# Hydrothermal synthesis of ordered corkite PbFe3(PO4)(SO4)(OH)6, a S = 5/2
kagomé antiferromagnet

Austin M. Ferrenti  Department of Chemistry, The Johns Hopkins University,
Baltimore, Maryland 21218, USA<EMAIL_ADDRESS>Vanessa Meschke  Department
of Physics, Colorado School of Mines, Golden, Colorado 80401, USA  Shreenanda
Ghosh  Institute for Quantum Matter, William H. Miller III Department of
Physics and Astronomy, The Johns Hopkins University, Baltimore, Maryland
21218, USA  Jackson Davis  Institute for Quantum Matter, William H. Miller III
Department of Physics and Astronomy, The Johns Hopkins University, Baltimore,
Maryland 21218, USA  Natalia Drichko  Institute for Quantum Matter, William H.
Miller III Department of Physics and Astronomy, The Johns Hopkins University,
Baltimore, Maryland 21218, USA  Eric. S. Toberer  Department of Physics,
Colorado School of Mines, Golden, Colorado 80401, USA  Tyrel M. McQueen
Department of Chemistry, The Johns Hopkins University, Baltimore, Maryland
21218, USA<EMAIL_ADDRESS>
###### Abstract

Corkite, PbFe3(PO4)(SO4)(OH)6, an understudied relative of the jarosite family
of Heisenberg antiferromagnets, has been synthesized and its magnetic
properties characterized for the first time. Relative to natural samples,
synthetic corkite displays signatures in both infrared and Raman spectra of a
more ordered arrangement of polyanion groups about the kagomé sublattice that
retains inversion symmetry. Magnetic susceptibility measurements reveal that
dried corkite undergoes a transition to a long-range, antiferromagnetically-
ordered state below $T_{N}$ = 48 K, lower than that observed in the majority
of jarosite phases, and indicative of further spin frustration. Curie-Weiss
fitting of the measured magnetic susceptibility yields an effective magnetic
moment of peff = 6.29(1) $\mu_{B}$/Fe3+ and $\theta_{CW}$ = -526.0(1.1) K,
analogous to that observed in similar high-spin Fe3+ systems, and indicative
of strong antiferromagnetic coupling. Estimation of the change in magnetic
entropy as a function of temperature from T = 0 K to T = 195 K, $\Delta
S_{mag}$ = 14.86 $J\cdot mol_{Fe^{3+}}^{-1}K^{-1}$, is also in good agreement
with the $\Delta S_{mag}$ = Rln(2S+1) = 14.9 $J\cdot mol^{-1}K^{-1}$ expected
for a S = 5/2 system. In comparison to the pure jarosites, where both
structure and magnetism remain largely invariant upon a variety of chemical
substitutions, the replacement of one sulfate group per formula unit with a
higher-valent phosphate group applies additional steric and electronic
pressure on the kagomé lattice in corkite, further frustrating the magnetic
ground state of the material. Corkite thus represents both an outlier in the
known body of jarosite-type materials, and an illustration of how existing
structures may be further strained in the development of highly frustrated
magnetic systems.

Institute for Quantum Matter, William H. Miller III Department of Physics and
Astronomy, The Johns Hopkins University, Baltimore, Maryland 21218, USA
Institute for Quantum Matter, William H. Miller III Department of Physics and
Astronomy, The Johns Hopkins University, Baltimore, Maryland 21218, USA
Department of Materials Science and Engineering, The Johns Hopkins University,
Baltimore, Maryland 21218, USA

##  1 Introduction

Materials containing the kagomé lattice, consisting of a two-dimensional
network of corner-sharing triangles, have long been studied for the novel
phenomena they are often host to, including strongly correlated topological
states1, superconductivity,2 and most notably high degrees of magnetic
frustration. 3, 4, 5 One such family of materials, the jarosites,
(AB3(XO4)2(OH)6, where A = a monovalent or divalent cation, B = a trivalent
cation, and XO4 = a divalent polyanion group) have long been studied for the
impact of kagomé-sublattice vacancies on their magnetic properties 6, 7. The
majority of the jarosite phases possess metal-ligand-metal (M-L-M) bridging
angles on the order of 130-140°, are known to order antiferromagnetically
between _T_ = 55-65 K, and are considered near-ideal Heisenberg
antiferromagnets, with weakly ferromagnetic intralayer coupling and
antiferromagnetic interplanar coupling 8, 9. The strongly anisotropic
magnetism in these materials arises from the Dzyaloshinskii-Moriya (DM)
interaction, whereby the tilting of the Fe octahedra which comprise the kagomé
lattice produces a canting of the spins about each triangular hole into a spin
umbrella configuration 10. Despite the high degree of variability in both the
cation and capping groups which have been successfully substituted into the
jarosite structure, the tilt angle and degree of distortion of the kagomé-
layer FeO6 octahedra remains relatively consistent across the family.

However, several jarosite phases have been reported which contradict this
observed trend. The hydronium and deuterated jarosite analogues, owing to the
favorability of interactions between H3O+ and OH- ions in the structure,
instead exhibit a spin glass transition around $T_{g}$ = 15 K 11. In order to
maintain charge neutrality, jarosite phases containing divalent cations have
only a 1/2 A-site occupancy, but otherwise possess quite similar structural
and magnetic properties to those possessing full A-site occupancy. The Pb-
analogue plumbojarosite, Pb_0.5Fe3(SO4)(OH)6, is observed to order
antiferromagnetically below $T_{N}$ = 56.4 K, which is slightly lower than
other members of the family, and likely due to increased steric effects from
the larger Pb^2+ cation.12. Beudantite, PbFe3(SO4)(AsO4)(OH)6, containing
alternating sulfate and larger arsenate groups, has only been characterized
from a heavily H3O+-substituted natural sample, and was observed to order at
around $T_{N}$ = 45 K, indicating that the additional steric pressure on the
kagomé plane further frustrates the spins, thereby suppressing long-range
order 13. A recent computational study suggested that corkite,
PbFe3(PO4)(SO4)(OH)6, a related mineral with a similar calculated ground state
Ising spin configuration to the jarosites, would benefit from magnetic
characterization to determine whether it behaves as a Heisenberg
antiferromagnet, or if it exhibits more interesting magnetism at low
temperatures 5. The structure of corkite has been previously probed in both
natural and synthetic samples (Figure 1a, 1b), however its magnetic properties
have not yet been reported.14, 15 While pure phase specimens of corkite would
not contain the rather reactive H3O+ species, it is not known whether the
substitution of one sulfate group per formula unit with a higher-valent,
larger phosphate group could disrupt the interlayer couplings observed in the
pure jarosites, producing a more ideal Heisenberg antiferromagnet.

Here we report the hydrothermal synthesis of synthetic corkite, as well as the
characterization of its structural and magnetic properties. A combination of
powder x-ray diffraction and IR and Raman spectroscopy measurements are used
to confirm prior structural assignments of both natural and synthetic corkite
samples in the _R $\bar{3}$m_ space group (166). Magnetic susceptibility and
heat capacity measurements are also performed and show synthetic corkite to
order antiferromagnetically below TN = 48 K, with an effective magnetic moment
on the order expected for an Fe3+, S = 5/2 system. Our results show corkite to
be a frustrated Heisenberg antiferromagnet, and provide a deeper understanding
of how the split polyanion site occupancy, and subsequently larger steric and
electronic strain, impacts the magnetic behavior of corkite, relative to the
broader family of jarosite materials.

##  2 Results and discussion

###  2.1 Synthesis

Corkite samples were synthesized in Teflon-lined hydrothermal vessels at
elevated temperatures. Heating and cooling ramp rates for all syntheses were
set at 100°C per hour. The precursor Pb5(PO4)3Cl was synthesized in two steps.
First, stoichiometric quantities of PbO (Alfa Aesar, $\sim$99.9%) and NH4H2PO4
(VWR, ACS Grade) were thoroughly ground and heated in an uncovered platinum
crucible for 2 days at 1000°C. Platinum was chosen due to its lower reactivity
with PbO, compared to alumina. The product Pb3(PO4)2 powder was then ground in
a 3:2 molar ratio with PbCl2 (Alfa Aesar, 98%) and heated for 2 days at 850°C
in a covered alumina crucible. FePO4 was also synthesized in two stages.
First, equimolar amounts of NH4H2PO4 and Fe(NO3)3 $\cdot$ 9 H2O (Aqua
Solutions, ACS Grade) were dissolved in deionized water, stirred until clear,
then boiled until dry. The resulting powder was dried for 24 hours at 400°C,
and then for an additional 24 hours at 700°C. Corkite samples were produced by
the combination of a ground mixture of 0.95 mol Pb5(PO4)3Cl, 2.00 mol FePO4,
and 1.67 mol Fe2(SO4)3 $\cdot$ 9 H2O (Aqua Solutions, Reagent Grade) with 9.67
mol FeCl3 $\cdot$ 6 H2O (Alfa Aesar, 98%). All powders were added to the
Teflon liner, filled to $\sim$70% with distilled H2O, further acidified with 3
drops of $\approx$5M H2SO4 solution, sealed, and heated for 4 days at 98°C.

Samples produced by the above method consistently resulted in large
Pb5(PO4)3Cl and PbSO4 impurities, the latter of which could be removed by
stirring the product mixture in a concentrated 12.2M sodium acetate solution
in 6 minute increments. Due to the higher stability of Pb5(PO4)3Cl in acidic
solution relative to corkite, this impurity could only be partially removed
with a 1.0 M KOH solution without decomposition of the target phase and
formation of amorphous iron oxide/hydroxide species. As Pb5(PO4)3Cl was
present in the system both pre- and post-reaction, several syntheses were
attempted with reduced precursor concentrations. However, all resulted in both
a reduction in corkite yield and the formation of additional impurity phases.
A 5% deficiency of Pb5(PO4)3Cl was ultimately found to produce the purest
product mixture ($\sim$91% corkite). For this reason, and owing to the non-
magnetic nature of Pb5(PO4)3Cl, all characterization measurements have been
performed on representative samples containing both phases.

Reaction times shorter than 4 days typically yielded only minimal quantities
of the target corkite phase, instead consisting largely of Pb5(PO4)3Cl and
PbSO4. Here, the lower solubility of Pb5(PO4)3Cl limits the incorporation of
Pb2+ ions into the corkite structure and thus also its formation, but also
prevents the immediate precipitation of large quantities of PbSO4. Reaction
times longer than 4 days ultimately resulted in less corkite in the product
mixture and no Pb5(PO4)3Cl precipitation, as well as the formation of various
lead and iron phosphate impurity phases. When the precursor Pb5(PO4)3Cl has
fully dissociated, PbSO4 precipitation becomes more favorable relative to
corkite, leaving fewer sulfate groups to stabilize the corkite structure, and
ultimately producing only a S-deficient, more-heavily disordered analogue. The
formation of corkite is therefore most favorable only for reaction times close
to 4 days, where the relative solubilities of Pb5(PO4)3Cl, PbSO4, and corkite
are more balanced, and sufficient concentrations of all constituent ions are
present in solution.

