# Size invariant twisted optical modes for efficient generation of higher
dimensional quantum states

Ali Anwar1,2,∗, Shashi Prabhakar3, R. P. Singh1 1Quantum Science and
Technology Laboratory, Physical Research Laboratory, Ahmedabad, Gujarat, India
2Centre for Quantum Technologies, National University of Singapore, Singapore
3Photonics Laboratory, Physics Unit, Tampere University, Tampere, FI-33720,
Finland  ∗Email<EMAIL_ADDRESS>
###### Abstract

Optical vortex beams are profiled as helical wavefronts with a phase
singularity carrying an orbital angular momentum (OAM) associated with their
spatial distribution. The transverse intensity distribution of a conventional
optical vortex has a strong dependence on the carried topological charge.
However, perfect optical vortex (POV) beams have their transverse intensity
distribution independent of their charge. Such ‘size-invariant’ POV beams have
found exciting applications in optical manipulation, imaging and
communication. In this article, we investigate the use of POV modes in the
efficient generation of high dimensional quantum states of light. We generate
heralded single photons carrying OAM using spontaneous parametric down-
conversion (SPDC) of POV beams. We show that the heralding efficiency of the
SPDC single photons generated with POV pump is greater than that with normal
optical vortex beams. The dimensionality of the two-photon OAM states is
increased with POV modes in the pump and projective measurements using Bessel-
Gaussian vortex modes that give POV, instead of the Laguerre-Gaussian modes.

##  I Introduction

Light carrying OAM, which corresponds to beams with a helical twist of phase,
has gained interest due to its applications in various fields of optics.
Generally termed as ‘Optical Vortex’ in classical optics, light having phase
singularity has been used in the generation of bright optical solitons
Swartzlander Jr and Law (1992); Tikhonenko  _et al._ (1995), study of optical
chronograph Swartzlander  _et al._ (2008); Berkhout and Beijersbergen (2008)
and trapping of particles Gahagan and Swartzlander (1996). The transverse
intensity distribution of an optical vortex mode is radially symmetric in
nature that originates from the azimuthal dependence of the phase. Among such
classes of phase singular modes, Laguerre-Gaussian (LG) modes are most
commonly used in practical applications due to their profile stability upon
free space propagation. Since the size of a normal optical vortex strongly
depends on its topological charge Reddy  _et al._ (2015), they have their
limitations in applications involving transmission of OAM modes through
optical fibers Li and Wang (2013); Gregg  _et al._ (2013) in communication
Willner  _et al._ (2015). Due to this, projective measurements based on
‘phase-flattening’ technique become difficult for higher-order orbital angular
momentum (OAM) modes Qassim  _et al._ (2014). Bessel-Gaussian (BG) modes are
another class of structured light modes carrying OAMs, which come from the
non-diffracting solution of paraxial wave equation Gori  _et al._ (1987);
McGloin and Dholakia (2005). The transverse field distribution of BG modes is
described by a Bessel function Korenev (2002). They exhibit non-diffracting
nature Durnin  _et al._ (1987); Cruz-Ramírez  _et al._ (2012) as well as self-
healing property Litvin  _et al._ (2009); McLaren  _et al._ (2014) when
propagated in space. In most of the practical cases, Bessel-Gaussian modes are
generated using axicons Arlt and Dholakia (2000) and spatial light modulators
(SLM) Leach  _et al._ (2006). Fourier transformation of BG modes gives a class
of size invariant modes known as Perfect optical vortex (POV) Vaity and Rusch
(2015). They are a new class of ring-shaped beams carrying OAM. The size of a
POV modes does not change with respect to the change in OAM values.

The concept of OAM has been well established in quantum information too.
Access to higher dimensions in Hilbert space makes OAM suitable for encoding
higher amounts of information per photon. SPDC is the most common workhorse
for the generation of single photons carrying OAM Boyd (2008); Mair  _et al._
(2001). The paired photons generated in SPDC are inherently correlated in
their OAM values Walborn  _et al._ (2004). It has been experimentally verified
that the amplitude, as well as the helical phase of an optical vortex pump,
gets transferred to SPDC photons Vicuña-Hernández  _et al._ (2014); Anwar  _et
al._ (2018). Based on the OAM selection rule in SPDC process, single photons
carrying OAM are generated in ‘heralding’ configuration by pumping an optical
vortex beam into a nonlinear crystal and projecting one photon from the
generated pair to ‘zero OAM’ (Gaussian) mode Lal  _et al._ (2020). Single-
photons carrying OAM find potential applications in quantum information Erhard
_et al._ (2018), quantum gates Babazadeh  _et al._ (2017), quantum memories
Nicolas  _et al._ (2014) etc.

Apart from the Gaussian modes, OAM correlations among the SPDC photon-pairs
are explored using structured pump modes Romero  _et al._ (2012). Such systems
can be used to generate versatile high-dimensional quantum states by careful
shaping of pump beam having additional spiral modes using different pump
engineering techniques Liu  _et al._ (2018); Kovlakov  _et al._ (2018); Anwar
_et al._ (2020). Also, it was shown that the use of BG modes in photon-pair
detection using OAM projection improved the dimensionality of the OAM state
generated. However, the potential of the class of such modes in further
enhancement of the heralded single-photon detection and dimensionality of the
quantum state is yet unexplored. In this article, we discuss the use of POV
modes as the pump in a parametric down-conversion process to generate heralded
single photons carrying OAM. By exploiting the OAM independence on the size of
POV modes, we show that the heralding efficiency of the single photons with
POV pump is higher than that with a NOV pump. We also show that a
configuration with POV pump beams with generated SPDC photon-pairs projected
to BG modes give improved fidelity than the configuration with an LG mode
projection.

##  II Helical modes and their generations

Depending on the transverse intensity distribution, there are three main
classes of optical modes carrying azhimuthal phase. All those modes have phase
singularity at the center where the intensity is zero. Here we introduce
different helical modes and an experimental method to generate them.

Figure 1: Experimental schemes to generate NOV, then BGV, and then POV: (a)
Conversion of a Gaussian beam to helical modes Using spiral phase plate,
axicon lens and a spherical lens. (b) Generation of helical modes by
diffraction of fork grating hologram imprinted on a SLM.

###  II.1 Size-variant: Normal optical vortex (NOV)

A typical field distribution of a normal optical vortex (NOV) mode of
topological charge $\ell$ in polar coordinates $(r,\theta)$, is given by

$E_{\text{NOV}}^{\ell}(r,\theta)=\sqrt{\frac{2^{\absolutevalue{\ell}+1}}{\pi
w^{2}|\ell|!}}\left(\frac{r}{w}\right)^{|\ell|}\exp\left(-\frac{r^{2}}{w^{2}}\right)\exp(i\ell\theta),$
(1)

where $w$ is the radius of the mode. As shown in Fig. 2, the size of NOV modes
increases with OAM due to the OAM-dependant radial term
$r^{\absolutevalue{\ell}}$ in the above expression.

Figure 2: (a) Theoretical (top) and experimental (bottom) intensity
distributions of NOV and POV beams of order varying from $\ell=1$ to $25$. (b)
and (c) compares the intensity distribution of a NOV and POV with $\ell=1$ and
$\ell=25$.

###  II.2 Size-invariant: Perfect optical vortex (POV)

A new class of optical vortex beam, termed as ‘perfect optical vortex’, was
introduced by Ostrovsky et. al. that solves size effects of a normal vortex
Ostrovsky  _et al._ (2013). Conventionally, POV of order $\ell$ are formed as
a Fourier transform of Bessel-Gaussian vortex (BGV) of order $\ell$ Vaity and
Rusch (2015). The normalized expression for a BGV mode of azhimuthal order
$\ell$ is written as

$E_{\rm{BG}}^{\ell}(r,\theta)=\sqrt{\frac{2e^{\nicefrac{{1}}{{4}}}}{\pi
w^{2}I_{\ell}(\nicefrac{{1}}{{4}})}}J_{\ell}(k_{r}r)\exp\left(-\frac{r^{2}}{w^{2}}\right)\exp(i\ell\theta).$
(2)

The Fourier transform of the above equation leads to the field of a typical
perfect optical vortex of order $\ell$ is given as

$E_{\rm{POV}}^{\ell}(r,\theta)=i^{\ell-1}\frac{w_{g}}{w_{o}}\exp(i\ell\theta)\exp\left(-\frac{(r^{2}+r_{r}^{2})}{w_{o}^{2}}\right)I_{\ell}\left(\frac{2r_{r}r}{w_{o}^{2}}\right),$
(3)

where $w_{g}$ is the waist radius of the initial Gaussian beam,
$w_{o}=2f/kw_{g}$ is half of the ring width and $r_{r}$ is radius of the ring.
Here $f$ is the focal length of the Fourier lens and $k$ is the magnitude of
the wave-vector of the light beam.

