# The Electromagnetic Signature of Outward Propagating Ion-Scale Waves

Trevor A. Bowen Space Sciences Laboratory, University of California, Berkeley,
CA 94720-7450, USA Trevor A. Bowen<EMAIL_ADDRESS>Stuart D. Bale Space
Sciences Laboratory, University of California, Berkeley, CA 94720-7450, USA
Physics Department, University of California, Berkeley, CA 94720-7300, USA
The Blackett Laboratory, Imperial College London, London, SW7 2AZ, UK  School
of Physics and Astronomy, Queen Mary University of London, London E1 4NS, UK
J. W. Bonnell Space Sciences Laboratory, University of California, Berkeley,
CA 94720-7450, USA  Davin Larson  Space Sciences Laboratory, University of
California, Berkeley, CA 94720-7450, USA  Alfred Mallet  Space Sciences
Laboratory, University of California, Berkeley, CA 94720-7450, USA  Michael D.
McManus  Space Sciences Laboratory, University of California, Berkeley, CA
94720-7450, USA  Physics Department, University of California, Berkeley, CA
94720-7300, USA  Forrest Mozer  Space Sciences Laboratory, University of
California, Berkeley, CA 94720-7450, USA  Physics Department, University of
California, Berkeley, CA 94720-7300, USA  Marc Pulupa Space Sciences
Laboratory, University of California, Berkeley, CA 94720-7450, USA  Ivan Vasko
Space Sciences Laboratory, University of California, Berkeley, CA 94720-7450,
USA  J. L. Verniero Space Sciences Laboratory, University of California,
Berkeley, CA 94720-7450, USA

###### Abstract

First results from the Parker Solar Probe (PSP) mission have revealed
ubiquitous coherent ion-scale waves in the inner heliosphere, which are
signatures of kinetic wave-particle interactions and fluid-scale
instabilities. However, initial studies of the circularly polarized ion-scale
waves observed by PSP have only thoroughly analyzed magnetic field signatures,
precluding a determination of solar-wind frame propagation direction and
intrinsic wave-polarization. A comprehensive determination of wave-properties
requires measurements of both electric and magnetic fields. Here, we use full
capabilities of the PSP/FIELDS instrument suite to measure both the electric
and magnetic components of circularly polarized waves. Comparing spacecraft
frame magnetic field measurements with the Doppler-shifted cold-plasma
dispersion relation for parallel transverse waves constrains allowable plasma
frame polarizations and wave-vectors. We demonstrate that the Doppler-shifted
cold-plasma dispersion has a maximum spacecraft frequency $f_{sc}^{*}$ for
which intrinsically right-handed fast-magnetosonic waves (FMWs) propagating
sunwards can appear left-handed in the spacecraft frame. Observations of left-
handed waves with $|f|>f_{sc}^{*}$ are uniquely explained by intrinsically
left-handed, ion-cyclotron, waves (ICWs). We demonstrate that electric field
measurements for waves with $|f|>f_{sc}^{*}$ are consistent with ICWs
propagating away from the sun, verifying the measured electric field. Applying
the verified electric field measurements to the full distribution of waves
suggests that the vast majority of waves propagate away from the sun (in the
plasma frame), indicating that the observed population of coherent ion-scale
waves contains both intrinsically left and right hand polarized modes.

Introduction

Understanding observational signatures of energy transfer between
electromagnetic waves and particle distribution functions will constrain the
kinetic processes contributing to heating and acceleration in coronal and
solar wind plasma. Instabilities in the solar wind are thought to drive
coherent circularly polarized waves at both ion and electron scales (Gary,
1992, 1993; Gary et al., 2000; Kasper et al., 2002; Hellinger et al., 2006;
Podesta & Gary, 2011b; Verscharen et al., 2016; Yoon, 2017; Klein et al.,
2018; Verscharen et al., 2019; Tong et al., 2019a, b; Verniero et al., 2020).
At ion scales, the cold plasma approximation allows two circularly polarized
transverse electromagnetic (EM) modes: left-hand polarized ion cyclotron waves
(ICWs) and the right-handed polarized fast-magnetosonic waves (FMWs) (Stix,
1992; Gary, 1993). However, spacecraft observations of transverse waves are
often limited to single point magnetic field measurements, precluding
determination of intrinsic plasma-frame polarization. Single point magnetic
field measurements only estimate the handedness of fluctuations in the
spacecraft frame (Narita et al., 2009; Howes, & Quataert, 2010).

In the solar wind, large (supersonic) flow speeds significantly modify
spacecraft frame observations of ion-scale electromagnetic waves through
Doppler shift (Fredricks, & Coroniti, 1976; Klein et al., 2014). While the
spacecraft frame polarization may be determined through wavelet or Fourier
methods, the wave propagation direction in the plasma frame may be difficult
to infer from the Doppler-shifted observations (He et al., 2011; Podesta &
Gary, 2011a; Klein et al., 2014b; Wicks et al., 2016; Woodham et al., 2019).
Wave vectors of transverse waves are commonly determined using a minimum
variance analysis (MVA) of the magnetic field, which gives the direction of
minimum variance as the wave-vector propagation direction $\hat{{\bf{k}}}$
(Sonnerup, & Cahill, 1967; Means, 1972; Santolík et al., 2003; Jian et al.,
2009; Verniero et al., 2020). However, MVA analysis, and
eigenvector/eigenvalue determinations, cannot distinguish wave propagation
direction parallel or anti-parallel the minimum variance direction (i.e.
$\hat{{\bf{k}}}$ from -$\hat{{\bf{k}}}$) through magnetic field observations
alone. Together, the degenerate determination of sunward/anti-sunward
propagation direction and the presence of large Doppler shifts preclude
knowledge of the plasma frame polarization from single point magnetic field
measurements. Rigorous constraint of dynamical processes, which generate and
govern the evolution of these waves, inevitably requires an ability to discern
FMW from ICW modes as well as the wave-vector propagation direction.

Including electric field measurements significantly increases the feasibility
of wave-mode identification (Santolík et al., 2003; Bellan, 2012, 2016). Bale
et al. (2005) use electric field measurements to demonstrate the Alfvénic
nature of solar wind turbulence. Salem et al. (2012) additionally use the
anisotropy of the spacecraft frame measurements of $E/B$ to suggest that solar
wind turbulence is consistent with a cascade of quasi-perpendicular Alfvénic
turbulence. Stansby et al. (2016) use electric field measurements to determine
wave-vectors and frequencies in the solar wind frame to provide an empirical
determination of the whistler dispersion relation. In the magnetosphere,
observations of $E$ and $B$ have long been used simultaneously to study
coherent wave phenomena (Cattell et al., 1991; Chaston et al., 1998, 2002).

First results from PSP have revealed the presence of quasi-parallel
propagating ion-scale waves with both left and right hand polarizations, which
are preferentially observed with radial alignments of the magnetic field (Bale
et al., 2019; Bowen et al., 2020a). Verniero et al. (2020) show the
simultaneous existence of coherent circularly polarized waves with unstable 3D
ion-distributions in the PSP/SPAN-ion data. While numerical solutions to warm-
plasma dispersion suggest that both FMW and ICW are potentially driven by
instabilities, the specific mode composition of the observed waves has yet to
be rigorously constrained (Verniero et al., 2020).

The circular polarized events in the PSP data share qualities with wave
phenomena observed in the solar wind at 1 AU by the STEREO spacecraft: quasi-
parallel propagation, mixed handedness at ion scales, and preference for
radially aligned mean fields (Jian et al., 2009). Previous inner-heliosphere
measurements of coherent ion-scale waves by MESSENGER and Helios revealed
significant scaling in the amplitudes and occurrence rates of ion-scale waves
(Jian et al., 2010; Boardsen et al., 2015).

Jian et al. (2010) propose that populations of left and right-handed waves
correspond to ICW modes propagating in opposite directions; the transit time
difference between inward (sunward) and outward (anti-sunward) propagating
waves are due to solar wind expansion, resulting in different radial scalings
for each polarization signature. Boardsen et al. (2015) note that left-handed
ion-scale waves are more common than the right-handed waves, and that the
amplitude of left-hand waves scales with $\delta B^{2}\sim r^{-3}$, consistent
with WKB-like propagation suggested by Hollweg (1974). Meanwhile, the right-
handed waves show significantly shallower radial scaling than the left-handed
waves (Boardsen et al., 2015).

Coherent waves at ion-scales are generated by instabilities associated with
deviations of particle velocity distribution functions from a Maxwellian
distribution (Marsch, 2006). The generation of left-handed ion-cyclotron waves
are commonly associated with a $T_{\perp}/T_{\parallel}>1$ anisotropy, which
drives the Alfvén/ion cyclotron instability. On the other-hand, the
$T_{\parallel}/T_{\perp}>1$ anisotropy typically generates FMW waves by
driving the firehose instability (Gary, 1993). Woodham et al. (2019) assume
that waves with left-hand spacecraft frame polarization propagate uniformly
outward, and demonstrate agreement with a $T_{\perp}/T_{\parallel}>1$
anisotropy; their results suggest that the firehose instability, associated
with $T_{\perp}/T_{\parallel}<1$, drives intrinsically right-handed waves in-
wards (Podesta & Gary, 2011b). Telloni et al. (2019) show strong correlations
with the proton temperature anisotropy and polarization, suggesting that
damping of MHD turbulence leads to anisotropic distributions, which
subsequently emit parallel propagating ICWs.

Temperature anisotropy alone may not produce coherent wave signatures seen in
the solar wind (Gary et al., 2016; Verniero et al., 2020). The presence of
proton (or $\alpha$-particle) beams introduce extra asymmetry into the
velocity distribution, and subsequently more free energy, which may drive
waves (Gary, 1991; Marsch, 1991, 2006; Podesta & Gary, 2011a). Differential
flow speeds between the beam and the core (as well as density ratios and beam
temperature anisotropies) provide additional constraints on the instability
thresholds leading to wave generation (Verscharen et al., 2013; Verniero et
al., 2020).

There is significant observational evidence for the impact of the proton beam
on wave growth. Gary et al. (2016) demonstrate that theoretical growth rates
from observed proton distribution functions may be dominated by the effect of
the beam. Wicks et al. (2016) show that the beam drift correlates well with
the amplitude of coherent-wave “storms” at 1 au. Zhao et al. (2019) suggest,
based on statistical observations, that both temperature anisotropy and core-
beam drift contribute to the growth of left-handed waves at 1 au. In PSP data,
Verniero et al. (2020) demonstrate correlations between the presence of a
proton beam and significant ion-scale wave activity (Bowen et al., 2020a).

PSP provides electromagnetic measurements through the FIELDS instrument (Bale
et al., 2016). Magnetic fields measurements are made through both a low-
frequency flux-gate magnetometer and a high frequency search coil magnetometer
(Bowen et al., 2020b). Additionally, the Digital FIELDS Board (DFB) samples
the four FIELDS antenna in the plane of the spacecraft heat-shield, producing
both single ended and differential voltage waveforms at a survey cadence of up
to 292.969 Sa/Sec (Malaspina et al., 2016). Mozer et al. (2020) outline the
calibration of the electric field ${\bf{E}}_{sc}$ from differential
measurements, where a frequency dependent effective length $\alpha(f)$ with
${\bf{E}}_{sc}=\Delta V_{ij}/\alpha$ is constructed using observations of
coherent waves at ion-scales and above. However, knowledge of the wave phase
speed, and thus a specific wave mode and polarization, is required to perform
a precise correction.

