Directed Acoustic Assembly in 3D

Kai Melde1,2, Minghui Shi1,2, Heiner Kremer3, Senne Seneca2,4, Christoph
Frey2,4, Ilia Platzman2,4,
Christian Degel5, Daniel Schmitt5, Bernhard Schölkopf3 and Peer Fischer1,2*

1 Micro, Nano and Molecular Systems Group, Max Planck Institute for Medical
Research, Jahnstr. 29, 69120 Heidelberg, Germany
2 Institute for Molecular Systems Engineering and Advanced Materials,
Heidelberg University, Im Neuenheimer Feld 225, 69120 Heidelberg, Germany
3 Empirical Inference Department, Max Planck Institute for Intelligent
Systems, Max-Planck-Ring 4, 72076 Tübingen, Germany
4 Department of Cellular Biophysics, Max Planck Institute for Medical
Research, Jahnstr. 29, 69120 Heidelberg, Germany
5 Technical Ultrasound Department, Fraunhofer Institute for Biomedical
Engineering, Ensheimer Straße 48, 66386 St. Ingbert, Germany
* corresponding author<EMAIL_ADDRESS>

## Abstract

The creation of whole 3D objects in one shot is an ultimate goal for rapid
prototyping, most notably biofabrication, where conventional methods are
typically slow and apply mechanical or chemical stress on biological cells.
Here, we demonstrate one-step assembly of matter to form compact 3D shapes
using acoustic forces, which is enabled by the superposition of multiple
holographic fields. The technique is contactless and shown to work with solid
microparticles, hydrogel beads and biological cells inside standard labware.
The structures can be fixed via gelation of the surrounding medium. In
contrast to previous work, this approach handles matter with positive acoustic
contrast and does not require opposing waves, supporting surfaces or
scaffolds. We envision promising applications in tissue engineering and
additive manufacturing.

Keywords: one-step, directed assembly, acoustic hologram, radiation force

## Introduction

One-shot fabrication of polymer structures has recently been proposed as a
manufacturing concept, in which a whole three-dimensional object is formed in
one shot [1]. In this case the object’s 3D shape has to be defined by a
suitable intensity distribution of the generating light field, e.g. by a
photo-initiated polymerization reaction. However, it is difficult to shape
optical fields in a compact 3D volume at once using conventionally available
devices and spatial light modulators. We refer to _compact_ 3D fields as
those, which exhibit feature sizes over similar length scales in all three
dimensions. For a projected image the achievable lateral resolution scales
with $1/\mathrm{NA}$ and the axial resolution with $1/\mathrm{NA}^{2}$, where
$\mathrm{NA}$ is the numerical aperture [2]. Thus, to create compact 3D
holographic images the optimal working distance should be about half the
aperture diameter, so that $\mathrm{NA}\approx 1$. Such high numerical
apertures are thus far only practicable for illuminating microscopic target
volumes. Examples for optical computer generated holography (CGH) at larger
dimensions have correspondingly lower NAs and the resulting images are
stretched by several orders of magnitude in the beam propagation direction [3,
4], which limits their use for fabrication purposes of compact objects.
Proposed solutions to this problem are thus far based on the nonlinear
activation of a photoinitiator. One promising approach is the sequential
illumination from varying angles into a rotating chamber to obtain projected
tomography [5, 6, 7]. In this case it is necessary that the chemical
crosslinker activates only when the accumulated light dose exceeds a certain
threshold [7]. Recently, this concept has been demonstrated to also work
within lightly scattering media such as cell suspensions and using a
cytocompatible photoinitiator [8]. An alternative approach, called Xolography
[9], is based on two-photon excitation with different wavelengths and
consequently allows separation into two beams: one illuminating a plane with a
focused lightsheet and the second beam projecting the corresponding cross
section of the object on that plane. Since both excitation wavelengths are
necessary, the polymerization reaction is only initiated in one plane at a
time. It stands to reason that both methods, however, still generate the 3D
fields in a serial manner and not in a single step.

An alternate approach to light is to use sound fields that exert acoustic
forces for assembly. This approach has the distinct advantage that the
material of interest, such as particles or cells, is directly manipulated.
Furthermore, the application of ultrasound at the intensities used here is
cytocompatible and does not warrant the use of chemical additives such as
photoinitiators. Indeed, acoustic particle assembly shows promise for rapid
prototyping [10] and application in cell cultures [11], but has thus far been
limited to 2D assembly close to boundaries [12, 13] or point-like tweezing in
air [14, 15] and water environments [16, 17]. These tweezing methods require
phase discontinuities in the focal region, which complicates their extension
towards extended traps. Alternatively, standing waves can be used to assemble
cells in 2D into regular patterns [18, 19], or colloidal microparticles in 3D
[20], but here the patterns are generally highly symmetric. Prisbrey et al.
have examined the use of a closed cavity whose surface is fully lined by
transducer elements to assemble particles in a fluid volume [21]. They found
solutions for a selection of 3D traps using the boundary element method.
However, the geometric constraints imposed by opposing transducers and
inevitable reflections in this closed environment limit the attainable shapes
to mode geometries highly dependent on the container shape, i.e. cubic
symmetries.

In this article, we present a method to realize the first one-step 3D assembly
of matter into arbitrary shapes using ultrasound by combining multiple
acoustic holograms. In particular, we show that neither counterpropagating
waves nor phase discontinuities are required to trap particles in 3D. Our
method thus not only allows assembly into arbitrary 3D forms, but also confers
flexibility in the experimental setup as all acoustic sources can be placed on
one side of the projected field. That leaves room for experimental
intervention, i.e. the addition of sample material, the integration with other
processes, and access for optical characterization and imaging. We demonstrate
the practicability of our compact 3D fields by assembling different classes of
matter, namely solid particles, biological cells and hydrogel beads, into 3D
shapes in one shot.

## Results

### Computation of compact 3D holographic fields

Figure 1: Concept to form compact acoustic 3D pressure images. The target 3D
shape (a) is scaled and converted into a voxel representation (b). The number
of necessary holograms as well as their position and orientation relative to
the target are chosen as part of the design process. Each hologram is placed
on an ultrasound transducer that emits a plane wave at a fixed frequency. (c)
Our computer generated holography (CGH) routine finds the optimal phase maps
for each hologram. (d) Reconstruction of the ultrasound pressure amplitude
emanating from the transducer-hologram sources (1,2 and 3) and projecting into
the 3D volume shows good results, displayed here as isosurfaces at 40% of the
maximum pressure.

The wavelengths of waterborne ultrasound at MHz frequencies are about three
orders of magnitude larger than those of visible light. The experimental
dimensions and aperture sizes we use in practice, however, are of same order.
For this reason the axial and radial resolution scale favorably and we achieve
similar resolution in all three spatial directions. Superposition of
concurrent fields from different angles leads to high-fidelity acoustic fields
due to a wider range of accessible wave vectors (essentially a much larger
numerical aperture). Figure 1 shows the general concept and workflow. The
target object (Fig. 1(a)) is first converted and transferred to the
computation volume (Fig. 1(b)). Depending on the problem at hand the number
and orientation of the holograms and transducers are chosen and placed
relative to the target. In practice, we found orthogonal placement the most
versatile. Then we compute phase maps for all holograms using an optimization
algorithm (Fig. 1(c)). In this exemplary simulation three transducers with a
frequency of $3.5\text{\,}\mathrm{MHz}$ and a diameter of
$50\text{\,}\mathrm{mm}$ are employed. Results for the generated 3D pressure
fields in the shape of the target are shown in Fig. 1(d). The pressure images
in the shape of the 3D target will drive the assembly process.

Figure 2: (a) Exemplary 3D arrangement of focal points at the vertex locations
of an icosahedron and (b) its spatial frequency spectrum. (a,b) Are linked via
the (inverse) 3D Fourier transform, indicated by FT (or iFT). (c) Here, the
spectrum is limited to the Ewald surface of a monochromatic wave with
wavenumber $k=2\pi/\lambda$ and two sources with limited aperture size
(opening angle $\alpha$). (d) Shows the resulting field with close-up view on
a single spot.

The synthesis of three-dimensional fields forming a target volume or shape is
challenging and an active research topic in computer generated holography
(CGH) [22, 4]. The goal of CGH is to find the spatial phase and amplitude
distribution of an incident wavefront that will—upon diffraction—reconstruct
to form the target image. The goal is achieved by optimizing the output
acoustic field under the experimental constraints, such as the incident beam
profile and wave shaping modalities. The latter indicates whether the
amplitude or the phase (or both) can be controlled. CGH methods for 2D images
are well advanced, however accessing the third dimension poses the additional
problem that often the optical or acoustic waves have to propagate across
regions with highly varying intensities while energy conservation has to be
preserved. Some desired 3D target images may thus not be physically
realizable, only via a different geometry. As we show here, an effective way
to quickly infer the feasibility of volumetric fields without having to
undertake time-consuming volumetric wave propagation calculations is to
analyze the 3D spatial frequency spectrum of the target object [23]. Figure
2(a) shows an exemplary field of focal spots (regions where high intensity is
desired) arranged so as to form the vertices of a regular icosahedron. Its 3D
Fourier transform (FT) ( Fig. 2(b)) corresponds essentially to a decomposition
into plane waves, each represented by their wave vector $\mathbf{k}$. Thus for
monochromatic illumination (insonification) with wavelength $\lambda_{0}$ this
spectrum has to be limited to the surface of a sphere—the Ewald sphere—whose
radius is equal to the wavenumber $|\mathbf{k}|=k_{0}=2\pi/\lambda_{0}$.
However, sources of light or sound are restricted in space. The finite
aperture of a real source thus further constrains the wave vectors to a cone
of opening angle $\alpha$ which is given by the dimensions of the source (Fig.
2(c)). Back-transformation of this limited spectrum using the inverse Fourier
transform (iFT) reveals one possible solution to the initial field. We call
this approach the Fourier Constraint Method (FCM). Alternatively, solutions
can be found by varying the complex phases of points in the spatial field,
which then determines the hologram that is to be used in conjunction with the
source. The same procedure can be extended to multiple transducers with their
respective holograms in arbitrary (source) orientations and configurations. An
example for the spectrum limited to two holograms along the x and y direction,
respectively, can be seen in Fig. 2(c). The inverse FT reveals the resulting
field in Fig. 2(d), where characteristic intensity (pressure) nodes around the
foci are visible.