###  II.3 Experimental generation of NOV and POV modes

Figure 1(a) shows a simple method to generate POV beams using a spiral phase
plate (SPP), an axicon, and a lens. A Gaussian beam passing through an SPP of
topological order $\ell$ generates an NOV of OAM $\ell$. This beam then
passing through the axicon becomes a BGV, and Fourier transformation of the
BGV using lens generates a POV beam. Alternatively, these helical modes can
also be generated from the diffraction of a laser beam using a fork-grating
pattern imprinted onto SLM, as shown in Fig. 1(b). The first diffracted order
is imaged in the far-field plane will contain the desired helical mode, or NOV
modes. A Gaussian light incident on a fork-grating hologram obtained through
interference converts into BG modes in the immediate plane after SLM, and the
corresponding POV mode is generated at the far-field plane.

In Fig 2(a), the experimental intensity distribution of normal optical vortex
pump is compared with the perfect optical vortex pump of different orders
$\ell$=1, 5, 10, 15, 20 and 25. For a NOV pump, Eqn.1 shows an $\ell$
dependence on the radial terms. So, the size of the OAM modes increases with
$\ell$. However, for a POV pump, radial terms are independent of $\ell$ ((2)),
and thus the size of modes remains the same for higher OAM values. The spatial
profiles are shown in Fig 2(b) and (c) for NOV and POV, respectively. From the
intensity profile, one can observe that the spatial profile change
considerably for a NOV, while remains unchanged for POVs.

##  III Parametric down conversion of helical modes

First, we will discuss the theory of SPDC with NOV and POV pump beams. A
comparative study of down-converted photons with these pump modes is important
to understand their two-photon modal spectra.

###  III.1 Theory

In the perturbative treatment of spontaneous parametric down-conversion
process, interaction of pump ($p$), signal ($s$) and idler ($i$) modes in a
medium (Non-linear $\chi^{(2)}$ crystal) is represented by an interaction
Hamiltonian $\mathcal{H}_{I}$. The initial state is a vacuum state
$|0\rangle_{s}|0\rangle_{i}$, therefore the output state of SPDC is
approximated as

$|\Phi\rangle\approx\left(1-\frac{i}{\hbar}\int_{0}^{\tau}\mathcal{H}_{I}(t)dt\right)|0\rangle_{s}|0\rangle_{i}.$
(4)

The biphoton mode function of the generated twin photons in transverse
momentum coordinates ($\mathbf{k}$) is obtained as

$\Phi(\mathbf{k}^{\perp})=\langle\mathbf{k}^{\perp}_{s}|\langle-\mathbf{k}^{\perp}_{i}|\Phi\rangle,$
(5)

where $\mathbf{k}^{\perp}_{s}$ and $-\mathbf{k}^{\perp}_{i}$ represents the
transverse position in the momentum coordinates for signal and idler
respectively. On simplification, the biphoton mode function in transverse
momentum coordinates is given by

$\Phi(\mathbf{k}_{s}^{\perp},\mathbf{k}_{i}^{\perp},\Delta
k)=E_{p}(\mathbf{k}_{p}^{\perp})L\text{sinc}\left(\dfrac{\Delta
kL}{2}\right)\exp\left(i\dfrac{\Delta kL}{2}\right),$ (6)

where $E_{p}(\mathbf{k}_{p}^{\perp})$ represents the pump transverse amplitude
distribution,
$\mathbf{k}_{p}^{\perp}(=\mathbf{k}^{\perp}_{s}+\mathbf{k}^{\perp}_{i})$ is
the angular coordinates of the pump, $L$ is the thickness of the crystal and
the exponential factor in the Eqn. 6 is a global phase term. $\Delta k$ is the
longitudinal phase mismatch given by

$\Delta
k(\mathbf{k}_{s}^{\perp},\mathbf{k}_{i}^{\perp})=k_{p}^{z}(\mathbf{k}_{s}^{\perp},\mathbf{k}_{i}^{\perp})-k_{s}^{z}(\mathbf{k}_{s}^{\perp})-k_{i}^{z}(\mathbf{k}_{i}^{\perp}).$
(7)

Consider a Type-0/I SPDC using a $\chi^{(2)}$ crystal of thickness $L$. The
magnitude of the wave-vector of the interacting fields under phase-matching
condition are given by

$\displaystyle k_{p,s,i}^{z}(\mathbf{k}_{p,s,i}^{\perp})$
$\displaystyle=\sqrt{k_{p,s,i}^{2}-|\mathbf{k}_{p,s,i}^{\perp}|^{2}}.$ (8)

Here $k_{j}$=$n_{j}\omega_{j}/c$ $(j=p,s,i)$ are the magnitudes of wave-
vectors of the fields. Here, $n_{j}\equiv n_{j}(\omega_{j})$ $(j=p,s,i)$ are
the refractive indices and $\omega_{j}$ are the frequencies of the pump,
signal and idler respectively. $c$ is the speed to light in vacuum. For near-
collinear SPDC, the z-component of the wave-vector is approximated by
$k-|\mathbf{k}^{\perp}|^{2}/2k$. So, (7) becomes

$\displaystyle\Delta
k(\mathbf{k}_{s}^{\perp},\mathbf{k}_{i}^{\perp})=\frac{|\mathbf{k}_{s}^{\perp}-\mathbf{k}_{i}^{\perp}|^{2}}{2k_{p}}.$
(9)

Also, an additional term of $-2\pi/\Lambda$ will be added to the phase-
mismatch in (9), in case of a quasi phase-matched crystal, where $\Lambda$ is
the grating period.

###  III.2 Angular spectrum of SPDC

The generated signal and idler photons in SPDC propagate in space according to
the phase-matching condition of SPDC. The locus of all points in space that
satisfies phase-matching conditions comes out to be an annular ring
distribution where the signal and idler photons in a pair are present. The
angular spectrum of the down-converted signal photons for frequency
$\omega_{s}$ is obtained by tracing the biphoton mode function overall idler
photons Jeronimo-Moreno and Jáuregui (2014)

$R_{s}(\mathbf{k}^{\perp}_{s})=\int
d\mathbf{k}_{i}^{\perp}|\Phi(\mathbf{k}_{s}^{\perp},\mathbf{k}_{i}^{\perp},\Delta
k)|^{2}.$ (10)

Figure 3: Experimental setup to record the angular spectrum of SPDC photons
generated by pumping a non-linear crystal with different pump beams. Figure in
inset is the SPDC annular distribution in the angular spectrum. Two
diametrically opposite points (circled in red) on the distribution are chosen
to collect the signal-idler pairs.

The experimental setup to record the angular spectra of SPDC photons with
different helical pump modes is given in Fig. 3. A pump beam of wavelength
405$\pm$2 nm from a continuous-wave diode laser (TOPTICA iBeam Smart) of 50 mW
power is incident on a $\chi^{(2)}$ crystal. The dashed box corresponds to the
case-by-case method to prepare NOV or POV helical beams using the method given
in Fig. 1(a). A half-wave plate (HWP) is used to orient the pump polarization
along the optic axis of the crystal. As discussed earlier, we use Type-I BBO
crystal to generate SPDC photon-pairs. The down-converted photons, signal &
idler, are generated in a non-collinear fashion at diametrically opposite
points of the SPDC ring. A bandpass filter (BPF) of width 10 nm centered at
810 nm is used to filter down-converted photons and block the unconverted pump
beam after the crystal. A $2f$-imaging configuration with a plano-convex lens
of focal length 50 mm is used to image the SPDC in $k$-space. The angular
spectrum of SPDC is recorded using an electron-multiplying CCD (EMCCD) camera
with a gain $\times$100 and the addition of 100 frames, each having an
exposure time of 0.5 s. The EMCCD camera has an imaging area of 512$\times$512
pixels with a pixel size of 16 $\mu$m.