The linearized Faraday equation provides the fundamental relation between
${\bf{E}}$ and ${\bf{B}}$ for electromagnetic oscillations. In the solar wind
plasma frame,

${\bf{E}}_{sw}=-{{v}}_{ph}\hat{{\bf{k}}}\times{\bf{B}}$ (1)

and ${{v}}_{ph}=\omega/|{\bf{k}}|$, such that measurements of ${\bf{B}}$ and
the ${\bf{E}}_{sw}$ components perpendicular to ${\bf{k}}$ uniquely determine
both the wave phase speed and propagation vector. However, in a frame moving
relative to the solar wind, i.e. the spacecraft frame, the electric field
${\bf{E}}_{sc}$ includes contribution both from the solar wind electric field
${\bf{E}}_{sw}$ and convection:

${\bf{E}}_{sc}={\bf{E}}_{sw}-{\bf{V}}_{sw}\times{\bf{B}}.$ (2)

For a transverse quasi-parallel electromagnetic wave, the spacecraft frame
electric field is given by,

$\displaystyle{\bf{E}}_{sc}=-{{v}}_{ph}\hat{{\bf{k}}}\times{\bf{B}}-{\bf{V}}_{sw}\times{\bf{B}}$
(3)
$\displaystyle{\bf{E}}_{sc}=-({{v}}_{ph}\hat{{\bf{k}}}+{\bf{V}}_{sw})\times{\bf{B}}$
(4) $\displaystyle{\bf{E}}_{sc}=-{\bf{V}}_{eff}\times{\bf{B}},$ (5)

with the effective velocity ${\bf{V}}_{eff}$ corresponding to the total wave
speed (convected plus phase speed) measured in the spacecraft frame.

In addition to in situ electromagnetic field measurements, PSP makes
measurements of the plasma distribution functions with the Solar Wind Electron
Alpha and Proton (SWEAP) instrument suite (Kasper et al., 2016). SWEAP can
determine the bulk plasma flow ${\bf{V}}_{sw}$ as well as thermodynamic
moments of the plasma, such as density $n$ and temperature $T$. Integrating
measurements from FIELDS and SWEAP provide a set of measurements capable of
constraining the observed distributions of wave-modes and propagation
directions of coherent polarized waves at ion scales.

##  2 Data

Data is obtained from PSP FIELDS and SWEAP on Nov 04, 2018. A continuous
wavelet transform, similar to Bowen et al. (2020a), is used to locate
circularly polarized ion-scale events. Magnetic and electric fields are
convolved with a set of wavelets $\psi(\xi)$ normalized to unit energy

$W_{j}(s,\tau)=\sum_{i=0}^{N-1}\psi\left(\frac{t_{i}-\tau}{s}\right)B_{j}(t_{i})\\\
$ (6)

with the un-normalized Morlet wavelet $\psi(\xi)$ defined as

$\psi(\xi)=\pi^{-1/4}e^{-i\omega_{0}\xi}e^{\frac{-\xi^{2}}{2}}$ (7)

where $\omega_{0}=6$, with the relationship between wavelet scale and
spacecraft frequency approximated as $f\approx\frac{\omega_{0}}{2\pi s}f_{s}.$
The local, scale dependent, mean magnetic field is computed using a Gaussian
envelope of each wavelet (Bowen et al., 2020a; McManus et al., 2020). The
scale dependent mean field determines a local right-handed field aligned
coordinate system for each scale $\hat{W}=(\hat{B}_{\perp 1},\hat{B}_{\perp
2},\hat{B}_{0})$.

The polarization relative to ${\bf{B}}_{0}$ is computed as

$\sigma_{B\perp}=\frac{\langle-2\text{Im}(B_{\perp 1}B_{\perp
2}^{*})\rangle}{\langle B_{\perp 1}^{2}+B_{\perp 2}^{2}\rangle}$ (8)

where $\langle...\rangle$ denotes time averaging over a period defined by the
$e$-folding of the wavelet (i.e. the cone of influence (Torrence & Compo,
1998)).

Measurement of a left-hand circularly polarized wave gives
$\sigma_{B\perp}=1$, while a right-hand circularly polarized wave gives
$\sigma_{B\perp}=-1$. For each wavelet scale, we identify a polarized event if
$|\sigma_{B\perp}|\geq 0.95$, where the start and end times of the event are
defined at the transition $|\sigma_{B\perp}|=0.8$.

The FIELDS electric field measurements are only available in the spacecraft
$x$-$y$ plane–where $z$ is approximately sun pointing. Once wave events are
identified in the magnetic field, in the mean-field aligned system
($\perp_{1}$,$\perp_{2}$,$\parallel$), the subsequent analysis is performed in
the spacecraft coordinate system ($x$,$y$,$z$) in order to facilitate
comparison with electric field measurements.

As electric field measurements are limited to the spacecraft $x\text{-}y$
plane, we recompute the magnetic helicity measured along the spacecraft $x$
and $y$ axes, $\sigma_{Bxy}$, for each interval where $\sigma_{B\perp}$ meets
the condition for circular polarization. Differences in $\sigma_{B\perp}$
between $\sigma_{Bxy}$ result from variations in the angle between the
magnetic field and solar wind flow $\theta_{BV}$; accordingly, not all
intervals which meet the $\sigma_{B\perp}$ polarization condition maintain
helicity in $\sigma_{Bxy}$. To account for this difference, we only consider
intervals with at least half the magnetic field measurements with a helicity
of $|\sigma_{Bxy}|>0.85$. The electric field polarization $\sigma_{Exy}$ is
additionally computed.

Figure 1 shows four sample events (one event per row) with helical
electromagnetic fields. The first column shows hodograms of the wavelet
filtered magnetic field in the $x-y$ plane. The second column shows the
corresponding wavelet filtered electric field in the $x-y$ plane. The third
and fourth columns show measured $E_{x}$ vs $B_{y}$ and $E_{y}$ vs $B_{x}$.
For transverse circularly polarized waves travelling in the $z$ direction,
Equation 1, indicates that a linear relationship with
$E_{x}/B_{y}\approx{{{v}}_{ph}}_{z}$ and
$E_{y}/B_{z}\approx-{{{v}}_{ph}}_{z}$, where ${{{v}}_{ph}}_{z}$ is in the
measured (spacecraft) frame; $v_{ph(x,y)}$ and $B_{z}$ are taken as small
terms. Respective root-mean-square (rms) ratios $E^{rms}_{x}/B^{rms}_{y}$,
e.g. $E^{rms}_{x}=\sqrt{\langle{E_{x}}^{2}\rangle}$, are plotted, in addition
to the line corresponding to ${{v}}_{ph}={{V}}_{sw}$. In these several cases,
the measured phase speed of the waves is larger than the local solar wind
speed. Additionally the sign of slope of the line indicates outward
propagation in the spacecraft frame.

Figure 1: Four example intervals with significant helical signatures. Each row
shows one interval. Left-most (first) column shows the hodogram of magnetic
fluctuations in $x\text{-}y$ plane. The wavelet frequency and time-span of
each event are noted in the first column. Second column shows corresponding
hodogram of electric-field fluctuations in $x\text{-}y$ plane. Third column
shows phase diagrams of $E_{x}$ and $B_{y}$; rms $E_{x}/B_{y}$ is shown in a
dashed green line, with the total rms $E/B$ shown as solid purple line. Right-
most (fourth) column shows phase diagrams of $E_{y}$ and $B_{x}$; the measured
rms $E_{y}/B_{x}$ is shown in a dashed green line, with the total rms $E/B$ in
purple. The measured solar wind speed along the $z$-direction is plotted in
yellow.

We note that by measuring helicity in the spacecraft coordinate system, we
have removed the dependence on ${\bf{B}}_{0}$, which is necessary to determine
the plasma-frame polarization. However, during the first perihelion, PSP was
connected to a small coronal hole of negative polarity, with a field in the
$+{\bf{\hat{z}}}$ spacecraft direction (the solar wind flow is approximately
in the $-{\bf{\hat{z}}}$ direction) (Bale et al., 2019; Badman et al., 2020).
Since the measurements are taken with a single (positive) background-field
direction, no rectification is necessary to account for sector structure
associated with heliospheric polarity (Woodham et al., 2019; McManus et al.,
2020; Badman et al., 2020). However, deviations from a $+{\bf{\hat{z}}}$
oriented mean field occur during large-scale magnetic field switchbacks, in
which the mean field deflects, and the polarization plane of the quasi-
parallel waves is no longer aligned with the $x\text{-}y$ plane: i.e.
$\sigma_{Bxy}$ significantly deviates from $\sigma_{B\perp}$ (Dudok de Wit et
al., 2019). Circularly polarized events occurring within switchbacks are then
typically excluded by the condition $\sigma_{Bxy}>0.85$, as the polarization
plane is then perpendicular to the spacecraft $z$-axis.

Figure 2: (a) Distribution of phase difference between $B_{x}$ and $B_{y}$ for
circularly polarized intervals. (b) Distribution of phase difference between
$E_{x}$ and $E_{y}$ for circularly polarized intervals. Black curve shows the
total measured distribution for $\sigma_{Bxy}>0.85$, while the red and blue
distributions show the respective conditions of $|\pm\sigma_{Exy}|>0.5$ where
$\pm$ corresponds to left/right (red/blue) helicity measured in the spacecraft
frame.

Figure 2(a) shows the distribution of measured phase difference between
spacecraft $x$ and $y$ magnetic field components for events with
$\sigma_{Bxy}>0.85$. A value of $+90^{\circ}$ corresponds to left-handed
polarization and $-90^{\circ}$ corresponds to right-handed polarization.
Figure 2(b) shows the corresponding distribution for electric field
measurements. While the electric field measurements are similarly peaked at
$\pm 90^{\circ}$, corresponding to circular polarization, the respective peaks
are much broader. There are a significant number of measurements that have
helical magnetic fields, but no helical signature in the electric field. This
is possibly a result of increased noise in the electric field measurements
arising from either the innate non-orthogonality of the sensors, or the
optimization routines which generate electric field from the measured
differential potentials (Mozer et al., 2020). Additionally, we do not exclude
the possibility that physical dynamics other than parallel propagating
transverse waves, e.g. an electrostatic component or non-parallel propagation,
may be relevant in some cases. We limit our study to measurements with
$|\sigma_{Exy}|>0.5$, respectively shown in red (blue) for left (right) handed
polarizations, such that our discussion of circular polarization maintains
connection with physically realized measurements. In the magnetic field 7153
events with $\sigma_{Bxy}>0.85$ (over all scales) were measured,
$|\sigma_{Exy}|>0.5$ is measured in 5191 of these events.

Identifying this subset of events with circular polarization in both magnetic
and electric fields provides an empirically observed distribution, which may
be compared with theoretical behavior of parallel propagating electromagnetic
cold-plasma waves under the effect of Doppler shift.