In order to better approximate the non-magnetic contribution to the measured
heat capacity, the Ga-analogue of corkite (PbGa3(PO4)(SO4)(OH)6) was
synthesized under conditions identical to those described for the parent
compound. Precursor GaPO4 was synthesized by heating a well-ground,
stoichiometric mixture of Ga2O3 (NOAH, 99.995%) and NH4H2PO4 (VWR, ACS Grade)
in air for 24 hours at 800°C. A ground mixture of 3 mol Pb5(PO4)3Cl, 6 mol
GaPO4, 5 mol Ga2(SO4)3 $\cdot$ H2O (Thermo Scientific, 99.999% (metals
basis)), and 14.5 mol Ga2O3 (NOAH, 99.995%) was added to a Teflon-lined
hydrothermal vessel, sealed, and heated for 4 days at 98°C. The resulting
product was stirred for 6 minutes in a concentrated 12.2M sodium acetate
solution to remove the PbSO4 impurity phase. Relative to the majority of
single-step precipitation Fe-corkite synthesis trials, the Pb5(PO4)3Cl
impurity in all Ga-corkite trials was found to be significantly reduced
without a secondary washing step, comprising only 8-9% of the sample by
weight. As a result, the product mixture was only stirred in 12.2M sodium
acetate solution once for 6 minutes. Unlike the majority of reported jarosite
systems, Al-corkite could not be synthesized in any quantifiable amount, at
least under the chosen reaction conditions.

Synthetic jarosites initially produced via the traditional single-step
precipitation method were frequently found to yield samples with up to a 15%
Fe deficiency on the kagomé sublattice, as well partial A-site substitutions
by H3O+.9 Later methods employing a slower, two-step precipitation method
ultimately produced synthetic jarosites with full B-site occupancies and
negligible A-site substitutions.16 Due to the strong dependence of the
magnetic properties of other jarosite-family phases on the occupancy of both
the A- and B- sites, the latter synthesis route was also explored in an
attempt to discourage substitution by H3O+. When Fe metal was instead used as
the sole source of magnetic cations, the speed at which Fe3+ entered the
solution was reduced and the amount of corkite produced decreased
significantly relative to trials employing more soluble species. This, as well
as the observed time-dependence of corkite formation via the single-step
precipitation method, suggests that corkite is the metastable product in this
reaction system, in contrast with the more thermodynamically-stable jarosites
produced by the same method.

###  2.2 Characterization Methods

Powder x-ray diffraction (pXRD) patterns were collected on a laboratory Bruker
D8 Focus diffractometer with LynxEye detector and Cu K$\alpha$ radiation in
the 2$\theta$ range from 5-120 degrees. Rietveld refinements on pXRD data were
performed using Topas 5.0 (Bruker). Structures were visualized with the Vesta
3 program. 17

Magnetization data was collected on a Quantum Design Magnetic Property
Measurement System (MPMS3). Magnetic susceptibility was approximated as
magnetization divided by the applied magnetic field ($\chi\approx M/H$).
Representative dry magnetization data was collected from synthetic corkite
samples dried overnight at 150°C in air. Magnetization data for K,S-jarosite,
K,Se-jarosite, and Pb-jarosite was extracted from previous studies (Ref. 9 and
Ref. 12), and $\chi_{0}$ values were chosen such that each exhibited ideal
Curie-Weiss behavior at high temperatures [K,S-jarosite: $\chi_{0}$ = 0.0007,
K,Se-jarosite: $\chi_{0}$ = -0.0060, Pb-jarosite: $\chi_{0}$ = -0.0092]. The
magnetization of both K,Se-jarosite and Pb-jarosite was also directly scaled
by a factor of 0.44 and 0.37 respectively to allow for direct comparison.

Heat-capacity data was collected on a Quantum Design Physical Property
Measurement System (PPMS) using the semi-adiabatic pulse technique with a 1%
temperature rise and over three time constants. The change in magnetic entropy
as a function of temperature, $\Delta S_{mag}$, was approximated as $\Delta
S_{mag}$ = $\int C_{p}/T\,dT$ of the measured $C_{p,mag}$ for corkite
($C_{p,total,Fe-corkite}$ \- $C_{p,total,Ga-corkite}$), from _T_ = 0–212 K.
The entropy rise from T = 0 K to 2 K was estimated from linear extrapolation
over this range.

Simultaneous thermogravimetric analysis/differential thermal analysis
(TGA/DTA) was performed using a TA Instruments Q600 SDT. The samples were
loaded into pre-dried alumina pans and ramped quickly to 50°C. After holding
for several minutes, the system was then ramped to 600°C at a rate of
10°C/minute under inert N2 flow (100 mL/minute). The sample was then allowed
to cool to room-temperature under N2 gas flowing at the same rate.

Fourier-transform infrared (FTIR) measurements of synthetic corkite powder
were conducted using a Thermo Nicolet Nexus 870 ESP FTIR with a Golden Gate
KRS5 ATR accessory from 500 to 4000 cm-1. Micro-Raman spectra of synthetic
corkite powder were collected using a Horiba JY T64000 spectrometer equipped
with an Olympus microscope in single monochromator mode under a 50x objective
magnification. The samples were excited using the 514.5 nm line of a Coherent
Innova 70C Spectrum laser, and the spectra collected using a Horiba Symphony
II CCD detector. No signs of laser-induced degradation of the sample were
observed over the course of each measurement. Representative dry sample
spectra were collected from corkite samples dried overnight at 200°C in air.

###  2.3 Structural characterization

To determine the phase purity of the synthesized polycrystalline corkite
samples, pXRD measurements and subsequent Rietveld refinements were performed.
The pXRD pattern of a synthetic corkite sample of representative purity
(Figure 1c) shows the presence of two distinct phases, as indicated by the
contrasting peak widths observed. Broader reflections could be attributed to
the target corkite phase crystallizing in the trigonal _R $\bar{3}$m_ space
group (166), while the sharper reflections were attributed to a hexagonal
Pb5(PO4)3Cl impurity. In several samples, a weak reflection also appeared at
around $2\theta$ = 8.8 degrees, which seemed to correspond to a complex
Pb-/Fe-phosphate phase and could not be removed via washing in either strongly
acidic or basic media. The two-phase refinement of the purest corkite samples
indicates that the non-magnetic Pb5(PO4)3Cl impurity is present at 9.5(1)% by
weight (Tables 1-2). The refined corkite lattice parameters obtained were a =
b = 7.3078(3) Å, c = 16.855(2) Å, $\alpha$ = $\beta$ = 90°, $\gamma$ = 120°.
Slight deficiencies observed in refined site occupancies, as well as
relatively large thermal displacement parameters, can likely be attributed to
slight disorder in the material, as has been commonly observed in other
synthetic members of the jarosite family.6, 7 Drying of the samples produced
no clear change in the measured diffraction pattern, indicating that no
significant structural changes occur upon heating to 200°C.

Significant disagreement has thus far accompanied determination of the true
structure of corkite, with the majority of recent studies on natural samples
concluding that the centrosymmetric _R $\bar{3}$m_ space group (166) is a
better fit than the noncentrosymmetric _R3m_ group (160). 18, 19, 20 However,
the elemental composition of natural corkite samples typically differs from
the nominal stoichiometry, often possessing significant disorder on the
phosphorus and sulfur sites. The sole report of structural studies on a
sulfate-rich synthetic sample of the material only notes that the _R3m_ space
group would be appropriate if the phosphate and sulfate groups occupy
alternate sites in an ordered manner.21 As the two space groups differ only by
the absence and presence of a net electric dipole moment, respectively, pXRD
is not a viable means of distinguishing between them.

Refinement of the experimental diffraction pattern in both the _R $\bar{3}$m_
and _R3m_ structure types resulted in visibly similar fits to the experimental
data. The refinement shown in Figure 1 was carried out in the nonpolar _R
$\bar{3}$m_ space group ($R_{wp}$ value 2.61%), as would be expected for
synthetic corkite, with ordered alternating phosphate and sulfate groups about
the kagomé plane. However, refinement in the polar _R3m_ space group ($R_{wp}$
value 2.51%) also resulted in analogous calculated site occupancies and
thermal displacement parameters. The conflicting structural assignments made
in the literature can be interpreted as arising from varying degrees of
disorder on the [PO4]^3- and [SO4]^2- sites bounding each kagomé layer,
producing highly variable distributions of disordered and ordered regions
which could average to produce a net dipole of variable magnitude and a
subsequently more polar structure.

In order to draw further comparisons to previous spectroscopic studies of
natural corkite, infrared and Raman spectra were collected from representative
synthetic corkite samples, both before and after drying. The majority of
features in the measured infrared spectra, shown in Figure 2a and summarized
in Table 3, are significantly sharper than those observed for the natural
specimens, which is indicative of a more ordered structure and subsequently
fewer distinct coordination environments. This is also evident in the OH-
stretching region (here from 2800-3500 cm-1), where two broad humps are
observed. In natural samples only one broader feature appears, due to the
wider distribution of OH-stretching frequencies which arise from the higher
degree of disorder. The majority of band assignments were made in agreement
with those made by Frost _et al_15 based on the symmetry of the ordered
structure. Although drying of the synthetic sample would be expected to result
in the disappearance of the water stretching band at $\sim$1630 cm-1, the
feature is present in both spectra, possibly due to either the incomplete
removal of water from the sample upon drying, or to minor hydration of the
sample over the course of the measurement. This band is common among other
jarosite materials, both natural and synthetic. However as the ordered corkite
structure does not possess a monovalent cation which can be as easily
substituted, significant water incorporation is less likely 22, 23. A
background measurement performed without a sample present suggests that all
bands not directly attributable to the corkite structure arise due to
atmospheric contributions, instrumental error, or a combination thereof.
Standard and subsequent subtractions of this background from the acquired
spectra could not be reliably performed without significant distortion of the
experimental data, and thus they are presented together for comparison.

The frequencies of Raman active vibrations observed for the representative
undried corkite sample agree with those reported previously on natural samples
(Figure 2b).15, 24 As in the measured IR spectum, Raman bands in the synthetic
sample are visibly narrower than in the naturally occurring analogue. The
increased width of the vibrational bands in the natural samples is typically
evidence of higher disorder in the crystal structure,25 in agreement with the
conclusion of Frost _et al_ that natural corkite samples deviate strongly from
the ideal composition.15 Band assignments are summarized in Table 4 and agree
well with those made by Frost _et al_.15 The singular discrepancy between the
Raman spectrum measured from natural and synthetic samples is the band
centered around 3069 cm-1, which is not present in measurements of any
jarosite-family material. Similar to the additional, lower energy absorption
band present in the infrared spectrum, this feature can be understood as a
consequence of the more ordered synthetic corkite structure. In natural
samples, one would expect a more diffuse distribution of hydrogen bond
distances throughout the structure, and thus only a single, broad band in the
O-H stretching region would be expected. However, with a more ordered
arrangement of alternating phosphate and sulfate groups about the kagomé
sublattice, fewer distinct stretching frequencies should be represented.
Application of the Libowitzky correlation, derived from a subset of bond
distances in mineralogical samples, allows for an approximation of the Raman
shift at which a hydroxyl stretch would be expected, given the O-O or O-H
distances between the atoms participating in hydrogen bonding26. By this
approximation, a band centered at 3069 cm-1 would correspond to an O-O
distance of 2.67 Å, close to the 2.70 Å observed by Giuseppetti _et al_ in the
_R3m_ structure.14 This distance corresponds to the H-bonding occurring
between the hydroxide doubly-coordinated by Pb and Fe and the oxygen triply-
coordinated by Pb, Fe, and P. As the P-site is significantly more disordered
in natural samples, hydrogen bonding of this type would occur over a more
variable range of distances, and the band centered at 3069 cm-1 would be
expected to be less intense, if not absent.