The numerical angular spectra of SPDC with different pump OAM modes with
similar experimental conditions obtained from Eqn. (10) is shown in Fig.
4(top). This is well-matched with their corresponding recorded intensity
distributions of SPDC photons with NOV and POV pumps of different OAMs, as
shown in Fig. 4(bottom). The asymmetric intensity distribution for all the
angular spectra is mainly due to the spatial walk-off of the thick crystal
used. For a NOV pump, the SPDC annular distribution broadens with an increase
in the topological order. However, the signature of ’size-invariance’ of POV
pump beams is observed in their corresponding angular spectra, where the
intensity distribution remains the same for higher OAM values.

Figure 4: Numerical (top) and experimental (bottom) angular spectra of SPDC
generated using NOV, and POV pump modes of various orders.

For a non-collinear down-conversion, the signal and idler are located at
diametrically opposite points of the annular distribution (Fig. 3). If one
chooses to place a collection optics setup each at such two points and detect
the photons, then we could observe a decrease in the number of detected
photons with the increase in OAM of an NOV pump. There will not be a
significant change in the number of detected photons in the case of a POV
pump. To achieve the full advantage of our study, we preferred the pump
generation method illustrated in Fig.1(a) over Fig.1(b) due to the limitation
of spiral-phase plates of order more than 3. As the low down-conversion rates
in BBO crystal restricted us to investigate further the size effects on
photon-pair detection, we choose periodically poled KTP (ppKTP) crystal, which
not only provides higher photon-pairs production rate, however also removes
the asymmetry in the SPDC distribution. Pumping the crystal with a beam
carrying OAM $l_{p}$ and the collection of idler photon through a single-mode
fiber leads to a conditional detection of signal photon (heralded) carrying an
OAM identical to that of the pump, which is discussed below.

###  III.3 Heralded twisted single photons

Equation (6) gives the two-photon modal distribution for a pump with electric
field $E_{p}(\mathbf{k}_{p}^{\perp})$. According to conservation of OAM in
SPDC process Walborn  _et al._ (2004), for a pump carrying an OAM $l_{p}$ and
the idler photon projected to a zero-OAM mode ($l_{i}=0$), the corresponding
heralded signal photon will have the same OAM as that of the pump
($l_{s}=l_{p}$). This heralded single-photon carrying OAM can be
experimentally generated by coupling one photon in a pair to a single-mode
fiber and the other one to a multimode fiber. So, the coincidence counts can
be calculated from the modal distribution as

$C\propto\int_{0}^{a_{s}}\int_{0}^{a_{i}}\Phi(\mathbf{k}_{s}^{\perp},\mathbf{k}_{i}^{\perp},\Delta
k)\xi_{s}^{*}(\mathbf{k}_{s}^{\perp})\xi_{i}^{*}(\mathbf{k}_{i}^{\perp})d\mathbf{k}_{s}^{\perp}\mathbf{k}_{i}^{\perp},$
(11)

where $\xi_{s}$ and $\xi_{i}$ are the characteristic functions of spatial mode
projectors in signal and idler arms respectively. The fibers in each arm is
represented by a Gaussian function with mode-field diameters $2a_{s}$ and
$2a_{i}$, given by

$\displaystyle\xi_{s,i}^{*}(\mathbf{k}_{s,i}^{\perp})=\sqrt{\frac{a_{s,i}^{2}}{2\pi}}\text{exp}\left(-\frac{a_{s,i}^{2}}{4}|\mathbf{k}_{s,i}^{\perp}|^{2}\right).$
(12)

The experimental setup to generate heralded twisted single-photons from SPDC
is given in Fig. 5. Here, we have used a blue diode laser (TopMode) of
wavelength 405 nm and power 10 mW with a spectral bandwidth of 0.1 nm, as the
pump beam. NOV and POV beams are generated by illuminating the Gaussian laser
mode onto the corresponding grating holograms imprinted on a SLM (Hamamatsu),
and the first order far-field diffracted beam after Fourier transformation
using a 750 mm lens is incident on a Type-0 ppKTP crystal of thickness 30mm
and transverse dimensions of 1 mm$\times$2 mm. The HWP allows us to vary the
pump beam polarization along the crystal axis.

Figure 5: Experimental setup for generating heralded twisted single photons.
BPF - Band pass filter; PM - Prism mirror; SMF - Single mode fiber; MMF -
Multimode fiber; FC - Fiber coupler; SPCM - Single photon counting module; CC
- Coincidence counter.

The down-converted photons (signal & idler) of wavelength 810 nm each
(degenerate) are generated in a non-collinear fashion at diametrically
opposite points of the SPDC ring. To measure the number of generated photon
pairs, two diametrically opposite portions of the SPDC ring at a given plane
were selected using apertures (not shown in the setup) and the photons coming
out of each aperture were collected using fiber collimators (CFC-2X-B,
Thorlabs), of focal lengths 2 mm each. The fiber collimator in the idler arm
is attached to a SMF (P1-780A-FC-2, Thorlabs) having a numerical aperture of
0.13 and a mode field diameter of $5\pm 0.5$ $\mu$m, and that in the signal
arm is attached to a multi-mode fiber (M43L02, Thorlabs). The fibers are
connected to the single-photon detectors SPCMs (SPCM-AQRH-16-FC, Excelitas).
The detectors have a timing resolution of 350 ps with 25 dark counts per
second. To count the correlated photon-pairs, the two detectors are connected
to a coincidence counter (IDQuantique-ID800), having a time resolution of 81
ps.

We recorded the coincidences of signal and idler for NOV and POV pumps
starting from $\ell=1$ to 25. Figure 6(a) shows the coincidence counts for NOV
and POV pump beams with different OAM. For a pump OAM up to 10, the
coincidence counts of SPDC photon pairs is considerably higher for a POV pump
than that with a NOV pump beam.

Figure 6: Plot of (a) measured coincidence counts for different values of OAM
of pump, and (b) corresponding heralding efficiency, for NOV and POV pumps.

The difference in coincidences corresponding to the two pump modes of the same
order decreases for higher orders. The robustness of the source with POV pump
mode can be more understood by comparing the heralding efficiencies. Figure
6(b) shows the variation of heralding efficiency of SPDC single photons with
pump OAM for NOV and POV pumps. From the graph, it is clear that, for up to a
pump OAM of 15, the efficiency of heralded single photons carrying OAM with
POV pump is considerably greater than that corresponding to the NOV pump.
Ideally, the coincidences for POV pump should be independent of the order.
However, we observe a drop in their coincidences for higher OAM values. This
is due to the fact that the ring diameter of the POV mode gradually increases
with order and the difference is more exposed for orders starting from $l=15$,
which causes a considerable drop in counts.

##  IV Two-photon OAM states using POV pump

###  IV.1 Theory

Parametric down-converted photons generated using a pump beam of any spatial
profile will have an incoherent sum of different OAMs. Based on the OAM
conservation in SPDC process Walborn  _et al._ (2004), the photon-pairs
represent a joint signal-idler OAM state in the allowed OAM subspace, given as

$\ket{\psi}=\sum_{\ell_{i}=-\infty}^{\infty}C_{\ell_{p}-\ell_{i},\ell_{i}}\ket{\ell_{p}-\ell_{i}}_{a}\ket{\ell_{i}}_{b},$
(13)

where $l_{i}$ is the OAM of idler photon, $C_{l_{p}-l_{i},l_{i}}$ is the
probability amplitude for occurrence of the state
$\ket{l_{p}-l_{i}}_{a}\ket{l_{i}}_{b}$. The subscripts $a$ & $b$ represents
signal and idler modes respectively. In the experiment, the probability
amplitude corresponds to the coincidence counts obtained by projecting
conjugate azimuthal modes in signal and idler. The coincidence counts
representing each projection is given by

$C_{\ell_{s},\ell_{i}}\propto_{a}\langle\ell_{s}^{\text{(proj)}}|_{b}\langle\ell_{i}^{\text{(proj)}}|\psi\rangle.$
(14)

Here, $\ell_{s}^{\text{(proj)}}$ and $\ell_{i}^{\text{(proj)}}$ are the OAM
values of azimuthal modes considered in projective measurement.