Iterating through these steps to compute a hologram has been termed a 3D
variant of the Gerchberg-Saxton (GS) algorithm in optical CGH [3, 24].
However, this is not directly applicable in the case of ultrasound, which we
will use for assembly. In ultrasound, the near-field needs to be considered in
contrast to the farfield typically encountered in optics, where the source and
target are related by a simple Fourier transform. Before source constraints
can be applied it becomes necessary to propagate the wave from the image to
the hologram plane. This propagation is accomplished in a straightforward
manner using the angular spectrum method applied to the limited spectrum,
which is contained on the section of the Ewald surface with regards to the
opening angle of the source. The FCM delivers very good results for fields
that consist only of collections of focal points. However, one difficulty for
it (and including GS and its derived iterative angular spectrum approach,
IASA[25, 12]) is to achieve uniform amplitude over extended regions, which is
particularly desirable for acoustic traps making up lines, surfaces or
volumes. To improve the homogeneity of amplitudes over extended traps we
therefore resorted to parallel computation of the volumetric field using the
angular spectrum method combined with a gradient-based optimization procedure.
Non-convex optimization has been employed previously in this context with
pioneering work done by Zhang et al. [26]. In summary, our new approach is
suitable for holographic reconstruction in the acoustic near-field and allows
for arbitrary numbers of holograms at user-defined positions and orientations
to generate complex 3D pressure patterns in space.

### Directed 3D assembly of microparticles

Figure 3: Particle trapping potential around the mutual focal spot of two,
respectively, three focused beams. Beam directions are indicated by red arrows
in the circular insets. (a) Normalized intensity distribution at the focus of
two beams pointed in X and Z direction (isosurfaces at
$0.25\cdot\mathrm{max}\\{I\\}$), and (b) the particle trapping sites indicated
by the negative Gor’kov potential $-\Upsilon$ (isosurfaces at
$-0.25\cdot\mathrm{min}\\{\Upsilon\\}$). (c,d) Intensity distribution
(isosurfaces at $0.1\cdot\mathrm{max}\\{I\\}$) and negative Gor’kov potential
(isosurfaces at $-0.25\cdot\mathrm{min}\\{\Upsilon\\}$) for three focused
beams in X, Y and Z direction. All dimensions are normalized by wavelength
$\lambda$ and the potentials have been computed for a silica gel particle with
radius $a_{\mathrm{p}}=\lambda/10$.

Structured acoustic fields have long been used to trap and assemble
microparticles [27, 28]. A popular measure to gauge the attainable trapping
potential in acoustics is the Gor’kov radiation potential because of its
straightforward computation [29], which has been extended to complex
structured fields [30] and arbitrary shaped particles [31]. Figure 3
visualizes two examples of superimposed beams: two (Fig. 3(a)) or three (Fig.
3(c)) beams are focused to a point at the origin (0,0,0) coming from
orthogonal directions corresponding to the axes highlighted red in the insets.
The particles used in this work (e.g. silica gel particles) have positive
acoustic contrast against water and are consequently attracted to minima of
the Gor’kov potential. Therefore the opposite sign of the potential is
displayed as isosurfaces in Figs. 3(b) and (d). The closed surfaces visualize
the approximate form of particle aggregates in a superimposed focal spot. It
should be noted here that these traps do not constitute a single connected
potential well that the particles fall into, because the trapping regions span
multiple pressure nodes in proximity to the focal points, leading to a striped
or clustered appearance of the assembled structures. Using higher frequency
ultrasound will reduce the spacing between stripes.

Figure 4: Trapping of silica gel microspheres with $2.25\text{\,}\mathrm{MHz}$
ultrasound at specified points in 3D. (a) Schematic of the experimental setup
using two holograms and a removable particle container. After initial
resuspension by shaking the particles (white dots) sediment under gravity $g$
until they pass a trapping site (red cross). The trap locations represent
vertices of platonic solids: (b) octahedron, (c) icosahedron. Scale bar
$10\text{\,}\mathrm{mm}$.

Trapping of solid particles in specified locations in 3D has been demonstrated
by arranging trapping sites at vertices of the platonic bodies. In this
experiment two holograms were computed for two source beams intersecting at
$90\text{\,}\mathrm{\SIUnitSymbolDegree}$, as shown in the schematic in Figure
4(a). A removable cubic container was filled with a suspension of porous
silica gel spheres in water and placed so that it rested directly on the
holograms. Its faces were covered with acoustically transparent windows and
therefore allowed the ultrasound field to propagate through the container with
minimal reflection. The holograms were computed so that an acoustic image
forms at the center of the container. At the start of the experiment the cube
was shaken manually and then placed on the rig. The microparticles quickly
sedimented under the influence of gravity, however, particles that traversed
the trapping sites around each focal point were retained by the acoustic
radiation force. Figures 4(b, c) show photographs of the final trapping state
for an octahedron and an icosahedron, respectively.

Figure 5: 3D holographic assembly of cells and microgel beads. (a) The compact
3D sound image formed by three $2.25\text{\,}\mathrm{MHz}$ ultrasound
transducers and associated holograms is focused into a standard cuvette. The
inset shows the target field, in this example an extended spherical volume,
where panels (b,c) show the simulated sound intensity I (isosurface at
$0.09\cdot\mathrm{max}\\{I\\}$) and a 3D fluorescence image stack of assembled
C2C12 mouse myoblasts, respectively. (d) Microscope image of fluorescent
microgels. (e) Size distribution of microgels before and after curing. (f,g)
Simulated sound intensity field of a distorted figure eight curve shown in
front view and turned $90\text{\,}\mathrm{\SIUnitSymbolDegree}$ around the
vertical axis. Sound intensity isosurfaces at $0.16\cdot\mathrm{max}\\{I\\}$.
(h,i) Assembly of microgels shown according to views in panels (f,g). Scale
bar of (d) is $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$. All other scale
bars $1\text{\,}\mathrm{mm}$

The combination of beams from orthogonal directions (causing highly varying
wave vectors) causes interference patterns with pressure nodes that can be
used to trap and assemble matter with a positive acoustic contrast. This is
useful as most materials, including cells, have a positive acoustic contrast
in water (PDMS is an exception). Figure 5 shows assemblies of biological cells
(C2C12 mouse myoblasts) as well as microscopic hydrogel beads (gelatin
methacrylate, GelMa). The assembled particles and cells were fixed in a slow-
curing hydrogel medium, which leaves enough time for excess particles to
settle before solidification. The experiments were performed inside square
cuvettes, which were aligned to the transducers so that the target field is
projected towards the center of the medium (Fig. 5(a)). The first
demonstration is the assembly of biological cells to an extended spherical
volume. Panels (b,c) show the simulated sound intensity field and the final
cell assembly, respectively. We recorded and confirmed the 3D structure of the
sample using a custom-built laser-sheet fluorescence imaging setup. Hydrogel
beads labelled with a fluorescent dye were fabricated on a microfluidic
platform. A fluorescence microscope image and their size distribution before
(droplets) and after curing (Gels) can be seen in Figs. 5(d,e), respectively.
We then computed the holograms to obtain a compact figure eight curve and used
the same experimental setup to assemble the hydrogel beads. Figs. 5(f,g)
illustrate the simulated sound intensity fields in front view and side view,
respectively. Laser-sheet images of the same sample are shown corresponding to
the views in Figs. 5(h,i).

## Discussion and outlook

In summary, we introduced a concept for generating compact three-dimensional
acoustic pressure shapes through superposition of fields from multiple
acoustic holograms. To compute the fields in 3D we initially used a Fourier
constraint method (FCM), which is sufficient to generate point-like targets.
Further, FCM is useful as a fast and intuitive way to gauge the feasibility
whether certain target shapes can be generated with a particular configuration
of holograms. In order to obtain traps beyond points that extend along lines
in space, one needs to resort to a gradient-based optimization of the phase
profile. To this aim, we devised an alternative computational method based on
non-convex optimization, that minimizes the mean squared error between the
generated field and the target field via a projected gradient descent scheme.
Note that closely related approaches to non-convex optimization have been
proposed by [26] and subsequently been taken up by [32]. However, these works
consider standard single source optical holograms. Here, we show how to obtain
complex 3D pressure shapes using multiple acoustic sources and holograms. The
compact acoustic fields we created experimentally using the computed holograms
allowed us to instantly assemble matter into arbitrary 3D shapes for the first
time. We have demonstrated this concept with silica gel particles, biological
cells and also hydrogel beads in setups consisting of two or three
transducers, where each is fitted with one hologram. The assemblies were fixed
in a hydrogel phase for subsequent analysis. Our compact 3D acoustic fields
trap and assemble particles around specified points or along curved lines in a
bulk suspension and even inside conventional sample tubes and cuvettes.

