet $i^{n+1}$. For any other
cadet $i\in I\setminus\Big{(}J^{n}\cup
I^{0}\cup\big{(}I^{1}\setminus\\{i^{1},\ldots,i^{n+1}\\}\big{)}\Big{)}$ with
$(i,t^{+})\;\omega^{+}_{b}\;(i^{n+1},t^{0})$, we must have
$\emptyset\succ_{i}(b,t^{+})$ since $J^{n}\supseteq J^{0}$. Therefore none of
these individuals can receive an assignment of $(b,t^{+})$ under a Nash
equilibrium strategy, and hence the number of cadets who can have higher
$\pi^{+}_{b}(s^{*})$-priority than cadet is $i^{n+1}$ is at most
$q-(n+1)+n=q-1$ under any Nash equilibrium strategy. That is, cadet
$i^{n+1}\in I^{+}(s^{*})$ regardless of her submitted strategy, and therefore,

$\varphi^{2020}_{i^{n+1}}(s^{*})=(b,t^{0}),$ (38)

since her best response $s^{*}_{i^{n+1}}$ to $s^{*}_{-i^{n+1}}$ results in an
assignment of $(b,t^{0})$. Moreover, Lemma 1 and relation (38) imply that, for
any cadet $i\in
I^{0}\cup\big{(}I^{1}\setminus\\{i^{1},\ldots,i^{n+1}\\}\big{)}$,

$\varphi^{2020}_{i}(s^{*})=(b,t^{0}).$ (39)

Hence Lemma 2 and relations (38), (39) imply
$\varphi^{2020}(s^{*})=\phi^{BR}(\succ)$.

Finally, the strategy profile $s^{\prime}$ where $s^{\prime}_{i}=b$ for any
cadet $i\in J^{n}$ and $s^{\prime}_{j}=\emptyset$ for any cadet $j\in
I\setminus J^{n}$ is a Nash equilibrium, with an outcome
$\varphi^{2020}(s^{\prime})=\phi^{BR}(\succ)$, showing that there exists a
Nash equilibrium completing the proof for Case 2. $\blacksquare$

Case 3. $n=q^{+}_{b}$

Since at most $q^{+}_{b}$ positions can be assigned at the increased cost
$t^{+}$, Lemma 1 and Lemma 2 immediately imply
$\varphi^{2020}(s^{*})=\phi^{BR}(\succ)$.

Finally the strategy profile $s^{\prime}$ where $s^{\prime}_{i}=b$ for any
cadet $i\in J^{q^{+}_{b}}\cup I^{0}$ and $s^{\prime}_{j}=\emptyset$ for any
cadet $j\in I\setminus\big{(}J^{n}\cup I^{0}\big{)}$ is a Nash equilibrium,
with an outcome $\varphi^{2020}(s^{\prime})=\phi^{BR}(\succ)$, showing that
there exists a Nash equilibrium completing the proof for Case 3, and the proof
of the proposition. $\blacksquare$ ∎

Proof of Theorem 2: Fix $(\pi_{b})_{b\in B}\in\Pi^{|B|}$ and
$(\omega^{+}_{b})_{b\in B}\in\prod_{b\in B}\Omega^{+}_{b}$.

For any cadet $i\in I$, branch $b$, and preference
$\succ_{i}\;\in\mathcal{Q}$, by assumption we have
$(b,t^{0})\succ_{i}(b,t^{+})$. Therefore, since cadet proposals to branches
follow their submitted preferences, the set of contracts available to any
branch at any stage of the cumulative offer process is viable. That is,
whenever the increased cost contact $(i,b,t^{+})$ of a cadet $i\in I$ is
available for a branch $b\in B$, so is her base cost contract $(i,b,t^{0})$.

We first show that the mechanism $\phi^{COM-BR}$ satisfies the five axioms.
For the proofs of individual rationality, non-wastefulness, lack of priority
reversals, and enforcement of BRADSO policy, fix
$\succ\,\in\mathcal{Q}^{|I|}$.

Individual rationality: No cadet $i\in I$ ever makes a proposal to a branch
$b$ at the increased cost $t^{+}$ under the cumulative offer process, unless
her preferences are such that $(b,t^{+})\succ_{i}\emptyset$. Hence the
mechanism $\phi^{COM-BR}$ satisfies individual rationality.