For a nearly collinear phase-matched SPDC, the probability amplitude in Eqn.
(14) is rewritten in terms of an integral in polar coordinates representing
the field overlap of pump, projected signal and idler modes

$C_{\ell_{s},\ell_{i}}\propto\int_{0}^{2\pi}d\theta\int_{0}^{\infty}rE_{p}(r,\theta)E_{s}^{*}(r,\theta)E_{i}^{*}(r,\theta)dr,$
(15)

where $E_{p}$ is the pump mode and $E_{s}$ & $E_{i}$ are the projected signal
and idler modes respectively. Here, we consider a POV pump represented by (3)
and NOV or BGV modes in both signal and idler arms.

###  IV.2 Experiment

The schematic of the experimental setup to measure in higher dimension is
shown in Fig. 7. In this case, the combination of a SLM and a SMF was used as
the detection system, instead of the heralding configuration using fibres as
discussed earlier. The combination was added in both the signal & idler arms.
The SLM is used to generate either the NOV or POV modes at the plane of the
crystal. The crystal is then imaged on the two SLMs using two lenses and then
imaged again to the SMF. The photon is then sent to the APDs, and the signal
is analysed using the coincidence logic (CC).

Figure 7: Experimental setup for generating OAM entangled photon-pairs with
different helical pump modes.

We measured OAM spectra of SPDC photon-pairs with a POV pump and estimated
their bandwidths using LG and BG mode projections in signal/idler. The OAM
bandwidth is quantified as Schmidt number, which estimates the dimensionality
of the state generated Pors  _et al._ (2008); Pires  _et al._ (2010); Straupe
_et al._ (2011). Fig. 8 shows the OAM spectra of photon-pairs for POV pump of
order 0 & 1.

Figure 8: OAM spectrum of SPDC photon-pairs generated using POV pump of OAM
(a) $\ell_{p}=0$ and (b) $\ell_{p}=1$.

For $\ell_{p}=0$ (Fig. 8(a)), the Schmidt numbers are 3.8 & 8.7 for LG & BG
mode projections respectively. For $\ell_{p}=1$ (Fig. 8(b)), the Schmidt
numbers are 6.3 & 11.6 for respective LG and BG mode projections. This shows
that projecting the photon-pairs generated from a POV pump to a BG mode will
effectively bring down the ’size effects’ that causes a reduction in OAM
bandwidth. Studies had already shown that the bandwidth could be further
increased by increasing the radial wavenumber $k_{r}$, which broadens the
spectrum further. McLaren  _et al._ (2012).

Figure 9: The density matrices of the two-photon OAM entangled states. (a)-(c)
correspond to the estimation by projecting photon-pairs on LG basis, and
(d)-(f) are the estimations in BG projection. The holograms or the true state
and the fidelity of the estimated state are given in the inset of each density
matrix. (a) & (d) corresponds to a POV pump of $\ell_{p}=0$ and the other
density matrices correspond to a POV pump of $\ell_{p}=1$.

We also studied the effect of zero-order mode projection in idler on the
fidelity of generated quantum states using POV pump. For simplicity, we
consider two-dimensional OAM states in the form of Bell states
$(\ket{\ell_{s}}\ket{\ell_{i}}+\ket{\ell_{i}}\ket{\ell_{s}})/\sqrt{2}$, with
different $\ell_{s}$ and $\ell_{i}$ values. Figure 9(a)-(c) show the density
matrices of OAM states estimated with LG mode projection and Fig. 9(d)-(f)
gives the corresponding density matrices with BG mode projections. In all
these cases, the pump OAM $\ell_{p}$ is the sum of OAMs of signal and idler,
$\ell_{p}=\ell_{s}+\ell_{i}$. Here, we observe that the state fidelity is
lower for the case of a zero-OAM mode in signal or idler, which is the setting
for the generation of heralded twisted single photons, as discussed in the
earlier section. So, there is a trade-off in using a two-photon SPDC system
with complete mode projection capabilities and using heralding with the fiber
projective systems.

We also observed that the fidelity of the state is improved with the use of BG
projection. Efficient OAM detection based on phase-flattening of POV modes in
classical light source has been demonstrated Pinnell  _et al._ (2019a, b).
Thus, a robust source of high dimensional quantum states in OAM degree of
freedom can be implemented by using a size-invariant twisted optical mode like
a perfect optical vortex as a pump, as well as by performing projection of
photon-pairs in Bessel-Gaussian modes.

##  V Conclusion

In conclusion, we have experimentally demonstrated that the conditional
coupling efficiency of the heralded twisted single photons for higher OAM
values can be improved by using a perfect optical vortex beam as the pump. We
also showed that the dimensionality of the two-photon OAM states is increased
with the use of POV modes in the pump, as well as projective measurements
using Bessel-Gaussian vortex modes that give POV, instead of the Laguerre-
Gaussian modes. The presented results may be utilized for the practical
realization of efficient higher dimensional OAM entangled photon-pair sources.

## Disclosures

The authors declare that there are no conflicts of interest related to this
article.

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# Pre- and Post-Auction Discounts in First-Price Auctions

Miguel Alcobendas Eric Bax Yahoo Research Yahoo Research<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

One method to offer some bidders a discount in a first-price auction is to
augment their bids when selecting a winner but only charge them their original
bids should they win. Another method is to use their original bids to select a
winner, then charge them a discounted price that is lower than their bid
should they win. We show that the two methods have equivalent auction
outcomes, for equal additive discounts and for multiplicative ones with
appropriate adjustments to discount amounts. As a result, they have
corresponding equilibria when equilibria exist. We also show that with the
same level of multiplicative adjustments, bidders with discounts should prefer
an augmented bid to a discounted price. Then we estimate optimal bid functions
for valuation distributions based on data from online advertising auctions,
and show how different discount levels affect auction outcomes for those bid
functions.

††copyright: none

##  1\. Introduction

First-price auctions are now the norm in online advertising exchanges (Bigler,
2019; Cox, 2019; Akbarpour and Li, 2020; Despotakis et al., 2021): bidders
enter bids, the bidder with the highest bid wins the right to display an
advertisement, and they pay their bid. An exchange or a seller may offer some
bidders discounts for a variety of reasons, including funding new bidders’
learning on the exchange, increasing demand and competition in auctions, or in
fulfillment of contract conditions. In online advertising, as in other types
of media like TV or radio, it is common for publishers to offer auction
discounts to ad agencies or advertisers with large budgets, hoping that the
discount attracts more ad spending from the buyer. Outside of online
advertising, discounting in auctions has been a way to increase participation
from underrepresented or disadvantaged groups of bidders (Marion, 2007;
Krasnokutskaya and Seim, 2011).

A discount may be applied either pre-auction or post-auction. With a pre-
auction discount, which we refer to as an augmented bid, the bid is increased
for the purpose of determining the auction winner, but not for determining the
winner’s price. With a post-auction discount, the winner is selected based on
the original bids, but if a bidder with a discount wins, then their winning
price is a discounted version of their bid. We will refer to a pre-auction
discount as bid augmentation and a post-auction discount as price reduction.

An additive discount is a fixed amount added to a bid when selecting a winner
or subtracted from a bid when charging the bidder should they win. A
multiplicative discount is a fraction of the bid that is either added to the
bid or subtracted from the price – for example, a 5% bid augmentation or
discount. In this paper, we consider both additive and multiplicative
discounts. Additive discounts are simpler to analyze; multiplicative discounts
occur more often in practice.