##  3 Dispersion & Doppler Shift

The cold plasma approximation, commonly used to model transverse
electromagnetic plasma waves, has a dispersion-relation with two branches
corresponding to different plasma-frame polarizations of the electric field
relative to the mean background field:

$\left(\frac{\omega^{\pm}}{\Omega_{i}}\right)^{2}=\left[\frac{kd_{i}}{2}\left(\sqrt{k^{2}d_{i}^{2}+4}\pm
kd_{i}\right)\right]^{2}$ (9)

where $\Omega_{i}=eB_{0}/m_{i}$ is the ion gyrofrequency and
$d_{i}={{V}_{A}}/\Omega_{i}$ is the ion inertial length. The branch with
${{\omega^{FM}}}=\omega^{+}$ corresponds to the FMW wave with an intrinsic
right-handed polarization, and the ${{\omega^{ICW}}}=\omega^{-}$ branch
corresponds to the left-handed ion-cyclotron wave; Figure 3(a) shows the two
branches of the dispersion relation. Note that we take an approximation with
$\omega<\Omega_{e}$, which is quite appropriate for the low-frequency waves
under consideration.

In the spacecraft frame, these polarized transverse electromagnetic waves
appear at the Doppler-shifted frequency

$2\pi f=\omega(k)+{\bf{k}}\cdot{\bf{V}}_{sw}.$ (10)

For quasi-parallel propagation, characteristic of the transverse ion-scale
waves measured in the solar wind (Jian et al., 2010; Boardsen et al., 2015;
Bowen et al., 2020a), Equation 10 reduces to

$2\pi f=\omega(k)\pm k{{V}}_{sw}cos{\theta_{BV}},$ (11)

where ${\theta_{BV}}$ is the angle between the mean magnetic field and solar
wind flow.

Observations from the solar wind in the inner-heliosphere and at 1 AU reveal a
preference for occurrence of waves when the mean field is aligned with the
solar wind flow direction, i.e. ${\theta_{BV}}\sim 0$ or
${\theta_{BV}}\sim\pi$ (Jian et al., 2010; Boardsen et al., 2015; Bale et al.,
2019). Few waves are observed when $\pi/4<{\theta_{BV}}<3\pi/4$, though this
is likely a sampling effect due to single spacecraft measurements of quasi-
parallel waves in anisotropic turbulence Bowen et al. (2020a).

The spacecraft frame frequency $f_{sc}$ (in units of Hz) is a positive
definite quantity such that the observations at Dopper-shifted frequencies
$f$, or $-f$, are both observed at $f_{sc}=|f|$. The plasma frame frequency,
$\omega(k)$, is intrinsically positive; however, the Doppler-shifted frequency
of a wave to a negative value of $f$, causes a change of sign in the measured
helicity in the Doppler-shifted (spacecraft) frame. For the two parallel
propagating ICW and FMW modes, observation of a wave at the spacecraft
frequency $f_{sc}$ most generally corresponds to one of eight cases

$\displaystyle\pm 2\pi f_{sc}$ $\displaystyle={{\omega^{ICW}}}\pm
k{{V}}_{sw}cos{\theta_{BV}}$ (12) $\displaystyle\pm 2\pi f_{sc}$
$\displaystyle={{\omega^{FM}}}\pm k{{V}}_{sw}\cos{\theta_{BV}},$ (13)

where each $\pm$ sign on either side of the equations correspond to a set of
two equations.

Several cases with no real solutions and can be discarded a priori: with the
convention of positive frequency $\omega$ (${\bf{k}}\cdot{\bf{V}}_{sw}<0$
corresponds to backwards propagation), the addition of $k$ cannot produce a
negative spacecraft frequency. Thus, outward propagating FMW and ICW
(${\bf{k}}\cdot{\bf{V}}_{sw}>0$) are never Doppler-shifted to negative
frequency, and always appear in the spacecraft frame with their intrinsic
plasma-frame polarization. Additionally, consideration of the ICW dispersion
in the low $k$ limit gives a phase speed ${{v}}_{ph}^{ICW}={{V}_{A}}$, with
${{v}}_{ph}$ monotonically decreasing with increasing $k$. Under the condition
${{V}_{A}}<{{V}}_{sw}$, inward propagating ICWs are observed at negative
spacecraft frequencies–with an observed right-hand polarization in the
spacecraft frame (we note that in future perihelion encounters, the
${{V}_{A}}<{{V}}_{sw}$ ordering may not hold and a more general approach must
be taken).

Figure 3(b) shows the Doppler shifted cold plasma ICW and FMW dispersion
equations with measured $\Omega_{i}$, $d_{i},$ and ${\bf{V}}_{sw}$, from an
interval with significant signature of left-hand polarization at 09:28 on
November, 04 2018. For a wave at a given spacecraft frequency $f_{sc}$ there
are five possible combinations of Doppler shifted wave-modes: outward (inward)
propagating ICW appearing at a positive (negative) $f$. Outward propagating
FMW waves appear at a positive $f$ and inward propagating FMW waves may appear
at either positive or negative $f$. From Equation 10, the inward propagating
FMW appears right-handed as long as its plasma frame frequency is larger than
the Doppler shift, i.e. ${{\omega^{FM}}}>|{\bf{k}}\cdot{\bf{V}}_{sw}|$. The
inward propagating FMW will occur as a left hand wave in the spacecraft frame
at a frequency if ${{\omega^{FM}}}<|{\bf{k}}\cdot{{\bf{V}}_{sw}}|$.

Clearly, measurements of the spacecraft-frame helicity of the magnetic field
do not uniquely determine wave-mode and propagation direction. However, in
some cases the cold-plasma dispersion can constrain wave-modes within a range
of frequencies. Specifically, only two possible cases lead to left handed
spacecraft-frame polarization, e.g. Figure3(b): outward propagating ion
cyclotron waves and inward propagating FMW waves with
$|{\bf{k}}\cdot{{\bf{V}}_{sw}}|>{{\omega^{FM}}}$. However, outside of the
limit $k_{di}<<1$, ${{\omega^{FM}}}$grows faster than $k$ (e.g. Equation 9)
while the Doppler shift term is proportionate with $-k$. Continuity of the
dispersion relation imposes a critical wave number at which the FMW phase
speed is greater than the solar wind speed, such that Doppler shift cannot
change the sign of the handedness in the spacecraft frame. Thus a minimum
value exists $f^{*}$, i.e. a maximum spacecraft frequency $f^{*}_{sc}=|f^{*}|$
at which, an inward propagating FMW can appear left-handed, e.g. Figure 3(b).
Figure 3(c) shows the curve of $f^{*}_{sc}$ as a function of
${{V}}_{sw}\text{cos}\theta/{{V}_{A}}$, determined through Equations 9 and 10.
A left-handed wave observed at $|f|>f^{*}_{sc}$ is out of the possible
frequency range for an inward propagating, Doppler shifted, FMW and is thus
uniquely explained by an outward propagating ICW.

Figure 3: (a) ICW (solid) and FMW (dashed) cold-plasma dispersion curves in
the solar wind frame. Red and Blue correspond to the left/right handed
transverse wave polarization. (b) Doppler shifted cold plasma ICW and FMW
dispersion into the spacecraft frame using measured $d_{i}$ and
${\bf{V}}_{sw}$; four curves are shown, outward propagating FMW and ICW
(i,ii), and inward propagating FMW and ICW (iii,iv), each curve has been
Doppler shifted outward by the solar wind. The minimum spacecraft frequency of
the inward propagating FMW is shown in orange. For $|f|<f_{sc}^{*}$ there are
at most six possible intersections of the Doppler shifted dispersion relations
with $f_{sc}$. For $|f|>f^{*}_{sc}$ there are four possible intersections
between $|f_{sc}|$ and the Doppler shifted Dispersion relation; of the four,
only one is left hand polarized. (c) Shows the curve of $f_{sc}^{*}/f_{\Omega
i}$ as a function of the ratio
${{V}}_{sw}$$\text{cos}{\theta_{BV}}$/${{V}_{A}}$ for a parallel propagating
wave; the corresponding curve in the plasma frame is shown in purple
$\omega_{sc}^{*}/\Omega_{i}$. (d) Shows a coherent circularly polarized wave
packet with left-hand polarization $|f|>f_{sc}^{*}$. (d) Wavelet spectrogram
of circular polarization shows left-hand polarization with $|f|>f_{sc}^{*}$
indicating that the wave must be an ICW. Figure 4: (a) Distribution of
measured ion-scale waves on 04 November, 2018 (black). The subset of measured
events which occur at a frequency $|f|>f_{sc}^{*}$ is shown in blue.
Additionally, the subset of waves with frequency $|f|>f_{sc}^{*}$ and have a
left-handed polarization is shown in red. (b) Mean ratio between measured rms
electric fields for left-handed waves with $|f|>f_{sc}^{*}$ and theoretical
rms electric fields, assuming outward propagating ICW. At each wavelet scale;
error bars show one standard deviation. The $x$ and $y$ antennas are shown in
purple/orange.

As a specific example, the curves in Figure 3(b) show Doppler shifted cold
dispersion equations with measured $\Omega_{i}$, $d_{i},$ and ${\bf{V}}_{sw}$.
The minimum negative frequency for a FMW with left-handed spacecraft frame
polarization is approximately 2 Hz. However, left-hand circularly polarized
power extends up to $\sim 6$ Hz Figure 3(d-e); at these frequencies the
outward propagating ICW is the only permitted left-handed mode in the
spacecraft frame.

Left-handed waves with $|f|>f^{*}_{sc}$ allow for a direct comparison of
FIELDS observations of ${\bf{E}}_{sc}$, ${\bf{B}}$, and the derived
${\bf{V}}_{eff}$ with the values predicted by cold plasma dispersion. Figure
4(a) shows the number of circularly polarized measurements on Nov, 04 2019 at
each wavelet scale. For each interval, the maximum spacecraft frequency that a
FMW wave can appear left handed ($f^{*}_{sc}$) is determined using measured
values of $d_{i}$, ${\bf{V}}_{sw}$, and the cold plasma dispersion equation
(Equation 9). The distribution of all waves with $|f|>f^{*}_{sc}$ is shown in
blue, and the distribution of left-hand polarized waves with $|f|>f^{*}_{sc}$
is additionally shown in red–we take this as a measured distribution of
outward propagating ICWs.

