$\mu^{-}+N(A,Z)\rightarrow\gamma^{(*)}+\nu_{\mu}+N(A,Z-1)$, is an intrisic
background at Mu2e and Mu2e-II. It is similar to RPC, where the internal pair
production spectrum used for RPC [136] is also applicable to describe internal
pair production for nuclear RMC, as discussed in Ref. [89]. RMC measurements
historically focused on the total rate, while the spectrum endpoint region is
less well understood. Most studies of nuclear RMC rely on approximations, such
as the closure approximation, which are unreliable near this endpoint [140,
141]. In order to understand any potential signals and describe the background
electron and positron spectra at Mu2e-II it is important understand the RMC
(on- and off-shell) photon spectrum in the high energy region. This is
especially important for the $\mu^{-}N\to e^{+}N^{\prime}$ search, as the
maximum kinematically allowed photon energy is $\sim$10 MeV above the
conversion positron signal region. For aluminum, the maximum allowed electron
energy from an RMC interaction is $\sim$2.5 MeV below the signal region, and
so the background contribution in this region for the $\mu^{-}\to e^{-}$
search is negligible, as discussed in Ref. [142]. A measurement of the on- and
off-shell RMC photon spectrum on alumimum (and potentially other medium heavy
nuclei) is necessary to understand the electron and positron background
contributions from RMC. Data from Mu2e can help understand these aspects of
RMC in aluminum.

###  XI.5 Beam Backgrounds

Beam-related background contributions arise in our detectors when beam
particles do not stop in the stopping target. These particles originate from
the initial protons arriving at the production target in between the proton
pulses and are suppressed by the proton beam extinction. Beam related
backgrounds are expected to be very small:

  * •

Beam electrons result from muons decaying in the beamline. If they reach the
stopping target with a momentum close to 105 MeV/c they could fake a signal.
As there is only a small probability of a large angle scattering in the
stopping target and the beam extinction factor is small ($10^{-11}$) the
expected contribution from beam electrons is $<0.004$ events.

  * •

Decay in Flight of muons and pions in the DS can produce electrons with the
momentum above 100 MeV/c which could get reconstructed without scattering in
the stopping target. The estimated contribution from decays in flight is
expected to be very small, at $<0.007$ events.

###  XI.6 Stopping target studies

The muon stopping target provides the nuclear target for the $\mu^{-}\to
e^{-}$ conversion process to take place; it also provides the normalization
signal in the form of the muon capture process as well as contributes to the
signal electron energy losses before reaching the detectors. The choice of
target design can impact momentum resolution through energy straggling and
multiple scattering of the electron as it leaves. Consequently, the muon
stopping target can significantly affect the achievable sensitivity of the
overall Mu2e-II experiment. Its design is crucial to achieving the physics
goal.

In Mu2e the stopping target consists of 37 aluminum foils, 105 $\mu$m thick
and 75 mm in radius, placed 22.22 mm apart and suspended in a frame. Each foil
has a 21.5 mm radius hole in the center to help reduce the effects of beam
flash.

In this section preliminary studies are presented which look at optimization
of the Mu2e-II muon stopping target design in terms of improving the single
event sensitivity (SES) and achievable branching fraction upper limit
(assuming no signal). Further R$\&$D will be required to finalize the Mu2e-II
stopping target.

####  XI.6.1 Single Event Sensitivity Equation

The single event sensitivity (SES) is the signal rate corresponding to an
expectation of 1 event over the course of the experiment. The SES can be
written as:

$SES=\frac{1}{POT\times\frac{stops}{POT}\times\frac{capture}{stop}\times\frac{N_{CE}^{reco}}{N_{CE}^{gen}}}$
(11)

where:

  * •

$POT$ \- number of protons at the production target;

  * •

$\frac{stops}{POT}$ \- number of stopped muons at the stopping target per
number of simulated protons at the production target;

  * •

$\frac{captures}{stop}$ \- capture rate of the target nuclei, e.g. for
aluminum this is 61 $\%$;

  * •

$\frac{N_{CE}^{rec}}{N_{CE}^{gen}}$ \- the reconstruction efficiency.

Fig. 33 shows three effects and how their relative importance changes as we
increase the target mass:

  * •

Number of stopped muons - the more massive a target, the more muons we expect
will be stopped;

  * •

Multiple Scattering and energy loss of the outgoing electron - the more
massive the target the more scattering it experiences as it attempts to leave;

  * •

DIO background - the effects of the DIO background increase with mass due to
the broadening of the signal energy spectrum after energy losses.

The studies aim to find a “sweet spot” in the target design where the number
of stopped muons is large but where multiple scattering and energy losses do
not overwhelm the conversion electron search.

Figure 33: Relative changes in three processes as function of target mass.
Here the green indicates a improvement in sensitivity and red a worsening
sensitivity.

####  XI.6.2 Design Features

There are a number of design features of the stopping target which may require
optimization specifically for the Mu2e-II beam:

  * •

Target Material - see Sec. II.3. Depending on the results of Mu2e an
alternative material, which complements aluminum may be used to help elucidate
physics signal;

  * •

Target Mass - the choice of mass is expected to be a compromise. Larger target
mass means higher stopped muon rate, but also increased scattering of the
outgoing electron;

  * •

Target Geometry - for the same mass the physical design of the target can
affect the amount of material the outgoing electron traverses, and both impact
the SES and increase the background due to the broadening of the signal
distribution;

  * •

Incoming Muon Momentum Structure - only around 37$\%$ of muons which enter the
DS are stopped in the target. These are typically the lower momenta particles.
If the incoming muon beam momenta distribution were altered, this could affect
the momentum space of the muons stopped, and thus affect the yield.

####  XI.6.3 Alternative Materials

As discussed in Sec. II.3, in the event that Mu2e has measured muon-to-
electron conversion in Al, Mu2e-II should use an alternative target material.
Measuring the relative conversion rates in two complementary materials can
help elucidate the underlying physics responsible for the conversion [65, 72].
A preliminary study looked at titanium, vanadium and lithium foil style
targets and attempted to measure achievable SES in each. High-Z materials such
as gold have the advantage of larger capture rates, but the muonic atom
lifetime is too short, and this makes the delayed live-gate highly
inefficient, thus beam-related backgrounds become an issue.

Theoretical work has suggested a low-Z material, such as lithium, could
complement aluminium [143, 72]. From a practical point of view, a foil
lithium-based target is feasible. Lithium has been used in foil form by a
previous experiment; Ref. [144] details the practicalities of fabricating the
foils. From a physics perspective, the muonic lifetime in lithium is close to
the free muon lifetime and the capture rate is small at just a few percent
(see Tab. 2). Consequently, to reach the same upper limit as aluminum would
require a much longer running time. So, choosing a lithium target would depend
what conversion rate had been observed in aluminum; if a rate of
$\mathcal{O}(10^{-14}-10^{-16})$ had been measured at Mu2e or COMET, lithium
could indeed be a good alternative material in Mu2e-II.

Another practicality is that, due to lithium’s much lower density ($0.523$
g/cm3), a very large number of foils would be required to get the same number
of stopped muons as in aluminum. Since there is limited space in the detector
solenoid, there is a limit to how big the target can be in terms of horizontal
length. The dimensions of the foils were kept the same as those for the
aluminium with the separation between foils being adjusted accordingly as more
foils are added. The minimum SES from those geometries simulated came from the
$\sim$ 400 foil design which gave an SES of $\mathcal{O}(10^{-17})$.

Titanium has often been tipped as one of the best alternatives to aluminium.
Titanium has a larger capture rate at $\sim$ 85% but a shorter lifetime of
$\sim 329$ ns (see Tab. 2); however, natural titanium has a mixed composition
of isotopes. Ref. [145] discusses the difference between potential spin-
dependent and spin-independent contributions. Knowing the spin composition of
the target might be useful, meaning a pure titanium target of a specific
isotope would be necessary. Obtaining pure samples of titanium is possible,
but to get $\mathcal{O}(100g)$ of a given isotope would be difficult and
costly. Vanadium has an atomic number close to titanium and thus offers the
same benefits but has just one main isotope. A titanium and a vanadium target
was studied using the same foil design as used for the aluminum target. The
optimum mass (with the lowest SES) was found for both elements to be $\sim$
160 g. The SESs with this mass were found to be slightly better than that
achievable with aluminum due to the larger capture rate and were
$\mathcal{O}(10^{-18})$.