Non-wastefulness: For any branch $b\in B$, unless there are already $q$
contracts with distinct cadets on hold, it is not possible for all contracts
of any given cadet to be rejected at any stage of the cumulative offer process
under the choice rule $\mathcal{C}^{BR}_{b}$. Hence the mechanism $\phi^{COM-
BR}$ satisfies non-wastefulness.

Lack of priority reversals: Suppose that $\phi_{j}^{COM-
BR}(\succ)\succ_{i}\phi_{i}^{COM-BR}(\succ)$ for a pair of cadets $i,j\in I.$
Since the mechanism $\phi^{COM-BR}$ is individually rational, $\phi_{j}^{COM-
BR}(\succ)\not=\emptyset$. Let branch $b\in B$ and cost
$t\in\\{t^{0},t^{+}\\}$ be such that $\phi_{j}^{COM-BR}(\succ)=(b,t)$. Let $k$
be the final step of the cumulative offer process. Since $\phi_{j}^{COM-
BR}(\succ)\succ_{i}\phi_{i}^{COM-BR}(\succ)$, cadet $i$ has proposed the
contract $(i,b,t)$ to branch $b$ at some step of the cumulative offer process,
which is rejected by branch $b$ (strictly speaking for the first time) either
immediately or at a later step. Since the proposed contracts remain available
until the termination of the procedure under the cumulative offer
process,343434It is this feature of the cumulative offer process that is
emphasized in its name. the contract $(i,b,t)$ is also rejected by branch $b$
at the final Step $k$ of the cumulative offer process. In contrast, since
$\phi_{j}^{COM-BR}(\succ)=(b,t)$, contract $(j,b,t)$ is chosen by branch $b$
at the final step $k$ of the cumulative offer process. If the contract
$(j,b,t)$ is accepted as one of the first $q^{0}_{b}$ positions under the
choice rule $\mathcal{C}^{BR}_{b}$, then $(j,b,t)\;\omega^{0}_{b}\;(i,b,t)$.
Otherwise, if the contract $(j,b,t)$ is accepted as one of the last
$q^{+}_{b}$ positions under the choice rule $\mathcal{C}^{BR}_{b}$, then
$(j,b,t)\;\omega^{+}_{b}\;(i,b,t)$. In either case we have $j\;\pi_{b}\;i$,
proving that the mechanism $\phi^{COM-BR}$ has no priority reversals.

Enforcement of the BRADSO policy: First suppose that cadets $i,j\in I$ are
such that $\phi^{COM-BR}_{i}(\succ)=(b,t^{+})$ and
$(b,t^{0})\succ_{j}\phi^{COM-BR}_{j}(\succ)$. The relation
$(b,t^{0})\succ_{j}\phi^{COM-BR}_{j}(\succ)$ implies that cadet $j$ has
proposed the contract $(j,b,t^{0})$ to the branch $b$ at some step of the
cumulative offer process, which is rejected by branch $b$ either immediately
or at a later step. Let $k$ be the final step of the cumulative offer process.
Since the proposed contracts remain available until the termination of the
procedure under the cumulative offer process, the contract $(j,b,t^{0})$ is
also rejected by branch $b$ at the final Step $k$ of the cumulative offer
process. More specifically, it is rejected by the choice rule
$\mathcal{C}^{BR}_{b}$ at the final Step $k$ both for the first $q^{0}_{b}$
positions using the native priority order $\omega^{0}_{b}$ and for the last
$q^{+}_{b}$ positions using the BRADSO policy $w^{+}_{b}$. In contrast,
contract $(i,b,t)$ is chosen by branch $b$ at the final Step $k$ of the
cumulative offer process using the BRADSO policy $w^{+}_{b}$. Therefore,

$\left.\begin{array}[]{l}\phi^{COM-BR}_{i}(\succ)=(b,t^{+}),\;\mbox{ and}\\\
(b,t^{0})\succ_{j}\phi^{COM-
BR}_{j}(\succ)\end{array}\right\\}\quad\implies\quad(i,t^{+})\;\omega_{b}^{+}\;(j,t^{0}).$
(40)