For each of these ICW events, the minimum variance direction of ${\bf{B}}$ is
determined and taken as the wave propagation direction (Sonnerup, & Cahill,
1967; Means, 1972; Santolík et al., 2003; Verniero et al., 2020). The
observation of $|f|>f^{*}_{sc}$ and left handed polarization breaks the
degeneracy of the MVA determination, and the minimum eigenvector corresponding
to outward propagation is chosen as ${\bf{k}}$. The cold plasma phase speed
corresponding to the intersection of a Doppler shifted ICW with the wave
frequency $f^{*}_{sc}$ is determined through a Newton-Raphson root finding
algorithm. Estimation of the phase speed allows construction of a synthetic
solar wind frame electric field ${\tilde{\bf{E}}_{sw}}$ obtained from
${\bf{B}}$ as

${{\tilde{\bf{E}}_{sw}}}^{ICW}=-v_{ICW}{\hat{{\bf{k}}}}\times{\bf{B}}.$

Subsequently, a model spacecraft frame electric field is given as

$\displaystyle{{\tilde{\bf{E}}_{sc}}}$
$\displaystyle={{\tilde{\bf{E}}_{sw}}}^{ICW}-{\bf{V}}_{sw}\times{\bf{B}}$ (14)
$\displaystyle{{\tilde{\bf{E}}_{sc}}}$
$\displaystyle=-(v_{ICW}{\hat{{\bf{k}}}}+{\bf{V}}_{sw})\times{\bf{B}}$ (15)
$\displaystyle{{\tilde{\bf{E}}_{sc}}}$
$\displaystyle=-{{\tilde{\bf{V}}}_{eff}}^{ICW}\times{\bf{B}}$ (16)

Knowledge that waves with left hand polarization in the spacecraft frame with
$|f|>f^{*}_{sc}$ must be on the ICW branch, allows for comparison of the
theoretical ${\tilde{\bf{E}}_{sc}}$ to the measured ${\bf{E}}_{sc}$. Equation
16, predicts the rms quantities measured in the spacecraft frame,

$\tilde{V}_{z+}=\sqrt{\frac{\langle\tilde{E}_{scx}^{2}\rangle}{\langle{B_{scy}^{2}\rangle}}}$
$\tilde{V}_{z-}=\sqrt{\frac{\langle\tilde{E}_{scy}^{2}\rangle}{\langle{B_{scx}^{2}\rangle}}}.$

Additionally, we test the frequency dependent effective length factors applied
in Mozer et al. (2020) by comparing the theoretical ICW electric fields with
the empirical measurements through calculation of

using

$\displaystyle\gamma_{x}=\sqrt{\frac{\langle{E}_{x}^{2}\rangle}{\langle\tilde{E}_{x}^{2}\rangle}}$
(17)
$\displaystyle\gamma_{y}=\sqrt{\frac{\langle{E}_{y}^{2}\rangle}{\langle\tilde{E}_{y}^{2}\rangle}}$
(18)

Figure 4(b) shows the mean $\gamma_{x}$ and $\gamma_{y}$ measured at each
wavelet scale, with error bars given by one standard deviation; only scales
with more than 10 counts were evaluated. The correction factor $\gamma_{x}$ is
systematically larger than $\gamma_{y}$ indicating that measured electric
fields are systematically larger in the $x$ direction than the $y$ direction.
The larger uncertainties at higher frequencies are likely the result of
measuring low amplitude polarized signals at the edge of broadband wave peak:
Bowen et al. (2020a) show that the proton-gyroscale ($\sim$4-5 Hz) typically
limits the distribution of circularly polarized waves.

Incorporating $\gamma$ into the effective antenna lengths give corrections

$\alpha^{\prime}_{x}=\alpha\gamma_{x}$

and

$\alpha^{\prime}_{y}=\alpha\gamma_{y}.$

A scale-by-scale gain correction is then applied to the measured electric
fields. As these correction factors are of order unity, their omission does
not significantly change the subsequent results–or more importantly, our
conclusions.

Application of $\gamma_{x}$ and $\gamma_{y}$ to waves with a left-hand
spacecraft frame polarization enables measurements of rms speeds

${V}_{z+}=\sqrt{\frac{\langle\tilde{E}_{scx}^{2}\rangle}{\langle{B_{scy}^{2}\rangle}}}$
${V}_{z-}=\sqrt{\frac{\langle\tilde{E}_{scy}^{2}\rangle}{\langle{B_{scx}^{2}\rangle}}}.$

which constrains the mode and wavevector composition of the wave population.
Figure 5 shows distributions of synthetic $\tilde{V}_{z\pm}$ and measured
$\tilde{V}_{z\pm}$, normalized to ${{V}}_{sw}$, with correction factors
$\gamma_{x}$ and $\gamma_{y}$ applied; as the correction factors are of order
unity, their omission does not significantly change the subsequent results.

Figure 5(a) shows the distribution of measured ${V}_{z+}/V_{sw}$ for waves
with $|f|>f_{sc}^{*}$, along with the synthetic distribution
$\tilde{V}_{z+}/V_{sw}$ constructed from the theoretical properties for
outward propagating ICWs. Figure 5(b-c) shows the distribution for left handed
waves, but with $|f|<f^{*}$, for which inward propagating FMW may appear left-
handed. In Figure 5(b) we compare the measured distribution to synthetic
quantities $\tilde{V}_{z+}/V_{sw}$ corresponding to outward propagating ICW;
Figure 5(c) shows synthetic quantities corresponding to inward propagating FMW
waves (two roots are generally available). Notably, the distribution of
measured ${V}_{z+}/V_{sw}$ for frequencies $|f|<f^{*}$ is similar to the
synthetic distribution of outward-propagating ICW.

Similar distributions for measured ${V}_{z-}/V_{sw}$ and synthetic
$\tilde{V}_{z-}/V_{sw}$ are computed with identical results as presented for
${V}_{z+}/V_{sw}$ in Figure 5(a-c). Due to the redundancy in results producing
an additional figure, we elect not to display both sets of distributions.
Analysis of ${V}_{z+}$ and ${V}_{z-}$ both strongly suggest that observed
distribution of left-hand population of waves have an effective speed similar
to outward propagating ICWs.

Figure 5(d-f) show measured distributions of ${V}_{z+}/V_{sw}$ for all
observed right-handed waves. Synthetic distributions $\tilde{V}_{z+}/V_{sw}$
for outward propagating FMW waves are shown in Figure 5(d), while synthetic
distributions for inward propagating ICW are shown in Figure 5(e). Synthetic
data corresponding to an inward- propagating FM, which maintains right-hand
polarization is shown in Figure 5(f). The measured distribution of data again
largely has ${V}_{z+}/V_{sw}>1$, consistent with a dominant population of
outward propagating FMW. Careful inspection of the measured distributions in
Figure 5(d-f) reveals the presence of two peaks, which we identify as
corresponding to two separate wind streams in the data. Separate populations
are plotted for ${{V}}_{sw}>320$ km/s and ${{V}}_{sw}<320$ km/s. The synthetic
data does not resolve two separate distributions. In either case, both
distributions are roughly consistent with outward propagating FMW waves.
Interestingly, no bi-modality associated with stream-speed is observed in the
left-handed waves.

Again, data for ${V}_{z-}/V_{sw}$ were similarly analyzed. However, identical
results are obtained as that from Figure 5(d-f), suggesting that the right-
hand population of waves are propagating in the spacecraft frame with
effective speeds similar to outward propagating FMW. To avoid redundancy, the
distributions of ${V}_{z-}/V_{sw}$ are not shown.

Figure 5: (a) Distributions of measured ${V}_{z+}$ (black) and synthetic
$\tilde{V}_{z+}$ (red) normalized to $V_{sw}$ for left hand wave events with
$|f|>f_{sc}^{*}$, where synthetic data is constructed for outward propagating
ICW. (b) Measured (black) and synthetic (red) normalized to $V_{sw}$ for
events with $|f|<f_{sc}^{*}$, where synthetic data is constructed for outward
propagating ICW. (c) Measured and synthetic distributions for events with
$|f|<f_{sc}^{*}$ where synthetic data is constructed for inward propagating
FMW waves; two roots are possible for the synthetic cata (red and orange). (d)
Distributions of measured ${V}_{z+}$ (black) and synthetic $\tilde{V}_{z+}$
(blue) normalized to $V_{sw}$ for right-hand events; synthetic data is
constructed for outward propagating FM. (e) Distributions of measured (black)
and synthetic (blue) data for right-hand events; synthetic data is constructed
for inward propagating ICW. (f) Distributions of measured and synthetic data
normalized for right-hand events with synthetic data corresponding to an
inward propagating FMW. The measured distribution shown in (d-f) has two
peaks, which correspond to two individual wind streams; the distribution is
broken into data with ${{V}}_{sw}<320$ km/s (green) and ${{V}}_{sw}>320$ km/s
(teal)

##  4 Discussion

PSP observations of circularly polarized waves in the spacecraft frame suggest
that effective wave speeds are typically larger than the bulk solar wind flow,
indicating that most waves are likely outward propagating. While sub-dominant
populations are present with effective speeds less than the solar wind speed,
Figure 5(a-c), it is not clear whether this occurs as the result of
statistical noise, or whether a subdominant population of inward propagating
FMW is actually present. Further case studies may reveal the existence of
individual inward propagating transverse waves.

Outward propagating waves, which are Doppler-shifted to larger spacecraft
frequencies by the solar wind, (with our convention of positive plasma frame
frequency $\omega$), cannot occur at a negative frequency in the spacecraft
frame. This suggests that, typically, no inversion of helicity occurs due to
Dopper shift: e.g. left-hand ICWs observed in the spacecraft frame retain a
left handed polarization. Thus, for the vast majority of measured waves, it is
likely that the measured helicity corresponds to their intrinsic plasma-frame
polarization.

Doppler shifting the cold-plasma dispersion relation reveals a maximum
frequency, $f^{*}_{sc}$ for which inward-propagating right-hand FMWs appear
left handed in the spacecraft frame. Left-handed waves above this frequency
are uniquely described by ICWs propagating outward. The measured electric
field of waves which are left-handed and have $f>f^{*}_{sc}$, are to within
our certainty, consistent with theoretical predictions for ICW made by the
cold plasma dispersion. This analysis is made possible by an empirical
verification of electric field calibration presented in Mozer et al. (2020) in
the range of ion-scales.

Many previous studies utilize the low wave-number approximation $kd_{i}\ll 1$,
in which ${{v}}_{ph}\sim{{V}_{A}}$, in order to study ion scale waves (Jian et
al., 2009, 2014; Gary et al., 2016; Wicks et al., 2016; Woodham et al., 2019;
Bowen et al., 2020a). However, the low wave number approximation is not
broadly applicable, as these waves occur at ion-scales with $kd_{i}\sim 1$;
our consideration of the full plasma dispersion presents a significant
advancement in understanding ion-scale waves. Verniero et al. (2020) similarly
pursued a more explicit estimate of $\mathbf{k}$, through Doppler-shifting the
warm-plasma dispersion relation to determine the intersection with spacecraft
frame frequency. The method of Doppler-shifting the dispersion relation curves
is a promising method to infer intrinsic wave handedness, and has the
advantage of being independent of the validity of the Taylor hypothesis.

The FMW and ICW dispersion relations are both non-linear at $kd_{i}\sim 1$,
e.g. Figure 3, implying that the observed ion-scale waves should be subject to
dispersive effects. However, broadband wave packets are commonly observed,
which contain a range of wave-phase speeds, e.g. Figure 3(e). Over relatively
short time-scales, broadband wave-packets at ${{k_{di}}}\sim 1$ can disperse
significantly. The lack of observations of waves with dispersive
characteristics suggests that waves are driven locally, and may be quickly
damped. In contrast, if the waves are subject to WKB like transport, the
dispersion of packets should be evident. It may be possible to constrain wave-
packet life times, and thus damping, and associated heating rates, by
thoroughly considering the dispersive effects on broadband waves.

We note that ICW and FMW wave phase speeds computed with warm-plasma
dispersion relations will deviate from values derived under the cold plasma
approximation. However, over the range of observed waves around
$kd_{i}=[0.3,1]$, Verniero et al. (2020) show that for core $\beta=0.43$,
beam-to-core drift speed of 1.5${{V}_{A}}$, and $\alpha$-to-core drift speed
of 0.7${{V}_{A}}$, the ICW warm plasma phase speed is 9%-33% lower than the
ICW cold plasma phase speed; likewise, the FMW phase speed is 1.3%-3.6% faster
the FMW cold plasma phase speed (see Figure 7(c) of Verniero et al. (2020)).
These perturbations are not significant enough to affect our interpretation of
Figure 5; however, warm plasma effects may contribute to the dispersion in the
measured distribution of ${V}_{z+}/V_{sw.}$ We suggest that while solutions to
the warm-plasma dispersion are needed to understand growth and damping rates
of these waves, propagation remains well described by the cold-plasma
approximation.