Due to the versatility and ability to accommodate different materials our
method shows promise for fabrication of scaffolds or directly assembling
biological tissue. We foresee applications of compact high-fidelity three-
dimensional ultrasound fields in medical therapy, targeted drug delivery and
neuro-stimulation. The effects of ultrasound in these emerging fields have
already been shown and are investigated by many research groups [33]. Further,
the directed 3D assembly of matter is a promising avenue for rapid prototyping
and in particular for tissue engineering.

## Acknowledgements

The authors thank Dr. Sadaf Pashapour and the microfluidic core facility at
the Institute for Molecular Systems Engineering and Advanced Materials
(IMSEAM) at Heidelberg University for the microfluidic device fabrication. The
research was in part supported by the European Research Council under the ERC
Advanced Grant Agreement HOLOMAN (No. 788296); the Max Planck Society and the
Fraunhofer Society (AKUSTOGRAMME); as well the German Research Foundation
(Deutsche Forschungsgemeinschaft, DFG) under Germany’s Excellence Strategy via
the Excellence Cluster 3D Matter Made to Order (EXC-2082/1-390761711)..

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Analyses of voting algorithms often overlook informational externalities shaping individual votes.
For example, pre-polling information often skews voters towards candidates who may not be their top choice, but who they believe would be a worthwhile recipient of their vote.
In this work, we aim to understand the role of external information in voting outcomes.
We study this by analyzing (1) the probability that voting outcomes align with external information, and (2) the effect of external information on the total utility across voters, or social welfare.
In practice, voting mechanisms elicit coarse information about voter utilities, such as ordinal preferences, which initially prevents us from directly analyzing the effect of informational externalities with standard voting mechanisms.
To overcome this, we present an intermediary mechanism for learning how preferences change with external information which does not require eliciting full cardinal preferences.
With this tool in hand, we find that voting mechanisms are generally more likely to select the alternative most favored by the external information, and when external information reflects the population's true preferences, social welfare increases in expectation.

§ INTRODUCTION

Classical results in voting theory prove that no voting mechanism, which is both non-dictatorial and anonymous, can be entirely immune to strategic manipulation [10, 17].
This phenomenon is often evident in real-world scenarios: for instance, a voter who supports candidate $a$ might opt to vote for candidate $b$ instead if candidates $b$ and $c$ are leading in the polls (see, for example, [20]).
This strategic behavior often appears to be “directed” in practice, influencing voters to align their preferences with a piece of external information.
We explore how this external information, which we call anchoring information, (1) affects the probability its most favored alternatives are selected by different voting mechanisms.
When external information is in line with societal preferences, we additionally examine (2) the impact of external information on total utility over the population, measured in social welfare.

Understanding the effects of anchoring information on voting outcomes is not straightforward: most voting mechanisms only elicit ordinal preferences from voters, but cardinal preferences conveying the strength of preferences are typically needed to know when a voter will respond to anchoring information.
In practice, cardinal preferences are often unknown and impractical to elicit from voters.
However, without access to cardinal utilities, it is unknown how to reason about the effect of anchoring information because we have no information about the strength of preferences within the available ordinal preferences.
Therefore, we must find some way to understand the effect of anchoring information without eliciting cardinal utilities.
Only once we have done this can we begin to study the impact of anchoring information on voting outcomes in expectation.

To this end, we introduce an intermediate mechanism in  <ref> that enables us to analyze the effect of anchoring information in order to answer the following three questions:

* In  <ref>, we obtain bounds on the probability an alternative is selected by a voting rule; do these bounds tighten when voters have anchored preferences? (Corollaries <ref> and <ref>)

* When does the expected social welfare increase with anchored preferences? (Theorem <ref>)

* Can we bound the probability that social welfare decreases on any given voting instance? (Theorem <ref>)

§.§ Related work

Anchored preferences
Recently, 9 use implicit utilities (term from [6]) to examine the distortion that emerges under public-spirited voting.
Their model of public-spirited utilities is a direct analog of the model presented by 8, which studies the price of anarchy in congestion games with altruistic players.
Our model of utilities is very similar to the previous models, but is slightly different in that 9 and 8, inspired by <cit.>, model players and voters as moving towards the optimal social outcome, while we model voters that move towards a given social outcome.
Their models have slightly more flexiblility on the weight of social outcomes as they analyze worst-case (distortion, price of anarchy) results in their respective settings, while we focus on average-case outcomes, which requires the additional assumption that all voters weigh the external information equally.[Average-case voting analyses are based on the premise that votes are i.i.d. samples from some distribution; changing $\alpha$ induces some unique distribution shift per voter, and therefore samples are no longer identically distributed.]
4 use the same model of voting as 9 to bound the distortion of participatory budgeting mechanisms with public-spirited voters.

Average-case voting
While the majority of the computational social choice literature focuses on worst-case impossibilities and tradeoffs, a solid line of work has examined average-case voting outcomes.
These works tend to examine voting outcomes in expectation over i.i.d. votes from some density $\mu$.
When $\mu$ is the uniform density, this yields the impartial culture assumption, which is notably unrealistic, but an important starting point for understanding average-case voting [14, 21].
5 and 2 both study the expected social welfare in average-case voting under slightly more general assumptions than impartial culture.
Both show that, if the impartial culture assumption is satisfied, then Borda voting maximizes the expected social welfare conditioned on the fixed ordinal votes.
Our approach to studying average-case voting is quite different; instead of examining expected social welfare conditioned on a voting profile, we are interested in changes to expected social welfare under mild, but directed, perturbations to voters' utilities.

Metric voting
Finally, our voting model is similar to, but subtly different from metric voting, in which voters and alternatives are embedded into $\reals^d$, then preferences are computed by evaluating relative distances to alternatives.
Distortion bounds in metric voting models are well understood [1, 19, 7].
In our model, instead of voters and alternatives being embedded into $\reals^d$, as in metric voting, we assume voters and reports are embedded into $X \subseteq \reals^m$, and voters submit the report they are closest to in distance.

§ VOTING MODEL

Suppose $n$ voters are working together to select an alternative in $[m] := \{1,2,\ldots, m\}$.
Voter $i$ has some normalized utility $u_i \in \simplex$, where $\simplex = \{u \in \reals^m_+ : \|u\|_1 = 1\}$ denotes the probability simplex.
We suppose utilities are drawn from some density $\mu$ defining the probability space $([m], \simplex, \mu)$.
Voters must select some score vector $r \in \R$ to assign to alternatives such that $|\R| = R < \infty$, where $\R \subset c \simplex$ for some $c > 0$ is a subset of a general simplex.
(Up to scaling, we can intuitively think of $\R$ as being a subset of $\simplex$.)
At times, we abuse notation and write $\R$ as a matrix in $\reals^{m \times R}$.
A set of votes is then input to the voting rule $f : \N^R \to [m]$ which maps frequencies of assigned score vectors to one selected alternative.
If $f$ is anonymous, then for each $r \in \R$, every permutation $\sigma(r) \in \R$ as well.
We often work with two $\R$: for Plurality, we let $\R = \Rplurality := \{e_i : i \in [m]\}$ be the standard basis, and when eliciting fully ordinal rankings, we let $\R = \Rborda := \{\sigma([m])\}$ be all permutations of $[m]$.

We suppose voter $i$ has preferences $u_i \sim \mu$, and voter $i$ chooses to report the score $r \in \R$ minimizing $d(u_i, r)$, where $d(\cdot, \cdot)$ is Euclidean distance.
The “scoring menu” $\R$ induces the function $\GammaR: u \mapsto \argmin_{r \in \R} d(u,r)$ mapping real-valued utilities to reports.
If $\R$ is understood from context, we simply denote this function $\Gamma$.
We suppose ties are broken uniformly at random between relevant parties.
Often, we work with the level set $\Gamma_r = \{u \in \simplex : r \in \Gamma(u)\}$, which is the set of cardinal preferences leading to a voter submitting $r \in \R$.
Each kite in Figure <ref> represents a different level set for $\Rplurality$.

[Plurality voting]
Consider the Plurality voting rule over $m = 3$ alternatives: $a$, $b$, and $c$.
A vote for alternative $a$ ascribes scores $(1,0,0)$ to alternatives $(a,b,c)$ respectively; a vote for alternative $b$ ascribes $(0,1,0)$, and a vote for $c$ score $(0,0,1)$.
The level set $\Gamma_{(1,0,0)} = \{u \in \Delta_3 \mid u_1 \geq u_2 \text{ and } u_1 \geq u_3\}$ is then the set of utility profiles representing voters who most prefer alternative $a$.
For a voter with cardinal utility $(\frac 1 5, \frac 1 2, \frac 3 {10})$, we have their vote $\Gamma((\frac 1 5, \frac 1 2, \frac 3 {10})) = \{(0,1,0)\}$, meaning they vote for alternative $b$.
Equivalently, their utility $(\frac 1 5, \frac 1 2, \frac 3 {10}) \in \Gamma_{(0,1,0)}$ is in the $(0,1,0)$ level set.
See Figure <ref> for depictions of the level sets for plurality and Figure <ref> for Borda and other voting rules that elicit full ordinal rankings over outcomes.