We analyze equilibria for first-price auctions with discounts, comparing
equilibria under pre- and post-auction discounts. There is a rich literature
on equilibria in first-price auctions. Under continuity and independence
constraints on bidder valuations, it is possible to show that first-price
auctions have unique equilibria and to characterize the equilibria (Milgrom
and Weber, 1982; Lebrun, 1999; McFadden, 2003; Levin, 2004; Reny and Zamir,
2004; Milgrom, 2004) in terms of conditional second values (CSV), with each
bidder bidding the expectation of the runner-up valuation conditioned on their
own valuation being the highest. Some challenges for more general methods
include dependence among valuations (common values), bidders having different
valuation distributions (asymmetry), and the fact that there are simple
examples of non-continuous valuation distributions for which equilibria do not
exist (Maskin and Riley, 2000b; Wang et al., 2020). More recently, there have
been advances in computing estimated equilibria via an alternative tie-
breaking method and discrete approximations (Maskin and Riley, 2000b, 2003),
indirect methods based on payoff ratios (Kirkegaard, 2009), dynamical systems
methods (Fibich and Gavish, 2012), polynomial basis functions (Hubbard and
Paarsch, 2012; Hubbard et al., 2013), and fictitious play (Heymann and
Mertikopoulos, 2021). There is also a method to characterize Bayesian Nash
equilibrium bidding strategies for bidders with some discrete valuation
distributions (Wang et al., 2020).

In Section 2, we show that the same additive discounts applied pre- and post-
auction lead to the same incentives and outcomes for bidders and the seller,
implying the same equilibria. Section 3 presents an equivalence between (non-
equal) multiplicative discount levels applied pre- and post-auction, shows how
equal pre- and post-auction discounts impact bidders and the seller, and
considers how discounts can produce symmetry in a specific case. Section 4
presents results based on data from online advertising auctions, explaining
how to estimate bid functions under different levels of discounting and
examining auction outcomes based on those bid functions. Section 5 concludes
with some potential directions for future work.

##  2\. Additive Discounts and Matching Equilibria

In this section, we show that for additive discounts, bid augmentations and
price reductions at the same levels lead to equivalent equilibria, in the
sense that they have the same outcomes for all participants. Let vectors
$\mathbf{v}$ be the bidders’ valuations, $\mathbf{b}$ be their bids, and
$\mathbf{a}$ their bid augmentations (the amounts added to the bids solely for
the purpose of selecting the winner). Let $v_{i}$, $b_{i}$, and $a_{i}$ be
bidder $i$’s valuation, bid, and bid augmentation. (If bidder $i$ has no
discount, then $a_{i}=0$.) With bid augmentations, bidder $i$ has utility
function

(1)
$u^{+}_{i}(\mathbf{v},\mathbf{b},\mathbf{a})=p_{i}(\mathbf{b}+\mathbf{a})(v_{i}-b_{i})$

where $p_{i}()$ is the probability that bidder $i$ wins, which is one if
bidder $i$ has the greatest augmented bid:

(2) $\forall j\not=i:\hbox{ }b_{i}+a_{i}>b_{j}+a_{j},$

zero if some other bidder has a greater augmented bid, and $\frac{1}{k}$ if
$k$ bidders, including bidder $i$, tie for maximum augmented bid.

With price reductions, let $\mathbf{v}$ and $\mathbf{b}$ be valuations and
bids as before, and let $\mathbf{r}$ be the vector of price reductions, with
$r_{i}$ the price reduction for bidder $i$. Then bidder $i$ has utility
function:

(3)
$z^{+}_{i}(\mathbf{v},\mathbf{b},\mathbf{r})=p_{i}(\mathbf{b})(v_{i}-b_{i}+r_{i}).$

###### Theorem 2.1.

For additive discounts, first-price auctions with bid augmentation and those
with price reductions have corresponding Nash equilibria (if either has an
equilibrium), with each bidder having that same utility under both forms of
discounting, if the bid augmentations are equal to the price reductions.

###### Proof.

Let $\mathbf{u}^{+}$ and $\mathbf{z}^{+}$ be the vectors of utility functions
for bid augmentation and price reduction, respectively. Note that

(4)
$\mathbf{z}^{+}(\mathbf{v},\mathbf{b},\mathbf{r})=\mathbf{u}^{+}(\mathbf{v},\mathbf{b}-\mathbf{r},\mathbf{r})$

since, for each bidder $i$

(5)
$u^{+}_{i}(\mathbf{v},\mathbf{b}-\mathbf{r},\mathbf{r})=p_{i}(\mathbf{b}-\mathbf{r}+\mathbf{r})(v_{i}-(b_{i}-r_{i}))=p_{i}(\mathbf{b})(v_{i}-b_{i}+r_{i})=z^{+}_{i}(\mathbf{v},\mathbf{b},\mathbf{r}).$

Prove by contradiction. Let $\mathbf{b}^{*}$ be any Nash equilibrium bids for
price reductions. Now suppose $\mathbf{b}^{*}-\mathbf{r}$ are not Nash
equilibrium bids for bid augmentation. Then some bidder $i$ can change their
bid and increase their own utility. Let $\mathbf{b}^{+}-\mathbf{r}$ be the
boosting bids after bidder $i$ changes their bid to one that increases their
utility. Then

(6)
$z^{+}_{i}(\mathbf{v},\mathbf{b}^{*},\mathbf{r})=u^{+}_{i}(\mathbf{v},\mathbf{b}^{*}-\mathbf{r},\mathbf{r})<u^{+}_{i}(\mathbf{v},\mathbf{b}^{+}-\mathbf{r},\mathbf{r})=z^{+}_{i}(\mathbf{v},\mathbf{b}^{+},\mathbf{d}),$

but
$z^{+}_{i}(\mathbf{v},\mathbf{b}^{*},\mathbf{r})<z^{+}_{i}(\mathbf{v},\mathbf{b}^{+},\mathbf{r})$
implies that $\mathbf{b}^{*}$ is not a Nash equilibrium. So
$\mathbf{b}^{*}-\mathbf{r}$ are Nash equilibrium bids for bid augmentation.
Since

(7)
$\mathbf{z}^{+}(\mathbf{v},\mathbf{b}^{*},\mathbf{r})=\mathbf{u}^{+}(\mathbf{b}^{*}-\mathbf{r},\mathbf{r}),$

each bidder’s utility is the same for bid augmentation as for price reduction,
for these corresponding equilibria.

Similar reasoning shows that each Nash equilibrium for bid augmentation
corresponds to a Nash equilibrium for price reduction, with augmentations
equal to reductions, and with the same utilities for each bidder. ∎

##  3\. Multiplicative Discounts and Corresponding Equilibria

In this section, we explore how multiplicative discounts affect auctions. We
show that there are price reductions that produce auctions with equilibria
that have the same utilities for all participants as those for bid-
augmentation auctions. Next, we compare price-reduction and bid-augmentation
auctions that use the same discount rates. Then, we show how discounts can be
set to remove asymmetry among bidders with some uniform valuation
distributions.

Under multiplicative discounts, let $a_{i}$ be the bid augmentation rate for
bidder $i$, so that their bid is treated as $\hat{b}_{i}=b_{i}(1+a_{i})$ when
determining the winner, but they pay only their bid $b_{i}$ should they win.
Define vector $\mathbf{\hat{b}}$ to be the vector of augmented bids
$\hat{b}_{i}$. Then bidder $i$ has utility function

(8)
$u_{i}(\mathbf{v},\mathbf{b},\mathbf{a})=p_{i}(\mathbf{\hat{b}})(v_{i}-b_{i}).$

Note that

(9)
$b_{i}=\frac{\hat{b}_{i}}{1+a_{i}}=\frac{\hat{b}_{i}+\hat{b}_{i}a_{i}-\hat{b}_{i}a_{i}}{1+a_{i}}=\hat{b}_{i}-\hat{b}_{i}\left(\frac{a_{i}}{1+a_{i}}\right).$

So bidder $i$’s utility function

(10)
$u_{i}(\mathbf{v},\mathbf{b},\mathbf{a})=p_{i}(\mathbf{\hat{b}})(v_{i}-b_{i})=p_{i}(\mathbf{\hat{b}})\left[\left(v_{i}+\hat{b}_{i}\left(\frac{a_{i}}{1+a_{i}}\right)\right)-\hat{b}_{i}\right].$

This is equal to a utility function without an augmented bid ($a_{i}=0$),
except with bidder $i$’s valuation not $v_{i}$ but rather a “virtual
validation” of
$\hat{v}_{i}=v_{i}+\hat{b}_{i}\left(\frac{a_{i}}{1+a_{i}}\right)$.