Further study is ongoing and a subsequent publication is expected which will
detail the studies described here.

####  XI.6.4 Alternative Geometries

In this study the mass of the aluminium target was kept at a constant value of
171g. This is the approximate mass of the current Mu2e target. Several
alternative geometries were studied:

  * •

Screen Thick Mesh (SD) - circular screen made of mesh wires of 0.1143 mm
thick. Fig. 34 shows an example screen design.

  * •

Screen Thick Mesh with Hole (SDH) - same as the SD design, but with a hole of
21.5 mm radius.

  * •

Screen Fine Mesh (SM) - similar to the SD design but with 0.02665 mm thick
wires

  * •

Screen Fine Mesh with Hole (SMH) - the same as the SM design but with a 21.5
mm hole in the center.

Figure 34: The screen designs are constructed from arrays of thin wires, with
two commercially available wire sizes. The designs consist of grids of
aluminum wires arranged in disks.

  * •

Hexagonal Mesh (HM) - array of cylinders in hexagonal cross-section in XY.

Figure 35: Design with hexagonal cross section, design comprises of screen
based cylinders arranged in hexagonal shape.

  * •

Cylinder Thick Mesh (CTM) - cylinder but made of mesh of 0.1143 mm thick
strings.

  * •

Cylinder Fine Mesh (CM) - cylinder but made of mesh of 0.02665 mm thick
strings.

(a)

(b)

Figure 36: Longitudinal (top) and transverse (bottom) view of cylindrical
design made of strings of 0.02665 mm arranged in a mesh.

Table 6: Relative changes (as fraction of what can be achieved with the Mu2e target design) in SES and median upper limit on $R_{\mu e}$ at 90% CL ($R_{\mu e}$(90% CL)) (BFUL) for alternative geometries with respect to the Mu2e era geometry. Design Name | SES | BFUL (90$\%$ CL)
---|---|---
Mu2e Default Design | 1 | 1
CTM | 1.03 | -
HM | 1.06 | -
SD | 0.93 | 0.96
SDH | 0.95 | 0.96
SM | 0.99 | 1.0
SMH | 1.03 | 1.02

Table 6 lists the ratio of SES (relative to the default) for each alternative
design. The screen designs are shown to improve the SES by $\sim 5\%$ with the
thinner strings having the better performance. The shift in projected
branching fraction upper limit is calculated using a background only
hypothesis and the work of Feldman-Cousins [146] in a counting experiment. DIO
and RPC are the only background processes considered.

Since the improvements seen are minimal, in the remainder of this analysis the
37 foil, Mu2e target design is used.

###  XI.7 Experimental Assumptions

The simulation and reconstruction software used in this study was taken from
the framework developed for the Mu2e experiment, where the geometry was
updated for the Mu2e-II design. The Mu2e simulation framework is based on
Geant4 (see Appendix A.2). It takes into consideration cross sections and time
dependencies of the physics processes as well as the responses of the sub-
detectors, and effects of hit readout and digitization. The Mu2e software
environment is described in detail in Ref. [142]. The reconstruction
algorithms are also described in this reference. We do not update, or re-
optimize, these algorithms for the Mu2e-II environment, meaning our
reconstruction efficiency with pileup will likely be improved with further
study.

The geometry includes:

  * •

A carbon conveyor production target with 28 spheres, each 0.75 cm in radius;

  * •

No antiproton windows in the TS since the beam energy is too low to produce
antiprotons;

  * •

An aluminium stopping target with design identical to that used in Mu2e;

  * •

A straw tube tracker similar to the Mu2e tracker with thinner straws with 8
$\mu$m thick walls (compared to 15 $\mu$m in Mu2e) and reduced mass through
the removal of the gold layer in straws;

  * •

The crystal calorimeter has the same number of crystals as in Mu2e and they
have the same geometry ($34\times 34\times 200$ mm3). The major difference is
that they are made of barium fluoride instead of cesium iodide. The waveform
response of the SiPMs with the barium fluoride is an educated guess. Further
study is required to understand the actual response function.

The extinction factor (i.e., the ratio of out of time to in time protons) is
assumed to be $<1\times 10^{-11}$.

###  XI.8 Selection Criteria

The following selection cuts are applied in this study:

  * •

N(hits) $\geq 20$: Requires a sufficient number of hits along the track for
the momentum to be reconstructed well;

  * •

$|{\rm D}_{0}|<100$ mm: Requires the helical track to be consistent with the
75 mm in radius stopping target, where the 100 mm threshold accounts for
tracks with energy loses before reaching the tracker;

  * •

${\rm R_{max}}<680$ mm: Ensures tracks are well contained within the 700 mm
radius tracker;

  * •

$0.5<\tan({\rm dip})<1$: Requires the topology to be consistent with helical
tracks originating from the stopping target;

  * •

$\sigma_{\rm T_{0}}<0.9$ ns: Rejects tracks with high $\rm T_{0}$ uncertainty,
which are often tracks with the calorimeter cluster not included in the track
reconstruction;

  * •

$E_{\rm cluster}>10$ MeV and $E_{\rm cluster}/P_{\rm track}<1.05$: Requires a
calorimeter cluster associated with the track with a $E/P$ ratio consistent
with an electron;

  * •

Track quality ANN selection: Rejects poorly reconstructed tracks;

  * •

$T_{0}<1650$ ns: Avoid early flash from the next beam pulse.

###  XI.9 Efficiencies

For a pure conversion electron sampled (without pileup) the reconstruction
efficiency per simulated event is 36.7$\%$ and the selection efficiency using
the cuts defined in Sec. XI.8 is 72.7$\%$, meaning the total efficiency is
26.7$\%$. Current studies indicate the trigger efficiency at Mu2e for the
electron signal will be higher than 98%. We therefore assume a trigger
efficiency of 95% at Mu2e-II. The dominant effect of pileup on the simulated
track reconstruction is the worsening of the track momentum resolution due to
both background hits added along the reconstructed track trajectory and the
loss of true track hits due to tighter hit selection requirements. The
reconstruction efficiency decreases by 2.5% after the introduction of pileup
at Mu2e, where we assume a reconstruction efficiency drop of 5% at Mu2e-II.

###  XI.10 Tracker Resolution

The tracker momentum resolution is crucial to discriminating the conversion
signal from DIO background. The distribution of the difference between the
reconstructed momentum and the MC true momentum of the electron at the front
of the tracker is studied to understand the tracker reconstruction resolution.
Furthermore, the effect of energy losses before entering the tracker is also
important as this can broaden the signal momentum distribution, so the full
width at half maximum (FWHM) of the reconstructed momentum spectrum for the
conversion electron signal is another important metric for how well the signal
can be separated from the rapidly falling DIO background.

Figs. 37 and 38 show how the resolution and conversion electron momentum
distributions look before and after updating the tracker geometry to use
thinner straw walls. The thinner straw tracker has a better core resolution
than the Mu2e era tracker design. The FWHM of the reconstructed momentum
spectrum is not significantly improved due to the large contribution from the
energy losses in the stopping target and the inner proton absorber, which
protects the tracker from the significant flux of protons emitted in muon
captures in the stopping target.

Fig. 39 shows the comparison between the resolution of the tracker for the
Mu2e-II era tracker geometry (with 8 $\mu$m, lower mass straws) before and
after the selection cuts outlined in Sec. XI.8 were applied. The right-side
tail of the resolution distribution is the most dangerous, as it leads to
lower momentum DIO electrons entering the signal window due to a higher
reconstructed track momentum. The selection cuts, mostly the ANN based track
quality selection, significantly suppress these poorly reconstructed events.

Figs 40 and 41 show how the resolution and electron momentum distributions
changed with pileup. The impact on the resolution and FWHM is minor, with a
small degradation of both after pileup is introduced to the simulation. This
shows the robustness of the track reconstruction algorithms to changes in the
detector occupancies.