Since preferences are drawn from some unknown $\mu$, preferences are typically approximated by measuring $\frac{|\{i \mid \Gamma(u_i) = r\}|}{n} \approx \mu(\Gamma_r) = \int_{\Gamma_r} d\mu$.
Throughout, the measure of these level sets serve as a central piece of our analysis, as external information shifts the distribution $\mu$ in some directed way.
In general, we denote the probability distribution over reports with $p := \{\mu(\Gamma_r)\}_{r\in\R}$.

Plurality: Voters whose utilites lie in the red region (lower left) vote for alternative $a$ by ascribing score $(1,0,0)$, and similar for yellow voters voting for alternative $b$, and blue voting for alternative $c$.

Fully ordinal prefs: Voters whose utility lies in the orange region vote $a \succ b \succ c$ by ascribing score $(2,1,0)$ and so on by permutations of $[m]$.

To reason about anchoring information, we suppose a mechanism designer knows this information $w \in \simplex$ and voters are influenced by weight $\alpha \in [0,1]$ such that, instead of voting according to their standard utilities $u$, voters instead vote for the report $r \in \R$ minimizing $d((1-\alpha) u_i + \alpha w,r)$.
That is, each voter shifts their preferences $\alpha$ towards $w$.
We are interested in understanding the role of $w$ in changing voting outcomes, particularly its effect on social welfare.

[Changing vote via $w$]
Suppose a voter has utility $(1/2, 9/20,1/20)$, meaning they very slightly prefer the first outcome the most, then the second, and have almost no utility for the third.
If external information suggests $w = (0, 1/2, 1/2)$ and they are influenced with weight $\alpha = \frac 1 {10}$, then their updated preferences are $(0.45, 0.455, 0.095)$, and they switch their vote from candidate $a$ to candidate $b$.

Given a profile of $n$ utilities $\bld u \in \simplex^n$, we consider the induced histogram of votes $\#\Gamma(\bld u) \in \N^R$, which are determined by cardinal utility profiles $\bld u$ and the (property induced by the) report set $\GammaR$.
The $r^{th}$ entry of the histogram represents the number of voters submitting score $r$.[We generally assume preferences are drawn from a smooth enough distribution $\mu$ that the probability of a voter being exactly indifferent between two reports is $0$.]
We denote a general histogram of votes by $h \in \mathbb{N}^R$.

[Induced histograms]
Consider the Borda voting rule with the report set $\R$ being all $m!$ permutations of $\{0, 1, \ldots, m-1\}$.
If $m = 3$ and $n = 15$, a voting profile $h = (1,4,2,2,5,1) \in \mathbb{N}^{m!}$ implies one voter prefers $a \succ b \succ c$, four voters prefer $a \succ c \succ b$, etc., fixing the order of $\R$.
If the voting rule is Plurality, then $h \in \mathbb{N}^m$, and $h = (5,4,6)$ means $5$ voters have $a$ as their most preferred candidate, $4$ have $b$, and $6$ have $c$ as their most preferred.

In  <ref>, we examine positional scoring rules $f : \mathbb{N}^R \to [m]$, which are those that select the alternative with highest score given a histogram of score assignments $h \in \mathbb{N}^R$, denoted $f(h) = \argmax_{a} (\inprod{\R}{h})_a$.
If $|f(h)| >1$, there is a tie, which we assume are broken uniformly at random, as is common in the literature (cf. [5]).
Given a report set $\R$ and $n$ voters, let $f_a = \{h \in \mathbb{N}^R \mid a \in f(h), \|h\|_1 = n\}$, which is the set of histograms which lead to alternative $a$ being selected by voting rule $f$.

[Determining the winner of an election]
Suppose $m = 3$ and fix $n$.
For alternative $a$, the set of histograms selecting alternative $a$ according to Plurality voting, $\fplurality_a = \{h \in \mathbb{N}^m \mid h_{(1,0,0)} \geq h_{(0,1,0)} \text{ and } h_{(1,0,0)} \geq h_{(0,0,1)}\}$ is the set of histograms where $a$ receives more votes than $b$ and $c$.
If the voting rule is veto, then the histograms selecting alternative $a$ $\fveto_a = \{h \in \mathbb{N}^R \mid h_{(1,0,1)} \geq h_{(0,1,1)} \text{ and } h_{(1,1,0)} \geq h_{(0,1,1)} \}$ are those that veto $a$ less than $b$ and $c$.

§ TECHNICAL TOOLS TO ANALYZE HOW VOTING OUTCOMES CHANGE

In order to understand the effect of anchoring information on voting outcomes, we need two technical tools: first, we need to derive bounds on the probability any fixed alternative is selected when voters submit reports according to standard preferences, so that we can later juxtapose these bounds with anchored preferences ( <ref>).
This tool is used in  <ref> in conjunction with the mechanism in  <ref> in order to show that both upper and lower bounds on probability of the alternative $\astar$ most preferred by $w$ is selected increase.
The second technical tool we need is an intermediary mechanism that elicits anchored preferences ( <ref>).
This mechanism modifies the report scores $\R$ into a hypothetical set of scores $\M$ so that voting under $u$ with options in $\M$ always yields the same vote as voting under $(1-\alpha) u + \alpha w$ with options in $\R$.
This enables us to examine and compare the utility profiles $u$ with the submitted report changes with the introduction of anchoring information.

§.§ Bounding the probability alternative $a$ is selected via balls-and-bins

We first examine the probability of different outcomes being reached by a few standard voting rules.
We do this by bounding the probability any given alternative $a$ is selected as the outcome.
If, for a fixed voting rule $f$, there is some function $c : \mathbb{N} \to \mathbb{N}$ such that, for all alternatives $a \in [m]$, there exists a set of reports $Q_a \subseteq \R$ such that at least $c(n)$ reports belonging to the set $Q_a$ implies that $a$ selected by the voting mechanism.
That is, $\sum_{r \in Q_a} h_r > c(n) \implies f(h) = \{a\}$ is sufficient for a vote to be determined solely by the finite set $Q_a$ of reports, rather than searching over the entire space of votes.
(For example, if $c(n) = \frac n 2$, this reduces to the majority criterion <cit.>.)
In particular, with the report set $\Rborda$, we often use the set $Q^\sigma_a := \{r \in \Rborda : a \text{ is highest ranked alternative}\}$.

With this sufficient condition, we bound the probability alternative $a$ is selected by a reduction to the nonasymptotic balls and bins setting[Throwing $n$ balls into $R$ bins that land in each bin with probability according to $p$, what is the probability the max-loaded bin has $\geq k$ balls inside? e.g., [15]] studied by 18. In  <ref>, we show that if voters have anchored preferences, these lower bounds tighten for $w$'s most preferred alternative $\{\astar\} = \argmax_{a'}w_{a'}$, which demonstrates the influence of the anchoring point $w$.
Throughout this section, we let $p := \left\{\mu(\Gamma_r)\right\}_r$ be the probability distribution over coarse reports $\R$.

First, we use the fact that plurality satisfies the majority criterion as a sufficient condition for an alternative $a$ to be selected by $\fplurality$, which we reduce to balls and bins.
Moreover, by the Mean Value Theorem, we know that some alternative must receive $n / m$ votes in Plurality voting, with which we obtain upper bounds on the probability of any alternative $a$ winning an election.

Given a density $\mu$, consider $p~:=~\left\{\mu(\Gamma_r)\right\}_r$.
For any $a \in [m]$, we have

\begin{multline*}
\min\left({n \choose \frac{n}{m}}\|p_a\|^{n/m}_{n/m}, 1 - \sum_{b \neq a} \left(\Pr[Binom(n, p_b) > n/2]\right) \right)\\ \geq \Pr_{\bld u}[\fplurality(h) = a]
\geq \Pr[Binom(n, p_a) \geq n/2].
\end{multline*}

For the lower bound,

\begin{align*}
\Pr_{\bld u}[\fplurality(h) = a] &\geq \Pr_{\bld u}[h_a > n/2] + \frac 1 2 \Pr_{\bld u}[h_a = n/2] & \text{Majority criterion} \\
&\geq \Pr_p[Binom(n, p_a) > n/2] + \frac 1 2 \Pr_p[Binom(n, p_a) = n/2] & \text{[18]}~.
\end{align*}

For the upper bound, observe that the second term is simply $1-$ the probability any other alternative is selected.

The first term leverages the mean value theorem.
In plurality voting, precisely $n$ points are distributed among the $m$ alternatives, so some alternative must receive at least $\frac n m$ votes.
This also reduces to the nonasymptotic balls-and-bins, and we leverage the upper bound from <cit.>.

Proposition <ref> gives both upper and lower bounds on the probability that the Plurality voting rule selects alternative $a$.
Lower bounds are determined by the probability a Binomial sample of $n$ votes returns at least $c(n) := n/2$ “yes” results, and the upper bound is given by either the probability some other alternative receives at least $n/2$ votes or the probability that alternative $a$ receives at least $\frac n m$ votes; a necessary condition for winning the election.

We derive a similar bound for Borda, though this bound is strictly looser than the bound for Plurality, as the number of votes required for the sufficient condition is $c(n) = \frac{n(m-1)}{m}$: strictly greater than the $\frac n 2$ required for Plurality.
For large $m$, this bound loses its meaning as $c(n) \overset{m \to \infty}{\to} n$, but it is helpful for small $m$.