For price reductions, let $r_{i}$ be the reduction rate for bidder $i$, with
their original bid $b_{i}$ determining whether they win, but their price being
$b_{i}(1-r_{i})$ should they win. Then bidder $i$ has utility function

(11)
$z_{i}(\mathbf{v},\mathbf{b},\mathbf{r})=p_{i}(\mathbf{b})(v_{i}-b_{i}(1-r_{i})).$

Note that this is

(12) $=p_{i}(\mathbf{b})\left[(v_{i}+b_{i}r_{i})-b_{i}\right].$

This is equal to a utility function without a price reduction ($r_{i}=0$), but
with bidder $i$’s valuation a “virtual valuation”
$\tilde{v}_{i}=v_{i}+b_{i}r_{i}$ rather than $v_{i}$.

###  3.1. Corresponding Equilibria

Setting price reduction levels $r_{i}$ to make virtual valuations the same for
a price-reduction auction as those for a bid-augmentation auction produces
auctions with corresponding equilibria:

###### Theorem 3.1.

Setting $r_{i}=\frac{a_{i}}{1+a_{i}}$ for each bidder $i$ creates
corresponding equilibria (if any) for a first-price auction with bid
adjustments as for one with price reductions, assuming each bidder $i$ has the
same distribution of valuations for both auctions. The corresponding
equilibria have the same expected utilities for all participants.

###### Proof.

With bid augmentation, the utility function for bidder $i$ is

(13)
$u_{i}(\mathbf{v},\mathbf{b},\mathbf{a})=p_{i}(\mathbf{\hat{b}})(v_{i}-b_{i}),$

with $\mathbf{\hat{b}}$ the vector of augmented bids
$\hat{b}_{i}=b_{i}(1+a_{i})$. For a price-reduction auction, suppose each
bidder bids $\hat{b}_{i}$. Then bidder $i$ has utility

(14)
$z_{i}(\mathbf{v},\mathbf{\hat{b}},\mathbf{r})=p_{i}(\mathbf{\hat{b}})(v_{i}-\hat{b}_{i}(1-r_{i}))=p_{i}(\mathbf{\hat{b}})(v_{i}-b_{i}(1+a_{i})(1-r_{i})).$

But this is equal to $u_{i}(\mathbf{v},\mathbf{b},\mathbf{a})$ if

(15) $b_{i}=b_{i}(1+a_{i})(1-r_{i}).$

Setting $1=(1+a_{i})(1-r_{i})$ and solving for $r_{i}$:

(16) $r_{i}=\frac{a_{i}}{1+a_{i}}.$

So those price reductions $r_{i}$, with bidders augmenting their own bids to
$\hat{b}_{i}$, produce the same utility function for each bidder under price
reductions as they would have with bids $b_{i}$ under bid augmentations
$a_{i}$, with the same bids $\mathbf{\hat{b}}$ determining allocations. So the
auctions are equivalent, in the sense that bidders have an equivalent set of
choices with the same impact on each other’s outcomes and their own, and hence
they have corresponding equilibria with the same expected utility for each
bidder.

We can prove that by contradiction. Suppose $\mathbf{b}^{*}()$ is a vector of
equilibrium bid functions: $[b_{1}^{*}(),\ldots,b_{n}^{*}()]$, for bid
augmentation with augmentation $a_{i}$ for each bidder $i$. Let
$r_{i}=\frac{a_{i}}{1+a_{i}}$ for each bidder $i$. Then, by Equation 14, all
bidders receive the same expected utilities for price reduction as for bid
augmentation if they use the vector of bid functions
$\mathbf{\hat{b}}^{*}()=[(1+a_{1})b_{1}^{*}(),\ldots,(1+a_{n})b_{n}^{*}()]$.
Now suppose these are not equilibrium bid functions. Then some bidder can
improve their expected utility for the price-reduction auction by using a
different bid function. Without loss of generality, let it be bidder 1, and
let the bid function that improves their expected utility be
$\tilde{b}_{1}()\not=(1+a_{1})b_{1}^{*}()$. Then, they can use bid function
$\tilde{b}_{1}()/(1+a_{1})$ to improve their expected utility for the bid-
augmentation auction. But that contradicts $\mathbf{b}^{*}()$ being a vector
of equilibrium bid functions. ∎

###  3.2. Bid Augmentation vs. Price Reduction with Equal Rates

Based on Theorem 3.1, changing from multiplicative bid augmentations $a_{i}$
to price reductions at the same level, $r_{i}=a_{i}$, has the same effect on
auction equilibria and outcomes as lowering the bid augmentations to

(17)
$\frac{a_{i}}{1+a_{i}}=\frac{a_{i}+a_{i}^{2}-a_{i}^{2}}{1+a_{i}}=\frac{a_{i}(1+a_{i})-a_{i}^{2}}{1+a_{i}}=a_{i}-\frac{a_{i}^{2}}{1+a_{i}},$

which is lowering them by $\frac{a_{i}^{2}}{1+a_{i}}$. Since it is equivalent
to lowering the bid augmentation rates, changing from bid augmentations to
equal price reduction rates is a disadvantage for bidders with discounts, and
a greater disadvantage for bidders with greater discounts. For small bid
augmentation rates, the disadvantage is mild. For example, with a bid
augmentation rate of $a_{i}=10\%$, changing to a price reduction of
$r_{i}=10\%$ is equivalent to lowering the bid augmentation rate by
$\frac{1}{110}$, or about 1%. For a 5% discount, the discount rate loses about
a quarter of a percent, while a 25% discount becomes equivalent to a 20%
discount instead.

These changes are small relative to the discount amounts, but they may still
be large relative to the bidder’s expected profit, since the difference
between a bidders price and their valuation may be significantly smaller than
their price. Hence, changing from one type of discount to the other may affect
bidder’s expected surplus (profit) significantly, even to the point of
determining whether participation produces a positive profit.

###  3.3. Optimal Bidding for Uniform Value Distributions with Compensating
Discounts

Assume $n$ bidders each have independent value distributions $U[0,1]$. Without
discounts, Bayes-Nash equilibrium bids for this case are (Vickrey, 1961;
Myerson, 1981; Riley and Samuelson, 1981; Levin, 2004; Milgrom, 2004)

(18) $b_{i}=\frac{n-1}{n}v_{i},$

where $v_{i}$ is the value for bidder $i$, and $b_{i}$ is their optimal
(equilibrium) bid. Since the conditions for the revenue-equivalence theorem
(Vickrey, 1961; Myerson, 1981; Riley and Samuelson, 1981; Maskin and Riley,
2000a; Milgrom, 2004) hold for this case – including having independent and
identical (also called symmetric) value distributions – this result can be
proven by computing conditional second values (CSV) (Milgrom, 2004) (Thm.
4.6): conditioned on $v_{i}$ being the top value, the other $n-1$ bidders’
values are independently distributed over $U[0,v_{i}]$, so their top value as
a fraction of $v_{i}$ has the same distribution as the top value among $n-1$
independent draws from $U[0,1]$. That has a $\mathrm{Beta}(n-1,1)$
distribution (David and Nagaraja, 2003), which has mean $\frac{n-1}{n}$.

Now consider the impact of price reductions for independent uniform
distributions of values. For each bidder $i$, a price reduction of $r_{i}$
boosts their equilibrium bid by a factor of $\frac{1}{1-r_{i}}$, for the case
of independent uniform value distributions $U[0,1-r_{i}]$ rather than
$U[0,1]$:

###### Theorem 3.2.

If each bidder $i$ has price reduction $r_{i}$ and $n$ bidders each have
independent value distributions $U[0,1-r_{i}]$, then

(19) $b_{i}=\left(\frac{n-1}{n}\right)\left(\frac{1}{1-r_{i}}\right)v_{i}$

are Bayes-Nash equilibrium bids.