Figure 37:  Tracker resolution for Mu2e-II (in red) compared to the Mu2e (in
blue) era tracker simulation. The reconstructed track momentum at the front of
the tracker is compared to the MC true momentum at the front of the tracker,
without any selection cuts applied. The distributions are normalized to per
simulated conversion electron event.  Figure 38:  Reconstructed track momentum
for the conversion electron signal for Mu2e-II (in red) compared to the Mu2e
(in blue) era tracker simulation, without any selection cuts applied. The
distributions are normalized to per simulated conversion electron event.
Figure 39:  Tracker resolution for the conversion electron signal at Mu2e-II.
The reconstructed track momentum at the front of the tracker is compared to
the MC true momentum at the front of the tracker, before (in blue) and after
(in red) selection cuts are applied. The distributions are normalized to per
simulated conversion electron event.  Figure 40:  Tracker resolution for the
conversion electron signal at Mu2e-II before (in blue) and after (in red) the
introduction of pileup. The reconstructed track momentum at the front of the
tracker is compared to the MC true momentum at the front of the tracker, after
selection cuts are applied. The distributions are normalized to unity.  Figure
41:  Reconstructed track momentum for the conversion electron signal at
Mu2e-II, before (in blue) and after (in red) the introduction of pileup, after
selection cuts are applied. The distributions are normalized to unity.

###  XI.11 DIO background estimate

The DIO background drives the choice of the low edge of the signal window, as
the background rapidly increases with decreasing momentum. The DIO background
is sensitive to low probability tails in the tracker resolution, which can
lead to lower momentum DIO tracks entering the signal window. Due to the
computational requirements to generate a sufficiently large DIO dataset with
pileup and in the interest of time, we estimate this background using a toy MC
technique. A DIO electron dataset was generated without pileup, and the true
electron momentum at the tracker was convolved with the tracker resolution to
estimate the reconstructed DIO spectrum. The high momentum tail of this
reconstructed spectrum was fit with an exponential, which was used to estimate
the background for the signal window optimization. The final fit is shown in
Figure 42, where the convolved spectrum agrees well with the lower statistics
full MC simulation.

Figure 42:  The top plot compares the MC true DIO electron momentum to the
reconstructed momentum using the full MC simulation as well as to a
convolution of the true momentum at the tracker with the tracker resolution.
The right side tail of the convolution is fit with an exponential. The bottom
plot is a ratio of the full simulation over the expectation from the
convolution.

###  XI.12 Sensitivity Estimate and Justification

To estimate the Mu2e-II sensitivity to $\mu^{-}\rightarrow e^{-}$, the two-
dimensional time and momentum signal window is first optimized assuming the
two-dimensional expected background and signal distributions. The optimization
requires a metric to optimize, where the mean discovery potential of the
experiment is used, as was done in Ref. [142]. The mean discovery potential is
the value of $R_{\mu e}$ corresponding to an average deviation of $5\sigma$
from the background-only hypothesis. The discovery threshold is defined as at
least a $5\sigma$ deviation (p = $2.867\times 10^{-7}$) from the background-
only hypothesis, or $P(n\geq n_{\rm obs}|\mu_{\rm bkg})<2.867\times 10^{-7}$.
This is not as clear of a sensitivity value as the median discovery potential,
which is the minimum value of $R_{\mu e}$ such that at least 50% of identical
experiments would claim a discovery, but has the advantage of being less
sensitive to the discrete nature of the measurement and avoids known
pathologies of optimizations using median values.

The optimized signal window is $104.05<p<104.90$ MeV/c and $690<T_{0}<1650$
ns. For the 5-year Mu2e-II data taking run, the total expected background in
the signal window is 0.47 events. The SES in this window is $R_{\mu
e}=3.3\times 10^{-18}$, over five orders of magnitude below the current limit
from SINDRUM II, $R_{\mu e}<7\times 10^{-13}$ at 90% CL. For an expectation of
0.47 events, the median expected 90% CL upper limit on the mean expected
number of signal events using the Feldman-Cousins confidence belt construction
[146] is 1.97 events, or $R_{\mu e}<6.4\times 10^{-18}$. This is a five order
of magnitude improvement over the SINDRUM II limit and an order of magnitude
improvement on the sensitivity of the Mu2e experiment. For the Mu2e-II
background expectation of 0.47 events, a discovery requires an observation of
at least 8 events, and the median discovery potential is $R_{\mu e}=2.3\times
10^{-17}$.

###  XI.13 Summary table

Table 7 gives a break-down of the expected yields for each background process
for Mu2e (Runs I and II combined) and the Mu2e-II study presented here, with
the assumptions described above. In this table we assume 5 years of data
taking, equal to $6\times 10^{22}$ protons on target and $5.5\times 10^{18}$
stopped muons. The upper and lower edges of the momentum and time selection
windows are optimized using the mean discovery potential, as described
previously. The optimal window is determined to be $104.05<p<104.90$ MeV/c and
$690<T_{0}<1650$ ns.

Table 7: Summary of the expected background rates for the Mu2e experiment compared to the expected background rates for the Mu2e-II experiment. The Mu2e signal window is $103.85<p<104.90$ MeV/c and $700<t<1695$ ns and the Mu2e-II signal window optimized for the 5-year running plan is $104.05<p<104.90$ MeV/c and $690<t<1650$ ns. It is important to note that the Mu2e-II numbers are for the geometry described which includes the carbon ball production target and aluminum foil stopping target designs. Results | Mu2e | Mu2e-II (5-year)
---|---|---
Backgrounds |  |
DIO | $0.144$ | $0.263$
Cosmics | $0.209$ | $0.171$
RPC (in-time) | $0.009$ | $0.033$
RPC (out-of-time) | $0.016$ | $<0.0057$
RMC | $<0.004$ | $<0.02$
Antiprotons | $0.040$ | $0.000$
Decays in flight | $<0.004$ | $<0.011$
Beam electrons | $0.0002$ | $<0.006$
Total | 0.41 | 0.47
N(muon stops) | $6.7\times 10^{18}$ | $5.5\times 10^{19}$
SES | $3.01\times 10^{-17}$ | $3.25\times 10^{-18}$
$R_{\mu e}$(discovery) | $1.89\times 10^{-16}$ | $2.34\times 10^{-17}$
$R_{\mu e}$(90% CL) | $6.01\times 10^{-17}$ | $6.39\times 10^{-18}$

###  XI.14 Discussion and Future Improvements

We have presented a preliminary, but detailed, study to determine the
estimated sensitivity to neutrinoless muon to electron conversion in the
Mu2e-II experiment. We have assumed a carbon based, conveyor style production
target, and an aluminum, foil stopping target. The tracker mass has been
reduced by using thinner straws of just 8 $\mu$m thick walls. A barium
fluoride calorimeter replaces the Mu2e cesium iodide calorimeter. The study
took advantage of the sophisticated simulation and reconstruction software
devised for the Mu2e experiment, further optimization of the reconstruction is
required to deal with tracker occupancies seen at Mu2e-II. The estimated SES
is found to be $3.25\times 10^{-18}$ for a 5 year running period ($6\times
10^{22}$ POT and $5.5\times 10^{18}$ stopped muons). This is a factor of
$10^{5}$ below the current upper limit, and at least an order of magnitude
lower than the Mu2e projection. There is still a large amount of R&D and
software optimization to be done, and it is likely that this is a conservative
estimate.

Further optimization of the production and stopping targets could improve the
stopped muon yields.

A wedge absorber could be deployed in the transport solenoid to effectively
cool the incoming muons, pushing them to lower momentum, and increasing the
fraction which are stopped in the stopping target. Around 37$\%$ of muons
entering the DS are stopped in the target. These are the lower momentum muons
($<50$ MeV/c). One way to improve the achievable SES is to increase the number
of stopped muons. A low-Z disk absorber, made of lithium hydride (LiH) or
beryllium, within the TS or at the entrance to the DS could distort the
spectrum such that higher momentum muons are “pushed” into the $<50$ MeV/c
region and thus are stopped. Preliminary studies showed that although higher
momentum muons could be pushed to lower momenta, the effect was counteracted
by the lower momentum muons being lost to the new absorbers. At too low
thickness the effect on the incoming momenta is too small to make any
significant change in the number of stops and for anything $>0.1$ mm too many
lower momentum muons are lost, decreasing the overall number of stopped muons.

Further optimization and studies are required to understand how best to alter
the incoming momentum distribution to achieve the desired improvement in
number of stops. Wedge absorbers have been used in other muon experiments:
MICE [147] and the muon g - 2 experiment [148]. This design takes advantage of
the correlation in the muon position in the XY plane and its momentum after
traversing the TS. The wedge shape can be used to slow the higher momentum
muons, which pass through the thickest part, while the lowest momentum muons
pass through the thinnest region and thus experience little energy loss.

The momentum resolution could be improved through removal of the inner proton
absorber within the detector solenoid. The occupancies are highly impacted by
beam electrons, shielding in the transport solenoid could reduce the flux of
these electrons, again improving our resolution. There’s also substantial
tracker R&D where alternative gases, for example, could be used within the
straws, again improving resolution through better timing separation.