For any $a \in [m]$, density $\mu$, and $c(n) := \frac{n(m-1)}{m}$, then

\begin{align*}
1 - \sum_{b \neq a} \left(\Pr[Binom(n, p_{Q^\sigma_b}) > c(n)] \right) &\geq \Pr_{\bld u}[\fborda(h) = a] \\
&\geq \Pr_{\bld u}[Binom(n, \|\sum_{r \in Q^\sigma_a}p_r\|_{c(n)}) > c(n)]~.
\end{align*}

The proof follows in the same manner as the proof of Proposition <ref>.
Intuitively, because Borda rewards alternatives for being highly, but not highest, ranked.
This makes it easier for polarizing alternatives “to lose” an election than plurality, and Borda makes it easier “to win” by generally appealing to most voters for less polarizing alternatives.
This means the bounds are weaker than those given by plurality since these sufficient conditions are on “winning” and election, rather than “not losing.”

Other voting rules
Like Plurality, Copeland and Instant Runoff Voting (IRV) also satisfy the majority criterion.
Therefore, if, for all $a$, there is a set $Q_a \subseteq \Rborda$ such that $\sum_{r \in Q_a} h_r > \frac n 2 := c(n)$, then $f^{Copeland}(h) = f^{IRV}(h) = \{a\}$, then similar bounds can be derived.

Above, we give bounds based on sufficient conditions for an alternative to be declared the winner.
However, in Veto voting, we can intuitively think of the winner as being the “least disliked” alternative, which does not lend itself to (nontrivial) sufficient conditions that are independent of other alternatives.

§.§ Eliciting anchored preferences

In practice, algorithm designers do not know the density of voters' utilities $\mu$, and instead approximate the probability a voter submits report $r$ by considering the measure of the report set $\mu(\Gamma_r)~=~\int_{\Gamma_r}~d\mu~\approx \frac{|\{i : \Gamma(u_i) = r\}| }{n}$ with only access to $n$ reports $\{\Gamma(u_i)\}_{i \in [n]}$ instead of cardinal utilities $\{u_i\}_{i \in [n]}$.
This yields a more practical representation of the density $\mu$, but sacrifices the granular information needed to understand the impacts of the anchoring point $w$.

In Theorem <ref> show there is another “scoring menu” $\M$ inducing a new property $\Gaw$ and bijection $\phi : \R \to \M$ such that $\M$ simulates the role of anchoring information, demonstrated with the equivalence $(1-\alpha) u + \alpha w \in \Gamma_r \iff u \in \Gaw_{\phi(r)}$ for all $r \in \R$.
That is, for every report $r \in \R$, an anchored vote with the standard mechanism yields the report $r$ if and only if a non-anchored vote under the perturbed menu also produces vote $r$.
This (non-anonymous) scoring rule emulates each voter moving $\alpha$ towards the anchoring point $w$ without eliciting their full cardinal utility $u_i$.
We elicit slightly more granular information than standard ordinal preferences, but do not require full cardinal preferences.

The proof of Theorem <ref> is inspired from previous results on property elicitation [11, 12].

Given a scoring menu $\R$ inducing $\GammaR$, a voter with anchored preferences (parameterized by $w, \alpha$) will vote the same way under the property $\Gaw$ induced by the scoring menu $\M := \{\frac{r - \alpha w}{1-\alpha}\}_{r}$.
That is, $(1-\alpha)u + \alpha w \in \Gamma_r \iff u \in \Gaw_{\phi(r)}$ for all $r \in \R$, where $\phi : r \mapsto \frac{r-\alpha w}{1-\alpha}$.

Let $v = (1-\alpha)u + \alpha w$.
For a score $r$, we have

\begin{align*}
v \in \Gamma_r &\iff d(v, r) \leq d(v, {r'}) \qquad \forall r' \neq r \\
&\iff \|r - v\|^2 \leq \|r' - v\|^2 \qquad \forall r' \neq r \\
&\iff \|\frac{r - \alpha w}{1- \alpha} - u\|^2 \leq \|\frac{r' - \alpha w}{1-\alpha} - u\|^2 \qquad \forall r' \neq r \\
&\iff \|u - \frac{{r} - \alpha w}{1-\alpha}\|^2\leq \|u - \frac{{r'} - \alpha w}{1-\alpha}\|^2 \qquad \forall r' \neq r\\
&\iff u \in \Gaw_{\phi(r)}~.
\end{align*}

Theorem <ref> shows that we can use $\M$ as a set of scoring vectors which simulates the addition of external information.
This allows us to conceptually replace the shift in votes with a shift in the voting rule, and avoid trying to compute a distribution shift for a distribution we do not even know.
We benefit as we can now equivalently use the histogram of standard utilities when voting according to $\M$, the histograms $\#\Gaw(\bld u)= \#\Gamma^{\phi(\R)}(\bld u)$, which is equal to the histogram of anchored preferences according to standard scores $\#\GammaR((1-\alpha) \bld u + \alpha w)$ by Theorem <ref>.
Deriving $\M$ then allows us to empirically estimate the distribution $q := \{\mu(\Gaw_s)\}_{s \in \M}$ when $\mu$ is not known exactly, and does not require eliciting voters' cardinal preferences.
In the sequel, we slightly abuse notation and let $\Gaw_r$ denote what is technically $\Gaw_{\phi(r)}$, which is the set of utilities which would lead voters to submit report $r$ under anchored preferences.
Since $\phi$ is defined by $\alpha$, the mechanism designer is the entity imparting the value of anchored preferences $\alpha$ rather than the voters.[If $\alpha$ varies for each voter and menus were customized, we can no longer use the assumption that voters are identically distributed. Intuitively, we think of $\alpha$ as an average of individual $\alpha_i$s if $\alpha_i$ is drawn from a single-peaked and roughly symmetric distribution.]

§.§.§ Anchored votes align more closely with $w$ on the individual level

We now ask, if one elicits anchored preferences, how might an individual's vote change, given a general anchoring point $w \in \simplex$.
Perhaps unsurprisingly, we show that individual votes only move to reports that “better align” with $w$ in this model.

Fix $w \in \simplex$ and $\alpha \in [0,1)$, and consider the function $\phi : r \mapsto \frac{r - \alpha w}{1-\alpha}$.
For any two reports $s,t \in \R$, for voter utility $u \in \simplex$ such that $d(u,s) \leq d(u,t)$ and $\langle w, s \rangle \geq ]langle w, t \rangle$, then $d(u, \phi(s)) \leq d(u, \phi(t))$.

It suffices to show the inequality on squared distance.
Observe that we can re-write the squared distance $d(u, \phi(s))^2 =  d(u, \frac s {1-\alpha})^2 + \sum_{i \in [m]} \frac{\alpha w_i}{1-\alpha} (\frac{\alpha w_i + 2 u_i -2 s_i}{1-\alpha})$.

\begin{align*}
\sum_{i \in [m]} w_i (t_i - s_i) &\leq 0  \\
\sum_{i \in [m]} \frac{\alpha w_i}{1-\alpha} \left(\frac{\alpha w_i + 2u_i -2 s_i}{1-\alpha} - \frac{\alpha w_i +2u_i-2 t_i}{1-\alpha}\right) &\leq 0 \\
\sum_{i \in [m]} \frac{\alpha w_i}{1-\alpha} \left(\frac{\alpha w_i + 2u_i -2 s_i}{1-\alpha}\right) &\leq \sum_{i \in [m]} \frac{\alpha w_i}{1-\alpha} \left(\frac{\alpha w_i +2u_i-2 t_i}{1-\alpha}\right) \\
\implies d\left(u, \frac{s}{1-\alpha}\right)^2 + \sum_{i \in [m]} \frac{\alpha w_i}{1-\alpha} \left(\frac{\alpha w_i + 2u_i -2 s_i}{1-\alpha}\right) &\leq d\left(u, \frac{t}{1-\alpha}\right)^2 + \sum_{i \in [m]} \frac{\alpha w_i}{1-\alpha} \left(\frac{\alpha w_i + 2u_i -2 t_i}{1-\alpha}\right) \\
d(u, \phi(s))^2 &\leq d(u, \phi(t))^2
\end{align*}

Proposition <ref> suggests that reports “aligning with $w$” increase in probability: if an agent's vote moves from $s$ to $t$, then $w$ “prefers $s$ to $t$”, given by $\langle w, s \rangle < \langle w, t \rangle$.
Wquivalently, no player will move from $s$ to $t$ if  $w$ prefers $s$ to $t$, again given $\inprod w s > \inprod w t$.
This phenomena is demonstrated in Figure <ref>, where the red cell, corresponding to the vote for alternative $a$, which is most preferred by $w$, is bigger under anchored voting.
Proposition <ref> immediately implies Corollary <ref>, which states that the set of cardinal utilities yielding the report $r^\star$ that “would be submitted by $w$” strictly increases.

Let an anchoring point $w \in \simplex$ and weight $\alpha \in (0,1]$ be given and consider the report $\{r^\star\} = \argmax_{r' \in \R} \langle w, r' \rangle$.
Then $\GammaR_{r^\star} \subsetneq \Gaw_{r^\star}$.