Next suppose that cadets $i,j\in I$ are such that $\phi^{COM-
BR}_{j}(\succ)=(b,t^{0})$, $(b,t^{+})\succ_{i}\phi^{COM-BR}_{i}(\succ)$,
$(i,t^{+})\;\omega_{b}^{+}\;(j,t^{0})$, and moreover, let cadet $j$ be the
lowest $\pi_{b}$-priority cadet with an assignment of $\phi^{COM-
BR}_{j}(\succ)=(b,t^{0})$. The relation $(b,t^{+})\succ_{i}\phi^{COM-
BR}_{i}(\succ)$ implies that cadet $i$ has proposed the contract $(j,b,t^{+})$
to the branch $b$ at some step of the cumulative offer process, which is
rejected by branch $b$ either immediately or at a later step. Let $k$ be the
final step of the cumulative offer process. Since the proposed contracts
remain available until the termination of the procedure under the cumulative
offer process, the contract $(j,b,t^{+})$ is also rejected by branch $b$ at
the final Step $k$ of the cumulative offer process. More specifically, it is
rejected by the choice rule $\mathcal{C}^{BR}_{b}$ at the final Step $k$ even
for the last $q^{+}_{b}$ positions using the BRADSO policy $w^{+}_{b}$.
Therefore, since by assumption we have $(i,t^{+})\;\omega_{b}^{+}\;(j,t^{0})$,
cadet $j$ must have received one of the first $q^{0}$ positions using the
native priority order $\omega^{0}_{b}$. But since cadet $j$ is the lowest
$\pi_{b}$-priority cadet with an assignment of $\phi^{COM-
BR}_{j}(\succ)=(b,t^{0})$, that means no cadet has received any of the last
$q^{+}_{b}$ positions at the base cost of $t^{0}$. Therefore, since
$\phi^{COM-BR}$ satisfies non-wastefulness,

$\left.\begin{array}[]{l}\phi^{COM-BR}_{j}(\succ)=(b,t^{0}),\\\
(b,t^{+})\succ_{i}\phi^{COM-BR}_{i}(\succ),\;\mbox{ and}\\\
(i,t^{+})\;\omega_{b}^{+}\;(j,t^{0})\end{array}\right\\}\quad\implies\quad\big{|}\big{\\{}i^{\prime}\in
I:\phi^{COM-BR}_{i^{\prime}}(\succ)=(b,t^{+})\big{\\}}\big{|}=q^{+}_{b}.\\\ $
(41)

Relations (40) and (41) imply that mechanism $\phi^{COM-BR}$ satisfies
enforcement of the BRADSO policy.

Strategy-proofness: Our model is a special case of matching problems with
slot-specific priorities by Kominers and Sönmez (2016). Hence strategy-
poofness of the mechanism $\phi^{COM-BR}$ is a direct corollary of their
Theorem 3, which proves strategy-proofness of the cumulative offer mechanism
more broadly for matching problems with slot-specific priorities.

Uniqueness: We prove uniqueness via two lemmata.

######  Lemma 3.

Let $X,Y\in\mathcal{A}$ be two distinct allocations that satisfy individual
rationality, non-wastefulness, enforcement of BRADSO policy, and have no
priority reversals. Then there exists a cadet $i\in I$ who receives non-empty
and distinct assignments under $X$ and $Y$.

Proof of Lemma 3: The proof is by contradiction. Fix
$\succ\;\in\mathcal{Q}^{|I|}$. Let $X,Y\in\mathcal{A}$ be two distinct
allocations that satisfy individual rationality, non-wastefulness, enforcement
of BRADSO policy, and have no priority reversals. To derive the desired
contradiction, suppose that, for any cadet $i\in I$,

$X_{i}\not=Y_{i}\quad\implies\quad X_{i}=\emptyset\;\mbox{ or
}\;Y_{i}=\emptyset.$ (42)

Pick any branch $b\in B$ such that $X_{b}\not=Y_{b}$. Let $j\in I$ be the
highest $\pi_{b}$-priority cadet who is assigned to branch $b$ either under
$X$ or under $Y$ but not both. W.l.o.g., let cadet $j$ be assigned to branch
$b$ under allocation $X$ but not under allocation $Y$. By relation (42),