###### Acknowledgements.

We are grateful for the vital contributions of the Fermilab staff and the
technical staff of the participating institutions. This work was supported by
the US Department of Energy; the Istituto Nazionale di Fisica Nucleare, Italy;
the Science and Technology Facilities Council, UK; the Ministry of Education
and Science, Russian Federation; the National Science Foundation, USA; the
National Science Foundation, China; the Helmholtz Association, Germany; and
the EU Horizon 2020 Research and Innovation Program under the Marie
Sklodowska-Curie Grant Agreement Nos. 101006726, 101003460, 734303, 822185,
and 858199. Members of the Mu2e Collaboration were involved in the preparation
of this document, using the resources of the collaborating institutions,
including the Fermi National Accelerator Laboratory (Fermilab), a U.S.
Department of Energy, Office of Science, HEP User Facility. Fermilab is
managed by Fermi Research Alliance, LLC (FRA), acting under Contract No. DE-
AC02-07CH11359.

*

##  Appendix A Software Tools

Several different software simulation tools were used in the studies done for
this paper. We discuss these tools here.

###  A.1 FLUKA

The FLUKA [149, 150, 151] radiation transport and interaction code is a fully
integrated particle physics Monte Carlo simulation package containing
implementations of sound and modern physical models. FLUKA simulates the
interaction and transport of 60 different particles, and also enables an
estimate of the residual radiation and activation. Editable FORTRAN routines
allow including user-defined settings and modifications. A powerful graphical
interface [152] facilitates the editing of FLUKA input, execution of the code.
and visualization of the output. The Mu2e magnetic field maps have also been
interfaced to FLUKA and allow a realistic modeling of the trajectories of
charged particles.

###  A.2 GEANT4

Geant4 is a toolkit for simulating the passage of particles through matter
[153]. It includes particle tracking and hits, provides for complex
experimental geometries, and offers a variety of physics models that cover
many orders of magnitude of energy. Geant4 serves as the simulation tool for
the Mu2e/Mu2e-II particle event chain starting with particle production
resulting from the proton beam interacting with the production target,
continuing with particle transport through the solenoids, and lastly modeling
interactions of particles in the stopping target and the subsequent
interactions in the detectors. In the Geant4 studies done for this paper, we
used version 10.7 patch-02. The hadronic interactions are modeled according to
the ”ShieldingM” model provided by Geant4.

###  A.3 MARS15

The MARS15 code [154, 155] is a fully integrated particle transport and
interaction code consisting of a set of Monte Carlo programs (written in
Fortran and C++) that allow modeling and simulation of interactions of all
types of leptons, hadrons, photons, and heavy ions with matter. Interactions
in the energy range spanning from a fraction of an electronvolt up to hundreds
of TeV can be simulated. A model geometry can be described in several ways
including a built-in extended geometry description language, ROOT geometry,
and the import of external gdml files. MARS15 incorporated CEM and LAQGSM
[156] generators for the description of particle-nuclei and particle-particle
interactions above 100 MeV. Also, MARS15 includes the DeTra [157] code to
model nuclear decays and transmutations as well as EGS5 [158] code to model
low-energy electromagnetic showers.

###  A.4 TrackToy

The TrackToy hybrid Monte Carlo is used to quickly estimate the relative
importance of different tracker and tracker-region parameters on the Mu2e-II
signal sensitivity. In TrackToy, muon particle 4-vectors from the Geant4
simulation (see Appendix A.2) are propagated as quasi-helical trajectories
through a simplified model of the Mu2e Detector Solenoid (DS) magnetic field
to the stopping target, using algorithms for helix propagation in non-uniform
fields from the KinKal package. The stopping target is modeled as a hollow
cylinder with an adjustable but uniform mass density. The stopping position
and time of each beam muon is modeled from its predicted helical path through
the target material, using the muon stopping ranges tabulated by the Particle
Data Group (PDG) [159]. The TrackToy stopping rate, times, and positions were
found to be in good agreement with predictions from the Geant4 simulation.

Daughter electrons from stopped muons which Decay in Orbit (DIO) or convert
are generated from the muon stop positions and times assuming an isotropic
angular distribution, and a decay time distribution sampled from the material-
dependent muon lifetime exponential. Conversion electrons are given the target
material endpoint energy, DIO electron energies are generated by sampling the
distribution tabulated in [133]. The average daughter electron energy loss in
the target material is estimated using the EStar [160] tables from NIST.
Electrons emitted backwards (away from the tracker) may reflect in the
magnetic gradient and thus re-enter the target, resulting in additional energy
loss. The actual daughter electron energy loss is modeled by sampling a Moyal
distribution with average and RMS set to the EStar table values.

A thin polyethylene cylindrical shell between the target and the tracker is
used to stop or degrade protons emitted following nuclear muon capture in Mu2e
before they reach the tracker. This proton absorber (IPA) protects the tracker
sense wires from the charge load and large pulses generated by highly-ionizing
protons. It as assumed this shield, or something like it, will also be needed
for Mu2e-II. Several models of the IPA geometry and material are supported in
TrackToy. Daughter electron energy loss in the IPA is modeled the same way as
target energy loss.

The tracker active volume is modeled as a hollow cylinder. The density of
cells, the cell mass, and the hit efficiency and resolution properties are all
parameters. Daughter electrons trajectories are propagated geometrically
through the tracker active volume to predict the number of cells traversed.
The straw azimuthal distribution is selected randomly, based on the
distribution of straws angles in the Mu2e tracker. The straw material
traversed is predicted by sampling a flat impact parameter distribution. The
energy loss and multiple scattering angle mean and RMS from the straw material
are predicted using the KinKal material model. Tracker hit times are simulated
assuming a constant drift velocity and signal propagation velocity, smeared
using a Gaussian resolution model.

Daughter electrons exiting the Tracker intersect a 2-disk Calorimeter, with
the disk geometry taken from Mu2e. Daughters trajectories that traverse more
than a few mm of a disk generate a signal at shower max (or the disk exit
position for short paths), with position and time Gaussian smeared according
to spatial and timing resolution parameters. These calorimeter signals are
used as time constraint hits in the KinKal track fit described below.

Simulated hits are fit using a multi-stage configuration of the KinKal fit.
The initial stage uses hits as digital signals, constraining the track
according to the straw size and the signal time. The final stage uses the
drift information to constrain the track, assigning left-right ambiguities
according to the sign of the tracks estimated angular momentum about the wire,
for hits at least 0.5 mm from the wire. Simulated annealing is used to find
the global optimum ambiguity assignment.

Figure 43 plots compare the TrackToy conversion electron track fits with
tracks reconstructed in Mu2e Geant4 simulations, with TrackToy configured to
emulate the Mu2e experiment. The TrackToy simulation with KinKal fit runs at
approximately 100 Hz on a macbook pro.

Figure 43:  Comparison of conversion electron tracks simulated by the TrackToy
hybrid Monte Carlo and the Mu2e Geant4 Monte Carlo. The top row of plots
compares the energy loss predictions in the passive material before the
tracker (left) and the tracker itself (right). The bottom row of plots compare
the reconstructed track momentum resolution (left) and the reconstructed
momentum distribution (right). The vertical axes are scaled to the number of
Protons On Target (POT) simulated.

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# Ironing allocations††thanks: I am grateful to Ilya Segal, Piotr Dworczak and
Ben Brooks for helpful comments.

Filip Tokarski
Stanford GSB

###### Abstract

I propose a new approach to solving standard screening problems when the
monotonicity constraint binds. A simple geometric argument shows that when
virtual values are quasi-concave, the optimal allocation can be found by
appropriately truncating the solution to the relaxed problem. I provide a
simple algorithm for finding this optimal truncation when virtual values are
concave.

## 1 Introduction

This note revisits the problem of finding optimal menus in standard single-
agent screening environments. When agents’ payoffs are quasi-linear and
satisfy single-crossing, the standard approach involves writing the objective
as an integral of ‘virtual values’, which depend only on one type’s
allocation, and separately choosing each type’s allocation to maximize its
corresponding virtual value. Incentive compatibility, however, requires that
the allocation be increasing in type. If monotonicity binds, however, it is
standard to transform virtual values so that after the transformation, point-
wise maximization yields and increasing solution. This transformation, known
as ironing, was described by Myerson (1981) and subsequently generalized by
Toikka (2011). A different approach uses optimal control methods (Guesnerie
and Laffont, 1984; Hellwig, 2008; Ruiz del Portal, 2011). These, however,
typically require assuming piece-wise differentiability of the control
variable, and thereby imposing restrictions on endogenous objects.111Ruiz del
Portal (2011) formulates a necessary condition for optimality without this
assumption; Hellwig (2008) formulates a version of the maximum principle
without it.