Our measurements of effective phase speeds suggest that the vast majority of
waves are outward propagating. Observations of left and right handed waves
have at times been interpreted as observations of counter propagating waves,
e.g. Jian et al. (2009). The observed radial scaling of wave properties (e.g.
amplitudes and occurrence rates) has been attributed to differences in transit
time (Jian et al., 2010; Boardsen et al., 2015). We argue that radial scaling
is more likely related to the heliospheric evolution of the ion distribution
function (Hellinger et al., 2013; Huang et al., 2020).

Observations and theoretical analysis of quasi-parallel propagating ion-scale
waves, suggest three general types of ion-driven instabilities that result in
either left or right-hand wave generation (Jian et al., 2009, 2010, 2014;
Wicks et al., 2016; Gary et al., 2016; Verniero et al., 2020): (1) proton
anisotropies with $T_{\perp}>T_{\parallel}$ drive the ion-cyclotron
instability Gary et al. (2001), resulting in left-handed ICWs, (2) proton
anisotropies with $T_{\parallel}>T_{\perp}$ drive the firehose instability,
generating right-handed FMWs Hellinger et al. (2006), and (3) relative drift
speeds between ion components drive magnetosonic instabilities, also producing
right-handed FMWs (Goldstein et al., 2000; Gary et al., 2000, 2016). Though
Podesta & Gary (2011b) suggest that FMW driven through temperature anisotropy
should be inward propagating, observations of proton distribution functions at
1 au suggest that both temperature anisotropy and core-beam drift
instabilities, are capable of driving outward propagating FMW (Gary et al.,
2016).Furthermore, Verniero et al. (2020) show that waves observed by PSP are
likely correlated with a proton beam. Our observation of a mixed distribution
of outward propagating wave-modes suggests that a variety of instabilities are
important in the young solar wind.

Generally, Klein et al. (2018) suggest that the solar wind at 1 au is unstable
approximately half of the time, with a vast range of both fluid and kinetic
scale instabilities capable of converting free energy into electromagnetic
waves (Yoon, 2017; Verscharen et al., 2019). In the inner-heliosphere, these
instabilities are ever-more present (Klein et al., 2019). Our constraint of
mode composition and wave vector distribution of ion scale waves in the solar
wind through electric field measurements, provides a key step in understanding
how energy in the wave-field is redistributed to the plasma.

##  5 Acknowledgements

The FIELDS instrument was designed and developed under NASA contract
NNN06AA01C.

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# Success functions in large contests

Yaron Azrieli and Christopher P. Chambers Department of Economics, The Ohio
State University, 1945 North High street, Columbus, OH 43210,
<EMAIL_ADDRESS>of Economics, Georgetown University, ICC 580 37th
and O Streets NW, Washington DC 20057<EMAIL_ADDRESS>
###### Abstract

We consider contests with a large set (continuum) of participants and
axiomatize contest success functions that arise when performance is composed
of both effort and a random element, and when winners are those whose
performance exceeds a cutoff determined by a market clearing condition. A co-
monotonicity property is essentially all that is needed for a representation
in the general case, but significantly stronger conditions must hold to obtain
an additive structure. We illustrate the usefulness of this framework by
revisiting some of the classic questions in the contests literature.

##  1 Introduction

Models of contests, in which agents compete for awards by making irrevocable
investments, have been used in the literature to analyze a wide variety of
phenomena. Starting from the seminal works of Tullock (1980, 2001) and Lazear
and Rosen (1981), many of these models assume that the identity of the
winner(s) is not a deterministic function of agents’ choices, i.e., that the
outcome has some randomness to it. The mapping from strategy profiles to the
probability of winning of each participant is known as a _Contest Success
Function (CSF)_ ; it is a key ingredient of the model determining agents’
incentives and consequentially the equilibria of the game.

We study CSFs when the set of participants is large (a continuum). As
demonstrated in the literature (e.g., Olszewski and Siegel (2016, 2019, 2020),
Morgan et al. (2018), Adda and Ottaviani (2023), Azrieli (2024)), as well as
in the last section of this paper, working “in the limit” with an infinite set
of contestants is often more tractable than with a large but finite set.
However, specifying a CSF in this context is not as straightforward as in the
finite case, and the properties of such functions are not well understood. The
main contribution of this work is to axiomatize a class of CSFs that we refer
to as _Random Performance Functions (RPFs)_. Roughly speaking, in an RPF the
effort choice of an agent is randomly mapped into performance, and the agent
wins if their performance exceeds a cutoff set by the principal; the cutoff is
adjusted based on the distribution of efforts in the population to satisfy a
market clearing condition, namely, that the total mass of winners is equal to
the budget.111Morgan et al. (2018), Adda and Ottaviani (2023) and Azrieli
(2024) all use RPFs in their models.

The first main result characterizes RPFs without assuming any particular
structure of the mapping from effort to performance; the only restriction is
that higher effort implies higher distribution of performance in the sense of
first-order stochastic dominance. We show that the key property needed for
such a representation is Co-monotonicity, which means that if an agent
exerting effort $e$ has a higher winning probability when facing competition
(distribution of efforts) $p$ than when facing competition $p^{\prime}$, then
the same is true for any other effort level $e^{\prime}$. In other words,
under Co-monotonicity one can rank effort distributions $p$ based on how
competitive they are in an unambiguous way. Theorem 1 shows that this axiom,
along with standard continuity and monotonicity properties, are necessary and
sufficient for a CSF to be an RPF.

In Theorem 2 we add more structure and characterize the popular specification
where performance is the sum of effort plus a noise term. As can be expected,
this is much more demanding as it requires invariance of the winning
probability to certain shifts of the effort distribution. We further show that
in this case the distribution of noise can be identified from the CSF up to
translations.

We view these two theorems as providing a foundation for the use of RPFs in
models of large contests. Isolating the necessary and sufficient conditions
clarifies the structure of agents’ incentives under this type of CSFs.
Alternatively, in environments where the CSF can be estimated from observable
data our results facilitate testing whether an RPF is an appropriate modeling
choice. As an example, consider competition among scientists for research
grants: The quality distribution of submissions in a given cycle can be
estimated ex-post by the impact of each proposal, and so does the rate of
success of each quality level; one can then test whether Co-monotonicity holds
to determine if an RPF captures the evaluation process of proposals.

In the finite-agent case, the CSF proposed by Tullock (1980) and its
generalizations are by far the most popular. Skaperdas (1996) provides the
first axiomatization of this class by using a population consistency
condition. Clark and Riis (1998) provide an asymmetric generalization of the
homogeneous special case of Skaperdas (1996), interpreting the consistency
condition as an analogue of the choice axiom of Luce (1959). Other
generalizations include Myerson and Wärneryd (2006) for the case of unknown
population size, Münster (2009) who considers contests between groups, and Rai
and Sarin (2009) where agents’ investments are multidimensional. Berry (1993)
was the first to propose an extension of Tullock’s CSF to the case of multiple
prizes as we have here; a critique of Berry’s extension and an alternative
specification were suggested by Clark and Riis (1996). See Sisak (2009) for a
survey of the literature on multi-prize contests.

Recently there have been several attempts to extend Tullock-type CSFs to
contests with a continuum of agents, see for example Lahkar and Mukherjee
(2023); Lahkar and Sultana (2023) and Doğan et al. (2023). The goal of these
papers is to obtain explicit characterization of the equilibrium in cases
where this is not possible with a finite set of contestants. However, the
range of these proposed CSFs is not bounded above by one, so it is hard to
interpret them as winning probabilities or shares of a prize. The RPFs
axiomatized in this paper to dot have this issue. While they do not yield
closed form expressions for equilibria, we illustrate in Section 5 that they
can be used to address some of the classic questions in the contests
literature.

##  2 Contest success functions

There is a continuum of agents with total mass one that can choose whether to
participate in a contest and if so how much effort to exert. The contest
designer’s budget is sufficient to give prizes to a mass of $k\in(0,1)$
agents. We refer to $k$ as the budget and fix it until Section 5.222We assume
that all winners receive the same prize so that there are only two possible
outcomes – win or lose – for each agent.

Let $E=(0,\infty)$ be the set of possible effort levels agents can choose
from. A _measure on $E$_ is a non-negative, $\sigma$-additive function defined
over the Borel sets in $E$. We denote by $\Delta_{k}$ the set of measures $p$
on $E$ with $p(E)\in(k,1]$. Each $p\in\Delta_{k}$ is interpreted as a
potential ‘distribution of efforts’ corresponding to some strategy profile. We
sometimes refer to $p$ as the _competition_. We allow for $p(E)<1$ to capture
situations where not all agents choose to participate in the contest, namely,
$p(E)$ is the fraction who participate and $1-p(E)$ stay out. The restriction
to $p(E)>k$ guarantees that there is not enough budget to reward all
participants; if $p(E)\leq k$ we simply assume that any effort level
guarantees a win.

It will be convenient to use the following notation. For any $e\in E$ the
Dirac measure $\delta_{e}\in\Delta_{k}$ is defined by $\delta_{e}(A)=1$ if
$e\in A$ and $\delta_{e}(A)=0$ otherwise. If $p\in\Delta_{k}$ and $\bar{e}\in
E$ then $p\dotplus\bar{e}$ is the ‘right-shift’ of $p$ by $\bar{e}$, namely,
the measure defined by $(p\dotplus\bar{e})(A)=p(\\{e-\bar{e}:e\in
A\cap(\bar{e},+\infty)\\})$. Finally, we write $\int$ rather than $\int_{E}$
whenever the domain of integration is $E$.

The object of interest is a mapping $W:E\times\Delta_{k}\to[0,1]$. Here
$W(e,p)$ is interpreted as the probability of winning for an agent that exerts
effort $e$ given that they face competition $p$.333Note that $W(e,p)$ is
defined even if $e$ is not in the support of $p$. This allows one to determine
whether a given strategy profile is a Nash equilibrium by considering all
possible effort deviations. Following the literature, we refer to $W$ as a
_Contest Success Function_.

######  Definition 1.

A Contest Success Function (CSF) is a mapping $W:E\times\Delta_{k}\to[0,1]$
such that, for any fixed $p\in\Delta_{k}$, $W(e,p)$ is measurable in $e$ and
satisfies

$\int W(e,p)dp(e)=k.$ (1)

Equation (1) is a market clearing condition requiring that for any $p$ the
total mass of winners is equal to $k$. In other words, the designer spends
their entire budget irrespective of the distribution of efforts.444This is
similar to the constraint that the sum of winning probabilities of all agents
is 1 in the classic finite setup, see for example property A1 in Skaperdas
(1996).

##  3 Random Performance Functions

We are interested in characterizing those CSFs $W$ that arise as a result of
the following process: The performance of an agent that exerts effort $e$ is
randomly drawn according to a distribution that depends on $e$, where higher
effort is more likely to result in better performance; the designer sets a
threshold performance value $s$ and the winners are those whose performances
exceed this cutoff; the threshold $s$ is adjusted based on the competition $p$
so that (1) is satisfied. This is formalized in the next definition.

######  Definition 2.

A CSF $W$ is a _Random Performance Function (RPF)_ if there exists a
collection of cumulative distribution functions $\\{F_{e}\\}_{e\in E}$ such
that the following hold:

  1. 1.

Each $F_{e}$ is continuous and strictly increasing.

  2. 2.