###### Proof.

If bidder $i$ bids $b_{i}$ and bidder $j$ bids
$b_{j}=\left(\frac{n-1}{n}\right)\left(\frac{1}{1-r_{j}}\right)v_{j}$ then
$b_{j}\leq b_{i}$ iff

(20) $\left(\frac{n-1}{n}\right)\left(\frac{1}{1-r_{j}}\right)v_{j}\leq
b_{i}.$

Equivalently,

(21) $\left(\frac{1}{1-r_{j}}\right)v_{j}\leq\left(\frac{n}{n-1}\right)b_{i}.$

Since each $\left(\frac{1}{1-r_{j}}\right)v_{j}\sim U[0,1]$ and
$\left(\frac{n}{n-1}\right)b_{i}\leq 1$, the probability that $b_{j}\leq
b_{i}$ is

(22) $\left(\frac{n}{n-1}\right)b_{i}.$

Since the product is the probability that bidder $i$ has a top bid, and
$v_{i}-b_{i}(1-r_{i})$ is bidder $i$’s surplus if they win, bidder $i$ has
expected utility:

(23)
$\mathrm{E}z_{i}=\prod_{j\not=i}\left[\left(\frac{n}{n-1}\right)b_{i}\right]\left[v_{i}-b_{i}(1-r_{i})\right].$

Simplifying the product,

(24)
$\mathrm{E}z_{i}=\left(\frac{n}{n-1}\right)^{n-1}b_{i}^{n-1}\left[v_{i}-b_{i}(1-r_{i})\right],$

To show that
$b_{i}=\left(\frac{n-1}{n}\right)\left(\frac{1}{1-r_{i}}\right)v_{i}$ is an
optimal bid, take the derivative:

(25) $\frac{\partial}{\partial
b_{i}}\mathrm{E}z_{i}=\left(\frac{n}{n-1}\right)^{n-1}\left[(n-1)b_{i}^{n-2}\left[v_{i}-b_{i}(1-r_{i})\right]-b_{i}^{n-1}(1-r_{i})\right].$

Then set it to zero:

(26) $(n-1)b_{i}^{n-2}\left[v_{i}-b_{i}(1-r_{i})\right]=b_{i}^{n-1}(1-r_{i}).$

Equivalently,

(27) $(n-1)\left[v_{i}-b_{i}(1-r_{i})\right]=b_{i}(1-r_{i}),$

and

(28) $(n-1)v_{i}=nb_{i}(1-r_{i}).$

Solve for $b_{i}$:

(29) $b_{i}=\left(\frac{n-1}{n}\right)\left(\frac{1}{1-r_{i}}\right)v_{i}.$

∎

Theorem 3.2 is based on uniform value distributions, but over domains
$U[0,1-r_{i}]$ rather than $U[0,1]$ – distributions with values reduced to
match the price discounts. In equilibrium, bidders boost their bids (relative
to the symmetric $U[0,1]]$-distribution case) enough to compensate for their
price discounts, making the winner’s price $\frac{n-1}{n}$ times their post-
discount values. So discounts even the playing field, giving the asymmetric
$U[0,1-r_{i}]$ auction dynamics similar to a symmetric $U[0,1]$ auction.

##  4\. Bid Functions and Auctions Under Realistic Valuations

In online advertising, as in other type of media like TV or radio, ad agencies
or advertisers with large budgets may benefit from discounts offered by
publishers designed to attract ad spending. This section explores auction
outcomes under discounting for bidders with realistic valuation distributions
using online advertising data from Yahoo! ad exchange. Amounts (revenue,
valuations, bids, costs, and surpluses) are all multiplied by the same secret
constant to obscure actual amounts but preserve relative relationships.
Original bids are in US dollars per 1000 impressions – cost per mille (CPM).

###  4.1. High-Level Process

We consider an asymmetric first price auction model with two types of buyers:
a bidder that benefits from discounts, and the rest. We use the following
process to recover the valuation distribution of bidders from bid data,
estimate the bidding functions, and estimate expected auction outcomes when
one of the buyers benefits from a discount:

  1. (1)

Collect bid data for an ad position on the Yahoo! ad platform that has a 5%
price reduction discount for some bidders. The data is for about 200 000
auctions, drawn from a day in February 2023. The average number of
participants in the auction data is around five. For our analysis, we assume
that only one of the five bidders receives a discount.

  2. (2)

Assuming that valuation distributions follow a log-normal distribution and
valuation distributions are i.i.d. (no affiliation between bidders), apply an
approach by Guerre, Perrigne, and Vuong (Guerre et al., 2000; Ma et al., 2019)
to the historical bids, then use maximum likelihood to estimate the shape and
scale parameters for two valuation distributions: one for discounted bidders
and the other for other bidders. The resulting valuation distributions are
similar to each other, having shape and scale parameters that each differ by
less than 3%.

  3. (3)

For a discounted bidder with the discounted bidders’ valuation distribution
and four other bidders with the other valuation distribution, numerically
estimate the bid functions for the discounted bidder and for the other
bidders. To do this, we apply Euler’s method to a pair of differential
equations to compute estimated inverse bid functions, based on ideas from
Maskin and Riley (Maskin and Riley, 2000a) and differential equations from
(Gentry et al., 2018), modified to reflect a price reduction. Then we invert
to produce estimated bid functions. (For details, see Subsection 4.2.)

  4. (4)

Based on those estimated bid functions, compute auction outcomes for auctions
with a discounted bidder and four other bidders, under different price
reduction rates. The outcomes include seller revenue and bidder surplus for
the discounted and other bidders. We present those results in Subsection 4.3.

###  4.2. Estimating Bid Functions

To estimate bid functions for a single discounted bidder and four undiscounted
bidders, we use differential equations similar to those from (Gentry et al.,
2018). We modify them for a bid reduction, and, instead of fitting a
polynomial, we compute based on Euler’s method. Following Maskin and Riley
(Maskin and Riley, 2000a), we first estimate inverse bid functions –
valuations as functions of bids that are optimal for the valuations – then
invert to produce bid functions. Also based on ideas from Maskin and Riley
(Maskin and Riley, 2000a), we “guess and check” to identify a maximum bid that
implies a feasible boundary condition for the system of differential
equations. We assume the undiscounted bidders have identical valuation
distributions and, by symmetry, equal bid functions, so we only explicitly
solve for one of them.

Let bidder 1 be a discounted bidder, with a price reduction that is $r$ times
their bid. Let bidders 2 to 5 be undiscounted. Let $F_{1}$ and $f_{1}$ be the
cdf and pdf, respectively, of the valuation distribution for bidder 1. Let
$F_{2}$ and $f_{2}$ be those functions for the valuation distributions for
bidders 2 to 5. Let $v_{1}$ and $v_{2}$ be the valuations for bidders 1 and 2
to 5, respectively, for which $b$ is the optimal bid. Use $F_{i}$ and $f_{i}$
as shorthand for $F_{i}(v_{i})$ and $f_{i}(v_{i})$. Use an apostrophe to
denote a derivative with respect to $b$, and use $n$ for the number of bidders
without discounts. (For us, $n=4$.)

The utility function for bidder 1 is

(30) $u_{1}=[v_{1}-(1-r)b]F_{2}^{n},$

since all $n$ undiscounted bidders must have valuations below $v_{2}$ to bid
less than $b$. (Assume continuous valuation distributions to ignore ties.) The
utility function for bidder 2 is

(31) $u_{2}=(v_{2}-b)F_{1}F_{2}^{n-1},$

since the discounted bidder and the other $n-1$ undiscounted bidders must all
bid less than bidder 2 for bidder 2 to win.