I propose an alternative approach that involves solving the relaxed problem
without the monotonicity constraint and transforming the resulting allocation
to satisfy monotonicity. Theorem 1 says that whenever an optimal allocation
rule exists, it can be found by optimally truncating the solution to the
relaxed problem. Moreover, the optimal truncation is pinned down by the
allocations of types at which the solution to the relaxed problem changes
monotonicity. These observations generalize insights about the structure of
solutions from the literature—they require no continuity or differentiability
assumptions, do not need virtual values to be concave, and do not assume that
an agent’s allocation is chosen from a compact interval. Therefore, in
contrast to previous work, my results also apply when the planner can only
assign discrete allocations.

I subsequently use these insights to develop a simple algorithm for finding
the optimal allocation rule under the additional assumptions that virtual
values are concave and that each agent’s allocation is chosen from a compact
interval.

## 2 Problem

Agents with types $\theta\in[0,1]$ are allocated objects from a compact set
$\mathcal{X}\subset\mathbb{R}$. The planner chooses an increasing allocation
rule $x:[0,1]\to\mathcal{X}$ to maximize the following objective:

$F[x]=\int_{0}^{1}J(x(\theta),\theta)d\theta.$

I refer to $J:\mathcal{X}\times[0,1]\to\mathbb{R}$ as the virtual value.

######  Assumption 1.

The virtual value satisfies the following properties:

  1. 1.

$J(\cdot,\theta)$ is weakly quasi-concave for every $\theta\in[0,1]$.

  2. 2.

$J(x(\theta),\theta)$ is integrable in $\theta$ on $[0,1]$ for every piece-
wise monotonic allocation rule $x\in\mathcal{X}^{[0,1]}$.

Throughout, I will refer to the problem of maximizing $F[x]$ over the space of
increasing allocation rules $x\in\mathcal{X}^{[0,1]}$ as the planner’s
problem. I will call the version of this problem where $x$ need not be
increasing the relaxed problem. I also assume that the relaxed problem has a
well-behaved solution:

######  Assumption 2.

There exists a solution to the relaxed problem, $x_{R}$, that is piece-wise
monotonic.

## 3 Structure of the solution

In this section I present and prove Theorem 1 describing the structure of the
solution to the planner’s problem. To formulate the result, I first introduce
the following definition:

######  Definition 1.

A critical point $i\in[0,1]$ is a point where $x_{R}$ changes monotonicity,
that is, $x_{R}$ is monotonic on $(i-\epsilon,i)$ for some $\epsilon>0$ but is
not monotonic on $(i-\epsilon,i+\delta)$ for any $\delta>0$. By convention, I
also call $0$ and $1$ critical points. I use $i_{n}$ to denote the $n$-th
critical point and $\mathcal{I}$ to denote the set of critical points.

######  Theorem 1.

Let $x^{*}_{v}:[0,1]\to\mathcal{X}$ parametrized by
$v\in\mathcal{X}^{|\mathcal{I}|-1}$ be such that:

$x^{*}_{v}(\theta)=\begin{cases}\min\left\\{v_{n+1},\max\left\\{v_{n},x_{R}(\theta)\right\\}\right\\}&\text{if
}\theta\in[i_{n},i_{n+1})\text{ where $x_{R}$ is increasing},\\\
v_{n}&\text{if }\theta\in[i_{n},i_{n+1})\text{ where $x_{R}$ is
decreasing},\end{cases}$ (1)

where $v_{|\mathcal{I}|}=x^{*}_{v}(1)=\max_{x\in\mathcal{X}}x$. Then, if the
planner’s problem has a solution, it has a solution of the form $x^{*}_{v}$
for some $v\in\mathcal{X}^{|\mathcal{I}|-1}$. This solution can be recovered
by solving:

$\max_{v\in\mathcal{X}^{|\mathcal{I}|-1}}F[x^{*}_{v}]\quad\text{subject
to}\quad v_{1}\leq v_{2}\leq\dots\leq v_{|\mathcal{I}|-1}.$ (I)

Theorem 1 says that when the planner’s problem has a solution, we can find it
by optimally truncating the solution to the relaxed problem, $x_{R}$.
Moreover, the optimal truncation is pinned down by the allocations of types at
which $x_{R}$ changes monotonicity.

The remainder of this section is devoted to proving Theorem 1. I first show
three lemmas:

######  Lemma 1.

If $x_{1},x_{2}$ are allocation rules s.t. $x_{2}$ lies point-wise between
$x_{1}$ and $x_{R}$, then $F[x_{2}]\geq F[x_{1}]$.

###### Proof.

$J(x_{R}(\theta),\theta)\geq J(x_{1}(\theta),\theta)$ for almost all $\theta$;
else we could strictly improve upon $x_{R}$ by changing it to $x_{1}$ wherever
the inequality fails. Now, by quasi-concavity of $J(\cdot,\theta)$:

$\displaystyle F[x_{2}]=\int_{0}^{1}J(x_{2}(\theta),\theta)d\theta$
$\displaystyle\geq\int_{0}^{1}\min\\{J(x_{1}(\theta),\theta),J(x_{R}(\theta),\theta)\\}d\theta$
$\displaystyle=\int_{0}^{1}J(x_{1}(\theta),\theta)d\theta=F[x_{1}].$

∎

Lemma 1 tells us that the objective always increases when we move the
allocation rule point-wise closer to the first-best one. This property
underpins the proofs of Lemmas 2 and 3.

######  Lemma 2.

Suppose $x_{R}$ is decreasing on $[a,b)$. Then any increasing
$x\in\mathcal{X}^{[0,1]}$ can be weakly improved upon by some increasing
$x^{*}\in\mathcal{X}^{[0,1]}$ that is constant on $[a,b)$ and coincides with
$x$ elsewhere.

###### Proof.

Fix any increasing $x\in\mathcal{X}^{[0,1]}$. Consider
$x^{*}\in\mathcal{X}^{[0,1]}$ that coincides with $x$ on $[0,1]\setminus[a,b)$
and takes the following values for $\theta\in[a,b)$:

$x^{*}(\theta)=\begin{cases}\lim_{\theta\to a^{+}}x(\theta)&\text{if
}x_{R}(\theta)\leq x(\theta)\text{ for all }\theta\in(a,b),\\\
x(\sup\\{\theta\in(a,b):x_{R}(\theta)\geq x(\theta)\\})&\text{if
}\sup\\{\theta\in(a,b):x_{R}(\theta)\geq x(\theta)\\}\in(a,b),\\\
\lim_{\theta\to b^{-}}x(\theta)&\text{if }x_{R}(\theta)\geq x(\theta)\text{
for all }\theta\in(a,b).\end{cases}$

$x^{*}$ is increasing and point-wise between $x_{R}$ and $x$, so $F[x^{*}]\geq
F[x]$ by Lemma 1. ∎

######  Lemma 3.

Suppose $x_{R}$ is increasing on $[a,b)$. Then any increasing
$x\in\mathcal{X}^{[0,1]}$ can be weakly improved upon by an increasing
$x^{*}\in\mathcal{X}^{[0,1]}$ s.t. $x^{*}(\theta)=\min\left\\{x(b),\
\max\left\\{x(a),\ x_{R}(\theta)\ \right\\}\right\\}$ on $[a,b)$ and coincides
with $x$ elsewhere.

###### Proof.

Fix any increasing $x\in\mathcal{X}^{[0,1]}$ and consider:

$x^{*}(\theta)=\begin{cases}\min\left\\{x(b),\max\left\\{x(a),x_{R}(\theta)\right\\}\right\\}&\text{if
}\theta\in[a,b),\\\ x(\theta)&\text{elsewhere.}\end{cases}$

$x^{*}$ is point-wise between $x$ and $x_{R}$, so $F[x^{*}]\geq F[x]$ by Lemma
1. Moreover, $x^{*}$ is increasing on $[a,b)$ as $x_{R}$ is increasing on
$[a,b)$ by assumption. ∎

$x(\theta)$$x^{*}(\theta)$$x_{R}(\theta)$$a$$b$$\sup$$\\{\theta\in(a,b):x_{R}(\theta)\geq
x(\theta)\\}$ (a) Constructing an improvement in the proof of Lemma 2.