Anchoring Plurality votes in the point $w$ yields a menu $\M$ with the level sets of $\Gaw$ laid over the level sets of $\GammaR$. Utilities in red and yellow intersection (hatched) will vote for candidate $b$ under standard preferences and candidate $a$ with the anchoring information. They are close to indifferent between $a$ and $b$ under standard preferences.

Under the common impartial culture assumption that $\mu$ is the uniform density $m$, this further implies that the probability of any anchored preference is ordered according to $w$.
(Recall that we do not generally requite the impartial culture assumption, unlike much of the previous literature.)

Let $f$ be an anonymous voting rule.
If $m$ is the uniform density, then for all $s,t \in \R$, the probability an agent reports $s$ is greater than the probability they report $t$ if and only if $w$ prefers $s$ to $t$.
That is, $m(\Gaw_{s}) \geq m(\Gaw_t) \iff \langle w, s \rangle \geq \langle w, t \rangle$.

This implies that, when $\mu$ is uniform, incorporating external information anchored in $w$ leads to the probabilities of alternatives being elected to follow the same order as $w$ if we recursively apply this argument to all pairs of reports $s,t$.
While mathematically tractable, it is well-known that impartial culture assumptions are often unrealistic [14].

§ ANCHORED PREFERENCES CAN INCREASE THE PROBABILITY $W$'S PREFERRED ALTERNATIVE IS SELECTED

Recall that in  <ref> we reduce voting in expectation to a balls-to-bins problem to obtain lower and upper bounds on the probability an alternative $a$ is selected.
We naturally wonder if and when anchored preferences tighten these lower bounds or loosen the upper bounds.
We show in Corollary <ref> that, for the alternative $\astar$ such that $\{\astar\} = \argmax_{a'}w_{a'}$, the lower bounds do indeed tighten when the voting rule is Plurality, and give conditions on $w$ that tighten the lower bounds for Borda.
Moreover, under certain smoothness conditions on $\mu$, we also show the upper bounds loosen (increase) in Corollary <ref>.

Since the bounds in  <ref> are given by the binomial distribution, we first study the monotonicity of the binomial distribution in the probability of a positive event.
We show in Lemma <ref> that if the probability of a positive instance increases, then the probability that a binomial sample has at least $k$ positive instances (meaning $r \in Q_a$) also increases.
For plurality, this suffices to tighten bounds, and in Lemma  <ref>, we give a sufficient condition on $w$ for the probability that $r \in Q_{\astar}$ to increase with anchored preferences.

[Monotonicity of binomial bounds in $p$][end,restate,text link]
Suppose $p \leq q$.
Then for any $k \in [n]$, we have $\Pr[Binom(n, p) \geq k] \leq \Pr[Binom(n,q) \geq k]$.

It suffices to show $\frac d {dp}\Pr[Binom(n, p) \geq k] \geq 0$ for all $p \in [0,1]$.
We recursively derive the closed form of $\frac d {dp}\Pr[Binom(n, p) \geq n - k] = (n-k) {n \choose k} p^{n-k-1} (1-p)^k$, which is nonnegative for all $p \in [0,1]$, yielding the result.
In general, we have the form

\begin{align*}
\frac d {dp}\Pr[Binom(n, p) \geq n - k]
&= \sum_{\ell=0}^k\frac d {dp} \Pr[Binom(n,p) = n-\ell] \\
&= \sum_{\ell = 0}^k {n \choose \ell} p^{n-\ell - 1} (1-p)^{\ell-1} (n-\ell - np)~.\\
\end{align*}

First, let us consider the base case: $k = 0$.

\begin{align*}
\frac d {dp}\Pr[Binom(n, p) \geq n] &=  p^{n- 1} (1-p)^{-1} n (1-p)\\
&= n p^{n-1}\\
&= (n-0) {n \choose 0} p^{n-1} (1-p)^0
\end{align*}

Now assume this holds for $k$. We show it holds for $k+1$.

\begin{align*}
& \frac d {dp}\Pr[Binom(n, p) \geq n-(k+1)] \\
&= \frac d {dp}\Pr[Binom(n, p) \geq n-k] + \frac d {dp}\Pr[Binom(n, p) = n-(k+1)] \\
&= \overbrace{{n \choose k}(n-k)p^{n-k-1} (1-p)^k} + \overbrace{{n \choose k+1}p^{n - (k+1)-1} (1-p)^k (n - (k+1) - np)} \\
&= {n \choose k+1}p^{n-k-2} (1-p)^k (p (k+1)) + {n \choose k+1}p^{n - (k+1)-1} (1-p)^k (n - (k+1) - np) \\
&= {n \choose k+1}p^{n- k- 2} (1-p)^{k} (p(k+1) + (n - k - 1 - np)) \\
&= {n \choose k+1}p^{n- k- 2} (1-p)^{k} ((1-p) (n-k - 1)) \\
&= {n \choose k+1}(n-k-1)p^{n- k- 2} (1-p)^{k+1}
\end{align*}

One corollary of Proposition <ref> is that when the report set is the set of elementary basis vectors, $\R = \Rplurality$, then $\mu(\GammaR_{\astar}) \leq \mu(\Gaw_{\astar})$ for all $w$ and $\alpha$. In turn, Lemma <ref> tightens the lower bounds from Proposition <ref> for all $w \in \simplex$.

However, when $\R = \Rborda$, this is not necessarily the case.
For intuition, if $w$ is close to indifferent between two alternatives as its preferred, then there might be some report $s \not\in Q_{\astar}$ that does not rank $\astar$ as its top choice, yet because of utility over non-top outcomes, we might have $\inprod w s \geq \inprod w t$ for some $t \in Q_{\astar}$.
This opens up the possibility that the probability a voter ascribes the highest score to alternative $\astar$ decreases, i.e., $\sum_{r \in Q_{\astar}} \mu(\Gamma_r) > \sum_{r \in Q_{\astar}} \mu(\Gaw_r)$, as votes might move from $t$ to $s$.
In Lemma <ref>, we give a sufficient condition on $w$ for the binomial lower bound to be tightened, illustrated with $m=3$ alternatives in Figure <ref>.
Intuitively, if $w$ favors one alternative enough, then the reports $r \in \R$ that most align with $w$ are those that assign the highest score to $a$.
In the sequel, let $k = (m-1)!$.

[Sufficient condition on $w$ for probability of a report in $Q^\sigma_{\astar}$ to increase][end,restate]
Let $\R = \Rborda$.
If $Q^\sigma_{\astar} = \topk{k}(\langle w, r\rangle_{r \in \R})$, then $\sum_{r \in Q^\sigma_{\astar}}\mu(\Gaw_{\phi(r)}) \geq \sum_{r \in Q^\sigma_{\astar}}\mu(\GammaR_r)$, and in turn, $q_{Q^\sigma_{\astar}} \geq p_{Q^\sigma_{\astar}}$.

By Proposition <ref>, for any $r \in Q_{\astar}$, if $u \not \in \Gaw_{\phi(r)}$, it must be in $\Gaw_{\phi(t)}$ for some other $t \in Q_{\astar}$.

\begin{align*}
\cup_{r \in Q_{\astar}} \GammaR_r &= \cup_{r \in Q_{\astar}, s \in \R} (\GammaR_r \cap \Gaw_{\phi(s)}) & \text{disjoint sets} \\
&= \cup_{r \in Q_{\astar}, s \in Q_{\astar}} (\GammaR_r \cap \Gaw_{\phi(s)}) &\text{if $s \not \in Q_{\astar}$, then $\GammaR_r \cap \Gaw_{\phi(s)} = \emptyset$} \\
&\subseteq \cup_{s \in Q_{\astar}, r \in \R} (\GammaR_r \cap \Gaw_{\phi(s)}) & Q_{\astar} \subseteq \R \\
&= \cup_{r \in Q_{\astar}} \Gaw_r &
\end{align*}

Since $\cup_{r \in Q_{\astar}} \GammaR_r \subseteq \cup_{r \in Q_{\astar}} \Gaw_{\phi(r)}$, we have $\sum_{r \in Q_{\astar}} \mu(\GammaR_r) = \mu(\cup_{r \in Q_{\astar}} \GammaR_r) \leq \mu(\cup_{r \in Q_{\astar}} \Gaw_{\phi(r)}) = \sum_{r \in Q_{\astar}} \mu(\Gaw_{\phi(r)})$.

Lemma <ref> shows that if reports assigning score $m-1$ to alternative $\astar$ are the top-$k$ preferred by $w$, then the probability that any one vote assigns score $m-1$ to alternative $a$ increases.

[][all end,restate]
Fix $w \in \simplex$.
Let $\astar = \argmax_{a'} w_{a'}$.
If $w_a \geq (m-1) w_{[2]} + \sum_{i=3}^m w_{[i]} (m - 2(i-i))$, then $Q_a = \topk{k}(\{\langle w, r\rangle\}_{r \in \R})$.

For ease of exposition, let $x_{[i]}$ denote the $i^{th}$ largest element of $x$.
It suffices to show

\begin{align*}
\min_{r \in Q_a} \inprod w r &\geq \max_{s \not \in Q_a} \inprod w s \\
w_a (m-1) + \sum_{i=2}^m w_{[i]} (i-2) &\geq w_a (m-2) + w_{[2]} (m-1) + \sum_{i=3}^m w_{[i]} (m-i) \\
w_a &\geq w_{[2]}(m-1) + \sum_{i=3}^m w_{[i]} (m-2i + 2)~,
\end{align*}

which concludes the proof.