Drying of synthetic corkite samples is observed to produce significant changes
in the measured Raman spectra. As shown in Figure 2b, the intensity of the two
hydroxyl vibrational modes decreases to barely above the baseline, indicative
of dehydroxylation of the structure. All bridging hydroxide ligands present in
undried specimens of corkite would thus become oxygen bridges upon drying. A
clear broadening of the $\nu_{3}$ antisymmetric (PO4)^3-/(SO4)^2- modes is
also observed, possibly due to changes in the coordination of the
phosphorus/sulfur atom, as opposed to the more uniform environment in the less
disordered, hydroxide-containing structure. Both the observed broadening of
vibrational modes and higher Raman background are attributed to intrinsic
photoluminescence from defects introduced as the material undergoes
dehydroxylation.27

Direct comparison of the measured infrared and Raman spectra for undried
corkite, shown in Figure 2c, provides additional evidence for the retention of
an inversion center. By the rule of mutual exclusion, the presence of an
inversion center in a structure precludes the coincident infrared and Raman
activity of normal vibrational modes.28 If the structure lacked inversion
symmetry, all normal modes would be expected to appear in both the infrared
and Raman spectra, however little of this is observed for our synthetic
samples. In order for pure corkite to crystallize in the non-centrosymmetric
_R3m_ space group, the phosphate and sulfate groups were assumed to occupy
discrete layers between adjacent kagomé layers. However, in both natural, non-
stoichiometric samples and ordered synthetic analogues, it is much more likely
that the polyanion groups instead more randomly occupy each of these layers,
thereby retaining the inversion center and supporting the _R $\bar{3}$m_
structural assignment.

TGA-DTA analysis of a representative synthetic corkite sample (Figure 3) was
also performed to determine its stability upon heating. At around 143°C, the
mass of the sample begins to decrease (up to a 2 wt% loss), and the system is
observed to absorb heat, as indicated by the endothermic peak centered around
173°C. This first set of features is most likely indicative of water being
driven from the structure, as any significant decomposition would result in a
much larger observed mass loss. Up to 175°C, this corresponds to a loss of
approximately 41 $\mu$mol H2O from 58.3 $\mu$mol of total sample. The second,
sharper decrease in mass (totaling an additional $\sim$7.7% loss), as well as
a large endotherm, both begin above 330°C and likely correspond to
decomposition of the corkite phase. This is also supported by pXRD analysis of
the sample after heating to 600°C, which revealed only the presence of Fe2O3.
This trend agrees well with DTA data reported for several other synthetic
jarosites by Baker, however not with that claimed to represent the behavior of
synthetic corkite 21. Their corkite DTA measurements show no corresponding
absorption of heat for the removal of water or decomposition at lower
temperatures, but rather an exotherm at around 640°C, and a decomposition-type
endotherm at around 950°C. This result appears rather unphysical, particularly
in comparison to the typical lower thermal stability of hydrothermally-
synthesized phases as well as to the data reported for other synthesized
jarosites.

###  2.4 Magnetic characterization

Temperature-dependent magnetic susceptibility of a representative synthetic
corkite sample shows a single, broad antiferromagnetic transition at around
$T_{N}$ = 48 K, below those observed in the other non-hydronium Fe-jarosite
phases ($T_{N}$ = 60-65 K) (Figure 4a). 9 Several samples were observed to
possess a second transition at around $T_{N}$ = 58 K, but as the lower-
temperature transition was consistently present and the higher-temperature one
was not, the latter was attributed to largely amorphous impurity phases not
present in the measured pXRD pattern. When the as-synthesized sample mass is
adjusted to account for the $\sim$ 9.0% non-magnetic Pb5(PO4)3Cl impurity,
Curie-Weiss fitting of the inverse susceptibility at high temperature (_T_ =
150 K – 300 K) suggests that the iron species present is largely high-spin
Fe3+, as indicated by the calculated effective magnetic moment of p${}_{eff}=$
6.01(1) $\mu_{B}$. When dried overnight at 150°C, the measured magnetic
susceptibility at high-temperatures (_T_ $\geq$ 280 K) was more linear,
consistent with water being driven from the structure (Figure 4b). Curie-Weiss
fitting for the dried sample should thus be more representative of synthetic
corkite and yields an effective magnetic moment of p${}_{eff}=$ 6.29(1)
$\mu_{B}$, higher than would be expected for a high-spin Fe3+ system, but
analogous to what has been observed in other water-free synthetic jarosites
16. Tables comparing the magnetic properties of jarosite-family phases
synthesized by both the method described in this work and by the slower,
redox-based method can be found in Ref. 12 and Ref. 9, respectively. The
calculated Weiss constant and frustration parameter for the dried sample,
$\theta_{CW}$ = -526.0(1.1) K and _f_ = 10.96(3), indicate both strongly
antiferromagnetic interactions and a high degree of frustration amongst the
Fe3+ ions on the kagomé lattice 29.

Normalization of the magnetization as a function of temperature to the Curie-
Weiss law, C/[($\chi-\chi_{0}$)$|\theta|$] = T/$|\theta|$ \+ 1, (shown in
green in Figure 4d), exhibits ideal behavior from high temperatures to TN, and
antiferromagnetic deviations from TN to very low temperatures, where
ferromagnetic deviations are instead observed. These deviations correspond to
the onset of long-range, antiferromagnetic order and weak, intralayer
ferromagnetic coupling, respectively, as has been observed in other jarosite
phases. Magnetization data for synthetic KFe3(SO4)2(OH)6 (K, S-jarosite -
black), KFe3(SeO4)2(OH)6 (K, Se-jarosite - yellow), and Pb_0.5Fe3(SO4)2(OH)6
(Pb-jarosite - blue) is also presented for comparison and agrees well with the
trends observed in synthetic corkite. As in the case of corkite, the
substitution of both sulfate groups in potassium jarosite for the larger
selenate produces ferromagnetic deviations from the ideal Curie-Weiss behavior
at low temperatures. However, these deviations are less significant in the
unsubstituted KFe3(SO4)2(OH)6 data and are barely present in that of
Pb_0.5Fe3(SO4)2(OH)6. This suggests that while the long-range magnetic
properties of jarosite-type compounds are largely invariant upon substitution
on the A-, B-, or X-site, the strength of the intralayer Fe-Fe coupling is
less consistent and more heavily dependent upon the steric and electronic
pressure applied on the kagomé plane by the A- and X-site cations.

Representative magnetization measurements collected for the undried sample
from $\mu_{0}$H = –7 T to 7 T at _T_ = 2 K, 44 K, 52 K, and 300 K show no sign
of magnetic hysteresis or saturation, and only a very slight curvature at _T_
= 2 K from $\mu_{0}$H = –2 T to 2 T, supporting the absence of ferromagnetic
impurities in the sample (Figure 4c). The slight curvature observed in the _T_
= 2 K measurement can likely be attributed to weak, ferromagnetic intralayer
coupling between adjacent Fe^3+ atoms, as has been observed in several
vanadium-jarosite phases, which is fully suppressed by _T_ = 35 K 30. After
drying, no change was observed in the measured magnetization at any
temperature, aside from a slight increase in magnitude. The magnetic structure
of pure corkite can thus likely be considered to be directly analogous to that
observed across the entire family of jarosite materials, owing to the relative
rigidity of the intralayer Fe^3+_3($\mu$-OH)_3 triangular units upon
substitution of the $A^{1+}$-/$A^{2+}$-, $B^{3+}$-, or
$\ch{XO4}^{2-}$-/$\ch{XO4}^{3-}$-sites. Even substitution of the bounding
[SO4]^2- groups with higher-valent [PO4]^3- produces only a 16.4° tilt of the
Fe-octahedra, higher than that of the smaller A-site cation-containing
jarosite phases, but slightly lower than that of the Pb-, Ag-, and Tl-
jarosites. As such, the DM antisymmetric interaction which dominates the
magnetic behavior of these materials is expected to produce a
ferromagnetically-coupled spin-umbrella structure above each
Fe^3+_3($\mu$-OH)_3 triangle. The canted spins are then coupled
antiferromagnetically between layers, producing the long-range
antiferromagnetic order observed below $T_{N}$.

Temperature-dependent heat capacity measurements of a representative synthetic
Fe-corkite sample, shown divided by temperature in Figure 5a, exhibit a peak
at around _T_ = 45.5 K, followed by a broader hump spanning the _T_ = 50-125 K
range. The measurement of less pure specimens resulted in the appearance of an
additional broad feature, visible around _T_ = 4-5 K, which did not correspond
to any local maximum in either DC or AC susceptibility measurements. As
corkite samples of representative purity did not exhibit this feature, it was
attributed to the complex Pb-/Fe-phosphate impurity phase observed in some
pXRD patterns. The higher-temperature fluctuations observed largely arise due
to the low point density above _T_ = 200 K, with the small bump at around _T_
= 275 K corresponding to the freezing of the Apiezon N grease used for sample
mounting.

In order to subtract the phononic contribution to the measured heat capacity,
the Ga-corkite analogue was also synthesized, with a similar calculated
Pb5(PO4)3Cl impurity by mass ($\sim$9.0%). The measured heat capacity for both
samples was scaled by a factor of 0.91 to account for the phononic
contribution of the impurity phase. Subtraction of the measured heat capacity
of Ga-corkite from that of Fe-corkite was thus approximated as the magnetic
contribution, $C_{mag}$, to the total heat capacity $C_{Total}$ of the
magnetic phase. The main peak observed in both $C_{mag}$ and $C_{Total}$
agrees well with the cusp in the DC susceptibility, and is thus attributed to
AFM ordering in pure corkite. Integration of $C_{mag}$ from _T_ $\sim$ 0-195 K
yields a rise in magnetic entropy of $\Delta S_{mag}$ = 14.86 $J\cdot
mol_{Fe^{3+}}^{-1}K^{-1}$ in good agreement with that expected for high-spin
Fe^3+, where $\Delta S_{mag}$ = Rln(2S+1) = 14.9 $J\cdot mol^{-1}K^{-1}$ for a
S = 5/2 system (Figure 5b). Integration of the as-substracted Fe- and Ga-
corkite heat capacities produced an unphysical upward trend in $\Delta
S_{mag}$ up to _T_ = 195 K, as opposed to the saturation expected where the
thermal energy present in the system exceeds that of magnetic correlations in
the structure. The inclusion of a 4.9% mass error of the Ga-corkite sample in
the calculation was found to result in the anticipated trend. Approximately
49% of the total magnetic entropy is released below the observed ordering
temperature, suggesting that short-range correlations at higher temperatures
also contribute significantly to the magnetic character of stoichiometric
corkite.