$a$$b$$x^{*}(\theta)$$x_{R}(\theta)$$x(\theta)$ (b) Constructing an
improvement in the proof of Lemma 3.

Observations similar to Lemmas 2 and 3 were independently made by Sandmann
(2022) who studies the optimality of sparse menus in a price discrimination
problem.

Given the above lemmas, the proof of Theorem 1 is straightforward. By
Assumption 2, $[0,1)$ can be partitioned into finitely many intervals
$[i_{n},i_{n+1})$ where $x_{R}$ is monotonic. Then, by Lemmas 2 and 3, if the
planner’s problem has a solution, it has one of the form in (1) with
$v_{1}\leq v_{2}\leq\dots\leq v_{|\mathcal{I}|-1}$. Solving problem (I)
recovers this solution.

## 4 Solution algorithm

I now provide a simple algorithm that solves planner’s problem under the
following additional assumption:

######  Assumption 3.

The virtual value satisfies the following properties:

  1. 1.

The planner chooses allocations from a closed interval: $\mathcal{X}=[l,h]$.

  2. 2.

$J(\cdot,\theta)$ is weakly concave for every $\theta\in[0,1]$.

  3. 3.

$J(x,\theta)$ is uniformly bounded across all $x\in[l,h]$ and
$\theta\in[0,1]$.

The algorithm uses the following transformation
$T:[l,h]^{[0,1]}\times[l,h]\times\mathbb{N}\to{[l,h]}^{[0,1]}$:

$\operatorname{T}[x,v,n](\theta):=\begin{cases}\min\\{v,x(\theta)\\}&\text{if
}\theta<i_{n},\\\ v&\text{if }\theta\in[i_{n},i_{n+1})\text{ where $x_{R}$ is
decreasing},\\\ \max\left\\{v,x(\theta)\right\\}&\text{if
}\theta\in[i_{n},i_{n+1})\text{ where $x_{R}$ is increasing},\\\ h&\text{if
$\theta=i_{n+1}$},\\\ x(\theta)&\text{if }\theta>i_{n+1}.\end{cases}$

When $T[\cdot\ ,v,n]$ is applied to $x$, the allocation rule is truncated from
above by $v$ before the $n$th critical point and set equal to $v$ or truncated
by it from below between the $n$th and $n+1$st critical points. It is also set
to $h$ at the $n+1$st critical point and remains unchanged beyond it. Note
that if $x(\theta)$ was increasing on the interval $[0,i_{n})$, then
$\operatorname{T}[x,v,n](\theta)$ will be increasing on $[0,i_{n+1})$. By
iteratively applying this transformation at subsequent critical points, the
algorithm expands the interval on which the resulting allocation rule is
increasing.

The following algorithm takes in the solution to the relaxed problem, $x_{R}$,
and the set of critical points $i\in\mathcal{I}$ as inputs and produces the
solution to planner’s problem:

Algorithm 1

$r\leftarrow x_{R}$

$n\leftarrow 1$

while $n\leq|\mathcal{I}|-1$ do

$v^{*}\leftarrow\operatorname{argmax}_{v\in[l,h]}\int_{0}^{i_{n+1}}J(\operatorname{T}[r,v,n](\theta))d\theta$

$r\leftarrow\operatorname{T}[x,v^{*},n]$

$n\leftarrow n+1$

end while

return $r$

Intuitively, each iteration of Algorithm 1 starts with the optimum subject to
monotonicity from 0 until the $n$th critical point and ‘irons out’ this
partial solution further to produce a solution subject to monotonicity until
the $n+1$st critical point (Figure 2).

$i_{1}$$i_{2}$$i_{3}$$i_{4}$$i_{5}$ (a) $n=1$

$i_{1}$$i_{2}$$i_{3}$$i_{4}$$i_{5}$ (b) $n=2$

$i_{1}$$i_{2}$$i_{3}$$i_{4}$$i_{5}$ (c) $n=3$

$i_{1}$$i_{2}$$i_{3}$$i_{4}$$i_{5}$ (d) $n=4$

Figure 2: Algorithm 1 recursively transforming $x_{R}$ (blue) into subsequent
$\operatorname{T}[x,v^{*},n]$ (red).

######  Theorem 2.

Under Assumption 3, the output of Algorithm 1 solves the planner’s problem.

The remainder of this section is devoted to proving this result. We begin by
defining some useful objects and establishing their properties. Let $F_{a}$ be
the value of the objective evaluated only until type $a$, that is:

$F_{a}[x]:=\int_{0}^{a}J(x(\theta),\theta)d\theta.$

Since $J(\cdot,\theta)$ is concave for every $\theta$, $F_{a}[x]$ is concave
in the allocation rule $x$ for every $a\in[0,1]$.

Now, consider the set of allocation rules that are increasing on $[0,a)$ and
below $k$ everywhere on that interval. Let $\mathcal{O}^{k}_{a}$ contain the
allocation rules that maximize $F_{a}$ over this set:

$\mathcal{O}^{k}_{a}:=\operatorname{argmax}\left\\{F_{a}[x]:\
\text{$x\in[l,h]^{[0,1]}$ is increasing on $[0,a)$ and $x(\theta)\leq k$ for
all $\theta\in[0,a)$}\right\\}.$

I now show that $\mathcal{O}^{k}_{a}$ is non-empty for any $a\in(0,1]$ and
$k\in[l,h]$. Let $\mathcal{A}$ be the set of functions $x\in[l,k]^{[0,a]}$
that are increasing. It suffices to show that $F_{a}$ attains a maximum on
$\mathcal{A}$. $\mathcal{A}$ is homeomorphic to a Helly space and so is
compact. Hence, by Weierstrass’ theorem, it suffices to show that $F_{a}$ is
continuous on this space. Since $\mathcal{A}$ is homeomorphic to a Helly
space, it is first-countable. Thus, $F_{a}$ is continuous on it if for any
sequence in $\mathcal{A}$ such that $x_{n}\to x$ we have $F_{a}[x_{n}]\to
F_{a}[x]$. Fix any such $x_{n}$. Then $x_{n}(\theta)\to x(\theta)$ for every
$\theta\in[0,a]$. Since $J(\cdot,\theta)$ is concave, it is continuous, which
implies $J(x_{n}(\theta),\theta)\to J(x(\theta),\theta)$ for every
$\theta\in[0,a]$. The Dominated Convergence Theorem then implies that
$F_{a}[x_{n}]\to F_{a}[x]$.

Finally, the following proposition is shown in the appendix. It says that when
we have an $x$ in $\mathcal{O}_{a}^{h}$ and impose the additional constraint
that $x$ be below $k$ on $[0,a)$, we need not resolve the problem, but can
simply truncate the solution without this constraint by $k$.

######  Proposition 1.

If $x_{a}\in\mathcal{O}^{h}_{a}$, then
$\max\\{k,x_{a}\\}\in\mathcal{O}^{k}_{a}$.

I now present an inductive proof of Theorem 2. Recall that $|\mathcal{I}|\geq
2$ since $0,1\in\mathcal{I}$. The base case demonstrates that the first
iteration of the algorithm produces $r\in\mathcal{O}^{h}_{i_{2}}$. The step
shows that when the $n$th iteration starts with $r\in\mathcal{O}^{h}_{i_{n}}$,
it produces $r\in\mathcal{O}^{h}_{i_{n+1}}$. The base and the step thus imply
that the algorithm will produce $r\in\mathcal{O}^{h}_{i_{|\mathcal{I}|}}$
solving the planner’s problem in $|\mathcal{I}|-1$ steps.

Base. Since $\mathcal{O}_{i_{2}}^{h}\neq\emptyset$, Lemmas 2 and 3 tell us
there exists $x^{*}\in\mathcal{O}_{i_{2}}^{h}$ for which:

$x^{*}(\theta):=\begin{cases}v_{1}&\text{if }\theta\in[i_{1},i_{2})\text{
where $x_{R}$ is decreasing},\\\
\min\left\\{v_{2},\max\left\\{v_{1},x_{R}(\theta)\right\\}\right\\}&\text{if
}\theta\in[i_{1},i_{2})\text{ where $x_{R}$ is increasing}.\end{cases}$

Moreover, by Lemma 1 we can without loss set $v_{2}=h$. The first iteration of
the algorithm will recover such an $x^{*}$ when optimizing over $v$.