$Y_{j}=\emptyset.$

Since allocation $Y$ satisfies non-wastefulness, there exists a cadet $k\in I$
who is assigned to branch $b$ under allocation $Y$ but not under allocation
$X$. By relation (42),

$X_{k}=\emptyset,$

and therefore, by choice of cadet $j$, cadet $k$ has lower $\pi_{b}$-priority
than cadet $j$. Moreover, since allocation $Y$ has no priority reversals and
$Y_{j}=\emptyset$, we have

$Y_{k}=(b,t^{+}),$ (43)

and since allocation $Y$ satisfies (condition 1 of) the axiom enforcement of
BRADSO policy, we have

$(k,t^{+})\;\omega^{+}_{b}\;(j,t^{0}).$ (44)

Also relation (43) and individual rationality allocation $Y$ imply

$(b,t^{+})\,\succ_{k}\;\emptyset.$ (45)

Define

$I^{*}\equiv\\{i\in I\,:\,X_{i}=(b,t^{+})\\}.$

Since allocation $X$ satisfies (condition 2 of) the axiom enforcement of
BRADSO policy, the assumption $X_{j}\in\\{(b,t^{0}),(b,t^{+})\\}$ and relation
(44) imply

$|I^{*}|=q^{+}_{b},$ (46)

and since allocation $X$ has no priority reversals and $X_{k}=\emptyset$, for
any $i\in I^{*}$,

$i\;\pi_{b}\;k.$ (47)

But since $Y_{k}=(b,t^{+})$ by relation (43) and $|I^{*}|=q^{+}_{b}$ by
relation (46), there exists a cadet $\ell\in I^{*}$ with
$Y_{\ell}\not=(b,t^{+})=X_{\ell}$, and therefore by relation (42) we have,

$Y_{\ell}=\emptyset.$ (48)

Since $X$ satisfies individual rationality and $\ell\in I^{*}$, we have

$(b,t^{+})\;\succ_{\ell}\;\emptyset,$

and therefore relations (43), (47), and (48) imply allocation $Y$ has priority
reversals, giving us the desired contradiction and completing the proof of
Lemma (3).  $\diamondsuit$

######  Lemma 4.

There can be at most one direct mechanism that satisfies individual
rationality, non-wastefulness, enforcement of BRADSO policy, strategy-
proofness, and has no priority reversals.

Proof of Lemma 4: The proof of the lemma is inspired by a technique introduced
by Hirata and Kasuya (2017). Towards a contradiction, suppose there exists two
distinct direct mechanisms $\varphi$ and $\psi$ that satisfy individual
rationality, non-wastefulness, enforcement of BRADSO policy, strategy-
proofness, and have no priority reversals. Let the preference profile
$\succ^{*}\in\mathcal{Q}^{|I|}$ be such that,

  1. 1.

$\varphi(\succ^{*})\not=\psi(\succ^{*})$, and

  2. 2.

the aggregate number of acceptable contracts between all cadets is minimized
among all preference profiles $\widetilde{\succ}\in\mathcal{Q}^{|I|}$ such
that $\varphi(\widetilde{\succ})\not=\psi(\widetilde{\succ})$.

Let $X=\varphi(\succ^{*})$ and $Y=\psi(\succ^{*})$. By Lemma 3, there exists a
cadet $i\in I$ such that

  1. 1.

$X_{i}\not=\emptyset$,

  2. 2.

$Y_{i}\not=\emptyset$, and

  3. 3.

$X_{i}\not=Y_{i}$.