##  3 Conclusion

In conclusion, we report the hydrothermal synthesis and magnetic
characterization of corkite, PbFe3(PO4)(SO4)(OH)6, an understudied relative of
the jarosite family of materials. Relative to previous studies on natural
samples of this phase, the structure appears to have a more even and ordered
distribution of phosphate and sulfate groups bounding the kagomé sublattice,
as supported by the presence of additional features in and overall higher
resolution of the measured infrared and Raman spectra. Magnetic susceptibility
measurements of synthetic corkite reveal a transition to a long-range AFM
state below $T_{N}$ = 48 K, lower than that observed in the majority of
jarosite phases, indicating that the alternating bounding polyanion groups do
indeed further suppress magnetic ordering in the system. The calculated
effective magnetic moment and observed saturation of magnetic entropy are both
on the order of what is expected for a S = 5/2 system, indicating that Fe3+ is
the main species present on the kagomé sublattice, and that corkite’s Fe-site
occupancy is analogous to jarosite phases synthesized by similar methods.
TGA/DTA measurements also show that the structure incorporates a small amount
of water under atmospheric conditions, producing a small suppression of the
calculated effective magnetic moment. Our results highlight the importance of
steric strain in the development and design of novel frustrated magnetic
systems, and contribute to a more complete understanding of the jarosite
family of Heisenberg antiferromagnets.

This work was supported by the Institute for Quantum Matter, an Energy
Frontier Research Center funded by the U.S. Department of Energy, Office of
Science, Office of Basic Energy Sciences, under Grant DE-SC0019331. The
TGA/DTA data was collected within the National Science Foundation Materials
Innovation Platform PARADIM, under cooperative agreement #2039380. The MPMS3
system used for magnetic characterization was funded by the National Science
Foundation, Division of Materials Research, Major Research Instrumentation
Program, under Grant #1828490.

Figure 1: Crystal structure of corkite projected a) along the _a_ axis and b)
along the _c_ axis. Each lead (black) and iron (gold) atom is 12-fold and
6-fold coordinate, respectively, with 2/3 hydroxide and 1/3 oxygen (red)
bridging ligands. The kagomé plane (emphasized in green) is bound by both
highly-coordinated lead atoms and alternating phosphate (purple) and sulfate
(yellow) capping groups. c) pXRD pattern of a representative synthetic corkite
sample (black). Two-phase simulated Rietveld refinement profile for
PbFe3(PO4)(SO4)(OH)6 (space group _R $\bar{3}$mH_) and Pb5(PO4)3Cl (space
group _P63/m_) (red). Difference curve between the experimental and simulated
patterns (blue). Refinement of the experimental pattern indicates that the
best samples possess approximately 91.0% of the corkite phase by weight.

Table 1: Unit-cell parameters for PbFe3(PO4)(SO4)(OH)6 at T = 298 K, as obtained from Rietveld refinement of a pXRD pattern ($\lambda$ = 1.5406 Å). Errors representing statistical uncertainties are shown in parentheses. Parameter | Value
---|---
Chemical Formula |  PbFe3(PO4)(SO4)(OH)6
Source | Cu K$\alpha$ (1.5406 Å)
Formula weight (g/mol) | 667.8
Temperature (K) | 298
Crystal system | Rhombohedral
Space group |  _R $\bar{3}$m_ (No. 166)
a = b (Å) | 7.3078(3)
c (Å) | 16.855(2)
$\alpha$ = $\beta$ (deg) | 90
$\gamma$ (deg) | 120
Volume (Å3) | 779.53(8)
Z | 1
$\rho_{calc}$ (g/cm3) | 6.9243(7)
R-factors (%) | RP = 1.93, Rwp = 2.61, Rexp = 1.63
Table 2: Atomic coordinates, occupancies, and atomic displacement parameters for PbFe3(PO4)(SO4)(OH)6 at T = 298 K, as obtained from Rietveld refinement of a pXRD pattern ($\lambda$ = 1.5406 Å). All occupancies besides that of iron were fixed at unity, and isotropic displacement parameters Beq were refined together by element type. Errors representing statistical uncertainties are shown in parentheses. Parameters listed without associated errors could not be refined without worsening of the fit to the experimental data, and thus are unchanged from previous reports. Atom |  _a_ (Å) |  _b_ (Å) |  _c_ (Å) | Occupancy |  $B_{eq}$ (Å2)
---|---|---|---|---|---
Pb | 0 | 0 | 0 | 1 | 0.54(10)
Fe | 0.5013 | 0.0026 | 0.5142(7) | 0.830(6) | 0.54(10)
S | 0 | 0 | -0.3070(10) | 1 | 1.8(2)
P | 0 | 0 | 0.3178(11) | 1 | 1.8(2)
O1 | 0 | 0 | -0.3900(21) | 1 | 0.7(2)
O2 | 0 | 0 | 0.4099(21) | 1 | 0.7(2)
O3 | -0.2191 | 0.2191 | 0.0477(18) | 1 | 0.7(2)
O4 | 0.2216 | -0.2216 | -0.0581(19) | 1 | 0.7(2)
O5 | -0.1264 | 0.1264 | -0.1206(14) | 1 | 0.7(2)
O6 | 0.1264 | -0.1264 | 0.1374(16) | 1 | 0.7(2)
H1 | -0.0162(867) | -0.0176(671) | -0.0018(1073) | 1 | 1
H2 | 0.3333(150) | -0.2097(72) | 0.1551(125) | 1 | 1
Table 3: Assignment of observed infrared stretching frequencies Band center(s) (cm-1) | Band strength | Vibrational mode(s)
---|---|---
591, 624 | Weak |  $\nu_{4}$ (SO4)^2- bending modes
725 | Broad |  $\delta$-FeOH deformation modes
955 | Strong |  (PO4)^3- symmetric stretch
1060, 1174 | Strong, Medium |  (PO4)^3-/(SO4)^2- antisymmetric stretch
1637 | Weak | Water
3315 | Broad |  OH- stretch
Table 4: Assignment of observed Raman stretching frequencies Band center(s) (cm-1) | Band strength | Vibrational mode(s)
---|---|---
110, 145, 229 | Medium, Strong | FeO stretch
346 | Medium |  $\nu_{2}$ (PO4)^3-
437 | Strong |  $\nu_{2}$ (SO4)^2-
555-571 | Medium |  $\nu_{4}$ (PO4)^3-
622 | Medium |  $\nu_{4}$ (SO4)^2-
1005 | Strong |  $\nu_{1}$ (SO4)^2-
1103-1162 | Medium |  $\nu_{3}$ (PO4)^3-/$\nu_{3}$ (SO4)^2-
3069, 3390 | Medium, Broad |  $\nu$ OH-
Figure 2: a) Infrared spectra of representative undried (red) and dried (blue)
synthetic corkite powder samples. The collected background is shown in black
for comparison. b) Raman spectra of undried (red) and dried (blue) synthetic
corkite samples, as well as a representative natural corkite sample (RRUFF™
Project, Sample 050522 - black). c) Comparison of the infrared (green) and
Raman (gold) spectra collected on a representative undried synthetic corkite
powder sample. The black dotted line highlights the most intense mode in the
IR spectrum, which has no corresponding mode in the Raman spectrum of the same
sample. No normal vibrational modes are observed to overlap, indicative of a
structure with inversion symmetry. d) OH$\mathord{\cdot\cdot\cdot}$O hydrogen
bond attributed to the peak observed at 3069 cm-1 in the undried synthetic
Raman spectrum. Figure 3: Thermogravimetric analysis (black) and differential
thermal analysis data (red) as a function of temperature for a representative
synthetic corkite powder sample. Clear signs of dehydration are present in
both measurements between 140–175°C, up to 2% of the total sample mass, or
roughly 41 $\mu$mol H2O from 58.3 $\mu$mol of total sample. The large
endotherm observed above 330°C corresponds to decomposition of the corkite
phase.

Figure 4: Temperature-dependent magnetic (black) and inverse magnetic (pink)
susceptibility of a representative synthetic corkite powder sample a) before
and b) after drying, from 2-300 K in an applied field of 1.0 T. The best fit
lines used for Curie-Weiss analysis from T = 150-300 K are shown in blue, and
extrapolated to T = 0 K. The calculated Curie-Weiss parameters for undried [C
= 4.51(1) emu K/mol${}_{Fe^{3+}}$ Oe, $\theta_{CW}$ = -544.0(6) K, peff =
6.01(1) $\mu_{B}$/Fe3+, $\chi_{0}$ = -0.001] and dried [C = 4.95(1) emu
K/mol${}_{Fe^{3+}}$ Oe, $\theta_{CW}$ = -526.0(1.1) K, peff = 6.29(1)
$\mu_{B}$/Fe3+, $\chi_{0}$ = -0.0025] corkite samples agree well with those
reported for other Fe-deficient jarosites. c) Field-dependent magnetization of
a representative undried corkite powder sample from $\mu_{0}$H = -7 to 7 T at
2 K (black), 44 K (yellow), 52 K (blue), and 300 K (green). The slight
curvature in the 2 K measurement is attributed to weak, ferromagnetic
intralayer coupling between adjacent Fe^3+ atoms. The 44 K and 52 K curves are
nearly identical, and show no sign of hysteresis directly below and above the
antiferromagnetic ordering temperature, TN = 48 K. Drying of the samples
produced only a slight increase in the magnitude of the magnetization, and no
change in the observed trend at each temperature. d) Normalized magnetization
of K,S-jarosite (black)6, K,Se-jarosite (yellow)16, Pb-jarosite (blue)16, and
corkite (green - this work), where C/[($\chi-\chi_{0}$) $|\theta|$] =
T/$|\theta|$ \+ 1. Magnetization data presented in previous works was
extracted and scaled to fit the expected Curie-Weiss behavior at higher T. The
inset shows the deviations of each phase from Curie-Weiss behavior from low T
to their respective TN; all four phases exhibit antiferromagnetic deviations
from ideal behavior below TN, and ferromagnetic deviations at very low T,
attributable to the weak intralayer Fe-Fe coupling. Figure 5: a) Heat capacity
divided by temperature as a function of temperature, CTotal/T, for synthetic
Fe-corkite (black) and Ga-corkite (blue). The measured heat capacity of Ga-
corkite was scaled to account for a 4.9% mass error. The magnetic contribution
to the heat capacity of Fe-corkite, Cmag/T, as a function of temperature
(red). b) Magnetic entropy rise of Fe-corkite as a function of temperature
from T = 0-200 K. The entropy rise from T = 0 K to 2 K was estimated from
linear extrapolation over this range.