Step. Since $\mathcal{O}^{h}_{i_{n+1}}\neq\emptyset$, Lemmas 2 and 3 tell us
there exists $x^{*}\in\mathcal{O}_{i_{n+1}}^{h}$ for which:

$x^{*}(\theta):=\begin{cases}v_{n}&\text{if }\theta\in[i_{n},i_{n+1})\text{
where $x_{R}$ is decreasing},\\\
\min\left\\{v_{n+1},\max\left\\{v_{n},x_{R}(\theta)\right\\}\right\\}&\text{if
}\theta\in[i_{n},i_{n+1})\text{ where $x_{R}$ is increasing}.\end{cases}$

Moreover, by Lemma 1 we can without loss set $v_{n+1}=h$. Let
$r\in\mathcal{O}^{h}_{i_{n}}$ be the allocation rule produced by the $n-1$st
iteration of the algorithm. By Proposition 1,
$\max\\{v_{n},r\\}\in\mathcal{O}^{v_{n}}_{i_{n}}$. Thus,
$F_{i_{n}}[\max\\{v_{n},r\\}]\geq F_{i_{n}}[x^{*}]$ and so the following
allocation rule weakly improves upon $x^{*}$:

$x^{**}(\theta):=\begin{cases}\min\\{v_{n},r(\theta)\\}&\text{if
}\theta<i_{n},\\\ v_{n}&\text{if }\theta\in[i_{n},i_{n+1})\text{ where $x_{R}$
is decreasing},\\\ \max\left\\{v_{n},x_{R}(\theta)\right\\}&\text{if
}\theta\in[i_{n},i_{n+1})\text{ where $x_{R}$ is increasing}.\end{cases}$

Moreover, $x^{**}$ is increasing and weakly below $k$ everywhere on $[0,a)$,
so $x^{**}$ belongs to $\mathcal{O}_{i_{n+1}}^{h}$. The $n$th iteration of the
algorithm will recover such an $x^{**}$ when optimizing over $v$.

## 5 Appendix

### 5.1 Proof of Proposition 1.

###### Proof.

If $x_{a}(\theta)\leq k$ for all $\theta\in[0,a)$, the result is trivial as
the additional constraint does not bind, so consider the case where
$x_{a}(\theta)>k$ for some $\theta\in[0,a)$. Let $b$ be the earliest point
where $x_{a}$ crosses $k$, so $b:=\inf\\{\theta:x_{a}(\theta)\geq k\\}$. We
can assume without loss that $x_{a}(b)=k$, as this does not violate
monotonicity on $[0,a)$ or alter the value of $F_{a}[x_{a}]$.

Recall that $\mathcal{O}_{a}^{k}$ is non-empty for all $a\in(0,1]$ and all
$k\in[l,h]$, and that all allocation rules in $\mathcal{O}_{a}^{k}$ are
increasing and below $k$ on $[0,a)$. Let $c$ be the earliest point in $[b,a]$
at which some allocation rule $x$ belonging to $\mathcal{O}_{a}^{k}$ takes the
value $k$:

$c=\inf_{\theta\geq b,\ x\in\mathcal{O}^{k}_{a}}\\{\theta:x(\theta)=k\\}.$

Notice that $c\leq a$. To see this, take any $x^{*}\in\mathcal{O}_{a}^{k}$ and
notice that setting $x^{*}(a)=k$ does not affect the value of $F_{a}$ or
violate the constraints, as these are only imposed on $[0,a)$. Thus, there
always exists some $x^{*}\in\mathcal{O}_{a}^{k}$ such that $x^{*}(a)=k$.

I now show that this infimum is attained, i.e. that there exists an allocation
rule $x\in\mathcal{O}^{k}_{a}$ such that $x(c)=k$. I already showed this for
the case of $c=a$, so suppose $c<a$. The following lemma will be useful:

######  Lemma 4.

Fix $z\in[b,a)$ such that some sequence of allocation rules
$f_{n}\in\mathcal{O}_{a}^{k}$ converges point-wise to $k$ on $(z,a)$. Take any
$x_{z}\in\mathcal{O}_{z}^{k}$. Then any allocation rule $x^{*}$ satisfying (2)
belongs to $\mathcal{O}^{k}_{a}$.

$x^{*}(\theta)=\begin{cases}x_{z}(\theta)&\text{if }\theta<z,\\\ k&\text{if
}\theta\in[z,a).\end{cases}$ (2)

###### Proof.

The allocation rule $x^{*}$ is increasing and below $k$ on $[0,a)$, so it
suffices to show that it guarantees a value of $F_{a}$ at least as large as
some other allocation rule in $\mathcal{O}_{a}^{k}$.

By assumption, $f_{n}(\theta)\to k$ for every $\theta\in(z,a)$.
$J(\cdot,\theta)$ is concave and hence continuous, so
$J(f_{n}(\theta),\theta)\to J(k,\theta)$ for all $\theta\in(z,a)$. By the
Dominated Convergence Theorem, we have:

$\int_{z}^{a}J(f_{n}(\theta),\theta)d\theta\to\int_{z}^{a}J(k,\theta)d\theta.$
(3)

Take any $\epsilon>0$. Then there exists $n^{*}$ for which:

$\int_{z}^{a}J(k,\theta)d\theta+\epsilon>\int_{z}^{a}J(f_{n^{*}}(\theta),\theta)d\theta.$
(4)

Note that $f_{n}(z)\leq k$, so allocation rules in $\mathcal{O}_{z}^{k}$
guarantee a weakly higher $F_{z}$ than those in $\mathcal{O}_{z}^{f_{n}(z)}$.
Hence:

$\int_{0}^{z}J(x_{z}(\theta),\theta)d\theta\geq\int_{0}^{z}J(f_{n^{*}}(\theta),\theta)d\theta.$
(5)

Adding together (4) and (5) gives:

$\int_{0}^{a}J(x^{*}(\theta),\theta)d\theta+\epsilon>\int_{0}^{a}J(f_{n^{*}}(\theta),\theta)d\theta.$

Recall that $\epsilon$ was arbitrary, so $F_{a}[x^{*}]\geq F_{a}[f_{n^{*}}]$.
Since $f_{n^{*}}\in\mathcal{O}^{k}_{a}$, we get that
$x^{*}\in\mathcal{O}^{k}_{a}$ too. ∎

Return to the case of $c<a$. By monotonicity and the definition of $c$, there
exists a sequence of allocation rules $f_{n}\in\mathcal{O}_{n}^{k}$ converging
point-wise to $k$ on $(c,a)$. Hence, Lemma 4 tells us there exists
$x^{*}\in\mathcal{O}_{a}^{k}$ for which $x^{*}(c)=k$.

Now, let $u:=\sup\\{x(b):x\in\mathcal{O}_{a}^{k}\text{ and }x(c)=k\\}$. In
what follows, I show that $u=k$.

Suppose towards a contradiction that $u<k$. Then it must be that $b<c$. By
Assumption 1, $x_{R}$ is monotonic on $[c-\epsilon,c)$ for $\epsilon$ small
enough. Fix such an $\epsilon$ small enough that $b<c-\epsilon$.

######  Lemma 5.

$x_{R}$ is decreasing on $[c-\epsilon,c)$.

###### Proof.

Suppose towards a contradiction that $x_{R}$ is increasing on $[c-\epsilon,c)$
and take $x^{*}\in\mathcal{O}_{a}^{k}$ such that $x^{*}(c)=k$ (by the above,
such an $x^{*}$ exists). Consider two cases:

Case 1: there exists $\epsilon^{\prime}\in(0,\epsilon)$ such that
$x^{R}(c-\epsilon^{\prime})\geq k$. Then $x_{R}(\theta)\geq k$ on
$[c-\epsilon^{\prime},c)$. By Lemma 1, we can improve upon $x^{*}$ by setting
it to $k$ on $[c-\epsilon^{\prime},c)$. Thus, there is some allocation rule
$x^{**}\in\mathcal{O}_{a}^{k}$ equal to $k$ at $c-\epsilon^{\prime}<c$. This
contradicts the definition of $c$.