Since both allocations $X$ and $Y$ satisfy individual rationality,

$X_{i}\;\succ^{*}_{i}\;\emptyset\quad\mbox{ and
}\;Y_{i}\;\succ^{*}_{i}\;\emptyset.$

W.l.o.g., assume

$X_{i}\;\succ^{*}_{i}\;Y_{i}\;\succ^{*}_{i}\;\emptyset.$

Construct the preference relation $\succ^{\prime}_{i}\in\mathcal{Q}$ as
follows:

If $X_{i}=(b,t^{0})$ for some $b\in B$, then

$(b,t^{0})\;\succ^{\prime}_{i}\;\emptyset\;\succ^{\prime}_{i}\;(b^{\prime},t^{\prime})\qquad\mbox{
for any }(b^{\prime},t^{\prime})\in B\times T\setminus\\{(b,t^{0})\\}.$

Otherwise, if $X_{i}=(b,t^{+})$ for some $b\in B$, then

$(b,t^{0})\;\succ^{\prime}_{i}\;(b,t^{+})\;\succ^{\prime}_{i}\;\emptyset\;\succ^{\prime}_{i}\;(b^{\prime},t^{\prime})\qquad\mbox{
for any }(b^{\prime},t^{\prime})\in B\times
T\setminus\\{(b,t^{0}),(b,t^{+})\\}.$

Since $X_{i}\succ^{*}_{i}Y_{i}\,\succ^{*}_{i}\emptyset$ and
$(b,t^{0})\succ^{*}_{i}(b,t^{+})$, the preference relation
$\succ^{\prime}_{i}$ has strictly fewer acceptable contracts for cadet $i$
than the preference relation $\succ^{*}_{i}$.

By strategy-proofness of the mechanism $\psi$, we have

$\underbrace{\psi_{i}(\succ^{*}_{i},\succ^{*}_{-i})}_{=Y_{i}}\;\succeq^{*}_{i}\;\psi_{i}(\succ^{\prime}_{i},\succ^{*}_{-i}),$

and since no branch-cost pair $(b^{\prime},t^{\prime})\in B\times T$ with
$Y_{i}\succ^{\prime}_{i}(b^{\prime},t^{\prime})$ is acceptable under
$\succ^{\prime}_{i}$, by individual rationality of the mechanism $\psi$ we
have

$\psi_{i}(\succ^{\prime}_{i},\succ^{*}_{-i})=\emptyset.$ (49)

Similarly, by strategy-proofness of the mechanism $\varphi$, we have

$\varphi_{i}(\succ^{\prime}_{i},\succ^{*}_{-i})\;\succeq^{\prime}_{i}\;\underbrace{\varphi_{i}(\succ^{*}_{i},\succ^{*}_{-i})}_{=X_{i}},$

which in turn implies

$\varphi_{i}(\succ^{\prime}_{i},\succ^{*}_{-i})\not=\emptyset.$ (50)

But then, by relations (49) and (50) we have

$\varphi(\succ^{\prime}_{i},\succ^{*}_{-i})\not=\psi(\succ^{\prime}_{i},\succ^{*}_{-i}),$

giving us the desired contradiction, since between all cadets the preference
profile $(\succ^{\prime}_{i},\succ^{*}_{-i})$ has strictly fewer acceptable
contracts than the preference profile $\succ^{*}$. This completes the proof of
Lemma 4.  $\diamondsuit$

Since we have already shown that $\phi^{COM-BR}$ satisfies all five axioms,
Lemma 4 establishes the uniqueness, concluding the proof of Theorem 2. ∎

Proof of Corollary 1: Since BRADSO-IC is implied by strategy-proofness,
Corollary 1 is a direct implication of Theorems 1 and 2. ∎

Table 1: Branches and Applications for Classes of 2020 and 2021

Notes. This table reports information on branches for the Class of 2020 and
2021. Number Assigned equals the capacity of the branch. Ranked First is the
number of cadets ranking the branch as their highest rank choice. BRADSO
Willing is the number of cadets who rank a BRADSO contract at the branch
anywhere on their rank order list. Explosive Ordnance Disposal was not a
branch option for the Class of 2020.