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11institutetext: University of Victoria, Victoria, Canada
11email<EMAIL_ADDRESS>
22institutetext: McGill University, Montreal, Canada
22email<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Pirates in Wonderland:
Liquid Democracy has Bicriteria Guarantees

Jonathan A. Noel 11 Mashbat Suzuki 22 Adrian Vetta 22

###### Abstract

Liquid democracy has a natural graphical representation, the delegation graph.
Consequently, the strategic aspects of liquid democracy can be studied as a
game over delegation graphs, called the liquid democracy game. Our main result
is that this game has bicriteria approximation guarantees, in terms of both
rationality and social welfare. Specifically, we prove the price of stability
for $\epsilon$-Nash equilibria is exactly $\epsilon$ in the liquid democracy
game.

##  1 Introduction

Liquid democracy is a form of direct and representative democracy, based on
the concept of delegation. Each voter has the choice of voting themselves or
transferring (transitively) its vote to a trusted proxy. Recent interest in
liquid democracy, from both practical and theoretical perspectives, was
sparked by the Pirate Party in Germany and its Liquid Feedback platform [2].
Similar initiatives have subsequently been undertaken by the Demoex Party in
Sweden, the Internet Party in Spain, and the Net Party in Argentina.

There are many potential benefits of a transitive delegation mechanism.
Participation may improve in quantity for several reasons. The system is easy
to use and understand, induces low barriers to participation, and is
inherently egalitarian: there is no distinction between voters and
representatives; every one is both a voter and a delegator. Participation may
also improve in quality due to the flexibility to choose different forms of
participation: voters can chose to be active participants on topics they are
comfortable with or delegate on topics they are less comfortable with.
Accountability may improve due to the transparent nature of the mechanism and
because there is a demonstrable line of responsibility between a delegated
proxy and its delegators. The quality of decision making may improve via a
specialization to delegated experts and a reduction in induced costs, such as
the duplication of resources.

Our objective here is not to evaluate such claimed benefits, but we refer the
reader to [1, 2, 4, 11, 13] for detailed discussions on the motivations
underlying liquid democracy. Rather, our focus is to quantitatively measure
the performance of liquid democracy in an idealized setting. Specifically, can
equilibria in these voting mechanisms provide high social welfare? That is, we
study the price of stability of liquid democracy.

###  1.1 Background

As stated, vote delegation lies at the heart of liquid democracy. Furthermore,
vote delegation in liquid democracy has several fundamental characteristics:
optionality, retractability, partitionability, and transitivity. So let us
begin by defining these concepts and tracing their origins [1, 2].

The notion of optional delegation proffers voters the choice of direct
participation (voting themselves/choosing to abstain) or indirect
participation (delegating their vote). This idea dates back over a century to
the work of Charles Dodgson on parliamentary representation [9].111Dodgson was
a parson and a mathematician but, as the author of “Alice in Wonderland”, is
more familiarly known by his nom de plume, Lewis Carroll.

Miller [17] proposed that delegations be retractable and partitionable. The
former allows for delegation assignments to be time-sensitive and reversible.
The latter allows a voter to select different delegates for different policy
decisions.222This option is particularly useful where potential delegates may
have assorted competencies. For example, Alice may prefer to delegate to the
Hatter on matters concerning tea-blending but to the Queen of Hearts on
matters concerning horticulture.

Finally, transitive delegation is due to Ford [11]. This allows a proxy to
themselves delegate its vote and all its delegated votes. This concept is
central to liquid democracy. Indeed, if an agent is better served by
delegating her vote to a more informed proxy it would be perverse to prohibit
that proxy from re-delegating that vote to an even more informed proxy.
Moreover, such transitivity is necessary should circumstances arise causing
the proxy to be unable to vote. It also reduces the duplication of efforts
involved in voting.

As noted in the sixties by Tullock [18], the development of the computer
opened up the possibility of large proxy voting systems. Indeed, with the
internet and modern security technologies, liquid democracy is inherently
practical; see Lumphier [16].

There has been a flurry of interest in liquid democracy from the AI community.
This is illustrated by the large range of recent papers on the topic; see, for
example, [5, 6, 7, 8, 12, 13, 14, 15, 19]. Most directly related to our work
is the game theoretic model of liquid democracy studied by Escoffier et al.
[10]. (A related game-theoretic model was also investigated by Bloembergen et
al. [3].) Indeed, our motivation is an open question posed by Escoffier et al.
[10]: are price of anarchy type results obtainable for their model of liquid
democracy? We will answer this question for a generalization of their model.

###  1.2 Contributions

In Section 2, we will see that vote delegation has a natural representation in
terms of a directed graph called the delegation graph. If each agent $i$ has a
utility of $u_{ij}\in[0,1]$ when agent $j$ votes as her delegate then a game,
called the liquid democracy game, is induced on the delegation graph. We study
the welfare ratio in the liquid democracy game, which compares the social
welfare of an equilibrium to the welfare of the optimal solution.

Pure strategy Nash equilibria need not exist in the liquid democracy game, so
we focus on mixed strategy Nash equilibria. Our main result, given in Section
3 is that bicriteria approximation guarantees (for social welfare and
rationality) exist in the game.

###### Theorem 1

For all $\epsilon\in[0,1]$, and for any instance of the liquid democracy game,
there exists an $\epsilon$-Nash equilibrium with social welfare at least
$\epsilon\cdot\operatorname{OPT}$.

Theorem 1 is tight: the stated bicriteria guarantees cannot be improved.

###### Theorem 2

For all $\epsilon\in[0,1]$, there exist instances such that any
$\epsilon$-Nash equilibrium has welfare at most
$\epsilon\cdot\operatorname{OPT}+\gamma$ for any $\gamma>0$.

Theorems 1 and 2 imply that the price of stability for $\epsilon$-Nash
equilibria is $\epsilon$. An important consequence of Theorem 1 is that strong
approximation guarantees can simultaneously be obtained for both social
welfare and rationality. Specifically, setting $\epsilon=\frac{1}{2}$ gives
factor $2$ approximation guarantees for each criteria.

###### Corollary 3

For any instance of the liquid democracy game, there exists a
$\frac{1}{2}$-Nash equilibrium with welfare at least
$\frac{1}{2}\cdot\operatorname{OPT}$.

##  2 A Model of Liquid Democracy

In this section, we present the liquid democracy game. This game generalizes
the game-theoretic model studied by Escoffier et al. [10].

###  2.1 The Delegation Graph

In liquid democracy each agent has three strategies: she can abstain, vote
herself, or delegate her vote to another agent. So we can represent an
instance by a directed network $G=(V,A)$ called the delegation graph. There is
a vertex in $V=\\{1,\cdots,n\\}$ for each agent. To define the sets of arcs,
there are three possibilities. First, if agent $i$ votes herself the
delegation graph contains a self-loop $(i,i)$. Second, if agent $i$ delegates
her vote to agent $j\neq i$ then there is an arc $(i,j)$ in $G$. Third, if
agent $i$ abstains then the vertex $i$ has out-degree zero.

Now, because the out-degree of each vertex is at most one, the delegation
graph $G$ is a $1$-forest. That is, each component of $G$ is an arborescence
plus at most one arc. In particular, each component is either an arborescence
and, thus, contains no cycle or contains exactly one directed cycle (called a
delegation cycle). In the former case, the component contains one sink node
corresponding to an abstaining voter. In the latter case, if the delegation
cycle is a self-loop the component contains exactly one voter called a guru;
if the delegation cycle has length at least two then the component contains no
voters. An example of a delegation graph is shown in Figure 1.

Figure 1: A delegation graph.

Observe that, by the transitivity of delegations, if an agent $i$ is in a
component containing a guru $g$ then that guru will cast a vote on $i$’s
behalf. On the other hand, if agent $i$ is in a component without a guru (that
is, with either a sink node or a cycle of length at least two) then no vote
will be cast on $i$’s behalf. We denote the guru $j$ representing agent $i$ by
$g(i)=j$ if it exists (we write $g(i)=\emptyset$ otherwise). Furthermore, its
easy to find $g(i)$: simply apply path traversal starting at vertex $i$ in the
delegation graph.

For example, in Figure 1 two components contain a guru. Agent $4$ is the guru
of agents $1,2,3$ and itself; agent $9$ is the guru only for itself. The
vertices in the remaining three components have no gurus. There are two
components with delegation cycles, namely $\\{(10,11),(11,10)\\}$ and
$\\{(5,6),(6,7),(7,5)\\}$. The final component also contains no guru as agent
$12$ is a sink node and thus abstains.

###  2.2 The Liquid Democracy Game

The game theoretic model we study is a generalization of the model of
Escoffier et al. [10]. A pure strategy $s_{i}$ for agent $i$ corresponds to
the selection of at most one outgoing arc. Thus we can view $s_{i}$ as an
$n$-dimensional vector ${\bf x}_{i}$. Specifically, ${\bf x}_{i}$ is a single-
entry vector with entry $x_{ij}=1$ if $i$ delegates to agent $j$ and
$x_{ij}=0$ otherwise. Note that if $x_{ii}=1$ then $i$ votes herself
(“delegates herself”) and that ${\bf x}_{i}={\bf 0}$ if agent $i$ abstains.

It immediately follows that there is a unique delegation graph $G_{{\bf x}}$
associated with a pure strategy profile ${\bf x}=({\bf x}_{1},{\bf
x}_{2}\cdots,{\bf x}_{n})$. To complete the description of the game, we must
define the payoffs corresponding to each pure strategy profile ${\bf x}$. To
do this, let agent $i$ have a utility $u_{ij}\in[0,1]$ if she has agent $j$ as
her guru. Because of the costs of voting in terms of time commitment,
knowledge acquisition, etc., it may be that $u_{ii}<u_{ij}$ for some $j\neq i$
trusted by $i$.333Indeed, if this is not the case then liquid democracy has no
relevance. We denote the utility of agent $i$ in the delegation graph $G_{{\bf
x}}$ by $u_{i}({\bf x})=u_{i,g(i)}$. If agent $i$ has no guru then it receives
zero utility.444We remark that our results hold even when agents who abstain
or have no guru obtain positive utility. It follows that only agents that lie
in a component of $G_{{\bf x}}$ containing a guru can obtain positive utility.

For example, in Figure 1 agent $9$ is a guru so receives utility $u_{9,9}$.
Each agent $i\in\\{1,2,3,4\\}$ has agent $4$ as its guru so receives utility
$u_{i,4}$. All the remaining agents have no guru and so receive zero utility.

An agent $i$ is playing a best response at a pure strategy ${\bf x}=({\bf
x}_{1},{\bf x}_{2},\cdots,{\bf x}_{n})$ if he cannot increase his utility by
selecting a different or no out-going arc. The strategy profile is a pure Nash
equilibrium if every agent is playing a best response at ${\bf x}$. Our
interest is in comparing the social welfare of equilibria to the optimal
welfare in the game. To do this, let the social welfare of ${\bf x}$ be
$\operatorname{SW}({\bf x})=\sum_{i\in V}u_{i}({\bf x})$ and let
$\operatorname{OPT}=\max\limits_{{\bf x}}\sum_{i\in V}u_{i}({\bf x})$ be the
optimal welfare over all strategy profiles. The price of stability is the
worst ratio over all instances between the best welfare of a Nash equilibrium
and the optimal social welfare.