Case 2: $x_{R}(\theta)<k$ everywhere on $[c-\epsilon,c)$. By the definition of
$c$, we have $x^{*}(c-\epsilon)<k$. Since $b<c-\epsilon$, we also know that
$x_{a}(c-\epsilon)\geq k$. Hence, there exists $\alpha\in(0,1]$ for which
$\alpha x_{a}(c-\epsilon)+(1-\alpha)x^{*}(c-\epsilon)=k$. Consider the
following allocation rule:

$x^{**}(\theta)=\begin{cases}\alpha
x_{a}(\theta)+(1-\alpha)x^{*}(\theta)&\text{if }\theta<c-\epsilon,\\\
k&\text{if }\theta\in[c-\epsilon,a).\end{cases}$

I will show that $x^{**}$ belongs to $\mathcal{O}_{a}^{k}$. The allocation
rules $x_{a}$ and $x^{*}$ are increasing on $[0,a)$, so their convex
combination is too. It is also weakly below $k$ on $[0,c-\epsilon)$ and so
$x^{**}$ is increasing and below $k$ on $[0,a)$. Therefore, it suffices to
show that $x^{**}$ guarantees a weakly higher value of $F_{a}$ than some other
allocation rule in $\mathcal{O}_{a}^{k}$. I will show that $F_{a}[x^{**}]\geq
F_{a}[x^{*}]$. Since $x_{a}\in\mathcal{O}_{a}^{h}$ solves a problem with laxer
constraints than does $x^{*}\in\mathcal{O}_{a}^{k}$, we have $F_{a}[x_{a}]\geq
F_{a}[x^{*}]$. Hence, by concavity of $F_{a}$:

$F_{a}[\alpha x_{a}+(1-\alpha)x^{*}]\geq F_{a}[x^{*}].$

Notice $x^{**}$ and $\alpha x_{a}+(1-\alpha)x^{*}$ coincide everywhere on
$[0,a)$ except for the interval $[c-\epsilon,c)$, where
$x^{**}(\theta)\leq\alpha x_{a}(\theta)+(1-\alpha)x^{*}(\theta)$. However,
recall that $x_{R}<k$ on $[c-\epsilon,c)$, and so $F_{a}[x^{**}]>F_{a}[\alpha
x_{a}+(1-\alpha)x^{*}]$ by Lemma 1. Therefore, $F_{a}[x^{**}]\geq
F_{a}[x^{*}]$ and so $x^{**}\in\mathcal{O}_{a}^{k}$. However,
$x^{**}(c-\epsilon)=k$, which contradicts the definition of $c$. ∎

In what follows, I obtain a contradiction by constructing an allocation rule
in $\mathcal{O}_{a}^{k}$ that attains a value above $u$ at $b$. Fix some
strictly positive sequence $\\{\delta_{n}\\}_{n}$ such that $\delta_{n}\to 0$.
By the definition of $u$, for any $n$ there exists
$g_{n}\in\mathcal{O}^{k}_{a}$ such that $g_{n}(c)=k$ and
$g_{n}(b)+\delta_{n}>u$.

Moreover, I show that for any $n$ there exists an allocation rule
$g_{n}^{*}\in\mathcal{O}^{k}_{n}$ such that $g_{n}^{*}(b)+\delta_{n}>u$,
$g_{n}^{*}(c)=k$ and $g_{n}^{*}(\theta)$ is constant on $[c-\epsilon,c)$.
Lemma 5 tells us that $x_{R}$ is decreasing on $[c-\epsilon,c)$ and so, by
Lemma 2, for every $n$ there exists $g_{n}^{*}\in\mathcal{O}_{a}^{k}$ that is
constant on $[c-\epsilon,c)$ and coincides with $g_{n}$ elsewhere. Since
$b,c\not\in[c-\epsilon,c)$, for any $n$ we still have
$g_{n}^{*}(b)+\delta_{n}>u$ and $g_{n}^{*}(c)=k$.

Furthermore, the sequence of allocation rules $\\{g_{n}^{*}\\}_{n}$ has to be
uniformly bounded away from $k$ on $[0,c)$. By the above, for any $n$ there
exists $\gamma_{n}\in\mathbb{R}$ s.t. $g_{n}^{*}(\theta)=\gamma_{n}$ for
$\theta\in[c-\epsilon,c)$. Suppose towards a contradiction that
$\sup\gamma_{n}=k$. Then there exists a subsequence of $\\{g_{n}^{*}\\}_{n}$
that converges point-wise to $k$ on $[c-\epsilon,c)$. Lemma 4 then tells us
there exists an allocation rule $x^{*}\in\mathcal{O}_{a}^{k}$ equal to $k$ on
$[c-\epsilon,c)$. This contradicts the definition of $c$.

Thanks to this uniform bound, we know there exists $\alpha\in(0,1)$ such that
for all $n$ and $\theta\in[0,c)$ we have $\alpha
x_{a}(\theta)+(1-\alpha)g_{n}^{*}(\theta)\leq k$. Let us then consider
$g_{n}^{**}$ defined as follows:

$g_{n}^{**}(\theta)=\begin{cases}\alpha
x_{a}(\theta)+(1-\alpha)g_{n}^{*}(\theta)&\text{if }\theta<c,\\\ k&\text{if
}\theta\geq c.\end{cases}$

I now show $g_{n}^{**}$ belongs to $\mathcal{O}_{a}^{k}$ for any $n$. Notice
$x_{a}(\theta)+(1-\alpha)g_{n}^{*}(\theta)$ is increasing on $[0,c)$ as
$g_{n}^{*}$ and $x_{a}$ were increasing on that interval. By the choice of
$\alpha$, $x_{a}(\theta)+(1-\alpha)g_{n}^{*}(\theta)$ is also below $k$ on
$[0,c)$. Thus, to show that $g_{n}^{**}$ belongs to $\mathcal{O}_{a}^{k}$, it
suffices to show that it guarantees a value of $F_{a}$ as high as another
allocation rule in that set. I will show $F_{a}[g_{n}^{**}]\geq
F_{a}[g_{n}^{*}]$.

First, note that $F_{c}[x_{a}]\geq F_{c}[g_{n}^{*}]$; otherwise $F_{c}[x_{a}]$
could be increased by replacing $x_{a}$ for $g_{n}^{*}$ on $[0,c)$. This
replacement would preserve monotonicity and produce an allocation rule that is
below $k$ on $[0,a)$. To see that, recall that $x_{a}$ equals to $k$ on
$[b,a)$ and $c>b$, so $x_{a}(c)=k$. Since $g_{n}^{*}(c)\leq k$ and $g_{n}^{*}$
is increasing on $[0,c)$, both constraints hold after the replacement.

By concavity of $F_{c}$ and the fact that $F_{c}[x_{a}]\geq F_{c}[g_{n}^{*}]$,
we have:

$F_{c}[\alpha x_{a}(\theta)+(1-\alpha)g_{n}^{*}]\geq F_{c}[g_{n}^{*}].$ (6)

Both $g_{n}^{*}$ and $g_{n}^{**}$ equal to $k$ on $[c,a)$, so (6) implies that
$F_{a}[g_{n}^{**}]\geq F_{a}[g_{n}^{*}]$.

Now, recall that $b<c$ and $x_{a}(b)=k$. Thus, for every $n$:

$g_{n}^{**}(b)=\alpha k+(1-\alpha)g_{n}^{*}(b)>\alpha
k+(1-\alpha)(u-\delta_{n}),$

where the latter inequality holds because $g_{n}^{*}(b)+\delta_{n}>u$ for
every $n$. Since we assumed that $u<k$, for sufficiently large $n$ we have
$g_{n}^{**}(b)>u$. Since $g_{n}^{**}\in\mathcal{O}_{a}^{k}$ for any $n$, this
contradicts the definition of $u$.

Thus, we have shown by contradiction that $u=k$. I will now apply Lemma 4 to
show that $\max\\{x_{a},k\\}\in\mathcal{O}_{a}^{k}$. First, note that by the
definition of $u$ there exists a sequence of allocation rules
$f_{n}\in\mathcal{O}_{a}^{k}$ converging point-wise to $k$ on $(b,a)$. Second,
observe that $x_{a}\in\mathcal{O}_{b}^{k}$; otherwise we could improve
$F_{a}[x_{a}]$ by replacing $x_{a}$ for some $x^{*}\in\mathcal{O}_{b}^{k}$ on
$[0,b)$. Since $x_{a}(b)=k$, this replacement would not violate either
constraint.

Finally, $x_{a}$ is below $k$ on $[0,b)$ and above $k$ on $[b,a)$, so Lemma 4
tells us $\max\\{x_{a},k\\}\in\mathcal{O}_{a}^{k}$. ∎

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