Figure 1: Comparison of Outcomes of the USMA-2006 and USMA-2020 Mechanisms

Notes. This figure reports Strategic BRADSOs, BRADSO-IC Failures, and
Detectable Priority Reversals under the USMA-2006 and USMA-2020 Mechanisms.
The leftmost three columns correspond to outcomes under the USMA-2006
Mechanism averaging over six classes who participated in the mechanism from
2014-2019. The rightmost three columns correspond to outcomes under USMA-2020
Mechanism for the Class of 2020. A cadet has a Strategic BRADSO if she is
assigned a branch at base cost and would have still received that position at
base cost if she did not indicate a willingness to BRADSO at the branch. A
cadet has a BRADSO-IC failure if she is assigned a branch at increased cost,
but would receive that branch at base cost if she did not indicate a
willingness to BRADSO at the branch. A cadet has an detectable Priority
Reversal if she is assigned a branch at base cost and another cadet with
higher priority either receives that branch with BRADSO or is assigned a
strictly less preferred branch.

Figure 2: BRADSO Ranking Relative to Non-BRADSO Ranking by Class of 2021

Notes. This figure reports where in the preference list a branch is ranked
with BRADSO relative to where it is ranked without BRADSO. A value of 1 (2 or
3) indicates that the branch is ranked with BRADSO immediately after (two
places or three places after, respectively) the branch is ranked at base cost.
4+ means that the a branch is ranked with BRADSO four or more choices after
the branch is ranked at base cost.

Figure 3: USMA-2006 and USMA-2020 Mechanism Performance under Truthful
Strategies Simulated from Preference Data from Class of 2021

Notes. USMA used the strategy-proof COM-BRADSO mechanism for the Class of
2021. This figure uses data from the Class of 2021 on cadet preferences,
branch priorities, and branch capacities to simulate the outcomes of the
mechanisms USMA-2006 and USMA-2020. Since the strategy spaces of the
mechanisms USMA-2006 and USMA-2020 differ from that of the mechanism COM-
BRADSO, cadet strategies that correspond to truthful branch-preferences and
BRADSO willingness are are simulated from cadet preferences over branch-cost
pairs under the COM-BRADSO mechanism. Truthful strategies for the mechanisms
USMA-2006 and USMA-2020 are constructed from Class of 2021 preferences by
assuming that a preference indicating willingness to BRADSO at a branch means
the cadet’s strategy under the USMA-2006 and USMA-2020 mechanisms has her
willing to BRADSO. Strategic BRADSOs, BRADSO-IC Failures, and Detectable
Priority Reversals are defined in  Figure 1. To compute Priority Reversals, we
compare a cadet’s outcome in the USMA-2006 or USMA-2020 mechanism to a cadet’s
preference submitted under the COM-BRADSO mechanism. If a cadet prefers a
higher ranked choice and has higher priority over a cadet who is assigned that
choice, then the cadet is part of a Priority Reversal.

Figure 4: USMA-2020 Mechanism Performance Under Indicative and Final
Strategies

Notes. This figure reports on the number of Strategic BRADSOs, BRADSO-IC
failures, Detectable Priority Reversals, and Priority Reversals under
indicative strategies submitted in a dry-run of the USMA-2020 mechanism and
final strategies of the USMA-2020 mechanism for the Class of 2020.

Figure 5: USMA-2020 Mechanism Performance under Truthful Strategies Simulated
from Indicative and Final Preference Data from Class of 2021

Notes. USMA used the strategy-proof COM-BRADSO mechanism for the Class of
2021. This figure uses data from the indicative and final rounds from the
Class of 2021 on cadet preferences, branch priorities, and branch capacities
to simulate the outcome of the USMA-2020 mechanism. Since the strategy space
of the mechanism USMA-2020 differs from that of the mechanism COM-BRADSO,
cadet strategies that correspond to truthful branch-preferences and BRADSO
willingness are are simulated from cadet preferences over branch-cost pairs
under the COM-BRADSO mechanism. Truthful strategies are constructed from Class
of 2021 preferences by assuming that a preference indicating willingness to
BRADSO at a branch means the cadet’s strategy under the USMA-2006 and
USMA-2020 mechanisms has her willing to BRADSO. USMA-2020 (Indicative) reports
outcomes using strategies constructed from preferences submitted in the dry-
run of COM-BRADSO. USMA-2020 (Final) reports outcomes using strategies
constructed from preferences submitted in the final run of COM-BRADSO.