The reader may ask why could an equilibrium have low social welfare. The
problem is that delegation is transitive, but trust is not. Agent $i$ may
delegate to an agent $j$ where $u_{ij}$ is large but $j$ may then re-delegate
to an agent $k$ where $u_{ik}$ is small. Worse, agent $i$ may receive no
utility if the transitive delegation of its vote leads to a delegation cycle
or an abstaining voter. Unfortunately, not only can pure Nash equilibria have
low social welfare in the liquid democracy game they need not even exist!

###### Lemma 4

There exist liquid democracy games with no pure strategy Nash equilibrium.

###### Proof

Let there be three voters with $u_{1}=(\frac{1}{2},1,0)$,
$u_{2}=(0,\frac{1}{2},1)$ and $u_{3}=(1,0,\frac{1}{2})$. (A similar instance
was studied in [10].) Assume ${\bf x}$ is a pure Nash equilibrium and let $S$
be the set of gurus in $G_{{\bf x}}$. There are two cases. First, if $|S|\leq
1$ then there exists an agent $i$ with zero utility. This agent can deviate
and vote herself to obtain utility of $\frac{1}{2}>0$, a contradiction.
Second, if $|S|\geq 2$ then one of the gurus can delegate its vote to another
guru to obtain a utility of $1>\frac{1}{2}$, a contradiction. Therefore, no
pure Nash equilibrium exists. ∎

###  2.3 Mixed Strategy Equilibria

Lemma 4 tells us that to obtain performance guarantees we must look beyond
pure strategies. Of course, as liquid democracy is a finite game, a mixed
strategy Nash equilibrium always exists. It follows that we can always find a
mixed Nash equilibrium in the liquid democracy game and compare its welfare to
the optimal welfare.

Here a mixed strategy for agent $i$ is now simply a non-negative vector ${\bf
x}_{i}$ where the entries satisfy $\sum_{j=1}^{n}x_{ij}\leq 1$. Note that
agent $i$ then abstains with probability $1-\sum_{j=1}^{n}x_{ij}$. However,
because abstaining is a weakly dominated strategy, we may assume no voter
abstains in either the optimal solution or the best Nash equilibria. Hence,
subsequently, each ${\bf x}_{i}$ will be unit-sum vector.

What is the utility $u_{i}({\bf x})$ of an agent for a mixed strategy profile
${\bf x}=({\bf x}_{1},{\bf x}_{2},\dots,{\bf x}_{n})$? It is simply the
expected utility given the probability distribution over the delegation graphs
generated by ${\bf x}$. There are several equivalent ways to define this
expected utility. For the purpose of analysis, the most useful formulation is
in terms of directed paths in the delegation graphs. Denote by
$\mathcal{P}(i,j)$ be the set of all directed paths from $i$ to $j$ in a
complete directed graph on $V$. Let $\mathbb{P}_{{\bf x}}(g(i)=j)$ be the
probability that $j$ is the guru of $i$ given the mixed strategy profile ${\bf
x}$. Then

$\displaystyle u_{i}({\bf x})$ $\displaystyle=\sum_{j\in V}\mathbb{P}_{{\bf
x}}(g(i)=j)\cdot u_{ij}$ $\displaystyle=\sum_{j\in V}\mathbb{P}_{{\bf
x}}(\exists\text{ path from }i\ \text{to}\ j)\cdot\mathbb{P}_{{\bf
x}}(\exists\text{ arc }(j,j))\cdot u_{ij}$ $\displaystyle=\sum_{j\in
V}\left(\sum_{P\in\mathcal{P}(i,j)}\prod\limits_{a\in P}x_{a}\right)\cdot
x_{jj}\cdot u_{ij}$

To understand this recall that ${\bf x}_{i}$ is a unit-sum vector. It follows
that each delegation graph generated by the mixed strategy ${\bf x}$ has
uniform out-degree equal to one. This is because we generate a delegation
graph from ${\bf x}$ by selecting exactly one arc emanating from $i$ according
to the probability distribution ${\bf x}_{i}$. Consequently, $j$ can be the
guru of $i$ if and only if the delegation graph contains a self-loop $(j,j)$
and contains a unique path $P$ from $i$ to $j$. The second equality above then
holds. Further, the choice of outgoing arc is independent at each vertex $i$.
This implies the third equality.

Regrettably, however, no welfare guarantee is obtainable even with mixed
strategy Nash equilibria.

###### Lemma 5

The price of stability in liquid democracy games is zero.

###### Proof

Consider an instance with three agents whose utility is given as
$u_{1}=(\delta,1,0),u_{2}=(0,\delta,1)$ and $u_{3}=(1,0,\delta)$. The optimal
welfare is $\operatorname{OPT}=1+2\delta$ obtained by the first and second
agent voting and the third agent delegates to the first agent.

By a similar argument to Lemma 4, we know that no pure strategy Nash
equilibrium exists. It is straightforward to verify that this game has a
unique mixed strategy Nash equilibria ${\bf x}$, where
$x_{1}=(\delta,1-\delta,0)$, $x_{2}=(0,\delta,1-\delta)$ and
$x_{3}=(1-\delta,0,\delta)$.

The expected utility of agent $1$ is then

$\displaystyle u_{1}({\bf x})$ $\displaystyle=$ $\displaystyle\sum_{j\in
V}\left(\sum_{P\in\mathcal{P}(i,j)}\prod\limits_{a\in P}x_{a}\right)\cdot
x_{jj}\cdot u_{ij}$ $\displaystyle=$ $\displaystyle x_{11}\cdot
u_{11}+x_{12}\cdot x_{22}\cdot u_{12}$ $\displaystyle=$
$\displaystyle\delta^{2}+(1-\delta)\cdot\delta$ $\displaystyle=$
$\displaystyle\delta$

The case of agents $2$ and $3$ are symmetric, hence the social welfare of this
equilibria is $3\delta$. Thus, the price of stability is
$\frac{3\delta}{1+2\delta}$ which tends to zero as $\delta\rightarrow 0$. ∎

Lemma 5 appears to imply that no reasonable social welfare guarantees can be
obtained for liquid democracy. This is not the case. Strong performance
guarantees can be achieved, provided we relax the incentive constraints.
Specifically, we switch our attention to approximate Nash equilibria. A
strategy profile ${\bf x}$ is an $\epsilon$-Nash equilibrium if, for each
agent $i$,

$u_{i}({\bf x})\geq\ (1-\epsilon)\cdot u_{i}(\hat{{\bf x}}_{i},{\bf
x}_{-i})\qquad\forall\hat{{\bf x}}_{i}$

Above, we use the notation ${\bf x}_{-i}=\\{{\bf x}_{i}\\}_{j\neq i}$. Can we
obtain good welfare guarantees for approximate Nash equilibria? We will prove
the answer is yes in the remainder of the paper. In particular, we present
tight bounds on the price of stability for $\epsilon$-Nash equilibria.

##  3 The Price of Stability of Approximate Nash Equilibria

So our task is to compare the social welfare of the best $\epsilon$-Nash
equilibrium with the social welfare of the optimal solution. Let’s begin by
investigating the optimal solution.

###  3.1 An Optimal Delegation Graph

By the linearity of expectation, there is an optimal solution in which the
agents use only pure strategies. In particular, there exists an optimal
delegation graph maximizing social welfare. Moreover, this optimal graph has
interesting structural properties. To explain this, we say that agent $i$ is
happy if she has strictly positive utility in a delegation graph $G$, that is
$u_{i,g(i)}>0$. A component $Q$ in $G$ is jolly if every vertex in $Q$
(except, possibly, the guru) is happy.

The key observation then is that there is an optimal delegation graph in which
every component is a jolly star.

###### Lemma 6

There is an optimal delegation graph that is the disjoint union of jolly
stars.

###### Proof

Let $Q$ be a component in an optimal delegation graph $G$. We may assume $Q$
contains a guru. To see this, suppose $Q$ contains an abstaining sink node
$j$. Then the graph $\hat{G}=G\cup(j,j)$ where $j$ votes herself is also
optimal. On the other hand, suppose $Q$ contains a cycle $C$ of length at
least $2$. Take an agent $j\in C$. Then the graph $\hat{G}$ obtained by $j$
voting herself instead of delegating her vote is also optimal.

So we may assume each component contains a guru. Further, we may assume each
component is a star. Suppose not, take a component $Q$ with guru $j$
containing an agent $i$ that does not delegate to $j$. But then the graph
$\hat{G}$ obtained by $i$ delegating her vote directly to $j$ is optimal.

Finally, we may assume each star is jolly. Suppose $i$ is not a happy agent in
a star $Q$ with guru $j$. Thus $u_{ij}=0$. But then if $i$ changes her
delegation and votes herself we again obtain an optimal solution. In this case
$i$ will form a new singleton component which is trivially a jolly star. ∎

Lemma 6 states that the optimal solution can be obtained by a pure strategy
${\bf x}^{*}$ whose delegation graph is union of jolly stars. The centre of
each star is a guru in the optimal solution and the leaves are the
corresponding happy agents who delegated to the guru. Denote the set of gurus
in the optimal solution by $D^{*}=\\{i\in V\ :\ x^{*}_{ii}=1\\}$, and let
$L_{j}=\\{i\in V\setminus D^{*}\ :\ x^{*}_{ij}=1\\}$ be the agents who
delegate to the guru $j$ as illustrated in Figure 2. It follows that the
optimal solution has welfare

$\operatorname{OPT}=\sum_{j\in D^{*}}\left(u_{jj}+\sum_{i\in
L_{j}}u_{ij}\right)$

Figure 2: Welfare optimal delegation graph that is disjoint union of jolly
stars.

###  3.2 A Stable Solution

In order to study the best $\epsilon$-Nash equilibrium we now show the
existence of a “potentially” stable solution ${\bf x}$ whose definition is
inspired by the set $D^{*}$ of gurus in the optimal solution. In Section 3.3
we will prove that ${\bf x}$ is indeed an $\epsilon$-Nash equilibrium and also
has high social welfare.

To obtain ${\bf x}$ we require the following definitions. Denote the standard
set of feasible mixed strategies for agent $i$ as

$\mathbb{S}_{i}=\\{{\bf x}_{i}\in\mathbb{R}^{n}_{+}\ :\ \sum\limits_{j\in
V}x_{ij}=1\ \\}$

Given a fixed strategy profile ${\bf x}_{-i}=\\{{\bf x}_{j}\\}_{j\neq i}$ for
the other agents, let the corresponding best response for agent $i$ be

$B_{i}({\bf x}_{-i})=\operatorname*{arg\,max}\limits_{\hat{{\bf
x}}\in\mathbb{S}_{i}}u_{i}(\hat{{\bf x}},{\bf x}_{-i})$

For each $i\in D^{*}$, we denote a restricted set of mixed strategies

$\mathbb{S}^{R}_{i}=\\{{\bf x}_{i}\in\mathbb{R}^{n}_{+}\ :\ \sum\limits_{j\in
V}x_{ij}=1\ ,x_{ii}\geq\epsilon\\}$

Then for a fixed strategy profile ${\bf x}_{-i}$, let

$B^{R}_{i}({\bf x}_{-i})=\operatorname*{arg\,max}\limits_{\hat{{\bf
x}}\in\mathbb{S}^{R}_{i}}u_{i}(\hat{{\bf x}},{\bf x}_{-i})$

be the best response for the agent $i$ from amongst the restricted set of
feasible strategies. Next recall Kakutani’s Fixed Point Theorem.