For each $x\in\mathbb{R}$, the mapping $e\mapsto F_{e}(x)$ is continuous,
strictly decreasing, and satisfies $\lim_{e\uparrow+\infty}F_{e}(x)=0$.

  3. 3.

For each $e\in E$ and $p\in\Delta_{k}$ it holds that $W(e,p)=1-F_{e}(s(p))$,
where $s(p)\in\mathbb{R}$ is defined as the unique solution to the equation
$\int[1-F_{e}(s(p))]dp(e)=k$.

The first condition in the definition is for technical reasons. The second
implies that the distribution of performance for higher effort levels first-
order stochastically dominates the distribution for lower effort levels, and
that very high effort is likely to result in high performance. Condition 3
means that $W$ can be represented by the collection $\\{F_{e}\\}_{e\in E}$.

######  Remark 1.

Existence and uniqueness of $s(p)$ that satisfies the equality in the above
definition is guaranteed by our assumptions. Indeed, by property 2 of the
definition $1-F_{e}(s)$ is a measurable function of $e$ so the integral
$\int[1-F_{e}(s)]dp(e)$ is well-defined, and this integral is continuous and
strictly increasing in $s$ by part 1 of the definition. Further, this integral
converges to zero as $s\uparrow+\infty$ and converges to $p(E)>k$ as
$s\downarrow-\infty$. Thus, there exists a unique $s$ such that
$\int[1-F_{e}(s)]dp(e)=k$ holds.

###  3.1 Characterization

To characterize RPFs we introduce the following properties.

$e$-Continuity: For any fixed $p\in\Delta_{k}$, $W(e,p)$ is continuous in $e$.

$p$-Continuity: For any fixed $e\in E$ and $p,p^{\prime}\in\Delta_{k}$,
$W(e,\alpha p+(1-\alpha)p^{\prime})$ is a continuous function of
$\alpha\in[0,1]$.

Monotonicity: For any fixed $p\in\Delta_{k}$, $W(e,p)$ is strictly increasing
in $e$ and $\lim_{e\uparrow+\infty}W(e,p)=1$.

Competitiveness: For any fixed $e\in E$,
$\lim_{\bar{e}\uparrow+\infty}W(e,\delta_{\bar{e}})=0$.

Co-monotonicity: For all $e\in E$ and $p,p^{\prime}\in\Delta_{k}$, if
$W(e,p)\geq W(e,p^{\prime})$ then $W(e^{\prime},p)\geq
W(e^{\prime},p^{\prime})$ for all $e^{\prime}\in E$.

######  Theorem 1.

The contest success function $W$ is an RPF if and only if it satisfies
$e$-Continuity, $p$-Continuity, Monotonicity, Competitiveness, and Co-
monotonicity.

###### Proof.

(Necessity) Suppose that $W$ is an RPF. First, $e$-Continuity is immediate
since $W(e,p)=1-F_{e}(s(p))$ and the mapping $e\mapsto F_{e}(x)$ is assumed to
be continuous for any fixed $x$. To show $p$-Continuity it is enough to prove
that $s(\alpha p+(1-\alpha)p^{\prime})$ is a continuous function of $\alpha$
for any given pair $p,p^{\prime}\in\Delta_{k}$. Consider a sequence
$\\{\alpha_{n}\\}$ with $\alpha_{n}\to\alpha\in(0,1)$. For each $n$ denote
$s_{n}=s(\alpha_{n}p+(1-\alpha_{n})p^{\prime})$. The sequence $\\{s_{n}\\}$ is
clearly bounded (e.g. $s_{n}$ is between $s(p)$ and $s(p^{\prime})$ for all
$n$), so it has a convergent subsequence $s_{n_{l}}\to s_{0}$. Note that for
any $n$ we have

$k=\int(1-F_{e}(s_{n}))d(\alpha_{n}p+(1-\alpha_{n})p^{\prime})(e)=\alpha_{n}\int(1-F_{e}(s_{n}))dp(e)+(1-\alpha_{n})\int(1-F_{e}(s_{n}))dp^{\prime}(e).$

Therefore, using the dominated convergence theorem,

$\displaystyle k$ $\displaystyle=$
$\displaystyle\lim_{l}\alpha_{n_{l}}\int(1-F_{e}(s_{n_{l}}))dp(e)+(1-\alpha_{n_{l}})\int(1-F_{e}(s_{n_{l}}))dp^{\prime}(e)=$
$\displaystyle\alpha\int(1-F_{e}(s_{0}))dp(e)+(1-\alpha)\int(1-F_{e}(s_{0}))dp^{\prime}(e),$

and therefore $s(\alpha p+(1-\alpha)p^{\prime})=s_{0}$. We have established
that $\\{s_{n}\\}$ has a subsequence converging to $s(\alpha
p+(1-\alpha)p^{\prime})$, but note that we could have applied the same
argument starting from any subsequence of $\\{s_{n}\\}$ rather than from the
original sequence. Thus, any subsequence of $\\{s_{n}\\}$ has a sub-
subsequence that converges to $s(\alpha p+(1-\alpha)p^{\prime})$, and hence
the entire sequence converges to this limit.

To show Monotonicity, fix some $p$ and suppose that $e<e^{\prime}$. Then
$W(e,p)=1-F_{e}(s(p))<1-F_{e^{\prime}}(s(p))=W(e^{\prime},p)$, where the
inequality is by condition 2 in Definition 2. Also, by the same condition we
have that
$\lim_{e\uparrow+\infty}W(e,p)=\lim_{e\uparrow+\infty}1-F_{e}(s(p))=1$.

Next, consider the function $g(\bar{e})=s(\delta_{\bar{e}})$. Applying (1) to
the measure $\delta_{\bar{e}}$ we have that $F_{\bar{e}}(g(\bar{e}))=1-k$.
Thus, by condition 2 in Definition 2, $g$ is an increasing function. If $g$
was bounded, say by $\bar{s}$, then we would have $F_{\bar{e}}(\bar{s})\geq
F_{\bar{e}}(g(\bar{e}))=1-k$ for all $\bar{e}$, which is impossible since
condition 2 in Definition 2 requires that
$\lim_{\bar{e}\uparrow+\infty}F_{\bar{e}}(\bar{s})=0$. It follows that
$\lim_{\bar{e}\uparrow+\infty}s(\delta_{\bar{e}})=+\infty$ and therefore that
$\lim_{\bar{e}\uparrow+\infty}W(e,\delta_{\bar{e}})=0$ for any fixed $e$,
proving Competitiveness.

Finally, Co-monotonicity is immediate as well since $W(e,p)\geq
W(e,p^{\prime})$ implies that $s(p)\leq s(p^{\prime})$, from which
$W(e^{\prime},p)\geq W(e^{\prime},p^{\prime})$ follows.

(Sufficiency) Suppose $W$ satisfies $e$-Continuity, $p$-Continuity,
Monotonicity, Competitiveness and Co-monotonicity. Fix some $e_{0}\in E$. We
break the proof into several steps.

Step 1: We claim that $\\{W(e_{0},p)~{}:~{}p\in\Delta_{k}\\}=(0,1)$. To see
this, denote by $A$ the set on the left-hand side of the equality. From
Competitiveness we immediately get that $\inf A=0$. Consider
$p=\alpha\delta_{e_{0}}$ for some $\alpha>k$. By (1) we have that $\alpha
W(e_{0},p)=k$. Taking the limit as $\alpha$ converges to $k$ shows that $\sup
A=1$. Finally, note that if for some $p,p^{\prime}$ we have $W(e_{0},p)\geq
W(e_{0},p^{\prime})$, then $W(e_{0},p)\geq W(e_{0},\alpha
p+(1-\alpha)p^{\prime})\geq W(e_{0},p^{\prime})$ for any $\alpha\in(0,1)$.
Indeed, suppose that was not the case, say $W(e_{0},p)<W(e_{0},\alpha
p+(1-\alpha)p^{\prime})$. Then by Co-monotonicity the same inequality would be
true for every $e$. But then

$\displaystyle k$ $\displaystyle=$ $\displaystyle\int W(e,\alpha
p+(1-\alpha)p^{\prime})d(\alpha p+(1-\alpha)p^{\prime})(e)=$
$\displaystyle\alpha\int W(e,\alpha
p+(1-\alpha)p^{\prime})dp(e)+(1-\alpha)\int W(e,\alpha
p+(1-\alpha)p^{\prime})dp^{\prime}(e)>$ $\displaystyle\alpha\int
W(e,p)dp(e)+(1-\alpha)\int W(e,p^{\prime})dp^{\prime}(e)=k,$

where the first and last equalities are from (1). A similar argument applies
if $W(e_{0},p^{\prime})<W(e_{0},\alpha p+(1-\alpha)p^{\prime})$. Combining
this with $p$-Continuity proves that indeed $A=(0,1)$.

Step 2: Let $\phi:(0,1)\to\mathbb{R}$ be a strictly increasing and continuous
function whose image is the entire real line. For any $p\in\Delta_{k}$ define
$s(p)=\phi(1-W(e_{0},p))$. Notice that $s(p)\geq s(p^{\prime})$ if and only if
$W(e_{0},p)\leq W(e_{0},p^{\prime})$. For any $x\in\mathbb{R}$ choose $p_{x}$
such that $s(p_{x})=x$; existence of $p_{x}$ is guaranteed by Step 1. Define
$F_{e}(x)=1-W(e,p_{x})$ for every $e\in E$.

Step 3: In this step we show that each $F_{e}$ just defined is a continuous
and strictly increasing cdf. Fix $e$. For monotonicity, let $x<x^{\prime}$ and
note that this implies $s(p_{x})<s(p_{x^{\prime}})$ and therefore
$W(e_{0},p_{x})>W(e_{0},p_{x^{\prime}})$. It follows from Co-monotonicity that
$W(e,p_{x})>W(e,p_{x^{\prime}})$ and hence $F_{e}(x)<F_{e}(x^{\prime})$.

Next, we claim that for any $p$ there is $x$ such that $W(e,p)=W(e,p_{x})$.
Indeed, by construction we have that $W(e_{0},p)=W(e_{0},p_{s(p)})$, so by Co-
monotonicity $W(e,p)=W(e,p_{s(p)})$ as well. Therefore,

$\\{W(e,p_{x})~{}:~{}x\in\mathbb{R}\\}=\\{W(e,p)~{}:~{}p\in\Delta_{k}\\}=(0,1),$

where the last equality is as in Step 1 of this proof. Together with the
monotonicity property just shown this implies that
$\lim_{x\downarrow-\infty}F_{e}(x)=0$ and $\lim_{x\uparrow+\infty}F_{e}(x)=1$.

Finally, for continuity, let $x_{n}\rightarrow x$. Then there are
$\underline{x}$, $\overline{x}$ for which $\underline{x}<x_{n}<\overline{x}$
for all $n$. Using the same argument as in Step 1, for each $n$ we can find
$\alpha_{n}\in(0,1)$ such that
$s(\alpha_{n}p_{\underline{x}}+(1-\alpha_{n})p_{\overline{x}})=x_{n}$. Take a
subsequence $\alpha_{n_{l}}$ converging to some $\alpha$ and observe that

$x=\lim x_{n_{l}}=\lim
s(\alpha_{n_{l}}p_{\underline{x}}+(1-\alpha_{n_{l}})p_{\overline{x}})=s(\alpha
p_{\underline{x}}+(1-\alpha)p_{\overline{x}}),$ (2)

where the last equality follows from $p$-Continuity. Consequently,

$F_{e}(x_{n_{l}})=1-W(e,p_{x_{n_{l}}})=1-W(e,\alpha_{n_{l}}p_{\underline{x}}+(1-\alpha_{n_{l}})p_{\overline{x}})\to
1-W(e,\alpha p_{\underline{x}}+(1-\alpha)p_{\overline{x}})=F_{e}(x),$

where the first equality is by definition, the next follows from
$s(\alpha_{n}p_{\underline{x}}+(1-\alpha_{n})p_{\overline{x}})=x_{n}=s(p_{x_{n}})$
and Co-monotonicity, the limit is by $p$-Continuity, and the last equality is
by (2). Since we could have applied the same argument starting from any
subsequence of $\\{x_{n}\\}$, it follows that $F_{e}(x_{n})\to F_{e}(x)$.