With $m=3$, Lemma <ref> shows that it suffices for $w_b \leq 1/3$ for all $b \neq a$ in order for the probability that alternative $\astar$ is the top-ranked to increase.
Lemma <ref> in  <ref> characterizes $w$ such that $q_{Q^{\sigma}_{\astar}} \geq p_{Q^{\sigma}_{\astar}}$, the condition of which is captured by Assumption <ref>.
Throughout, we let $w_{[i]}$ denote the $i^{th}$ largest component of the vector $w$.

Hatched regions subsets of $\Delta_3$ where any $w$ in the hatched region increases ensures that reports where $\astar$ is the favorite are precisely the top-$(m-1)!$ most preferred.
In turn, for any $w$ in the hatched regions, the bounds on the probability that $w$'s most favored alternative is selected increase.

For $m \geq 3$, let $w$ be an anchoring point such that $w_{[1]} \geq w_{[2]}(m-1) + \sum_{i=3}^m w_{[i]} (m-2i + 2)$.

In essence, we assume the anchoring point $w$ favors one alternative over others strongly enough.
If this is true, the probability that a voter ranks that alternative the highest increases.
This now enables us to observe that the lower bounds derived in  <ref> tighten under anchored preferences.

Let $w$ be an anchoring point with $\{\astar\} = \argmax_{a'}w_{a'}$.
The lower bound given by Proposition <ref> tightens for the alternative $\astar$.
Moreover, if $\R = \Rborda$ and $w$ satisfies Assumption <ref>, then the lower bound from Proposition <ref> tightens.

Corollary <ref> shows that anchored preferences can, but do not necessarily, tighten the lower bounds on the probability of a voting rule selects the alternative $\astar$ most preferred by $w$.
Moreover, the upper bound loosens if $\mu$ is sufficiently smooth.

Let $w$ be an anchoring point such that ${\astar} = \argmax_{a'}w_{a'}$.
Suppose $\astar$ is the only alternative that increases in probability (i.e., $p_{Q_b} \geq q_{Q_b}$ for all $b \neq \astar$).
Then the upper bounds in Propositions <ref> and <ref> loosen.

§ CHANGES IN SOCIAL WELFARE ARISING FROM ANCHORED VOTING

While Proposition <ref> implies that individuals will only vote more in line with $w$, it is well-known that Paretian and non-dictatorial voting rules cannot satisfy independence of irrelevant alternatives [3], and intuitively, this suggests that that voting according to anchored preferences may change votes to irrelevant alternatives.
While an individual vote might increase the score of an irrelevant alternative, it's unclear when and if that will actually change the alternative selected by the voting rule.
In order to understand this, we examine sets of voting profiles in histogram space.

For a fixed, finite set of $n$ voters, the set of possible induced histograms is finite, though large.
Given a density $\mu$ on individual utilities and the induced distribution $p := \left\{ \mu(\Gamma_r) \right\}$, weight $\alpha$, and anchoring point $w$, we examine $\Gaw$ and examine the histogram distributions $\nu$ and $\nusoc$ such that

\begin{align*}
\nu(f_a)
&= \sum_{h : a \in f(h)} \frac{1}{|f(h)|}\Pr_{\bld u}[\#\GammaR(\bld u) = h] \\
&= \sum_{h : a \in f(h)} \frac{1}{|f(h)|} (\frac{n!}{\prod_r h_r!}) \prod_r p_r^{h_r}\end{align*}

and likewise for $\nusoc(f_a)$, substituting $\Pr_{\bld u}[\#\GammaR((1-\alpha) \bld u + \alpha w) = h]$, which is a function of $q := \{\mu(\Gaw_{\phi(r)})\}$.
The $\frac 1 {|f(h)|}$ term breaks ties uniformly at random

§.§ Expected social welfare increases when external information reflects social preferences

We are now equipped to ask how the anchoring point $w$ affects social welfare?
We give a sufficient condition for social welfare to increase in expectation in Theorem <ref>.
Then, in Proposition <ref> we show that if the density $\mu$ is smooth enough, then an anchoring point $w$ roughly according to the “average utililty” $\E_\mu u$, this sufficient condition is satisfied.
Let $\sw(f(h), \bld u) := \sum_i u_i(f(h))$ be the total utility attained by the voting outcome $f(h)$.
Throughout, we let $\Delta \sw(\bld u) := \sw(f(\#\GammaR((1-\alpha)\bld u + \alpha w), \bld u) - \sw(f(\#\GammaR(\bld u), \bld u))$, and we ask when $\E [\Delta \sw \mid w, \alpha] \geq 0$.
That is, when does social welfare increase (in expectation) under anchored preferences?

Let $w, \alpha, \mu$ be given.
Suppose $v := \E_{u \sim \mu} u$, and let the set $\inc := \{a \in [m] : \nusoc(f_a) - \nu(f_a) \geq 0\}$ be the set of alternatives that weakly increase in probability of being selected under anchored preferences, and $\dec := [m] \setminus \inc$ be the set of alternatives that decrease.
If $\max_{a \in \dec} v_a \leq \min_{a \in \inc} v_a$, then $\E_{\bld u}[\Delta \sw(\bld u) \mid w, \alpha] \geq 0$.

First, observe that since social welfare is additive and preferences are drawn iid, we can restrict our focus to $v$ instead of $\bld u$.

\begin{align*}
\E_{\bld u \sim \mu^n} [\Delta \sw(\bld u) \mid w,\alpha] &= \E_{\bld u \sim \mu^n}[\sum_{i \in [n]} \sum_{a \in [m]} u_i(a) \left(\Pr_{\bld u}[f(h(\bld u, \Gaw)) =  a \mid w] -  \Pr_{\bld u}[f((h(\bld u, \Gamma)) = a]\right) \\
&= \E_{\bld u \sim \mu^n} [ \sum_{i \in [n]} \langle u_i, \nusoc(f) - \nu(f) \rangle] \\
&= n \E_{u \sim \mu} [ \langle u, \nusoc(f) - \nu(f) \rangle] \\
&= n \langle v, \nusoc(f) - \nu(f) \rangle  \geq 0 \iff
\langle v, \nusoc(f) - \nu(f) \rangle \geq 0
\end{align*}

Now we argue that under the assumptions, that $\langle v, \nusoc(f) - \nu(f) \rangle \geq 0$.

First, observe that since $\nu$ and $\nusoc$ are probability distributions (both subject to affine constraints), $\sum_a \nusoc(f_a) - \sum_a \nu(f_a) = 0$, and therefore

\begin{align}
\sum_{a \in \inc} (\nusoc(f_a) - \nu(f_a)) &= - \sum_{a \in \dec}(\nusoc(f_a) - \nu(f_a)) &= \sum_{a \in \dec}(\nu(f_a) - \nusoc(f_a)).\label{eq:order-diff}
\end{align}

This yields

\begin{align*}
\langle v, \nusoc(f) - \nu(f) \rangle \geq 0 &\iff
\sum_{a \in \inc} v_a (\nusoc(f_a) - \nu(f_a)) \geq \sum_{a \in \dec} v_a (\nu(f_a) - \nusoc(f_a))
\end{align*}

We can show the latter inequality, as

\begin{align*}
\sum_{a \in \inc} v_a (\nusoc(f_a) - \nu(f_a)) &\geq \min_{a \in \inc} v_a \sum_{a \in \inc} (\nusoc(f_a) - \nu(f_a)) \\
&= \min_{a \in \inc} v_a \sum_{a \in \dec} (\nu(f_a) - \nusoc(f_a)) \qquad \text{Eq.~\eqref{eq:order-diff}}\\
&\geq \max_{a \in \dec} v_a \sum_{a \in \dec} (\nu(f_a) - \nusoc(f_a)) \qquad \text{by assumption}\\
&\geq \sum_{a \in \dec} v_a (\nu(f_a) - \nusoc(f_a))
\end{align*}

Intuitively, Theorem <ref> shows that expected social welfare increases if the anchoring point $w$ is at least a good approximation of the true expected utility $v$.

Moreover, if $\mu$ is approximately uniform (Definition <ref>), having small total variation distance (TV) from the uniform density, then the assumptions of Theorem <ref> are satisfied, yielding Proposition <ref>.

A density $\mu$ is $\epsilon$-uniform if, $\sup_{A \in \mathcal{B}} \left| \int_A d\mu - \int_A dm \right| \leq \epsilon$.
That is, $TV(\mu,m) \leq \epsilon$, where $m$ is the uniform density.

Our notion of $\epsilon$-uniformness is a much stricter condition than necessary, but its generality allows us to draw conclusions about any $w$ that is ordered roughly according to the “average utility” $\E_\mu u$.

[][all end]
Let $f$ be an anonymous voting rule, and $\mu$ be $\epsilon$-uniform.
If $2 \epsilon \leq \min_{s,t \in \R} |m(\Gaw_t) - m(\Gaw_s)|$, then $\{\mu(\Gaw_t)\}_t$ preserves the same order as $\langle w, \R\rangle$.

Since $\mu$ is $\epsilon$-uniform, then $|\int_{\Gaw_r} d\mu - \int_{\Gaw_r} dm|  = |\mu(\Gaw_r) - m(\Gaw_r)|\leq \epsilon$ implies $\mu(\Gaw_r) \in [m(\Gaw_r) - \epsilon, m(\Gaw_r) + \epsilon]$.
As $\{m(\Gaw_r)\}$ has the same order as $\langle w, \R \rangle$ (by Corollary <ref>), the result follows immediately.