######  Theorem 7 (Kakutani’s Fixed Point Theorem)

Let $K$ be a non-empty, compact and convex subset of $\mathbb{R}^{m}$, and let
$\Phi:K\rightarrow 2^{K}$ be a set-valued function on $K$ such that:
(i) $\Phi(x)$ is non-empty and convex for all $x\in K$, and
(ii) $\Phi$ has a closed graph.
Then $\Phi$ has a fixed point, that is, there exists an $x^{*}\in K$ with
$x^{*}\in\Phi(x^{*})$.

Here a set-valued function $\Phi$ has a closed graph if
$(x^{k},y^{k})\rightarrow(x,y)$ and $y^{k}\in\Phi(x^{k})$ implies that
$y\in\Phi(x)$.

###### Theorem 8

There exists a strategy profile ${\bf x}$ such that:
(a) For all $i\in D^{*}$, we have ${\bf x}_{i}\in B^{R}_{i}({\bf x}_{-i})$,
and
(b) For all $j\notin D^{*}$, we have ${\bf x}_{j}\in B_{j}({\bf x}_{-j})$.

###### Proof

Let the feasible set of strategy profiles be $\Xi=\prod\limits_{i\in
D^{*}}\mathbb{S}^{R}_{i}\times\prod\limits_{j\notin D^{*}}\mathbb{S}_{j}$, a
subset of Euclidean space. Without loss of generality, let
$D^{*}=\\{1,2,\dots,k\\}$. Now define a set valued function
$\Phi\colon\Xi\longrightarrow 2^{\Xi}$ by

${\bf x}\mapsto(\underbrace{B_{1}^{R}({\bf x}_{-1}),\cdots,B_{k}^{R}({\bf
x}_{-k}),}_{D^{*}}\underbrace{B_{k+1}({\bf x}_{-(k+1)}),\cdots,B_{n}({\bf
x}_{-n})}_{V\setminus D^{*}})$

That is, for each ${\bf x}\in\Xi$ we have $\Phi({\bf x})\subseteq\Xi$. Note
the statement of the theorem is equivalent to showing that $\Phi$ has a fixed
point.

Observe that $\Phi$ satisfies the conditions of Kakutani’s Fixed Point
Theorem. Indeed $\Xi$ is nonempty, compact and convex, since it is a product
of non-empty, compact and convex sets $\mathbb{S}^{R}_{i}$ and
$\mathbb{S}_{j}$.

Next let’s verify that $\Phi({\bf x})\neq\emptyset$. This holds since, for
each agent $i$, we have $B^{R}_{i}({\bf x}_{-i})\neq\emptyset$ or $B_{i}({\bf
x}_{-i})\neq\emptyset$ by the continuity of $u_{i}(\ \cdot\ ,{\bf x}_{-i})$
and the Weierstrass Extreme Value Theorem.

Furthermore, for all ${\bf x}\in\Xi$ the set $\Phi({\bf x})\subseteq\Xi$ is
convex. This is because, for each $i\in D^{*}$ and $j\in V\setminus D^{*}$,
the sets $B^{R}_{i}({\bf x}_{-i})$ and $B_{j}({\bf x}_{-j})$ are convex, and
thus $\Phi({\bf x})$ is Cartesian product of convex sets. We must now show
that both $B_{j}({\bf x}_{-j})$ and $B^{R}_{i}({\bf x}_{-i})$ are convex. The
convexity of $B_{j}({\bf x}_{-j})$ follows immediately by the multilinearity
of $u_{i}$. Next take an agent $i\in D^{*}$. If ${\bf y}_{i},{\bf z}_{i}\in
B^{R}_{i}$ then, for all $\lambda\in[0,1]$ and any $\hat{{\bf
x}_{i}}\in\mathbb{S}^{R}_{i}$, we have

$\displaystyle u_{i}(\lambda{\bf y}_{i}+(1-\lambda){\bf z}_{i},{\bf x}_{-i})$
$\displaystyle\ =\ \lambda u_{i}({\bf y}_{i},{\bf
x}_{-i})+(1-\lambda)u_{i}({\bf z}_{i},{\bf x}_{-i})$ $\displaystyle\ \geq\
u_{i}(\hat{{\bf x}}_{i},{\bf x}_{-i})$

Observe $\lambda{\bf y}_{i}+(1-\lambda){\bf z}_{i}\in\mathbb{S}^{R}_{i}$ since
$\lambda
y_{ii}+(1-\lambda)z_{ii}\geq\lambda\epsilon+(1-\lambda)\epsilon=\epsilon$ and
$\lambda\sum_{j\in V}y_{ij}+(1-\lambda)\sum_{j\in V}z_{ij}=1$. Thus
$\lambda{\bf y}_{i}+(1-\lambda){\bf z}_{i}\in B^{R}_{i}({\bf x}_{-i})$ for any
$\lambda\in[0,1]$, which implies $B^{R}_{i}({\bf x}_{-i})$ is convex.

Finally, $\Phi$ has a closed graph because each $u_{i}({\bf x}_{i},{\bf
x}_{-i})$ is a continuous function of ${\bf x}_{i}$ for any fixed ${\bf
x}_{-i}$, and both sets $\mathbb{S}^{R}_{i}$ and $\mathbb{S}_{i}$ are compact.
Thus, by Kakutani’s Fixed Point Theorem, $\Phi$ has a fixed point ${\bf x}$.
Hence (a) and (b) hold. ∎

###  3.3 Bicriteria Guarantees for Liquid Democracy

We will now prove that the fixed point ${\bf x}$ from Theorem 8 gives our main
result: there is an $\epsilon$-Nash equilibrium with welfare ratio at least
$\epsilon$. First let’s show the incentive guarantees hold.

###### Lemma 9

The fixed point ${\bf x}$ is an $\epsilon$-Nash equilibrium.

###### Proof

Take an agent $j\notin D^{*}$. Then for any $\hat{{\bf
x}}_{j}\in\mathbb{S}_{i}$ we have

$u_{j}({\bf x}_{j},{\bf x}_{-j})\ \geq\ u_{j}(\hat{{\bf x}}_{j},{\bf x}_{-j})\
\geq\ (1-\epsilon)\cdot u_{i}(\hat{{\bf x}}_{i},{\bf x}_{-i})$

The incentive guarantee for $j$ follows immediately.

Next consider an agent $i\in D^{*}$. Take any $\hat{{\bf
x}}_{i}\in\mathbb{S}_{i}$ and define a new strategy ${\bf y}_{i}$ as follows:

$y_{ij}=\begin{cases}\epsilon+(1-\epsilon)\cdot\hat{x}_{ii}&\text{if}\ \
j=i\\\ (1-\epsilon)\cdot\hat{x}_{ij}&\text{if}\ \ j\neq i\\\ \end{cases}$

Observe that ${\bf y}_{i}\in\mathbb{S}^{R}_{i}$ because $y_{ii}\geq\epsilon$
and $\sum_{j\in N}y_{ij}=\epsilon+(1-\epsilon)\cdot\sum_{j\in
V}\hat{x}_{ij}=1$.

Now for any path $P=\\{a_{1},a_{2},\cdots,a_{k}\\}$ from $i$ to $j$ where
$a_{i}$ are the arcs, the probability of obtaining this path in the delegation
graph generated by the strategy profile $\\{{\bf y}_{i},{\bf x}_{-i}\\}$ is
exactly $y_{a_{1}}\cdot\prod\limits_{a\in P\setminus a_{1}}x_{a}$. Thus

$\displaystyle u_{i}({\bf y}_{i},{\bf x}_{-i})$ $\displaystyle\ =\ \sum_{j\in
V}\mathbb{P}_{{\bf y}_{i},{\bf x}_{-i}}(g(i)=j)\cdot u_{ij}$ $\displaystyle\
=\ u_{ii}y_{ii}+\sum_{j\in V\setminus
i}u_{ij}x_{jj}\cdot\left(\sum_{P\in\mathcal{P}(i,j)}y_{a_{1}}\prod\limits_{a\in
P\setminus a_{1}}x_{a}\right)$ $\displaystyle\ \geq\ (1-\epsilon)\cdot
u_{ii}\hat{x}_{ii}+(1-\epsilon)\cdot\sum_{j\in V\setminus
i}u_{ij}x_{jj}\cdot\left(\sum_{P\in\mathcal{P}(i,j)}\hat{x}_{a_{1}}\prod\limits_{a\in
P\setminus a_{1}}x_{a}\right)$ $\displaystyle\ =\
(1-\epsilon)\cdot\left(u_{ii}\hat{x}_{ii}+\sum_{j\in V\setminus
i}u_{ij}x_{jj}\cdot\left(\sum_{P\in\mathcal{P}(i,j)}\hat{x}_{a_{1}}\prod\limits_{a\in
P\setminus a_{1}}x_{a}\right)\right)$ $\displaystyle\ =\ (1-\epsilon)\cdot
u_{i}(\hat{{\bf x}}_{i},{\bf x}_{-i})$

But ${\bf x}_{i}\in B^{R}_{i}({\bf x}_{-i})$. Hence, $u_{i}({\bf x}_{i},{\bf
x}_{-i})\geq u_{i}({\bf y}_{i},{\bf x}_{-i})$ because ${\bf
y}_{i}\in\mathbb{S}^{R}_{i}$. It follows that $u_{i}({\bf x}_{i},{\bf
x}_{-i})\geq(1-\epsilon)\cdot u_{i}(\hat{{\bf x}}_{i},{\bf x}_{-i})$ and so
the incentive guarantee for $i$. Thus ${\bf x}$ is an $\epsilon$-Nash
equilibrium. ∎

Next let’s prove the social welfare guarantee holds for ${\bf x}$.

###### Theorem 10

For all $\epsilon\in[0,1]$, and for any instance of the liquid democracy game,
there exists an $\epsilon$-Nash equilibrium with social welfare at least
$\epsilon\cdot\operatorname{OPT}$.