Figure 6: Number of BRADSOs Charged Across BRADSO Policies and Cap Sizes

Notes. This figure reports on the number of BRADSOs charged for three BRADSO
policies: Ultimate BRADSO, BRADSO-2020, and BRADSO-2021 using data from the
Class of 2021. The BRADSO cap ranges from 5% to 75% of slots at each branch.
Each outcome is computed by running COM-BRADSO given stated cadet preferences
under different BRADSO policies and cap sizes.

##  Appendix B Online Appendix: Supplementary Material

###  B.1 Individual-Proposing Deferred Acceptance

The USMA-2020 mechanism was based on the individual-proposing deferred
acceptance algorithm (Gale and Shapley, 1962). Given a ranking over branches,
the individual-proposing deferred acceptance algorithm (DA) produces a
matching as follows.

> Individual-Proposing Deferred Acceptance Algorithm ($\mathbf{DA}$)
>
> Step 1: Each cadet applies to her most preferred branch. Each branch $b$
> tentatively assigns applicants with the highest priority until all cadets
> are chosen or all $q_{b}$ slots as assigned and permanently rejects the
> rest. If there are no rejections, then stop.
>
> Step k: Each cadet who was rejected in Step k-1 applies to her next
> preferred branch, if such a branch exists. Branch $b$ tentatively assigns
> cadets with the highest priority until all all cadets are chosen or all
> $q_{b}$ slots are assigned and permanently rejects the rest. If there are no
> rejections, then stop.
>
> The algorithm terminates when there are no rejections, at which point all
> tentative assignments are finalized.

###  B.2 Cadet Survey Questions and Answers

In fall 2020, the Army administered a survey of cadets. This survey asked two
questions related to assignment mechanisms, one on cadet understanding of
USMA-2020 and the other on cadet preferences over assignment mechanisms. This
section reports the questions and the distribution of survey responses.

Question 1. What response below best describes your understanding of the
impact of volunteering to BRADSO for a branch in this year’s branching
process?

  * A.

I am more likely to receive the branch, but I am only charged a BRADSO if I
would have failed to receive the branch had I not volunteered to BRADSO.
(43.3% of respondents)

  * B.

I am charged a BRADSO if I receive the branch, regardless of whether
volunteering to BRADSO helped me receive the branch or not. (9.5% of
respondents)

  * C.

I am more likely to receive the branch, but I may not be charged a BRADSO if
many cadets who receive the same branch not only rank below me but also
volunteer to BRADSO. (38.8% of respondents)

  * D.

I am more likely to receive the branch, but I do not know how the Army
determines who is charged a BRADSO. (6.7% of respondents)

  * E.

I am NOT more likely to receive the branch even though I volunteered to
BRADSO. (1.8 percent of respondents)

38.8% of cadets answered the correct answer (answer C). 43.3% of cadets
believed that the 2020 mechanism would only charge a BRADSO if required to
receive the branch (answer A)

Question 2. A cadet who is charged a BRADSO is required to serve an additional
3 years on Active Duty. Under the current mechanism, cadets must rank order
all 17 branches and indicate if they are willing to BRADSO for each branch
choice. For example:

  * •

Current Mechanism Example:

    * –

1: AV/BRADSO, 2: EN, 3: CY

Under an alternative mechanism, cadets could indicate if they prefer to
receive their second branch choice without a BRADSO charge more than they
prefer to receive their first branch choice with a BRADSO charge. For example:

  * •

Alternative Mechanism Example:

    * –

1: AV, 2: EN, 3: AV/BRADSO, 4: CY

When submitting branch preferences, which mechanism would you prefer?

  * •

A. Current Mechanism (21.4% of respondents)

  * •

B. Alternative Mechanism (49.7% of respondents)

  * •

C. Indifferent (24.2% of respondents)

  * •

D. Do Not Understand (4.8% of respondents)

Table B.1: Mechanism Replication Rate

Notes. This table reports the replication rate of the USMA assignment
mechanism across years. The USMA-2006 mechanism is used for the Classes of
2014-2019, USMA-2020 mechanism is used for the Class of 2020, and the COM-
BRADSO mechanism is used for the Class of of 2021. Number incorrect are the
number of cadets who obtain a different assignment under our replication.
Branch percent correct is the number of branch assignments that we replicate.
BRADSO percent correct is the number of BRADSO assignments we replicate.