Step 4: Here we argue that condition 2 of Definition 2 is satisfied. Fix
$x\in\mathbb{R}$. Since $F_{e}(x)=1-W(e,p_{x})$, and by $e$-Continuity, we
have that $e\mapsto F_{e}(x)$ is continuous. This mapping is strictly
decreasing by Monotonicity. And $\lim_{e\uparrow+\infty}F_{e}(x)=0$ also
directly follows from Monotonicity.

Step 5: We conclude by showing that the collection $\\{F_{e}\\}$ represents
$W$ as in condition 3 of Definition 2. Fix some $\bar{p}\in\Delta_{k}$ and
denote $\bar{x}=s(\bar{p})$. Then by construction we have that
$s(p_{\bar{x}})=\bar{x}=s(\bar{p})$ and therefore
$W(e_{0},\bar{p})=W(e_{0},p_{\bar{x}})$. By Co-monotonicity,
$W(e,\bar{p})=W(e,p_{\bar{x}})=1-F_{e}(s(\bar{p}))$ for all $e\in E$, and it
follows from (1) that

$k=\int W(e,\bar{p})d\bar{p}(e)=\int(1-F_{e}(s(\bar{p})))d\bar{p}(e).$

∎

##  4 Additive noise

The definition of an RPF allows for a very general relationship between effort
and performance. Indeed, the only substantial requirement is that higher
effort implies higher distribution of performance in the sense of first-order
stochastic dominance. In this section we add more structure and characterize
the case where performance is obtained by adding a random shock to the effort
choice. More explicitly, performance is equal to $e+X$, where $X$ is a random
variable representing the noise term.

######  Definition 3.

The contest success function $W$ is an Additive RPF (A-RPF) if there is a
continuous and strictly increasing cdf $F$ such that $W$ has an RPF
representation with $F_{e}(x)=F(x-e)$ for every $e\in E$.

###  4.1 Characterization

To characterize Additive RPFs we introduce the following two additional
properties.

Invariance to Common Shifts: For every $e,a\in E$ and $p\in\Delta_{k}$,
$W(e,p)=W(e+a,p\dotplus a)$.

Invariance to $p$ Shifts: For every $e,e^{\prime},a\in E$ and
$p,p^{\prime}\in\Delta_{k}$, if $W(e,p)=W(e^{\prime},p^{\prime})$, then
$W(e,p\dotplus a)=W(e^{\prime},p^{\prime}\dotplus a)$.

######  Theorem 2.

The contest success function $W$ is an A-RPF if and only if it satisfies
$e$-Continuity, Monotonicity, Invariance to Common Shifts, and Invariance to
$p$ Shifts.

###### Proof.

(Necessity) Suppose $W$ is an A-RPF with cdf $F$. Since $W$ is in particular
an RPF, $e$-Continuity and Monotonicity follow from Theorem 1. Fix some
$p\in\Delta_{k}$ and $a\in E$. Then

$\int[1-F(s(p)+a-e)]d(p\dotplus a)(e)=\int[1-F(s(p)-e)]d(p)(e)=\int
W(e,p)dp(e)=k,$

where the first equality is a change of variable, the second by the assumption
that $F$ represents $W$, and the last by (1). It follows that $s(p\dotplus
a)=s(p)+a$.

Now, for Invariance to Common Shifts, we have for every $e,a\in E$ and
$p\in\Delta_{k}$,

$W(e+a,p\dotplus a)=1-F(s(p\dotplus
a)-(e+a))=1-F(s(p)+a-(e+a))=1-F(s(p)-e)=W(e,p).$

Similarly, suppose that $W(e,p)=W(e^{\prime},p^{\prime})$ holds for some
$e,e^{\prime}\in E$ and $p,p^{\prime}\in\Delta_{k}$. Then
$1-F(s(p)-e)=1-F(s(p^{\prime})-e^{\prime})$, which, by the strict monotonicity
of $F$, implies that $s(p)-e=s(p^{\prime})-e^{\prime}$. But then for any $a\in
E$,

$s(p\dotplus a)-e=s(p)+a-e=s(p^{\prime})+a-e^{\prime}=s(p^{\prime}\dotplus
a)-e^{\prime}.$

This shows that $W(e,p\dotplus a)=W(e^{\prime},p^{\prime}\dotplus a)$ and
hence that $W$ satisfies Invariance to $p$ Shifts.

(Sufficiency) Fix some arbitrary $(e^{*},p^{*})\in E\times\Delta_{k}$. Define
$F$ by

$\displaystyle F(x)=\left\\{\begin{array}[]{ll}1-W(e^{*},p^{*}\dotplus
x)&\textit{ if }~{}~{}x\geq 0,\\\ 1-W(e^{*}-x,p^{*})&\textit{ if
}~{}~{}x<0.\\\ \end{array}\right.$ (5)

We start by showing that $F$ is strictly increasing. If $x_{1}<x_{2}\leq 0$
then by Monotonicity $W(e^{*}-x_{1},p^{*})>W(e^{*}-x_{2},p^{*})$, so
$F(x_{1})<F(x_{2})$. If $0\leq x_{1}<x_{2}$ then

$W(e^{*},p^{*}\dotplus x_{1})=W(e^{*}+(x_{2}-x_{1}),(p^{*}\dotplus
x_{1})\dotplus(x_{2}-x_{1})))=W(e^{*}+(x_{2}-x_{1}),p^{*}\dotplus
x_{2})>W(e^{*},p^{*}\dotplus x_{2}),$

where the first equality is by Invariance to Common Shifts, the next is just a
simplification, and the inequality is by Monotonicity. Thus,
$F(x_{1})<F(x_{2})$ and $F$ is strictly increasing.

Next, we argue that $F$ is continuous. For $x<0$ this follows immediately from
$e$-Continuity. For $x\geq 0$, fix some $\bar{e}>x$. Then by Invariance to
Common Shifts,

$W(e^{*},p^{*}\dotplus
x)=W(e^{*}+(\bar{e}-x),p^{*}\dotplus(x+(\bar{e}-x)))=W(e^{*}+\bar{e}-x,p^{*}\dotplus\bar{e}).$

Thus, by $e$-Continuity we get that $F$ is continuous at any $x\geq 0$ as
well.

It remains to show that $F$ represents $W$ as in Definition 3. Fix some
$p\in\Delta_{k}$. Our goal is to show that there is $s\in\mathbb{R}$ such that
$W(e,p)=1-F(s-e)$ for every $e\in E$. Indeed, if that is true then we would
have

$\int[1-F(s-e)]dp(e)=\int W(e,p)dp(e)=k,$

implying that $s$ is the performance cutoff for $p$ and therefore that $F$
represents $W$.

Take some $e>e^{\prime}$ two effort levels. By Monotonicity $W(e,p)\in(0,1)$,
so there is a unique $s$ such that $W(e,p)=1-F(s-e)$. We need to show that
$W(e^{\prime},p)=1-F(s-e^{\prime})$ and we distinguish between two cases.

Case 1 – $s\geq e$: Here we have
$W(e,p)=1-F(s-e)=W(e^{*},p^{*}\dotplus(s-e))$. Thus, applying the axiom
Invariance to $p$ Shifts with $a=e-e^{\prime}$ we obtain

$W(e,p\dotplus(e-e^{\prime}))=W(e^{*},p^{*}\dotplus(s-e+e-e^{\prime}))=W(e^{*},p^{*}\dotplus(s-e^{\prime})).$
(6)

We can now write

$W(e^{\prime},p)=W(e^{\prime}+(e-e^{\prime}),p\dotplus(e-e^{\prime}))=W(e,p\dotplus(e-e^{\prime})=W(e^{*},p^{*}\dotplus(s-e^{\prime}))=1-F(s-e^{\prime}),$

where the first equality is by Invariance to Common Shifts, the next is a
simplification, the next is from (6), and the last is from the definition of
$F$ and using the assumption that $s\geq e>e^{\prime}$.

Case 2 – $e>s$: In this case $W(e,p)=1-F(s-e)=W(e^{*}+e-s,p^{*})$, so applying
Invariance to $p$ Shifts with $a=e-e^{\prime}$ gives

$W(e,p\dotplus(e-e^{\prime}))=W(e^{*}+e-s,p^{*}\dotplus(e-e^{\prime})).$ (7)

Let $b>0$ be large enough so that $s-e+b>0$. Then

$\displaystyle W(e^{\prime},p)=W(e,p\dotplus(e-e^{\prime}))$ $\displaystyle=$
$\displaystyle W(e^{*}+e-s,p^{*}\dotplus(e-e^{\prime}))=$ $\displaystyle
W(e^{*}+e-s+(s-e+b),p^{*}\dotplus(e-e^{\prime}+s-e+b))=$ $\displaystyle
W(e^{*}+b,p^{*}\dotplus(s-e^{\prime}+b)),$

where the first equality is by Invariance to Common Shifts, the next is by
(7), the next is again by Invariance to Common Shifts, and the last is just a
simplification.

Now, if $s\geq e^{\prime}$ then this last expression is equal to
$W(e^{*},p^{*}\dotplus(s-e^{\prime}))$ by Invariance to Common Shifts, so we
obtain
$W(e^{\prime},p)=W(e^{*},p^{*}\dotplus(s-e^{\prime}))=1-F(s-e^{\prime})$ as
needed. If on the other hand $s<e^{\prime}$ then again by Invariance to Common
Shifts the last expression is equal to $W(e^{*}+e^{\prime}-s,p^{*})$ and hence
$W(e^{\prime},p)=W(e^{*}+e^{\prime}-s,p^{*})=1-F(s-e^{\prime})$. This
completes the proof. ∎

###  4.2 Uniqueness of the representation

The following result states that if $W$ is an Additive RPF then the noise
distribution is identified up to translations.

######  Proposition 1.

Suppose that $W$ is an Additive RPF that can be represented by the cdf
$F_{1}$, i.e., $W(e,p)=1-F_{1}(s_{1}(p)-e)$ for all $e,p$. Then $F_{2}$ also
represents $W$ if and only if there is $t\in\mathbb{R}$ such that
$F_{2}(x)=F_{1}(x+t)$ for all $x\in\mathbb{R}$.

###### Proof.

(If): Suppose $F_{1}$ represents $W$ and fix $t$. Define
$F_{2}(x)=F_{1}(x+t)$. Then for any $p$ we have

$\int[1-F_{2}(s_{1}(p)-t-e)]dp(e)=\int[1-F_{1}(s_{1}(p)-e)]dp(e)=k,$

where the first equality is by the definition of $F_{2}$ and the second from
the characterizing property of $s_{1}(p)$. It follows that
$s_{2}(p)=s_{1}(p)-t$. Thus, for all $e$ and $p$,

$W(e,p)=1-F_{1}(s_{1}(p)-e)=1-F_{1}(s_{2}(p)+t-e)=1-F_{2}(s_{2}(p)-e)$

as claimed.