###### Proof

It suffices to prove that ${\bf x}$ has social welfare at least
$\epsilon\cdot\operatorname{OPT}$. We have

$\displaystyle\operatorname{SW}({\bf x})$ $\displaystyle=\sum_{j\in
V}u_{j}({\bf x})$ $\displaystyle=\sum_{j\in D^{*}}u_{j}({\bf x})+\sum_{j\in
D^{*}}\sum_{i\in L_{j}}u_{i}({\bf x})$ $\displaystyle\geq\sum_{j\in
D^{*}}u_{jj}\cdot x_{jj}+\sum_{j\in D^{*}}\sum_{i\in L_{j}}u_{ij}\cdot x_{jj}$
$\displaystyle\geq\sum_{j\in D^{*}}\left(u_{jj}\cdot\epsilon+\sum_{i\in
L_{j}}u_{ij}\cdot\epsilon\right)$ $\displaystyle=\epsilon\cdot\sum_{j\in
D^{*}}\left(u_{jj}+\sum_{i\in L_{j}}u_{ij}\right)$
$\displaystyle=\epsilon\cdot\operatorname{OPT}$

The first inequality follows since each agent $i\in L_{j}$ satisfies
$u_{i}({\bf x}_{i},{\bf x}_{-i})\geq u_{i}(\hat{{\bf x}}_{i},{\bf x}_{-i})$
for all $\hat{{\bf x}}_{i}\in\mathbb{S}_{i}$. In particular, the deviation
$\hat{{\bf y}}_{i}$ of delegating to the guru $j$ with probability $1$ implies
$u_{i}({\bf x}_{i},{\bf x}_{-i})\geq u_{i}(\hat{{\bf y}}_{i},{\bf x}_{-i})\geq
u_{ij}x_{jj}$. Finally, the second inequality holds as we have ${\bf
x}_{j}\in\mathbb{S}_{j}^{R}$, for each $j\in D^{*}$. Therefore
$x_{jj}\geq\epsilon$ and the welfare guarantee holds. ∎

We can deduce from Theorem 10 that strong approximation guarantees can
simultaneously be obtained for both social welfare and rationality. In
particular, setting $\epsilon=\frac{1}{2}$ gives factor $2$ approximation
guarantees for both criteria.

###### Corollary 11

For any instance of the liquid democracy game, there exists a
$\frac{1}{2}$-Nash equilibrium with welfare at least
$\frac{1}{2}\cdot\operatorname{OPT}$.

###  3.4 A Tight Example

We now prove upper bounds on the welfare guarantee obtainable by any
$\epsilon$-Nash equilibrium. In particular, we show that the bicriteria
guarantee obtained in Theorem 10 is tight.

###### Theorem 12

For all $\epsilon\in[0,1]$, there exist instances such that any
$\epsilon$-Nash equilibrium has welfare at most
$\epsilon\cdot\operatorname{OPT}+\gamma$ for any $\gamma>0$

###### Proof

Take an instance with $n+2$ agents. Let agents $\\{1,2\\}$ have identical
utility functions with $u_{ij}=\delta$ if $j=1$ and $u_{ij}=0$ otherwise. The
remaining agents $i\in\\{3,\cdots,n+2\\}$ have utilities

$u_{ij}=\begin{cases}1&\text{if }j=2\\\ 0&\text{otherwise}\end{cases}$

Observe that $\operatorname{OPT}=\delta+n$ which is obtained by agents 1 and 2
voting while the remaining agents delegate to agent $2$. Now let ${\bf x}$ be
an $\epsilon$-Nash equilibrium. We claim that $x_{22}\leq\epsilon$. To see
this, note that $x_{11}\geq(1-\epsilon)$. If $x_{11}<(1-\epsilon)$ then
$u_{1}({\bf x})<(1-\epsilon)\delta$. But this contradicts the fact that ${\bf
x}$ is an $\epsilon$-Nash equilibrium, as agent $1$ can deviate and vote
herself to obtain a utility of $\delta$. Furthermore, $x_{11}\geq(1-\epsilon)$
implies $x_{21}\geq(1-\epsilon)$ by a similar argument. Since $\sum_{j\in
N}x_{2j}=1$, we do have $x_{22}\leq\epsilon$. The social welfare of ${\bf x}$
is then

$\operatorname{SW}({\bf x})\ \leq\ \delta+(1-\epsilon)\delta+\epsilon n\ =\
2(1-\epsilon)\delta+\epsilon\operatorname{OPT}$

Letting $\gamma=2(1-\epsilon)\delta$ gives the desired bound. Since $\delta$
can be made arbitrarily small, $\gamma$ can also be made arbitrarily small. ∎

Together, Theorems 10 and 12 imply that the price of stability of
$\epsilon$-Nash equilibrium is exactly $\epsilon$ in the liquid democracy
game.

##  4  Extensions and Computational Complexity

###  4.1 Model Extensions

Our bicriteria results extend to the settings of repeated games, weighted
voters, and multiple delegates.

$\bullet$ Repeated Games. Recollect that an underlying motivation for liquid
democracy is that it can be applied repeatedly over time with agents having
the option of delegating to different agents at different times. Evidently, by
repeating the delegation game over time, with different utility functions for
the topics considered in different time periods, the same bicriteria
guarantees holds.

$\bullet$ Weighted Voters. In some electoral systems, different agents may
have different voting powers. That is, the voters are weighted. In this case,
the number of votes a guru casts is simply the sum of the weights of all the
votes to which it was delegated. Our bicriteria guarantees then follow
trivially.

$\bullet$ Multiple Delegates. In some settings it may be the case that an
agent is allowed to nominate more than one delegate. That are two natural
models for this. First, an agent nominates multiple delegates but the
mechanism can use only one of them. Second, an agent splits its voting weight
up and assigns it to multiple delegates. In both cases our techniques can be
adapted and applied to give the same performance guarantees.

###  4.2 Computational Aspects

Let us conclude by discussing computation aspects in the liquid democracy
game. Recall, to find an $\epsilon$-Nash equilibrium with a provably optimal
social welfare guarantee we solved a fixed point theorem. Moreover, solving
the fixed point theorem requires knowledge of the optimal set $D^{*}$ of
gurus. But obtaining $D^{*}$ is equivalent to finding an optimal solution to
the liquid democracy game and this problem is hard.

###### Theorem 13

It is NP-hard to find an optimal solution to the liquid democracy game.

###### Proof

We apply a reduction from dominating set in a directed graph. Given a directed
graph $G=(V,A)$ and an integer $k$: is there a set $S\subseteq V$ of
cardinality $k$ such that, for each $i\notin S$, there exists an arc $(i,j)\in
A$ for some $j\in S$. This problem is NP-complete. We now give a reduction to
the liquid democracy game.

Given the directed graph $G=(V,A)$ let the set of agents in the game be $V$.
Define the utility function of an agent by:

$u_{ij}=\begin{cases}1&\text{if }(i,j)\in A\\\ 0&\text{otherwise}\end{cases}$

It immediately follows that there is a dominating set of cardinality at most
$k$ if and only if the liquid democracy game has a solution with social
welfare at least $n-k$. This completes the proof. ∎

So is it possible for the agents to compute a good bicriteria solution in
polynomial time? If we allow for sub-optimal approximation guarantees then
this is achievable. The idea is simple: the characteristics required of the
agents to ensure reasonable performance guarantees are narcissism and
avarice.555This is not an unreasonable assumption for both pirates and many of
the inhabitants of Wonderland! First, since $D^{*}$ is unknown, each agent $i$
narcissistically assumes he himself is an optimal guru. Consequently, he will
vote with probability $p$. Thus, he will delegate his vote with probability
$(1-p)$. This he will do avariciously, by greedily delegating to the agent
$i^{*}$ that gives him largest myopic utility, that is
$i^{*}=\operatorname*{arg\,max}_{j\in V}u_{ij}$.

###### Theorem 14

For any $\epsilon\in[\frac{3}{4},1]$, the narcissistic-avaricious algorithm is
linear time and produces an $\epsilon$-Nash equilibrium with social welfare at
least $(1-\epsilon)\cdot\operatorname{OPT}$.

###### Proof

Consider the incentive guarantee for agent $i$ for the strategy profile ${\bf
z}$ induced by the algorithm. As $i$ votes with probability $z_{ii}=p$ and
delegates to $i^{*}$ with probability $z_{ii^{*}}=(1-p)$, his utility is

$\displaystyle u_{i}({\bf z}_{i},{\bf z}_{-i})$ $\displaystyle\ \geq\
z_{ii}u_{ii}+z_{ii^{*}}z_{i^{*}i^{*}}u_{ii^{*}}$ $\displaystyle\ =\ p\cdot
u_{ii}+(1-p)\cdot p\cdot u_{ii^{*}}$ $\displaystyle\ \geq\ (1-p)\cdot p\cdot
u_{ii^{*}}$ $\displaystyle\ \geq\ (1-p)\cdot p\cdot u_{i}(\hat{{\bf
z}_{i}},{\bf z}_{-i})\qquad\forall\hat{{\bf z}_{i}}\in\mathbb{S}_{i}$

Hence ${\bf z}$ is an $\epsilon$-Nash equilibrium for

$\epsilon=(1-p(1-p))=1-p+p^{2}=\frac{3}{4}+(p-\frac{1}{2})^{2}$

It follows that the narcissistic-avaricious algorithm can provide incentive
guarantees only for $\epsilon\in[\frac{3}{4},1]$. Further, by solving the
corresponding quadratic equation, to obtain such an $\epsilon$-Nash
equilibrium we simply select
$p=\frac{1}{2}\left(1+\sqrt{1-4(1-\epsilon)}\right)$.

Now, let’s evaluate the social welfare guarantee for the narcissistic-
avaricious algorithm. As above, we have

$\displaystyle\operatorname{SW}({\bf z})$ $\displaystyle\ =\ \sum_{i\in
V}u_{i}({\bf z}_{i},{\bf z}_{-i})$ $\displaystyle\ \geq\ \sum_{i\in
V}(1-p)\cdot p\cdot u_{ii^{*}}$ $\displaystyle\ =\ (1-p)\cdot p\cdot\sum_{i\in
V}u_{ii^{*}}$ $\displaystyle\ \geq\ (1-p)\cdot p\cdot\operatorname{OPT}$

Since $p=\frac{1}{2}\left(1+\sqrt{1-4(1-\epsilon)}\right)$ this gives

$\operatorname{SW}({\bf z})\ \geq\
\frac{1}{4}\Big{(}1-(1-4(1-\epsilon)\Big{)}\cdot\operatorname{OPT}\ =\
(1-\epsilon)\cdot\operatorname{OPT}$

Therefore, as claimed, the narcissistic-avaricious algorithm outputs a
solution whose welfare is at least $(1-\epsilon)$ times the optimal welfare.

Finally, observe that implementing the narcissistic-avaricious strategy
requires that each agent $i$ simply computes
$i^{*}=\operatorname*{arg\,max}_{j\in V}u_{ij}$. This can be done for every
agent in linear time in the size of the input. ∎

We emphasize two points concerning Theorem 14. One, it only works for weaker
incentive guarantees, namely $\epsilon\in[\frac{3}{4},1]$. Unlike the fixed
point algorithm it does not work for the range $\epsilon\in(0,\frac{3}{4})$.
Two, the social welfare guarantee is $(1-\epsilon)$. This is a constant but,
for the valid range $\epsilon\in[\frac{3}{4},1]$, it is much worse than the
$\epsilon$ guarantee obtained by the fixed point algorithm. A very interesting
open problem is to find a polynomial time algorithm that matches the optimal
bicriteria guarantees provided by Theorem 10 and applies for all $\epsilon>0$.

Acknowledgements: We thank Bundit Laekhanukit for interesting discussions. We
are also grateful to the reviewers for suggestions which greatly improved the
paper.

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