For $b$ to be optimal for $v_{1}$ and $v_{2}$, the derivative of utility
functions with respect to $b$ must be zero. Since

(32)
$u_{1}^{\prime}=[v_{1}-(1-r)b]nF_{2}^{n-1}f_{2}v_{2}^{\prime}-(1-r)F_{2}^{n}$

and

(33)
$u_{2}^{\prime}=(v_{2}-b)\left[f_{1}v_{1}^{\prime}F_{2}^{n-1}+(n-1)F_{1}F_{2}^{n-2}f_{2}v_{2}^{\prime}\right]-F_{1}F_{2}^{n-1},$

we can set each RHS to zero, solve the first for $v_{2}^{\prime}$:

(34) $v_{2}^{\prime}=\frac{(1-r)F_{2}}{nf_{2}[v_{1}-(1-r)b]},$

and solve the second for $v_{1}^{\prime}$:

(35)
$v_{1}^{\prime}=\frac{F_{1}}{f_{1}}\left(\frac{1}{v_{2}-b}-\frac{(n-1)f_{2}v_{2}^{\prime}}{F_{2}}\right).$

Let $v_{1}^{*}$ and $v_{2}^{*}$ be the maximum values in the support of the
valuation distributions for bidders 1 and 2, respectively. Given a maximum bid
$b^{*}$ (as yet unknown), for which $v_{1}^{*}$ and $v_{2}^{*}$ are optimal,
we know $F_{1}(v_{1}^{*})=F_{2}(v_{2}^{*})=1$ and we know $f_{1}(v_{1}^{*})$
and $f_{2}(v_{2}^{*})$. Our method substitutes those values for $F_{1}$,
$F_{2}$, $f_{1}$, and $f_{2}$ and $v_{1}^{*}$ and $v_{2}^{*}$ for $v_{1}$ and
$v_{2}$, with $b$ set to $b^{*}$ in Equations 34 and 35. We then set the
inverse bid function for $v_{1}$ at $b^{*}-\Delta_{b}$ (for a small
$\Delta_{b}$) to $v_{1}^{*}-v_{1}^{\prime}\Delta_{b}$ and the inverse bid
function for $v_{2}$ at $b^{*}-\Delta_{b}$ to
$v_{2}^{*}-v_{2}^{\prime}\Delta_{b}$. In other words, we use Euler’s method,
computing the derivatives $v_{1}^{\prime}$ and $v_{2}^{\prime}$ at $b^{*}$ and
using a linear estimation for values of $v_{1}$ and $v_{2}$ of the inverse bid
functions at $b^{*}-\Delta_{b}$.

Then we repeat Euler’s method, using the values at $b=b^{*}-\Delta_{b}$ to
estimate those at $b=b^{*}-2\Delta_{b}$, and so on, until we reach the minimum
bid of zero. We use $\Delta_{b}=\frac{b^{*}}{10000}$, yielding ten thousand
and one pairs $(b,v_{i})$ in each estimated inverse bid function. To identify
$b^{*}$, we use a binary search over candidate values. The greatest value that
results in a solution with all successive values decreasing as bids decrease
in the inverse bid functions is $b^{*}$.

After inversion, we find that the bid functions are very close to optimal. To
test this, we identified the best responses for bidder 1 and bidder 2 to the
other bidders’ bids across the 10001 bids in each bid function. On average,
the difference between the best-response bid and the bid function bid was less
than 0.03% of the range of bids for bidder 1 and 0.002% for bidder 2.

Figure 1 shows the estimated optimal bid functions that result from Euler’s
method. Figure 2 shows more detail for the higher valuations. For no discount
($r=0.00$), the discounted bidder has almost the same bid function as the
other four bidders, because the valuation distribution for the discounted
bidder is nearly the same as that for the other bidders. For positive
discounts, the bidder receiving a discount (with bid function shown by a
dotted line) bids higher than the other bidders at the same valuations, since
the discounted bidder receives a price discount that is $r$ times their bid.
This gap increases with the discount. Also, as the discount increases, the
discounted bidder exerts more competitive pressure on the other bidders,
forcing them to increase their bids, which raises the bid functions and the
maximum bid.

Figure 1. Estimated optimal bids as a function of valuations for a bidder with
a price reduction rate $r$ (dashed line) and one of 4 undiscounted bidders
(solid line) in auctions with the discounted bidder. Figure 2. Estimated
optimal bids as a function of valuations for a bidder with a price reduction
rate $r$ (dashed line) and one of 4 undiscounted bidders (solid line) in
auctions with the discounted bidder, showing detail for higher values.

###  4.3. Auction Outcomes

Auction Outcome Statistics
---
|  |  | Win Rate | E Surplus | E Cost
r | E Rev. | Eff. | disc. | other | disc. | other | disc. | other
0.00 | 9.320 | 1.00 | 0.200 | 0.200 | 1.977 | 1.975 | 1.865 | 1.864
0.05 | 9.319 | 0.99 | 0.209 | 0.198 | 2.067 | 1.951 | 1.904 | 1.854
0.10 | 9.299 | 0.98 | 0.218 | 0.196 | 2.165 | 1.930 | 1.936 | 1.841
0.15 | 9.280 | 0.97 | 0.227 | 0.193 | 2.269 | 1.904 | 1.965 | 1.829
0.20 | 9.255 | 0.96 | 0.237 | 0.191 | 2.379 | 1.877 | 1.989 | 1.817
0.25 | 9.219 | 0.95 | 0.248 | 0.188 | 2.498 | 1.848 | 2.007 | 1.803
Table 1. Expected revenue for the seller ($E\;Rev.$), efficiency ($Eff.$), and
win rate, expected buyer surplus ($E\;Surplus$), and expected buyer cost
($E\;Cost$) for discounted ($disc.$) and other bidders, by price reduction
rate $r$. To condition surplus and cost on winning, divide by win rate.

Table 1 shows auction statistics based on the estimated bid functions for the
discounted bidder and the four other bidders. The values are from full
integration over the bid functions (not random samples from them). Because
bids are discretized in the estimated bid functions, ties become possible, and
they are considered in the computation of the statistics. (With 10 001
different bid values, ties have a very small impact on the results.)

As discount rates $r$ increase, there are mild declines in expected seller
revenue ($E\;Rev.$), since a portion of the price is refunded when the
discounted bidder wins. That is partially offset by the rise in bids shown in
Figures 1 and 2. The efficiency ($Eff.$) – the probability that the winner has
a highest valuation among the bidders – decreases because the discounted
bidder sometimes outbids the others even if some of them have a slightly
higher valuation. However, the impact on efficiency is mild, because the bid
curves for the discounted bidders are not far to the left of those for the
other bidders, as shown in Figure1.

The win rate for the discounted bidder increases smoothly as the discount
increases ($disc.$). The surplus ($E\;Surplus$) for the discounted bidder does
the same, due to their price reduction, but partially offset by their
increased equilibrium bids. The expected cost ($E\;Cost$) – price after
discount – for the discounted bidder increases, but that is because of their
increased win rate. To get expected cost given an auction win, divide the
expected cost in the table by the win rate. For the discounted bidder, the
expected cost given a win decreases as the discount rate increases. The
opposite is true for the other bidders.

##  5\. Conclusion

Comparing pre- and post-auction discounts, we showed that the same additive
discount level produces the same equilibria. For multiplicative discounts, we
showed that there are discounts that produce corresponding equilibria for pre-
and post-discounting, but they are unequal: price reduction rate $r$ and bid
augmentation rate $a$ have corresponding equilibria if $r=\frac{a}{1+a}$. So,
in shifting between pre- and post-auction discounts, the rates need to be
adjusted to maintain auction outcomes. We also explored how using the same
pre- and post-auction discount rates alters auctions and how discounts can be
set to compensate for asymmetry among bidders.

Using empirical data from an online advertising auction, we studied how
different discount rates would affect auction outcomes. To do that, we showed
how to accurately estimate optimal bid functions under discounting for
valuation distributions based on data. For that data, we found that
discounting, even up to a 25% price reduction, has only a mild negative impact
(about 1.1%) on revenue while shifting almost 5% of the auction wins to the
discounted bidder.

In the future, it would be interesting to apply the methods we use in this
paper to evaluate impacts on auction outcomes if discounted bidders have
valuation distributions that are very different from the other bidders’
distributions – either much lower or higher on average or with different
levels of variation. For those cases, it would be interesting to model
bidders’ decisions to enter or exit the auction and add that to the
computation of estimated auction outcomes. Also, for cases with complicated
valuation distributions, it could be useful to employ higher-order methods
rather than Euler’s method to estimate bid functions, or to use a curve-
fitting approach (Gentry et al., 2018).

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