(Only if): Suppose both $F_{1}$ and $F_{2}$ represent $W$. Thus,
$F_{1}(s_{1}(p)-e)=F_{2}(s_{2}(p)-e)$ for all $e$ and $p$. Fix some
$\bar{e}>0$ and recall that $\delta_{\bar{e}}$ is the Dirac measure on
$\bar{e}$. Then $1-F_{1}(s_{1}(\delta_{\bar{e}})-\bar{e})=k$ by condition (1),
so that $s_{1}(\delta_{\bar{e}})=F_{1}^{-1}(1-k)+\bar{e}$. By the same
argument, $s_{2}(\delta_{\bar{e}})=F_{2}^{-1}(1-k)+\bar{e}$. Therefore,
denoting $t=F_{1}^{-1}(1-k)-F_{2}^{-1}(1-k)$, we get
$s_{1}(\delta_{\bar{e}})-s_{2}(\delta_{\bar{e}})=t$. It follows that for every
$\bar{e}$ and $e$

$F_{2}(s_{2}(\delta_{\bar{e}})-e)=F_{1}(s_{1}(\delta_{\bar{e}})-e)=F_{1}(s_{2}(\delta_{\bar{e}})+t-e),$

where the first equality is by assumption and the second by the definition of
$t$ (note that $t$ is independent of $\bar{e}$).

To complete the proof we need to show that for any $x\in\mathbb{R}$ we can
find $\bar{e},e>0$ such that $x=s_{2}(\delta_{\bar{e}})-e$, or equivalently,
$x=F_{2}^{-1}(1-k)+\bar{e}-e$. This can be achieved by taking some
$\bar{e}>x-F_{2}^{-1}(1-k)$ and letting $e=\bar{e}-(x-F_{2}^{-1}(1-k))$. ∎

##  5 Applications

In this final section we illustrate the usefulness of the RPF framework by
revisiting some of the classic questions in the contests literature.

###  5.1 Equilibrium with identical agents

Consider a unit mass of agents, each of which chooses whether to participate
in a contest, and if so how much effort $e\in E$ to exert. Staying out costs
zero, while exerting effort $e$ has a cost $c(e)$, where
$c:E\to\mathbb{R}_{+}$ is twice differentiable and satisfies the standard
conditions $\lim_{e\downarrow 0}c(e)=\lim_{e\downarrow 0}c^{\prime}(e)=0$,
$c^{\prime}>0$, and $c^{\prime\prime}>0$. An agent who wins gets a prize of
$V>0$. The total utility of an agent that exerts effort $e$ is $u(V)-c(e)$ if
they win and $u(0)-c(e)$ if they do not, where the utility function $u$ is
strictly increasing, (weakly) concave, and satisfies $u(0)=0$.

Instead of explicitly considering strategy profiles, we focus on the
population distribution of efforts they induce; let $W(e,p)$ be the
probability of winning for an agent who exerts effort $e\in E$ given
$p\in\Delta_{k}$.555If $p(E)\leq k$ then set $W(e,p)=1$ for every $e\in E$.
The expected payoff of such agent is

$U(e,p):=W(e,p)u(V)-c(e).$

Say that $\bar{e}$ is a best response to $p$ if
$\bar{e}\in\operatorname*{arg\,max}_{e}U(e,p)$, and that $p$ is an equilibrium
distribution (an equilibrium, for short) if $p$-almost every $\bar{e}$ is a
best response to $p$.666Note that if $\bar{e}$ is a best response to $p$ then
in particular $U(\bar{e},p)\geq 0$, so staying out of the contest is not a
profitable deviation. This follows from $\lim_{e\downarrow 0}U(e,p)\geq 0$. We
have the following.

######  Proposition 2.

Suppose that $W$ is an Additive RPF with associated cdf $F$, that $F$ admits a
strictly positive and differentiable density $f$, and that
$c^{\prime\prime}(e)>-f^{\prime}(s)u(V)$ for all $e\in E$ and
$s\in\mathbb{R}$. Then the contest has a unique equilibrium
$p=\delta_{e^{*}}$, where $e^{*}$ solves

$c^{\prime}(e^{*})=f(F^{-1}(1-k))u(V).$ (8)

###### Proof.

Fix some $p\in\Delta_{k}$. We first argue that the best response to $p$ is
unique. Indeed, we have that

$U(e,p)=W(e,p)u(V)-c(e)=(1-F(s(p)-e))u(V)-c(e),$

so the condition in the proposition implies that $U(e,p)$ is strictly concave
in $e$. Further, the first order condition is given by

$f(s(p)-e)u(V)-c^{\prime}(e)=0.$

Since $f$ is strictly positive, and since $\lim_{e\downarrow
0}c^{\prime}(e)=0$, the difference $f(s(p)-e)u(V)-c^{\prime}(e)$ is positive
for all $e$ sufficiently close to zero; and this difference is clearly
negative for all $e$ large enough. Thus, a best response exists and is unique.

It follows that if $p$ is an equilibrium then $p=\delta_{\bar{e}}$ for some
$\bar{e}\in E$. By the market clearing condition (1) we have that
$1-F(s(\delta_{\bar{e}})-\bar{e})=k$, implying that
$s(\delta_{\bar{e}})=F^{-1}(1-k)+\bar{e}$. The first order condition from
above then becomes

$f(F^{-1}(1-k)+\bar{e}-e)u(V)-c^{\prime}(e)=0.$

For $\bar{e}$ to be a best response to $p=\delta_{\bar{e}}$ it is thus
necessary and sufficient that

$f(F^{-1}(1-k))u(V)-c^{\prime}(\bar{e})=0.$

This is condition (8) from the proposition, and clearly there is a unique
$\bar{e}$ that satisfies it.

∎

We note that the second order condition in the proposition holds for example
if $f^{\prime}$ is bounded and $c(e)=Ae^{2}$ for sufficiently large $A$. Of
course, even without global concavity $e^{*}$ from (8) may still be an
equilibrium.

###  5.2 The optimal prize structure

In the contest just described, suppose that the principal’s goal is to
maximize the equilibrium effort $e^{*}$. The principal has total budget $B$
and can choose the fraction of winners $k$ and the size of each award $V$
subject to the constraint $kV=B$. Krishna and Morgan (1998) study this
question in the same framework as ours but with a small number of agents (at
most four).777Other papers that study the optimal design of prizes in contests
include Moldovanu and Sela (2001), Fang et al. (2020) and Olszewski and Siegel
(2020), among many others. They assume that $F$ is symmetric around zero and
that the density $f$ satisfies $sf^{\prime}(s)\leq 0$ for all
$s\in\mathbb{R}$, and show that it is optimal to allocate the entire budget to
agents in the top half of the performance ranking.888Krishna and Morgan (1998)
allow for heterogeneous prizes, while we restrict the principal to give the
same prize to all winners. They also prove that with at most four risk neutral
agents it is optimal to give the entire purse to the top performer. They
conjecture that the same is true for larger contests. We now show that their
conjecture holds true in our setup with a continuum of agents.

######  Proposition 3.

Assume that $F$ is symmetric around zero and that its density $f$ satisfies
$sf^{\prime}(s)\leq 0$ for all $s\in\mathbb{R}$. Let $e^{*}(k)$ be the
solution to (8) with $V=\frac{B}{k}$. Then $e^{*}(k)$ is decreasing for
$k\in[0.5,1)$, so allocating awards to more than half of the population is
never optimal.

###### Proof.

Since $c^{\prime}$ is assumed to be increasing, we have that
$e^{*}(k_{1})>e^{*}(k_{2})$ if and only if
$f(F^{-1}(1-k_{1}))u\left(\frac{B}{k_{1}}\right)>f(F^{-1}(1-k_{2}))u\left(\frac{B}{k_{2}}\right)$.
The utility $u\left(\frac{B}{k}\right)$ is clearly decreasing in $k$. Now, if
$k>0.5$ then by symmetry $F^{-1}(1-k)<0$, and since $sf^{\prime}(s)\leq 0$ we
have that $f(F^{-1}(1-k))$ is decreasing for $k\in(0.5,1)$ as well. ∎

When agents are risk neutral, finding the optimal $k$ boils down to finding
the maximum of the hazard ratio $\frac{f}{1-F}$. Indeed, the right-hand side
of (8) in this case is given by $f(F^{-1}(1-k))\frac{B}{k}$, which after a
change of variable $s=F^{-1}(1-k)$ becomes $\frac{Bf(s)}{1-F(s)}$. Thus,
$e^{*}(k_{1})>e^{*}(k_{2})$ if and only if the hazard ratio $\frac{f}{1-F}$ is
larger at the point $F^{-1}(1-k_{1})$ than at the point $F^{-1}(1-k_{2})$. In
particular, if $F$ has an increasing hazard ratio, as is the case for the
normal distribution, then $e^{*}(k)$ is decreasing in $k$ so reducing the
fraction of winners increases equilibrium effort. But if the hazard ratio is
maximized at some finite $s$, e.g. in the family of Student’s $t$
distributions, then there would be an interior optimal $k\in(0,0.5)$. Figure 1
illustrates this for several distributions.

Figure 1: The figure shows the graphs of $\frac{1}{k}f(F^{-1}(1-k))$ as a
function of $k\in(0,1)$ for the cases of a standard normal distribution (left
panel), Student’s $t$ distribution with 3 degrees of freedom (middle panel),
and Student’s $t$ distribution with 1 degree of freedom (right panel). In the
first case the graph decreases in $k$, implying that a lower fraction of
winners yields higher equilibrium effort, while in the other two cases the
maximum is attained at an interior point (with higher optimal $k$
corresponding to less degrees of freedom).

###  5.3 Rent dissipation

A key issue in the contests literature has been the extent of rent dissipation
– the relationship between the rents obtained by winning and the total cost of
effort exerted in order to capture them.999See for example Nitzan (1994) for
an early survey in the case of a single prize, and Berry (1993) and Clark and
Riis (1996) for the case of multiple winners. Let us return to the contest
model described above with fixed $k$, and for expositional reasons suppose
that agents are risk neutral, $u(V)=V$, and that the cost function is
quadratic, $c(e)=Ae^{2}$. With these assumptions, the equilibrium effort
$e^{*}$ from Proposition 2 is given by $e^{*}=\frac{V}{2A}f(F^{-1}(1-k))$, and
therefore the equilibrium cost of effort is

$c(e^{*})=A(e^{*})^{2}=\frac{V^{2}}{4A}(f(F^{-1}(1-k)))^{2}.$

Total rents in the contest are $kV$, and it is straightforward to calculate
the dissipation of rents and how it is affected by the size of the award $V$,
the budget $k$, and the noise distribution $F$. In particular, the ratio of
costs to rents

$\frac{V^{2}}{4A}(f(F^{-1}(1-k)))^{2}/kV=\frac{V}{4Ak}(f(F^{-1}(1-k)))^{2}$

is less than one for all sufficiently small $V$ and becomes larger than one
when $V$ crosses a threshold. Thus, we can have either under-dissipation or
over-dissipation of rents depending on the size of the award, unlike in the
classic Tullock contest with linear costs where this ratio is independent of
$V$.

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