[][all end]
Let $f$ be an anonymous voting rule and $\mu$ be $\epsilon$-uniform.
If $2 \frac{\sum_{k=1}^n {n \choose k}(R\epsilon)^k}{m} \leq \min_{a,b} |\nu(f_a) - \nu(f_b)|$, then $\{\nusoc(f_a)\}_a$ preserves the same order as $\{\nusoc(f_a) - \nu(f_a)\}_a$.

Since $\mu$ is $\epsilon$-uniform, we have $|\mu(\Gamma_r) - m(\Gamma_r)| \leq \epsilon$, which implies $\mu(\Gamma_r) \in [\frac{1 - R\epsilon}{R}, \frac{1 + R \epsilon}{R}]$ for all $r \in \R$ since $f$ is anonymous.
This yields

\begin{align*}
\sum_{h \in f_a} \frac{1}{|f(h)|} \frac{n!}{\prod_r h_r!} \prod_r(\frac{1 - R\epsilon}{R})^{h_r} &\leq \nu(f_a) \leq \sum_{h \in f_a} \frac{1}{|f(h)|} \frac{n!}{\prod_r h_r!} \prod_r(\frac{1 + R\epsilon}{R})^{h_r} \qquad \forall a \in [m] \\
\frac{(1 - R\epsilon)^n}{m} &\leq \nu(f_a) \leq \frac{(1 + R\epsilon)^n}{m} \qquad \forall a \in [m] \\
\frac 1 m + \frac{\sum_{k=1}^n {n \choose k}(-R \epsilon)^k}{m} &\leq \nu(f_a) \leq \frac 1 m + \frac{\sum_{k=1}^n {n \choose k}(R \epsilon)^k}{m} \qquad \forall a \in [m]
\end{align*}

Therefore, $\nu(f_a) \in [\frac 1 m - \delta, \frac 1 m + \delta]$ for $\delta := \max\left(|\frac{\sum_{k=1}^n {n \choose k}(-R \epsilon)^k}{m}|, \frac{\sum_{k=1}^n {n \choose k}(R \epsilon)^k}{m}\right) = \frac{\sum_{k=1}^n {n \choose k}(R \epsilon)^k}{m}$.

If $2 \delta$ is smaller than the smallest gap $|\nusoc(f_a) - \nusoc(f_b)|$, then the order of $\nusoc(f_a)$ is preserved.

[][all end]
Let $f$ be an anonymous positional scoring rule and $\mu$ be a $\epsilon$-uniform density for $\epsilon$ sufficiently small.
Then if $\left\{\mu(\Gaw_r)\right\}_r$ has the same order as $\langle w, \R \rangle$, then $\left\{\nusoc(f_a) - \nu(f_a)\right\}$ has the same order as $w$.

By Lemma <ref>, $\{\nusoc(f_a)\}$ has the same order as $\{\nusoc(f_a) - \nu(f_a)\}$.
We show the contrapositive.
Suppose $\{\nusoc(f_a) - \nu(f_a)\}$ has a different order then $w$, then so does $\{\nusoc(f_a)\}$.
Then there exist $a,b \in  [m]$ such that $\nusoc(f_a) > \nusoc(f_b)$ but $w_a < w_b$.

$\nusoc(f_a) > \nusoc(f_b) \implies \sum_{h \in f_a}\frac{1}{|f(h)|}\frac{n!}{\prod_r h_r!} (\prod_r \mu(\Gaw_r)^{h_r} - \prod_r \mu(\Gaw_r)^{h_{\sigma(r)}}) > 0$ for the permutation $\sigma$ switching the relative position of $a$ and $b$ in each report.

This summation being positive implies the existence of an $h$ such that $f(h) = \{a\}$ and $\prod_r \mu(\Gaw_r)^{h_r} - \prod_r \mu(\Gaw_{\sigma(r)})^{h_r} > 0$.
Moreover, $\inprod h {\R}_a > \langle h, \R \rangle_b$ since $f(h) = \{a\}$ and $f$ is a positional scoring rule.
This implies the existence of a $t \in \R$ so that $\mu(\Gaw_t)^{h_t} - \mu(\Gaw_{\sigma(t)})^{h_t} > 0$ and $t_a > t_b$.The first statement implies $\mu(\Gaw_t) > \mu(\Gaw_{\sigma(t)})$, and the second implies $\sigma(t)_b > \sigma(t)_a$.

\begin{align*}
\langle w, \sigma(t)-t \rangle &= w_a (\sigma(t)_a - t_a) + w_b (\sigma(t)_b - t_b) \\
&= (w_b-w_a) (\sigma(t)_b - \sigma(t)_a) \geq 0~,
\end{align*}

which yields the contrapositive.

Suppose $f$ is an anonymous positional scoring rule, and $w$ and $v$ have the same order.
Moreover, suppose $\mu$ is $\epsilon$-uniform for $\epsilon \leq \min_{s,t \in \R} \frac{|m(\Gaw_s) - m(\Gaw_t)|}{2}$ and $\frac{\sum_{k=1}^n {n \choose k}(R \epsilon)^k}{m}\leq \min_{a,b} \frac{|\nu(f_a) - \nu(f_b)|}{2}$, then expected social welfare increases.

First, Lemma <ref> implies $\{\mu(\Gaw_{\phi(r)})\}$ has the same order as $\inprod w \R$, and therefore $\{\nusoc(f_a) - \nu(f_a)\}$ has the same ordering as $w$ by Lemma <ref>, which yields the conditions of Theorem <ref>.

While we show that expected social welfare increases if $w$ roughly aligns with social welfare, an adversarial anchoring point $w$ that is ordered in the reverse of $v$ will generally lead to a decrease in social welfare.
This demonstrates the influence exhibited by the anchoring information $w$.

Upon learning that welfare increases in expectation, a natural next question is to understand the probability the social welfare decreases.
Since political voting mechanisms are often run once every few years, understanding the probability that social welfare decreases is important, even when $w$ is perfectly aligned with expected utility $v$.

Given a density $\mu$ such that preferences are drawn i.i.d. from $\mu$, and suppose the anchoring point $w$ and weight $\alpha \in [0,1]$.
Then $\Pr_{\bld u}[\sw(f(\#\GammaR(\bld u)), \bld u) > \sw(f(\#\GammaR((1-\alpha)\bld u + \alpha w), \bld u)] \leq \E[\exp(\sw(f(\#\GammaR(\bld u)), \bld u) - \sw(f(\#\GammaR((1-\alpha)\bld u + \alpha w)), \bld u))]$.
That is, the probability that social welfare decreases under anchored preferences is upper bounded.

For notational simplicity, fix any $\epsilon \in (0, \min_{a, b : w_a > w_b} (w_a - w_b))$ and let $V := -\Delta \sw = \sw((1-\alpha)\bld u + \alpha w, \GammaR) - \sw(\bld u, \GammaR)$.

\begin{align*}
\Pr[\Delta \sw < 0] &= \Pr[-\Delta \sw \geq \epsilon] &\text{Finite alternatives $m$}\\
&\leq M_V(t) e^{-t \epsilon} \qquad \forall t > 0 &\text{Chernoff bound}\\
&= \E_V [\exp (tV)] \exp(-t\epsilon) \qquad \forall t > 0 &\text{definition of MGF} \\
&\leq \E_V [\exp(tV)] \qquad \forall t > 0 &\text{$-t\epsilon < 0 \implies \exp(-t\epsilon) \in (0,1)$}\\
&\leq \E_V [\exp(V)] &\text{choose $t = 1$}\\
&= \E_{\bld u} [\exp(\sw(f(\#\GammaR(\bld u), \bld u)) - \sw(f(\#\Gaw(\bld u)), \bld u)]
\end{align*}

This bound is only meaningful (i.e., it is tighter than $1$) if $\E [\Delta \sw \mid w,\alpha] > 0$; that is, if voting by anchored preferences increases social welfare in expectation.
Moreover, as $\E [\Delta \sw \mid w,\alpha] \to \infty$ (i.e. social welfare increases by a significantly large quantity), the probability that social welfare decreases tends to $0$.

§ CONCLUSIONS AND FUTURE WORK

This work provides methodologies to examine the outcomes of voting mechanisms in expectation when voters are influenced by external information.
We often focus on the case where $w$ is aligned with social welfare. Unsurprisingly, imposing an anchoring point aligned with social welfare tends to increase social welfare in expectation.
Moreover, we give upper and lower bounds on the probability that any one alternative is selected by the voting mechanism, and show that, for the alternative $a$ maximizing $w_a$, this lower bound tightens, and the upper bound can loosen when anchored preferences.

Many directions for future work lie ahead.
For example, voters often receive external information conveying varying messages, and it would be fruitful to understand the effect of external information when there are multiple, conflicting sources of information.
Furthermore, it would be interesting to use these tools to examine how other mechanisms respond to socially– or fairness–concerned players; do mechanisms with welfare or fairness guarantees lose those guarantees?

§ ACKNOWLEDGEMENTS

We would like to thank Bailey Flanigan for helpful comments in scoping out and situating this work.
This material is based upon work supported by the National Science Foundation under Award No. 2202898 and by National Science Foundation and Amazon under Award No. 2147187.

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§ OMITTED PROOFS